Probing the structure of mathematical writing

DAVID J. CLARKE, ANDREW WAYWOOD, AND MAX STEPHENS

PROBING THE STRUCTURE OF MATHEMATICAL WRITING 1

ABSTRACT. This paper examines one mode of mathematical communication: that of student journal writing in mathematics. The focus of the discussion is a study of four years' use of journal writing in mathematics involving approximately 500 students in Grades 7 through 11 in a particular Victorian secondary school. The evaluation of the experimental use in one school of journal writing in mathematics provides a powerful demonstration of the link between language and mathematics and suggests a relationship between students' mathematical writings and their perceptions of mathematics and mathematical activity.

INTRODUCTION

The learning of mathematics is fundamentally a matter of constructing mathematical meaning. The environment of the mathematics classroom provides experiences which stimulate this process of construction. While the mathematical knowledge of schoolchildren will incorporate visual imagery, both at the level of iconic thought and more elaborate visual representations (geometrical, graph- ical), mathematical meaning requires a language for its internalization within the learner's cognitive framework and for its articulation in the learner's interactions with others.

Communication is at the heart of classroom experiences which stimulate learning. The NCTM document Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) lists "Mathematics as Communication" as its second standard, preceded only by "Mathematics as Problem Solving". Classroom envi- ronments that place particular communication demands on students can facilitate the construction and sharing of mathematical meaning and promote student reflection on the nature of the mathematical meanings they are required to communicate. This paper examines one mode of mathematical communication: that of student journal writing in mathematics.

Since the 1960s with the birth and blossoming of the "writing across the curriculum" movement and its sharpening of focus in the language and learning movement (Britton, 1972), there has been a great deal of research and debate concerning the relationship between language and learning in mathematics. The various emphases in this research can be seen from reviews (Bell et al., 1983; Ellerton and Clements, 1991) and edited volumes (Bickmore-Brand, 1990; Con- nelly and Vilardi, 1989; Durkin and Shire, 1991). A number of writers have identified benefits that accrue with the use of writing, both for learning and for the classroom process that supports learning (Allen, 1991; Hoffman and Powell, 1989; Mett, 1989; Miller, 1991). That there is an intimate connection between

Educational Studies in Mathematics 25: 235-250, 1993. © 1993 Kluwer Academic Publishers. Printed in the Netherlands.

236 D.J. CLARKE ET AL.

writing and learning has been highlighted in the work of Vygotsky (1962), Emig (1977), and Gopen and Smith (1990). The view that emerges from the work of these researchers is that (1) the act of writing necessarily involves processes that are fundamental to learning that otherwise are not necessarily engaged (writing is an enactive, iconic, and symbolic act); and (2) that the process of writing mirrors the process of learning and can be seen as supportive of it. These authors have all conducted their reflections within the framework of natural language. The writing to learn mathematics movement appeals to these authors because there are no equivalent studies for mathematical communications. Current research is being undertaken into the structure of student writing in mathematics.

Borasi and Rose (1989) report a study of 29 students in a college mathematics course who were asked to keep a journal for the duration of the course. They begin by delineating journal writing as a particular type of"writing to learn", where they take the term "journal" to mean the "keeping of a log or personal notebook". They concluded a survey of the sparse literature on journal writing by saying:

... we are still lacking a convincing argument about how and why journal writing can contribute

uniquely to the improvement of mathematics instruction. With this paper, we aim to address such

a void by providing a comprehensive and articulated analysis of the potential benefits of journal

writing, based on an investigation combining both conceptual and empirical components. (Borasi

and Rose, 1989, p. 349)

The work by Borasi and Rose suffers from the same limitation as most work in this area to date - the sample of work is to small to yield stable results. The difficulty lies in the trade-offs that necessarily exist in the choice between a content-based approach to the data- the course taken by Borasi and Rose- which requires reducing the amount of data examined to make the analysis practicable, and a quantitative approach which is so coarse it hides the very things you might hope to find in the data. We have tried to avoid these pitfalls by constructing a study which focused on the perceptions of students concerning their experience of writing in the context of learning mathematics and relating this to an hypothetical construct, which we have designated "Mode of Use", of student mathematics journals.

