Teachers' Responses to an Investigative Mathematics Syllabus

Teachers' Responses to an Investigative MathematicsSyllabus: Their Goals and Practices

Author

Norton, Stephen John

Published

2002

Journal Title

Mathematics Education Research Journal.

DOI

https://doi.org/10.1007/BF03217115

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© 2002 MERGA. The attached file is reproduced here in accordance with the copyright policyof the publisher. Please refer to the journal's website for access to the definitive, publishedversion.

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Mathematics Education Research Journal 2002, Vol. 14, No. 1, 37 59

Teachers' Responses to an Investigative Mathematics Syllabus: Their Goals and Practices

Stephen Norton, Campbell J. McRobbie, and Tom J. Cooper Queensland University of Technology

Despite attempts to encourage teachers to adopt investigative teaching behaviours, there is strong evidence of the resilience of teacher centred school nladmmatics teaching. This study uses interpretive research methods to explore teachers' practices and relate these to their goals. Analysis of case studies indicates that syllabus documents have influenced teachers' choices of teaching strategies. Most teachers had calculation based goals for less able students and conceptual goals for more able students. Three distinct teaching strategies were identified and described. The relationships between teachers' goals, beliefs, and practices can guide the construction of teacher programmes that focus on student construction of knowledge.

In 1989, the N a t i o n a l Council for Teachers of M a t h e m a t i c s (NCTM) introduced its new s tandards (NCTM, 1989), a reform curriculum t h a t was i m i t a t e d by l a t e r Aus t r a l i an curricula. The NCTM document e m p h a s i s e d t h e impor tance of "doing" m a t h e m a t i c s r a t h e r t h a n "knowing t h a t " (p. 6), s t ressed t h a t m a t h e m a t i c a l knowledge "should grow out of p rob lem si tuat ions; and second, learning occurs th rough ac t ive as we l l as pass ive invo lvement w i t h m a t h e m a t i c s " (p. 8), and cr i t ic ised t r a d i t i o n a l t each ing pract ices because t h e i r " e m p h a s i s on prac t i se in man ipu l a t i ng expressions and prac t i s ing a l g o r i t h m s as a precursor to solving problems ignore the fact t h a t knowledge often emerges f rom the problems" (p. 8). Further , the document recommended t h a t t eachers become fac i l i t a to r s of learn ing r a t h e r t h a n impa r t e r s of informat ion .

This e m p h a s i s on p rob lem solving and f a c i l i t a t i o n continued into the second NCTM s ta t emen t oll s t andards (NCTM, 2000), where t eachers were encouraged to a l l ow t h e i r students to "make, refine, and explore" and to become "f lexible and resourceful p rob lem solvers" (p. 3). The e m p h a s i s was echoed by the na t i ona l s t a t emen t deve loped by the Aus t r a l i an Educat ion Council (AEC), in both r a t i o n a l e and recommendat ions about t each ing s t ra teg ies (AEC, 1990), and t h e 2000 d r a f t of the Queens land P 10 m a t h e m a t i c s syl labus (Queensland Schools Curriculum Council [QSCC], 2000). The AEC document recommended developing students ' c apac i ty to use m a t h e m a t i c s in "dea l ing w i t h non routine m a t h e m a t i c a l problems and u n f a m i l i a r s i tuat ions ... both i n d i v i d u a l l y and c o l l a b o r a t i v e l y " (p. 12) and a l lowing learners to construct t he i r own mean ings from, and for, the ideas , objects and events w h i c h t h e y experience" (p. 16). I t cr i t ic ised t r a d i t i o n a l t each ing pract ices and s t a t ed t h a t " m a t h e m a t i c s d e v e l o p s th rough the in teract ion of communities of people" (p. 13). The QSCC document reflects the strong influence of the NCTM (1989). S i m i l a r to the NCTM and AEC documents, the QSCC document e m p h a s i s e d the common themes of learner focus and ac t ive engagement, often in a social set t ing.

In th i s paper , the a p p r o a c h e s to m a t h e m a t i c s t each ing recommended by NCTM and AEC are ca l led i n v e s t i g a t i v e . The essence of i n v e s t i g a t i v e learning is contained in the recommendat ions of t h ree m a t h e m a t i c s curriculum documents: (a)

38 Norton, McRobbie & Cooper

Curriculum Council (1998) tha t stated tha t "mathematical learning is most successful when students actively engage in making sense of new information and ideas" (p. 1): (b) Australian Association of Mathematics Teachers (AAMT) Inc. (1996) tha t argued for "investigating mathemat ical processes situated wi th in meaningful contexts" (often derived from real world data) (p. 4): and (c) Anderson (1994) tha t concluded "effective learning and teaching requires active construction of meaning" (p. 1).

Terms other than invest igat ive have been used in other documents for the problem solving and faci l i ta t ive approach described above. For example, Senior Secondary Assessment Board of South Australia (2000) used the term "problem solving approach", recommending tha t students "construct mathemat ical models, typica l ly by establishing functional relationships, and use them to solve problems" (p. 5), whi le Carroll (1997) and Senger (1999) used the terms "Everyday Mathematics" and "reform curricula", respectively. A number of terms have been used to describe investigative teaching wi th its emphasis on problem solving, reasoning, communication, use of manipulat ive material, group work, and facilitation, where teachers see themselves as guides, listeners, and observers rather than authorit ies and answer givers.

Curriculum documents tha t are essentially investigative reflect theories of learning consistent wi th major elements of social constructivist theory. They acknowledge the importance of students act ively constructing their own knowledge from the environment through interaction wi th physical rea l i ty and through social interaction wi th peers and teachers (Cobb, Yackel, & Wood, 1992: Vygotsky, 1987). This social constructivist stance, tha t acknowledges the importance of both social and personal elements in learning, is referred to in this paper by the term constructivist.

In the USA, there has been resistance to the investigative approach and opponents have formed a movement called M a t h e m a t i c a l l y Correct. Members of Mathemat ica l ly Correct reject the idea tha t student construction of knowledge ought to be central to mathematics learning and believe tha t investigative approaches downgrade basic skills, par t ly because the time taken to "discover" means less material can be covered and par t ly because "memorisation" has not been developed and utilised (Allen, 1998). They acknowledge tha t mathematics includes some discovery and modelling of the real i ty of the world, but oppose student discovery. They recommend a return to teacher exposition or "direct instruction" wi th an emphasis on basic skills and practice, predominantly by students working individual ly (Allen, 1998). They believe tha t mathemat ica l concepts and understandings are better and more efficiently developed through teacher centred rather than learner centred pedagogy.

In Australia, there has been similar resistance to attempts by curricula writers to persuade teachers to adopt investigative approaches to teaching (McDonald & Ingvarson, 1995: McRobbie & Tobin, 1995). This is part icular ly the case in low socioeconomic urban schools where teachers tend to adopt directive and authori tar ian pedagogies (Perry, Howard, & Tracey, 1999).

Teachers' classroom practices are strongly connected to whether thei r teaching orientation is calculatiorlal or corJceptual (Thompson, Phi l l ip , Thompson, & Boyd, 1994). They argue tha t a calculational orientation is "driven

Teachers'Responses to an Investigative Mathematics Syllabus: Their Goals and Practices 39

by a fundamental image of mathematics as the application of calculations and procedures for deriving numerical results" (p. 86) and is strongly associated w i t h teaching practices tha t focus on procedures for "getting answers", usually numerical in nature. It is a common orientation in secondary mathematics teaching (Crawford, 1996: Gregg, 1995: Perry et al., 1999), is related to teaching behaviours described as content focused w i th an emphas i s on performance (Kuhs & Ball, 1986) and ins trumental (Skemp, 1978), and is linked to transmission images of learning (Atweh & Cooper, 1995; Perry et al., 1999). By contrast, a conceptual orientation is associated wi th teaching behaviours tha t "focus students' attention away from the thoughtless application of procedures and towards a rich conception of situations, ideas, and relationships among ideas" (Thompson et al., 1994, p. 86), tha t is, t ha t focus on solving problems by working from a deep understanding of the mathematics underlying the problems. A conceptual orientation is consistent wi th the Kuhs and Ball's (1986) category of teaching behaviour described as content focused w i th an emphas i s on understanding and with Perry et al.'s (1999) child centred approach to teaching.

Purpose This paper reports on a study to explore how Queensland secondary

mathematics teachers are responding to a syllabus tha t encourages i n v e s t i g a t i v e teaching and learning. It investigates reasons why teachers may resist such syllabi. The specific research objectives of this study are:

• to investigate how teachers are responding in their teaching practice to the recommendations of this syllabus; and

• to relate teachers' practices to interpretations of the reform Syllabus and their goals for students.

