# Fraction Error Analysis (aka “Help Jane, Peter, and Sam!”)

3November 20, 2013 by ray_emily

It’s been a minute since I’ve come up with something I really felt compelled to share here, so… guess this is a big day.

This year, my kids have math blogs – which they seem to be digging, which makes me very happy. I’m aiming to have them post about 10 times this year to write about how they are making use of mathematical habits (thanks, Bryan Meyer), among other things.

This week, I had two goals: I wanted my students to do some error analysis pertaining to fraction operations, and I wanted them to do some for-real reflecting about their understanding of fraction operations. Essentially, I hoped to cement student understanding of addition, subtraction, and fraction ‘basics’ before jumping into multiplying and dividing after Thanksgiving break.

I threw together this sheet:

And then this one:

The conversations that ensued were great. (Students discussed in groups, and then we shared out as a class, prior to any writing on that second worksheet.) I loved hearing kids verbalize their conceptual understanding of fraction operations as they remarked upon what Jane, Peter and Sam did correctly and incorrectly. Additionally, my students seemed to take great pride in being able to locate the patterns. (If you haven’t closely analyzed the first sheet: Jane, Peter and Sam each have unique misunderstandings that reoccur in the three problems that they complete.)

I was pretty pleased with the writing prompts, too—which ask kids to think about and propose theories for WHY Jane, Peter, and Sam made these mistakes. (Thanks for the inspiration, Michael Pershan!) So often, when I attempt to get kids analyzing wrong answers, they tell me, “Oh, it was just a careless mistake,” or, “That was a computational error – I need to slow down.” I heard nothing of the sort today, which made me happy happy happy.

Each student will author two blog posts as a result of this activity (alas, each student must choose to neglect either Jane, Peter, or Sam). I’m hoping for good results and good dialogue in the comments.

One improvement I’d like to make, next time, would be to use real student work, in actual student handwriting. If you have other suggestions, I’d love to hear them. I’m hoping to recycle this activity in the future.

PS – Come to think of it, this activity is kind of like an updated, much-improved version of another activity I wrote about here.

[…] did a second go-round of blog posts / error analyses using the same method as last time, but I offered up a different batch of […]

I think the only change I’d make is to have the students fill out those sheets for themselves. This could be done in stages: (1) what I’m doing right; (2) my biggest roadblock; (3) where I was going wrong.

The trick here is to get them to specify what that roadblock is. Don’t be satisfied if they write (for example): I don’t know how to divide fractions. There will be something about division of fractions that they remember & can state or do correctly. What you want them to be able to identify for you and for themselves is the point at which they either don’t know how to get from there to the next step or get confused about whether to go this way or that way. Use prompts to help them put words to their roadblocks, but refrain from using what you see to guide their insights. This is when you get to see the questions through their eyes.

When they find the bridge from what they know to what they want to be able to do, what they now do correctly will stick with them. You’ll have undone the incorrect learning (which is as or more difficult than breaking a bone that has healed the wrong way) and replaced it with correct understanding. You’ll also have gained a huge amount of insight into where their learning problems really lie — and that’s often very unexpected information.

This won’t be easy and you may get resistance, but if you periodically have them look back at all the roadblocks they’ve overcome, they will get a tremendous sense of progress.

[…] honing your skills with eliciting and extending students’ thinking and having students engage in error-analysis tasks aimed at addressing common mistakes or misconceptions. These activities get students communicating with and about fractions as they refine their […]