The focus of the discussion is a study of a four-year teaching experiment involving approximately 500 students in Grades 7 through 11 in a particular Vic- torian secondary school. The evaluation of the experimental use in one school of writing in mathematics provides a powerful demonstration of the link between language and mathematics and suggests a relationship between students' mathematical writings and their perceptions of mathematics and mathematical activity.

THE STUDY CONTEXT

This study explored the implications of the regular completion of student journats in mathematics. The study was undertaken at a Catholic secondary girls school in Melbourne with an enrollment of approximately 500 students from Year 7 to Year 12. The students are drawn from working- and middle-class families from across

MATHEMATICAL WRITING 237

the city. Of the school's population, 20 percent are of Asian background (Chinese and Vietnamese), 30 percent of Italian and Greek background, with the remaining 50 percent being predominant Anglo-Saxon.

In 1986, journal writing in mathematics was introduced as an experiment by several teachers in their classes. Starting the following year, the completion of journals in mathematics became progressively part of the school's policy. From 1988 onwards, for all students from Year 7 upwards, a central component of mathematical activity has been the regular completion of a student journal.

As a first introduction to journal writing, Year 7 students were supplied with a journal book in which each page was divided into three sections: "What we did", "what we learned", "examples and questions". These three headings were intended to introduce students to the three major tasks that underlie journal writing for all year levels: to summarize what has been taught; to discuss both the ideas and the learning; and, to formulate questions and collect examples. Students were expected to write in their journals after every mathematics lesson. Completion of the journal was seen to be part of home study.

The school has set out a history and rationale for students' mathematical journals. In this statement, the following aims are given:

By keeping a mathematics journal, we intend that students will formulate, clarify, and relate

concepts; appreciate how mathematics speaks about the world; think mathematically - that is,

practice the processes (e.g., problem solving) that underlie the doing of mathematics; formulate

physical relations mathematically.

It was further intended that, through their journal keeping activities, students would be introduced to describing what they had learned, summarizing key topics, and identifying appropriate examples and questions. Regular monitoring of the journals should inform teaching practice and provide the basis for discussion with students individually or as part of discussion with the whole class.

It is important to note the shift from journal keeping in mathematics as an experiment undertaken by several mathematics teachers in their own classrooms to its adoption by all teachers of mathematics as part of the school's policy. To achieve this, it was necessary for all teachers of mathematics to be involved in supporting the use of journals.

First, a clear understanding of what is intended by "keeping a journal" had to be communicated to teachers and students as well. Second, class and homework time had to be given over to supporting the use of journals. Students' journal work needed to be seen to be valued as highly as more traditional aspects of mathematics learning. Therefore, journals needed to be assessed and reported on.

At the time of the study, journals contributed 30 percent to the assessment of mathematics. When writing student reports, mathematics teachers in the school used the following framework for the assessment of student journals. This framework comprised a table of progress descriptors for student journal use (Table I). Table I makes explicit the components of the journal writing task. Student progress with respect to each of the four identified components: Summary, Exemplification, Questioning, and Application was considered independently.

238 D, J. CLARKE ET AL.

TABLE I

Table of progress descriptors for student journal use.

Summary Exemplification Questioning Application

Able to regularly

copy part or most of

the board notes into

the journal.

As above but also

able to describe

important aspects of

what was done in

class.

Able to record some

of the main ideas of a

lesson and able to

write some thoughts

about them.

Able to isolate and

record in own words

a sequence of

connected ideas from

a lesson, with an

emphasis on

expressing mathe-

matical learning.

Able to formulate and

state an overview of

material covered in a

lesson, text, or topic

with appropriate use

of formal language

and vocabulary.

Able to extrapolate

from material

presented in class, or

in a text, and reshape

it in terms of own

learning needs.

Includes

copies from board or

from exercises, but

unable to connect them

with the journal entry.

Able to choose

appropriate practice

exercises as examples

to illustrate the content

of the lesson.

Able to use examples

to show how a

mathematical procedure

is applied.

Able to choose

important examples

and show clearly how

the example illustrates

the mathematics being

used and how it

works.

Able to choose relevant

mathematical examples

to illustrate points in

the discussion of an

idea.

