Issues in Learning Relations and Operations
The Extensive Learning Path for Addition and Subtraction

The teaching-learning path we describe shows that even the most advanced solution strategies for adding and subtracting single-digit numbers have their roots before age 2 and may not culminate until Grade 1 or even Grade 2. The paths also illustrate how children coordinate several different complex kinds of understandings and skills beginning with perceptual subitizing through conceptual subitizing and then counting and matching to employ more sophisticated problem-solving strategies. This makes it clear that one cannot characterize the learning of single-digit addition and subtraction as simply “memorizing the facts” or “recalling the facts,” as if children had been looking at an addition table of numbers and memorizing these. Children do remember particular additions and subtractions as early as age 2, but each of these has some history as perceptually or conceptually subitized situations, counted situations over many examples, or additions/subtractions derived from other known additions/subtractions. It is therefore much more appropriate to set learning goals that use the terminology fluency with single-digit additions and their related subtractions rather than the terms recalled or memorized facts. The latter terms imply simplistic rote teaching/learning methods that are far from what is needed for deep and flexible learning.

The Mental Number Word List as a Representational Tool

We have demonstrated how children come to use the number word list (the number word sequence) as a mental tool for solving addition and subtraction problems. They are able to use increasingly abbreviated and abstract solution methods, such as counting on and the make-a-ten methods. The number words themselves have become unitized mental objects to be added, subtracted, and ordered as their originally separate sequence, counting, and cardinal meanings become related and finally integrated over several years into a truly numerical mental number word sequence. Each number can be seen as embedded within each successive number and as seriated: related to the numbers before and after it by a linear ordering created by the order relation less than applied to each pair of numbers (see Box 5-12). This is what Piaget (1941/1965) called truly operational cardinal number: Any number in the sequence displays both class inclusion (the embeddedness) and seriation (see also Kamii, 1985). But this fully Piagetian integrated sequence will not be finished for most children until Grade 1 or Grade 2, when they can do at least some of the Step 3 derived fact solution methods, which depend on the whole teaching-learning path we have discussed.



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