International Electronic Journal of Mathematics Education_Vol 8_N2-3

8/12/2019 International Electronic Journal of Mathematics Education_Vol 8_N2-3

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International Electronic Journal of Mathematics Education – IΣJMΣ Vol.8, No.2-3

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ARTICLES

Teachers’ eliefs aout mathematical !no"led#e for teachin#

definitions

$eidar Mos%old & Janne Fauskanger

'3

The de%elo(ment of students’ al#eraic (roficienc)

Irene %an *ti(hout, +aul ri%ers & oeno /ra%emeier

02

Varied "a)s to teach the definite inte#ral conce(t

Iiris 1ttor(s, ell r!, Mir!o $adic & Timo Tossa%ainen

84

The influence of elementar) (reser%ice teachers’ mathematical

e5(eriences on their attitudes to"ards teachin# and learnin#

mathematics

6ind) Jon# & Thomas E. 7od#es

4




Teachers’ Beliefs about Mathematical Knowledge for Teaching Definitions

Reidar Mosvold

University of Stavanger

Janne Fauskanger

University of Stavanger

Previous research indicates the importance of teachers’ knowledge of mathematical definitions—as

well as their beliefs. Much remains unknown, however, about the specific knowledge required doing

the mathematical task of teaching involving definitions and the related teacher beliefs. In this article,

we analyze focus-group interviews that were conducted in a Norwegian context to examine the

adaptability of the U.S. developed measures of mathematical knowledge for teaching. Qualitative

content analysis was applied in order to learn more about the teachers’ beliefs about mathematical

knowledge for teaching definitions. The results indicate that teachers believe knowledge of

mathematical definitions is an important aspect of mathematical knowledge for teaching, but they do

not regard it as important to actually know the mathematical definitions themselves.

Keywords: mathematical knowledge for teaching, teacher beliefs, mathematical definitions

In his presidential address at the 1985 Annual Meeting of the American Educational

Research Association, Lee Shulman presented his theories concerning the different aspects of

teachers’ professional knowledge (Shulman, 1986). A number of attempts have been made by

researchers afterwards to build upon these ideas (e.g., Graeber & Tirosh, 2008). In

mathematics education, the efforts of Deborah Ball and her colleagues at the University of

Michigan (see e.g., Ball, Thames, & Phelps, 2008) are among the most promising (Morris,Hiebert, & Spitzer, 2009). They have formulated a practice-based theory of what is often

referred to as ‘mathematical knowledge for teaching’ (MKT), and they have also created

measures of teachers’ MKT (e.g., Hill, Schilling, & Ball, 2004). The MKT measures—as

well as the MKT framework—have been developed from studies of mathematics teaching in

the U.S.

In the last couple of years, researchers have made attempts to translate, adapt and use

MKT items in other countries (for a review, see Blömeke & Delaney, 2012). Among the first

attempts was that of Delaney (2008), who adapted and used a set of MKT items for use in

Ireland. Researchers who have translated and used MKT items in other countries after this

normally build upon his results and suggestions (e.g., Mosvold, Fauskanger, Jakobsen, &Melhus, 2009). Several researchers have—in their attempts to analyze the challenges of

adapting MKT items for other countries—pointed at possible cultural differences in the tasks

of teaching. Since MKT is conceptualized in practice, Cole (2012) argued, the question of

whether or not the tasks of teaching are independent of environment and cultural context is a

logical one to ask. In their study of Norwegian teachers’ perceived difficulties with the

adapted MKT items, Fauskanger and Mosvold (2010) also indicated that there might be

cultural issues involved.



TEACHERS’ BELIEFS ABOUT MKT DEFINITIONS 44

One particular task of teaching that has received attention in previous research is that of

“choosing and developing useable definitions” (Ball, Thames, & Phelps, 2008, p. 400). In his

study of Indonesian teachers’ mathematical knowledge for teaching geometry, Ng (2012)

found that the MKT measures discriminated between teachers who adopted inclusive and

those who adopted exclusive definitions rather than between knowledgeable and less

knowledgeable teachers. He argued that there might be some cultural differences between theuse of inclusive geometric definitions between Indonesian teachers and U.S. teachers; he also

argued that using the measures were useful for providing a better understanding of what

teachers need to know in order to do the work of teaching in Indonesia (ibid.).

Mathematics teachers all over the world face demands related to choosing and developing

definitions that are appropriate for use among their students, and Zazkis and Leikin (2008)

suggest that teachers’ knowledge of mathematical definitions and their concept images affect

their instructional decisions, the explanations they provide in the classroom, the way they

guide their students, and how they conduct mathematical discussions. To plan future

professional development it is asserted that teachers’ beliefs about teaching knowledge may

influence their interpretation of their experiences (e.g., Ravindran, Greene, & Debacker,

2005). Given these results from previous research, we found it relevant to make an effort to

learn more about teachers’ beliefs about the mathematical knowledge needed for teaching

definitions in a different cultural context. With this as a background, we approach the

following research question: What do teachers’ reflections on MKT items reveal about their

beliefs concerning mathematical knowledge for teaching definitions?

In order to answer this question, we analyze the reflections given by Norwegian teachers

in focus-group interviews where MKT items were used to focus the discussions. Before we

approach this, however, we first need to make some clarifications about beliefs related to

teaching knowledge and how they relate to other types of beliefs. Then we need to discuss

how these beliefs relate to knowledge in general and MKT in particular. We also need to

elaborate on our focus on that particular task of teaching concerning definitions in relation to

the more general research on teachers’ knowledge of mathematical definitions. These issues

are addressed in the next section.

Theoretical Influences

Philosophers have pondered about beliefs and knowledge—and the connection between

the two—for centuries. The result of the philosophers’ reflections on these issues is manifest

in the branch of philosophy called epistemology—which has a particular focus on discussions

concerning knowledge and beliefs. Within the field of educational research in general and

mathematics education in particular, there has been a vast amount of research related tobeliefs. In his overview of research in this area, Philipp (2007) presented some of the terms

that have been used when these issues have been investigated in mathematics education

research: affect (including emotions, attitudes and beliefs), beliefs systems, conceptions,

identity, knowledge and values. All of these concepts—including that of beliefs—have been

used with various meanings by different researchers.



45 R. Mosvold & J. Fauskanger

Beliefs About...

Before approaching the concept of beliefs about teaching knowledge—which is our focus

in this article—we need to make some clarifications concerning the more general concept of

beliefs. Mathematics teachers’ beliefs have often been grouped into beliefs about the nature

of mathematics, about mathematics teaching and about mathematics learning—as presented

in Table 1.

Table 1

Categories of teachers' beliefs (adapted from Beswick, 2012, p. 130)

Beliefs about the nature of

mathematics

Beliefs about mathematics

teaching

Beliefs about mathematics

learning

Instrumentalist Content focused with an emphasis

on performance

Skill mastery, passive

reception of knowledge

Platonist Content focused with an emphasis

on understanding

Active construction of

understanding

Problem solving Learner focused Autonomous exploration of

own interest The three categories of Ernest (1989)—as presented in the left column of Table 1—have

been widely used as a description of beliefs about the nature of mathematics. In the

instrumentalist view, mathematics is seen as “an accumulation of facts, skills and rules to be

used in the pursuance of some external end” (Ernest, 1989, p. 250). The Platonist view sees

mathematics as a body of pre-existing knowledge. Finally, in the problem solving view,

mathematics is regarded as a dynamic human invention.

Almost three decades ago, Thompson (1984) claimed that the connection between

teachers’ beliefs about mathematics and their teaching practice had been largely ignored. Shecalled for research with a focus on this connection between beliefs and practice, and a

number of studies with such a focus subsequently emerged (e.g., Cooney, 1985; Raymond,

1997; Skott, 2001); several of these studies had a focus on inconsistencies between beliefs

and practice. Following Thompson’s initiative, there has been an increased interest in beliefs

about the nature of mathematics; there has also been a continually increasing focus on beliefs

about mathematics teaching and learning. Van Zoest, Jones and Thornton (1994)

distinguished between three important aspects in research on beliefs about mathematics

teaching (see the middle column of Table 1), whereas others (e.g., Ernest, 1989)

distinguished between beliefs concerning three aspects of mathematics learning (see the right

column of Table 1).

Beliefs and Knowledge

The relationship between knowledge and beliefs makes up a long-standing discussion

(Pehkonen, 2008), and a main difficulty has been to distinguish beliefs from knowledge

(Thompson, 1992). There appear to be differences as well as similarities between students’

knowledge and beliefs (Op’ Eynde, De Corte, & Verschaffel, 2002); research on teachers’

knowledge and beliefs indicates that this is also the case here (Forgasz & Leder, 2008). In her




attempt to sort out the connection between teacher knowledge and teacher beliefs, Thompson

(1992) pointed out that the difficulties involved in changing teacher performance are

intimately connected with what teachers believe and know. Her approach has had significant

impact on the direction of research in this area. Furinghetti and Pehkonen (2002) emphasized

the close connections between knowledge and beliefs, and they argued that beliefs should be

considered as part of teachers’ personal knowledge. In another attempt to clarify between theconcepts, Kuntze (2011) used the term ‘professional knowledge’—in which beliefs were

included. Many researchers distinguish between these two concepts, but some argue that

beliefs and knowledge are strongly related. Beswick (2011, 2012) argued for the equivalence

of beliefs and knowledge; she also suggested that beliefs about mathematical content and

pedagogy should be included in the MKT framework. Philipp (2007), on the other hand,

maintained that beliefs are closely related to knowledge, but a distinction should be made

between the terms. In this article, we follow Philipp’s suggestion and distinguish between

knowledge and beliefs. We focus on the beliefs teachers have about knowledge needed for

teaching, and we consider this to be an aspect of teachers’ personal epistemology.

Beliefs about Teaching Knowledge

Teachers’ personal epistemology includes beliefs about knowledge—commonly referred

to as epistemological beliefs (Hofer, 2002)—and these epistemological beliefs are considered

important; Schommer-Aikins and colleagues (2010) proposed that teachers’ epistemological

beliefs have a potential impact on students’ learning in all academic levels. Even though the

origin of studies concerning students’ epistemological beliefs can be traced four decades

back—Perry’s (1970) seminal work has often been referred to—the actual term

‘epistemological beliefs’ was first used by Schommer (1994). She used the term in reference

to “beliefs about the nature of knowledge and learning” (Schommer, 1994, p. 293). In

research regarding epistemological beliefs, there is, however, little agreement concerning the

actual construct. Some argue that epistemological beliefs are domain specific, and some

argue that they are not (e.g., Buehl, Alexander, & Murphy, 2002). There is also disagreement

about how the construct is connected with other related constructs (ibid.). Although several

competing models of the nature of epistemological beliefs have been proposed, general

epistemological beliefs seem to refer to “individuals’ belief about the nature of knowledge

and the processes of knowing” (Hofer & Pintrich, 1997, p. 112); sometimes it is also used

with reference to learning and teaching (Op’t Eynde et al., 2006). In their attempt to clarify

the research in this area, Hofer and Pintrich (1997) proposed that epistemological theories are

composed of “certainty of knowledge, simplicity of knowledge, source of knowledge, and

justification for knowing” (ibid., p. 133). Work on disciplinary beliefs indicates that

epistemological beliefs might vary from one discipline to another (e.g., Hofer & Pintrich,1997).

Despite the importance and amount of research related to teachers’ beliefs, relatively few

studies focus on teachers beliefs’ about teaching knowledge in general (Buehl & Fives, 2009;

Fives & Buehl, 2008); even fewer studies focus on teachers’ beliefs about the content of their

mathematical knowledge for teaching in particular. Consequently, the body of knowledge to

be considered is—due to the complexity and multidimensionality of teachers’ knowledge (see




next section)—of importance in studies of teachers’ beliefs about teaching knowledge (Buehl

& Fives, 2009). Prior research emphasized the importance of studying teachers’ beliefs about

teaching knowledge, because these beliefs may influence how and what they learn from

participating in professional development (e.g., Ravindran, Greene, & DeBacker, 2005);

beliefs about teaching knowledge may also influence teaching practices (e.g., Sinatra &

Kardash, 2004). Fives and Buehl (2010) proposed that teachers’ beliefs about what they needto know constitute a distinct domain. Bendixen and Feucht (2010) supported this, and they

maintained that this “provides additional depth to our understanding of teachers’ personal

epistemology” (p. 567).

Distinct beliefs about different aspects of teaching knowledge exists, such as the source of

teaching knowledge, the stability of teaching knowledge and the structure of teaching

knowledge (Buehl & Fives, 2009). In the present article we focus on practicing teachers’

beliefs about a fourth aspect: the content of teaching knowledge (as in Fives & Buehl,

2008)—in particular teachers’ beliefs about the knowledge needed to teach mathematical

definitions.

Mathematical Knowledge for Teaching

Several frameworks for teachers’ knowledge have been developed (e.g., Ball, Thames, &

Phelps, 2008; Blömeke, Hsieh, Kaiser, & Schmidt, 2014; Rowland, Huckstep, & Thwaites,

2009). For the purpose of this article, we focus on the MKT framework only. This has been

regarded as one of the most promising frameworks of teacher knowledge (Morris, Hiebert, &

Spitzer, 2009), and the items we used to focus the group discussions were developed within

this framework.

It is evident that teachers need to have some knowledge of the content they are supposed

to teach. It is also generally agreed upon that teachers’ knowledge need to go somewhat

beyond the content they teach; their knowledge must be deeper than a plain knowledge of the

content of the curriculum. The burning question is, however, what characterizes the content

knowledge needed for teaching a subject like mathematics. Building upon Shulman’s (1986)

ideas concerning the existence of a domain of content knowledge that is unique to the

teaching profession, Ball, Thames and Phelps (2008) made an effort to contribute to the

further development of our understanding of this particular kind of knowledge. Shulman and

his colleagues developed typologies to describe the various aspects of teachers’ professional

knowledge, and they focused in particular on what they referred to as “pedagogical content

knowledge”. This domain of knowledge connects the knowledge of content with teaching

practice, and this—Ball and colleagues (2008) argue—is why it is so popular.

At the University of Michigan, they started investigating the work of teaching

mathematics in the Mathematics Teaching and Learning to Teach project (MTLT). In thisproject, they started with practice in order to learn more about the knowledge needed by

teachers in order to teach mathematics. The results provided a foundation for what they refer

to as “a practice-based theory of mathematical knowledge for teaching” (Ball et al., 2008, p.

395). In these classroom studies, the researchers focused on the work of teaching

mathematics rather than on teachers. They also focused on the mathematical demands of

teaching, and these “tasks of teaching” are regarded as specific to the work of teaching




mathematics; the mathematical tasks of teaching are also strongly connected with the MKT

items. Hill and colleagues (2004) elaborated on this when they explained how item writing

served different purposes for the researchers in Michigan. Item writing served the purpose of

exploring the nature and composition of subject-matter knowledge of mathematics for

teaching and MKT in particular. The item writing process was used to develop the tasks of

teaching. On a more practical level, they hoped that the creation of these measures wouldlead to increased understanding of—and renewed interest in—the content knowledge of

teachers (ibid.).

Building upon the results from the MTLT project, the researchers at the University of

Michigan started developing survey measures of the content knowledge needed for teaching

mathematics as part of the Learning Mathematics for Teaching project (LMT). In Figure 1 an

example from the public released LMT items that focuses on definitions is presented. Among

the items that were discussed by the teachers in our study, one of the items had a focus on

whether or not 1 is defined as a prime number. The item in Figure 1 is not the exact same, but

we let it serve as an illustration since it also has a focus on the definition of prime numbers.

Figure 1. Item 2 from the released LMT items (Ball & Hill, 2008, p. 4).

Mathematical Knowledge for Teaching Definitions

Knowledge of mathematical definitions is part of MKT, and it is represented in a task of

teaching that Ball and colleagues (2008) formulated as “choosing and developing usable

definitions”. When reviewing an item like the one above, it becomes apparent that knowledge

of definitions might relate to all areas of the MKT framework. Mathematical definitions are

relevant for research in mathematics education in general, and the study of mathematical

definitions is strongly connected with that of mathematical proofs (Knapp, 2006; Leikin &Zazkis, 2010). In the TIMSS 1999 Video Study—an international comparison study of

mathematics teaching in seven countries—the results indicated cultural differences in the way

teachers focused on mathematical definitions (Hiebert et al., 2003). Hiebert and colleagues

(ibid.) found, among other things, that teachers from Hong Kong SAR had a stronger focus

on presenting definitions than teachers from other countries.

Definitions have developed throughout the history of mathematics, and it was on the basis

of the genetic approach—where a main idea is that learners should follow the path in which




discoveries were originally made—that De Villiers (1998) suggested that students should be

engaged in defining concepts rather than learning about definitions. Zazkis and Leikin (2008)

followed up on this when they argued that definitions of mathematical concepts as well as the

processes of defining are fundamental aspects of teachers’ subject matter knowledge. They

continued to argue that teachers’ knowledge of mathematical definitions and their concept

images affect their instructional decisions, the explanations they provide in the classroom, theway they guide their students, and how they conduct mathematical discussions (e.g., Zazkis

& Leikin, 2008). Leikin and Zazkis (2010) found that prospective mathematics teachers’

knowledge of definitions is situated in the content domain of mathematics. They claimed that

it reflects the nature of school mathematics textbooks and of the school curriculum and they

found a gap between the mathematics learned in university courses and school mathematics.

It is therefore not surprising that Ball, Thames and Phelps (2008), in their presentation of

mathematical tasks of teaching, listed “choosing and developing useable definitions” (p. 400)

as one of the challenges that are distinctive to the work of teaching mathematics. This goes

beyond the ability to recite the actual definitions and into the area of understanding variations

of definitions—whether congruent or non-congruent (Usiskin & Griffin, 2008)—and

understanding mathematically accurate yet useful definitions and its trajectory.

Methods

In our efforts to learn more about teachers’ beliefs about the content of their teaching

knowledge, we arranged focus-group interviews. Focus groups have the potential to initiate

“concentrated conversations that might never occur in the «real world»” (Morgan, 1998, p.

31). Such focused discussions could give realistic accounts of what teachers think about the

adapted MKT items “because they are forced to think about and possibly revise their views”

(Bryman, 2004, p. 348). The initial aim with these interviews was to investigate whether or

not our adaptation of the MKT measures was successful by bringing in the voices of the test-takers (Fauskanger, Jakobsen, Mosvold, & Bjuland, 2012). In our previous analyses of these

interviews (e.g., Fauskanger, 2012; Fauskanger & Mosvold, 2010), we learned that the

practicing teachers also discussed different aspects of the knowledge they found relevant and

irrelevant for their work as teachers—including aspects related to mathematical definitions

(Fauskanger, 2012). For the purpose of this article—and in order to learn more about the

Norwegian teachers’ beliefs concerning MKT definitions—we decided to make a new

analysis of the transcripts focusing on what was actually discussed related to definitions.

Participants

Fifteen teachers participated in seven semi-structured focus-group interviews, and theseteachers were selected from a convenience sample of schools and teachers. All the

participants had a special interest in mathematics and mathematics teacher education. The

first two interviews were held at the university, whereas the other five were held at the

teachers’ respective schools. The first group consisted of two experienced teachers, whereas

the second group consisted of three inexperienced teachers. The participants in these two

groups were selected on the basis of their level of experience and special interest in

mathematics education, and were all from different schools. In the next five interviews, pairs




of teachers from five schools were selected for participation in collaboration between the

school principals and the researchers; these five schools were selected out of the total sample

of 17 schools that participated in our pilot study.

In the first focus-group interview (FGI1), Eric and Eve participated. Both were

experienced mathematics teachers. In the second interview (FGI2), three inexperienced

teachers participated: Ingrid, Ingeborg and Ingfrid. In the third focus-group interview, the twoteachers from Beta School were both responsible for mathematics teaching in their school.

Betty was teaching mathematics in Grade 6 at the moment, whereas Benjamin had an

administrative position and was not teaching that year. Both teachers in the fourth

interview—at Zeta High—had finished their teacher education not long ago. The teachers

from Zeta High were given the following nicknames in our data: Zachariah and Zelda. In

Kappa High—which was where the fifth interview was held—Karen and Ken participated in

the interviews. Matthew was one of the participating teachers from Mu School in the sixth

interview, and he had lots of experience as a teacher. His colleague, Mary, was less

experienced. In the seventh and final focus-group interview—which was held at Nu High—

Nigel and Nora participated in the interview. Nigel had 15 years of experience as a teacher,

whereas Nora had been working as a teacher for four years. Both had taught mathematics

every year of their teaching careers.

Instrument and Procedure

Before the interviews, we used a form (Elementary form A, MSP_A04) of items from the

LMT project to measure the teachers’ MKT. These items had been translated and adapted for

use among Norwegian teachers (Fauskanger et al., 2012; Mosvold et al., 2009). The form

consisted of 30 item stems and 61 items and contained the following three sets of MKT

items: number concepts and operations (27 items), geometry (19 items), and patterns,

functions and algebra (15 items).

When they had finished the test, the teachers were given a short break. After this break,

the selected teachers were interviewed in focus groups of two or three teachers. The

interviews were designed to study our adaptation of the MKT measures, and questions were

asked about the following: a) teachers’ background, b) general considerations of the MKT

measures, c) particular considerations in relation to the MC format, d) comments on the

mathematical topic, structure and difficulty item by item, and e) comments and reflections

that supplement the other issues discussed in the interviews (Fauskanger et al., 2012).

Data Analysis

The focus-group interviews were recorded and transcribed verbatim; these transcriptions

were analyzed using a combination of two different approaches to qualitative content

analysis. As part of the data reduction—and in order to learn more about what the teachers

said about definitions—a summative qualitative content analysis was first applied to the data

(Hsieh & Shannon, 2005). We began by identifying all that was discussed related to MKT

items focusing on definitions, and all that was said related to definitions when discussing

other items as well. Both authors of this article carried out independent analysis of the data to

ensure reliability. One carried out content analysis with the aid of the computer software




NVivo10 (QSR International), whereas the other carried out his analysis using open source

tools for text analysis. Both authors searched the transcripts for occurrences of the word

‘definition’ and derived terms. In this part of the analysis, we defined the utterance as a

coding unit; the context unit was defined as two utterances before and after the utterance in

which the key word appeared (Krippendorf, 2004). When reading the transcripts, we

discovered that words like ‘concept’ and ‘formula’ were used more or less as synonyms of‘definition’. We therefore searched the transcripts for these terms as well. In our separate

analyses, we ended up with an almost perfect overlap of excerpts from the transcripts. These

excerpts (the context units) have been subject to further qualitative content analysis and

discussion below. In this second part of the data analysis, we used a more conventional

content analysis (Hsieh & Shannon, 2005), and categories were developed inductively. In the

results section some of the transcripts have been slightly adapted to avoid gap fillers and

repetitions.