Able to choose

examples that

summarize important

aspects of topic, idea,

or application. These

examples are fully

annotated to show their

relevance.

Asking question that Writes short entries

are unfocussed, about occasional

E.g., How do you do lessons.

algebra? How do you

do these things?

Able to ask focussed Writes brief entries

questions to get help regularly.

with particular

difficulties.

Able to ask questions Maintains regular

about mistakes or entries that give an

misunderstandings adequate coverage of

that lead to a the day's lesson.

discussion of the

underlying idea.

Able to ask questions Maintains regular

that explore entries that often

consequences or explore or extend the

extend ideas - material covered in

E.g., What if ....? class.

Able to ask questions

that are aimed at

linking one part of

mathematical learning

with another.

Able to pose clearly

mathematical

questions. That is,

questions that are

appropriate to the

discipline of

mathematics, in a

mathematical way.


As a minimum, a satisfactory journal entry was intended to reflect the intellec- tual involvement of the student in the day's lesson. The form taken by a particular journal entry should reflect what took place in that day's lesson and the level of sophistication and experience which the student could bring to the completion of the journal task.

Journal writing was intended to assist students to see themselves as active agents in the construction of mathematical knowledge (see Stephens, 1982). The school hoped that journal writing would assist students progressively to engage in an internal dialogue through which they reflected on and explored the mathematics they met. In this respect, journal writing has a focus on the development of metacognitive learning. Through their journal writing, it was also hoped that students would begin to see mathematical activity, not simply in terms of applying prescribed rules and procedures, but more in engaging in such activities as searching for patterns, making and testing conjectures, generalizing, asking Why?, trying to be systematic, classifying, transforming, searching for methods, deciding on rules, defining, a~eeing on equivalences, reasoning, demonstrating, expressing doubt, and proving (cf. Mason, 1984). In other words, journal writing was aimed at engaging students in a constructive dialogue for learning mathematics.

It was intended that this study should identify indicators of any developmen- tal process associated with student journal writing and document student beliefs regarding mathematics, mathematical activity, and the role of journals in mathematical learning.

METHOD

During 1988 and 1989, an evaluation was conducted of student journal use and its effects on the learning and teaching of mathematics. Consultation with school staff, perusal of a sample of student journals, and the selective interviewing of a cross-section of pupils led to the construction of a questionnaire which, after field testing, was administered to all students in Years 7 to 12. The questionnaire examined student use of journals and their perceptions of the purpose of journal communication and its contribution to their learning of mathematics. Students' conceptions of the nature of mathematics and of mathematical activity in schools were also addressed. A similar survey was conducted of school mathematics staff, with specific focus on the extent to which they valued and fostered students' journal communications and made use of students' journal communications in their classroom teaching and in their work with individual students.

While questionnaires were administered to every student, a sample of 150 students, a random sample of 25 at each year level, was chosen for statistical analysis. Three questionnaires were administered ("Mathematics", "Journals - Part A" and "Journals - Part B", in that order) and the sample selection procedure ensured that all students at a particular year level, who had completed all three questionnaires, had the same chance of appearing in the sample.

The student questionnaires were designed to generate data with respect to the


fo l lowing:

Journals Journal use - period of experience with journals, frequency of journal writing, weekly duration

of journal writing, and frequency of journal reading.

Journal purpose - motivation for journal writing, perceived purpose for journal writing.

Journal difficulties -difficulties experienced in journal writing.

Journal value - learning from journals, uses of journal writing, outcomes of journal writing.

Teacher actions - teacher actions related to journal writing.

Mathematics Perceptions of school mathematics - typical activities, attitudes, valued activities.

Perceptions of the discipline of mathematics - origin of mathematical ideas, everyday relevance,

character of mathematical expression.

RESULTS

A full statistical report was prepared for the use o f the school (Clarke et aI., 1989).

M u c h of this repor t was descr ip t ive and site-specific. The purposes o f this paper

are best served by a summary of significant findings. These are set out below, with

related conc lus ions appropr ia te ly clustered. It must be borne in mind that these

f indings are the results o f s tudents ' reports o f their behaviour, their perceptions,

their beliefs, and their teachers ' behaviour.

