EFFECTIVE STUDENT LEARNING IN THE MATHEMATICS CLASSROOM BY PETER CORKILL B.Sc. (Hons.) (Melb.), Dip. Ed. (Melb) AND CHRIS WILSON B.Sc. (Ed.) (Melb.) Commonly expressed concerns among many secondary mathematics teachers centre on the lack of real and lasting understanding of mathematical concepts exhibited by their students. This, together with the lack of motivation and self-esteem so evident in the mixed-ability classroom, has created challenges for teachers as they come to grips with the changing trends in the teaching of mathematics in recent years. In 1995, a group of ten practising mathematics teachers at Frankston High School in Melbourne, Australia embarked on a journey together with a view to addressing these problems. The group met once a week throughout the year to develop, implement and review teaching strategies which would encourage students to take control over their own learning and become metacognitive in their approach to the learning of mathematics. The group followed the PEEL model of operation, whereby teachers from a number of learning areas meet and share strategies to enhance the learning in their classrooms (Baird & Northfield, 1992). This was the first group of this kind to operate in only one area of learning. The group developed a model for "good learning" in mathematics based on the idea of good learning behaviours discussed by Baird and Northfield (1992). The strategies put into place by the teachers generated many instances of effective learning on the part of students. The students were handed greater control of the agenda and their response was to assume more responsibility for their own learning. The project has reinforced in the minds of the teachers the power of collaboration in the process of teacher reflection and change, and has left a lasting impression on their practice. In this presentation, the outcomes of the project will be shared by its leader and one of its most energetic participants. The presenters will share some of the strategies which were put into place, and discuss the positive learning outcomes which resulted. Peter Corkill is a practising teacher of 14 years experience, and has coordinated Mathematics Faculties in two schools. He has been involved in curriculum development and its implementation in Mathematics for many years, and currently leads the Middle School administrative team at Frankston High School. Peter led the Mathematics PEEL project there in 1995. Chris Wilson is a practising teacher of 25 years experience, with a long history of school leadership in curriculum development and gifted education. He was a member of the PEEL project at Frankston High School in 1995. Baird, J. R., & Northfield, J. R. (Eds.) (1992). Learning from the PEEL experience. Melbourne: Monash University. ENHANCING EFFECTIVE STUDENT LEARNING IN MATHEMATICS: A COLLABORATIVE APPROACH Peter Corkill (and Chris Wilson) Frankston High School It's different to any other Maths class I've ever been in. (Year 8 student, Nancy, 22/11/95) They just couldn't believe the amount of Maths that was coming out of one simple question. I thought hey, I've never seen kids like this before! (Interview, Peter, 4/12/96) Not only do you learn about Maths, but you learn about taking more of the responsibility for doing your own work. (Year 11 student, Michael, 8/11/95) Introduction A central issue in mathematics education concerns the identification of teaching strategies for enhancing student learning of mathematics. Of particular interest in this study was the importance of developing strategies for encouraging students to be more metacognitive and to take greater responsibility for their own learning. The Project for Enhancing Effective Learning (PEEL) provided a model of how teachers can work together to develop such strategies. Traditional PEEL groups consist of teachers from a range of learning areas who meet on a regular basis to discuss strategies to improve student learning outcomes. These strategies can often be applied in a number of subject areas. Unlike other PEEL networks, the teachers who participated in this research project were all mathematics teachers from the one school. This research involved a case study of a group of secondary mathematics teachers at Frankston High School, who made a conscious commitment to work collaboratively to respond to a range of concerns about the learning of their students, and to seek to develop strategies that would lead to positive learning behaviours and outcomes. Their collaborative efforts in 1995 provided a unique and rewarding experience for them. They felt strongly that despite the heavy demand on time and energy, it was a journey they wished to continue into 1996 and perhaps beyond. This summary of their findings and experiences, adapted mostly from the unpublished thesis (Corkill, 1996) written on the project, is a tribute to their commitment and professionalism. Review of the Literature The call for change to mathematics education has reverberated internationally in recent times. The National Council of Teachers of Mathematics produced several documents in the 1980's which called for a reform of traditional "transmissive" teaching approaches (Mitchell, 1994). The Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) stated that the emphasis should be on "doing" mathematics rather than "knowing" it, and stressed that mathematical "power" should be conferred on students which would give them the "ability to use information to reason and think creatively and to formulate, solve and reflect critically on problems" (p. 205). Documents such as A National Statement on Mathematics for Australian Schools (Australian Educational Council, 1991) and Victoria's Curriculum & Standards Framework: Mathematics (Board of Studies, 1995) have ensured that the reform process has also remained high on the agenda in Australian education. The aims of the mathematics PEEL project at Frankston High School were at the cutting edge of these reforms as it sought to have students become more aware of and responsible for their own learning, in other words to become more metacognitive. The work of Flavell (1976), Garofalo and Lester (1985), and Schoenfeld (1987, 1990, 1992) showed the importance of self-regulation in mathematics, particularly with regard to problem solving. It was clear that an increased emphasis on metacognition in the mathematics classroom had the potential to improve mathematical performance and understanding. The PEEL Model The Project for Enhancing Effective Learning (PEEL) reported successful efforts to foster secondary students' control over their own learning through the promotion of metacognition (Baird & Mitchell, 1986; Baird & Northfield, 1992). The project began at Laverton High School in Melbourne in 1985 and has since spread to many other schools, both nationally and abroad. Traditionally, PEEL groups have consisted of teachers from many learning areas who meet on a regular basis to discuss and implement teaching strategies which encourage students to be more reflective and to take greater control over their own learning. It is also true from my discussions with the originators of the project, Mitchell, Baird and Northfield, that few mathematics teachers had participated in PEEL groups prior to this, and there was a need for strategies to be developed which would be of benefit to both teachers and students in this discipline. The PEEL project provided a model for the Frankston teachers to follow. Despite the fact that all of the teachers belonged to the one learning area, mathematics, their collaboration and cooperation enabled them to develop these strategies in the same manner as more traditionally constituted PEEL groups. Concerns about the Teaching of Mathematics The teachers in the PEEL Mathematics Project at Frankston High School identified the following concerns regarding the teaching and learning of mathematics in the classroom: 1. The inability of students to retain knowledge Students have a poor retention of knowledge, both from topic to topic and from year to year. There is a lack of rigour in skill practice. There is a lack of clear expectations of the children - what skills did the teachers want them to attain? Teachers were disillusioned with the students' inability to retain the efficient use of basic skills, and questioned whether the teaching approaches placed enough emphasis on the basic skills of the subject. Their concerns included the students' inability to employ the correct skill to a given situation, their inability to retrieve knowledge despite clear cues, and the need for the teacher to communicate clearly to the students what they should be able to do at the end of a course of study. There was also concern at the apparent acceptance of this situation by schools who seemed to structure their courses on the assumption that students would not remember work from year to year. 2. The inability of students to use skills in their correct context. There is a lack of context in mathematics teaching - students often fail to see the relevance of mathematics There is a lack of opportunity to do cross-curriculum and co-operative teaching. There is not enough use made of technology. Students often display a lack of ability to transfer mathematical skills to other subject areas, or even to related mathematical topics - they apparently do not see mathematics as a set of tools or skills applicable in a wide variety of situations. Teachers were concerned with the lack of opportunities for contextual cross-curriculum teaching, and observed that the students have learned that mathematics, as taught in the classroom, is a matter of 'topics and tests' rather than 'context and understanding'. The more widespread use of technology was seen as a desirable step in enhancing both motivation and understanding. 3. The difficulties faced by teachers in catering for a wide range of abilities in the classroom, and the low self-esteem of learners in mathematics It is difficult to cater for the wide range of abilities in a class. Students generally have a low self-esteem in mathematics. Many teachers were frustrated with the difficulties of adequately catering for the learning needs of students with such a broad range of interests and degrees of motivation within the one classroom. Allied to this were the problems faced by those students who found mathematics difficult and at times impossible, who seemed threatened by the subject, and who had lost confidence in their ability to learn successfully. Teachers who had tried to retrieve this situation reported little success, and were concerned by the disturbing lack of self-esteem demonstrated by many students. Major Findings of the Project The PEEL Mathematics Project at Frankston High School had much to offer fellow teachers. The following discussion summarises its most important findings. 