Student mistakes are important. It’s important to know **what** mistakes students make and **why** they make those mistakes.

For example, last time, we looked at why simplifying expressions isn’t so simple for students and why they tend to make mistakes like 2a +5b=7ab.

*When we can anticipate what mistakes students commonly make and why they make them, we can began the hard work of figuring out how to develop understanding so these mistakes don’t make sense to students.* We can start to answer questions like the one below.

**How do we make simplifying expressions make sense to students?**

Here are 5 ideas from the research:

- Write out the hidden multiplication in algebraic expressions. We know that students often mistake the missing operation in
*2a*as addition. In order to avoid this, write out the expression in expanded form at first (*a x 2*or*2 x a*). Give students lots of experience simplifying expressions with this expanded notation at first and then gradually move away from it. - Develop meaning for the equal symbol. I’ve talked about how important this is before and suggested some ways to do this starting in early grades.
- Don’t use the fruit analogy. It isn’t helpful. Two apples plus five bananas equals seven apples-and-bananas makes perfect sense to students. Plus it creates confusion about a variable represents.
- Avoid using the same letter for the variable as the first initial of what it might represent. Although it appears makes sense to us to use
*a*to represent the number of apples, students see it as apples and they start to believe that the variable is a label for the object. Once students have a clear understanding of what a variable is, you can use the same letter, but try to avoid it at first. - Give students experience with substituting values for variables before and after simplifying expressions and encourage them to find counterexamples. Through continued experience, they will strengthen their understanding of why certain terms can be simplified and others can’t.

I’d love to hear what other suggestions you might have.

Want to know more? Read the articles below.

Booth, L. R. (1988). Children’s difficulties in beginning algebra. *The ideas of algebra, K-12*, 20-32.

Carpenter, T. P., & Levi, L. (2000). *Developing conceptions of algebraic reasoning in the primary grades**. *Madison, WI*: *National Center for improving student learning and achievement in mathematics and science.

MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation. *Educational Studies in Mathematics, 33*, 1-19