The Syllabus In this study the focus is upon the Senior Mathemat ics B syllabus (BSSSS,

1992) as an example of an i nve s t i gaHve syllabus. Mathematics B is an academic but not specialist course and was developed for students wi th aspirations to study a ter t iary course tha t has a mathematics component (there are less academic mathematics options for students without these aspirations). It was designed to be inves t iga t i ve in its approach to pedagogy, as is stated in its rationale:

Mathematics education must recognise the dynamic nature of mathematics and encourage an approach to its study through problem solving and applications in life related contexts. (p. 1)

In support of this rationale, the learning experiences in Mathematics B also emphasised an i n v e s t i g a t i v e approach. For example, for the topic area of applied geometry, the recommended teaching approach was based on "experiences which involve life re la ted applications of mathematics, including both real and simulated situations, use of instruments, and the opportunity for problem solving" (p. 11) and contained many recommendations to "design," "investigate", or "model" practical problems (pp. 12 13).

However, the assessment procedm'es for Mathematics B tended to work


against an invest igat ive approach. The importance of testing procedures in influencing teachers' behaviours has been well documented (Barnes, Clark, & Stephens, 1996; Gregg, 1995). Students are awarded one of five exit levels of achievement. They are assessed against three criteria: communication, "clear and concise use of appropr ia te terminology and appropria te forms of presentation, whi le consistently adhering to the conventions of language and mathematics" (BSSSS, 1992, p. 41); techniques, involving the recall of rules and procedures, use of instruments and the applicat ion of these in famil iar situations: and applications involving applying mathematics in unfamiliar situations. They are expected to consistently perform at h igh levels in all criteria in order to be awarded tile top achievement level. However, in order to achieve a middle level, students need to demonstrate "some knowledge of ma themat i ca l techniques" and be "generally accurate and proficient when applying them to famil iar situations" (BSSSS, 1992, p. 42); and when they only "rarely provided solutions in unfamiliar situations" (BSSSS, 1992, p. 42). Thus, it is possible for students to have the middle level of Sound Achievement in Mathematics B w i t h only procedural knowledge of techniques and without knowledge of applications. Students wi th Sound Achievement may be without conceptual understanding of mathematics because, as Tall and Thomas (1991) argued, students can carry out computations wi thout conceptual understanding but success on unfamil iar problems requires it. Moderation of Mathematics B does not seem to have taken into account t ha t the criteria for Sound Achievement do not require the understandings tha t are at the basis of an investigative approach to teaching mathematics.

Method The study was a series of embedded qual i ta t ive educational case studies

(Stenhouse, 1990) involving an interpret ive and constructivist approach (Denzin & Lincoln, 1994). In particular, the s tudywas interpret ive in tha t the goal was to understand the meaning of a social phenomena whi le giving recognition to the subjective nature of observation, and it was constructivist in tha t findings were al lowed to emerge from the data in an hernleneutic cycle which took account of perspective in relation to knowledge and recognised t ha t "truth" is created by the mind (Schwandt, 1994). The researcher employed a wide range of interconnecting research techniques.

Subjects and Contexts The study was carried out in two Brisbane suburban schools (denoted by

pseudonyms). The first school, Forest View, was a large (1400 students) coeducational government school located in a middle class suburb. The second school, Hil l View, was a mid sized (650 students) girls Catholic school also located in a middle class suburb. Forest View and Hi l l View were both regarded by the district Mathematics B panel review members as innovative schools w i t h h igh teaching standards. Further, tile researcher had developed a close rapport and mutual trust wi th several of the s taff in each school. This was critical in obtaining rich data (Fontana & Frey, 1994).


Six of the most senior ma thema t i c s teachers at Forest View and th ree at Mi l l View became the subjects of the study. These teachers were seen as experienced and competent teachers by o ther school staff. Present and previous computer and ma thema t i c s s t a f f coordinators at both schools confirmed these teachers as inf luent ia l in the teaching of ma thema t i c s in t h e i r schools. T h e y were t h e wri ters of the work programmes and internal assessment tasks t h a t s t ipu la ted (a) w h i c h concepts would be t augh t and when, and (b) w h i c h textbooks and o the r resources would be selected for the o ther teachers to use. The importance of textbook selection in influencing teaching practice has been recognised prev ious ly (Crawford, 1996: Ney land , 1996). In short, these teachers became pa r t i c ipan t s because t h e y h a d considerable influence on how ma themat i c s was t augh t in e ach school.

Data Sources Data were collected using a t eacher questionnaire, classroom observations

(using f ie ld notes), in terv iews (using audio recordings), and from students ' examinat ions and mark al locations. The questionnaire was admin is te red first and designed to collect d a t a on demographics and beliefs about m a t h e m a t i c s and teaching mathemat ics . Following tile emergent nature of the methodology, t h e observations and in terv iews were designed taking into account the results of t h e survey. The questionnaire contained 31 s ta tements to wh ich teachers responded on a five po in t Liker t scale. The work of Ernest (1989, 1991, 1996) and Cobb (1988, 1989) was used as a f ramework upon w h i c h to design questions about the nature of ma themat i c s .

The in te rv iew instrument was designed to probe for beliefs about the nature of ma thema t i c s teaching by seeking to iden t i fy w h a t aspects of ma thema t i c s were considered important . Items t h a t ref lected strong beliefs in the importance of performing procedures in ma thema t i c s study, included "There is considerable merit in getting back to basics." A more conceptual focus was ref lected by items such as "Basic skil ls are less impor tan t t h an the ab i l i t y to t ransfer problem solving strategies." The work of Kuhs and Ball (1986) provided a t h e o r e t i c a l f ramework for the structuring of questions probing teachers ' beliefs about t h e nature of teaching mathemat ics . For example, agreement w i t h the s ta tement "Listening to and th inking about m y explanat ions forms the basis of most of m y students ' learning" was considered to indica te a t eacher centred approach . By contrast, agreement to s ta tements such as "I sit students in groups to f a c i l i t a t e discussion" and "It is impor tan t for students to do solutions in t h e i r own way ," were considered to indicate a learner focused pedagogy based upon principles of constructivism.

Procedure The da t a explored in th is s tudy were collected at the two schools over a

twelve m o n t h per iod in the following four-stage sequence:

• p re l iminac7 in terviews w i t h the ma thema t i c s Heads of Depar tments (HODs);


• surveys of most of the ma themat i c s s ta f f in each school; • purposeful selection of the nine teachers using the responses from t h e

survey, and • observations of the nine teachers ' classes, in terv iews w i t h them, and

collections of students ' examinat ions and mark a l locat ions .

For each of the nine teachers , two to four lesson observations were undertaken. Field notes were taken of the classroom ac t iv i ty , the black board presentat ions, and the t eache r and student interactions, and audio-recordings were made of the teachers ' dialogues w i t h t h e i r students. These were transcribed.

Up to six in terviews of the selected teachers were made before and af te r t h e observations. During these interviews, the teachers were asked to explain t h e i r reasons for conducting classes in the way t h a t t h e y did. The researcher used t h e teachers ' responses to the survey instrument to probe t h e i r beliefs and pract ices and introduced contrasting and divergent views and images, giving the s tudy a d ia lec t ic nature (Guba & Lincoln, 1989), in order to probe reasons for beliefs and practices. Both direct and indirect questioning was used. For example, t h e teachers were asked w h a t t h e y t h o u g h t of the two textbook series, Investigating Change: An Introduction to Calculus for Australian Schools (Barnes, 1991) and Access to Algebra (Lowe, Johnston, Kissane, & Wil l i s , 1993). The t eachers ' responses were found to be very reveal ing of t h e i r beliefs. The two text series h a v e many inves t iga t ive ac t iv i t i es t h a t are learner focused and encourage conceptual deve lopment through phys ica l ac t iv i ty , discovering pat terns and then formalis ing the mathemat ics . Many of the ac t iv i t i es are embedded in a p rac t ica l and cultural context. Thus, rejection of these resources was in te rpre ted as a rejection of the pedagogy t h a t under lay them. Kagan (1992) argued t h a t t h i s indirect method of collecting d a t a reduces unintentional cues and the chances of leading tile subject. The teachers were also asked to explain the factors t h a t guided t he i r teaching pract ice of able and less able students, and to comment on the syllabus.

