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A few months ago, I swung by Justin Reich’s classroom and showed his undergrad some math mistakes. (Read about it here or here.) In planning the session, I practically begged Justin to let me use a class set of mistakes instead of individual pieces of interesting mistakes. Here’s what I wrote:

Your first question in the protocol is “Look at three problems on the board. Predict all of the mistakes that students might make.” I love this question. But I feel lingering guilt about how mathmistakes.org usually responds to this sort of question with a single example of student work. This has felt problematic in some of the conversations that I’ve had surrounding student work, because someone might be entirely right in their predictions in a way that isn’t affirmed by the chosen work. I worry that this feeling of “gotcha” sometimes kills discussion around student work since the initial predictions aren’t entirely engaged.
My proposal is to tweak the protocol a bit. Instead of showing kids what a single student actually did, what about showing them a class set of responses? Then we can better check our initial predictions and ask a whole host of other interesting questions. (e.g. What patterns do you see? Do you think you can tell how these kids were taught? Why do kids tend to make this mistake?)
I ended up using a class set of fraction-comparison work for Justin’s class. My experience cemented my opinion: if you want to talk about teaching or student thinking, it’s gotta be the class set. Why?
Reason #1: You can’t be right or wrong with a prediction about one piece of student work. 
This is what I mentioned in my email to Justin, but I want to expand on it here. A great way to learn something is to make a prediction, and then check it against reality. But knowledge about student thinking is most powerful either in the aggregate or in the very specific. Meaning, to know something about student thinking is either to know something about how kids, in the aggregate, often think about something, or it’s knowledge about how this kid, right here thinks about it.
This is a long way of saying that checking a prediction against one wacky error is inevitably a bit of a letdown:
  • OK people! Here’s this math problem. What thinking do you predict you’d see from kids here?”
  • You collect predictions
  • “OK here’s this piece of student work. Were you right?”
  • Umm. Yes? No?

It feels unfair to me to ask — as I have in the past — for people to invest themselves with a prediction that I can’t honor with a realistic response. Using a class set better respects other people’s predictions.

 

Reason #2: You can’t really make connections and form generalizations about student thinking from one example of student work.
When I worked with Justin’s students, I saw them making connections across different students. They noticed the prevalence of the area model in the student work, and that led to interesting observations, connections and questions on their part. Wouldn’t have happened if I just showed them the craziest one.
 
Reason #3: Math Mistakes discussions sometimes devolve into “Well this kid needs one-on-one tutoring/special attention/I would pull her aside and ask…” The class context nudges us away from that.
Look back on the comments from the first few years of this site, and you’ll see this line appearing over and over again. This imagines a context where you’ve got one student having a hard time, while the class as a whole mostly gets the math. This allows participants in a math mistakes discussion to shift the responsibility onto the individual student.
That’s a fine context to imagine, but offering a class set of math mistakes offers a much richer context for conversation. Here is a class, in the middle of learning something. On the whole, they know some things and struggle with others. What are those things? What could we do next?
 
Reason #4: It’s more authentic to the actual work of teaching.
Finally, the class set more closely resembles the actual work of teaching, so participants get to practice an aspect of that actual work. We can sort and categorize student work. We can talk about groups of students, and develop language to describe different sorts of students and different sorts of struggles.
I haven’t posted very many class sets on this site, though I’d like to do more. Feel free to continue submitting individual pieces of student work, but you’ll get a special high five if you send me a set of a class’ work ready to post.
Here are the class sets that I’ve posted so far:

http://mathmistakes.org/recursive-and-relational-thinking-and-the-feedback-each-deserves/

http://mathmistakes.org/do-these-properties-guarantee-congruence/

http://mathmistakes.org/which-fraction-is-larger/

http://mathmistakes.org/multiplication-strategies-my-students-are-starting-with/

http://mathmistakes.org/getting-better-at-multiplying-two-digit-numbers/

Guest post by Justin Reich, cross-posted at Justin’s blog at Education Week (here).

In my Introduction to Education class, one of my goals is for students to get a sense of the value of looking at student work. Not just glancing at it, reading it, or grading it, but really trying to understand what we can learn about students’ thinking by examining their performances. In this post, I want to share one lesson that I did with my pre-service teachers in the MIT STEP program, using resources from MathMistakes.org.

