“…some seemingly simple ideas are not always as simple for students as they seem to adults.” -Lesley Booth
How often have you seen students make the following type of mistake?
My guess is that if you teach algebra, you’ve seen this mistake more than once.
If we believe that students do things that make sense to them, we need to ask:
Why does 2a + 5b=7ab make sense to students?
Here’s what the research has to say:
- The nature of “answers” in algebra is different. From their prior experience in arithmetic, they have developed certain expectations of what answers look like. They assume that a “single term” answer is what is needed. There is also evidence that points to the idea that students give one term answers because they have difficulty accepting a “lack of closure.”
- Algebraic notation is confusing to students. They don’t necessarily see the invisible operation between the variable and the number as multiplication. If anything, they may assume the hidden operation is addition given their experience so far in math. For example, when we write 4½, it means 4 + 1/2 and when we write 43, it means 4 tens + 3 ones. It makes sense to them that 2+a+5+b would be equal to 7+a+b.
- Students sometimes view the variables as concrete objects. For example, they see a as apples instead of “the number of apples” and b as bananas instead of the “number of bananas.” Therefore, they justify 2a+5b=7ab by saying 2 apples plus 5 bananas is equal to 7 apples-and-bananas.
Here’s the more difficult question: After reading why this mistake makes sense to students, what might we do differently in our teaching?
I’ll talk about what the research suggests next time, but if you want to know now, check out the work below.
Booth, L. R. (1988). Children’s difficulties in beginning algebra. The ideas of algebra, K-12, 20-32.
Tirosh, D., Even, R., & Robinson, N. (1998). Simplifying algebraic expressions: Teacher awareness and teaching approaches. Educational Studies in Mathematics, 35(1), 51-64.