A f inal in te rv iew w i t h each t eache r (following the analys is of the da ta ) was conducted in w h i c h the major findings were reported for t h e i r comment and confirmation as a reasonable in te rpre ta t ion of t he i r beliefs and practices.

Analysis Each da t a collection source was used to inform the subsequent d a t a collection

and analysis . The d a t a were ana lysed using procedures t h a t involved "examining the meanings of peoples ' words and actions" (Maykut & Morehouse, 1994, p. 121). Combining the teachers ' responses to the survey instruments and in terviews w i t h the classroom observations developed a r ich description of each teacher ' s bel iefs and practices. Tr iangula t ion was ach i ev ed w i t h teachers ' reactions to the tex t resources and the researchers ' assertions. The descriptions were ana lysed for commonali t ies and th ree teachers were chosen as being typ i ca l of the dis t inct teaching preferences, goals and beliefs about teaching h e ld among the nine teachers . The results are presented as descriptions of these th ree t eachers fo l lowed by summaries and t he i r re la t ion to o ther s imi la r teachers .


Results The first t eache r (pseudonym Eva) was i l l u s t r a t ive of a p redominan t ly

ca lcu la t iona l or ienta t ion and her teaching app roach has been described as s h o w and tell, since the dominant teaching behav iour was one of modell ing correct app l i ca t ion of a lgor i thms. The second t eache r (David) i l lus t ra ted w h a t has been classif ied as the expla in t eaching approach . T h a t is, his dominat ing b eh av io u r was careful explanat ions of the logic behind m a t h e m a t i c a l concepts, p a r t i c u l a r l y when teaching able students. The f inal t eacher (Jan) most ly used w h a t has been classif ied as an i n ve s t i ga t i v e t eaching app roach in her teaching of more able students. D a v i d and Jan are not pseudonyms af te r member checking, these teachers asked to be referred to by t h e i r Chr i s t i an names.

Prior to considering each of the th ree selected teachers , the goals and teaching approaches of a l l tile nine teachers are summarised (see Table 1). The table shows t h a t the show and tell , expla in and inves t iga t ive pat terns of behav iour were common among the s t a f f in these two schools.

Table 1 Summary of Teachers' Goals and Favoured Pedagogy

Teache r Stated Goals for Pedagogy for

Able Students Less able Able Students Less able Students Students

Eva Conceptual Conceptual S h o w an d tell S h o w an d tell S a s h a Conceptual Ca lcu la t iona l S h o w an d tell S h o w an d tell W i l l Conceptual Ca lcu la t iona l S h o w an d tell S h o w an d tell D a v i d Conceptual Ca lcu la t iona l Expla in S h o w an d tell Peter Conceptual Ca lcu la t iona l Expla in S h o w an d tell Kurt Conceptual Ca lcu la t iona l Expla in S h o w an d tell Jan Conceptual Ca lcu la t iona l Inves t iga t ive S h o w an d tell MmTf Conceptual Ca lcu la t iona l Inves t iga t ive S h o w an d tell Simon Conceptual Conceptual Inves t iga t ive Inves t iga t ive

Findings for the Show and Tell Teacher Eva was 28 years of age and h a d t augh t in her school since she began

teaching six years ago. Eva h a d completed a Bachelor of Science majoring in ma thema t i c s and psychology. At tile t ime of the in terview, she was t each ing .junior" ma thema t i c s (Years 8, 9, & 10) and Mathemat i c s A (Years l l & 12). S h e h a d been a member on t h e Mathemat i c s A panel for one year . Two yeats prior to the intepeiew, Eva h a d been made coordinator of.junior" mathemat ics . Since th i s t ime, the frequency of open ended student assignments h a d been reduced from four a semester to one a semester, the use of the problem solving student centred and ac t iv i ty b a s e d text, Access to Algebra (Lowe et al., 1993), h a d been reduced, and

44 Norton, McRobbie & Coopor

a more t r ad i t i ona l text, MathemaHcs Year 10 for Queensland (Wasley, Manche & Winter , 1996), h a d been adopted. Her justif ication for discontinuing the use of Access to Algebra was t h a t the text was "too laborious for good students and too confusing for students who are slow readers."

Eva h a d s ta ted t h a t her' goal for both able and less able students was to h e l p them understand mathema t i c s concepts, but in her survey and in te rv iew responses, she indicated a narrow conception of understanding. For example, in her completion of the survey, she responded in a w a y t h a t indica ted she considered "getting back to the basics" very important . She s ta ted "I t h i n k eve ry th ing is based upon those a lgor i thms ... it is based upon laws and theorems and algori thms." Her beliefs about teaching and learning conformed to t he t ransmiss ion/absorpt ion model. She s ta ted t h a t she l iked to "show them and te l l them". She described her teaching of t r igonometry as follows:

Well with tan I always start, offwith SOHCAHTOA .... And from there I'd tell them that before we can even start looking at tan we need to do a bit on right angled triangles, because that's where the principle of SOHCATOA comes from.

She went o11 to describe how she would c lear ly tell students w h a t t h e y need to know and how to remember the procedure. It is interesting t h a t Eva describes S O H C A H T O A as a principle. At i1o stage during Eva's l eng thy explana t ion of her" pedagogy did she suggest t h a t students migh t be required to make the i r own conclusions about the meaning of S O H C A H T O A ,

Eva emphas i sed ind iv idua l r a the r t han cooperat ive student work. She explained:

I put them all in single file, because I've found in the past (that) putting them in groups, putting them in two by twos, .just does not work for some reason. I find them more focused when they are seated in single file.

In the classes observed, her students were indeed seated in rows. Student communication was l imi ted to surreptitious whispers . This was not surprising since Eva h a d s ta ted:

From the moment they meet me, every new class I have, I make it very clear to them what my expectations are. And I have only one simple rule, that is, when I am teaching please do not distract me or anyone around you. I set those rules and make those expectations.

Lesson observation confirmed the strict conformity of the class, suggesting t h a t Eva h a d commanded the i r compliance. The lessons observed were v e r y ordered and controlled. Students who were fast in putt ing the i r hand up, answered questions w i thou t providing explanat ion of the i r answers. In the l a t t e r par t of each lesson, students who f in ished the question t h a t was p laced on t he whi te boa rd were nominated to come out to the wh i t e board and i l lus t ra te a port ion of the i r answer. No student expla ined the complete answer or t h e i r reasons for each step in the solution. It appea red t h a t coming out to the w h i t e board was a reward for speed in completing the a lgor i thm. In addi t ion, i f a student made a mis take she would sit down and another student would h a v e a chance to correct the previous error. At no time were students asked to discuss a m a t h e m a t i c a l concept or' share the i r ideas w i t h fe l low students or work coopera t ive ly . The following dialogue was typ ica l of the sort of explanat ion, question and answer dialogue t h a t made up most of the lesson.

Teachers' Responses to an Investigative Mathematics Syllabus: Their Goals and Practices 45

[Eva had drawn and labelled a right angle triangle on the white board.l Eva: OK, girls pens down, looking at the board please. OK. what we are going

to do now is by using those preliminary exercises that we have just done, we are actually given a right angle and we have been given side lengths and an angle. And in these examples we have to find out one of the side lengths given the other side length the angle. OK. So let's look at the first one please. We're asked to find the value of x, where it's been labelled x. what side of the right triangle, which part is it labelling? Which side of the right triangle? Rebecca?

Rebecca: The opposite. Eva: It is labelling the opposite side. Now we are asked to find x, so therefore

which of our trig. rules are we going to use to find x in this case? Which trig. rule? Which trig. rule? Jackie?

Jackie: Sine. Eva: Why are we going to use sine? Jackie: Because ... (pause since the student was unable to answer). Eva: So. using the information, right, we have been given the opposite. In this

case it's been labelled x centimetres. And we have been given the hypotenuse which has been labelled 20. We need to substitute the information. Crystal, would you do that please.

Cwstal: OK. Sine of an angle equals x over ...(unable to continue). Eva: OK. How do we solve for x in this case? How do we solve for x? Sinead? Sinead: xover H. Eva: Well done. Will every one calculate please. It will equal 20 times sine 28.

Hands up when you have got an answer.

