Error Analysis, Decimal Operations, and Being Less Lame


September 10, 2012 by ray_emily

Yesterday’s call for help was met with one suggestion in the comments, and one suggestion via twitter. Both are lovely ideas (thanks @stephanie231333 and @reminoodle!) that I look forward to implementing. Even more so, however, I’m looking forward to continuing to grapple with this problem, on my own.

Why the change of heart? Happily (unsurprisingly?), after pinpointing and itemizing my gripes about this unit, I feel much more optimistic and ready to make a change. Now that I’ve moved from moping to actually determining what in particular was not working—a necessary step that I’d pathetically avoided, previously—it’s time to get to work. (And… Yay, blogging!)

Anyhow, my first attempt to be less lame was inspired by @misscalcul8, who has written a lot lately about her aspirations to be less talkative. (She does not need to work on reducing lameness, like I do.) In addition to providing a million examples of how she’s doing this, Elissa writes that, “A good teacher creates opportunities to learn but doesn’t necessarily lead them.” I read this, and I thought, YES!!!! And so, Elissa was in the corner of my brain as I attempted to find a better way to teach, reteach, and review (simultaneously!) decimal addition and subtraction.

I came up with a totally-not-attractive, handwritten worksheet. (I’m seriously ashamed that it is so sloppy. There was, alas, no time to prettify. It served its function – and that’s what matters most!) I passed it out, and explained to kids that I had culled the errors (in the left-hand column) from previous students. (That was a white lie.) I then told them it was their task to analyze the mistakes and write a super-helpful description to the confused student who made those mistakes. That confused student was counting on them!

Here’s one student’s work. (Hopefully you can tell the difference between kid handwriting and my handwriting…)

All the highlighting and arrows and explanation are the student’s!

The idea of helping a struggling student was quite motivating for my kids, and gave my classroom a certain spirit that honestly surprised me. Here’s what I mean: Kids were not just analyzing errors, but rather exhibiting their wisdom and generosity – and taking pride in doing so. They made efforts to present their ideas and their work more thoughtfully, thoroughly, and neatly than usual. (Until I saw the awesome work that kids were producing, it had not occurred to me what perfect training this activity provides for my error analysis sheet, which is a compulsory step for students who want to reassess in my class. It’s almost like I planned it! Except, no, I’m not so clever.)

There was also (yep) a handful of kids who behaved as though the mystery students’ errors were CRAZY, implausible, and beneath them. That, too, was actually sort of pleasant and amusing to watch. (I probably should have discouraged this sort of response. The kids responsible were being so goofy and having such a blast that I just shrugged and let them keep working.)

(Yes: This kid mixed up “bigger” and “smaller,” which makes his worksheet even more entertaining.)

Oh! I also explained to my kids that, in each group of four, I wanted to see all students on the same problem, at all times. I distributed mini whiteboards, so kids could teach one another improvisatory lessons, on the spot, in case anyone in their group could not locate the error or figure out the correct solution. (Speaking of which: I’ve been doing SO MUCH MORE group work this year, that I’m thinking it might be wise for me to nab Amy Gruen’s lovely idea, on helping students understand what it means to be a good coach.) I saw a lot of interactions that I liked, but some kids could to stand be way more supportive and helpful to their table-mates. I want to encourage that.

Anyhow – in case you hadn’t drawn these conclusions on your own, I liked this activity a lot! Here’s the rundown of how it met my students’ needs.

  • Kids who might have simply gone through the motions, in the past, were forced to write a little bit about why various algorithms work.
  • Kids who would have likely fallen into the usual traps were (a) spared the trouble (someone already made that mistake! No need to do it again!), and (b) challenged to articulate why various errors are tempting, but must be avoided.
  • Kids who would likely have been utterly lost received mini-lessons from peers. (Also, I could spot those kiddos and work with them fairly easily, given that I was NOT in front of the class blah-blah-blahing. Oh – in case it wasn’t clear: There was no blah-blah-blahing. I just let kids dive in.)
  • Kids received training and practice in analyzing errors, which is an all-around useful skill – one that I’ve always wished they knew how to do more effectively!

If I have a bit of time to make a less ugly version of this sheet that you can use, I promise I will update this post and add it.


