| Bowers (1999) reported on an instructional sequence aimed at promoting an understanding of increasingly sophisticated number concepts and place value. The learning took place over a nine-week period in a class of 23 third grade children and centred on an imaginary Candy Factory that the children were to control. The instructional sequence developed as follows: |
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| 1. | Setting the scene - the children were given a description of the Candy Factory and were asked how the workers might pack and keep track of candies. After a variety of suggestions had been made by the children, the teacher told them that the workers had decided to pack the individual candies in rolls of ten and to pack ten rolls into a box. | |||
| 2. | Exploring number - exploring ways of packing and unpacking the candies (using Unifix cubes) and working on activities that required them to make up set amounts and to record them in picture form, showing boxes, rolls and pieces. | |||
| 3. | Interpreting number - children were asked to interpret how many candies there might be in different drawings. | |||
| 4. | Creating / regrouping - the children created different ways of packing (regrouping) specified numbers of candies. They were introduced to a software programme, which enabled them to do this more quickly and easily than by using the Unifix cubes. This developed the idea of the conservation of quantity and led to the understanding that a number can be rearranged in many ways without changing its value. As the children's concepts developed, old concepts that previously had had to be justified came to be accepted as givens and no longer required justification. | |||
| 5. | Recording - moving towards symbolic representations of number. The children started to make use of digits instead of pictures. | |||
| 6. | Problem-solving with number - keeping track of the candies required the children to develop ways of solving addition and subtraction problems. | |||
When confusion occurred the teacher made use of the “folding back” strategy (Pirie and Kieran, 1994). A key part of the lessons was the use of a software programme that enabled the children to explore the packing and unpacking problems without having to go through the relatively time consuming task of physically changing pieces for rolls and rolls for boxes using the Unifix cubes. Using the computer the children could see, at a glance, the effect of rearranging the candies when they changed groupings. Place value relationships were revealed in the course of just “playing around” with the different options for packing and unpacking candies. Bowers (1999) maintains that the software, therefore, enabled the children to operate at a higher level of thinking about place value. Regrouping and use of symbols were significant ideas at the core of the developing understanding of place value. During the discussions that followed these activities the children's ideas were shared with those of their peers and, together, these contributed to developing communal understandings about the Candy Factory and the number system. The development of sociomathematical norms was seen as an important part of the developing learning culture. These included an awareness of what counted as a different mathematical solution, what counted as a clear explanation and what would be the most efficient solution. Because of the conventions established at the outset the children viewed the boxes and rolls as composite units and the children's understanding of this emerged from the activities and discussion. As the classroom mathematical practices evolved, a reflexive relationship was established with individual children contributing to the class understanding and, in turn, learning from the classroom culture. Children who thought in original and different ways and did not “follow the herd” had to justify their ideas and sometimes they were the ones who prompted the emergence of new practices within the classroom culture. This analysis shows the focus on the communal process - an alternative to idea that mathematics is only accessible to children via the curriculum. |
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