The s t r ik ing aspect of t h i s discourse is t h a t m a n y s tudents s h a r e in t h e process of ca r ry ing out t h i s s imple a p p l i c a t i o n of an a l g o r i t h m . H o w e v e r , no s tudent u l t i m a t e l y took r e s p o n s i b i l i t y for it. If a s tudent f a l t e r ed , as in t h e cases of C r y s t a l a nd Jackie, t h e y w e r e pas sed over. A s tuden t ' s l ack of k n o w l e d g e or misconcept ion w a s not p robed in t he two lessons a n a l y s e d . There w a s s u b s t a n t i a l ev idence t h a t Eva did not embar ra s s s tudents by confronting t h e m w i t h t h e i n a d e q u a c y of t h e i r conceptions. Eva ' s ea r l i e r s t a t e m e n t s on t h e i m p o r t a n c e of s tudent se l f -conf idence m a y h a v e been a r a t i o n a l e for t h i s t e a c h i n g b e h a v i o u r .

The ma in focus of Eva ' s lessons a p p e a r e d p rocedura l , even t h o u g h her c lass w a s a top s t r eam Year 10 class. For example , S O H C A H T O A h a d been w r i t t e n on t h e w h i t e boa rd pr ior to the girls en te r ing t he c lassroom and even before t he sine or cosine concepts w e r e exp la ined . Fur ther , t he sine question discussed in t h e e x a m p l e above w a s one of s e v e r a l of t he s a m e form. The r e sea rche r e x a m i n e d t w o s tudent workbooks a nd t h e r e w a s no ev idence t h a t t he meanings beh ind t h e S O H C A H T O A rule h a d been t a u g h t . It a p p e a r e d t h a t in past lessons, as in t h i s lesson, t h e r e l a t i o n s h i p s be tween angles and s ide l eng ths w e r e p re sen ted as rules. Thus , t h i s lesson suppor t s Eva ' s s t a t e m e n t to t he researcher" t h a t " m a t h e m a t i c s learn ing is based upon a l g o r i t h m s " .

P e r h a p s t he most i l l u s t r a t i v e c lassroom e x a m p l e suppor t ing Eva ' s c l a im t h a t she l i k e d to s h o w a nd te l l s tudents h o w to do m a t h e m a t i c s w a s seen in a n o t h e r of her classes. This class w a s being t a u g h t l i nea r equa t ions and Eva w a n t e d to s h o w t h e m t h a t t h e g rad ien t s of p e r p e n d i c u l a r lines h a d n e g a t i v e r ec ip roca l g rad ien t s . On the w h i t e b o a r d w a s s k e t c h e d t he lines y 2 x l and y 1 / 2 x + 1, w h i c h w e r e l abe l l ed . The s tudents h a d .just spen t abou t 15 minutes revis ing t h e g r a d i e n t a nd y in t e rcep t concepts and h a d been shown t h a t p a r a l l e l lines h a d t h e s a m e g rad ien t . The s tudents h a d been asked to look for a r e l a t i o n s h i p be tween t h e g rad ien t s of t h e t wo lines. T h e y h a d descr ibed t h e g r a d i e n t in t e rms of


n e g a t i v e a n d p o s i t i v e g r a d i e n t s a n d d e s c r i b e d t h e m in t e r m s of s t e e p n e s s of s l ope , but t h i s w a s not w h a t Eva w a n t e d .

Eva: No, no, no, is anyone going to say that they are perpendicular lines? All right so they're perpendicular lines, now what do you think would make them perpendicular? The fact that I've drawn in those little boxes there? OK, pretend that I forgot to draw the boxes in to make it more obvious to you. What do you think actual ly makes graph three (y = 2x 1) perpendicular to graph one (3I = - 1 / 2 x + l ) . Just think of the equation or gradient or y intercept, do a little comparison.just the information that ' s there. Kristen.

Kristen: Their gradients are opposite. Eva: Their gradients are opposite, not so much opposite ... yeah go on. Sally: Inverted of . . . Eva: What 's another word that we can actual ly use for this idea of inverted?

One gradient is positive wha t ' s the other? Toni: Negative Eva: So put all of that new found knowledge together. One gradient is the

negative reciprocal. The gradient for graph three is the negative reciprocal of the gradient of graph one. The gradient of graph three is negative a half and the gradient of graph one is two.

By p l a c i n g t h e square a t t h e p o i n t of i n t e r sec t i on of t h e t w o l ines a n d d r a w i n g s t u d e n t s ' a t t e n t i o n to i t , Eva w a s d e n y i n g t h e m t h e o p p o r t u n i t y to d e t e r m i n e t h a t t h e t w o l ines w e r e p e r p e n d i c u l a r . In t h e end , she t o l d t h e m t h a t t h e p e r p e n d i c u l a r l ines w o u l d h a v e n e g a t i v e r e c i p r o c a l g r a d i e n t s of e a c h o t h e r . T h i s t e a c h i n g b e h a v i o u r h a s been d e s c r i b e d as " T o p a z e " l i ke (Brousseau , 1984) in t h a t t h e s tuden t s a r e a s k e d e x p l i c i t ques t ions but t h e t e a c h e r t a k e s c h a r g e of t h e m a i n p a r t of t h e w o r k to t h e ex ten t t h a t t h e l e a r n i n g s i t u a t i o n is e m p t i e d of c o g n i t i v e conten t . Lesh a n d K e l l y (1997) d e s c r i b e d s i m i l a r t e a c h i n g b e h a v i o u r s as "bug r e p a i r " t u t o r i n g b e h a v i o u r t y p i c a l of nov ice tu to rs , t h a t is, t h e t e a c h e r focuses on gu id ing s tuden t s a l o n g p a t h s in w h i c h e r ro r s a re a v o i d e d .

Eva s e e m e d s o m e w h a t u n a w a r e of t h e s h o w a n d t e l l n a t u r e of he r t e a c h i n g . For e x a m p l e , in t h e m e m b e r check ing , w h e n t h e r e s e a r c h e r a s k e d Eva to c o n f i r m t h e f o l l o w i n g a s s e r t i o n , she d i d not d i r e c t l y a n s w e r t h e ques t ion but h o n e d in on one a s p e c t she cons ide r ed i m p o r t a n t :

Res.: In teaching in secondary school, you are concerned that students understand the rules and know how to apply them. Thus, your teaching is dominated by teacher explanation and modelling (of solutions and procedures). Students need to attend carefully and be able to fol low your demonstrations.

Eva: ... well in the sense that I really believe that the foundations have to be there and I demonstrate very clearly, and it has to be taught.

C l e a r l y , Eva b e l i e v e d t h a t t e a c h i n g s t r a t e g i e s b a s e d upon d i r e c t i n s t r u c t i o n w e r e a c ruc ia l componen t of her" t e a c h i n g . However ' , she d i d not r e in fo rce t h e " d o m i n a t e " p a r t of t h e a s s e r t i o n . T h i s is a l so e v i d e n t in her" s t a t e m e n t a b o u t s t u d e n t d i scuss ion :

It [discussion] is very important in terms of discovery. If you give them a task and they try to work together" to nut it out and to verbal ly express or explain things to others, that is very important. I try to do quite a bit of that within each lesson, even if it is a minute or two. I would say to the students explain that to me. which means they're explaining to the whole class.

N e i t h e r c o m p l e t e e x p l a n a t i o n s nor s t u d e n t d i scuss ion w e r e e v e r o b s e r v e d in a n y of he r lessons .


Eva did not a p p e a r to fu l ly a p p r e c i a t e the dist inctions be tween ca l cu la t iona l and conceptual goals. Evidence to support th i s was observed in her mark ing of s tudent examina t ions where she often did not give p a r t marks for answers (both .junior m a t h e m a t i c s and senior m a t h e m a t i c s ) . T h a t is, she gave zero to s tudents w h o m a d e a ca lcu la t iona l error in the beginning of a th ree m a r k question even i f subsequent processes and calculat ions were correct.

In t e rms of assessment, Eva was very a w a r e of the impor tance of examina t ions in m o t i v a t i n g students. She used s t a t ements such as, "If you can do th i s question you wi l l do we l l on the exam." She described her role as one of bui lding confidence and "he lp ing t hem to learn and understand the rules of m a t h e m a t i c s . " Classroom observations, and her descript ions of her t each ing and of her goals, c rea ted an image of a caring t eache r w h o would h e l p students to do we l l on examinat ions , p a r t i c u l a r l y those t h a t e m p h a s i s e d the "basics." An examina t ion of the school 's assessment i tems confirmed t h a t t h e i r assessment was domina ted by "technique" questions. Eva 's rejection of the i n v e s t i g a t i v e resources and her reduction of the quan t i ty of i n v e s t i g a t i v e assessment required in t h e .junior (Years 8 10) work p rogram support the conclusion t h a t she s aw meri t in emphas i s i ng " the basics (which) you get by clear exp lana t ion in the c lassroom" in assessment as we l l as t each ing .