1. The collaborative approach can help address the shared concerns of teachers. The shared concerns of the teachers centred around their belief that many students did not demonstrate a thorough understanding of the mathematical concepts which they had been taught, even though they might have used these skills in several topics throughout their schooling. In conjunction with this, they were also concerned that many students were not able to remember how to use mathematical concepts in anything other than the short term, and that many others were unable to recognise or use these skills in other contexts or subjects. It was of particular concern that the self-esteem of many students was low, and that the large range of abilities present in a typical high school mathematics class made it difficult to meet the needs of all learners. The collaborative efforts of the teachers enabled many of these concerns to be discussed and addressed. It was of comfort to the teachers that their individual concerns were usually shared by their colleagues, and the PEEL structure gave them a forum for discussing these concerns. Too often teachers feel that they are the only ones not coping, or the only ones experiencing frustration with their students' learning. This project clearly showed this not to be the case. The teachers often discussed issues such as the self-esteem of their students, and became more sensitive to the needs and feelings of individual students as the project developed and more sharing took place. Many of the strategies attempted to address these and other concerns. Indeed, all of the 48 strategies sought to make students more aware of and responsible for their own learning, and to encourage students to reflect on their own understanding. Particular strategies such as the "Mathematics Smorgasbord" gave students greater choice and enabled them to work at a level of performance at which they were comfortable and to which they were suited. Many of the strategies were both enjoyable for the teachers to implement and for the students to work through, judging from the many anecdotes from the PEEL meetings. The teachers felt that the confidence of many of their students had grown, and their enthusiasm for mathematics had increased. This was supported by the comments of students in interviews and the observations of many of the project teachers. Many students had appreciated the change. 2. The teachers developed a collection of desirable learning behaviours which were indicators of effective student learning. Using the work of Baird and Northfield (1992) as a guide, the teachers developed a model of good learning which further informed their search for teaching strategies. The list of 34 "Good Learning Behaviours" (see figure 1) combined Baird and Northfield's (1992) indicators of good learning with those suggested as important by the teachers. The teachers decided that the first 17 of these good learning behaviours could be taught to and encouraged in students, for example "keeps a record of work done". The second 17 were more "elusive" in that they occurred less frequently and were indicative of effective learning on the part of the students, for example "anticipates and predicts possible outcomes". Students who independently asked questions which sought further insight, or voluntarily made links to alternative situations, were obviously thinking on a deeper level than the norm. This occurred quite often, and was particularly evident during problem solving sessions involving open-ended questions. Once this list had been compiled, the teachers directed their efforts to developing teaching strategies which would encourage students to display this good learning. They postulated that if students were to adopt some of the first 17 strategies as part of their normal approach to learning, then the more sophisticated learning behaviours would follow as a consequence. No data were obtained in support of this hypothesis during the project, but it could form the base of future study. 3. Teaching strategies were developed which enhanced the development of good learning behaviour. The teachers reported several occasions when their students displayed good learning behaviour as a consequence of the strategies put into place. For many of the teachers it was the first time that they had seen such behaviour in students. It was clear that the learning outcomes for the students had improved as a result of the project as so many became more involved in their learning and were prepared to contribute more willingly and often in class. After analysing all of the data collected in the project, in particular the transcripts of the weekly meetings, I identified 48 different procedures which had been used by the teachers, and six broad categories into which they could be placed. Each procedure was placed into only one category. A summary of each of these categories is reported here. Group A Procedures for Generating Discovery Learning in Mathematics These procedures encouraged students to either discover new concepts, or consolidate existing ones, through exploration. Students also gained deeper understanding and greater ownership. For example, students in a year 11 class were asked to use calculators to evaluate the following