Introduction

The title of this presentation carries implications about beliefs and value judgements that must be made explicit. Firstly, if the need is felt for information about students’ mathematical reasoning, it implies the conviction that mathematics learning consists of a series and combinations of students’ processes of personal and social constructions that build directly on previous constructions, that are constrained by the quality of previous constructions, by the beliefs of the student about the nature of mathematics and what type of behaviour is required from her, and by the types of tasks and learning culture to which she is exposed. Secondly, it implies a willingness from the educational system both at macro (educational design) and at micro (classroom) level, to adjust learning programmes according to the information obtained on students’ thinking, whether it be from minute to minute in classroom situations or in four or five year cycles by official policy makers. Thirdly, it implies that understanding and sense-making are involved: "Understanding involves the construction of knowledge by individuals through their own activities so that they develop a personal investment in building knowledge. They cannot merely perceive their knowledge simply as something that someone else has told them or explained to them; they need to adapt a stance that knowledge is evolving and provisional" (Carpenter & Lehrer, 1999:23). But, and this is the important issue, this understanding does not only apply to students’ (and teachers’) understanding of mathematics, but to teachers’ understanding of students’ thinking, "…so that teachers have conceptions of typical trajectories of student learning and can use this knowledge to recognize landmarks of understanding in individuals" (p. 31).

Information on students’ reasoning makes it possible to:

• describe levels of development in specific concepts
• identify emerging ideas (e.g. theorems in action) in students’ thinking so that they can be clarified, shared and formalised
• identify misconceptions, inconsistencies, and limiting constructions in students’ conceptions of a particular topic
• identify likely reasons for the above problems
• identify mathematical structures (problem types) that are essential for students’ constructions of powerful mathematical ideas
• identify suitable starting situations for learning trajectories
• investigate the effects of different contexts, classroom culture, and wider cultural environments on students’ understanding and sense-making.

This type of information has the following four broad categories of use:

• to shift the teacher’s focus away from herself and the content to be "covered" and towards her students’ development
• to help teachers to continuously adjust the classroom events and learning programmes according to their students’ needs (diagnostic teaching)
• to help teachers, materials designers, and education authorities in curriculum design efforts
• to develop models and theories about how concepts develop and how understanding grows, or to refine or discard existing models and theories.

These categories are of course never mutually exclusive; for example, a model of learning or development may have a severe impact on the other three categories, as for example the model which we will describe in the next section. Also, in the context of teacher professional development, one simple act of sensitising a teacher to her students’ thinking may in the end affect her and her planning much more widely and deeply than was originally hoped for (e.g. Fennema, Franke & Carpenter, 1993).

It is crucial to realise that simply making available to teachers, education authorities, and the community information on students’ reasoning processes does not necessarily mean that they will value, respect, or build on students’ thinking (Murray, Olivier & Human, 1998). Students’ thinking can only be valued if it is understood to be mathematically sound. If a community (of teachers or parents or mathematicians) only accept the set (prescribed) ways of solving problems as "mathematically correct", they will benefit very little from exposure to students’ thinking. Over the years, many lower elementary teachers told us that yes, they knew children used other methods to reach the correct answer, but that no, these methods were not mathematically correct because they were not sanctioned by the establishment.

When we probe students’ reasoning processes, we therefore need enough understanding of mathematics to appreciate the students’ inventions and the reasoning on which they are based, but we also need to distinguish between different causes for incorrect (and correct) responses and explanations, otherwise we are still not able to interpret and respond appropriately to students’ thinking.

Models and theories

Models and theories about learning, about learning mathematics, about learning particular mathematical concepts, etc. have direct impact on the mathematics curriculum at all levels, even when such models have not been made explicit by the person or institution who is guided by them, and might therefore rather be classified as a belief.

A good model or theory can be of enormous practical value. Pirie and Kieren (1994) comment as follows on their own theory of how mathematical understanding develops: "The theory has enabled us to comment closely on the levels at which different students are making sense of the mathematical activities and thoughts … Such insight into students’ understandings have been used to provide a frame for planning and engaging in mathematics lessons and, in addition, to make observations about curriculum development" (p. 78).

The following is an example of how a model for the development of number concept based on data of how young children think about number, had a major impact both at macro (policy design) and at micro (classroom) level.