Results

When analyzing our interview data to investigate what teachers’ reflections on MKT

items reveal concerning their beliefs about MKT definitions, we ended up with two partially

overlapping categories. Some teachers seemed to believe that knowledge of definitions is an

important part of their MKT. Other teachers seemed more reluctant, and—although they

might believe that knowledge of definitions is important—they argued that teachers do not

actually need to remember the mathematical definitions or formulas in order to be good

teachers. Below is a presentation and discussion of the results from our analysis.

Knowledge of Definitions is an Important Part of Teachers’ MKT?

The teachers discussed definitions, concepts as well as formulas, and algorithms in all the

interviews. There were negative statements concerning definitions in all the interviews, and

there were positive statements about definitions in all but one interview. Further analysis of

these statements revealed different aspects of teachers’ beliefs concerning MKT definitions.

1) Definitions are important. In most of the interviews, teachers made statements

indicating a belief that knowledge of definitions is an important aspect of teachers’

knowledge. In the discussions between the interviewer and the two teachers from Beta

School—they discussed a testlet item focusing on non-existing geometrical figures (testlet 17

in our form)—we can see how one of the teachers emphasizes knowledge of definitions:

153. Interviewer: You suggest, in a way, more of the kind of tasks that focus on definitions,

and less of the kind of tasks that focus on calculations, then?154. Betty: Yes, I think that is correct.

155. Benjamin. Definitions are incredibly important as a prerequisite, because if you don’t

have clear definitions and know a little about it, then you will easily be out of track.

156. Betty: And, what was said after the TIMSS study, what I have heard anyway, is that we

score low on concepts. So, I believe it is more important to be clear about this than to be able

to calculate correctly. [FGI3, Beta School, March 2, 2009]




Just prior to this, the teachers have been discussing the previous couple of items. In

relation to an item focusing on special cases in geometry (testlet 15), the teachers have just

argued that knowledge of definitions and concepts are important. When discussing item 17 in

the dialogue above, Benjamin argues that a teacher will be out of track if he does not have

clear definitions (155). These teachers’ beliefs seem to include knowledge related to defining

concepts as a prerequisite for teaching “on the track”.Benjamin here—arguing that knowledge of definitions is important—appears to be in line

with research on mathematical definitions (e.g., Zazkis & Leikin, 2008). He contends that

knowledge of definitions is an important prerequisite for teachers, and this is also in

concurrence with the way Ball and colleagues (2008) present the task of teaching related to

definitions. The actual task of teaching is formulated as “choosing and developing useable

definitions”. In order for a teacher to be able to do this, knowing the actual definitions is

necessary.

2) Remembering definitions is not important. Although teachers in all the interviews

appeared to believe that knowledge of definitions is important for mathematics teachers, not

everyone seemed to agree with Benjamin’s views. Several teachers maintained that

remembering the actual definition is less important for them, and some of the teachers said

explicitly that knowing the formula or definition is not an important aspect of teachers’

knowledge.

When discussing a testlet focusing on student-made definitions and how they would meet

the students’ suggestions, the teachers from Zeta High said:

193. Zachariah: Yes, there you have definitions again (…). How do you define polygons and

parallelograms versus rectangles [inclusive definitions] (…) What is the established

[definition]?

194. Interviewer: Mmm.

195. Zachariah: The point is, I do not have [know the definition] (…).196. Interviewer: So you are uncertain about the definition (…) Like, what is the formal

definition?

197. Zachariah: Some [definitions] are OK (…), like equilateral right-angled triangle…

198. Zelda: When I, yes… If I study these students’ proposals [presented in the MKT items

discussed] to plan my teaching the next day, I would have looked it [the definitions] up in a

book (…) I do not go round remembering this. Maybe when I have taught for 20 years I will

have looked it up enough times to remember it, but right now I do not have room for this

information. [FGI4, Zeta High, March 5, 2009]

The teachers at Zeta High seem to believe that remembering definitions is not an

important part of their MKT (198), and Zelda’s apparent base for this argument is that she

can always look up the definitions in books when preparing her lessons (198). On the other

hand, Zachariah seems to believe that it is beneficial to remember some definitions—like that

of the equilateral right-angled triangle (197). A possible explanation might be that

Zachariah’s belief that it is not important to remember the definitions is related to his lack of

knowledge on this—and the belief might then be interpreted as a kind of defense mechanism.

Another possible explanation is that he says: “some are OK” because they are easy to

remember or because they are relevant for his students.




This brings us into a discussion concerning the nature and properties of knowledge (e.g.,

Hofer & Pintrich, 1997), and it initiates a discussion of whether or not it is possible to know a

definition without actually remembering it (Zazkis & Leikin, 2008). Some of the other

teachers in our study had a clear opinion about this.

To be able to engage students in defining concepts rather than learning about

definitions—as emphasized by De Villiers (1998)—teachers need to know definitions ofmathematical concepts as well as the processes of defining (Zazkis & Leikin, 2008). If

teachers hold the belief that knowing definitions is not an important part of their MKT, they

might struggle to learn the definitions and engaging students in this particular way might be

impossible.

Choosing and Developing Useable Definitions

Ball and colleagues (2008) formulate the task of teaching that relates to definitions by

using the keywords: choosing, developing and useable. In our interviews, the teachers made

some statements that are related to this. It appears from our analysis of the interview data,

that some teachers believe mathematical definitions are more important in the highergrades—and that the mathematically correct definitions could be confusing to their younger

students.

1) Adjusting to different groups of students. In their discussion of a testlet related to

definitions of quadrangles—the same item that was discussed by Zachariah and Zelda

above—the teachers from Kappa High said:

76. Karen: I think they [the MKT measures] should have been differentiated... As an example

if one can have a rectangle that is not a parallelogram and that stuff [definitions of

quadrangles]. (...). But we do not have [teach] it [definitions of different quadrangles] for the

younger ones [students] we teach.

77. Ken: No, exactly. [FGI5, Kappa High, March 9, 2009]

This statement from Karen (76)—when seen in its context—can be interpreted as an

argument against a focus on definitions in the lower grades. Leikin and Zazkis (2010)

described as part of teachers’ pedagogical content knowledge their “ability to match the

teaching of definitions and defining with a particular classroom and to attend to students’

ability levels, affective needs and motivation” (p. 454). In the excerpt above, however, it

seems more like Karen argues that they do not have to teach definitions of different

quadrangles with their students, and the teacher therefore does not need to know about this.

With reference to the MKT framework, however, one might argue that teachers need to know

the mathematical definitions if they are going to be able to choose and develop definitions

that are appropriate for their students. The teachers’ knowledge does, however, have to go

beyond the content of the particular grade level they are teaching (Ball, Thames, & Phelps,

2008).

In this connection, it should also be brought into discussion that the demand for teachers’

knowledge concerning mathematical definitions needs to be seen in relation to possible

cultural differences in teachers’ emphasis on learning definitions by heart. Mathematics

curricula vary in their emphasis on knowing and remembering definitions across countries




(e.g., Ng, 2012), and such cultural differences in the content domain might also be reflected

in cultural differences regarding teachers’ beliefs about the content of teaching knowledge.

2) Inclusive definitions are confusing. When discussing whether or not the suggested

definitions of quadrangles would be useable among their students, the teachers from Zeta

High argued:

204. Zachariah: Hmm, in the case of our students, I would never have said that a

parallelogram could—in any kind of definition—be mixed with a rectangle. When I

immediately say that they’d be completely confused. Whether that is the right definition, I

don’t know that. I don’t know the answer to that right now. But when I explain what a

rectangle is, then I say that: this is a rectangle where you have two sides/edges that are equally

long, two [more] sides/edges that are equally long, but the ratio between the two are not

always the same. In a parallelogram you have the shift (…) If I start to bring in definitions

claiming it might be like this, and it might be like that—but not always like that—but if we

touch it from this angle....

205. Interviewer: Yes. Do you agree with what he said?

206. Zelda: Yes, I have skimmed the cream a little now, no need to go deeper into it than what

is usually needed to solve the tasks. That might be something you explain individually to

those who handle it... [FGI4, Zeta High, March 5, 2009]

Zachariah—when discussing the inclusive definition of quadrangles above—seemed to

believe that one particular definition is correct (194). In this excerpt, however, the same

teacher appears to open up to the possibility that there are cultural differences when it comes

to mathematical definitions (204). This might be interpreted as a belief concerning the nature

of mathematics, but it might also be interpreted as an indication of cultural differences

concerning the use of definitions. In any case, this is only a minor observation and it was the

only occurrence of such a discussion in our interviews. Since the knowledge required for

teaching seems be more culturally based than pertaining simply to mathematical knowledge(Stylianides & Delaney, 2011), however, cultural aspects related to MKT definitions are

important to study further.

Concluding Discussion

Research on mathematics teachers’ knowledge has been thriving for decades, and a large

amount of studies build upon the foundations laid by Shulman (1986). The attempt by Ball

and her colleagues at the University of Michigan to develop a practice-based theory of

mathematical knowledge for teaching (Ball, Thames, & Phelps, 2008) is widely

acknowledged, and their theory represents an important extension of our understanding of

mathematics teachers’ knowledge. The theory has been criticized, however, and one issue

that has received criticism is the lack of inclusion of beliefs (e.g., Beswick, 2011, 2012).

Despite the large amount of research concerning beliefs and knowledge, researchers have still

not reached a consensus regarding the relationship between the two. Some argue that the two

are closely connected (e.g., Furinghetti & Pehkonen, 2002), whereas others propose that a

distinction should be made between the two (e.g., Philipp, 2007). In this article, we have

followed Philipp’s (ibid.) advice and distinguished between beliefs and knowledge.




When regarding beliefs and knowledge as two distinct categories, it makes sense to

investigate beliefs about knowledge. Beliefs about knowledge—often referred to as

epistemological beliefs—have been studied by researchers for a long time (e.g., Perry, 1970;

Schommer, 1994). We build upon the suggestion by Buehl, Alexander and Murphy (2002)

that epistemological beliefs are domain specific, and we thus argue that it makes sense to

study teachers’ beliefs about MKT. Fives and Buehl (2010) proposed that teachers’ beliefsabout knowledge they needed as teachers represented a distinct domain of teacher beliefs. We

support that, and we have tried to take this idea one step further in this article.

Previous research on mathematics teachers’ beliefs have often focused on teachers’

beliefs about: i) the nature of mathematics, ii) mathematics teaching, or iii) mathematics

learning (Beswick, 2012). In this article, we propose an extension of these categories, and we

suggest that beliefs about the knowledge needed for teaching mathematics should also be

included (see table 2).

Table 2

Extension of Beswick’s (2012) categories of teacher beliefs

Beliefs mathematics Beliefs about

mathematics teaching

Beliefs about

mathematics learning

Beliefs about MKT

Instrumentalist Content for performance Mastery of skills Remembering content

Platonist Content with

understanding

Construction of

understanding

Understanding content

Problem solving Learner focused Autonomous

exploration

Adjusting and

differentiating In our analysis, we have focused on teachers’ beliefs about the mathematical knowledge

needed to teach definitions. Most of the teachers in our study expressed beliefs about theimportance of such knowledge. In their discussions, however, differences appeared regarding

their understanding of what this meant. One teacher, Benjamin, argued that teachers need to

“have clear definitions and know a little about it”. “Have” and “know” means different things

for different teachers, and this relates to Ernest’s (1989) categories of beliefs about

mathematics learning (second column from the right in table 2). Some teachers expressed

beliefs supporting the idea that knowledge of definitions includes remembering them,

whereas others, like Zachariah, did not seem to believe that knowing the actual definitions is

important.

Teachers like Zachariah might hold beliefs that indicate an emphasis on understanding the

content more than simply mastering the skills and remember facts. Zachariah and his

colleague Zelda also seemed to be more concerned about adjusting the definitions to their

particular groups of students. Zachariah argued that some definitions—like inclusive

definitions—can be confusing for students, and we can interpret this as a belief concerning

MKT that implies a focus on adjusting and differentiating the content. Ball, Thames and

Phelps (2008, p. 400) presented “choosing and developing useable definitions” as a

mathematical task of teaching, and this might include adjusting them in order to be more

appropriate to students. This also fits well with the beliefs expressed by Karen and Ken. They




argued that inclusive definitions—although mathematically correct—are not necessarily

appropriate to introduce to students’ in lower grades. Both of these examples also indicate a

connection between teachers’ beliefs about teaching and their beliefs about MKT.

Based on the results from our analysis of these teachers’ beliefs concerning this specific

aspect of MKT, we suggest that a more general category of teacher beliefs should also be

considered for inclusion in an extended version of Beswick’s (2012) table. We have labeledthe category “Beliefs about MKT”, and we propose a set of sub-categories (see the right

column of table 2). We suggest that the beliefs in the same row are still theoretical consistent

across the table, and we suggest that the beliefs in the same column constitute a continuum.

This does not imply, however, that individual teachers’ beliefs are consistent across

categories (Beswick, 2012).

In this study, we analyzed data from focus-group interviews with Norwegian teachers

who had been measured with a set of adapted MKT items. This approach differs from a

traditional use of MKT items, and it also differs from a more traditional approach to

investigating teachers’ epistemological beliefs (e.g., Fives & Buehl, 2008). We suggest,

however, that such an approach might be useful to investigate further. When asking teachers

to comment on items that have been developed to measure MKT, the context for discussing

beliefs about MKT has been clearly defined. The discussions that naturally emerge in such a

context—e.g. the discussions about definitions in particular—can, we argue, provide

interesting information about the teachers’ beliefs concerning these particular issues. There

is, however, a need for further research in order to investigate whether or not the more

generalized categories that we have suggested can also be found when analyzing beliefs

about other aspects of MKT. Such studies can also delve deeper into the discussions

concerning the role of beliefs in relation to MKT.

Finally, we want to make a comment regarding the cultural issue. This study was made in

a Norwegian context, and other researchers, like Ng (2012) and Cole (2012), have suggested

that there are cultural differences in the use of definitions and in how student developed

algorithms are emphasized. Such differences might also influence teachers’ beliefs about

MKT, and further research is needed in order to learn more about the influence of such

cultural differences in teaching practice on teachers’ beliefs about MKT. This is also related

to an even bigger question about possible cultural differences in MKT as such.

References

Ball, D. L., & Hill, H. C. (2008). Mathematical knowledge for teaching (MKT) measures -

Mathematics released items 2008 . Ann Arbor: University of Michigan.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What

makes it special? Journal of Teacher Education, 59(5), 389–407.

Bendixen, L. D., & Feucht, F. C. (2010). Personal epistemology in the classroom: theory,

research, and implications for practice. Cambridge: Cambridge University Press.

Beswick, K. (2011). Knowledge/beliefs and their relationship to emotion. In K. Kislenko

(Ed.), Current state of research on mathematical beliefs XVI: Proceedings of the MAVI-




16 conference June 26-29, 2010 (pp. 43–59). Tallinn, Estonia: Institute of Mathematics

and Natural Sciences, Tallinn University.

Beswick, K. (2012). Teachers’ beliefs about school mathematics and mathematicians’

mathematics and their relationship to practice. Educational Studies in Mathematics,

79(1), 127–147.

Blömeke, S., & Delaney, S. (2012). Assessment of teacher knowledge across countries: a

review of the state of research. ZDM – The International Journal on Mathematics

Education, 44(3), 223–247

Blömeke, S., Hsieh, F. J., Kaiser, G., & Schmidt, W. H. (Eds.) (2014). International

perspectives on teacher knowledge, beliefs and opportunities to learn. TEDS-M results.

Dordrecht, The Netherlands: Springer.

Bryman, A. (2004). Social research methods (2nd Ed .). New York: Oxford University Press.

Buehl, M. M., Alexander, P. A., & Murphy, P. K. (2002). Beliefs about schooled knowledge:

Domain specific or domain general? Contemporary Educational Psychology, 27 (3), 415–

449.

Buehl, M. M., & Fives, H. (2009). Exploring teachers' beliefs about teaching knowledge:

Where does it come from? Does it change? The Journal of Experimental Education,

77 (4), 367–408.

Cole, Y. (2012). Assessing elemental validity: The transfer and use of mathematical

knowledge for teaching measures in Ghana. ZDM – The International Journal on

Mathematics Education, 44(3), 415–426.

Cooney, T. J. (1985). A beginning teacher’s view of problem solving. Journal for Research

in Mathematics Education, 16 (5), 324–336.

De Villiers, M. (1998). To teach definitions in geometry or teach to define? In A. Olivier &

K. Newstead (Eds.), Proceedings of the Twenty-second International Conference for the

Psychology of Mathematics Education (Vol. 2, pp. 248–255). Stellenbosch: University of

Stellenbosch.

Delaney, S. F. (2008). Adapting and using U.S. measures to study Irish teachers’

mathematical knowledge for teaching. Unpublished doctoral dissertation, University of

Michigan, Ann Arbor.

Ernest, P. (1989). The impact of beliefs on the teaching of mathematics. In P. Ernest (Ed.),

Mathematics teaching: The state of the art (pp. 249–253). New York, NY: Falmer.

Fauskanger, J. (2012). “For norske lærere har stort sett en algoritme” – om

undervisningskunnskap i matematikk [“Because Norwegian teachers normally have an

algorithm” – about mathematical knowledge for teaching]. In F. Rønning, R. Diesen, H.

Hoveid & I. Pareliussen (Eds.), FoU i Praksis 2011. Rapport fra konferanse om

praksisrettet FoU i lærerutdanning [Report from the conference on practice-based

research and development in teacher education] (pp. 129–141). Trondheim: Tapir

Akademisk Forlag.




Fauskanger, J., Jakobsen, A., Mosvold, R., & Bjuland, R. (2012). Analysis of psychometric

properties as part of an iterative adaptation process of MKT items for use in other

countries. ZDM – The International Journal on Mathematics Education 44(3), 387–399.

Fauskanger, J., & Mosvold, R. (2010). Undervisningskunnskap i matematikk: Tilpasning av

en amerikansk undersøkelse til norsk, og læreres opplevelse av undersøkelsen

[Mathematical knowledge for teaching: Adaptation of an American instrument intoNorwegian, and teachers' experience of the instrument]. Norsk Pedagogisk Tidsskrift,

94(2), 112–123.

Fives, H., & Buehl, M. M. (2008). What do teachers believe? Developing a framework for

examining beliefs about teachers’ knowledge and ability. Contemporary Educational

Psychology, 33(2), 134–176.

Fives, H., & Buehl, M. M. (2010). Teachers’ articulation of beliefs about teaching

knowledge: conceptualizing a belief framework. In L. D. Bendixen & F. C. Feucht (Eds.),

Personal epistemology in the classroom (pp. 470–515). New York, NY: Cambridge

University Press.Forgasz, H. J., & Leder, G. C. (2008). Beliefs about mathematics and mathematics teaching.

In P. Sullivan & T. Wood (Eds.), The international handbook of mathematics teacher

education (Vol. 1 - Knowledge and Beliefs in Mathematics Teaching and Teaching

Development, pp. 173–192). Rotterdam, The Netherlands: Sense Publishers.

Furinghetti, F., & Pehkonen, E. (2002). Rethinking characterizations of beliefs. In G. C.

Leder, E. Pehkonen & G. Törner (Eds.), Beliefs: A Hidden Variable in Mathematics

Education? (pp. 39–57). The Netherlands: Kluwer Academic Publishers.

Graeber, A., & Tirosh, D. (2008). Pedagogical content knowledge. Useful concept or elusive

notion. In P. Sullivan & T. Wood (Eds.), The international handbook of mathematics

teacher education (Vol. 1 - Knowledge and beliefs in mathematics teaching and teaching

development, pp. 117–132). Rotterdam/Taipei: Sense Publishers.

Hiebert, J., Gallimore, R., Garnier, H., Givving, K. B., Hollingsworth, H., Jacobs, J., et al.

(2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 video

study, NCES (2003-013). U.S. Department of Education. Washington, DC: National

Center for Education Statistics.

Hill, H. C., Schilling, S. G., & Ball, D. L. (2004). Developing measures of teachers'

mathematical knowledge for teaching. The Elementary School Journal, 105(1), 11–30.

Hofer, B. K. (2002). Personal epistemology as a psychological and educational construct: An

introduction. In B. K. Hofer & P. R. Pintrich (Eds.), Personal epistemology: The

psychology of beliefs about knowledge and knowing (pp. 3–14). Mahwah, NJ: Lawrence

Erlbaum Associates Publishers.

Hofer, B. K., & Pintrich, P. R. (1997). The development of epistemological theories: Beliefs

about knowledge and knowing and their relation to learning. Review of Educational

Research, 67 (1), 88–140.




Hsieh, H.-F., & Shannon, S. E. (2005). Three approaches to qualitative content analysis.

Qualitative Health Research, 15(9), 1277–1288.

Knapp, J. (2006). A framework to examine definition use in proof. In S. Alatorre, J. L.

Cortina, M. Sáiz, and A. Méndez (Eds.), Proceedings of the 28th annual meeting of the

North American Chapter of the International Group for the Psychology of Mathematics

Education (Vol. 2, pp. 15–22). Mérida, México: Universidad Pedagógica Nacional.

Krippendorff, K. (2004). Content analysis: An introduction to its methodology (2nd Ed .).

Thousand Oaks, CA: Sage Publications.

Kuntze, S. (2011). Pedagogical content beliefs: Global, content domain-related and situation-

specific components. Educational Studies in Mathematics, 79(2), 1–20.

Leikin, R., & Zazkis, R. (2010). On the content-dependence of prospective teachers’

knowledge: A case of exemplifying definitions. International Journal of Mathematical

Education in Science and Technology, 41(4), 451–466.

Morgan, D. L. (1998). The focus group guidebook - Focus group kit 1. Thousand Oaks,

California: Sage Publications.

Morris, A. K., Hiebert, J., & Spitzer, S. M. (2009). Mathematical knowledge for teaching in

planning and evaluating instruction: What can preservice teachers learn? Journal for

Research in Mathematics Education, 40(5), 491–529.

Mosvold, R., Fauskanger, J., Jakobsen, A., & Melhus, K. (2009). Translating test items into

Norwegian – without getting lost in translation? Nordic Studies in Mathematics

Education, 14(4), 101–123.