logarithms all to base 10: log2, log3, log4, log5, log6, log8, log9, log16, log25, log30, log48, log50, log64. They were then asked to 'play' with the results and try to establish as many patterns as they could. This led to the discovery of the three Laws of Logarithms by the class. Group B Procedures for Finding Links and Contexts Through these procedures students were able to see links between mathematical concepts, and also to other subject areas. They were able to see how a range of mathematical skills can be important in solving the same problem. One approach here was to write mathematical investigations which required the students to progress to higher levels of thinking, based on Bloom's Taxonomy. Some of the higher thinking levels (synthesis, for example), and other strategies such as concept mapping, enabled students to see links and contexts. The teachers used a table of "trigger words" when designing assignments which enabled them to produce questions which progressed students through a sequence of thinking levels. This is reproduced in table 1. Table 1. Trigger Words Which Activate Different Levels of Thinking Group C Procedures for Communicating Mathematical Ideas These procedures were intended to address the problem of students memorising rules without learning how to use them. They were encouraged to challenge their beliefs about mathematics, and to take up and defend a position on a mathematical conjecture. In one strategy, the teacher presented the class with a range of provocative mathematical assertions and challenged them to either find and justify an answer, or to prove or disprove the assertion. Examples were: 'Two negatives always make a positive' 'Perimeter is always bigger than area' 'When you divide a decimal by a decimal you get a smaller number' Group D Procedures for Developing Independence and Awareness Students were encouraged to take responsibility for their own learning by reflecting on their understanding, identifying areas of difficulty, and making decisions about the best course of action. One approach which was of great benefit in allowing students to make sensible decisions about their learning was the 'Mathematics Smorgasbord'. It involved giving students a range of activities through which they could work at their own pace, or a collection of activities at different levels of ability from which they could make a suitable selection. Students could do several shorter problems, or fewer longer ones. Group E Procedures to Enhance Learning for Understanding and Retention of School Mathematical Knowledge These procedures were designed to enhance students' understanding of mathematical knowledge, and to enable them to retain their ability to apply their skills over longer periods of time. Many of the teachers began lessons with twenty (or ten) short questions involving work which had been covered at some stage during the year. They found that it was a good strategy for focussing the class quickly on the lesson, and very useful for reinforcing and revising concepts. Group F Procedures in Organisation and Motivation Students were encouraged to examine their work practices with a view to improving their efficiency, and also to be ready to 'think mathematically' when they entered the classroom. One of the simplest strategies was 'Operation: Clear the Desks!', in which a concerted effort was made to have students work in a comfortable and uncluttered workspace by clearing bags and all unnecessary books from their desks. Many teachers reported an instant effect on the class, with many students adopting the strategy as soon as they came into class. 4. The collaborative environment of the project supported teachers in the process of teacher reflection and change. It was clear from the data that the collegial support provided to the teachers in the PEEL group had encouraged them to examine their own beliefs about and practices in teaching. The sharing of ideas with like-minded colleagues in a set time came to be greatly valued by the teachers. Many rued the lack of opportunity to do this as a part of their normal practice, and it suggested to me that this could well become accepted practice in the future. Teachers' comments to me both in and out of meetings indicated that a strong cooperative bond had developed in the group, and they were prepared to share ideas and resources with each other on an ongoing basis. Not all of the teachers took up strategies at the same rate. Some waited until a particular strategy had been successfully trialed in class before they would contemplate attempting it. One teacher expressed the concern to me that he hoped that "[I] didn't think everyone was running around and trying all the strategies in their classes, because it just wasn't the case". It needs to be emphasised that this was perfectly legitimate - the teachers appreciated being able to progress at their own rate, as well as being completely autonomous in deciding which strategies they would adopt and in what way. The fact that the teachers were comfortable with each other's contributions and rate of change meant that each teacher could take from the project that which was most relevant to them. This was highlighted by the fact that the teachers remained with the project for the whole year, and attended meetings regularly. Once the group had focussed its direction and its aims were clear to