F ina l ly , Eva a p p e a r e d not to d is t inguish be tween able and less able s tudents in describing her goals and in her teaching . She c l a imed the same for a l l students: to t each understanding by emphas i s i ng the rules and a lgo r i t hms t h a t were t h e foundat ions of m a t h e m a t i c s . She m a y h a v e be l i eved she h a d conceptual goals for a l l s tudents , but observat ions of her classes suppl ied no support for a conceptual a p p r o a c h to t each ing .

Summary and Relations to Other Show and Tell Teachers In th i s s tudy, t h ree t e a c h e r s (Eva, W i l l and Sasha see Table l) were found

to e m p l o y p r e d o m i n a n t l y show and tel l behav iou r s in t h e i r t each ing of both able and less able students. In observat ions of t h e i r classes, s tudent learning a p p e a r e d to be by imi ta t ion ; the t eachers demons t ra ted some questions and the s tudents i m i t a t e d tile t eachers ' b e h a v i o u r and per fo rmance on s im i l a r questions. Being able to a p p l y technique and procedure on closed book examina t ions was the major goal t h a t was a r t i cu l a t ed by each of these t eachers .

These t e a c h e r s opted not to use the resources t h a t were domina ted by i n v e s t i g a t i v e approaches . The i r exp lana t ions included "too s low ... i t t akes too long ... i t is a f e m a l e a p p r o a c h to it", "for good students i t ' s so laborious ... for t h e s low readers the questions .just confuse t hem" , and not sure " t h e y would be as eff ic ient" . The i r comments ind ica ted t h a t rejection of or reluctance to use i n v e s t i g a t i v e texts was s t rongly r e l a t ed to percept ions of how long it would t a k e to cover the content m a t e r i a l , w h i c h h a s wide r imp l i ca t ions for textbook choice considering t eachers ' percept ions about the crowded nature of the sy l labus .

A l t h o u g h W i l l and S a s h a demons t ra t ed t each ing s t ra teg ies t h a t h a d much in common w i t h those em p l oyed by Eva, t h e y did a r t i cu la t e d i f fe ren t goals for able and less able students. T h a t is, a l t h o u g h t h e y s t a t ed t h a t "br ight" s tudents could be " t augh t to understand the logical th ink ing behind geome t ry or a lgeb ra" ,


t h e y be l ieved t h a t less able students h a d "to be d r i l l ed in the skills." Thus, t h e i r in te rv iew responses a p p e a r e d to indicate ca lcu la t iona l goals for less able students and conceptual goals of more able students.

However , in a l l t h ree teachers ' lessons, l i t t l e difference was noted in t h e i r teaching approaches for the two ab i l i t y groups. Classroom observations of a l l t h ree teachers r evea l ed t h a t s h o w and t e l l t eaching approaches were r egu la r ly used w i t h al l students and included the following behaviours: sealing students in rows or i nd iv idua l ly , short w a i t t ime, use of questions to main ta in discipl ine, use of extrinsic mo t iva t i ona l techniques such as expectat ions of better p a id jobs, r e a d i l y reducing the cognitive load, r a re ly asking students to justify t h e i r answers, teaching from the front of the room, preferr ing to use texts t h a t contained many exercises, and r a r e ly using d i ag rammat i c models. In a l l t h e classes observed th is teaching behav iour resulted in order ly compliant s tudent behav iour .

Findings for the "Explain" Teacher D a v i d was a 43 y e a r old male who h a d been teaching ma thema t i c s for 21

years . He was h i g h l y qua l i f ied w i t h a Bachelor of Economics, Masters in Regional Science Degree, Graduate Dip loma in Teaching and a Gradua te D ip loma in Education (Computer Education). He was an Advanced Ski l ls Teache r and was tile t eacher charged w i t h representing the school on the Mathemat i c s A regional panel . He also t augh t Mathemat i c s B.

D a v i d exp la ined t h a t he h a d two dist inct goals for his students. He t r ied to he lp his better students understand m a t h e m a t i c s concepts. This was expressed in his comments about l iking "to pursue h i g h e r objectives" a f te r t h e y h a d deve loped "a solid understanding of the basics". However , he h a d more l imi ted goals for h is less able students. D a v i d s ta ted t h a t he wanted them to pass skills b a sed examinat ions and "not get r ipped off for t h e i r change.'" As he expla ined:

I think with the lower level groupings we get, you've either" got to make a decision as to ... well ... you've got to see what your obiective is. And your objective. I think is to get them through the examination. That's the thing they are interested in. That's the thing their" parents are interested in.

T y p i c a l l y D a v i d began lessons w i t h a r i t h m e t i c questions to revise basic skills. This was usually fo l lowed by modell ing procedures and subtly v a r i e d subsequent questions. He encouraged es t imat ion and student discussion during bookwork act iv i t ies , and gave students some autonomy in deciding w h a t ac t iv i t i es t h e y could do and how much time t h e y could spend on them. In t h e observed lessons, he spent most of the t ime trying to lead students to an understanding of w h y the procedure worked. He spent more t ime than Eva w i t h i nd iv idua l students and t a i l o r e d his questioning to correct specif ic misconceptions. He based his teaching oll reasonably soph i s t i ca ted questioning skil ls and analys is of students' misconceptions.

The dialogue below i l lus t ra tes the assertion t h a t Dav id ' s pedagogy was based on explanat ions and shows how he exp la ined concepts to students. In th i s lesson, the Year l0 students were s tudying discounts and were asked to find t h e or iginal price i f the discount was given as a percentage and in do l la r terms.

Teachers' Responses to an Investigative ]V[athematics Syllabus: Their Goals and Practices 49

D a v i d h a d modelled a problem on the board and h a d asked students to complete s imi lar examples fi'om the i r textbook.

Amy: I couldn't do that one. David: See the ad. (advertisement) here, you're getting up to $35 off. OK? It also

tells you it is 20% off the marked price. Amy: Yeah. David: So it is very similar to the ones you.just had a look at. You're getting a

discount of $35 and that is the same as 20%. OK? Amy: Art? David: Well put it another way. Suppose we had something that was worth

$1007 Amy: So it means we would have to work out what it would really be? (marked

price) David: Yes, what it was originally. Amy: I don't get it! David: Let's put it this way. Let's suppose you go into a shop and there is a maths

textbook you really want to buy and it is worth $100 and this shop is having a sale. They are going to give you a discount of 30%.

Amy: HnHn. David: How nmch would you get off?. How many dollars would you get off?. Amy: Hmmm, $30. David: Yeah, you would get $30 off, wouldn't you? So in other words, that $30 is

equal to 30% of the marked price, which was $100. Wasn't it? Amy: Yes.

Dav id ' s questions to A m y were explici t in trying to lead her to an understanding of the concepts. In doing this, he increasingly s impl i f ied the t a sk and g r a d u a l l y removed the cognitive content from the problem consistent w i t h the Topaze effect (Brousseau, 1984). In the example, it was D a v i d who took charge of the main par t of the work. Later in this discourse he h a d even "worked out the rule" and directed A m y t o fol low his model. I n h i s careful leading of A m y through simple steps, D a v i d was implementing his educat ional goals for less able students; t h a t is, he t r ied to lead these students to understanding, but i f th i s did not succeed he supplied models for them to imi ta te .

Like Eva, Dav id ' s questioning emptied the m a t h e m a t i c a l a c t i v i t y of most of the cognitive challenge. However, Eva 's s h o w and t e l l questioning was directed at tile who le class. In contrast, Dav id ' s questioning was usually t a i lo red to ind iv idua l students. However, both tile s h o w and t e l l of Eva and the questioning of D a v i d focused on he lp ing the students to do instrumental mathemat ics . W h e n Dav id ' s leading of students by careful questioning and explanat ions fa i led he showed students how to preform the procedures.

Because his focus was on explaining, D a v i d h a d different goals and strategies for more able students. For example, he did not s tar t lessons to able students w i t h mental a r i t hme t i c questions as he a l w a y s did for less able students. As he explained:

Oil/It's an academically much brighter class. And I don't think they need to practise their mental arithmetic as such. They're bright kids and they focus pretty quickly on what's going on. So that is mostly for the junior grades, particularly those that are a bit slower.