Implementing the Journal Program

The preced ing discuss ion set out the goals and structure of the school journal writ-

ing program. Student responses documented the inevitable discrepancy be tween

the in tended p rog ram and the p rogram as realized through student and teacher

pract ices.

The majority of students (54 percent) reported that they wrote in their math journals "after every

lesson". Three quarters of the sample reported writing in their mathematics journals at least twice

a week.

A similar majority (53 percent) estimated the time spent on journal writing in one week as less

than one hour. However, one quarter of the sample reported that they devoted more than two

hours per week to journal writing.

Ninety percent of students reported reading their journals either occasionally or often.

The most common student estimate of the frequency of teachers' reading of journals was "once

a month". However, one third of the sample reported that their teachers read the journals less

frequently than this.

A Teacher Action measure was constructed from a cluster of related questionnaire items. The

variation in Teacher Action with Teacher Identity was statistically significant; that is, the differ-

ences which students saw in the action which particular teachers took in relation to journal use

were consistent and significant.


It is clear that the practices associated with the journal writing program were not uniform across either students or teachers and, like other educational programs, the journal writing program varied in the details of its implementation by the individual teachers. However, the above findings demonstrate a sufficient fidelity to the structure and goals of the program for the consequent student beliefs and behaviours to be associated with a common experience of mathematics journal writing.

Perceptions of Purpose

It was necessary to address the question of whether the task of journal writing in mathematics was perceived as purposeful and reasonable.

Sixty percent of students gave as the main reason for writing in their journal, "because it helps me". In another item, the most popular justification for journal use was "To help me learn". Most students (75 percent) found the act of journal writing "mostly" or "always" easy. However, students were evenly divided over whether or not they found it difficult to put their mathematical thinking into words. In this regard, it is worth noting that half of the student sample reported

that the most important thing learned from journal completion was "To be able to explain what I think".

These responses link purpose to outcome, and suggest that student motivation for journal completion derived in part from the perceived value of the activity.

Asked to identify "the most important thing for me to do in my journal", students indicated "To summarize what we did in class", "To write down what I understand" and "To write down examples of how things are done" in that order. In response to the item "I think of my mathematics journal as ...", the most frequent student responses were "As a summary for me to study from later" and "As a record of the things I have learnt in math".

These perceptions of the nature of journal use relate quite closely to the stated goals of the school program and suggest both cognitive and metacognitive consequences.

Journal Text

In the process of implementing and integrating journal writing into the mathematics curriculum, the school developed a categorization of the types of use students made of journals. In evaluating journals teachers categorized them as either Nar- rative, Summary, or Dialogue. While "Narrative" encompasses the characteristics we would identify with that mode of writing, "Recount" is probably the technical term closest to our meaning. In evaluating students' perceptions of journal writing this categorization assumed the status of an hypothesis, and provided much of the structure for the initial data analysis. School sources asserted that a major aim of journal writing was to facilitate student development in question-asking, and that questioning reflects the dialectic of Recount, Summary, and Dialogue (Waywood, 1988).


T h e ca t ego r i za t i on o f s tuden t j o u r n a l use into Recoun t , Summary , and Di-

a l o g u e w a r r a n t s m o r e de ta i l ed exp lana t ion . T he e x a m p l e s w h i c h fo l low were

o f fe red as b o t h i l lus t ra t ive in s t ances o f e ach r e s p o n s e ca tegory ( E x a m p l e s 1, 4

and 6) and also as e x a m p l e s o f " t r an s i t i on" r e s p o n s e s f r o m s tudents w h o s e jour -

nal en t r i es sugges t tha t they are in t r ans i t ion b e t w e e n categor ies . Seen in this

l ight , E x a m p l e 2 s h o w s a s tuden t m o v i n g f r o m a s imp le r e c o u n t i n g o f c l a s s r o o m

e x p e r i e n c e s to the r e s t ruc tu r ing o f con t en t and e x p e r i e n c e r equ i red for e f fec t ive

summary . E x a m p l e s 3 and 5 s h o w two s tuden t s ' ini t ia l e x p e r i m e n t s wi th a n e w

f o r m o f j o u r n a l entry. In each o f these two cases, the excerp t s r ep re sen t e m b r y o n i c

i n s t a n c e s o f the s u m m a r y and d i a logue ca tegor ies , respect ively .

