“…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?
2a +5b=7ab
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.
I have always had a very easy time figuring out what errors are being made by students. However, finding ways to instruct students to help them stop making the same errors is difficult, unless Ivan reconstruct how they were taught the concept, and thereby understand the reason for the error.
Your comments are the egg white to my consumme – the error looks reasonable to the student!
This opens up a host of possibility in remediation because it allows me to look at the errors and the student’s understanding in a whole new light. Thank you.
I’m glad you found it helpful. Moving from knowing what mistakes students made to understanding why they made them was a huge shift in my teaching. The next shift was figuring out how to remediate–which was and is still the hardest part for me.
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Menu Math: Brad Fulton http://www.tttpress.com/uploads/2/0/4/2/20424731/menu_math.pdf
This is great and from my experience really works with students. Starting with the menu, display h + f and students will add their values easily, move to c + m + h, then to 4f, and you’ve got em! 4f will mean 4 times fries without even thinking about it. Good stuff there in Menu math.