Ng, D. (2012). Using the MKT measures to reveal Indonesian teachers’ mathematical

knowledge: challenges and potentials. ZDM – The International Journal on Mathematics

Education, 44(3), 401–413.

Op’t Eynde, O., De Corte, E., & Verschaffel, L. (2002). Framing students’ mathematics-

related beliefs. In G. C. Leder, E. Pehkonen & G. Törner (Eds.), Beliefs: A hidden

variable in mathematics education? (pp. 13–37). The Netherlands: Kluwer Academic

Publishers.

Op’t Eynde, P., De Corte, E., & Verschaffel, L. (2006). Epistemic dimensions of students’

mathematics-related belief systems. International Journal of Educational Research,

45(1–2), 57–70.

Pehkonen, E. (2008). State-of-the-art in mathematical beliefs research. In M. Niss (Ed.),

ICME-10 Proceedings and Regular Lectures (pp. 1–14). Copenhagen: ICME-10.

Perry, W. G. (1970). Forms of intellectual and ethical development in the college years: A

scheme. New York: Holt, Rinehard, and Winston.

Philipp, R. A. (2007). Mathematics teachers’ beliefs and affect. In F. K. Lester (Ed.), Second

handbook of research on mathematics teaching and learning (pp. 257–315). Charlotte,

NC: Information Age Publishing.




Ravindran, B., Greene, B. A., & DeBacker, T. K. (2005). Predicting preservice teachers’

cognitive engagement with goals and epistemological beliefs. Journal of Educational

Research, 98 (4), 222–232.

Raymond, A. M. (1997). Inconsistency between a beginning elementary school teacher’s

mathematics beliefs and teaching practice. Journal for Research in Mathematics

Education, 28 (5), 550–576.

Rowland, T., Huckstep, P., & Thwaites, A. (2009). Developing primary mathematics

teaching. London: Sage Publishers.

Schommer, M. (1994). An emerging conceptualization of epistemological beliefs and the role

of learning. In R. Garner & P. A. Alexander (Eds.), Beliefs about text and instruction with

text (pp. 25–40). Hillsdale, NJ: Erlbaum.

Schommer-Aikins, M., Bird, M., & Bakken, L. (2010). Manifestations of an epistemological

belief system in preschool to grade twelve classrooms. In L. D. Bendixen & F. C. Feucht

(Eds.), Personal epistemology in the classroom (pp. 31–54). New York, NY: Cambridge

University Press.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational

Researcher, 15(2), 4–14.

Sinatra, G. M., & Kardash, C. A. M. (2004). Teacher candidates’ epistemological beliefs,

dispositions, and views on teaching as persuasion. Contemporary Educational

Psychology, 29(4), 483–498.

Skott, J. (2001). The emerging practices of a novice teacher: The roles of his school

mathematics images. Journal of Mathematics Teacher Education, 4(1), 3–28.

Stylianides, A. J., & Delaney, S. (2011). The cultural dimension of teachers’ mathematical

knowledge. In Rowland T. & Ruthven K. (Eds.), Mathematical knowledge in teaching

(pp. 179–191). London: Springer.

Thompson, A. G. (1984). The relationship of teachers’ conceptions of mathematics and

mathematics teaching to instructional practice. Educational Studies in Mathematics,

15(2), 105–127.

Thompson, A. G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In

D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A

project of the National Council of Teachers of Mathematics (pp. 127–146). New York:

Macmillan.

Usiskin, Z., & Griffin, J. (2008). The classification of quadrilaterals: A study in definition.

Charlotte, NC: Information Age publishing.

Van Zoest, L., Jones, G., & Thornton, C. (1994). Beliefs about mathematics teaching held by

pre-service teachers involved in a first grade mentorship program. Mathematics

Education Research Journal, 6 (1), 37–55.

Zazkis, R., & Leikin, R. (2008). Exemplifying definitions: A case of a square. Educational

Studies in Mathematics, 69(2), 131–14.




Authors

Reidar Mosvold, Associate Professor, Department of Education and Sports Science,

University of Stavanger, 4036 Stavanger, Norway; [email protected]

Janne Fauskanger, Assistant Professor, Department of Education and Sports

Science, University of Stavanger, 4036 Stavanger, Norway;

[email protected]




The Development of Students’ Algebraic Proficiency

Irene van Stiphout

Eindhoven School of Education

Paul Drijvers

Utrecht University

Koeno Gravemeijer

Eindhoven School of Education

Students’ algebraic proficiency is debated worldwide. To investigate the development of algebraic

proficiency in Dutch secondary education, we set up a study, in which 1020 students in grades 8 – 12

took four algebra tests over a period of one year. Rasch analysis of the results shows that the students

do make progress throughout the assessment, but that this progress is small. A qualitative analysis of

test items that invite structure sense reveals that students’ lack of structure sense may explain the

results: the majority of the students were not able to deal flexibly with the mathematical structure of

expressions and equations. More attention to structure sense in algebra education is recommended.

Keywords: algebra; algebraic skills; Rasch scale; secondary education; structure sense; symbol sense

Student achievement in algebra is a worldwide concern. International comparative studies

such as the Trends in International Mathematics and Science Study (TIMSS) and the

Programme for International Student Assessment (PISA) induced review studies on how to

improve students' algebraic proficiency (e.g., National Mathematics Advisory Panel, 2008). It

is widely accepted that this proficiency includes not only procedural skills but also involvesalgebraic insights.

In the Netherlands, the discussion on algebraic proficiency focused on the level of basic

algebraic skills in the transition from secondary education to higher education. Complaints

were heard that students are not proficient in basic algebraic algorithms and cannot apply

them correctly. As a result, educators and politicians called for a stronger emphasis on

procedural skills (Van Gastel et al., 2007). The following examples from McCallum (2010),

however, illustrate that the demands in higher education exceed the level of superficial

procedural fluency:

• recognizing that ⋅ (1 + ) is linear in P (finance);

• identifying ()() as being a cubic polynomial with leading coefficient

(calculus);

• observing that 1 − () vanishes when v = c (physics);

• understanding that

√ halves when n is multiplied by 4 (statistics).



63 I. Van Stiphout, P. Drijvers & K. Gravemeijer

To address this worldwide and national debate on algebra achievement, and on procedural

fluency and conceptual understanding in particular, we decided to investigate the

development of algebraic proficiency in Dutch secondary education.

What is Algebraic Proficiency?

As an introduction to our research, we will first discuss what we understand by

proficiency. In doing so, we focus on two aspects, namely the relation between procedural

fluency and conceptual understanding, and the notion of structure sense.

The Relation between Procedural Fluency and Conceptual Understanding

The distinction between procedural fluency and conceptual understanding is central in

discussions on algebraic proficiency. Skemp (1976) distinguished knowing how to apply the

rules and algorithms correctly (instrumental understanding) and knowing both what to do and

why (relational understanding). Kilpatrick et al. (2001) see procedural fluency and conceptual

understanding as two of five strands of mathematical proficiency, along with strategic

competence, adaptive reasoning, and productive disposition. To Hiebert and Lefevre (1986),

procedural knowledge comprises the formal language (including the symbols), and

algorithms and other rules.

It is widely accepted that procedural fluency and conceptual understanding have to go

hand in hand: algebraic expertise encompasses a continuum which ranges from basic skills

such as procedural work for which a local focus and algebraic calculations suffice, to

strategic work which requires a global focus and algebraic reasoning and conceptual

understanding. The latter aspects are probably the hardest to learn and to teach, but at the

same time, the above examples of McCallum (2010) show the importance of flexible skills

such as the ability to read through symbolic expressions and to cleverly select and use

symbolic representations. In line with this, Sfard and Linchevski (1994) argue that flexible

manipulation skills can be seen as a function of the versatility of available interpretations, and

the adaptability of the perspective. In their view, these abilities are part of a structural mode

of thinking.

Symbol Sense and Structure Sense

To capture the flexible skills that are involved in algebraic proficiency, Arcavi (1994,

2005) introduced the notion of symbol sense. He defines symbol sense as a complex feel for

symbols that includes a positive attitude towards symbols and a global view (or Gestalt view)

of expressions. Part of this global view is the ability to read through symbols. As an example

of this, Arcavi (1994) discusses the equation (2x+3)/(4x+6) = 2. Reading through the symbols

reveals that the left-hand side of the equation equals for all x ≠ −1

, because the numerator

equals half the denominator. Inspecting the equation before starting to solve it with the

purpose of gaining a feeling for the meaning of the problem is seen as an instance of symbol

sense.

As a second example, Arcavi (1994) discusses Wenger’s equation, v√ u = 1 + 2v√ 1 + u,

which is to be solved for v (Wenger, 1987). The difficulty here is to recognize this equation



THE DEVELOPMENT OF STUDENTS’ ALGEBRAIC PROFICIENCY 64

as linear in v and to overcome the visual salience of the square roots in the equation (Kirscher

& Awtry, 2004). This requires identifying parts of the expression as units, an ability that is

referred to as the Generalized Substitution Principle (Wenger, 1987).

To solve Wenger’s equation, the ability to recognize its – in this case linear – structure is

crucial. This ability is labeled as structure sense. The term structure sense is introduced by

Linchevski and Livneh (1999, p. 191) to describe the ability “to use equivalent structures ofan expression flexibly and creatively.” In high school algebra, structure sense encompasses a

collection of abilities, such as: recognize a structure, see a part of an expression as a unit;

divide an expression into meaningful sub-expressions; recognize which manipulation is

possible and useful to perform; and choose appropriate manipulations that make the best use

of the structure (Hoch & Dreyfus, 2004, 2006). Novotná and Hoch (2008) define structure

sense as students’ ability to (1) recognize a familiar structure in its simplest form, (2) deal

with a compound term as a single entity and through an appropriate substitution recognize a

familiar structure in a more complex form, and (3) choose appropriate manipulations to make

best use of a structure.

We believe that structure sense, as defined by Novotná and Hoch (2008), is such an

important aspect of algebraic proficiency that it is worth studying in more detail. In contrast

to Novotná and Hoch, we consider structure sense to be part of symbol sense rather than

being separated ability, namely the part of symbol sense that involves seeing structures and

patterns in algebraic expressions and equations, which is needed while carrying out algebraic

manipulations such as simplifying expressions and solving equations.

Research Questions

In light of the above, we have formulated the following research questions.

1. How does students’ algebraic proficiency develop from a cross-sectionalperspective?

2. How does students’ algebraic proficiency develop from an individual perspective?

3. How does students’ algebraic proficiency develop in terms of structure sense?

How to Investigate Research Questions?

To address these questions, a set of test items was designed. Four tests consisting of

subsets of these items were administered over a calendar year in a partly cross-sectional and

partly longitudinal design. Within the cross-sectional perspective, we first determined thedistributions of the scores of groups of students at a given times, and then compared those

distributions. Within the longitudinal perspective, we looked at how the scores of individual

students developed.




Test Design

Because the complaints made by students as well as educators primarily concern

algebraic skills taught at the lower secondary level, the algebra test items focus on algebraic

skills taught in grades 7 – 9 (Van Gastel et al., 2007, 2010). These items are based on the

attainment targets formulated by the Dutch Ministry of Education, Culture and Science,

combined with the theoretical considerations discussed above, and are inspired by Dutchtextbook series and literature. In Dutch upper secondary education, students choose one of

four different streams, each with a different mathematics curriculum. Because of these

different programs, the items cover only the common algebra topics, which include

expanding brackets, simplifying expressions, and solving equations. The algebra items

included range from basic skills to symbol sense in general, and structure sense in particular.

The complete list of algebra tasks can be found in the Appendix. In addition, numerical tasks

are included that relate to the transition from arithmetic to algebra. These items are not

addressed in this article; they are, however, included in the Rasch scales discussed below.

From the set of items, four tests were composed. Each of these tests consisted of 12 to 16

items and was designed to be completed in half an hour, in order not to overburden thestudents and teachers. The four tests consisted of open questions, to be worked out with paper

and pencil. In this way, we avoided students’ guessing answers. During the tests, calculators

or notes were not allowed.

Test Administration

We assessed students in March 2008, May 2008, October 2008, and February 2009.

Students of grades 8, 9, 10 and 11 (ages 13-16) participated in the first and second

assessment. After the summer vacation, these students were in grades 9 up to 12 in October

2008 and February 2009. To be able to follow individual students, we have used similar items

in different assessments to enable an anchor design.Table 1

Number of students taking the tests

Grade March May October February Part. Part.

2008 2008 2008 2009 4 × 1≥

8/9 164 227 173 171 94 266

9/10 163 160 129 114 56 217

10/11 243 185 163 144 90 268

11/12 244 204 188 72 37 269

Total 814 776 653 501 277 1020

Table 1 provides an overview of the numbers of students who took the tests. Four schools

participated, all making use of one of the two mostly used Dutch text book series and in that




sense representative. From each school, two classes of each grade were involved. In total,

1020 students participated at least once and 277 students took all four tests.

During data collection a curriculum reform took place. The new curriculum for the grade

10/11 cohort pays more attention to algebraic skills than the old one for cohort grade 11/12,

most notably: solving equations, simplifying and calculating with fractions and square roots

(Ctwo, 2009). In the interpretation of the findings, this curriculum change will need to betaken into account.

Rasch Scales for Algebraic Proficiency

After data collection, the students’ written answers were coded 1 for correct and 0 for

incorrect. Doubtful cases were discussed with colleagues. To analyse the test results, we used

the Rasch model, a one parameter item response model (Rasch, 1980; Bond & Fox, 2007;

Linacre, 2009). With a Rasch analysis, one linear scale is created on which both persons are

situated according to their ability and items according to their difficulty. On this scale, not

only the order but also the distances between the items and the students have meaning. Rasch

theory supposes that the probability of a person giving a correct answer on an item is alogistic function of the difference between that person's ability and the difficulty of the item.

The probability niP of person n with ability n

B to correctly answer item i with difficulty i D

is given by =1

B Dn i

ni B Dn i

eP

e

−

−

+

.

Both the ability of the persons and the difficulty of the items are measured in so-called

units of log odds ratios, or logits. The local origin of the Rasch scale is usually situated in the

center of the range of item difficulties. If the ability equals the item difficulty, that is, if

=n i B D , then

0

0

1= = =

1 21

B Dn i

ni B Dn i

e eP

ee

−

−

++

.

For each assessment, including the arithmetic assignments, we created its corresponding

Rasch scale. Next, we connected the four Rasch scales by using anchor items, i.e., similar

items in the different assessments. As a result, items of all four assessments are placed on one

scale. Also, students of different assessments have a Rasch measure on one scale.

To determine which items students master, we need to decide what we view as mastery.

We consider a probability of 80% of answering an item correctly as an expression of

mastering that item. From the Rasch model it follows that a probability of 0.8 of person n

answering item i correctly corresponds to an ability n B which is 1.39 logit higher than the

difficulty i D of item i because if =1.39n i

B D− , then1.39

1 1.39= = 0.8

11

B Dn i

ni B D

n i

e eP

ee

−

− ≈

++

. As a

consequence, we consider that students with a measure at least 1.39 higher than the measure

of the item master that particular item. The fit of the Rasch model to the data was checked.

With respect to the reliability, we found values of .70 , .70 , .66 and .68 for assessments 1,

2, 3 and 4, respectively. These reliability scores can be compared to Cronbach’s alpha.




Differences between Student Cohorts

To address the first question on the development of the different cohorts of students, we

performed a cross-sectional comparison of grades 8 through 12. Figure 2 shows the

percentiles of Rasch measures in logits of the four cohorts of students. As the number of

participating students varied over the assessments, the bars between the dashed lines only

partly represent the same students.

Figure 1. Cross-sectional percentiles of all grades in all assessments.

Figure 1 shows that the averages of the different assessments generally increase with the

grades. For the central 50% of the students (the white parts of the bars), the difference

between the lowest average (grade 8, May 2008) and the highest average (grade 12, February

2009) is approximately two logits. Thus, the dispersion within the twelve or sixteen different

assignments is rather small with regard to the difference between the worst and the best

scoring student (the length of the whole bar). If we focus on 80% of the students (i.e., leaving

out the best and worst 10%), we see that they are within a range of at most 3.8 logits of each

other. Grade 9/10 performs better than grade 8/9 and grade 10/11 performs better than grade

9/10. The difference between grade 10/11 and grade 11/12 is less obvious. Here we note that

the curriculum of students in grade 11/12 differs from the curriculum of students of the other

grades on algebraic skills, which makes it hard to interprete this lack of difference between

grades 10/11 and grades 11/12.

To sum up, the cross-sectional analysis showed that there is progress and there is only

little dispersion among the central 50% of the students. There seems to be a growth in ability

between the cohort of grade 10/11 and that of grade 11/12, which might be a positive effect

of the curriculum changes in the Netherlands.




Individual Proficiency Development

Figure 2. Rasch measures of students in the first and fourth assessment ( N =390).

To address the second question on the development of individual students’ algebraic

proficiency over time, Figure 2 shows the Rasch measurements in logits of individual

students in the first assessment (horizontally) against the similar results of the fourth

assessment (vertically). Each dot represents one student who participated in both the first and

the fourth assessment. Students above the dashed line improved their performance, whereas

students below the dashed line show a decreasing ability. The open dots represent students

who did not make significant difference; the filled dots represent students who made

significant progress or retrogress. Based on the 95% confidence intervals provided by the

Rasch analysis, the results show that the majority of the students (358 out of 390) did not

make significant progress. Some students (30 out of 390) did make progress, whereas 2

students retrogressed.Summarizing, this analysis showed that individual students make progress during a

calendar year, but that only few students made significant progress.




A Closer Look on Structure Sense Items

Figure 3. February 2009 student percentiles and structure sense tasks on the Rasch scale.

To investigate students’ proficiency development in a more qualitative way, we analyze

in greater detail the tasks that are tailored to structure sense. We selected ten tasks from the

four assessments for which recognizing the algebraic structure of an expression or seeing a

part of an expression as a unit really pays off. Figure 3 provides an overview of student

measures of the fourth assessment in February 2009, and the task difficulties. For clarity’s

sake, in case of similar tasks, we included only one task in the Figure, resulting in seven

tasks: one involving simplification an expression, the other six on solving equations.

Central in Figure 3 is the horizontal axis with logits as units. The gray scaled bars above

the axis represent student ability in percentiles in the fourth assessment in February 2009.

The bars below the horizontal axis represent the difficulty of the tasks on the Rasch scale.

The left-hand side of a bar corresponds to a probability of 0.50 of answering the

corresponding task correctly, which is usually referred to as the Rasch measure of the task;

the right-hand side of the bar corresponds to a probability of 0.80 of answering that task

correctly. Tasks for which the corresponding bars lie on the left-hand side of the dashed line

are mastered by at least 50% of the grade 12 students. Figure 3 shows that all seven tasks are

mastered by less than 25% of the grade 12 students.

Below, we discuss the students’ performance on each of the seven tasks.

Items A1 and A2: Fraction in Equation

In the first assessment (March 2008), students were asked to report the first clue they

would provide a classmate to help solve the equation15

= 36

41 x

++

. The Rasch measure of

this task, A1, is 0.31− logit (probability of success 0.50 ). The way the task was formulated




left room for answers such as “I would call the teacher for help.” This kind of answer

revealed problems in the dichotomous coding of student answers. To avoid this kind of

answer, a formal version, A2, was included in the third assessment. The Rasch measure of the

latter version is 0.01 logit, so the tasks did not differ much.

To solve these equations, different strategies can be used. For example, students may use

the cover-up method and cover the denominator. In this way, the equation becomes 15 = 3W

,

which is easily solved, providing ≠ 0. Following this line of thought, the next step would

be 4 = 5+ ∆ which implies6

= 11 x+

. This equation in turn could be solved by covering the

denominator, thus yielding6

= 1◊

, which implies 1 = 6 x+ . An example of a more formal

strategy is to multiply the numerator and the denominator both with 1 x+ , or multiply both

sides of the equation with the denominator6

41 x

++

. These strategies have in common that

students have to identify a part of the equation (the denominator, or a part of thedenominator) as an object, which is seen as an expression of structure sense.

Item A3: Simplification

In the second assessment (May 2008), we asked students to simplify the expression2 2

2

5 10 2(2 4).

2

x x

x

+ − +

+

The Rasch measure of this task is 0.22− logit (probability of success 0.50 ). A similar

task, A4, with Rasch measure 0.50− , was included in the fourth assessment (February 2009).

This formula has the structure 5 4−W WW

which leads to

2

222

x x

+

+. But simply expanding the

brackets and taking similar terms together yields the same. In both cases the next step is to

recognize that this fraction consists of two similar expressions that are divided, so the fraction

yields one.

Translation:Vereenvoudig zo ver mogelijk: Simplify

Figure 4. Work of a grade 8 (left) and a grade 10 (right) student.




An example of the difficulties students experienced is presented in Figure 4. The grade 8

student rewrote the numerator2 25 10 2(2 4) x x+ − + to

2 25 8(2 4) x x+ + by erroneously taking

10 and 2− together. To this student, the algebraic structure of the numerator was not clear.

Another type of error is shown in the right screen of Figure 4. This grade 10 student correctly

simplified the expression to

2

2

2

2

x

x

+

+ . Then he concluded that

2

2

2

2

x

x

+

+ equals zero instead of

one. This error might stem from an inability to see the algebraic structure of the expression,

but other explanations such as a deep misunderstanding of fractions or the student being used

to the form “expression = 0” seem also reasonable.

Item A5: Factorization

In the second assessment, students were asked to solve the equation

( 5)( 2)( 3) = 0. x x x− + −

This requires students to identify the underlying algebraic structure, which involves three

factors on the left side of the equation, and the product of these three factors equals zero. The

structure is = 0 A B C ⋅ ⋅ , which implies = 0 A or = 0 B or = 0C . The Rasch measure of this

task is 0.68 logit (probability of success 0.50 ). A similar task, A6, was included in the fourth

assessment. This task was perceived as less difficult, with a Rasch measure of 0.60− logit,

probably due to a test-retest bias. The difficulty of these tasks can be explained by the

students' tendency to expand the brackets, after which they could not find the factorization.

Some of the students came to see the error and recovered by giving the correct answer.

Translation: Los op: Solve

Figure 5. Work of grade 12 student.