Recount

Example 1. "Today was the day that Mr. Waywood was absent and set us work to do that gave

me a lot of thinking to do. I don't think that it was very hard but you had to think about what to

write for the answer to the questions".

Example 2. "I think today I began to understand that math is a way of describing things in reality.

A great example is that a ball flying through the air travels the path of a parabola. Because there is

an infinite number of ways for the ball to travel there is an infinite number of possible parabolas.

Because parabolas can be written mathematically there would be a mathematical function to

describe every arc in the world"

Summary

Example 3. "Logarithms are an index which are used to simplify calculations. The whole number

part of a logarithm is called the characteristics. The decimal part of a logarithm is called the

mantissa".

Example 4. "Equations ... the main word here is to solve. Equations have an unknown - there

is an answer to the problem. Linear techniques revolve around inverse operations, and quadratic

equations, different from the above, require different techniques to solve them, such as factorisa-

tion .... You can't solve all the equations the same way, because they are all different, and that is

why we have to learn different techniques".

Dialogue

Example 5. [Having studied the effects of transformations on linear and quadratic functions, this

student began to investigate the effects of the same transformations on sine functions.] "I know

what the 'sine' [graph] looks like .... I 'm surprised to find that the rules are similar to those for a

quadratic function .... As I was unsure whether these rules apply to all or some functions, I went

on to find evidence to support this claim".

Example 6. "Transposition and substitution really show you the quality of operations. Like, di-

vision is sort of a secondary operation, with multiplication being the real basis behind it... Which

came first, multiplication or division? It would have to be multiplication. They are so similar,

no, that's not what I mean. I mean they are so strongly connected. But it's like division does not

really exist, multiplication is more real. The same with subtraction. Addition and multiplication

are the only real operations".

T h e s tudy d e s i g n p r o v i d e d a d ivers i ty o f da ta sources by wh ich the va l id i ty

o f the c a t e g o r i z a t i o n c o u l d be assessed. S t uden t in te rv iews , s tuden t and t eacher


questionnaires, teacher interviews, and the study of journal entries represented a substantial body of data by which both the individual validity of each category could be judged and any patterns of individual development identified.

An analysis of the student survey data suggested that journal writing leads to a progressive refinement of purpose from an initial Recount state of simply listing events in the mathematics classroom to summarizing work done and topics covered. With this stage, we noted a move away from a simple summary of items of mathematical work covered to a more personal summary of mathematical activity in terms of developing understanding and addressing problems. Finally, some students appeared to move beyond this to an internal dialogue, where they began to pose questions and hypotheses concerning the mathematics in which they were engaged (e.g., "I wonder whether this works for other graphs as well", and "So, why is it that...").

More specifically, the Recount descriptions of what was done on a particular day, so prevalent in Years 7 and 8, appeared to be progressively enriched by the inclusion of reflective writing in which the students discussed how they went about an investigation and how the work in hand related to work they had previously covered. This reflective process of review, together with requests for teacher help and indications of things they would like to find out, resembled the responses solicited through the IMPACT program (Clarke, 1987; Clarke, Stephens, and Waywood, 1992). Journal entries of some students occasionally took on the aspect of Dialogue. Our research suggests that through the process of their journal writing students increasingly interpret mathematics in personal terms: constructing meanings and connections.

The Nature of Journal Use

By clustering student responses to particular items on the questionnaires, it was possible to construct indices associated with the hypothesized taxonomy of writing modes: Recount, Summary, and Dialogue.

Item 8D on the Journals B questionnaire, "l think of my mathematics journal as a report to my teacher of what I have done", was associated with the Recount mode; Item 11C on the Journals A questionnaire, "Writing in the journal is most use~l to me when 1 write summaries", was associated with the Summary mode; Item 1D on the Journals B questionnaire, "I use my journal to practise thinking", was associated with the Dialogue mode.

The total number of positive repsonses made by an individual student to those items associated with the Recount mode, constituted the Recount index rating for that student.