all participants, the teachers worked together to develop the teaching strategies. Many formed part of their existing practice, but it was the first time for many of them that they were able to share their wisdom with their colleagues in a formal and ongoing way. The list of strategies grew quickly, and the fact that there were 48 by the end of the year is a tribute to their commitment to the task and indicative of the power of collaborative professional networks. Many of the teachers gained new insights into teaching and learning as a result of their involvement in collaborative ongoing research. Some of them may have had a sense of these insights from their previous experiences in teaching, and it is not the intention of this report to claim that these insights have all broken new ground. However it is true to say that the practice of many of the teachers has been informed by their experiences in the project classes. Teachers' comments indicate that this change has occurred in a manner and in a climate which shows little likelihood of a return to previous practice. Some Further Perspectives on the Project from a Participant Chris Wilson, a year 8 mathematics teacher, was a member of the project group through 1995. His involvement was also related to his position as 'Extension and Enrichment Co-ordinator', and a desire to explore ways in which mathematics learning in the regular, mixed-ability classroom could extend and enrich all students. For Chris, many important lessons emerged from his involvement in the project, and have continued to be powerful influences in his teaching and his continued involvement with PEEL in 1996. His insights, from his perspective of participant, add much to the project's findings from my perspective as its coordinator. The project clearly had a far reaching impact on all of us ! Chris felt the project impacted in six distinct ways: 1. The teacher becomes the learner. I learned much during the 1995 PEEL project, but not just about mathematics and how to teach it better. I learned a lot about learning. Two moments stand out in my mind. One was during a discussion with two wonderful year 8 students who had been frustrated by the normally slow pace in mathematics classes. I asked them how it felt for them, and one of them answered "We've got so much in our brains, but there's nowhere for it to go". The second was some time later when I set up a wonderfully intricate system for the students to learn some geometry through 'learning centres'. It involved student choice, group work with clear roles, interdependence within and between groups, learning about the learning process and about thinking levels, open ended questions, visual representation of the steps through the learning centre, and more. It was brilliant, and it was a disaster! It did not work, and we were all relieved when it was over. However I learned so much from each of these situations, and not only am I a different teacher for the experience, but my eyes are now open all of the time to how much we can learn by watching and talking to kids while they are learning, or while they are thinking, talking and reflecting about their learning experiences. We cannot expect students to do what we are not prepared to do. We must be learners too, and this involves risk-taking, discovering, enthusing, sharing, and honest reflection on what we have done. Through PEEL, we become the learners. 2. Awareness of 'poor learning behaviours'. Teachers are often concerned about 'poor behaviour' in their classrooms, and the solution may be difficult to find. The classroom is such a busy and complex place that it is usually very difficult to understand clearly what is happening and why. We all have an array of strategies we use as appropriate, but our task is made no less difficult by the fact that the students quickly learn each teacher's modus operandi. The PEEL project revealed to me a new way of looking at this problem. When we focused instead on 'poor learning behaviours', the solution often presented itself as a change in learning strategy. Of course there was no single, simple solution, but what this did was to ensure that our attention remained on our principal purpose - the learning which was occurring in our classrooms. We were also sending a clear message to our students - "My concern is for you and your learning, and I will not let other things interfere with this." Students appreciated this concern for their development. 3. Support between colleagues. PEEL meetings were a wonderful sharing time, and an opportunity to discuss what most of us felt were the real issues in teaching and learning, without the usual encumbrance of administrative detail present in most meetings. The support for and between colleagues extended well beyond the meeting itself. The PEEL project members would often pass each others' classrooms and knew just what was happening in there (and why), and would continue to share ideas in the staffroom throughout the week between meetings. There was a certain bond developed between us through the year, and this remains even for those who have moved into other PEEL groups this year. 4. Important principles. The quality of the discussion which took place at our meetings cannot be overstated. We were friends, and felt comfortable to make any comment or suggestion which seemed appropriate at the time. We explored a wonderful spectrum of issues and ideas, and in doing so developed a catalogue of approaches to learning which we would not have considered possible at the beginning of the year. The weekly discussions did much more than build up a rich resource of teaching and learning strategies. They clarified for each of us what were the important principles on which the learning experiences in our classrooms should be based. Although these might have been very different for each individual, it left each of us better equipped and more confident to manage spontaneously any situation as it arose in our classrooms. At any given time in any given class, I believe I was (and still am) aware of, and to a large extent guided by, the following principles: 1. That students must process information - they must not just absorb it, they must do something with it - their learning must in some way be active. 2. That no student is incapable in mathematics or in anything else - they are all capable in different ways - in different skill areas, and in different kinds of learning experiences. 3. That self-esteem is central and critical - every child has a right to feel good about him/herself, and will achieve much more when this happens. 4. That my relationship with the children in my classroom is the basis of the learning experiences which happen there, and that I will know when this relationship is good because the students will be working with me. 5. That the students are well aware when I try to put them through contrived learning experiences - they want real learning to happen. Why should they tell me an answer if they know I already know it ? 6. That I am learning also - about me, about kids, about learning, about maths. 7. That awareness is important - awareness of what we are doing, of why we are doing it, of how it fits into the big picture, of how each individual learns best, of what we are trying to gain from this work we are doing - and this awareness is just as important for both teacher and student. 8. That I know when learning is happening well because it just 'feels right' - and the kids know it too. 9. That no matter how open or individual or exploratory the learning becomes, I am still the person with the responsibility for its continued effectiveness. 10. That it is learning which matters, not teaching. (More learning would happen in a school without teachers than in a school without students!) 5. Self-awareness and self-confidence. In our PEEL group, everyone's contribution was valued. Our leader was sensitive to our needs, our workload, and our stresses through the year, and this became the dominant feeling between all members of the group. We soon realised that we all had an abundance of good, innovative ideas and that others did want to share them. We were also comfortable in using the group as a sounding board for an idea we were not too sure about, or to share a disappointment or concern if we felt our classes were not going too well. We knew the response would be positive, supportive and non-judgemental. As this confidence in the group grew, we become more aware of our own strengths as teachers, and were able to focus on these in their own classrooms. We learned much about ourselves as people, and also about how best to relate to our students and involve ourselves in the learning process with them. 6. Professional Development by Learning Much of the professional development to which teachers have been exposed has been top-down, teacher-centred, theoretical, just-in-case instruction, the very approach to teaching which most teachers are moving away from in their own classrooms. It makes little sense to tell teachers how to teach - firstly because we are all so different, and secondly because it models what we don't want teachers to do in their classrooms. Our PEEL group, on the other hand, was small group, practice based, learner-centred, bottom-up, just-in-time learning. It modelled for its participants the exact experiences which it sought to promote in the classroom. Most important, though, was that it placed the teacher back in the role of learner. Apart from the fairly obvious advantage that through PEEL we were learning new ideas and attitudes and skills, it also put us back in touch with what learning feels like. It is a very good feeling, and from our meetings perhaps the most powerful memory of all for me was that this was the feeling, the experience, I wanted to introduce my students to, just in case they had never experienced it before. And when I went into my classroom with that enthusiasm, guess who noticed it? Conclusion As a result of their experiences in 1995, some of the teachers felt that the project needed to be explained in greater detail to the whole staff of Frankston High School, and interested teachers from all learning areas could be asked if they wished to join. This could mean an evolution of the mathematics group into a more "traditional" PEEL group. It could also mean the creation of a number of PEEL groups which would focus on areas of common interest to the teachers involved. Some of those suggested included the class teachers of one particular class, the teachers of a particular level (eg. Junior or Middle or Senior), the teachers of a particular subject such as Mathematics or English, and the teachers of related subjects such as Mathematics/Science or English/History. Any