Observations of Dav id ' s teaching of his more able Year 11 Mathemat ics B class i l lus t ra ted subtle differences in his pedagogy from t h a t used w i t h less able students. This is i l lus t ra ted in the dialogue below. This is from a lesson in stat ist ics to a class of 26 students, who were seated in four groups. It was t he


second in a sequence of lessons , a n d to s t a r t t h e l e s son D a v i d h a d w r i t t e n on t h e w h i t e b o a r d :

8A: 1, 2, 98, 99: 8B: 48, 49, 51, 52.

H e e x p l a i n e d t h a t t h e s e numbers r e p r e s e n t e d t h e e x a m i n a t i o n r e su l t s of t w o c lasses a n d t h a t t h e h e a d m a s t e r w a n t e d to know t h e a v e r a g e m a r k for e a c h c lass . The s tuden t s c a l c u l a t e d t h e a v e r a g e , w h i c h w a s 50 for bo th c lasses . D a v i d t h e n began to focus t h e c l a s s ' s a t t e n t i o n onto o t h e r a s p e c t s of t h e t w o c lasses w i t h ques t ions :

David: What sort of conclusion do you think the headmaster would draw about the two classes?

Students: They're both the same. David: They're both the same. They're both the same types of classes. And yet, if

you look at the results that I 've there, for those two classes. I think you 'd agreewi th me that they aren't the same types of classes. How would you describe Class 8B?

Student: All similar. David: Yeah, they're all of a fair ly similar s tandard, a ren ' t they? OK? Anything

else you could say about them? They're all pretty much middle of the range students aren' t they? Nobody par t icular ly bright there but nobody real ly weak either. OK? They're a pretty average lot of students. I suppose. 8A if you had to put some words on that class to describe them. How would you describe those students (lower marks in 8B)? Dumb? We've got the two extremes haven' t we? OK? We've got a couple of genius at the top and a couple at the far end of the scale that haven' t got a clue wha t ' s going on with their mathematics. So, when the headmaster says how are those classes going in Year 8, and if we just tell him an average mark. it 's real ly quite misleading, isn't it, because it doesn ' t tell us much about how the classes in general are going because there's such a wide range of students in each class. Apar t from giving all the individual marks themselves, if I .just had to give some summary results, is there anything else that I could give. apart from the mean that might help this, convey what sort of students we've got in these classes?

Student: Median? David: Alright, well wha t ' s the median for Class 8A? Well, there's two middle

numbers, isn't there? What do we do when there's an even number of scores, there?

Student: Add the middle ones.

D i a l o g u e of t h i s n a t u r e con t inued for much of t h e c lass as D a v i d a t t e m p t e d to s h o w h i s s t uden t s t h e r e l e v a n c e a n d p rocedures a s s o c i a t e d w i t h t h e m e d i a n , r ange , a n d i n t e r q u a r t i l e range . In t h e l a t t e r p a r t of t h e lesson, s t uden t s w o r k e d on p r o b l e m s c o n t a i n e d in a t ex tbook . S o m e s tuden t s chose to w o r k c o o p e r a t i v e l y a n d t h i s w a s a l l o w e d . D a v i d h a d d e s c r i b e d t h e c lass as a c a d e m i c a l l y b r i g h t a n d t h e r e w a s e v i d e n c e t h a t he e x p e c t e d t h e m to u n d e r s t a n d t h e concepts b e h i n d t h e p rocedures : t h a t is, w i t h more a b l e s t u d e n t s , t h e e x p l a n a t i o n s of concepts w e r e more d e t a i l e d as t h e y focused on w h y as w e l l a s how, a n d he a p p e a r e d to expec t t h e s e s tuden t s to t a k e g r e a t e r c o g n i t i v e s teps .

D a v i d j u s t i f i e d h i s t e a c h i n g p r a c t i c e s in r e l a t i o n to h i s p e r c e p t i o n t h a t t h e cur r icu lum w a s c r o w d e d . D a v i d w a s f a m i l i a r w i t h t h e i n v e s t i g a t i v e tex t s a n d unde r s tood t h e t e a c h i n g a p p r o a c h t h e y s u p p o r t e d . D a v i d e x p l a i n e d t h a t one of t h e m a i n r e a s o n s w h y he used h i s explain p e d a g o g y r a t h e r t h a n i n v e s t i g a t i v e a p p r o a c h e s found in t h e s e tex t s w a s t h a t :

I think it is the speed at which you can get through the work. I guess one of the


overriding considerations are that you get through the work program ... (after all) you have only so many lessons.

Summary and Relations to Other Explain Teachers Two other teachers Peter and Kurt (see Table l) adopted s t ra tegies s imi la r to

D a v i d in t he i r teaching of able students and used s h o w and te l l approaches w i t h less able students. For them, the exp la in t eaching approaches a p p e a r e d to work on the assumption t h a t student mis in te rpre ta t ion could be reduced i f the t eachers were p a r t i c u l a r l y ski l ful in explaining and questioning. But as demonst ra ted by Da v id in his teaching of statist ics, a very good knowledge of the content was necessary. As has been noted (e.g., Crawford, 1996) teachers need to h a v e a deep re l a t iona l understanding of m a t h e m a t i c a l concepts in order to construct logical explanat ions. In th is study, a l l the teachers who used p redominan t ly exp la in approaches in t h e i r teaching were very experienced and m a t h e m a t i c a l l y knowledgeable . T h e y a l l h e l d positions of responsibi l i ty w i t h i n t h e i r schools. T h e y tended not to re ly upon a textbook as the main source of a u t h o r i t y and resource for classroom act iv i t ies ; t h e y were able, and preferred, to construct t h e i r own explanat ions .

As for s h o w and tell, exp la in t eaching approaches were teacher centred. This is to be expected since, as Peter said, "The knowledge is coming from me." The app roach also exh ib i t ed the following behaviours :

• careful explana t ion of the logic under lying rules, step by step "Topaze'" s tyle questioning, construction of personal explanat ions and ac t iv i t i e s ( ra ther t han relying on the textbooks), some use of modell ing ac t iv i t i e s , and a l lowing peer tutoring during the book work phase of lessons for more able students: and

• reversion to s h o w and te l l behav iour for less able students.

Findings for the Investigative Teacher Jan was a 49 yea r o ld female t eacher w i t h 25 years of secondary t each ing

experience. She was the chai rperson of a zone modera t ion panel and h a d a v e r y thorough knowledge of the senior syllabus. Jan acted as H e a d of Depa r tmen t when the incumbent was e lsewhere . She t augh t both .junior and senior ma thema t i c s including Mathemat i c s B.

Jan h a d two dist inct sets of goals for able and less able students. S h e exp la ined "The kids t h a t are not going to universi ty , you .just t each them enough so t h a t t h e y can get th rough t he i r examinat ions and t h a t is it." She s ta ted t h a t she t augh t able students "conceptual ly so t h a t t h e y can understand." She used d i f ferent teaching s trategies in a t t empt ing to ach ieve these goals, but favoured inves t iga t ive ac t iv i t i es and students working in groups to construct m a t h e m a t i c a l meaning when teaching new ma te r i a l .

The following description of a Year 9 lesson on quad r i l a t e r a l s i l lus t ra tes h e r inves t iga t ive approach . It was t augh t to 31 students sea ted in groups of th ree or four. Jan s ta r ted the lesson w i t h quick questions of a computat ional nature l i ke "e igh t minus sixteen" and "one t h i r d of six.'" The students worked i n d i v i d u a l l y on


t h e s e t a s k s a n d w r o t e t h e i r a n s w e r s in t h e i r exerc i se book. S h e e x p l a i n e d e a c h a n s w e r a t t h e end of t h e tes t a n d a c c o m p a n i e d t h e e x p l a n a t i o n w i t h m o d e l s of p rocedu re on t h e w h i t e boa rd . S h e s t a t e d l a t e r to t h e r e s e a r c h e r t h a t one r e a s o n she used ques t ions a t t h e beg inn ing of t h e l e s son w a s to "k i ck s t a r t t h e lesson. '"

The m a i n p a r t of t h e l e s son i n v o l v e d i n v e s t i g a t i n g t h e f e a t u r e s of q u a d r i l a t e r a l s i n c l u d i n g p a r a l l e l o g r a m s . S h e began t h i s a c t i v i t y in t h e m a n n e r d e s c r i b e d be low:

Right, quadri laterals , quad Q.U.A.D, four babies, quads. Lateral refers to the number of sides. OK. Underneath I want you to make a list of as many special ones as you can without talking to any one. Don' t draw them for the moment.just make a list. Has any one got five? Has any one got three or four? OK, Juliet would you like to read yours out.