Two hypotheses underlie the use of these indices: first, that student perceptions of their journal use are an indication of actual mode of use; second, that the three hypothesized modes of use represent an hierarchy. Both these hypotheses were addressed in the subsequent analysis.

It was expected that for some students the ratings might be spread between two adjoining categories, for example between Summary and Dialogue, but we


did not expect that students would show high ratings across all three categories, or that they would have a high rating on Recount and Dialogue but with a low rating on Summary. These expectations were borne out in the analysis.

Of the sample of 150 students, 65 could be identified as predominantly em- ploying one of the three modes of journal use. This enabled statistical analyses to be carried out for this subset of students incorporating a categorization of mode of use. A "Modal Rating" on a seven-point uni-dimensional scale: R, R+, S - , S, S+, D - , D was successfully applied to the responses of 123 of the 150 students. This scale enabled the classification of those students in transition between modes. The conclusions which follow held true for both measures.

Year Level was more decisive in determining the frequency of journal use than was a student's experience with journal use. However, experience with journal use was more significant in accounting for mode of use. This justifies the conclusion that it is the experience of using journals

that promotes more sophisticated modes of use rather than simply student maturation. This finding also supports the interpretation of the three modes of use as levels in a learning hierarchy.

In addition to indices reflecting mode of use, other indices were constructed by clustering questionnaire items which addressed particular issues such as the extent of journal use by students (User Index); the difficulty experienced by students in using journals (Difficulty Index); the extent to which the journal had assisted students (Assistance Index); the frequency of teacher actions related to student journal use (Teacher Action Index).

Multiple regression analysis revealed that mode of use made the most significant contribution to

the variation evident in the three key indices User Index, Difficulty Index, and Assistance Index. A clear, and statistically-significant trend emerged on consideration of mode of use in relation to other critical indices. The more sophisticated the mode of journal use the more likely a student

w a s to:

* make more use of journals, * find journal completion less difficult, * express greater appreciation of journal completion, * report positive, rather than negative, outcomes of journal use. These results may not be surprising, but the consistency in the direction of the trend and in the statistical significance strongly supports the interpretation of mode of use as a meaningful structure for the analysis of student journal writing. It also highlights the fact that the use of informal writing empowers change independent of maturation.

The consistency of the trend noted above Suggests a robustness in the mode of use categories which encourages their interpretation as truly indicative of the character of student journal writing. However, it must be borne in mind that these conclusions regarding the modes of journal use are the result of the self-reporting by students of their perceptions of journal use. As such, they require corroboration from another source if more general implications are to be inferred regarding the nature of student mathematical writing. Such corroboration could be provided by a detailed content analysis of grammatical forms to establish the generic forms of the modes of use. Work is presently underway on the refinement of the methods and theory of such text analyses (Waywood, in preparation).


Linking Questionnaire Responses to Student Text

For the purposes of this study, corroboration was sought in a validation exercise involving the blind coding of samples of student text taken from the journals of students whose questionnaire responses generated modal ratings suggesting archetypal mode of use characteristics. That is, students were identified who scored highly on the Summary mode of use index, for instance, and near zero on the other two mode of use indices. In this fashion, three sets of students were identified whose journal writing might be expected to display characteristics distinctive of each of the three modes of journal use.

Ten students were selected whose questionnaire responses were such that they could be considered archetypal in one category. Samples of the students' current writing were then collected and looked at independently by two of the authors. Each writing sample was rated on the seven-point Model Rating Scale. The criteria applied in locating student writing samples on the seven-point scale resembled the structure outlined in Table I. For example, student progress within the 'Questioning' component was a key discriminator between the Summary and Dialogue modes . The two independent assessments of the text showed a high degree of rater agreement. In the majority of cases, the authors gave the same classification to a student's text and in the remaining cases differed by only one point on the seven-point scale.

Significantly for this study, the author's classification of the text in the sample of students' work investigated confirmed the archetypal category in which students had been placed on the basis of their responses to the questionnaires. In view of these results, we believe that we are justified in identifying the categories used to describe students' questionnaire responses with those used to discuss the linguistic forms by which students communicated what they had learnt through their journals and how they had gone about it.