of these would constitute a further evolution of the original PEEL concept. I will allow one of the teachers in the project the final word on its effect on the participating teachers. Despite the energy needed to reflect on teaching practice and induce change in the classroom, there was a definite sense that there was to be no turning back - the teachers had come too far to return to their textbooks and past practices: It becomes a self-promoting idea, doesn't it? It's not as though we've finished the year and we've done PEEL for a year and now we'll go into Year 8 and "Oh, if we've got the energy we'll do it again". I mean you couldn't go back to "Here's the textbook, today we're doing Exercise 13.4". You couldn't do it. So you find the energy, and if the kids respond well, that's energising in itself, so it's a self-feeding thing. Even though it does take more energy, and I agree it does get very tiring, in reflection it's worth it a hundred times over. Personally it has been worth it many times over. I would never have thought at the beginning of 1995 that the collaborative efforts of a group of teachers could have produced the memories I now have of this project. My classroom became richer for my involvement with these dedicated and inspiring practitioners, a place where students began to take risks and to share their mathematical insights. There will be no turning back for me. References Australian Education Council. (1991). A national statement on mathematics for Australian schools. Carlton: Curriculum Corporation. Baird, J. R., & Mitchell, I. J. (Eds.) (1986). Improving the quality of teaching and learning: an Australian case study - the PEEL project. Melbourne: Monash University. Baird, J. R., & Northfield, J. R. (Eds.) (1992). Learning from the PEEL experience. Melbourne: Monash University. Board of Studies. (1993). Victorian certificate of education 1993 - space and number common assessment task 3: analysis task. Carlton: Board of Studies. Board of Studies. (1995). The mathematics curriculum & standards framework P-10. Carlton: Board of Studies. Corkill, P. J. (1996). In search of effective student learning in mathematics: A case study. Unpublished M. Ed. thesis (Australian Catholic University). Flavell, J. H. (1976). Metacognitive aspects of problem solving. In L. Resnick (Ed.), The nature of intelligence (pp. 231-236). Hillsdale, NJ: Erlbaum. Garofalo, J., & Lester, F. K. (1985). Metacognition, cognitive monitoring, and mathematical performance. Journal for Research in Mathematics Education, 16(3), 163-176. Mitchell, I. J. (1994). School-tertiary collaboration: a long term view. International Journal of Science Education, 16(5), 599-612. Mitchell, I. J., & White, R. T. (in press). Good learning behaviours. American Educational Research Journal. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, Va: Author. Schoenfeld, A. H. (1985). Mathematical problem solving. New York: Academic Press. Schoenfeld, A. H. (1987). What's all the fuss about metacognition? In A. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 189-215). Hillsdale, NJ: Lawrence Erlbaum. Schoenfeld, A. H. (1992). Learning to think mathematically: problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334-370). New York: Macmillan. Winebrenner, S. (1993). Teaching gifted kids in the regular classroom. Melbourne: Hawker Brownlow Education. Figure 1. A list of good learning behaviours in mathematics Has an Organised Approach 1.Allows an adequate, uncluttered workspace. 2.Has all necessary equipment. 3.Sets out work in an organised manner. 4.Maintains a record of work done. 5.Listens attentively. 6.Is willing to contribute a response. Displays Confidence 7.Makes an immediate, independent start to work. 8.Is able to stay on task, concentrate . Monitoring Behaviours 9.Takes pride in presentation of work. 10.Prepared to share success publicly with peers. 11.Is positive about learning mathematics . 12.Tells teacher when they don't understand. Seeks Assistance 13.Confers with other students. 14.Keeps a checklist of problems which cause difficulty. 15.Ask teacher questions looking for specific information. 16.Checks work against instruction/notes/examples. Checks Personal Progress 17.When stuck refers to earlier work before asking the teacher. 18.Checks personal comprehension of instruction and material. Requests clarification if needed. 19.Seeks reasons for why chosen strageties or steps are used. 20.Explains results in terms of mathematical reasonableness and commonsense. 21.Seeks further challenge/direction on completion/mastery of a task. 22.Checks teachers work for errors; offers corrections. Reflects on the Work 23.Anticipates and predicts possible outcomes. 24.Seeks links between adjacent activities, ideas. 25.Seeks links to other maths topics. 26.Seeks relevance to other subjects, real life. Constructing / Reconstructing Behaviours 27.Independently seeks further information, following up ideas raised in class. 28.Asks probing questions, seeking further insight "WHAT IF". 29.Offers personal examples which are generally relevant. 30.Suggests alternative procedures, solutions. 31.Is prepared to challenge sanctioned answers. Assumes a Position 31.Is prepared to challenge sanctioned answers. 32.Offers new ideas, new insights, alternative explanations. 33.Justifies opinions. 34.Reacts and refers to comments of other students.