Figure 5 shows such a work of a grade 12 student who started to solve the equation by

expanding the brackets. In the second line of this work, he correctly wrote that3 2 2 2

4 3 12 4 3 12 = 0. x x x x x x x− + − − + − + Moving to the next line, he forgot the factor 12 x− , so erroneously concluded that

3 22 12 = 0 x x x− + + . In the next line, which is scratched, he started to factor out x .

Apparently, then he realized that the solutions can be found more easily and wrote the correct

solutions down.

The students' tendency to expand the brackets indicates the visual salience of the brackets.

The underlying structure of = 0 A B C ⋅ ⋅ is overlooked.




Item A8: Arcavi's Equation

In the section on structure sense, we addressed the equation (2 3) / (4 6) = 2 x x+ + . Arcavi

(1994) argues that resisting the impulse to immediately solve the equation, and instead to try

to read meaning into the symbols, is an expression of symbol sense, and in our opinion of

structure sense in particular. In this example, it requires that students recognize the form 2

A

A in the expression (2 3) / (4 6) x x+ + . To do so, it is necessary to see 2 3 x + as a unit.

However, realizing that (2 3) / (4 6) x x+ + equals1

2 is not necessary for the solving process.

Just solving the equation and finding the solution1

= 12

x − and realizing that the denominator

equals zero, is a correct procedure for solving this equation. So, reading through the equation

and seeing the structure is practical, but not necessary.

The ability to see2

A

A manifests more structure sense than finding the solution for which

the denominator equals zero and then concluding that the equation does not have a solution.

Since more structure sense is supposed to be an expression of a higher level of algebraic

proficiency, we expect students with more symbol sense to have a higher Rasch measure, and

thus we expect students who see2

A

A to have a higher Rasch measure than students who

found the solution for which the denominator equals zero. To investigate this relation

between these strategies and students’ proficiency, we performed a qualitative analysis of the

written answers of the students. As we were interested in the strategies used to solve this

equation, we restricted the analysis to the correct answers. The analysis yielded three

categories of students. First, students that argued that the quotient equals

1

2 . These students

recognized the structure2

A

A in the equation. The second category contains students who

found the solution for which the denominator equals zero. These students correctly argued

that this solution is no solution. The third category contains students who gave another

argument. For example, these students only wrote “no.” Another example is a student who

argued that (2 1) / (4 2) x x+ + equals a fraction between 0 and 1. From the 501 participating

students in the fourth assessment, 105 gave a correct solution. Half of these students used the

strategy of the first category. The other students were nearly evenly distributed between the

other categories.

Figure 6 shows the relation between strategy and ability. Again, the central line represents

the Rasch scale of proficiency. The gray scaled bars below the axis represent percentiles of

students' Rasch measures in the fourth assessment in February 2009. The dots above the axis

represent students in a particular grade with a particular strategy.

From the figure we see that, although more students solve the task correctly using the

strategy “quotient equals1

2” than using the strategy “denominator zero,” the Rasch measure




of the students using the former strategy is not higher than those using the latter. In other

words, the strategy that is supposed to be a manifestation of more symbol sense does not

imply a higher Rasch measure.

Figure 6. Students’ ability combined with the strategy used.

In our view, this means that either the relation between strategy and structure sense is not

as strict as the literature suggests, or the relation between structure sense and the underlying

latent variable of the Rasch scale is weak. In the former case, the strategy students use might

depend on the strategy they think they are expected to use based on the recipes Dutch

textbooks provide. Students might think that they have to use these recipes as part of an

(implicit) didactical contract (Brousseau, 1990). From this point of view, the more broadly

applicable strategy “denominator zero” might be viewed as more valuable, because this

strategy could also be used in case the equation had been1

(2 1) / (3 2) =2

x x+ + . However,

this single case cannot justify radical conclusions; further research on the relation between

strategies, structure sense and algebraic proficiency seems appropriate.

Items A9 and A10: Wenger's Equation

The two items with the highest difficulty, A9 and A10, are adapted versions of Wenger’s

equation. Figure 3 shows that these two tasks were far beyond the ability of all students. In

the first assessment, we included Wenger's equation in which we replaced the letters u and v

for readability with the letters a and b . We did not ask students to solve =1 2 1a b a b+ +

for a, because Dutch students are not familiar with this kind of question. Instead, we asked

students to rewrite =1 2 1a b a b+ + as an expression of the form a = …. The analysis

revealed that this way of asking is ambiguous to students. For example, students divided both

sides of the equation by b , which yields to1 2 1

=a b

ab

+ +. This is not the kind of answer

we intended to see, but somehow meets the purport of the question. Exactly two students out




of the 650 of grades 9, 10 and 11 were able to solve this equation. These two students

recognized the linear form of the equation and gave the correct answer,1

=2 1

ab b− +

.

In the third assessment, an adapted version of the equation was included. In this version,

we substituted = 2b , so the equation to be solved became 2 =1 2 3a a+ . Only 19 out of

the 653 participating students were able to solve this task.

The difficulty of Wenger’s equations is explained by students’ inability to recognize the

linear form of the equation. This requires students to sense the symbols as arranged in a

special pattern, which is an expression of structure sense. Wenger (1987) found that students

are able to perform manipulations correctly, but these manipulations do not lead to a solution.

Rather, students go round in circles, create more complex expressions and then reduce these

terms. The square roots that serve as coefficients in this equation may be interpreted by the

students as a signal to square both sides of the equation. The unfamiliarity of students with

square roots as coefficients can enhance the visual salience of square roots (Kirscher &

Awtry, 2004).

We agree that recognizing the linear form is crucial for solving the equation. However, inour view, there are other hurdles. First, the unconventional sequence of symbols in

2 =1 2 3a a+ . In this equation, the variable is followed by the numerical coefficient in the

left-hand side of the equation. In the right-hand side, the variable is in the middle of the

numerical coefficient. This is an unusual sequence of symbols for Dutch students, because in

Dutch textbook series, the numerical coefficient usually precedes the variable. In this way,

the equation would have been 2 =1 2 3a a+ . We realize that this is only a slight difference

for an expert. But we believe that to students, this change of order perhaps makes the

difference between being able or not being able to solve the equation. The role of the order of

symbols in exercises would be an interesting topic of further studies.

The second hurdle concerns the different roles of the variables a and b in the equation.

The a in the equation serves as unknown, whereas the b serves as a variable. The versatility

of the use of variables is a well known difficulty in mathematics and has been studied by

many researchers (e.g., Matz, 1982; Janvier, 1996; Rosnick, 1981; Wagner, 1983; Trigueros

& Ursini, 1999; Drijvers, 2003; Ursini & Trigueros, 2001; Schoenfeld & Arcavi, 1988). The

skill of flexibly dealing with the different roles of variables can be seen as part of a broad

view on equations. The results for these tasks suggest that students do not have such a broad

view.

Summarizing, our findings confirm that Wenger’s equation presents students with several

difficulties. These difficulties all concern the ability to recognize the linear structure of the

equation which can be seen as a part of structure sense. The students’ performance on the two

linear equations does not allow us to conclude which of these difficulties is paramount, but in

our view, solving these equations requires structure sense −which the students apparently

lack.




Conclusion and Discussion

In this article we set out to answer the following research questions.

1. How does students’ algebraic proficiency develop from a cross-sectional perspective?

2. How does students’ algebraic proficiency develop from an individual perspective?

3. How does students’ algebraic proficiency develop in terms of structure sense?

In answer to the first question, we found that the Dutch student cohorts involved in the study

made some progress. In general we concluded that students mastered simple tasks, but tasks

become too complicated rather quickly. The difference between grade 10/11 and grade 11/12

is small. This might be explained by the aforementioned curriculum change: students in grade

8/9, grade 9/10 and grade 10/11 followed a curriculum that includes more algebra than the

program of students of grade 11/12. This curriculum change might have had a positive effect

on students' algebraic proficiency of the students from grade 8/9 through grade 10/11.

Furthermore, there is little dispersion among the middle 50% of the students.

Related to the second question, the analysis of the results yields that the majority of the

individual participants did make progress from the first to the fourth assessment, so in a

period of one year. However, this progress was not significant for the majority of the

students. The answer to the third question is that there are only few tasks that were mastered

by the majority of the students. Although students showed progress both cross-sectionally

and longitudinally, this progress did not encompass the majority of the items. In other words,

the majority of the items were too difficult for students of grade 8, and were still too difficult

for students of grade 12. Furthermore, the range of items that the majority of the students

mastered did not include tasks that involve conceptual aspects of algebraic proficiency, and

for structure sense in particular. In answering the third research question, we found that the

majority of the students was not able to deal flexibly with the mathematical structure of

expressions. Also, the notion of structure sense proved valuable to interprete student results

and to explain student difficulties. For example, the inability to solve the equation( 5)( 2)( 3) = 0 x x x− + − can be understood as the inability to recognize the mathematical

structure = 0 A B C ⋅ ⋅ that implies = 0 A or = 0 B or = 0C , which is a lack of structure

sense.

To put those results in perspective, we have to take into account some limitations of the

study. First, because we did not want to place too heavy a load on the teachers and the

students, we chose to keep the number of test items relatively low. As a consequence,

differences would have had to have been quite large to be significant. Second, a curriculum

change took place during the data collection. The new curriculum pays more attention to

algebra. We took this change into account by concluding that the curriculum change might

have a positive effect if the growth continues.If we reflect on the study’s results, we consider them as disappointing in that the students

hardly develop structure sense. The results suggest that Dutch students hardly develop

structure sense as is evident from the students’ inability to

• see that 15/(4 + (6/(1 + x))) = 3 can be read as 15/W = 3, which means that W=15:3

(providing W ≠ 0);

• observe that the expressions 5 x2

+ 10 and 2 x2

+ 4 are multiples of x2

+ 2;

• recognize the expression ( − 5)( + 2)( − ) as a multiplication;




• recognize the equation of Wenger as linear.

These examples reflect the demands of higher education—as expressed by the examples of

McCallum (2010) mentioned in the introduction—which exceed the level of procedural

fluency. Students tend to choose the routine way for solving problems, and do not manage to

step out of the procedure in order to reconnect to the underlying meaning when needed

(Arcavi, 1994).The students’ tendency to focus on routine procedures might be a consequence of the

didactical contract to which the textbooks contribute. These textbooks tend to focus on the

procedures and not so much on symbol sense and structure sense (Van Stiphout, Drijvers &

Gravemeijer, 2013).

A limited feel for the structure of expressions, and for (sub)expressions as objects (Sfard

1991) is an obstacle for reaching a higher level of conceptual understanding in which the

structure and ambiguous nature of the algebraic objects are central. Reaching this higher level

is inherently difficult and involves a shift of thinking. How to reach this higher level is a core

concern of the mathematics education research community. We may conclude that the call

from educators and politicians for more attention to routine and procedural skills will not

solve the students’ problems, because the problems with the more difficult items do not

primarily stem from a lack of procedural skills, but more from a lack of conceptual

understanding.

Acknowledgements

This article is based on Chapters 2 and 3 of the first author’s PhD study: Van Stiphout, I.

M. (2011). The development of algebraic proficiency (Doctoral dissertation). Eindhoven

School of Education, Eindhoven, The Netherlands.

References

Arcavi, A. (1994). Symbol sense: Informal sense-making in formal mathematics. For the

Learning of Mathematics, 14(3), 24-35.

Arcavi, A. (2005). Developing and using symbol sense in mathematics. For the Learning of

Mathematics, 25(2), 42-47.

Bond, T.G., & Fox, C.M. (2007). Applying the Rasch model: Fundamental measurement in

the human sciences (Second Edition). Mahwah, NJ: Lawrence Erlbaum.

Brousseau, G. (1990). Le contrat didactique: le milieu. [The didactical contract: The setting].

Recherches en didactique des mathématiques, 9(3), 309-336.

Ctwo. (2009). Rapport tussenevaluatie van de 2007-programma’s wiskunde havo/vwo.

[Report on the intermediate evaluation of the 2007 programs mathematics havo/vwo].

Retrieved 15 December 2013 from http://www.ctwo.nl/.

Drijvers, P. (2003). Learning algebra in a computer environment: Design research on the

understanding of the concept of parameter (Doctoral dissertation). University of Utrecht,

The Netherlands.




Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An

introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The

case of mathematics (pp. 1-27). Hillsdale, NJ: Lawrence Erlbaum.

Hoch, M., & Dreyfus, T. (2004). Structure sense in high school algebra: The effect of

brackets. Paper presented at the 28th Conference of the International Group for the

Psychology of Mathematics Education. Bergen, Norway: PME.Hoch, M., & Dreyfus, T. (2006). Structure sense versus manipulation skills: An unexpected

result. Paper presented at the 30th Conference of the International Group for the

Psychology of Mathematics Education. Prague, Czech Republic: PME.

Janvier, C. (1996). Modeling and the initiation into algebra. In N. Bednarz et al. (Ed.)

Approaches to algebra (pp. 225-236). Dordrecht: Kluwer Academic Publishers.

Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up. Washington, D.C:

National Academy Press.

Kirshner, D., & Awtry, T. (2004). Visual salience of algebraic transformations. Journal for

Research in Mathematics Education, 35(4), 224-257.

Linacre, J. M. (2009). A user's guide to Winsteps. Retrieved August 31, 2012, from

http://www.winsteps.com/winsteps.htm

Linchevski, L., & Livneh, D. (1999). Structure sense: The relationship between algebraic and

numerical contexts. Educational Studies in Mathematics, 40(2), 173-196.

Matz, M. (1982). Towards a process model for high school algebra errors. In D. Sleeman, &

J. Brown (Eds.), Intelligent Tutoring Systems (pp. 25-50). London: Academic Press.

McCallum, W. (2010). Restoring and balancing. In Z. Usiskin, K. Andersen, & N. Zotto

(Eds.), Future curricular trends in school algebra and geometry: Proceedings of a

conference (pp. 277-286). Charlotte, NC: Information Age Publishing Inc.

National Mathematics Advisory Panel. (2008). Foundations for success: The final report of

the national mathematics advisory panel. Washington, D.C.: U.S. Department of

Education.

Novotná, J., & Hoch, M. (2008). How structure sense for algebraic expressions or equations

is related to structure sense for abstract algebra. Mathematics Education Research

Journal, 20(2), 93-104.

Rasch, G. (1980). Probalistic models for some intelligence and attainment tests. Chicago:

University of Chicago Press.

Rosnick, P. (1981). Some misconceptions concerning the concept of variable. Mathematics

Teacher , 74(6), 418-420.

Schoenfeld, A., & Arcavi, A. (1988). On the meaning of variable. Mathematics Teacher,

81(6), 420-427.

Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes

and objects as different sides of the same coin. Educational Studies in Mathematics,

22(1), 1-36.

Sfard, A., & Linchevski, L. (1994). The gains and pitfalls of reification: The case of algebra.

Educational Studies in Mathematics, 26 (2-3), 191-228.




Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics

Teaching, 77 , 1-7.

Trigueros, M., & Ursini, S. (1999). Does the understanding of variable evolve through

schooling? Paper presented at the 23rd Conference of the International Group for the

Psychology of Mathematics Education. Haifa, Israel: Technion.

Ursini, S., & Trigueros, M. (2001). A model for the use of variable in elementary algebra .Paper presented at the 25th Conference of the International Group for the Psychology of

Mathematics Education. Utrecht: Freudental Institute.

Van Gastel, L., Cuypers, H., Jonker, V., Van de Vrie, E., & Van der Zanden, P. (2007).

Eindrapport nationale kennisbank basisvaardigheden wiskunde. Amsterdam: Consortium

NKBW.

Van Gastel, L., Cuypers, H., Grift, Y., Kaper, W., Van der Kooij, H., Tempelaar, D. et al.

(2010). Aansluitmonitor Wiskunde VO-HO. Amsterdam: Consortium NKBW.

Van Stiphout, I. M. (2011). The development of algebraic proficiency (Doctoral dissertation).

Eindhoven School of Education, Eindhoven, The Netherlands.

Van Stiphout, I. M., Drijvers, P. & Gravemeijer, K. (2013). The implementation of contexts

in Dutch textbook series: a double didactical track? In Lindmeier, A.M. & Heinze, A.

(Eds.). Proceedings of the 37 th

Conference of the International Group for the Psychology

of Mathematics Education, Vol. 4, pp. 337-344. Kiel, Germany: PME.

Wagner, S. (1983). What are these things called variables? Mathematics Teacher, 76 , 474-

479.

Wenger, R. (1987). Cognitive science and algebra learning. In A. Schoenfeld (Ed.), Cognitive

science and mathematics education (pp. 217-251). Hillsdale, NJ: Lawrence Erlbaum.

Authors

Irene van Stiphout, PhD., Eindhoven School of Education, Cito Amsterdamseweg 13, P.O.

Box 1034, 6801 MG Arnhem, The Netherlands; [email protected]

Paul Drijvers, Associate Professor, Freudenthal Institute for Science and

Mathematics Education, Faculty of Science, Utrecht University, Princetonplein

5, Office 367, PO Box 85170, 3508 AD Utrecht; [email protected]

Koeno Gravemeijer, Full Professor Emeritus, Eindhoven School of Education, Technische

Universiteit Eindhoven P.O. Box 513 5600 MB Eindhoven; [email protected]




Appendix: Test Items

Tasks with an A in the measure are tasks that have served as anchor items in the Rasch

analysis. Tasks are arranged by increasing measure. The Rasch measure of a task corresponds

to a probability of 0.50 to answer that task correctly.

Test Item Included in testversion

RaschMeasure

(logits)

Expand the brackets: −"(# + $) = March 2008 -3.46 A

Expand the brackets: −5(2 % + &) = May 2008 -3.46 A

Expand the brackets: −(" % + &) = October 2008 -3.46 A

Expand the brackets: −"(5 % + &) = February 2009 -3.46 A

Simplify: −2( − ') + (−"' − 2) = March 2008 -1.47 A

Simplify: −(2# − $) + "(−2$ − ) = May 2008 -1.47 A

Simplify: −"(2 − ') + 2(−' − ") = October 2008 -1.47 A

Simplify: −2(" − ') + (−2' − ") = February 2009 -1.47 A

You know the operations plus, minus, multiplication and

division. We introduce an operation, diamond, and # $ is

defined as follows. For two numbers # and $, we say # $ =

#

− # ⋅ $. Does

# $ = $ # hold for all numbers

# and

$?

October 2008 -1.24

Simplify (# + 2# + )(# + *). Show your work. March 2008 -0.85 A

Simplify (2 + + ")( + ). Show your work. May 2008 -0.85 A

Simplify (2# + "# + )(# + 5). Show your work. October 2008 -0.85 A

Simplify (# + "# + 5)(# + ). Show your work. February 2009 -0.85 A

Solve: ( − 1)( + )( − ") = 0. February 2009 -0.60

Simplify: ,-.,/(-.)-. = February 2009 -0.50

Solve: ( − 5)( − ) = . March 2008 -0.48

A classmate asks for your help in solving

34

= . He does

not know how to start. Describe what you would do to help your

classmate.

March 2008 -0.31




Test Item Included in test

version

Rasch

Measure

(logits)

Simplify:-./(-.)

-. = May 2008 -0.22

Solve:

634

= . October 2008 0.01

Martijn claims that 7 = √ − 2 implies = 7 + 2. Explain

why you do or do not agree with Martijn.

May 2008 0.07

Solve: − 5 + 38 = 0 March 2008 0.33

Solve: ( − 5)( + 2)( − ) = 0 May 2008 0.68

Substitute # = −1 and $ = −2 in −(#$) − 2(#$). October 2008 0.72

Substitute # = −2 and $ = −1 in −(#$) − 2(#$). March 2008 0.91

Martijn claims that √# = # holds for all numbers, #. Explain

why you do or do not agree with Martijn.

March 2008 1.00

Is there any for which-- = 2? If so, calculate ; if not,

explain why such an does not exist.

February 2009 1.34

Is there any for which-- = 2? If so, calculate ; if not,

explain why such an

does not exist.

March 2008 1.37

Rewrite the formula = 9 + 5 as a formula of the form 7 = …

something with …

March 2008 1.43

Solve: 2( + 2) = (2 − 1) + . October 2008 2.01

Solve: #√ 2 = 1 + 2#√ . The square roots may remain. October 2008 3.94

If #√ $ = 1 + 2#√ 1 + $, then # = March 2008 6.13




Varied Ways to Teach the Definite Integral Concept

Iiris Attorps, Kjell Björk, Mirko Radic

University of Gävle, Sweden

Timo Tossavainen

University of Eastern Finland, Finland

In this paper, we report on a collaborative teaching experiment based on the Learning Study model (LS

model) which grounds on the Variation Theory. Until today, most of such studies have focused on the

teaching and learning of elementary school mathematics; ours was carried out in undergraduate

mathematics education. In the following, we discuss how we managed to promote students’ conceptual

learning by varying the treatment of the object of learning (the concept of definite integral and the

Fundamental Theorem of Calculus) during three lectures on an introductory course in calculus. We

also discuss the challenges and possibilities of the LS model and the Variation Theory in the

development of the teaching of tertiary mathematics in general. The experiment was carried out at a

Swedish university. The data of the study consists of the documents of the observation of three lectures

and the students’ answers to the pre- and post-tests of each lesson. The analysis of learning results

revealed some critical aspects of the definite integral concept and patterns of variations that seem to be

effective to a significant degree. For example, we found several possibilities to use GeoGebra to enrich

students’ learning opportunities.

Keywords: definite integral, GeoGebra, learning study, tertiary education, variation

In Sweden, like in many other countries (Artigue, 2001), the concept of definite integral

is first met during the last two years of the upper secondary school. The integral function is

usually introduced using the notion of anti-derivative, along the Fundamental Theorem of

Calculus connecting the concept of the definite integral with the intuitive idea of area. The

theory of integration and the Riemann integrals are systematically discussed only in

universities.

Several studies have highlighted difficulties that students encounter with the integral

concept. In early studies carried out by Orton (1983, 1984), it was noticed that some students

have difficulties in solving problems that require capacity to see integration as a limit process

of sums. Orton’s studies also showed that students interpret the integral sign as a signal “to

do something” (cf. Attorps, 2006). Like Orton (1984), also Artigue (2001) found out that

although some students’ technical ability to calculate definite integrals can be quite

impressive, their conceptual understanding of the concept itself may be poor. Similarly,Rasslan and Tall (2002) verified that a majority of the students cannot write meaningfully

about the definition of the definite integral. Also many recent studies (e.g., Attorps, 2006;

Rösken & Rolka, 2007; Viirman, Attorps & Tossavainen, 2011; Tossavainen, Haukkanen &

Pesonen, 2013) concerning the learning of other concepts of calculus have verified that the

formal definitions only play a marginal role in students’ learning; intuition and non-formal

representations dominate their concept learning. For example, Attorps, Björk, Radic and

Tossavainen (2010), Blum (2000), Calvo (1997) and Camacho, Depool and Santos-Trigo



VARIED WAYS TO TEACH THE DEFINITE INTEGRAL CONCEPT 82

(2010) have verified that students have a strong intention to identify the definite integral with

the area of a domain restricted by the integrand and the coordinate axes.