The s tuden t s t h e n o f fe red t h e i r q u a d r i l a t e r a l s A t t h i s s t age , t h e p e d a g o g y s h i f t e d f rom one t h a t focused upon in t roduc ing t h e c on t e n t to an a l t e r n a t i v e i n v e s t i g a t i v e m o d e l as i l l u s t r a t e d b y t h e i n t r o d u c t i o n be low:

What is special about each of these? Draw me a sketch of each one. Yes, you can draw. Think about wha t ' s special about each one. Then label them please. Any sides that are equal, any angles that are right angles, any sides that are parallel mark them on your diagram.

Dur ing t h i s process , s t uden t s e n g a g e d in c o n s i d e r a b l e d i scuss ion w i t h t h e i r peers . J an ' s comment on t h i s was :

I honestly think that kids need to talk to each other. Because I think they learn from feeding off their ideas and talking to other people and putting in a bit and coming back. You know, if I sit in an isolated room by myself and try to do something, I'm real ly limited by what I've got up here (pointing to head).

T h i s d e s c r i p t i o n i n d i c a t e d t h a t Jan b e l i e v e d t h a t do ing m a t h e m a t i c s w a s a s o c i a l a c t i v i t y a n d w a s s u p p o r t e d b y her responses on t h e s u r v e y .

L a t e r in t h e lessons , t h e s t u d e n t s w e r e a s k e d to t r ace t r a p e z i u m s in to t h e i r books, d r a w in d i a g o n a l s a n d m e a s u r e t h e b i s e c t e d l e n g t h s of e a c h d i a g o n a l ; s h e i n s t r u c t e d t h e m :

You have to do it for the two diagonals for each one. And write down the ratio. Right, obviously we ' re looking for a pattern. See if you can find a pattern between those? ... OK, take your ruler and draw in the diagonals and measure them. And get your calculator and use it.

In g i v i n g t h e in s t ruc t ions , Jan p r o v i d e d c o g n i t i v e s c a f f o l d i n g a n d i l l u s t r a t e d t h e i n t e g r a t e d use of c a l c u l a t o r s to p e r f o r m t h e r a t i o c o m p u t a t i o n s . A f t e r f i v e more minu t e s of t h i s a c t i v i t y , t h e f o l l o w i n g d iscuss ion occurred b e t w e e n Jan a n d a f e m a l e s t u d e n t .

Student: Jan: Student: Jan:

Student: Jan: Student: Jan:

Student: Jan:

This one divide by this one? You don' t notice anything? Is it this one divided by this one? Yeah, OK. So you divided the largest one by the smallest one (portion of diagonal) each time? All right? Is there a pattern coming out? Yeah. Is there a pat tern? The opposite ones there are a round the same number. So what are they around? When we do the ratios what are you getting roughly? Yeah. (pointing to her page) So what you are saying is that one is to that one is the same ratio as that


one is to that one. Is that what you are telling me? Student: Yeah.

The la t t e r par t of this dialogue i l lustrates Jan's tendency to increasingly t ake the responsibi l i ty for students ' th inking if t h e y experienced diff icul ty. Jan's acceptance, even encouragement, of approximat ion, and her persistent questioning ("Is there a pat tern?") are consistent w i t h an i n v e s t i g a t i v e approach to mathemat ics . A l t h o u g h Jan led the students w i t h structural and procedural h in ts and, in tile case above, a r t i cu la ted tile idea t h a t a constant ra t io existed, she did not give the students the rule. S imi la r discussions occurred w i t h other groups. At the end of the lesson, Jan summarised the i r work and made tile conclusion for t he class.

It may be argued t h a t Jan's conclusion of the lesson undermined t he oppor tuni ty for some of the students to construct the i r own conclusions. However , Jan a l lowed the students to invest 30 minutes to find a re la t ionsh ip t h a t she could h a v e expla ined on the black board in much less time. This is strong evidence t h a t she was p repared to teach in an inves t iga t ive way . This tended to ma in ly occur at tile beginning of new work. Observations indicated t h a t when Jan used i n v e s t i g a t i v e approaches to teach a ma themat i c s topic, she employed more direct strategies, such as expla in or s h o w and tell, i f students did not grasp t he ideas in the a l loca ted t ime:

Jan: What don't you know how to do? What's the gradient of the tangent at the point where x is equal to three? Any tangent? A tangent? When x equals three on the curve, what's the gradient of the tangent?

Joanna: I don't know. Jan: How have we been doing these problems? How have we found the

gradients of the tangent? Don't we find the first derivative and substitute x 3 into it? OK, do it. And then do it for the one where x 3 and if the tangents have the same gradient what do you know about them? If over here at x 3, I find that I have got a tangent that has the same gradient, what do I know about them? (The curve and the tangents have been modelled on the board for Joanna, Jan pointed to them and waited about 10 seconds.)

Joanna: Parallel. Jan: They are parallel, all right. Are you still trying to draw it? What have you

done? Right leave it and do three (Question 3) for me. All right. Three is exactly the same but you've got to find the value of y yourself. Right, you're given x, you're given x, you've got to find the value ofy.

Joanna: Right, I can dothat.

This dialogue and the subsequent observations supported Jan's s ta tement to t he researcher t h a t she was "more directed in her teaching" for slower students.

Jan used s imi lar teaching techniques w i t h both the Year 9 students and t he top group of Year 12 Mathemat ics B students who were working from an inves t iga t ive resource. The researcher suggested to Jan t h a t she adop t ed exp la in and s h o w and te l l teaching strategies when students fa i led to demonstrate understanding and t h a t she real ised t h a t this migh t lead to less s tudent conceptual isat ion but at least students would be he lped in passing t he examinations. Jan responded w i t h a laugh, then said "I don ' t l ike the last bit, but yes it is true. T h a t is how I feel even if I don ' t l ike it. T h a t is a rea l i ty ." This acceptance of the researcher 's assertions reflects the mult i tude of pressures t h a t are p laced upon teachers. Jan tr ied to give students the oppor tuni ty to construct understanding. However, i f this did not happen in the a l lo t t ed time, she


emphasised procedure.

Summary and Relations to Other Investigative Teachers Two other teachers in this study, Simon and Mary, exhibited investigative

approaches tha t were similar to Jan for more able students (but not for the less able). When inte[~ciewed, they spoke, like Jan, in terms of the greater importance of understanding and thinking over procedural skill.

Jan, Simon, and Mary all appeared to dislike having students conclude a task and remain doubtful of the meaning of the mathematics under" study at the conclusion of a lesson. At the end of teaching sequences, they managed class discussions where inadequacies in student thinking were analysed and explanations consistent wi th accepted mathemat ica l knowledge were articulated. However, if students did not come to construct understandings t h a t were consistent wi th the shared meanings of the discipline before the lesson was due to end, these teachers adopted expla in teaching behaviours. If students st i l l struggled to understand they adopted s h o w and te l l strategies. That is, they gave recognition to the time saving advantages of these approaches to teaching. In this way, the three teachers lowered their expectations for student understanding from conceptual to calculational. The teachers explained t h a t they did this to ensure tha t students' "needs were been catered for" and t h a t students "would be able to, a t least, pass the semester examinations."

In summary, the behaviour exhibited by the investigative teachers in this study encouraged students to explore mathemat ica l phenomena using the stimulus materials tha t they supplied. However, if students did not demonstrate mathemat ica l understanding, they provided structural support in the form of guiding questions and, in some cases, this emptied the act ivi ty of cognitive challenge. Overall, the following behaviours were demonstrated by investigative teachers: using structural or scaffolding hints to guide students, using questions to develop meaning and to search for patterns and not just to maintain discipline, using a long wai t time after questions, extensively using act ivi ty based problems (that often included the manipulation of materials), confronting students wi th the paucity of their knowledge, requiring students to .justify their answers and to report their progress on tasks, using "Topaze like" leading questions some of the time, and using drill type questions for homework.

Discussion In this study, the focus has been upon the Queensland Senior Mathematics

Syllabus for Mathematics B. This syllabus is representative of reform syl labi tha t recommend an investigative approach to teaching mathematics. It has been implemented in the schools for over five years.