MathMistakes.org is a project by one of my favorite bloggers, Michael Pershan, a recent winner of the Heinemann Teacher Fellowship. The URL says it all, MathMistakes.org is a collection of annotated math mistakes, submitted by teachers with comment threads that attempt to get inside student thinking and propose teaching solutions. Michael was a huge help in finding some great content for my lesson, including three juicy problems (natural logsexponents, and simple equations) with really interesting conversations in the forums.

Before showing the student work, I give my pre-service teachers a framework for thinking about looking at student work. In particular, I encourage them to start low on the “ladder of inference.” It’s very easy when looking at student work to jump to judgments and conclusions, and then make observations about work that support your first hunch. I think a more valuable approach is to start by making observations, keeping an open mind, and then moving towards conclusions.

 

Looking at Student Work

To help model that kind of thinking, I shared a Looking at Student Work Protocol published by ATLAS, that I think does a nice job guiding students towards that kind of thinking. A “protocol” in this context means a set of steps for addressing a situation. The ATLAS protocol is probably more designed for looking at more in-depth performances than answers to a few math problems, but it works as a good foundation.

In the ATLAS LASW protocol, people start by examining student work and just noticing facts about it, trying to avoid making any kind of judgments or inferences. Ideally, observers assume that the student producing the work is making their best effort in good faith. When looking at student work, it’s usually a distraction to assume that kids are being lazy or obstinate. Better to assume that they are putting forth their best effort.

The next step is to start asking questions about the work. What do you think the student is working on here? From the student’s perspective, what are they trying to do? Then, observers start making some judgments about the work and suggesting changes to the instructional environment or approach that might address issues that appear in the work. So eventually, we get to making judgments and proposing solutions, but we get there slowly.

In-Class Protocol with Math Mistakes

I modified that protocol to take advantage of the great resources at MathMistakes.org. Here’s what we did:

1)   Look at three problems on the board. Predict all of the mistakes that students might make.

Michael gave me three problems to work with, and before showing any student work, I showed my pre-service teachers the original questions and problems. I asked my students to predict the different kinds of mistakes young mathematicians might make. I put students into three groups (of about 8 teachers each), and I ask them to consider all three problems.

2)   Look at what a student actually did. Make observations about their work, first. Then, start to ask questions about what you see. Then, start to make some predictions about what they may have been thinking.

For this section, I assigned each group to look at one problem. One issue that emerged here was that different Math Mistakes have different richness of student output. For instance, one problem just showed that a student wrote the number 0. Not much to observe there. Another problem showed several steps of work, including some non-standard notation that lends itself wonderfully to close parsing. So some groups raced through the step of making observations, whereas other groups needed more time.

3)   Enrich your conversations by bringing in voices of expert teachers from MathMistakes. What new ideas emerge here? What is the range of possibilities of what the student may have been thinking? What is the range of ways to respond?

When group conversation started to slow (pretty soon for the group whose student answered “0”), I gave them each a printed copy of the relevant comment thread from MathMistakes.org. I printed them in part for logistical reasons (the class didn’t need computers for anything else), and in part because I wanted them to be impressed by the heft of the discussion. The comments for the three problems I shared run 10 pages long, filled with insightful observations about student thinking, analogous mistakes, and instructional approaches. My sense was that students were quite impressed that a single mistake on one worksheet could generate so much thoughtful reflection from experienced educators.

To wrap up, we shared a bit about what we thought was happening in each problem, what students might be thinking, and how we might remediate. Mostly, we reflected on how a single problem could be such a deep window into student mathematical thinking and the complexity of teaching responses. 

Thanks again to Michael for helping me pull these materials together!

We’ve clearly got some work to do here to make the materials on this site more helpful for teachers, pre-service teachers, and students, and Justin has helped tremendously. Take to the comments with ideas on how to use student mistakes to even greater effect.

complex numbers

Help, if you don’t mind!

I’d like to flesh out our collection of complex numbers mistakes. If you’re teaching this topic in the next few months, could you send me some pictures of mistakes? I’m looking for all sorts of mistakes — either common or uncommon errors.

In the meantime, would you take a second and comment about a complex numbers mistake that you’re used to seeing from your students?

