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Students begin to move from arithmetic to algebra (in the structural sense) when they start thinking about properties of operations rather than properties of numbers. This happens quite early. For example, "missing addend" problems (such as 4 + ? = 9) can be solved one at a time, each as a special case, by any number of techniques (counting up, counting back, even subtracting). But when your students start saying things like "subtraction is the opposite of addition" or "subtraction undoes addition," they are starting to realize a structural relationship between two operations rather than a collection of relationships between pairs of numbers. Note 2
We took an initial look at algebra from a structural approach when we examined the concept of doing and undoing in Session 3. At that point, we were looking at relationships between operations with a focus on undoing, or inverting, operations. Another hallmark of a move to algebra as structure is a focus on comparing algorithms. For example, consider the following two algorithms:
Algorithm A:
• | Take a number |
• | Add 1 |
• | Double your answer |
Algorithm B:
• | Take a number |
• | Double it |
• | Add 2 to your answer |
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