An example: the development of number concept

Starting in the middle 80’s, we conducted several hundred task-based interviews with young children during which we posed problems involving whole-number calculations, but encouraged the children to solve them in any way they found convenient. They were also requested to explain their thinking as well as they could. An analysis of these results led us to propose a model for the development of number concept (Murray, 1988; Murray & Olivier, 1989), similar to the models proposed by Kamii (1985) and Resnick (1983).

This model describes the development of the child’s concept of number starting from the point where she can already count out correctly and recognize the numerals.

The first level

At the first level the child has not yet acquired the numerosities of two-digit numbers in a given range, and can therefore only use the pre-numerical strategy of counting all for computations in this range. The child knows the number names of the two-digit numbers and their associated numerals, and associates the whole numeral with the number it represents, but assigns no meaning to the individual digits.

The second level

At this level the child has acquired the numerosities of the two-digit numbers in a given range, which implies that she can utilize numerical computational strategies like counting-on for computations in that range.

Whereas it is sufficient for a child to use only counting on strategies for smaller numbers, counting on becomes very tedious and also prone to error when used with larger two-digit addends.

The third level

At this level the child sees a two-digit number as a complete unit, and can decompose or partition the number into other numbers that are more convenient to compute with, e.g. to replace 34 with 30 and 4. This provides the child with the conceptual basis to use heuristic strategies.

The heuristic strategies used by students in our research are almost always based on decimal decomposition, i.e. a decomposition into a multiple of ten and some units, e.g. 67 as 60 + 7. But the tens are most emphatically not treated as "so many tens"; they are called by their full number names, e.g. sixty-seven becomes "a sixty and a seven", and not "six tens and seven units". Students then use their knowledge of adding multiples of ten to obtain answers, e.g. Chris does 23 + 12 by saying: "Take the three away, add the twelve to the twenty, then add the three again", partitioning 23 into twenty and three. If both numbers are large, he partitions both: e.g. 36 + 27 is solved as "thirty plus twenty is fifty; now add six, then add seven."

The fourth level

At this level the child is truly able to think of a two-digit number as consisting of groups of tens and some units, i.e. the child can conceptualize ten as a new iterable unit, without losing the meaning of the number as a number. Whereas at level 3 the child works with ten as a number that is no different from any other number, at level 4 she is able to work with ten as an iterable unit, a thing that can be counted as a unit, so that the number 23 is conceptualized as "two tens and three ones".

Concerning computation, level 4 understanding of numeration facilitates a progressive schematization ("shortening" and abstraction) of the level 3 heuristic strategies.

Level 4 understanding of numeration is a prerequisite for the meaningful execution of the standard written algorithms. A further abstraction allows one to operate on the digits of numbers&endash;e.g. in the number 56, the meaning of fifty as fifty or five tens can temporarily be suspended to work with 5 as a object for the sake of convenience and the further progressive schematisation of computational strategies.

At micro-level, the model had a great impact on teachers. At the very least, they realized that different children think differently, that their thinking must be accepted and respected, and that number concept, as the other mathematical concepts and skills, is developed over time. Activities should also allow for children to respond at different levels. Many teachers also realized that the model explained why some skills and techniques were so difficult to teach, and why many children made particular kinds of mistakes when implementing the standard algorithms, in spite of what the teachers believed to be "good" teaching, with a plethora of different apparatus and a friendly, supportive classroom atmosphere. The answer supplied by the model is that you cannot expect children to make sense of a technique or method which is based on levels of abstraction which they have not yet reached. "Children seem to take in only those new ideas that they are prepared to ‘hear’ at that moment" (Davis, 1992: 235).

Furthermore, it was realized that the reproduction of taught algorithms, whether successfully or not, was no guide to children’s understanding, and may mask serious problems that children have. We will elaborate on this in the section about misconceptions.

Finally, teachers could now select activities and problems for their students that would help them to develop their number concept from the levels where they are, without requiring types of responses from them which they cannot yet give. For example, increasing number sizes may encourage students to develop more efficient strategies: "…Thys abandoned his direct representations when the numbers became larger. To solve 338 ÷ 13, Marianne started subtracting small multiples of 13, decided it would take too long, and added larger multiples of 13 instead." (Murray, Olivier & Human, 1992).

At macro-level, the above model had a major effect on the curriculum of the provincial Departments of Education who were introduced to it. Since it cannot be assumed that all children will have reached level 3 number concept before the end of Grade 3, the official recommendation was that the introduction of the standard (vertical) algorithms for the four basic operations should be delayed until the upper elementary grades. Even more, when these education authorities saw the results of a teaching approach based on this model, both in success rates for written tests and in the quality of students’ constructions, they gave the schools the freedom to choose whether to teach the standard algorithms in the upper elementary grades or not at all. (Murray, Olivier & Human, 1998).

Planning learning sequences

When a learning sequence for a particular topic is planned, the designer should, apart from the content analysis, also obtain clarity about the following issues:

• What relevant intuitive knowledge may children possess which can be built on and developed?
• What intuitive knowledge and previous experiences leading to limiting constructions may children have had that will hamper their sense-making of the topic?
• Which (unintended) misconceptions and limiting constructions can form in the course of the learning sequence (for example, "multiplication makes bigger")?

When successful mathematics learning is regarded as the rote application of algorithms and techniques, assessment instruments which test superficial compliance with what has been taught are sufficient; with the above view of mathematics education such instruments are not only insufficient, but can be harmful because they may mislead teachers, parents, and policy decision makers about the nature and quality of students’ mathematical knowledge (cf. Hunting, 1997), yet simply adding the question "Why?" to a test can elicit a wealth of useful information even though the assessment instrument is a written test.

An example: decimal fractions

The following test item (Brekke, 1996) scored a success rate of 78% among the Grade 8 students of one of our academically superior local high schools:

Draw a circle around the biggest number 3,75 3,521 3,6
Why do you think it is the biggest number?

The teacher was fairly satisfied with the success rate until she saw some of the reasons given by the students for their choices.

Some examples:

"3,521 because it has more numbers."

"3,6 because tenths are bigger than 100ths and 1000ths."

"3,6 because the smaller the decimal number the bigger he is."

"3,6 because it is = ." (Murray, in press).

These reasons indicate severe limiting constructions from the misapplication of whole number schemes and misunderstood rules from common fractions, lack of number sense for decimal fractions, and confusion about the notation.

In other classrooms where students completed the same test, students also gave reasons for correct choices which reveal their lack of understanding, thus rendering worthless the correctness of the choice.

These reasons not only indicate the misconceptions arising from previous learning and teaching, but also the lack of success of a teaching strategy which emphasizes place value (tenths, hundredths, …) without giving students opportunities to construct the full meaning of a decimal fraction (refer to the choice of 3,6 as the biggest number, "because tenths are bigger than 100ths and 1000ths", which is a perfectly correct statement, but ignores the role of the numerator).

The test item referred to above was part of a four-item questionnaire which only assessed students’ understanding of the concept of a decimal fraction, yet the questionnaire revealed information which disturbed the teachers, not because the students did badly, but because their reasons revealed very poor understanding of the basic concept, and confusion about the notation. We are therefore not interested in the correct answer, but why that particular answer was chosen or arrived at.

The information culled from this simple but revealing probe led this particular teacher to reflect on the following major issues in the school’s approach to teaching decimal fractions:

• What did we do so that we inadvertently allowed/encouraged our students to remain functioning under whole number "rules"? For example, we dealt only with tenths for a long time before introducing hundredths and thousandths, where 0,5 is obviously smaller than 0,7. Should this sequence therefore not be redesigned?
• Do we give the rules for the operations on decimal fractions much too soon ("remove the comma and multiply, then count the places, then put back the comma", etc.) so that students focus on the familiar whole-number meanings of the numbers and not on the decimal fraction meaning?
• Are our usual assessment instruments masking problems in students’ understanding? Should we not improve our assessment instruments?

Identifying emerging mathematical ideas

In many classrooms, students are neither expected to explain their thinking nor to justify it. In the high school, students are indeed expected to justify their reasoning, but usually only in a prescribed fashion, using the previously taught theorems.

Can students present their reasoning processes clearly and can they perhaps justify them as well? If so, of what use will this be?

The answer is that yes, indeed, even young students can present their thinking processes quite clearly once they are encouraged to do so and they know that their thinking is valued. Older students (Grade 4 and beyond) may find this much more difficult than younger students if they had never been encouraged to show their thinking. Even so, Louis (fifth grade) not only solves 784 ÷ 16, but also justifies his steps in a fairly formal fashion:

Louis 784 ÷ 160 = 4 and 144 is left
144 ÷ 16 = 9 because 10 _ 16 = 160
and 160 &emdash; 16 = 144
4 _ 10 = 40 40 + 9 = 49
(Murray, Olivier & Human, 1994).

Requiring students to present their reasoning processes and where possible to justify them help students to reflect on their thinking, to identify mistakes and improve their strategies. It also makes communication and discussion possible among students, and it enables the teacher to identify the emergence of powerful mathematical ideas among her students, so that these can be clarified and nurtured.

One set of these powerful ideas are what Vergnaud (1988) called "theorems-in-action". Our informal observation is that students with a weaker number sense show great ingenuity and understanding of the properties of whole numbers and their operations by their (the students’) use of theorems-in-action to make a calculation easier. For example, Niel (Grade 3) calculates 470 _ 7 as follows:

10 _ 470 ® 4700 ÷ 2 ® 2350 + 940 ® 3290
10 5 2 7

Claire-Anne (Grade 5) calculates 27 _ 35:

27 _ 10 = 270
27 _ 10 = 270
27 _ 10 = 270
27 _ 5 = 270 ÷ 2 = 135
270 + 270 + 270 + 135 = 945

(Both of these students are obviously already at what we describe as level 3 number concept, but it is important to realise that level 3 methods are not invented if students are not confronted with numbers big enough to create a need for these methods. However, the teacher cannot force or demonstrate such methods; students produce them when they are able to do so.)

Anticipatory transformations. The above two examples clearly illustrate the ability of students to transform a given task into equivalent sub-tasks that they know they can manage.

The essential nature of any non-counting computational algorithm is that it is a set of rules for transforming a calculation into a set of easier calculations the answers of which are already known or can easily be obtained. This process of changing the task to an equivalent but easier task involves three distinguishable sub-processes, illustrated here with reference to a procedure to calculate 17 _ 28 (Murray, Olivier & Human, 1994):

• Transformation of the numbers to more convenient numbers, e.g.

28 = 30 - 2

• Transformation of the given computational task to a series of easier tasks, e.g.

17 _ 28 = 17 _ 30 - 17 _ 2

• Calculation of the parts, e.g.

17 _ 28 = 17 _ 30 - 17 _ 2
= 510 - 34
= 476

A third-grader calculates half of 237 as follows:

This can be written more conventionally as

which clearly shows use of the distributive property of division over addition.

Yet the very power and popularity of the distributive property frequently leads to its misapplication, for example:

79 ÷ 13 70 ÷ 10 = 7
9 ÷ 3 = 3
7 + 3 = 10

therefore believing that

This misconception has different causes, but it is possible that the main cause is that the teacher (or previous teachers) had not encouraged sufficient discussion and reflection when the distributive property was first used in calculations, and why it worked.

Multiplicative decomposition of one number in a multiplication or division calculation by repeated doubling or halving is also extremely popular, but is also frequently misapplied when students have not reflected on exactly what they are doing.

For example, the student may reason that that, to multiply a number by four, you have to double twice because four is 2 + 2. Therefore, to multiply by six, you have to double three times because six is 2 + 2 + 2. If the teacher has not tried to discover why (on what grounds) the student doubled twice to multiply by four, she cannot address her misconception. The problem does not lie with the first incorrect answer, but with the earlier calculations where he obtained the correct answer for the wrong reason. We have referred to the dangers of obtaining the right answer for the wrong reason earlier; as Curtis (quoted later on) implied, it is better to be "rong" and find out, than not knowing that you were wrong.

The theorems-in-action found in children’s spontaneous constructions can therefore impress and intrigue the adult, but we can only be sure of their continued value to the child if we can make explicit to the students the reasoning on which they depend.

Some of the reasons may be applicable only to special cases, or even worse, produce the correct results by accident. It is therefore necessary to probe even more deeply than requiring a description of the method; we (the teacher) need to know why the student believes her invention (heuristic) works. This seems like a tall order for young children, yet the following record of their thinking by two third-graders give at least some indication of their reasoning process.

Both are solving 470 _ 7:

The theorems-in-action spontaneously used by elementary school students in classrooms which encourage students’ own heuristics for solving computational problems are regarded as important and valuable for their later algebra, yet we have found that there seems to be little transfer, even after a programme during which their computational theorems-in-action were identified and discussed in Grade 7 (Vermeulen, Olivier & Human, 1996). It is possible that this failure of our elementary school computational-based theorems-in-action to serve as springboards for the more formalised versions in algebra, may be caused by the fact that students had never had to justify their initial applications of the theorem-in-action. Then, by the time the awareness programme was implemented in Grade 7, these to them very old behaviours were such an organic part of their thinking that the formalised explications of these theorems-in-action could not be related to their previous, totally unclarified and never justified knowledge.

Informing daily practice

Teachers who have information on children’s thinking can use this in unexpected ways. Fennema, Franke and Carpenter (1993) report that the information they made available to teachers on children’s thinking in their Cognitively Guided Instruction Project not only enabled teachers to value their students’ thinking, organise their classrooms differently, structure their learning episodes differently, but also, as regards the teacher reported on "… to expand her expectations of children." "… the more problems I asked the better they got." (p. 579). If teachers never find out what children can do, they cannot give them appropriate tasks to challenge them.

It is, however, not enough for teachers to be sympathetic to and invite children’s comments and explanations. They also need mathematical and pedagogical knowledge to interpret students’ responses.

Four students’ responses

Thys (Grade 4). He had to do the following problem for homework: Peter buys 3 kilograms of sausage at R5,15 per kilogram. He pays for it with R19. How much change will he receive?

He stated firmly that he could not solve the problem. After much patient questioning from his extremely puzzled parents, it was revealed that nobody in his right mind would offer R19 in the South African coinage system to pay a bill of R15,45. Thys could perform the calculations easily, but because the question did not make sense to him, he could not proceed. If that question had been posed to him as part of a written test, he would have lost out.

Nomsa and Yolandi (Grade 3). Their class had to answer the following question: What would you rather have, a third of a bar of chocolate or a fifth of the same bar of chocolate?

Both Nomsa and Yolandi chose a fifth. For the second part of the question they had to explain why they made the choice.

Nomsa’s reason: "because it is bigger".

Yolandi’s reason: "because I get sick from chocolates" [she is not supposed to eat chocolate] (Fraser, 1999).

Annes (Grade 1). Annes shared 18 sweets equally among 3 friends as follows:

555 = 15 + 111 = 18 Answer: 6 each

To interpret and respond appropriately to the above students’ responses, teachers need pedagogical knowledge which goes beyond simply appreciating students’ thinking.

For example, they need to know that context plays a major part in children’s sense-making of the mathematical problem. Thys could not complete the calculation because the situation did not make sense to him; Yolandi gave an answer which was correct for her because the question posed did not make it clear that the answer was supposed to be the bigger fraction.

Teachers also need to know that there are different types of mathematical knowledge with different sources and that students’ problems need to be identified correctly as to which type of knowledge they need to overcome a particular problem. Annes solved the problem correctly but recorded it in an unacceptable fashion. His teacher would therefore offend or confuse him if she implied that what he wrote was wrong. She could help him appropriately by suggesting that he might write something like

5 + 5 + 5 = 15
15 + 1 + 1 + 1 = 18
5 + 1 = 6	Answer: 6 each

There are different ways of classifying mathematical knowledge, but we find Piaget’s classification a useful tool for elementary school teachers. Piaget described three types of mathematical knowledge (Kamii, 1985):

Physical knowledge, which is obtained through interaction with and observing the physical environment. This can be called externally-based knowledge.

Social knowledge, which makes it possible for us to communicate with others and with ourselves, and which includes terminology, symbols, and conventions. These may differ slightly from community to community. For example, in South Africa we use the "=" sign much more loosely than in the USA. This type of knowledge can also be called externally-based knowledge.

Logico-mathematical knowledge is the knowledge constructed by organizing, interpreting, and extending existing knowledge structures. This can be called internally-based knowledge.

The teacher who is aware of this classification may realize that Annes’ written solution of

555 = 15 + 111 = 18 Answer: 6 each

can be broken down into its logico-mathematical knowledge component (correct) and its social knowledge component (insufficient). Annes therefore needs some advice on acceptable recording.

For some problem situations, asking children to make a drawing or a diagram of their solution processes may provide valuable information for the teacher on how they think.

This is Sedick’s work:

Sedick’s drawing shows clearly that his problem lies with the social knowledge aspect and not with his understanding.

In contrast, this is Carla’s work:

Carla only produced the drawing after the teacher had asked her to, because the teacher suspected that the numerical solution was simply copied from a neighbour. This indeed proved to be the case, and the teacher realized that not only did Carla need help, but that she and her neighbours needed to be reminded of the classroom norms that governed interaction and collaboration.

Assessing attitudes and beliefs

Carla’s work showed that the teacher should not only be concerned about Carla’s mathematical knowledge, but also about her attitude towards learning. Attitudes towards and beliefs about the nature of mathematics and how mathematics is learnt may be regarded as forming no part of mathematical reasoning, yet such beliefs have a severe impact on children’s mathematical reasoning, or rather, on the reasoning children believe to be appropriate for the mathematics classroom.

Some children quite happily inhabit two mutually contradictory worlds, the "real" world and the school world. In the course of an interview, Mickey mentally solved problems involving 27 + 35 and 78 + 14 correctly using level 3 heuristics, yet he completed the following calculations as alongside.

He explained: "First I add the units, then I add the tens". This sounds correct, but then he carries on to explain for 27 + 6: "First I add the units which is seven plus six. But that is 13, which is two numbers (sic), and there is only a place for one number, so I add the tens and then I write it down, because there are now two places." He calculates the others in exactly the same way, feeling no discomfort when 27 + 6 and 34 + 17 produce the same answer. The fact that he got 23 + 12 correct therefore does not mean anything. His problem does not lie with poor place value understanding either; remember that he effortlessly solved similar problems using level 3 heuristics. His problem rather lies with his interpretation of the classroom situation and of the behaviour expected from him.

The author interviewed four third graders just after their teacher had "taught" them how to divide. She asked them to share 27 pizzas equally among the four Ninja turtles. The children produced two solutions:

In exasperation she said: "I don’t believe this! There are four of you&endash;if I put a box with 27 pizzas down in front of you, are you each going to get 21 pizzas, or then perhaps only one pizza each?"

The children laughed and explained that no, the 21 and the one were simply the answers to the sum; they were not really the answers to the real situation!

Tirosh (1990) discusses the problem of inconsistencies in students’ mathematical constructs, and identifies the above responses as only one of several types of inconsistencies. "If mathematics is viewed as as set of disconnected rules, there is no motivation to anticipate consistency" (p. 119) and there is no expectation for (school) mathematics to match daily experience. The issue here is that if this belief about mathematics and mathematics learning is not brought to the surface and addressed by the teacher, children’s learning is severely hampered. "Thus, it is reasonable to assume that instruction that takes into account the mathematical knowledge of the students, that attempts to build instruction upon this knowledge, and that strives to find ways to test whether the newly acquired knowledge is in line with previously existing knowledge, is more likely to prevent the creation of stable inconsistencies than instruction that does not view the learner’s world of mathematics as a crucial factor to be considered in instruction." (Tirosh, 1990: 122)

It is easy to say that the learning environment in which students learn and do mathematics influences their beliefs about the nature of mathematics and about doing and understanding mathematics.

Nothing is ever quite simple.

Even classrooms which value sense making and understanding, and where discussion, questioning, and argument are encouraged, still need to monitor individual students’ beliefs. Anthony (1996) describes Gareth who has simply adapted to the superficial aspects of social interaction in his classroom: "… his activity is structured in a social classroom environment rather than a mathematical learning environment" (p. 356).

In other cases, students do the calculations correctly and with understanding, but then produce a completely senseless answer based on their perception of the requirements of school mathematics. For example, some sixth-grade students worked out the correct amounts for the different ingredients needed to make five cakes, given the ingredients for one cake, then added all the ingredients together. When challenged whether this was the sensible thing to do, the response was "no, but in mathematics we have to give only one answer to a sum".

(Murray, Olivier & De Beer, 1999:3&emdash;310)

How do we assess beliefs?

An assessment instrument which is well-suited to probe beliefs is the student’s journal, but journals are not always successful or suitable, especially with younger children. A related instrument which we have used with great success is the occasional informal essay in which students are invited to write whatever they want to about their mathematical experiences (Murray, Olivier & Human, 1994). The teacher can occasionally structure the students’ comments by asking them to write down the most important thing they had learnt during the past week. Even first and second graders can provide the teacher with valuable insights into her students’ learning, and into the type of learning environment students find supportive.

For example, Keba (third grade) wrote: "If you working in groups you must not just copy and you must think of different ways".