Mathematics and the School Mathematics Program

Since data were collected regarding student conceptions of the nature of mathematics and mathematical activity, it was possible to examine particular student conceptions of the nature of mathematical activity.

Students reported that their most common experience of mathematics at school was "listening to

the teacher", closely followed by "writing numbers", "listening to other students", and "working

with a friend".

Students rated aspects of their mathematics course in order of importance. By far the most

important was "The teacher's explanations". Other important aspects, in order, were "The help

my teacher gives me", "Working with others", "My math journal", and "The textbook".

The role of communication in the learning of mathematics and in the performance of mathematical activities was given considerable prominence by a significant majority of students.

246 D.J. CLARKE ET AL

O f part icular interest was the student response to a quest ionnaire i tem con-

cerning the origins o f "ma themat i ca l ideas". Both the i tem and a graph of the

mean student response to each category are g iven in Figures l a and lb.

Item. (Circle "Agree" or "Disagree" depending on whether you think the statement is true or not.)

The ideas of mathematics

A. Have always been true and will always be true Agree Disagree B. Were invented by mathematicians Agree Disagree C, Were discovered by mathematicians Agree Disagree D. Developed as people needed them in daily life Agree Disagree E. Have very little to do with the real world Agree Disagree F. Are most clearly explained using numbers Agree Disagree G. Can only be explained using

mathematical language and special terms Agree Disagree H. Can be explained in everyday words

that anyone could understand Agree Disagree

Fig. la. Item probing student conceptions of the origins and nature of mathematical ideas.

.8.

.el

.41

.21 tO

= 0

" . 2

-.41

-.61

-.E

Fig. lb.

The ideas of mathemat ics ....

l

U l l " ,, LI i i I i I 6 I J

Always Discovered Unreal Special terms true Invented Developed Numbers Everyday words

Student responses to an item on the origins of mathematical ideas (n = 150). (Note: +1 on

the vertical axis represents total agreement, - 1 total disagreement).

The mean responses displayed in Figure lb suggest a student body which genera l ly holds a concept ion o f mathemat ics consistent with the tenets o f social

cons t ruc t iv ism, which sees mathemat ics as having arisen f rom societal need,

and which holds the ideas o f mathemat ics to be expressible in everyday, rather


than specialized, language. It remains to be seen whether this conception of mathematics is more widely held among secondary students or whether such beliefs are peculiar to students whose mathematics program requires the regular written articulation of mathematical meanings. Comparative studies are underway to resolve this question.

DISCUSSION

Progression in Journal Writing

Teacher interviews, together with an examination of students' journals, served to elaborate the character of the categories which had been used to classify student mathematical journal writing. The three categories, Recount, Summary, and Dialogue, as employed to categorize questionnaire responses showed a marked consistency with the linguistic forms by which students communicated what they had learned and how they had gone about it.

In the Recount stage, students' frames of reference for their journals are defined, in the main, by tasks which make up the mathematics lesson and by the chronology of the mathematics classroom. In some instances, the description may be as bald as, "Today we did the pink sheet". Students seem satisfied, at this stage, to describe themselves as "doing fractions", or "in the middle of chapter 3", and to comment on their learning in general terms, such as "'It was easy", or "I finished all the work and got most of it right". Examples seem chosen to do no more than illustrate the work done.

Many students, as they begin to use journals, may be expected to be at this Recount stage. A teacher of middle secondary classes described many students at this level as still coming to terms with journal writing. They are either still at a recount stage or just beginning to move into a summary stage. To use this teacher's own words:

It is a case of knowing that they have to write something, but many have difficulty knowing what

to do. At the be~nning of the year, these students are saying what they did in the mathematics

class . . . . They are able to describe what they did and the types of things they did.

Unlike students' writing in the Recount mode, in the Summary mode there is a deliberate effort to delineate key features of the territory. The mathematics may still be "out there", but students give greater attention to describing key steps in their work. It is no longer sufficient for students to describe in very general terms what they are doing, journal writing provides an opportunity for them to "map" the territory in some detail, and to record their progress. However, their describing is done almost without personal commentary or reflection. In this mode, their frame of reference is restricted to recording, "in very basic terms", the mathematics that has been covered in class.