When you’re done with that, check out some of the new mistakes that I just posted…

Thanks for all the great mistakes and comments lately, guys. Keep up the great work!

Hey everybody! It’s so nice of you to visit here.

I recently posted a bunch of mistakes, and I know how easy it is to get slogged down in the internet, but I think that you’ll like these. So go check out…

… a mini-essay I wrote about some 3rd Grade Exponent mistakes.

… a student who thinks that ln(5)-ln(4)=ln.

… some very non-standard fraction notation that my 4th Graders started using.

… all of the answers my kids gave to Andrew Stadel’s Black Box lesson.

… a meditation on the spoken of language of fractions, inspired by a kid who calls three quarters “a third.”

… a confusion a Geometry student of mine had about “scale factor.”

 

Also, from other people’s classrooms, we’ve got…

… a problem with equivalent fractions in algebra. Or, does squaring the top and the bottom of a fraction not do the trick?

… a division mistake, and what’s going on with it anyway? (I thought it was a calculator mistake.)

… a mysterious claim made from a sort of geometric scatterplot question that I’ve never seen before.

 

Come comment and join the fun!

At the one-year anniversary of this site, it’s time to take stock.

We’ve got a bunch of mistakes — a couple hundred, at this point. A ton of Trigonometry mistakes. A bunch of Algebra material. Not much from middle school or elementary school, though maybe that’ll change next year. (I’d like it to.)

I’ve had moments of doubt over the last few months, wondering whether this project might have run out of steam. Already in late November, I came to think that the site wasn’t serving its original purpose of challenging teachers to engage deeply with perplexing student work. Over the course of the year I’ve heard from some of you who use the mistakes on the site to create materials for students, and I appreciate that among other things, this site serves as a filing cabinet for student work. (Truth be told, I’ve only used mistakes from this site in my own classroom once or twice. It’s not an important part of my repertoire.)

To my mind, the biggest success of this project is the work that I did with exponents. And when I’ve angsted about the future of the site to other teachers, the thing that they’ve tried to tell me is that the sort of large-scale, synthetic analysis is what they value the most.

The problem is that this sort of big analysis is difficult. I don’t know how to reproduce it on anything like a consistent basis.

More and more, though, I’m wondering if I just need to dive into it more consistently. If I make a project of closely analyzing mistakes, maybe I can find patterns and trends that will tie some of these things together. If things go well, maybe I can inspire some of you to take the helm of the blog to expand on your own theories and ideas.

For what it’s worth, I think that this sort of deep, synthetic analysis of trends in student work can be especially powerful for teachers. In particular, the exponents work wasn’t just interesting, but it was fruitful — a lot of good lesson ideas emerged out of that analysis. Ultimately, I think that those lesson ideas are what most of us are chasing.

For the rest of the summer, you can expect new mistakes to be posted sporadically, at best. Instead, you can expect at least one blog post of deep analysis a week.

(Interested in writing a guest post? Drop a line!)

Hey all,

This is just a note to make something official that I’ve been doing unofficially for a few weeks.

I’m now posting 3 mistakes a week instead of 5. I think that’s more keeping with people’s ability to process this stuff and comment intelligently. Bonus: it gives me a bit more time, a bit less pressure.

Think this is the wrong move? Let it rip in the comments.

Thanks all for all your help and support and things.

-Michael

From Math Teachers at Play:

Analyzing mistakes is a great way to learn. Can your algebra students explain what went wrong in these Algebra 1 Math Mistakes? Can your geometry students make sense of these Geometry Math Mistakes?

From the Math Forum:

Every day since last summer, New York City high school math teacher Michael Pershan has posted a photo of a math error made by a student — and invited other teachers to come analyze and discuss the misunderstandings behind those mistakes.

These two postings represent the dual uses of the site: student and teacher analysis. As we compile more errors, this site is becoming increasingly valuable as a source for student analysis.

We here at MathMistakes are ready for vacation. Unfortunately, our school doesn’t have vacation this week. (I know!) But yours’ does.

Plus, we’re feeling pretty burnt out and we need a break to figure out what’s next for this site other than compiling and tagging hundreds of mistakes. (We’re up to 132, by the way.)

We’ll be back next week. Or later this week if someone sends me a really juicy piece of student work.