On the other hand, it seems that students’ learning of the definite integral can be

supported by using graphing calculators in classroom (Touval, 1997). Also Machín and

Rivero (2003) noticed that students may benefit from ICT in tasks which concern the graphic

and procedural aspects of the definite integral. Nevertheless, the research reports cited abovereveal the limitations of standard teaching methods. Although some students become

reasonably successful in standard tasks and develop in procedural skills, most of them have

difficulties in developing a solid conceptual understanding about the topics itself (Artigue,

2001).

The aim of this study is to investigate whether it is possible, by using technology-assisted

teaching (in this case, the dynamic geometric software GeoGebra), to design such teaching

sequences of the definite integral concept that help us to improve university students’

conceptual understanding of the concept. The theoretical framework for our experiment is

based on the Variation Theory which is described in the next section. In its terminology, we

seek an answer to the following questions: Which critical aspects of the definite integralconcept arise during the lectures? How can we compose effective patterns of variation (of the

object of learning) that support students to discern these critical aspects and learn from

them?

From a practical point of view, the design of our teaching experiment is that of the Lesson

Study model (LS model). The LS model is a synthesis of the Japanese Lesson Study (Lewis,

2002; Stigler & Hiebert, 1999) and Design Experiments (Brown, 1992; Cobb et al., 2003;

Collins, 1992). The LS model goes beyond the Japanese Lesson Study in two major aspects.

The first is its theoretical basis: the design of teaching is based on the Variation Theory

(Marton et al., 2004). Researchers and teachers work together to establish a framework for

the joint inquiry. The second is its method for the evaluation of learning. In the Japanese

version, the learners’ understanding is evaluated as a long developing process. In the LS

model, pre- and post-tests are made before and after every intervention in order to get an

immediate conception of what students have learned (see e.g. Runesson, 1999; Häggström,

2008).

The LS model (Marton et al., 2004) makes up a cyclic process as follows:

• A learning study group of teachers determines a common object of learning (in our case

the definite integral concept). Previous teaching experiences, theories of concept

learning (e.g., Tall & Vinner, 1981) and results from prior research on the teaching and

learning of the object are taken as a starting point for the design of a pre-test.

• Basing on the results of the pre-test, the learning study group plans the first lecture. TheVariation Theory is used as a theoretical framework for designing the lecture.

• One of the teachers conducts the first lecture. The lecture is video recorded or observed

by the other teachers (in our case, the teacher group made observations). The students’

learning is tested in a post-test designed collaboratively.

• Both the test results and the video recordings or the documented observations are

analysed by the learning study group. If the students’ learning results are not sufficient



83 I. Attorps, K. Björk, M. Radic & T. Tossavainen

with respect to the goals, the group revises the plan for the same lecture for the next

group of students.

• A teacher of the group implements the new plan in another class. In an ideal setting, the

cyclic process continues until the students’ learning results are optimal.

In our experiment, altogether three researchers participated in the design and analysis of

three lessons, a fourth researcher in the analysis of the results.

Theoretical Framework

The Variation Theory is a theory of learning which is based on the phenomenographic

research tradition (Marton & Booth, 1997). The main idea in the phenomenography is to

identify and describe qualitatively different ways in which people experience certain

phenomena in the world, especially in an educational context (Marton, 1993).

A significant feature of The Variation Theory is its strong focus on the object of learning.

A central assumption is that variation is a prerequisite for discerning different aspects of

object of learning. Hence the most powerful didactic factor for students’ learning is how theobject of learning is represented in a teaching situation. In order to understand what enables

learning in one teaching situation and not in another, a researcher should focus on discerning

what varies and what remains invariant during a lesson (Marton & Morris 2001). Marton et

al. (2004) have identified four patterns of variation or approaches to discuss the object of

learning: contrast, generalization, separation and fusion. The following excerpts illuminate

the essence of them:

Contrast : … in order to experience something, a person must experience something else

to compare it with.

Generalization: … in order to fully understand what ‘‘three’’ is, we must also experience

varying appearances of ‘‘three’’.

Separation: … in order to experience a certain aspect of something, and in order to

separate this aspect from other aspects, it must vary while other aspects remain invariant.

Fusion: If there are several critical aspects that the learner has to take into consideration at

the same time, they must all be experienced simultaneously. (Marton et al., 2004, 16).

According to Leung (2003), these patterns of variation create opportunities for the

students to understand the underlying formal abstract concept.

The object of learning can be seen from various different perspectives: that of a teacher, a

student or a researcher. The intended object of learning refers to the object of learning seen

from the teacher’s perspective. It includes what the teacher says and wants the students to

learn during the lecture. The students experience this in their own ways and what they

recognize and learn is called the lived object of learning. Obviously, what students’ really

learn does not always correspond to what the teacher’s intention was. The enacted object of

learning is observed from the researcher’s perspective and it defines what is possible to learn

during the lecture, to what extent and in which forms the necessary conditions of a specific

object of learning actualize in classroom. The enacted object of learning describes the space

of learning that students and teacher create together, i.e., the circumstances for discerning the

critical aspects of the object of learning. (Marton & Tsui, 2004).




In the Variation Theory, the necessary conditions for learning are the experiences of

discernment , simultaneity and variation. Variation is the primary factor to support students’

learning. In order to understand what variations a teacher should use, he or she must first

become aware of the varying ways students may experience the object of learning. This

information is needed for identifying potential ways to help students to discern those aspects

of the learning object they have not previously noticed (Marton, Runesson & Tsui, 2004).Every concept, situation and phenomenon has particular aspects of their own. If one

aspect is varied and others are kept invariant, the varied aspect should arise and be discerned.

The thorough understanding of the object of learning, e.g., a mathematical concept, requires

the simultaneous discernment of all critical aspects of the object of learning. (Marton &

Morris, 2001; Marton, Runesson & Tsui, 2004). Consequently, the triangle of discernment,

simultaneity and variation can be used also as a framework for analyzing teaching (ibid).

Although the theoretical framework in our study is mostly based on the Variation Theory,

we also acknowledge the theory of concept image and concept definition. Tall and Vinner

(1981, 152) use the term concept image “to describe the total cognitive structure that is

associated with the concept, which includes all the mental pictures associated properties andprocesses”. They suggest that when we think of a mathematical concept, something is evoked

in our memory. Often these images do not relate to the formal definition of a concept, i.e., the

concept definition, but students prefer to focus, for instance, on the archetypical examples

discussing a concept (e.g., Tall, 1994; Viirman, Attorps, Tossavainen, 2011; Tossavainen,

Haukkanen & Pesonen, 2013).

Vinner (1991) claims that the role of definition in mathematical thinking is also neglected

in the teaching of mathematics, textbooks and even in the documents about the goals of

teaching mathematics. He encourages teachers not only to discuss definitions with students

but to train them to use definitions as an ultimate criterion in mathematical reasoning (ibid).

The Variation Theory implies that, in addition to typical examples, it is useful also to pay

attention to nonexamples of mathematical concepts, even weird ones.

Method

The study took place at a Swedish university. Altogether 85 first-year undergraduate

students (engineering and teacher students) and four university teachers participated in the

study. The data consists of photos, observations, notebooks and the video recordings of three

lectures in an introductory calculus course. The students’ learning was measured using

written pre- and post-tests and interviews.

The interviews focused on the participants’ understanding about the concept of the

definite integral. They were first transcribed and then analysed following aphenomenographic research tradition (Marton, 1993): the main goal is to describe how many

qualitatively different conceptions from the certain phenomenon appear rather than to

determine how many people who have a certain conception. In our case, the analysis should

result in a number of the categories of description, i.e., categories representing the

qualitatively different ways in which students comprehend the definite integral concept.

(Booth, 1992).




The pre- and post-tests for measuring students’ knowledge about the definite integral and

the Fundamental Theorem of Calculus consisted of six problems; the items will be given

below. In both tests, the same set of questions was used in order to make the learning

outcomes statistically comparable. The maximum of points in each problem was three. To get

three points, the answer needed to be correct and well motivated. For minor faults in

calculations, we deducted one point. For a correct but not satisfactory motivated answer, weawarded one point. An empty or a meaningless answer resulted in zero points.

Students were given 25 minutes to do the test. The use of any technical facilities like

graphing calculators was not allowed. The results were analysed by using a statistic program

Minitab.

One can obviously ask whether the observed improvements in the post-tests are due to the

familiarity of problems and not a consequence of the implementation of the design of

lectures. In order to minimize this effect, we did not reveal the answers or the results of the

pre-test to the students. Moreover, they did the post-test without any notice about it in

advance. Furthermore, the participating groups were equivalent with respect to their

preliminary education; all students were first-year undergraduates from the engineering or

teacher programme studying the same introductory course in calculus.

A more detailed description of how we designed and implemented each lesson will be

given together with the report on our findings since the design of subsequent lectures was

based on the analysis of the previous one(s). The first lecture is to be considered as a

reference one. It was prepared without any knowledge of the pre-test results.

The pre- and post-test questionnaire was originally in Swedish. The translations of the

items in English are as follows:

Question 1: If you want to calculate the area between the curve and the x-axis and the lines

x=0 and x=5 (see the graphs below), you can get an approximate value of this area by

calculating and summing the area of each column.a) Which of the following graphs should you choose in order to make the error as small

as possible?

b) Explain your answer.

The aim of the first question was to test a student’s intuitive conception or concept image

of the exact area as a result of a limiting process (of the upper Riemann sums). By observing

Graph 1 Graph 2 Graph 3




that the width of each column is halved as we move from the graph 1 to the graph 3, a student

should be able to discern that the area representing the error of approximation also decreases.

Question 2. What does ∫b

a

dx x f )( mean?

The second question aims at measuring whether a student is familiar with the symbol of

the definite integral and, if so, what this symbol evokes in his or her concept image of the

definite integral.

Question 3. There are some approximate values of x and )( xF given in the table below:

x 1 2 3 4 5

)( xF -1 -0.61 0.30 1.55 3.05

Assume now that .ln)´( x xF = Approximate the value of ∫5

3

.ln dx x

The purpose of the third question was to test whether this kind of a problem evokes a linkto the Fundamental Theorem of Calculus in a student’s concept image of the definite integral.

Question 4. Suppose that ∫−

=

5

1

2)( dx x f and ∫−

−=

7

1

.1)( dx x f Evaluate ∫7

5

.)( dx x f

This question tests whether a student can apply the additive properties of the definite

integral.

Question 5. Can you find any error in the following reasoning?

21

1

1

1

1

1

1

11

1

2

1

1

2 −=

−

−−

−=

−

==

−

−

−

−

− ∫∫

x

dx x x

dx

The aim of the fifth question was to examine whether a student have a correct conception

about the prerequisites for applying the Fundamental Theorem of Calculus.

Question 6 . Find the area of the region limited by the functions f ( x) = 0.5 x2 and g ( x) = x3.

Give the exact value of it.

The idea of the last question was to test the students’ procedural skills in applying the

Fundamental Theorem of Calculus.

In the next section we are going to present the results of our study which consisted of

three lectures on the same topic. The first lecture is to be considered as a reference one. It wasprepared without any knowledge of the pre-test results. The second and the third lectures

were designed on basis of the information of the post-test results of the first and the second

lecture respectively. Having this information available, we revised the patterns of variation

of the observed critical aspects of the object of learning in lecture two and three.




Results

The analysis of our findings follows the hypothesis of Marton and Morris (2002) and

Marton and Tsui (2004) that different patterns of variation create different learning

opportunities. Therefore, we begin by illuminating the progression of each lecture.

Lecture One

The first lecture (LS1) was designed by the first lecturer alone, without having any prior

knowledge of the pre-test results. Two researchers observed the lecture. The first group is

therefore to be considered as a reference group; it consisted of engineering students only.

The lecture started with a discussion about the area concept and how to calculate the area

of common figures such as rectangles, triangles and parallelograms. For example, the area of

a circle was estimated by transforming the circle into a parallelogram. It was done by cutting

the circle into wedges which were then organized into the shape of a parallelogram. As the

number of wedges increases, the area of the parallelogram approaches to the area of the

original circle.

Figure 1. The transformation of a circle into a parallelogram.

The lecture continued with a discussion about how to calculate areas for irregular regions

such as an area between an arbitrary continuous function and the x-axis. In this context, the

sigma symbol (summing) and the concepts of Lower and Upper Riemann sums were

introduced. The end of the lecture was spent on demonstrating how to proceed when

calculating an area of the plane region lying above the x-axis and under the curve y = e x , i.e.,

e x 0

1

∫ dx .

The problem was studied first in terms of Lower and Upper Riemann sums and the

limiting process and then solved by applying the Fundamental Theorem of Calculus. In

discussion, the conditions for applying the theorem were not mentioned explicitly. After the

lecture, the students answered the post-test anonymously.




Figure 2. The calculation of area using the Riemann sums and the limiting process.

Lecture Two

Before designing the second lecture, we decided that, in order to improve the precision of

our statistical evaluation, we should compare the results of the pre- and post-test in the

subsequent learning studies LS2 and LS3 at the individual level instead of the group level as

was the case in LS1. Furthermore, we decided to videotape our next lectures.

Before the second lecture, we carefully analysed the observations and the results of the

post-test. The results in Table 1 summarize the learning results of the first group. In a more

thorough inquiry to LS1, we could identify the following three critical aspects.First, we noticed that most of the students, who answered the second question, had

interpreted the definite integral in Question 2 merely as an area and not as a real number that

can have negative, zero or positive value. Second , the results both in the pre- and post-tests

indicated that the students have difficulties in discerning the correct conditions for applying

the Fundamental Theorem of Calculus, especially in the case when it is not possible

(Question 5). Third , a large majority of the students failed in solving the ordinary routine

exercise (Question 6). For example, they could not decide which one of the functions

represents the upper or lower function or determine the intersection points between the

functions. Some of them even had problems with the arithmetic of fractions.

Having this information available, we revised the patterns of variation of these threecritical aspects in the next lectures so that the correct aspect should be easier to discern. For

example, we decided to emphasize the formal definition of the definite integral and the fact

that it cannot always be interpreted as an area. Further, students should pay more attention to

the conditions of theorems to be applied.

The second lecture was carried out by a teacher in the research group to a mixed group of

engineering and teacher students. The second lecture started with a discussion about the

concept of area and regular (polygonal) and irregular regions in the plane. After that, the




definite integral concept was introduced and discussed through a typical example from upper

secondary school: dx x x )2(2

0

2

∫ − . Also the geometric interpretation of the problem was

illustrated and the problem was solved using the Fundamental Theorem of Calculus

emphasizing the conditions for applying it. Then another variant of the same problem was

discussed graphically by studying the functions x x f 2)( = and 2)( x xg = , see Figure 3.

Further, using two different approaches to solve the same problem, we especially aimed at the

experiences of generalization and separation.

Figure 3. The illustration of the upper and lower functions.

The concepts of the upper ( x x f 2)( = ) and lower ( 2)( x xg = ) functions were introduced

in this connection. We also recalled how to find the intersection points of the functions. After

that, the second lecture continued similarly as the first one with discussions about how to find

the area by using estimation (Lower and Upper Riemann sums) and the limiting process for

arbitrary irregular regions above the x-axis. However, in order to show how to interpret the

definite integral in the general case (i.e. not only as an area), the following example wasconsidered thoroughly.

Figure 4. ∫b

a

dx x f )( = area R1 - areaR2 + areaR3.

By constructing an example where the definite integral of a function had a negative value, we

emphasized the experience of contrast.




In the end of the lecture, the example of dx x∫ −

2

01

1 was examined graphically reflecting on

the necessary conditions for applying the Fundamental Theorem of Calculus.

Figure 5. The graph of the function1

1)(

−

=

x x f .

In order to stress the importance of the necessary conditions for applying the Fundamental

Theorem of Calculus, here we emphasized the experiences of separation and fusion.

Lecture Three

The test results (Table 2) for the second group of engineering and teacher students

revealed that students’ understanding about the concept of definite integral was still

inadequate although some statistically significant improvements were observed. Most of the

students interpreted the definite integral again only as an area. Similarly, the problems related

to the conditions for applying the Fundamental Theorem of Calculus (Question 5) remained

actual; likewise the problems in solving the ordinary routine exercise (Question 6). In order

to gain a more detailed view of students’ conception of the definite integral, we interviewed

five students from the second group. The analysis of the interviews revealed three different

categories of description: the definite integral is seen as 1) a limiting process, 2) an area or 3)

a procedure.

The first category represents those students whose conceptions of the definite integral

focus on a limiting process, the approximation of the area of a curvilinear region by breaking

it into thin vertical rectangles. One of the students describes the process in the following way:

“The error decreases the closer the infinite the number of columns are nearing. The columns

will look like the curve more and more.” This excerpt and the test results from lectures one

and two indicated that some students have a relatively good intuitive understanding about the

definite integral as a limiting process.

For the students in the second category of description, the definite integral ∫

b

adx x f )(

stands for the area between f(x) and x-axis. “ It is an area between y=0 and y=f(x) in the

interval [a, b]” as one of the students explained in the interview. Most of the students in this

study described the definite integral in the pre- and post-tests in this or a similar way.

The students belonging to the third category viewed the definite integral as a procedure.

For them, the definite integral seems to be merely a formula and they use procedures without

considering definitions and theorems when solving problems related to that. One of the




interviewed students described his conception in the following way: “This I had to learn in

upper secondary school. You write down the primitive function with brackets. I take the

values of the end point minus the starting point, then it's just a simple subtraction”. Another

student said, when looking at Question 5, “It looks like an ordinary integral calculation. That

is correct… “

The weakest students of the study fell typically into this category. These studentsmentioned in interviews that theorems were not much discussed from a theoretical point of

view in upper secondary school. Theorems were applied more like formulas.

Taking into account the results from the pre- and post-tests and the interviews we again

revised our plan for the next lecture. The most notable difference between the third and the

previous lectures is that we decided to use the free dynamic mathematics software GeoGebra

for the illustration of critical aspects.

The third lecture was given by the same teacher as the second one but now to a new group

consisting of only engineering students. It began with a short discussion about how to find an

area for a (polygonal) regular and an irregular region lying above the x-axis.

The first exercise with GeoGebra (see Figure 6) focused on the numerical approximation

of the area as the Lower and Upper Riemann sums and the definition of the definite integral

as the limiting process. In Figure 6, two points, a and b, are shown and they can be moved

along the x-axis in order to modify the investigated interval. The values of the Upper and

Lower sums together with their difference are displayed as a dynamic text automatically

adapting to the modifications. In this exercise, we kept f ( x) and the interval invariant and

varied the number of subintervals. Our intention was to show that, by increasing the number

of subintervals, the difference between the lower and upper sums can be made to decrease,

suggesting that the lower and upper sums eventually coincide with the value of the definite

integral. By utilizing GeoGebra we created the pattern of generalization dynamically.

Figure 6. The Lower and Upper Riemann sums and the inherent infinite processes.

After this, the same problem as shown in Figure 3 was solved. It was also highlighted that

when applying the Fundamental Theorem of Calculus for dx x x )2(2

0

2

∫ − , the

function 22)( x x x y −= must satisfy the following assumptions: it must be a defined,

continuous and nonnegative function on the closed interval [a, b]. The following two figures

demonstrate how we illustrated the conflict between the definition of the definite integral




concept and the area interpretation of it. GeoGebra gave us a good opportunity to

dynamically demonstrate contrast, which was one of the patterns of variation.

Figure 7. The value of the definite integral is now identical to the area between function and

x-axis in the interval [a, b].

Figure 8. The definite integral results in a real number which can be positive, zero ornegative.

In the second GeoGebra application related to Figures 7 and 8, two points, a and b, are

shown so that they can be moved along the x-axis. The area and the value of the definite

integral are displayed as a dynamic text. In this exercise, we kept only f ( x) invariant and

varied both the length of the interval and the upper and lower limit points in order to show

that the values of the area between the function and the x-axis and the definite integral do not

always coincide. Our goal with the third exercise (Figure 9) was to help the students to

discern situations where it is possible to apply the Fundamental Theorem of Calculus and to

notice when it is not.




Figure 9. The illustration of the conditions of the Fundamental Theorem of Calculus.

By moving the point A along the x-axis, we can vary the position of the investigated

interval. In this exercise, we kept the length of the interval and the functions f ( x) and g( x)

invariant and varied the location of the point A. By using the dynamic nature of GeoGebra

we were able to demonstrate all the aspects of variation, i.e. contrast, generalization,

separation and fusion. In the end of the third lecture, the same problem ( dx x∫ −

2

01

1) as shown

in Figure 5 was studied.

Quantitative Analysis of the Pre- and Post-Tests

We analysed the scores of the pre- and post-tests with the Minitab software using both the

independent, two-sided, two-sample t-test (Lecture 1) and dependent, two-sided, t-test for

paired samples (Lectures 2 and 3) at the significance level of 5% (0.05). In the pre- and

post-test of the first lesson, the number of participants was 28 and 24, respectively. The

results of the pre- and post-tests were recorded on each item only at the group level, which

explains why we use the different t-test for this group. Concerning the following lessons, we

compared the means of the test results on each item at the individual level. The second group

(18/18 students) consists of both engineering and teacher students and the third group (39/39

students) only of engineering students. Tables 1 and 2 show the results of the analyses.

Table 1

The quantitative results of the pre- and post-tests (unpaired t-test) of the first lecture.

Problem no. Learning

study no.

Pre-test

mean

Post-test

mean

p Maximum

scores

1a 1 0.93 1.00 0.16 1

1b 1 1.07 1.00 0.67 2

2 1 0.43 0.46 0.88 3

3 1 0.68 0.88 0.59 3

4 1 1.54 1.75 0.60 3

5 1 0.00 0.00 0.91 3

6 1 0.04 0.25 0.18 3




In Table 1, we see that there are no statistically significant differences in learning results

concerning the first lecture. The results related to the second and third lectures are given

together in Table 2.