With further refinement, students begin to include themselves in their summary of the mathematics covered. Not only is there more detail about what has been covered in class, but students are beginning to locate themselves in relation to


the mathematics being taught. They begin to identify "problems" in their own learning and to describe how they achieved a solution. It is common, in this mode, for students to illustrate their work by reference to several examples and by commenting on them. Yet, there is little discussion of why these problems arose or an analysis by the students of their own thinking.

In the Dialogue mode, students begin to focus on the "ideas" being presented. This term ("ideas") marks a significant transition in the frame of reference for students' journal writing. It marks a point when students begin to relate the mathematics being taught to what they are learning and to their ability to connect new ideas with what they already know. One does not simply record or summarize "ideas". One has to try to make sense of, or come to terms with, them. They can be illustrated by, but are not identical with, examples. Ideas make up the territory, but no longer is the territory seen as fixed and unchanging. The student is part of the territory and can change the way it looks. Communicating ideas and connecting them to what is already known now become key features of students' writing.

In this mode, students are able to identify and analyze their difficulties, suggesting reasons why they are thinking in a certain way. According to teachers, students begin to question what they are doing, and show increasing confidence in using their own words to link ideas. They are able to make suggestions about possible ways to solve problems, even if these approaches may not prove to be successful. They begin to ask and to pursue questions of their own. The increasing use of the first person singular, "I", appears to be a crucial linguistic cue to the commencement of this mode of journal use. Through their writing, they show that they are actively constructing mathematics.

Articulating their own thinking, in their own terms, appeared to be both chal- lenging and empowering to students. As they move into this mode of journal writing, teachers reported that many students comment on realizing "just how valuable the journal has become". Helping students to achieve this level of development in their journal writing was a goal to which all teachers in the study expressed genuine commitment.

Some Broader Issues

Teachers can play a critical role in helping students to assume control over their learning. Getting students to articulate their own thinking at the point where they are coming to terms with a new idea, or meeting difficulty, is essential to helping many to move into the more reflective mode of writing, characterized as "dialogue". At the heart of the progression which a journal writing program seeks to facilitate is the shift from a Summary mode in which the focus is the compilation of cognitive knowledge and skills to the Dialogue mode involving the acquisition and utilization of metacognitive skills. The key appears to be to encourage students to question themselves when they do not understand rather than be dependent upon their teacher to tell them whether they understand. This requires an internalization of authority, responsibility, and control. External accountability


was provided through the obligation to communicate. One teacher wrote in a student's journal, "Unless you can explain it to me, you don' t really understand". In one form or another, the journal writing program reiterates to students the link between communication and learning.

The integrated development of communication skills and mathematical thinking was central to the aims of the school's program. For some teachers, the ultimate goal of journal writing was to equip students to use mathematical forms and structures to describe their everyday world. However, the nature of journal writing derives from classroom purposes, and this close connection with schoolwork may not offer students the opportunity to extend their growing confidence in mathematical language by applying it to situations outside schoolwork.

Communication in mathematics is not a simple and unambiguous activity. The significance of this study is that it points to modes of communication as indicative of stances towards learning mathematics and ultimately how students see mathematical knowledge.

The categories which we have employed serve a dual purpose: as a description of students' p e r c e p t i o n s of their learning of mathematics and, in the second instance, as a progression in student mathematical activi ty . When students write in the Recount mode, they see mathematical knowledge as something to be described. In the Summary mode, students are engaged in integrating mathematical knowledge, now conceived of as a collection of discrete items of knowledge to be collected and connected. When writing in the Dialogue mode, students are involved in creating and shaping mathematical knowledge.

NOTE

1 The research reported in this paper was funded in pan by a grant from the National Center for Research in Mathematical Sciences Education, University of Wisconsin-Madison.

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Dav id J. Clarke , A n d r e w W a y w o o d

Mathematics Teaching and Learning Centre Australian Catholic University (Victoria) - Christ Campus

1 7 Castlebar Road, Oakleigh, Victoria 3166, Australia

M a x Stephens

Victorian Curriculum and Assessment Board

First floor, 15 Pelham Street, Carlton, Victoria 3053, Australia

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Probing the structure of mathematical writing