Why did this student think that this verified the identity?

(Thanks Michelle!)

A note from the submitter:

Along the lines of one I sent you awhile back. This is one of my best students, and several other students gave answers with similar misconceptions. I pretty much ignored it last time it came up, thinking that it was an anomaly, but I think it’s a significant hole in my students’ understanding. Students were using calculators today.

What’s going on here in the student work? What’s the connection to the earlier post?

What you need to know is that the student work is the stuff typed in red, and that this came on a take-home quiz:

Why does this student think that ln(e) = x^1. Or did I get that wrong? Are there ideas that “feel similar” that he’s confusing, or is it something else?

(Thanks Taylor!)

A reflection from the submitter:

I think this 10th grader is saying .174>.34>.5. I wonder what she would have concluded if she’d followed the directions and rounded to 3 decimal places? Many kids were tripped up by the .5, maybe she’d say they were increasing except the .5?

Do you agree with the submitter’s assessment? How do you help a student learning trigonometry nail this down?

I think it’s important to say something more subtle than “this kid doesn’t understand decimals.” One thing that this site has documented is that kids can understand something at 1:00 and then do something entirely different at 1:01. It’s best to see this not as a failure of decimal knowledge, but maybe a failure to use decimal knowledge in this situation. (Some people would say this kid’s knowledge of decimals in a certain context failed to transfer to this problem.) The difference is in how we respond. This kid probably doesn’t need the “basics” of decimals. We just need to make a connection to somewhere where she knows about decimals, I’m speculating.

*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 logs, exponents, 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.*

Previously, I shared my 4th Graders strategies for multiplying two-digit numbers. That work was taken at the beginning of our unit, and it’s interesting to me to follow up that post with an update of how their multiplication strategies developed over the course of the past two weeks.