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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