As Table 1 shows, all but two of the nine teachers observed had essentially calculational goals for less able students, and all had conceptual goals for more able students, while eight of the teachers stated tha t they mostly used s h o w and te l l approaches wi th less able students but differed in their approaches for the more able. It also indicates tha t the teachers felt t ha t they were catering for the students' needs by doing this, and it is argued tha t teachers' interpretations of


the syllabus provided a r a t iona le for th is . For the less able, the argument is f a i r l y s t r a igh t fo rward . The teachers were

f a m i l i a r w i t h the syllabus documents, knew t h a t the students (and t h e i r parents) wanted passing grades, and rea l i sed t h a t students did not need to demonstra te the deep conceptual understandings required in solving unfami l ia r app l i ca t i on questions in order to pass. Thus, since m a t h e m a t i c a l procedures can often be completed w i t h o u t conceptual understanding, the syllabus encouraged teachers to adopt l imi ted goals for less able students. O the r authors h a v e noted t h a t t h e routine app l i ca t ion of procedure does not necessar i ly ref lect conceptual understanding (e.g., Gregg, 1995; Ney land , 1996).

For the more able students, tile ana lys is is more complex. All the t eachers c la imed to h a v e conceptual goals for more able students and were concerned as to how t h e y could teach these students to pass on the unfami l ia r app l i ca t i on assessment items where conceptual understanding was necessary. However , t h e r e were th ree d i f ferent approaches used to ach ieve this . A first group of t eachers un iversa l ly adopted a show and fell approach , demonstra t ing a l im i t ed collection of teaching behaviours . A second group of teachers used the explain pedagogy where conceptual isat ion was a major goal and show and fell approaches othe~wvise, p a r t i c u l a r l y to revise work, when t h e y fel t pressured by t ime constraints, or concluded t h a t the understandings were beyond the s tudents at th is t ime. A t h i r d group used the most v a r i ed approaches to teaching the more able. T h e y used invest igat ive approaches to t each new conceptual work, and show and tell and explain approaches to revise concepts. T h e y a r t i cu la ted a grea ter conviction t h a t inves t iga t ive learning ac t iv i t i es were ap p ro p r i a t e for developing student conceptual isat ion. A l though t h e y s ta ted t h a t getting s tudents to pass examinat ions was important , th is seemed not to dominate t h e i r r a t i ona l e for adopt ing a pa r t i cu la r pedagogy as much as the teachers who were res t r ic ted to the more teacher cent red teaching show and tell and explain behaviours . In fact, a l t hough t h e y somet imes fel t constrained by assessment and were uncomfortable w i t h its dominant influence, t h e y pers is ted w i t h invest igat ive approaches for more able students.

There a p p e a r e d to be two possible reasons for not using invest igat ive approaches , the pedagogy most in tune w i t h unfami l ia r app l i ca t ion assessment items. The first reason seemed to be t h e i r be l ie f t h a t such approaches were "too slow" as most of the teachers struggled to complete the m a t h e m a t i c a l content recommended by the syllabus and c la imed t h a t th is l imi ted t h e i r choice of teaching s t ra tegy. Interest ingly, a l t h o u g h the crowded curriculum was impor t an t in influencing teachers ' pedagogy, it was found not to be a cri t ical factor for some teachers .

The second reason a p p e a r e d to be t h a t the th ree teachers who favoured invest igat ive approaches to teaching ma themat i c s were, or h a d been, Heads of Depar tments and thus h a d an in d e p t h understanding of the intentions of t h e Senior Syl labus and more experience teaching mathemat ics . The i r t each ing behav iour seemed to better ref lect the intentions of the senior syllabus. This finding supports t h a t of Perry et al. (1999) who found t h a t curriculum leaders in schools were more inclined to h a v e more child cen t red beliefs about m a t h e m a t i c s teaching t han ordinary classroom teachers . However , it has to be taken into

56 Norton, McRobbie & Coopor

account tha t two of the remaining teachers were also HODs and the rest were very experienced senior teachers, including Advanced Skill Teachers (recognised as being h igh ly skilled).

Only three teachers exhibited investigative approaches and all but one of the teachers tended to maintain teaching approaches more closely aligned to "direct instruction" for less able mathematics students. This is of concern. It indicates tha t the recommendations of the syllabus authors have not been universally enacted even among senior teachers. The majority of the teachers in this study had not adopted the invest igat ive teaching approaches recommended by Australian educational bodies (AAMT, 1996: AEC, 1990) and the State syllabus documents (BSSSS, 1992: QSCC, 2000), and reflected in constructivist theories (Cobb, Yackel, & Wood, 1992). This supports previous findings t h a t many teachers have t radi t ional approaches tha t are well entrenched (Crawford, 1996; Gregg, 1995; McDonald & Ingvarson, 1995; Perry et al., 1999; Senger, 1999), and the effect of the reform agenda will not be dramatic or quick (Perry et al., 1999). However, the evidence from this study differs fi'om Perry et al. 0999) in t ha t the teachers in their study workedunder a Syllabus tha t stated "the Board does not, however, either stipulate or evaluate specific teaching methods" (Board of Studies NSW, 1997, p. 7). In contrast the Queensland Syllabus has been more explicit in encouraging teachers to adopt reform approaches to teaching mathematics.

However, taking a different perspective, it could be considered encouraging tha t three of the nine teachers had adopted investigative approaches to teaching mathematics at least wi th able students. This supports the findings of Clarke (1999, p. 21) tha t over time teachers can become comfortable w i t h "resisting the temptation to tell."

Conclusions While most of the teachers in this study stated tha t they agreed wi th the

principles behind syllabuses based upon constructivist learning theory (at least for able students), most teachers found it difficult to implement these strategies in the classroom. Their articulated reasons for not doing so related in part to the i r perceptions of the assessment criteria. A number of teachers stated tha t t h e y were fulfilling their duties to both students and parents by getting them to pass examinations tha t could be passed wi th li t t le need to demonstrate underlying concepts.

The implications of this study for invest igat ive curriculum writers are t h a t assessment criteria ought not to contain such a "loophole." In particular, a l l students ought to be required to demonstrate conceptual understanding by applying mathematics concepts in unfamiliar situations. Clear reasons wily investigation is the best approach in the rationale of the syllabus, descriptions of how investigation can be enacted in the main body, and models of appropriate investigatory teaching activities, assessment criteria, and instruments in appendices, would seem to be advantageous. So also would be a comprehensive programme of professional development to accompany the syllabus. It was evident tha t some of the experienced teachers in this study did not have a clear


mental image of a l ternat ive pedagogies to s h o w and te l l and exp la in , nor did they have a var ied repertoire of teaching strategies to meet the i r calculational and conceptual goals. Professional development needs to address teachers' mental images and beliefs, recognise the calculational conceptual distinction, and l ink goals to pedagogy.

In addition, the study indicates tha t further research needs to be done to determine the nature of the relat ionships between teachers' interpretations of syllabus documents, the i r perceptions of students' abilities, and how this impacts upon thei r teaching practice. It may be tha t changing the syllabus documents, including assessment criteria, so t ha t conceptualisation is a prerequisite of passing, might bring about changes in teaching approaches of students classified as less ma themat i ca l ly able. The idea tha t teachers' deep values (the good of the child) and subsequent practices can be modified by changing teachers ' instrumental beliefs (what const i tu tes good mathematics teaching) has been noted previously (Senger, 1999).

Further study also needs to determine the nature of professional development tha t could faci l i ta te such a change in teachers' goals and approaches, in particular, assessment guidelines in influencing teacher change. Failure to take into account how teachers interpret and enact a reform syllabus ignores the wel l documented conservative nature of mathematics teaching in terms of embracing pedagogical reform (Cuban, 1984; Gregg, 1995). Thus, many students are l ike ly to predominantly experience "practice and explain instruction tha t has fai led to foster mathemat ica l achievement" (Lo, Whea t ley , & Smith, 1994, p. 30). It should be noted tha t other" authors have argued tha t careful explanation (wi th accompanied worked examples) can "be h igh ly effective at faci l i tat ing learning across a wide range of ma themat ica l ly based content" since the degree of cognitive load is r e la t ive ly small (Cooper, 1998, p. 18). Perhaps the argument in relation to i n v e s t i g a t i v e teaching practices is not so much about understanding mathematics but the degree to which the different teaching methods foster generic problem solving skills and autonomous learning behaviours?

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Authors Stephen Norton, School of Mathematics, Science and Technology Education, Queensland University of Technology, Kelvin Grove Q 4059, Email: <[email protected]>.

Campbell McRobbie, School of Mathematics, Science and Technology Education, Queensland University of Technology, Kelvin Grove Q 4059, Email: <[email protected]>.

Tom Cooper, School of Mathematics, Science and Technology Education, Queensland University of Technology, Kelvin Grove Q 4059, Email: <[email protected]>.

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Teachers' Responses to an Investigative Mathematics Syllabus