11 thoughts on “Error Analysis, Decimal Operations, and Being Less Lame

  1. I’d read your tweet for help and your previous blog post and I had been thinking about them because I also think many of my previous decimal lessons have been too dry. I love this activity — I’m totally stealing it when we get to decimals in a few weeks. How many mistakes did you give them to analyze?

    • ray_emily says:

      I gave them three addition problems and three subtraction problems. The whole worksheet took about 30 minutes (maybe a little more?) because there was lots of dialogue about each problem and how best to articulate the correct way to solve. Let me know if you use this idea! I bet we could perfect it, much more. I think I’m going to try similar activities with multiplying and dividing decimals, too.

  2. @aeakland says:

    Rachel, I LOVE this activity and I love the way it makes the kids think about their thinking! We’re getting to decimals in a couple of weeks and I can’t wait to try it!

  3. bstockus says:

    Excellent job! I’m happy to see that you had a problem, shared it, got feedback, and then made changes to your practice for the benefit of your students. That’s awesome! I know that teaching is a demanding job by itself, so it speaks volumes that you’re willing to take the time to reflect and make changes as you go. Kudos to you! :-)

  4. SDalton says:

    I loved what you came up with to review decimal operations. I have had the same feeling about teaching/reviewing fractions with 7th graders, that you described in your previous post about decimal operations. I will definitely try error analysis when we start working on fractions this year. One thing that I have used in my classes to review material taught in earlier grades is the Jigsaw technique. I break the topic into 4 subtopics and students leave their original group to form a new group that focuses on one topic where they work together to become specialists on their assigned topic. I give them materials to review and example problems. Then when they are ready they return to their original or jigsaw group and each student, who has become an expert on one of the four subtopics, makes a short presentation to their group.
    Thanks for sharing your ideas!

    • ray_emily says:

      Oooh!! YES! Definitely doing this, soon. We will jigsaw-it-up within the next week, I suspect. It’s funny / interesting to me how these solutions feel so obvious, once we’ve thought them up – you know? It’s just a matter of being will to let go of something that doesn’t work, I think.

      That said – teaching fractions is my favorite. =) Let me know how the error analysis goes! Hope you enjoy it.

  5. Sarah says:

    I love this idea! I have finally started using individual white boards in my classes – so far the kids love them. This week we did a short lesson ( mostly examples) on scientific notation, then I told them to try the practice either in their books or on whiteboards. I teach grade 7 and 8 gifted kids and try to avoid too much drill type homework. They respond really well to the idea that I care more about them spending a decent amount of time practicing a concept than whether they complete a certain number of problems.

  6. […] I wrote about my least favorite unit, as well as my first attempt to improve it. While working with kids on division with decimals, recently, I attempted once again to step up my […]

  7. Amy Zimmer says:

    Great post…sorry just catching up from Blogger’s Initiative. I am appreciative of your risk-taking and sharing the ups and downs with us! I getting more and more inspired to talk less!

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

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

My Web 2.0 Journey

Some thoughts from a middle school math teacher.


Some thoughts from a middle school math teacher.

Always Formative

Some thoughts from a middle school math teacher.

the radical rational...

in search of innovative ideas with a well-balanced approach for the math classroom

The Number Warrior

Assorted posts on mathematics and education

Some thoughts from a middle school math teacher.

Teaching Statistics

Some thoughts from a middle school math teacher.

square root of negative one teach math

Some thoughts from a middle school math teacher.

Pi Crust

An accomplished novice wonders about the challenges of depth over breadth. For musings on parenting and technology, check out

{Musing Mathematically}

Some thoughts from a middle school math teacher.


Some thoughts from a middle school math teacher.


Some thoughts from a middle school math teacher.

Megan Hayes-Golding

Still learning.

Mathy McMatherson

Bloggy McBloggerson

Math Mama Writes...

Some thoughts from a middle school math teacher.

Math Goes Pop

Some thoughts from a middle school math teacher.

Lines and Lines of Tangency

mathematical asides

Learning to Fold

a work in progress

I Speak Math

Integrating Technology and Mathematics

I Hope This Old Train Breaks Down...

Some thoughts from a middle school math teacher.

Some thoughts from a middle school math teacher.

Some thoughts from a middle school math teacher.


Some thoughts from a middle school math teacher.


Some thoughts from a middle school math teacher.

%d bloggers like this: