Mathematical conceptual thinking develops over time. Cross-cultural research, according to Ginsburg and Baron (1993) reveals a similar course of development in diverse cultures for early number procedural and structural understandings. All students appear to develop elementary mathematical abilities in a similar developmental progression irrespective of their backgrounds. However, whilst research has shown that all students exhibit the same basic pattern of development, this is not to suggest that all students in the same age group are identical in their thinking. For a wide range of reasons – social, cultural, intellectual, and pedagogical – similar aged students perform at different levels on identical tasks and can explain and justify their reasoning with differing success. This next section case studies students’ responses located at varying developmental levels.

Cases of Differing Levels of Mathematical Development

This year 4 student read the cards at face value and proceeded to represent each number with a cube. Her attention was drawn to the order of the numbers on the first cards, rather than to the operation and the relationships which might ensue from that operation. The sociomathematical norms developed within a community of informed learners would enable a focus to be made on the operations. When we look at the conjectures and explanations the student offers in the questions concerning subtraction and multiplication we see they are not consistent with her earlier reasoning, despite the fact that she has again modeled the numbers on the cards: demonstrations and explanations are in conflict. Like a number of other Year 4 students she is not able to offer any solution to the identity problems.

In contrast, a number of Year 4 students focused on the number operations and the relationships between the numbers. The student in Case B, following, is in the process of making the transition between a procedural and structural focus. His initial cube demonstration for subtraction stands in opposition to his explanatory statements. At the time of the study he was in the process of learning which aspect of the problem takes precedence over other aspects and learning what constitutes an appropriate explanation.

Clearly the student knows multiplication facts. Although he readily demonstrates ‘two times five’ with the cubes he is not able to provide a representation for the problem ‘five times two’. Many other students in the study could not provide a model for the latter when a presentation of the former easily came to hand. The responses to the identity question appear to have no firm sociomathematical backing and could be regarded as ‘wild guesses’. It is likely that the student had worked with both ‘zero’ and ‘one’ in class but had not been confronted with thinking about those numbers in these terms before. There is too a possibility that the student was confused over the meaning of the question.

“Add to or take away from” may have signaled an operation followed by its reversal, e.g., “start with 7, add 6 then take away 6, still leaves 7”.

For a wide range of reasons – social, cultural, intellectual, and pedagogical – similar aged students perform at different levels on identical tasks and can explain and justify their reasoning with differing success. We consider now a Year 8 student struggling with mathematical conceptual understanding. Most students at Year 4 had surpassed this developmental level.

Mathematical understanding is a recursive phenomenon and thinking moves between levels of sophistication. The student is able to identify ‘one’ as the identity under multiplication and division and can provide an explanation by example. Carpenter and colleagues (2001) maintain that most students at the primary level justify by example. In relation to the other problems, however, the student does not have the mathematical language to convey his understanding, at whatever level that may be. We can only surmise from the demonstrations with the cubes that the student has not moved beyond the procedural level of understanding. In the following two case studies Year 8 students provide articulate explanations and offer a suitable demonstrations of their thinking.

Through their explanations and through the ways in which they represent the problems with the cubes, these students convey learning with understanding. They reveal that their focus is in making sense of the mathematical situations presented, and on the reasoning and thinking which leads to a solution, rather than on procedural rules to solve the problems. Eventually this engagement with the underlying structure and regularities of mathematical operations will enable the students to make links with the structure of algebra.

next page