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
Excellent five points…Jo Boaler mentions that our starting point for algebra should be a generalizing mindset whereas we start in a procedural mindset.
Interesting! That’s probably good advice for other topics too.
Interesting ideas. Are there specific links to those articles? I referenced this post in my own blog. Just FYI http://ontariomath.blogspot.ca/2013/10/math-links-for-week-ending-oct-11th-2013.html
Thanks for sharing David! I can’t share the specific links but hopefully you can find them at a library.
I completely agree with #4, especially. Another solution to #4 is to start developing the idea of a variable using lots of Jo Boaler/Fawnstyle visual patterns ( http://visualpatterns.org/ ). Visual patterns lead students to explaining reasoning like this: “well, Step 2 had 8 squares, and Step 3 had 12. So it seems like you just do 4 times the number, so my expression is 4N”. Students tend not to interpret 4N here as “4 numbers” but as 4 times the step number.
That sounds like a good solution to #4. I’ll have to take a look at Jo Boaler’s stuff, especially since you are the second one who mentioned her. I didn’t know she did research on algebra.
My concern in #5 is that students may overgeneralize the evaluation of expressions at a single number to prove equivalence of expressions. We know (and can prove) that for an nth degree polynomial we need n+1 points to determine it’s unique form (e.g., linear-degree 1-two points determine unique line; quadratic-degree2-three points determine unique curve, etc.). This has important implications for the numeric interpretation of equivalence. That is, students would need to verify at least n+1 values for the variable for an expression of degree n.
We’re really talking about “insertion equivalence” here (see Prediger and Zwetschler, 2013 who cite Malle, 1993) — in which expressions are equivalent if they have the same value for ALL inserted numbers. Depending on the age of the student, this approach may be best achieved through a functions approach in which the expressions are viewed as rules for functions from which tables of values and/or cartesian graphs can represent the equivalence or non-equivalence visually/numerically (Kieran & Sfard, 1999). Moreover, I often wonder about the introduction of symbolic equivalence prior to numeric/graphic equivalence. If numeric approaches are more intuitive and accessible to students, what is the motivation to use the symbolic approach to equivalence? Why not start with a numeric approach to equivalence and/or a graphic approach to equivalence. Yerushalmy (2006) is a good reference for this.
By the way, I’m really intrigued by your blog and the quest to link research and practice as described by Arbaugh and colleagues (2010). I hope to have more opportunities to interact on these topics and to share resources. This work is so important!
Some resources to consider:
Arbaugh, F., Herbel-Eisenmann, B., Ramierez, N., Knuth, E., Kranendonk, H., & Quander, J. R. (2010). Linking research and practice: The NCTM research agenda conference report.
Kieran, C., & Sfard, A. (1999). Seeing through symbols: The case of equivalent expressions. Focus on Learning Problems in Mathematics, 21(1), 1-17.
Malle, G. (1993). Didaktische Probleme der elementaren Algebra (Didactical problems in elementary algebra). Vieweg: Braunschweig.
Prediger, S., & Zwetschler, L. (2013). Topic-specific design research with a focus on learning processes: The case of understanding algebraic equivalence in grade 8. In T. Plomp & N. Nieveen (Eds.), Educational design research – Part B: Illustrative cases (pp. 407-424). Enschede, the Netherlands: SLO.
Yerushalmy, M. (2006). Slower algebra students meet faster tools: Solving word problems with graphing software. Journal for Research in Mathematics Education, 37(5), 356-387.
Thanks Nicole. That’s a great point about #5. I was thinking more about using that strategy to find counterexamples to disprove equivalence. You make good points for not using it to prove equivalence using single (or limited) examples. I also like the suggestion of using the graphical representation to demonstrate equivalence. I need to think more about the affordances and limitations of that approach for students. I’m interested to take a look at the references you listed, especially the Kieran & Sfard article. I hope we can continue the conversation!