We at the AIMS Center have been digging into research on how children come to know number. What I realize is that knowing the journey from perceptual to conceptual which children must take is important for teachers to understand, in order to effectively help children on their path to understanding number. We have discussed the first step children take as they traverse the gorge between these two ideas; the child imagines the perceptual material, using those images to count. For example, if you place eleven marbles in front of the child and then place a cloth in front of the child and tell them there are three marbles underneath, some children will imagine the three marbles, point to their imagined location and count these imagined items to determine the total. This is a sign of the student making a transition from perceptual to conceptual.
Children take these same steps while constructing composite units, which lead to multiplicative ideas. Suppose a child is posed with the task of finding how many stacks of ten blocks they could make with 87 blocks. Given perceptual material (blocks) a child could build the ten-block stacks and count the stacks to resolve the situation. The teacher can help the child in constructing the ideas needed to work through the situation without perceptual material by limiting or eliminating material. An example of this is seen in the work of Les Steffe with a student named Zachary.
Zachary was allowed to create only one stack of ten blocks and then posed the question.
After Zachary said that he couldn’t find a way to answer my question aside from actually stacking them up, I asked him if he could make another stack and if so, could he make 50 stacks. He thought he could make another stack but not 50 more. Moreover, he didn’t think that one more would be enough. Also, he said that he could put ten up again, and swept his finger vertically by the existing stack. He then said that he could find how many and swept his finger horizontally. However, he still had not established a scheme that he could use. So I reminded him that we did have a way (actually stack them up) and that seemed to activate his stacking actions because he said, “Oh yeah, we can stack the up” as if he had a way to proceed. I then asked him to pretend to stack them up (Steffe, 1992, p. 274).
Using his imagination and pretending stacks of ten were in front of him, Zachary was able to resolve the question. He took an important step from perceptual to conceptual material in regards to the development of composite units.
Zachary took a step further into the mathematical landscape of coming to know number and with continued guidance he will develop increasingly sophisticated numerical understanding. As adults we cannot go back and remember this journey we took, but through our interactions with students and attending to children’s mathematics, we can build second order models of their understanding and assist them in their important and necessary progression from perceptual to conceptual mathematical ideas.