Ashley (third grade): "We talk about other people’s answers".

Curtis (second grade): "I love (mathematics) because it is fon to be rong and to findout. I love the wird sum".

Contrast this with Marnelle (third grade): "If you make a mistake they laugh at you".

Inviting students to make a drawing of their mathematics period can also produce insight into how a particular student experiences her learning environment.

This drawing was made by a Grade 1 student to show him in the computer laboratory:

It would be presumptuous of us to analyse and describe students on a single drawing without knowing more about them, yet it is clear that such drawings can provide teachers with warning signals about possible problems.

The environment (classroom culture) in which mathematics learning is supposed to happen, influences the type of learning which takes place. In an authoritarian environment students are more likely to resort to behaviours which they hope may please the teacher, and tend to accept mathematical knowledge as Piaget’s "social knowledge" type, which means "doing sums in the way the teacher wants", as in the examples we have given.

Major problems

We have claimed that teachers need information on their students’ reasoning to inform their daily practice. It is not difficult to obtain this information: Much research has been done on children’s reasoning, intuitions and misconceptions on a variety of topics, and as we have discussed in this paper, there is a variety of ways in which the teacher can probe her own students’ thinking.

However, for the teacher to actually desire this information, she has to be convinced of several things:

• That children’s mathematical intuitions and reasoning processes can be sound and of great value
• That mathematics learning involves making sense, understanding, solving problems, explaining and justifying, and not imitating procedures
• That because of this, her minute-to-minute classroom decisions must take into account her students’ thinking.

She also needs knowledge of several kinds to provide her with a lens or lenses through which to view her own classroom, for example:

• Knowledge of models of development such as the model for the development of number concept
• Knowledge of typical intuitions, misconceptions, etc. about a particular topic
• Knowledge of the mathematical content from different perspectives, mathematical as well as pedagogical. We would cite Piaget’s three types of mathematical knowledge (physical, social, and logico-mathematical) as an analysis of the content from the pedagogical perspective.

Unless the teacher has this knowledge, her interpretations of and therefore the use that she can make of information on her students’ reasoning to inform her own practice will be limited.

Similarly, curriculum designers and material developers will only be willing and able to use the information available to inform their own processes if their perspectives on mathematics learning include the students as active sense-makers and thinkers whose personal and social constructions are the main components of the learning process.

References

Anthony, G. (1996). Active Learning in a Constructivist Framework. Educational Studies in Mathematics, 31, 349&emdash;369.

Brekke, G. (1996). A Decimal Number is a Pair of Whole Numbers. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the Twentieth Conference of the International Group for the Psychology of Mathematics Education, Vol. 2 (pp. 2-137&emdash;2-144). Valencia, Spain.

Carpenter, T.P., Fennema, E. & Franke, M.L. (1996). Cognitively Guided Instruction: A Knowledge Base for Reform in Primary Mathematics Instruction. The Elementary School Journal, 97(1), 3&emdash;20.

Carpenter, T.P. & Lehrer, R. (1999). Teaching and Learning with Understanding. In E. Fennema & T.A. Romberg (Eds.), Mathematics Classrooms that Promote Understanding (pp. 19&emdash;32). Mahwah, New Jersey: Lawrence Erlbaum.

Davis, R. (1992). Understanding "Understanding". Journal of Mathematical Behavior, 11, 225&emdash;241.

Fennema, E. Franke, M.L., Carpenter, T.P. & Carey, D. (1993). Using Children’s Mathematical Knowledge in Instruction. American Educational Research Journal, 30(3), 555&emdash;583.

Fraser, C. (1999). [Developing initial fraction concepts]. Unpublished raw data.

Gill, A.J. & Thompson, A. (1995). Bridging Second-Grade Children’s Thinking and Mathematical Recording. Journal of Mathematical Behavior, 14, 349&emdash;362.

Hiebert, J., Carpenter, T.P., Fennema, E., Fuson, K., Human, P., Murray, H., Olivier, A. & Wearne, D. (1996). Problem Solving as a Basis for Reform in Curriculum and Instruction: The Case of Mathematics. Educational Researcher, May, 12&emdash;21.

Hunting, R.P. (1997). Clinical Interview Methods in Mathematics Education Research and Practice. Journal of Mathematical Behavior, 16(2), 145&emdash;165.

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