Cognitive Psychology and Math Education

In the last three weeks I’ve read or skimmed about 2,000 pages of scholarly articles about math reform efforts, technology for teaching, innovations, change movements, faculty development, community college statistics, learning theories, and distance ed statistics. If you had any idea how many topics I want to blog about every day, but don’t have time for right now …well, just forgive me for lame posts for a little while, okay?

In the meantime, you can read this paper called “Applications and Misapplications of Cognitive Psychology to Mathematics Education” by Anderson, Reder, and Simon.  The full text is available at the link.  Interestingly, this was never published, although a similar article (not specific to math) was published.  I’ll warn you that I’ve read a few books on cognitive psychology, and it was still a difficult read because of all the terminology.  However, I assure you, it’s interesting reading.  If you find the first few paragraphs daunting, try skimming to where you read an applied example, then back up to read the section before it.  It’s easier to get the vocabulary when you have a concrete example in your head to pair with it.

As a math teacher, I use a lot of student-centered learning strategies.  I incorporate technology into my classes and emphasize the rule-of-four.  However, I still insist on students learning some procedural skills (like derivatives and integrals) because I think there’s value in learning how to sort problems by their structure and carefully pick apart which changes to structure mean a change in technique.  I found this paper validated that some of our “tried and true” teaching methods are not (according to cognitive learning theories) as “evil” as we may have been led to believe.  Nor does the paper discount newer techniques.  It simply brings new and old techniques into a sane balance.  Likely, this is a balance that many of us have reached on our own, but it’s nice to be given some practical examples and the proper vocabulary for discussing the pedagogical innovations.

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Teaching Math with Clickers

Today’s guest blogger is Derek Bruff, Assistant Director for the Center for Teaching at Vanderbilt University. Derek writes a blog you may have stumbled across called Teaching with Classroom Response Systems.

Here’s a question I ask the students in my probability and statistics course:

Your sister-in-law calls to say that she’s having twins. Which of the following is more likely? (Assume that she’s not having identical twins.)

A. Twin boys
B. Twin girls
C. One boy and one girl
D. All are equally likely

Since I ask this question using a classroom response system, each of my students is able to submit his or her response to the question using a handheld device called a clicker. The clickers beam the students’ responses via radio frequencies to a receiver attached to my classroom computer. Software on the computer generates a histogram that shows the distribution of student responses.


I first ask my students to respond to the question individually, without discussing it. Usually, the histogram shows me that most of the students answered incorrectly, which tells me that the question is one worth asking. I then ask my students to discuss the question in pairs or small groups, then submit their (possibly different) answers again using their clickers. This generates a buzz in the classroom as students discuss and debate the answer choices with their peers.

After the second “vote,” the histogram usually shows me that there’s some convergence to the correct answer, choice C in this case. This sets the stage for a great classwide discussion. I usually ask a student who changed his or her mind to share their reasoning with the class. I then invite other students to defend or question particular answer choices. I usually wrap things up by drawing the appropriate tree diagram on the board, which helps to explain this question and introduces to the students a visual tool they can use to analyze similar probability questions.

Why use clickers to ask a question like this one? There are several reasons. The histogram generated by the classroom response system I use gives me useful information on my students’ learning. If the histogram shows me that the students understand the question, then I can quickly move on to the next topic. If the histogram shows me that the students are confused, then I can drill down on the question at hand, using the popular wrong answers to guide the discussion.

Asking for a show of hands might provide similar information, but students answering a question by raising their hands don’t answer independently. They look to their peers as they answer, which means that the distribution of hands I see doesn’t accurately reflect my students’ understanding. Using clickers allows my students to answer independently, yielding more useful information I can use to make teaching decisions.

While individual student responses are visible to the students, they are visible to me. This means that I can hold students accountable for their responses by counting clicker questions as part of the students’ participation grades. Clickers provide me a way to expect each and every student to engage with the questions I ask them during class, not just the students who are quick or bold enough to volunteer answers verbally.

Moreover, showing the students the distribution of responses can enhance the classroom dynamic. When students see that two or more answers are popular, they become more interested in the question. When it is obvious from the histogram and from what I tell my students that most students answered a question incorrectly, students are more ready to hear the reasoning for the correct answer.

Classroom response systems can be very effective tools for engaging students during class and for gathering information on student learning useful for making “on the fly” teaching choices. Resources you may find helpful for using clickers in your mathematics courses include the following.

Project Math QUEST – This NSF-funded project out of Carroll College has generated clicker question banks for linear algebra and differential equations. Their Web site also includes question banks for precalculus and calculus courses. Their resources page features links to over a dozen published articles on teaching math with clickers.

• “Clickers: A Classroom Innovation” – Here’s a longer article on teaching with clickers I wrote for the NEA’s higher education magazine, Advocate.

Teaching with Classroom Response Systems Blog – I use my blog to discuss research on teaching with clickers, case studies of clickers in the classroom, conference sessions on clickers, and other resources. The blog is a companion of sorts to my book, Teaching with Classroom Response Systems: Creating Active Learning Environments, available from Jossey-Bass on February 17th.

Thanks to Derek for the post! With any luck, I’ll be back in the USA this afternoon and we will resume our regularly scheduled programming.

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Speed Rounds: My Test Review Game

Surprisingly, I find I haven’t ever blogged about this game we play on Test Review days. Here’s how it works. I count the students off into groups of 3-4 students. Each group gets an answer sheet for the game (a piece of colored paper with a letter, A, B, C, D, …) at the top. I make a “scoreboard” on the board to tally the results of the rounds (12 in this case). Here’s what that looks like:

Then we begin the game. Here’s a sample [Read more...]

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Activities for Algebra – Sample now available

I have always tried to teach algebra in a way that is understandable, interesting, and active. Although you can find activities for teaching algebra to younger students (6th and 7th graders), I have never found a good classroom resource to use with adult students. Even in the resources for younger students, the activities always seem more like busywork, when they could provide an opportunity for students to truly understand difficult concepts or explore the similarities and contrasts that abound in mathematical procedures and ideas.

Today, I can finally release some of the activities that I have been writing all summer that will be available from Cengage Learning (used to be Thomson/Brooks-Cole) to accompany the Tussy/Gustafson 4ed Algebra Series. These are activities to be used in the classroom [Read more...]

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