Celebrate the Errors in Math Practice

Jan 11, 2017 by

Dear math students,

As you work through your mathematics practice, I’m going to challenge you to embrace making errors in an entirely new way. 

Many students believe that every problem in math homework should be perfectly constructed with no errors. It might look something like this:

A nicely ordered problem solution with no mistakes.

 

But when it’s time to study after the initial problem run-through, what does this perfectly constructed problem say? Does it coach you on remembering how you struggled? Does it remind you where you made an error? No.

When you make an error as you’re working a problem, please don’t erase it from the face of the earth. Certainly you should learn from the struggle and complete a correct solution, but record your deviations from the straightforward solution path in another color. Leave yourself notes (also in a different color) to remind you what you should have paid more attention to the first time around. Maybe that would look something like this:

Worked problem with highlights and notes to self

 

 

Sometimes you’re going to recognize but you don’t have the right answer but you’re not going to be sure what’s gone wrong. You should always try first to figure it out yourself first. This process of error analysis in a variety of different situations is key to developing problem solving skills in mathematics. Without exploration of the problem space (which happens with error analysis), your brain is just recording rote procedures without the ability to transfer those procedures to new kinds of problems. It will, essentially, stunt your mathematical growth.

Now, I don’t want you to get to the point of tears or breaking your keyboard out of anger. If you get near that stage please just ask a question (email, discussions, chat-a-friend, etc), leave a sticky note on the page as a reminder to go back, and move on to the next problem or section of problems. Just switching to a slightly different problem can not only get you unstuck, but sometimes give you insights into the “stuck” problem. 

When you figure out how to do the problem you were stuck on, make sure to go find that flagged problem (remember the sticky note?) and annotate your corrections.

Worked math problem with cross-outs and restarts and notes to self.

 

Now you might be thinking “why do all these error corrections and problem annotating in another color?” When it comes time to study for your major assessments, you will be able to see the places where you stumbled the first time you tried the problem. These “notes to self” are the places where you’re most likely to make the same mistake again. They benchmark places to remember to be careful and show you problem types to repeat practice before an exam.

Celebrate your errors.

Embrace the messiness.

Learn from your mistakes.

Study from your struggle points.

And be great at mathematics!

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Algebra is Weightlifting for the Brain

Mar 29, 2010 by

This was my presentation on Friday in Austin, Texas at the Developmental Education TeamUp Conference.

The process of learning algebra should ideally teach students good logic skills, the ability to compare and contrast circumstances, and to recognize patterns and make predictions. In a world with free CAS at our fingertips, the focus on these underlying skills is even more important than it used to be. Learn how to focus on thinking skills and incorporate more active learning in algebra classes, without losing ground on topic coverage.

 

I’ve loaded the uncut, unedited video that I took of the presentation to YouTube.  I’m not going to claim the video recording is great (recorded with a Flip Video Camera sitting on a table), but you’ll get to hear the audio and more of the details.  View “Algebra is Weightlifting for the Brain” here.

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Hammer and Nail Problem

Aug 2, 2008 by

Believe it or not, I have been at MathFest for three days and have only managed to attend four sessions (besides the one I presented in). But I talk to a lot of people and talk to a lot of the exhibitors and because MathFest is a lot more research-oriented and focuses more on upper level math, there are actually not a ton of sessions that I am interested in – although, as is always the case, the sessions I did want to go to were all scheduled for the time I was speaking.
I haven’t seen Kien Lim since graduate school – I saw him walking down the street one night in Madison and recognized him. Kien and I used to be involved in some great discussions about how students learn. He invited me to his talk on the “Hammer-and-Nail Problem in Mathematics.” His talk (10 minutes) was briefly about his interest in math education and his experiences with students in Math for Elementary Ed. What he’s seen in teaching future teachers, is the same problem that I think we’ve all seen in different courses.

To a person with a hammer, everything looks like a nail.

Here are some examples that I’ve seen in math classes:

  • After you teach algebra students how to multiply polynomials, some can suddenly no longer add polynomials and will multiply an expression like (x + 3) + (x – 4).
  • In Calc II, I first teach how to use substitution to create new power series for functions. Later, I teach how to find a Taylor series. On the test, I always ask students to demonstrate finding some power series using the known series. But for many students, once they learn the Taylor series, it is the hammer they apply to everything.
  • In Calc I, students learn how to find derivatives using the power rule first. Later on, once they learn the quotient rule, they will try to find the derivative of 4/x using the quotient rule, rather than just rewriting the expression and using the power rule (which was the first, and simpler method).

For algebra, I’ve recorded hundreds of these types of hammer & nail issues in the Algebra Activities Instructor Resource Binders I recently authored. Once you know that the issues exist, you can try to change the hammer-and-nail thinking by continuing to present problems that make students think about alternate methods – or at least show them that using the most recent “hammer” is not the best way to do the problem. After seeing this enough times, the hope is that the students will begin learning to think for themselves.

One of the reasons that I love using the methods I outlined in “Back to the Board” is that I can instantly see which students that are trying to rotely apply the latest learned technique, and throw out new problems that use other techniques or tweaks until I see that they are no longer “thinking with a hammer.”

SIDE NOTE: These little 10-minute talks at conferences drive me crazy. Ten minutes is barely enough time to introduce a topic. If conferences are really going to use this format, then they should be sophisticated enough to have a website where presenters can link to a longer-format presentation that is hosted online. For some presenters, that might mean a set of slides from a longer presentation, and for others, it might be a recording (with slides) of the actual longer presentation. I just get annoyed with getting a 10-minute teaser and then not having instant access to the rest… which is why I always post my presentation prior to presenting. Just like our students, the best time to gain the attention of the audience is when they are interested in the topic – like when they go back to their room after the presentation.

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