As Table 2 shows, the third lecture seems to have succeeded best: statistically significant

improvements happened in many test items. The students’ scores in question 1 a) and b) show

that the students’ intuitive understanding about the definite integral concept as an infiniteprocess was quite good already at the beginning like their capacity to apply the additive

property of definite integrals.

Almost all students failed to give an adequate response to question 5; most of them could

not even find any errors at all. In Question 6, a majority of students could not discern which

of the functions represented upper and lower functions or that how to determine the

intersection points between the functions or how to calculate with fractions or how to give an

exact answer.

Table 2

The quantitative results of the pre- and post-tests (paired t-test) of the second and third

lectures.

Problem no.Learning

study no.

Pre-test

Mean

Post-test

meanp

Maximum

scores

1a2 0.94 1.00 0.33

13 0.92 1.00 0.08

1b2 1.11 1.11 1.00

23 0.82 0.82 1.00

22 0.83 0.44 0.09

33 0.51 0.97 0.00*

32 1.44 0.33 0.00*

33 0.13 0.92 0.00*

42 1.28 1.28 1.00

33 0.38 1.12 0.00*

52 0.00 0.00 --

33 0.00 0.46 0.01*

62 0.78 0.28 0.02*

3

3 0.10 0.31 0.02*--- = p-value could not be calculated (Minitab: all values in column are identical, * p < 0.05

Discussion

The purpose of this study was to find out whether university students’ learning can be

supported by finding suitable teaching sequences that help students to discern and experience

mathematical concepts from the meaningful points of view. Experiencing variations of




critical features of the object of learning should be, by the variation theory, a primary factor

in enhancing students’ learning (Marton & Booth, 1997; Marton & Morris, 2002).

In our study, two university teachers taught the definite integral concept for three student

groups on an introductory course in calculus. Two of the lectures were prepared and planned

with extraordinary care, taking into account the results from the written pre- and post- tests.

Although the study consisted only of three lectures, it revealed that different teachingapproaches had a significant influence on that how students’ learning outcomes developed

during the lectures.

We succeeded best in designing teaching sequences of the definite integral concept when

we used the GeoGebra software. We interpret this being mainly due to the fact that GeoGebra

is an effective tool for the illustration of dynamic processes, e.g., the limiting process of

Riemann sums, and it allows a learner to experience simultaneously many critical aspects,

e.g., how the area and the value of the definite integral are effected when the interval is

modified. Also earlier research (i.e., Leung, 2003) shows that GeoGebra is a suitable

pedagogical tool in creating the patterns of variation.

It is worth noticing that it did not provide a remarkable aid in Question 6 (although the

difference between the mean scores of the pre- and post-tests improved for the third group in

a statistically significant way). A plausible explanation is that GeoGebra or any other

software cannot be used to compensate the lack of fundamental arithmetic skills although it

often helps us to bypass challenging calculations and focus on the conceptual understanding

of a mathematical problem.

In this study, we observed three critical aspects of the definite integral that seem to be

important for the successful teaching of this concept and, consequently, for the design of the

relevant patterns of variations. All these aspects can be discussed using GeoGebra.

First, it is important to consider the definite integral as a real number (i.e. the result of a

limiting process) in a wider context and separate it from seeing it only as an area. This aspect

was not elaborated during the first lecture – which can also be seen in the results of Tables 1

and 2. The use of GeoGebra during the third lecture seemed to extend students’ possibilities

to experience the concept of the definite integral in this wider context.

In the teaching sequences related to Figures 7 and 8, the students were given

opportunities to experience an effective contrast , i.e., to discern the definite integral not only

as an area but, simultaneously, also as a real number. This allowed them to experience a

generalization, that is to say, to experience that the definite integral can be a negative

number, zero or a positive number.

Second, in spite of many efforts, it is plausible that many students’ concept images of the

definite integral will be based on the area interpretation (cf. Blum, 2000) and Tall and Vinner

(1981). To change this, it may require a thorough revision of mathematics textbooks in schoolsince they seem to emphasize this aspect. It is hard for an individual teacher to resist such a

tradition but as our third lecture verifies, it is possible in a technological environment.

Third, the results also indicated that most students have difficulties in applying the

Fundamental Theorem of Calculus, especially when the assumptions of the theorem are not

satisfied. During the first lecture, the theorem was only mentioned quite superficially. On

other lectures, the issue was given more attention; both examples and counter examples were

elaborated. In the teaching sequence particularly related to Figure 9, the students were given




an opportunity to experience a separation and a fusion. In order to experience a specific

aspect – when it is not possible to apply the Fundamental Theorem of Calculus – and in order

to separate this aspect from other aspects, the aspect must be varied while other aspects must

remain constant.

In our teaching sequence, we kept the length of the interval and the functions f ( x) and g( x)

invariant and by moving the point a along the x-axis we could vary the position of theinvestigated interval. The same sequence again gave the students an opportunity to

experience the pattern of variation called fusion, i.e. if there are several critical aspects as ‘it

is possible to apply the Fundamental Theorem of Calculus’, ‘it is not possible to apply the

Fundamental Theorem of Calculus’, ‘the function is defined and continuous in the closed and

bounded interval’, ‘the function is not defined and continuous in the closed and bounded

interval’ and so on , they must all be experienced simultaneously.

The students’ learning outcomes in Question 5 show that their conceptions of the

conditions for applying the theorem were not changed after the second lecture. Only after the

GeoGebra-based teaching sequence we could notice some statistically significant

improvements of their results. We agree with Vinner (1991) that the students should betrained to use definitions as an ultimate criterion in mathematical issues in teaching and

learning of mathematics. The students even mentioned in interviews that theorems were not

discussed from the theoretical point of view; they were used as formulas. Students use

procedures without considering definitions and theorems when solving problems. In order to

develop a deeper understanding about the definite integral concept it is therefore important

that the varying aspects of mathematical concepts are illuminated by using both examples and

non-examples of the concepts in teaching of mathematics.

Yet another critical aspect we found is that students’ poor arithmetic skills (Question 6)

prevent them from gaining a deeper conceptual understanding about mathematical

phenomena. Varying methods in order to solve this type of problem were applied during the

second and third lecture but with a vanishing effect.

All in all, we are not very satisfied with the students’ learning outcomes in this study.

Further studies need to be undertaken to identify which other factors than the integration of

technology and the LS model in the teaching and learning of mathematics can benefit both

mathematics educators and students. It must be stressed once again that teaching and learning

are very complex phenomena and the relation between them is not ‘one to one’. In a teaching

experiment like this, it would also be important to analyse what happens in the classroom in

the interaction between the teacher and the students and between the students. Not even a

good design of a lecture guarantees students’ learning but it can increase possibilities for

learning if students’ conceptions and misconceptions of mathematical concepts are taken into

account.

Finally, the study gave us a rare opportunity to collaborate with colleagues teaching and

preparing a lecture. It was a rewarding experience to reflect and analyse students’ learning

together. We all agree that the LS model and the Variation Theory are effective tools for

developing the teaching of mathematics and they provide a useful tool for increasing the

teachers’ awareness of the critical aspects of students’ learning and enhancing the learning of

mathematics in higher education.




References

Adams, R. A. (2006). Calculus a complete course. Toronto: Addison Wesley.

Artigue, M. (2001). What can we learn from educational research at the university level? In

D. Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI

study ( pp.207-220). Dordrecht: Kluwer Academic Publishers.

Attorps, I. (2006). Mathematics teachers’ conceptions about equations. Doctoral dissertation,

Research Report 266. Department of Applied Sciences of Education, University of

Helsinki, Finland.

Attorps, I., Björk, K., Radic, M., & Tossavainen, T. (2010). The learning study model and the

teaching of the definite integral concept. In M. Asikainen, P. E. Hirvonen, & K.

Sormunen (Eds.), Reports and Studies in Education, Humanities, and Theology, 77-86.

University of Eastern Finland, Joensuu, Finland.

Barton, B., & Thomas, M. (2010). Editorial. CULMS Newsletter , 1.

Burton, L. (1989). Mathematics as a cultural experience: Whose experience? In C. Keitel, P.Damerow, A. Bishop & P. Gerdes (Eds.), Mathematics education and society - Science

and Technologhy Education (Document Series 35, pp.16-19). Paris: Unesco.

Blum, W. (2000). Perspektiven für den Analysisunterricht. Der Mathematikunterricht ,

46 (4/5), 5-17.

Booth, S. A. (1992). Learning to program: A pehenomenographic perspective. Göteborg

Studies in Educational Sciences, 89. Göteborg: Acta Universitatis Gothoburgensis.

Brown, A. L. (1992). Design experiments: Theoretical and methodological challenges in

creating complex interventions in classroom settings. The Journal of the Learning Sciences,

2(2), 141-178.

Calvo, C. (1997). Bases para una propuesta didáctica sorbe integrales. Unpublished Master

Thesis.

Camacho, M., Depool, R. & Santos-Trigo, M. (2010). Student’s use Derive software in

comprehending and making sense of definite integral and area concepts. CBMS Issues in

Mathematics Education, 16 , 29-62.

Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Scauble, L. (2003). Design experiments in

educational recearch. Educational Researcher, 32(1), 9-13.

Collins, A.(1992). Towards a design science of education. In E. Scandlon & T. D. Shea(Eds.), New directions in educational technology. Berlin: Springer.

Emanuelsson, J. (2001). En fråga om frågor. Hur lärarens frågor i klassrummet gör det

möjligt att få reda på elevernas sätt att förstå det som undervisningen behandlar i

matematik och naturvetenskap. Göteborg Studies of Educational Sciences. Göteborg:

Acta Universitatis Gothoburgensis.




Gustavsson, L. (2008). Att bli bättre lärare. Hur undervisningsinnehållets behandling blir till

samtalsämne lärare emellan. Högskolan i Kristianstad, sektion för lärarutbildningen.

Häggström, J. (2008). Teaching systems of linear equations in Sweden and China: What is

made possible to learn? Göteborg Studies of Educational Sciences. Göteborg: Acta

Universitatis Gothoburgensis.

Leung, A. (2003). Dynamic geometry and the theory of variation. In N. Pateman, B. J.

Doughherty, & J. Zillox, Proceedings of the 27th International Conference for the

Psychology of Mathematics Education (Vol. 3, pp.197-204). Honolulu: University of

Hawaii.

Lewis, C. (2002). Lesson study: A handbook of teacher-led instructional change.

Philadelphia: Research for Better Schools Inc.

Machín, M. C., & Rivero, R. D. (2003). Using DERIVE to understand the concept of definite

integral. International Journal for Mathematics Teaching and Learning, 4, 1-16.

Marton, F. (1993). Phenomenography. In T. Husén & T. N. Postlethwaite (Eds.). The International Encyclopaedia of Education (2nd Ed) (pp.4424-4429). Oxford: Pergamon.

Marton, F., & Booth, S. (1997). Learning and awareness. Mahwah, N.J.: Law Earlbaum.

Marton, F., & Morris, P. (Eds.) (2001). What matters? Discovering critical conditions of

classroom learning. Göteborg Studies of Educational Sciences. Göteborg: Acta

Universitatis Gothoburgensis.

Marton, F., Runesson, U., & Tsui, A. (2004). The space of learning. In F. Marton and A. Tsui

(Eds.), Classroom discourse and the space of learning (pp. 3-40). New Jersey: Lawrence

Erlbaum Associates, INC Publishers.

Marton, F., & Tsui, A. (2004). Classroom discourse and the space of learning. Mahwah, NJ:Lawrence Erlbaum.

Orton, A. (1983). Student’s understanding of integration. Educational Studies in

Mathematics, 14(1), 1-18.

Pang, M. F., & Marton, F. (2003). Beyond “lesson study”: Comparing two ways of

facilitating the gasp of some economic concepts. Instructional Science, 31, 175-195.

Rasslan, S., & Tall, D. (2002). Definitions and images for the definite integral concept. In A.

Cockburn & E. Nardi (Eds.), Proceedings of the 26th International Conference for the

Psychology of Mathematics Education (Vol.4, pp.89-96). Norwich, UK.

Rovio-Johansson, A., & Johansson, I-L. (2005). Lärandets kontextualitet: Hur utvecklasekonomstudenters förståelse av ett grundläggande ekonomiskt begrepp under

utbildningen? Didaktiskt Tidskrift, 16 (2-3), 61-92.

Runesson, U. (1999). Variationens pedagogik. Skilda sätt att behandla ett matematiskt

innehåll. Göteborg Studies of Educational Sciences. Göteborg: Acta Universitatis

Gothoburgensis.




Rösken, B., & Rolka, K. (2007). Integrating intuition: The role of concept image and concept

definition for students’ learning of integral calculus. The Montana Mathematics

Enthusiast, 3, 181-204.

Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes

and objects as different sides of the same coins. Educational Studies in Mathematics, 22,

1-36.

Stigler, J. W., & Hiebert, J. (1999). The teaching gap. New York: The Free Press.

Tall, D. (1994). The transition to advanced mathematical thinking. Functions, limits, infinity

and proof. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and

learning (pp. 390-419). New York: Macmillan.

Tall, D., & Vinner, S. (1981). Concept image and concept definition with particular reference

to limits and continuity. Educational Studies in Mathematics, 12, 151-169.

Tossavainen, T., Haukkanen, P., & Pesonen, M. (2013). Different aspects of monotonicity.

International Journal of Mathematical Education in Science and Technology.

doi:10.1080/0020739X.2013.770088.

Touval, A. (1997). Investigating a definite integral – from graphing calculator to rigorous

proof. Mathematics Teacher, 90(3), 230-232.

Vinner, S. (1991). The role of definitions in teaching and learning. In D. Tall (Ed.). Advanced

mathematical thinking (Chapter 5, pp.65-81). Dordrecht: Kluwer Academic Publishers.

Viirman, O., Attorps, I., & Tossavainen, T. (2011). Different views – Some Swedish

mathematics students’ concept images of the function concept. Nordic Studies in

Mathematics Education, 15(4), 5-24.

Authors

Iiris Attorps, Associate Professor, Ph.D., Faculty of Engineering and Sustainable

Development University of Gävle , Sweden; [email protected]

Kjell Björk , University Lecturer, MSc, Faculty of Engineering and Sustainable


Mirko Radic, University Lecturer, Ph.D., Faculty of Engineering and Sustainable


Timo Tossavainen, Associate Professor, Ph.D., School of Applied Educational Science and

Teacher Education, University of Eastern Finland, Finland; [email protected]




The Influence of Elementary Preservice Teachers’ Mathematical Experiences on their

Attitudes towards Teaching and Learning Mathematics

Cindy JongUniversity of Kentucky

Thomas E. Hodges

University of South Carolina

This study examined how preservice elementary teachers’ perceptions of past schooling experiences

and their experience in a mathematics methods course influenced their attitudes about mathematics’

teaching and learning. Pre- and post-surveys were administered to preservice teachers (n = 75) enrolled

in a mathematics methods course at a university in the northeastern United States. The purpose of the

surveys was to understand entering attitudes about mathematics, whether those attitudes changed, and

why. Findings indicated that perceptions of prior schooling experiences influenced preservice teachers’

initial attitudes about mathematics. Over the course of a semester, however, significant positive

changes in preservice teachers’ attitudes and confidence to teach mathematics suggest that experiences

in the mathematics methods course were conducive to building on preservice teachers’ prior

experiences. We argue that regardless of the nature of preservice teachers’ prior experiences in

mathematics, those experiences can provide an effective backdrop for developing attitudes towards

mathematics teaching and learning aligned with reform recommendations. Recommendations are made

for mathematics teacher educators to build upon entering attitudes and experiences in their

mathematics methods courses.

Keywords: Mathematics Education; Preservice Teachers; Survey Research

It has long been argued that teachers’ affect is an important part of the way teachersunderstand mathematics (Ball, 1990; McLeod, 1994). At an international level, studies

examining affect have influenced the field of mathematics education and how it has been

conceptualized in teacher education (Leder & Grootenboer, 2005). In Philipp’s (2007) review

of literature on mathematics teachers’ beliefs and affect, he argues that “for many students

studying mathematics in school, the beliefs or feelings that they carry away about the subject

are at least as important as the knowledge they learn of the subject” (p. 257). Philipp defines

affect as “[a] disposition or tendency or an emotion or feeling attached to an idea or object,”

which is “comprised of emotions, attitudes, and beliefs” (p. 259). In this study we focus on

one aspect of affect, namely the attitudes that preservice teachers (PTs) develop through their

perceived experiences as K-12 learners of mathematics and their experiences in mathematicsmethods coursework. This article contributes to the literature on attitudes in mathematics

education research by quantitatively examining the connections among preservice teachers’

attitudes toward mathematics, perceived past schooling in mathematics, and the mathematics

methods course experience. This study also extends beyond descriptive statistics to examine

the factors that influence positive changes in attitudes along with a growth in PTs’ confidence

to teach mathematics. Ultimately, we argue that, regardless of the nature of prior experiences

in mathematics and whether or not they are oriented toward a reform view, teacher educators



101 C. Jong & T. Hodges

need to draw upon PTs’ entering attitudes and experience as resources to inform the

mathematics methods course instruction. This focus is significant because many students1

develop negative attitudes towards mathematics, seeing it as a source of frustration and

anxiety (Ignacio, Blanco Nieto, & Barona, 2006). These attitudes then become a part of the

apprenticeship of observation (Lortie, 1975), beginning with the thousands of hours spent as

a student in schools, which creates a “latent culture” that surfaces when one becomes ateacher.

Additional research has shown that this apprenticeship of observation is influential in

shaping preservice teachers’ ideas about teaching and learning (Ball & Cohen, 1999; Feiman-

Nemser, 1983; Grossman, 1990; Wideen et al., 1998). The lenses through which preservice

teachers make sense of these course and field experiences are shaped by prior knowledge and

experiences (Ball, 1989; Grossman, 1990). Adopting an asset view of teacher education is an

important step in building upon PTs prior experiences to understand the attitudes with which

PTs enter mathematics education coursework, how those attitudes are a reflection of prior

school experiences, and how attitudes change through participation in a mathematics

education course. Regardless of the nature of preservice teachers’ prior experiences in

mathematics, those experiences can provide an effective backdrop for developing attitudes

towards mathematics teaching and learning aligned with reform recommendations (Drake,

2006; National Council of Teachers of Mathematics (NCTM), 2000). Thus, the goal of this

study was to examine preservice teachers’ entering attitudes about mathematics teaching and

learning, whether those attitudes change, and the factors that might contribute to any changes

in attitudes. Correspondingly, our research question asked, How do elementary preservice

teachers’ perceptions of their past schooling and their mathematics methods course influence

their attitudes about the teaching and learning of mathematics?

Feiman-Nemser (1983) asserted that teacher educators often underestimate the effects of

past experiences on PTs and that these effects overshadow the role teacher education plays in

forming PTs attitudes about mathematics teaching and learning. While some (e.g. Wideen et

al., 1998) have argued that the prevailing aim in teacher education is to help PTs learn to

teach in ways that are essentially different from the way they have been taught and from what

they have observed, others (e.g. Ball, 1989) note that it is not necessary to completely change

teachers beliefs about teaching and learning, but to support PTs development, since many

enter the program with beliefs about mathematics teaching that can support student learning.

Our study follows a line of research that has attempted to examine PTs’ attitudes about the

nature of mathematics and whether they adopt a more reformed view of teaching

mathematics (Ebby, 2000; Eisenhart, Borko, Underhill, Brown, Jones, & Agard, 1993;

McGinnis, Kramer, Roth-McDuffie, & Watanabe, 1998; and MacNab & Payne, 2003).

Mathematics methods courses that expose teachers to reform practices tend to positivelyinfluence PTs’ attitudes towards mathematics teaching and learning. One approach teacher

educators have taken to understand and build upon PTs’ prior experiences is to examine their

mathematics autobiographies (Ellsworth & Buss, 2000; Drake, 2006; Harkness, D’ambrosio,

& Morrone, 2006). Ellsworth and Buss (2000) found that PTs’ past teachers had the most

1 We use students to refer specifically to K-12 pupils throughout this paper to avoid confusion with preservice

teachers who are college students.



PST’S MATHEMATICS EXPERIENCES AND ATTITUDES 102

salient effect, be it positive or negative, on their attitudes towards mathematics and science.

Harkness et al. (2006) found that PTs were highly motivated in methods courses that focused

on mastery goals by engaging them in problem solving. Harkness et al. (2006) and Drake

(2006) argued that mathematics autobiographies also provided a platform where PTs were

given a voice. Consistent across the aforementioned studies is the perspective that pre-service

teachers’ attitudes towards mathematics can provide an effective stage for developingattitudes towards mathematics teaching and learning aligned with reform recommendations.

In addition to coursework, PTs often engage in multiple field experiences (e.g.

observations, practica, internships, student teaching) that provide opportunities for the

evolution of attitudes about mathematics teaching and learning. While universally seen as

valuable, teacher educators and preservice teachers often face the dilemma of “bridging the

cultures of the school and the university” (Wideen et al., 1998, p. 156). PTs can be

overwhelmed with the practical demands of field experiences which may contribute to

feelings of frustration related to inadequate preparation in their coursework. Despite these

challenges, Feiman-Nemser and Remillard (1995) note that “powerful and innovative teacher

preparation can affect the way teachers think about teaching and learning, students, and

subject matter” (p. 65). While field experiences can contribute to PTs’ attitudes about

mathematics, we chose not to include this experience in our analysis for this paper. This was

due to the varied field experiences that would be difficult to examine with a survey. Thus, we

focus primarily on the mathematics methods course.

In the following sections, we describe our methods including the context of the study,

survey instrument, research design, and data analysis. Then we present results from our data

analysis. Lastly, we provide an interpretive summary of the findings and make

recommendations for teacher education and future research.

Methods

Context and Participants

Research was conducted at Hillside College2, a private university in the northeastern

United States. The teacher education program offered both a traditional four-year

undergraduate degree and a graduate degree that could be completed in a twelve-month

period. As part of the teacher education program, PTs were required to take one mathematics

methods course. This course was typically taken during the fall semester before student

teaching.

All participants were undergraduate or graduate level preservice elementary school

teachers enrolled in one of four sections of the mathematics methods course. Three professors

taught the four sections of the elementary mathematics methods course. The mathematics

methods courses at this university emphasized a reformed view of teaching mathematics

(NCTM), 2000) where the professors thoughtfully used the NCTM process standards as a

means for teaching the content standards. At least half of the class sessions used manipulative

materials where the professors emphasized a link between concrete models and abstract

mathematics concepts.

2 Pseudonyms are used throughout this study to maintain anonymity.




Participating PTs completed both a pre-survey, which was administered the first week of

the mathematics methods course, and a post-survey, administered during the last week of the

course during the same semester. The pre-survey sought to capture participants’ entering

attitudes about mathematics and perceived experiences as K-12 students of mathematics. The

purpose of the post-survey was to examine the exiting attitudes about mathematics, practicum

experience, and mathematics methods course experience. The population size was 102 andthe total sample size for those who completed both the pre- and post-surveys was 75, a 73.5%

response rate.

Instrumentation

To develop the pre-survey, we first searched educational research databases for existing

surveys pertaining to the teaching and learning of mathematics, attitudes towards

mathematics, and mathematics methods courses. We gathered 15 existing surveys that

overlapped with the purpose of this study. Then, we examined the surveys, highlighted items

that were possible candidates for the survey, and categorized the items. The pre-survey

included the following four sections about mathematics: attitude and past experiences,teaching and learning, methods course expectations, and diverse learners. The post-survey

included the following four sections about mathematics: attitudes and practicum experiences,

teaching and learning, diverse learners, and future teaching. Five drafts of the pre-survey

were constructed before the final version was drafted and agreed upon by the three

participating mathematics methods professors. The survey items were on a four-point Likert

scale including: SA = strongly agree, A = Agree, D = Disagree, and SD = strongly disagree.

Thus, some of the figures and tables include the abbreviations of the item responses, such as

SD for “strongly disagree.” In addition, a fifth option, “not applicable,” was included for

those who were not enrolled in a practicum or truly had no idea how to response to a

particular item. The fourth and fifth drafts were given to a group of mathematics educators to

pilot, examine, and provide feedback regarding the wording of items and item order. The

post-survey, constructed similarly to the pre-survey, was adapted once the pre-survey was

administered, and a factor analysis was completed. The post-survey included 31 items

identical to those in the pre-survey, except for changes in the stems of the items (see

Appendix for surveys). For example, questions pertaining to topics and strategies taught in

the mathematics methods course on the pre-survey were phrased in terms of what preservice

teachers expected and viewed as “important for [them] to learn.” The same items were

rephrased for the post-survey to ask whether “the methods course taught…” preservice

teachers a particular strategy such as “how to assess student learning in mathematics.” The

questions about PTs’ perceived past experiences were replaced with questions about

practicum experiences. For example, item 3 on the pre-survey stated, “I had several positiveexperiences with mathematics as a K-8 student.” The majority of PTs enrolled in a

mathematics methods course also has a field experience during that semester which consisted

of a weekly school visit for a 10 week period. One of the goals of the post-survey was to

capture these experiences. For example, item 5 on the post-survey stated, “My cooperating

teacher used a conceptual method (i.e., problem-solving, open-ended Qs) to teach math.”




The overall factor analysis of the pre-survey accounted for 79.3% of the total variance

among responses. Conceptually, the items fit into seven factors. When the instrument was

forced into seven factors, the analysis accounted for 66.8% of the variance. The rotated

component matrix and conceptual understanding were used to divide the items into seven

factors: 1. Attitude toward mathematics; 2. Negative experiences; 3. Procedural mathematics;

4. Conceptual mathematics; 5. Course expectations; 6. Confidence to teach; and 7. Social justice. Next, reliability tests for the pre-survey were completed to examine the scales as

indicated by Cronbach’s alpha, which examines the internal consistency of the scales within

an instrument. The alpha level for each factor is as follows: 1. attitude toward mathematics (α

= .912); 2. negative experiences (α = .780); 3. procedural mathematics (α = .612); 4.

conceptual mathematics (α = .626); 5. course expectations (α = .921); 6. confidence to teach

(α = .879); and 7. social justice (α = .648).

The attitudes toward mathematics factor included attitudinal items such as “I look

forward to teaching math” along with positively worded past experience items. The negative

experiences factor included items that were negatively worded about past experiences such as

“I have struggled with math in K-8” along with negatively worded attitude items. The

procedural and conceptual mathematics factors included items about the nature of

mathematics such as “Memorizing facts and formulas is essential,” to get a sense of PTs’

agreement with reform recommendations. The factor on course expectations included items

about what PTs viewed as important to address in the mathematics methods course, such as

“how students learn math developmentally.” The confidence to teach factor included items

related to teaching mathematics to different types of learning such as being “confident to

teach mathematics to English language learners.” The factor on social justice include items

about addressing equity in the mathematics classroom such as “math can help students

critically analyze the world.”

The overall psychometric properties of the instrument were sound. All seven factors had

adequate to high reliability levels (Nunnally, 1978). Two items did not load well onto the

factors where they fit conceptually; thus, we removed them from all analyses. Due to this,

more precise language was used in the post-survey, which defined terms and directly asked

participants whether they planned to teach mathematics in a traditional or conceptual manner.

For example, item 14 stated, “I plan on teaching math in a procedural way (facts, skills,

etc…).” . The post-survey instrument was divided into five factors and had similarly reliable

scales: 1. attitude toward mathematics (α = .709); 2. teaching practices (α = .751); 3.

practicum experiences (α = .696); 4. methods course experiences (α = .893); and 5.

confidence to teach (α = .888). The attitudes toward mathematics factor included all

attitudinal items as the pre-survey. The teaching practices factor included all the nature of

mathematics items, similar to the procedural and conceptual mathematics factors. The practicum and methods course experiences factors included items related to field and course

experiences such as “I had a positive practicum experience” and “my mathematics methods

course focused on how to assess student learning.” The confidence to teach factor included

the same items as the pre-survey to get a sense of any changes.

In addition to the pre- and post-surveys, which were administered during the first and last

weeks of the semester course, observations of one section of the mathematics methods course

were conducted and course artifacts (e.g. syllabi, assignments, and assessments) were




collected to provide contextual information related to survey and observation data associated

with the mathematics methods courses. However, this paper solely focuses on the survey

data.

Data Analysis

Survey data analyses were carried out with SPSS, a software package used for organizingdata, conducting statistical analyses, and generating tables and graphs that summarize data.

Our data analysis involved several steps. First, descriptive statistics were applied to analyze

overall item response percentages and note any possible trends in responses. Then, we used

correlations to examine the relationships among perceived past experiences, field

experiences, the mathematics methods course, attitudes about mathematics education, and

confidence to teach. Paired t-tests were then completed to compare the differences in

preservice teachers’ attitudes and perceived level of preparation between the pre- and post-

surveys. Lastly, a multiple regression model was created to examine how perceived past

schooling experiences and the mathematics methods course accounted for preservice

teachers’ a) attitude towards mathematics and b) perceived level of preparation to teachmathematics.

Results

Considering the influence past experiences have on PTs conceptions of teaching and the

desire of teacher education programs to help shape these conceptions (Ball, 1990; Cady,

Meier, & Lubinski, 2006; Lortie, 1975; Scott, 2005), we became interested in responses on

three unique items on the post-survey that directly asked preservice teachers about the

perceived impact of their past K-8 schooling, practicum, and mathematics methods course on

their future teaching practices (see Figure 1). The results across all three were similar,

suggesting that PTs perceptions of the role past schooling played in their future instructional

practices aligned with findings from prior research (Ellsworth & Buss, 2000; Drake, 2006;

Harkness, D’ambrosio, & Morrone, 2006). The percentages were based on the total n of 75 to

avoid an inflated percent due to missing data. The stem for the three items stated, “The

following will have a major impact on the way I teach math in the future.” The PTs were then

asked to respond to this statement specifically about their past K-8 schooling, practicum, and

mathematics methods course experiences on a four-point Likert scale, from strongly agree

(SA) to strongly disagree (SD).

Figure 1. Elements influencing preservice teachers’ anticipated practices by percentages

0

10

20

30

40

50

60

SA A D SD

Past

Schooling




To further examine the relationship between PTs attitudes and items related to prior

schooling experiences, bivariate two-tailed Pearson’s correlations were run at the 0.05 and

0.01 alpha levels. Table 1 displays results from the analyses among items pertaining to these

topics on the pre-survey. All correlations were significant at the 0.01 alpha level, indicating

very strong linear relationships between attitudes towards mathematics, experiences in

mathematics, and confidence in their ability to teach mathematics.Table 1

Relationships among attitudes and prior schooling experiences in mathematics

Pre-Survey Items 2 3 4 5 6

1. Positive math attitude .599** .719** .713** .661** .553**

2. Positive K-8 math .539** .526** .455** .368**

3. Positive 9-12 math .599** .508** .440**

4. Perceived Proficiency in

math

.563** .585**

5. Looking forward to teaching

math

.504**

6. Confidence in ability

** Correlation is significant at the p < 0.01 level (2-tailed)

To examine the relationships among preservice teachers’ experiences in the mathematics

methods course, attitudes about mathematics, anticipated approaches to teaching

mathematics, and perceived preparation, bivariate two-tailed Pearson’s correlations were run

at the .05 alpha level. Table 2 displays results from the analyses among items pertaining to

these topics on the post-survey. Results indicated a moderate positive relationship between

participants who had a more positive attitude towards mathematics and whether they learned

a variety of strategies in the mathematics methods course (r = .273, p < .05), planned to teach

mathematics in a conceptual manner (r = .326, p < .01), planned to require their students to

memorize facts (r = .274, p < .05), and agreed that the mathematics methods course would

have a major impact on their future teaching (r = .268, p < .05). Preservice teachers who

indicated that they learned a variety of strategies in the methods course showed an increased:

desire to teach mathematics (r = .371, p < .01), confidence (r = .277, p < .05), and belief that

the course would have an impact on their teaching practice (r = .440, p < .01). An increased

agreement that the mathematics methods course would have an impact on teaching practices

was also significantly related to an increase in looking forward to teaching mathematics ( r =

.360, p < .01) and confidence (r = .291, p < .05). Participants’ level of confidence was alsoassociated with whether they would encourage students to use multiple strategies (r = .279, p

< .05), a characteristic of teaching with a conceptual focus.




Table 2

Relationships among attitudes and the mathematics methods course experiences

Post-Survey Items 2 3 4 5 6 7 8 9 10

1. Positive math

attitude

.273* .306** .794** .566** .326** .168 .066 .274* .268*

2. Learned a variety

of strategies

.192 .371** .277* .149 .043 .142 -.013 .440**

3. Prepared to teach

math

.397** .438** .139 -.149 .227* .047 .210

4. Looking forward

to teaching math

.711** .311** .011 .140 .061 .360**

5. Confident in

ability

.412** 0.031 .279* .137 .291*

6. Teach conceptual

Math

.275* .382** .155 .014

7. Teach procedural

Math

.051 .601** -.083

8. Encourage

multiple strategies

.016 .119

9. Require students

to memorize facts

-.109

10. Methods course,

major impact

* Correlation is significant at the p < 0.05 level (2-tailed)

** Correlation is significant at the p < 0.01 level (2-tailed)

Results showed that whether PTs planned to teach in a conceptual manner related to

whether they would encourage students to use multiple strategies (r = .382, p < .01). This

positive relationship was stronger for those who planned to teach mathematics in a procedural

manner and planned to require their students to memorize facts (r = .601, p < .01). These

findings suggest that preservice teachers were familiar with characteristics commonly

associated with both approaches to teaching mathematics. A relationship between preservice

teachers’ plans to teach with both approaches was not surprising; there can be overlap among

strategies to teach mathematics where both conceptual and procedural knowledge are valued.

While examining relationships among items is interesting, it is also important to extendanalyses beyond correlations to further examine preservice teachers’ attitudes.

Paired t-tests

Paired t-tests were conducted to determine significant differences in the mathematics

attitude and confidence to teach over the course of the semester (see Table 3). The paired t-

test was carried out with a two-tailed 95% confidence interval. Results indicated that PTs in




the mathematics methods courses had statistically significant positive changes in their

attitudes towards mathematics. They also became significantly more confident in their overall

ability to teach mathematics.

Table 3

Overall statistically significant differences on pre- and post-survey results

Item Mean (pre to post) Test Results

Positive attitude towards math 2.076 to 2.280 t= 3.401, p < .01

Confident in ability to be a good math teacher 1.932 to 2.139 t= 3.110, p < .01

To examine preservice teachers’ anticipated teaching practices, we analyzed responses to

two items on the post-survey: “I plan on teaching math in a conceptual way (for

understanding, problem-solving)”and “I plan on teaching math in a procedural way (facts,

skills, etc…).” Figure 2 shows participants’ responses to these two items by percentages.

Results indicated that 100% of PTs teachers strongly agreed or agreed that they planned to

teach mathematics in a conceptual way. In contrast, only about 70% strongly agreed oragreed that they planned to teach mathematics in a procedural way. Paired t-tests showed a

statistically significant difference (p <.001) between PTs’ responses to the two items in favor

of a conceptual teaching method.

Figure 2. Participants’ planned approaches to teaching mathematics by percentages

Another finding showed that approximately 80% of preservice teachers strongly agreed or

agreed that: “As a K-8 student, I mostly learned mathematics in a traditional manner (i.e.,

textbooks, worksheets, rules, lectures).” However, the majority also disagreed or strongly

disagreed with the following statement: “I want to teach mathematics the same way I learned

it.” There was a statistically significant difference in responses to the two items (p <.0001),indicating that preservice teachers wanted to teach mathematics in a way that was different

from the way they learned it.

Regression Analyses

Ordinary least squares (OLS) hierarchical regression was completed to investigate the

extent to which past schooling experiences and the mathematics methods courses accounted

0

1020

30

40

50

60

70

SA A D SD

Conceptual

Procedural




for preservice teachers’ attitudes about teaching mathematics and their perceived preparation

to teach mathematics. The prep_lookforward served as the outcome variable. This variable

was computed by taking the mean of responses from items “I am prepared to teach” and “I

look forward to teaching.” The two items were selected because they provided a sense of

preservice teachers’ attitude and preparation to teach mathematics. The responses from the

two items were divided by two so the outcome variable was on 4-point scale, consistent withthe predictor variables. For the multiple regression model, the predictor variables were

entered as follows: positive K-8 math as the first predictor and math course strategies next.

First we entered positive K-8 math into the model; research suggests that perceptions on prior

schooling can have a strong influence on teachers due to their countless hours spent as

students observing their own teachers (Ball, 1989; Ellsworth & Buss, 2000; Lortie, 1975).

Participants also spent more time as K-8 students than as student teachers. Next math course

strategies was entered because the mathematics methods course was specifically designed to

prepare preservice teachers to teach mathematics, whereas teaching mathematics may not be

a focus of the practicum (Ebby, 2000). Following a confirmatory approach, we hypothesized

that the variation found in preservice teachers’ perceptions of preparation and anticipation to

teach mathematics after being in the mathematics methods course for one semester could be

explained in terms of the aforementioned variables. In statistical terms, the hypotheses were

expressed as:

0

0

8

80

≠=

==

math positiveK a

math positiveK

H

H

β

β

0

00

≠=

==

stratsmathcoursea

stratsmathcourse

H

H

β

β .

The significance level was set at the 0.05 two-tailed level. Prior to running this model, the

individual influence each predictor variable had on its own was examined, as described next.

Single Predictors

Before constructing the multiple regression models, two simple regression models were

carried out to examine the amount of variance of each predictor variable accounted for in

prep_lookforward . Table 4 shows a summary of each of the regression statistics and its

significance. The two predictors accounted for a significant portion of the prep_lookforward

on their own (p < .01). Positive K-8 math alone accounted for 12.5% of the variance in the

outcome variable (R2 = .125, F = 10.45, p < .01); the predictor variable math course

strategies alone explained 12.3% of the variance in prep_lookforward (R2 = .123, F = 10.23,

p < .01).

Table 4

Simple regression statistics (Prep_lookforward as outcome variable) Predictor Variable R Square Adjusted R

Square

Unstnd.

Coefficient

Standardized Coefficient F-value Sig.

Positive K-8 Math .125 .113 .457 .354 10.45 .002

Math Course

Strategies

.123 .111 .789 .351 10.23 .002




Multiple Regression Model with Two Predictors

The overall regression of prep_lookforward on positive K-8 math and math course

strategies was statistically significant [R2 = .208, F (2, 75) = 9.441, p< .001]. Overall, the

variance explained by the two predictors differed significantly from zero; thus, we rejected

the null. Table 5 shows the overall model summary and significance levels. This model had a

higher F-value and was statistically significant. The positive K-8 math variable accounted forapproximately 12.5% of the variance in prep_lookforward , while math course strategies

accounted for an additional 8.3% of the variance. Taken together, the predictor variables

could explain approximately 20.8% of the variance of prep_lookforward . Although the model

was significant, nearly 80% of the variance was unaccounted for in prep_lookforward , which

supports the argument that a multitude of variables influence preservice teachers’ attitudes

and preparation to teach mathematics. The unstandardized coefficients of the model were

.192 for positive K-8 math and .330 for math course strategies. The regression solution for

this model was:

stratsmathcoursemath positiveK d preplookfw X X Y 330.0192.0942.0 8 ++=∧

.

Similar to the first model, this means that if both predictor variables had a value of 0, there

would be a predicted prep_lookforward score of 0.942. However, a value of 0 is not possible.

Table 5

Model summary and significance of two predictors Predictors R 2 ∆ R

2 F P DW

1. Positive K-8 Math .125 .125 10.447 .002

2. Positive K-8 Math

Math Course

Strategies

.208 .083 9.441 .000 2.164

Note: Dependent Variable (constant): Prep_lookforward to teach Math

The values indicated that with every increased rating in positive K-8 math there was

almost a 0.192 increase (i.e. 1= .192, 2= .384) in prep_lookforward and approximately a .330

increase with increased ratings in math course strategies. Using the same example as the first

model, a participant who “agreed” to the two items on the survey would have a predicted

prep_lookforward score of 2.508 = .942 + (.192 x 3) + (.330 x 3). One who “agreed” to the

two items would yield an approximate score of 2.51 on prep_lookforward , indicating greater

feelings of preparation and anticipation to teach mathematics than one without positive

experiences and with a response of “disagree,” or a baseline outcome score of one out of four.

The Durbin-Watson statistic for this model was 2.164. The DW obtained was higher than

the upper limit of 1.68; therefore, we failed to reject the null or to accept 0 H and conclude

that there was no statistically significant autocorrelation in our regression model. Results

indicated that multicollinearity was at a minimum because the tolerance was a .962 when the

second predictor variable was added. Similarly, the Variance Inflation Factor (VIF) was

1.039 with two predictors, indicating a small amount of multicollinearity. Multicollinearity

occurs when two variables are related; thus, it is important to keep it at a minimum when




creating regression models. To evaluate the effect size of the regression model, we computed

a post-hoc power analysis. The model with two predictors (see Table 5) had a very high level

of power (1 - β = 0.97) with a medium effect size (f 2 = 0.26) at the alpha level of .05. The

next section situates survey results within the literature to consider implications.

Discussion

Our research question stated, How do preservice elementary teachers’ perceptions of their

past schooling and their mathematics methods course influence their attitudes about the

teaching and learning of mathematics? Results from the survey showed several statistically

significant positive relationships among preservice teachers’ attitudes towards mathematics,

confidence to teach mathematics, and the two predicting variables: perceived past schooling

and the mathematics methods course. While the paired t-tests showed a significant difference in

favor of participants planning to teach mathematics with a conceptual approach, there were

significant correlations between conceptual and procedural approaches, implying some overlap

in the way in which participants responded to the two approaches. Regression models

confirmed that past schooling experiences and the mathematics methods course were influential

in predicting a significant portion of preservice teachers’ preparation and attitude toward

teaching mathematics.

Descriptive statistics from the post-survey showed that more than 80% of participants

perceived their prior schooling and the mathematics methods course experiences as having a

major impact on their anticipated teaching practices. The multiple regression model

confirmed that the two variables accounted for a significant proportion of preservice teachers’

perceived level of preparation and their attitude towards teaching mathematics. Nevertheless,

the two factors combined accounted for only 20.8% of the desired outcome variable including

preservice teachers’ looking forward to teaching mathematics and viewing themselves as

prepared. Thus, other factors beyond those in our model account for almost 80% of PTs’

preparation to teach mathematics and attitudes about teaching mathematics. In this section we

present an interpretive summary of the two main themes from our data.

Evolution of Attitudes

Findings indicated a strong relationship between PTs’ attitudes about mathematics and

their prior schooling experiences. A positive increase in participants’ attitudes towards

mathematics was related to positive perceptions of experiences in K-8 prior schooling (r =

.599, p < .01) and to high school (r = .719, p < .01). Although both experiences had positive

relationships with PTs’ attitudes, their high school experiences in mathematics had a greater

shared variance with their attitudes, suggesting that high school experiences in mathematicsmay have a stronger influence on PTs’ attitudes. We conjecture that at the high school level,

mathematical content becomes more challenging and those with a more positive experience

were more likely to have experienced success in mathematics courses. Similarly, an increased

response to perceived proficiency in mathematics had a strong positive relationship with

attitude towards mathematics (r = .713, p < .01). In addition, participants’ perceived

proficiency was related to both their perceived prior K-8 (r = .526, p < .01) and high school (r

= .599, p < .01) experiences in mathematics. Thus, correlation results indicated that those




with more positive prior schooling experiences had more positive attitudes towards

mathematics and considered themselves as more proficient. These findings are consistent

with findings from a qualitative study conducted by Ellsworth and Buss (2000), who

examined preservice teachers’ attitudes towards mathematics by analyzing their

autobiographies. They found that past teaching models was the most salient theme because

preservice teachers’ commonly reported that their interest in or attitude towards mathematicswas positively or negatively affected by past teachers. However, elsewhere we have reported

PTs with relatively negative experiences in K-12 mathematics demonstrate significant gains

in attitudes over the duration of mathematics methods coursework (Hodges, Jong, & Royal,

2013). Consequently, by bringing past experiences to the surface, preservice teachers may be

more cognizant of how their own attitudes about mathematics are affected.

Attitudes about mathematics can also influence preservice teachers’ own confidence to

teach mathematics. Bursal and Paznokas (2006) suggest that PTs with more positive attitudes

towards mathematics also had greater confidence in their own ability to teach mathematics. In

addition, findings from paired t-tests indicated that preservice teachers had a significant

increase in both their attitude towards mathematics and confidence to teach mathematics over

the course of the semester. These results suggest that positive changes in PTs’ attitudes and

confidence can begin to grow over a semester long mathematics methods course. The

findings differed from those of Vinson, Haynes, Brasher, Sloan, & Gresham (1997), who

compared PTs’ mathematics anxiety before and after taking methods courses emphasizing the

use of manipulative materials. Pre- and multiple post-survey results indicated no significant

difference in the mathematics anxiety scale after the first quarter of classes in the fall;

however, significant differences showing a reduction of mathematics anxiety were evident

after the winter, spring, and summer quarter classes. Thus, although immediate changes

cannot always be detected, attitudes might be affected over time by learning opportunities in

the mathematics methods course.

Data suggested that preservice teachers who learned mathematics in a traditional manner

would like to teach it differently than the way in which they were taught. However, the desire

to teach in a reformed manner can be difficult to put into practice. Rasmussen and

Marrongelle (2006) argue that teaching in a manner consistent with NCTM reform

recommendations may be overwhelming for teachers, because part of the challenge includes

the ability to understand students’ thinking and use it to develop mathematical ideas. This can

be a struggle for beginning teachers, who in most cases already have feelings of uncertainty

about their teaching, due to their limited classroom experience. In addition, prior to teaching

in a reformed manner, a teacher must value the classroom characteristics of reformed

teaching and have explicit reformed goals as a part of their classroom practice (Remillard &

Bryans, 2004).

Influential Factors

It is particularly important to acknowledge that preservice teachers enter teacher education

programs with a wealth of knowledge from their prior schooling. Although in some cases, the

goal of a course is to change or challenge entering assumptions about the role of teaching,

PTs can also have positive perspectives about teaching upon which complementary ideas can




be built. Mathematics methods courses could be built upon PTs entering attitudes, which

could be more positive and fertile than expected. While the mathematics methods courses

observed in this study did not appear to have an overt agenda or strategy to build upon PTs’

past experiences, questions were raised about their view of the teaching and learning of

mathematics. As the instructors taught methods for different mathematics topics such as

multi-digit subtraction, PTs would use the standard algorithms in many cases and connecttheir prior knowledge about the procedure with concrete materials. It would have been

interesting for PTs to explicitly compare different strategies and discuss the benefits of

alternative algorithms.

While the two experiences we focused on, perceived prior schooling experiences and the

mathematics methods course, can be of great importance in preparing elementary teachers to

teach mathematics, our study found that the two only accounted for 20% of the variance of

preparation to teach mathematics. This suggests that past experiences that we often try to

work with may not account for as much as we thought. While past experiences are important,

they might play a smaller part than we expect. We suggest that explicit efforts still be made in

the mathematics methods courses to connect to PTs’ prior knowledge, in the same way that

teacher educators encourage PTs to build upon students’ prior knowledge. Many scholars

have made a strong case for the importance of adopting an asset model by using students’

prior experiences as resources (Cochran-Smith, 1999, 2004; Darling-Hammond, French, &

Garcia-Lopez, 2002; Ladson-Billings, 1995). In addition, there are many factors that need to

be explored over time, such as field experiences, student teaching, mentors, family members

who are educators, peers, mathematics methods course designs, and mathematics content

courses, which could potentially influence changes in participants’ attitudes and confidence.

Our survey results suggest that PTs had an ideological stance in favor of conceptual

approaches to teaching mathematics. If the goal is for teachers to adopt practices that

emphasize a conceptual understanding of mathematics, meaningful teacher learning

experiences need to foster such attitudes along with exposing teachers multiple strategies that

can be implemented in the classroom. Harkness et al. (2006) suggested that mathematics

methods courses should provide opportunities for PTs to engage in meaningful problem

solving tasks to make sense of the mathematics and make connections to improve upon their

future practices.

Recommendations for Future Research

Preservice teachers’ prior schooling experiences influence their attitudes towards

mathematics and perceptions of the teaching and learning of mathematics. Thus, it is

important that teacher educators learn about PTs’ entering attitudes and perceptions in orderto create learning experiences that connect their prior knowledge to new ideas. Although

several scholars have argued that beginning teachers’ socialization into teaching takes place

when they are students (Ball, 1989; Grossman, 1990; Peker & Mirasyedioglu, 2008; Scott,

2005; Wideen et al., 1998), more empirical work that explores the extent to which past

experiences influence preservice teachers is needed. This study explored that issue as it

pertains to mathematics teacher education and showed that perceived past experiences only

accounted for 12.5% of the explained variance in PTs’ attitudes and confidence to teach




mathematics. We believe that this could actually be a very encouraging finding. While past

schooling experiences are a significant factor and need to be taking into consideration, there

are many additional factors that account for and influence teachers’ attitudes. Thus, providing

teachers with meaningful mathematics experiences in methods courses, supportive field

experiences, and continued professional development may have the potential to account for a

greater portion of teachers’ attitudes and confidence to teach mathematics.Given that much of the influence on teachers’ attitudes toward mathematics teaching and

learning lie beyond prior experiences and methods courses, research is needed on other

possible contributing factors in the attitudes and practices that PTs develop over time. Future

research should follow PTs longitudinally across teacher education programs and their entry

into the profession. In this study, the pre- and post-surveys were confined to one semester-

long mathematics methods courses in one university. Based on the factor analysis, the

instrument also had room for improvement, as surveys do not fully capture the variables of

interest due to self-reporting and restricted Likert-scales. Multiple data sites over time and

across institutions would allow for stronger comparisons. In addition to survey results,

qualitative interviews that elaborate on these experiences would help us further investigate

preservice teachers’ attitudes.

Acknowledgements

An earlier concise version of this paper was published in the Proceedings of the 33rd

Annual Meeting of the North American Chapter of the International Group for the

Psychology of Mathematics Education. Reno, NV: University of Nevada, Reno.

References

Ball, D. (1989). Breaking with experience in learning to teach mathematics: The role of a preservice methods course (Issue Paper #89-10). East Lansing, MI: National Center for

Research on Teacher Education.

Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to

teacher education. Elementary School Journal, 90(4), 449-466.

Ball, D., & Cohen, D. (1999). Developing practice, developing practitioners: Towards a

practice-based theory of professional education. In L. Darling Hammond & G. Sykes

(Eds.), Teaching as the learning profession: Handbook of policy and practice (pp.3-32).

San Francisco: Jossey-Bass.

Bursal, M., & Paznokas, L. (2006). Mathematics anxiety and preservice elementary teachers’

confidence to teach mathematics and science. School Science and Mathematics, 106 (4),

173-180.

Cady, J., Meier, S. L., & Lubinski, C. A. (2006). The mathematical tale of two teachers: A

longitudinal study relating mathematics instructional practices to level of intellectual

development. Mathematics Education Research Journal, 18 (1), 3-26.

Cochran-Smith, M. (1999). Learning to teach for social justice. In G. Griffin (Ed.). The

education of teachers: Ninety-eighth yearbook of the national society for the study of




education (pp. 114-144). Chicago, IL: University of Chicago Press.

Darling-Hammond, L., French, J., & Garcia-Lopez, S. P. (Eds.). (2002). Learning to teach

for social justice. New York: Teachers College Press.

Drake, C. (2006). Turning points: Using teachers’ mathematics life stories to understand the

implementation of mathematics education reform. Journal of Mathematics Teacher

Education, 9(6), 579-608.

Ebby, C. B. (2000). Learning to teach mathematics differently: The interaction between

coursework and fieldwork for preservice teachers. Journal of Mathematics Teacher

Education, 3, 69-97.

Eisenhart, M., Borko, H., Underhill, R., Brown, C., Jones, D., & Agard, P. (1993).

Conceptualknowledge falls through the cracks: Complexities of learning to teach

mathematics for understanding. Journal for Research in Mathematics Education, 24(1),

8-40.

Ellsworth, J. Z., & Buss, A. (2000). Autobiographical stories from preservice elementary

mathematics and science students: Implications for K-16 teaching. School Science and

Mathematics, 100(7), 355-364.

Feiman-Nemser, S. (1983). Learning to teach. In L. S. Shulman & G. Sykes (Eds.) Handbook

of teaching and policy (pp. 150-170). New York, NY: Longman.

Feiman-Nemser, S., Remillard, J. (1995). Perspectives on learning to teach. In F. Murray,

(Ed.), Knowledge base for teacher educators. Oxford: Pergamon Press, 63-91.

Grossman, P. (1990). The making of a teacher . New York: Teachers College Press, Chapters

1 and 4.

Harkness, S. S., D’ambrosio, B., & Morrone, A. S. (2006). Preservice teachers’ voices

describe how their teacher motivated them to do mathematics. Educational Studies in

Mathematics, 65, 235-254.Hodges, T. E., Jong, C., & Royal, K. D. (2013). The development of attitudes about

mathematics during preservice teacher education. In Martinez, M. & Castro Superfine, A

(Eds.). Proceedings of the 35th Annual Meeting of the North American Chapter of the

International Group for the Psychology of Mathematics Education (pp. 785-788).

Chicago, IL: University of Illinois at Chicago.

Ignacio, N. G., Blanco Nieto, L. J., & Barona, E. G. (2006). The affective domain in

mathematics learning. International Electronic Journal of Mathematics Education, 1(1),

16-32.

Ladson-Billings, G. (1995). But that’s just good teaching: The case for culturally relevant

pedagogy. Theory into Practice, 34(3), 159-165.

Leder, G., & Grootenboer, P. (2005). Editorial: Affect and mathematics education.

Mathematics Education Research Journal, 17 (2), 1-8.

Lortie, D. C. (1975). Schoolteacher: A sociological study of teaching. Chicago: University of

Chicago Press.




MacNab, D.S., & Payne, F. (2003). Beliefs, attitudes and practices in mathematics teaching:

Perceptions of Scottish primary school student teachers. Journal of Education for

Teaching, 29(1), 55-69.

McGinnis, J. R., Kramer, S., Roth-McDuffis, A., & Watanabe, T. (1998). Charting the

attitude and belief journeys of teacher candidates in a reform-based mathematics and

science teacher preparation program. Paper presented at the Annual Meeting of theAmerican Educational Research Association. San Diego, CA.

McLeod, D.B. (1994). Research on affect and mathematics learning in the JRME: 1970 to the

present. Journal for Research in Mathematics Education, 25(6), 637-647.

National Council of Teachers of Mathematics. (2000). Principles and standards for school

mathematics. Reston, VA: Author.

Nunnally, J. C. (1978). Psychometric theory (2nd

ed.). New York: McGraw-Hill.

Peker, M., & Mirasyedioǧlu, S. (2008). Preservice elementary school teachers’ learning

styles and attitudes towards mathematics. Eurasian Journal of Mathematics, Science and

Technology Education, 4(1), 21-26.

Philipp, R. A. (2007). Mathematics teachers’ beliefs and affect. In F. K. Lester (Ed.), Second

handbook of research on mathematics teaching and learning (pp. 257-315). United

States: Information Age Publishing.

Rasmussen, C., & Marrongelle, K., (2006). Pedagogical content tools: Integrating student

reasoning and mathematics in instruction. Journal for Research in Mathematics

Education, 37 (5), 388-420.

Remillard, J. T., & Bryans, M. B. (2004). Teachers' orientations toward mathematics

curriculum materials: Implications for teacher learning. Journal for Research in

Mathematics Education, 28 (5), 550-576.

Scott, A.L. (2005). Pre-service teachers’ experiences and the influences on their intentions forteaching primary school mathematics. Mathematics Education Research Journal, 17 (3),

62-90.

Wideen, M., Mayer-Smith, J., & Moon, B. (1998). A critical analysis of the research on

learning to teach. Review of Educational Research, 68 (2), 130-178.

Vinson, B. M., Haynes, J., Brasher, J., Sloan, T., Gresham, R. (1997). A comparison of

preservice teachers’ mathematics anxiety before and after a methods class emphasizing

manipulatives. Paper presented at the Annual Meeting of the Midsouth Educational

Research Association, Nashville, TN.




Authors

Cindy Jong, Assistant Professor, Department of STEM Education, University of

Kentucky, 105 Taylor Education Building, Lexington, KY 40506, USA;

[email protected]

Thomas E. Hodges, Assistant Professor, Department of Instruction and Teacher

Education, University of South Carolina, Wardlaw 105, Columbia, SC 29208,

USA; [email protected]

APPENDIX A - Mathematics Education Pre-Survey

Using the scale 1=Strongly Agree, 2=Agree, 3=Disagree, 4=Strongly Disagree, or 5=Not Applicable (if

you absolutely do not know or the item does not apply to you), please respond to the following statements about

mathematics.

Attitude and Past Experiences

S A A D S D NA

1. I like mathematics. 1 2 3 4 5

2. I enjoy solving mathematical problems that challenge me to

think.1 2 3 4 5

3. I had several positive experiences with mathematics as a K-8

student.1 2 3 4 5

4. I had several positive experiences with mathematics as a 9-12

student.1 2 3 4 5

5. I am proficient in mathematics. 1 2 3 4 5

6. Mathematics is one of my favorite subjects. 1 2 3 4 5

7. I think mathematics is boring. 1 2 3 4 58. I have struggled with mathematics as a K-8 student. 1 2 3 4 5

9. I have struggled with mathematics as a 9-12 student. 1 2 3 4 5

10. I used hands-on materials to learn mathematics in either

elementary, middle school, or high school.1 2 3 4 5

11. The way mathematics is taught today is different from the way I

learned it as a K-8 student.1 2 3 4 5

12. The way mathematics is taught today is different from the way I

learned it as a 9-12 student.1 2 3 4 5

13. As a K-8 student, I mostly learned mathematics in a traditional

manner (i.e. textbooks, worksheets, rules, lectures).1 2 3 4 5

14. As a 9-12 student, I mostly learned mathematics in a traditional

manner (i.e. textbooks, worksheets, rules, lectures).

1 2 3 4 5




Teaching and Learning

SA A D SD N/A

15. I am looking forward to teaching mathematics.1 2 3 4 5

16. It is important to incorporate the use of technologies (e.g.

calculators, computers) when teaching mathematics.1 2 3 4 5

17. Using mathematics is essential to the every day life of K-12

students.1 2 3 4 5

18. I want to teach mathematics the same way I learned it.1 2 3 4 5

19. I am confident in my ability to be a good mathematics teacher.1 2 3 4 5

20. I plan to use hands-on materials to help my students learn

mathematics and solve problems.1 2 3 4 5

21. Memorizing facts and formulas is essential to learn mathematics.1 2 3 4 5

22. I will allow and encourage students to solve mathematical

problems in more than one way.1 2 3 4 5

23. I plan on integrating mathematics with different subjects (i.e.

science, literature, social studies).1 2 3 4 5

24. I am scared of teaching mathematics.1 2 3 4 5

Methods Course Expectations

It is important for me to learn…

SA A D SD NA

25. a variety of instructional strategies. 1 2 3 4 5

26. how to use technologies (i.e. calculators, computers) in

mathematics classrooms.

1 2 3 4 5

27. how students learn mathematics developmentally (i.e. age, grade

level).

1 2 3 4 5

28. how to use hands-on materials to teach mathematical concepts. 1 2 3 4 5

29. about national mathematics standards and state frameworks. 1 2 3 4 5

30. how to teach mathematics to a diverse student population. 1 2 3 4 5

31. how to assess student learning in mathematics. 1 2 3 4 5

32. about the role of standardized tests in mathematics. 1 2 3 4 5

33. about different mathematics curriculums used by districts across

the nation.

1 2 3 4 5

34. how to manage the mathematics classroom effectively (i.e.

behaviors, grouping, transitions).

1 2 3 4 5

35. how to integrate mathematics with science. 1 2 3 4 5

36. how to integrate mathematics with literature. 1 2 3 4 5

37. about a variety of mathematics games that can be used in the

classroom.

1 2 3 4 5




Diverse Learners

SA A D SD NA

38. I am confident in teaching mathematics to high achievers. 1 2 3 4 5

39. I am confident in teaching to students who do not have English

as their primary language.1 2 3 4 5

40. I am confident in teaching mathematics to students with special

needs. 1 2 3 4 5

41. I am confident in teaching mathematics to students of different

ethnic/racial/cultural backgrounds.1 2 3 4 5

42. Social justice plays an important role in the teaching and

learning of mathematics.1 2 3 4 5

43. Most students (who do not have severe special needs) can be

successful at learning mathematics.1 2 3 4 5

44. I am confident in teaching mathematics to students in an Urban

school.1 2 3 4 5

45. I am confident in teaching mathematics to students in a

Suburban school.1 2 3 4 5

46. I am confident in teaching mathematics to students in a Rural

school. 1 2 3 4 5

47. Mathematics can help students critically analyze the world. 1 2 3 4 5

48. Issues about equity should be addressed in the mathematics

classroom.1 2 3 4 5

Background Information

1. Gender: Male________ Female________

2. Degree: ______________________________ 3. Current Year:____________________

4. Major: _______________________________ Minor: ___________________________

5. If you are a Graduate Student, Undergraduate Major:______________________________

6. Course Professor:_________________________ Time: ___________________________7. Number of Math Content Courses Taken at the College Level: ______________________

8. Future Teaching Plans (check all that apply):

Suburban_________ Urban_________ Rural_________

Public_________ Private_________ Religious__________

Grade(s): ___________ Subject(s): ___________________________________________

9. Describe your ethnicity. ____________________________________________________

___________________________________________________________________________

10. How long have you (and your family) been in the U.S.A.?

Generation: 1st________ 2

nd _________ 3

rd _________ 4+ _________

11. Mother’s highest level of Education: _________________________________________Occupation: _____________________________________________________________

12. Father’s highest level of Education: __________________________________________

Occupation: _____________________________________________________________

13. Describe your previous teaching experience (if any).

___________________________________________________________________________

___________________________________________________________________________




APPENDIX B - Mathematics Education Post-Survey

Using the scale 1=Strongly Agree, 2=Agree, 3=Disagree, 4=Strongly Disagree, or 5=Not

Applicable (if you absolutely do not know or the item does not apply to you), please respond

to the following statements about mathematics.

Attitude and Practicum Experiences

S A A D S D NA

1. I like mathematics. 1 2 3 4 n/a

2. I had a positive practicum experience. 1 2 3 4 n/a

3. My cooperating teacher contributed greatly to my knowledge

about the teaching and learning of mathematics.1 2 3 4 n/a

4. My cooperating teacher used a traditional method (i.e. textbooks,

lectures, worksheets, rules) to teach math.1 2 3 4 n/a

5. My cooperating teacher used a conceptual method (i.e. problem-

solving, open-ended Qs) to teach math.1 2 3 4 n/a

6. The math curriculum used in my practicum focused on teaching

math in a conceptual manner.

1 2 3 4 n/a

7. The math curriculum used in my practicum focused on teaching

math in a traditional manner. 1 2 3 4 n/a

8. My practicum experience connected to my math methods course. 1 2 3 4 n/a

9. My practicum experience reinforced what I learned in my math

methods course.1 2 3 4 n/a

10. My practicum placement had a diverse student population. 1 2 3 4 n/a

11. I think math is boring. 1 2 3 4 n/a

Teaching and Learning

SA A D SD NA

12. I am looking forward to teaching mathematics. 1 2 3 4 n/a

13. I plan on incorporating the use of technologies (e.g. calculators,

computers, software) when teaching mathematics.1 2 3 4 n/a

14. I plan on teaching math in a procedural way (facts, skills, etc…). 1 2 3 4 n/a

15. I plan on teaching math in a conceptual way (for understanding,

problem-solving).1 2 3 4 n/a

16. I am confident in my ability to be a good mathematics teacher. 1 2 3 4 n/a

17. I plan to use manipulatives (hands-on materials) to help my

students learn mathematics and solve problems.1 2 3 4 n/a

18. I will require my students to memorize mathematical facts and

formulas.1 2 3 4 n/a

19. I will allow and encourage students to solve math problems in

more than one way.

1 2 3 4 n/a

20. I plan on integrating mathematics with different subjects (i.e.

science, literature, social studies).1 2 3 4 n/a

21. I am scared of teaching mathematics.1 2 3 4 n/a

22. I am prepared to teach mathematics. 1 2 3 4 n/a




Methods Course Evaluation

The math methods course taught me…SA A D SD NA

23. a variety of instructional strategies. 1 2 3 4 n/a

24. how to use technologies (i.e. calculators, computers) in

mathematics classrooms.1 2 3 4 n/a

25. how students learn mathematics developmentally (i.e. age, grade

level).1 2 3 4 n/a

26. how to use manipulatives (hands-on materials) to teach

mathematical concepts.1 2 3 4 n/a

27. about national mathematics standards and state frameworks. 1 2 3 4 n/a

28. how to teach mathematics to a diverse student population. 1 2 3 4 n/a

29. how to assess student learning in mathematics. 1 2 3 4 n/a

30. about the role of standardized tests in mathematics. 1 2 3 4 n/a

31. about different mathematics curriculums used by districts across

the nation.1 2 3 4 n/a

32. how to manage the mathematics classroom effectively (i.e.

behaviors, grouping, transitions).1 2 3 4 n/a

33. how to integrate mathematics with science. 1 2 3 4 n/a

34. how to integrate mathematics with literature. 1 2 3 4 n/a

35. about a variety of mathematics games that can be used in the

classroom.1 2 3 4 n/a

36. theories about the teaching and learning of mathematics. 1 2 3 4 n/a

Diverse Learners

SA A D SDNA

37. I am confident in teaching mathematics to high achievers.1 2 3 4 n/a

38. I am confident in teaching to students who do not have English

as their primary language. 1 2 3 4 n/a

39. I am confident in teaching mathematics to students with special

needs.1 2 3 4 n/a

40. I am confident in teaching mathematics to students of different

ethnic/racial/cultural backgrounds.1 2 3 4 n/a

41. I think social justice plays an important role in the teaching and

learning of mathematics.1 2 3 4 n/a

42. I am confident in teaching mathematics to students in an Urban

school.1 2 3 4 n/a

43. I am confident in teaching mathematics to students in a

Suburban school.1 2 3 4 n/a

44. I think issues about equity should be addressed in the

mathematics classroom. 1 2 3 4 n/a




Future Teaching

The following will have a major impact on the way I teach

mathematics in the future:SA A D SD NA

45. My past K-8 school experiences 1 2 3 4 n/a

46. My past 9-12 school experiences 1 2 3 4 n/a

47. Practicum experiences 1 2 3 4 n/a48. Math methods course 1 2 3 4 n/a

Background Information

Practicum

1. Grade level:____________________

Secondary, please specify content area(s)____________________________________

2. Setting: Urban_________ Suburban__________

3. Public __________ Private (religious)__________ Private (nonreligious)__________4. Math Curriculum used by Cooperating Teacher

_______________________________



Documents

International Electronic Journal of Mathematics Education_Vol 8_N2-3