Learning Math is Not a Spectator Sport

In November, I gave the keynote at the American Mathematical Association of Two-Year Colleges (AMATYC) Conference in Denver.

I have given versions of this talk that are not specific for mathematics, but I don’t have recordings of those. I promise that the math in this talk is not inaccessible and is used more for examples than a framework for the talk. In other words, don’t let the word “math” scare you away. The alternate version of the talk is “Learning is Not a Spectator Sport.”

Three triangles surrounding a central triangle with the letters C, I, and D
The first half of the video is the awards ceremony, so I’ve directed the embed link below to begin when the keynote actually begins at 45:48 (direct link to video on YouTube beginning at the keynote is here).


The talk emphasizes the importance of interaction, and as such, this talk has a lot of audience interaction in it near the beginning, so you may want to jump through some of that interaction as you watch (between 51:30 and 1:02:00).

At the end of the keynote, audience members are invited to participate in a Weekly Teaching Challenge to continue exploring the ideas and research in the talk. You’re invited too. Just sign up!

 

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Why high contextual interference?

This week I followed a hunch and, with the help of a friend who is a music educator, dug into some additional research around this idea of blocked and random practice. In music there are a few goals to achieve with any passage of music:
  • can you play a passage accurately by itself?
  • can you play the passage in the larger context of the piece?
  • can you play the passage to tempo?
  • can you play the passage with the right expression?

Think about these goals in your own subject area and see if you can find a similar set of goals. For example, here are some potential goals for solving a math problem:

  • can you find the correct solution?
  • can you solve the problem in an elegant way?
  • can you prove your solution is correct?
  • can someone else understand your solution?

The first research paper I looked at was When Repetition Isn’t the Best Practice Strategy (2001), by Laura A. Stambaugh. A short summary is available here, though the original paper is a bit harder to get ahold of. I’ll elaborate a bit on the summary with the relevant points to our study of learning design.

Students were asked to practice three passages (denoted below in three colors) in either blocked- or random-formatpractice sessions. The three practice sessions were covered on three different days (denoted 1, 2, and 3 in the diagram below). The performance during the last three trials of each practice session were used as the baseline measure of comparison for the retention measures.

music-research
In this experiment, practicing “randomly” meant practicing the same three passages in either At the end of the three sessions, there were no performance differences between the two groups. However, when tested for retention, the blocked-practice students’ performance began to slow to the level of early practice in the trials. While the accuracy of the two groups of students was still the same, the random-practice students could now play the passages faster than the blocked-practice students. Stambaugh also tested transferability of skills, but did not find any statistically significant differences from this experiment. One other variable that Stambaugh thought to test was attitude towards practice depending on the research treatment (maybe students will really dislike random practice or blocked practice?). Here too, there were no statistically significant differences in attitude towards practice between the two student groups.

One of the reasons I find this article interesting is that it discusses the idea of contextual interference, the amount of cognitive disruption the learner experiences during practice with multiple tasks. When the learner has to redirect attention as the tasks change, this results in a high degree of contextual interference. When the tasks don’t change much (blocked practice), the brain can go into a sort of “autopilot” and stop paying attention. At this point, there may not be much point to practicing more on that day. Practicing the same things on a different day would have positive effect (that’s spaced repetition).

 

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Recorded Webinar: Teaching Math in 2020

Just realized I never shared this webinar video from 2014 (you know, back when 2020 still seemed pretty far away).

What Does Teaching Math look like in 2020?

With every new iteration of technology, we create a generation of students whose primary media “language” for learning and interacting with the world is different than the one before it. In the last 5 years, technologies like free online videos, personalized learning software, and mobile devices, have been chipping away at the corners of education and traditional teaching. Technology-enhanced learning is here to stay, and it will alter the face of education, like it or not. This webinar is your guide to navigating and thriving in this new world.

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AMATYC Keynote Notes: Challenge and Curiosity

In the 2016 AMATYC keynote, I covered three main themes:

  1. Interaction & Impasse (last post)
  2. Challenge & Curiosity (this post)
  3. Durable Learning

Here are references and resources for Challenge & Curiosity:

First, I have to point you to one of my favorite books on the subject, A Theory of Fun for Game Design, by Raph Koster.

Quote from Game Design: “How do I get somebody to learn something that is long and difficult and takes a lot of commitment, but get them to learn it well?” – James Gee

How do players learn a game? 

  • They give it a try
  • They push at boundaries
  • They try over and over
  • They seek patterns

It looks something like this:

Shows web of many nodes and branches coming off a person, with bridges between branches and potential paths to expand knowledge.

How does a player learn a game?

How do we teach students?

  • We tell them what we’re going to tell them.
  • We tell them.
  • We tell them what we told them.
  • We have them practice repetitively.

It looks something like this:

Very few linear paths branching out from the person at the center. Few nodes and few places to expand on knowledge.

How do we teach students?

Reference: Productive Failure in Mathematical Problem Solving

There’s a much wider body of research on productive failure worth reading.

Video: Playing to Learn Math

Resource: Good Questions from Cornell

Resource: Classroom Voting Questions from Carroll College

Design more activities that let the student figure out the mathematical puzzle, instead of providing all the secrets yourself.

Shows the graph of a rational function with vertical asymptote at x=5 and horizontal asymptote at y=2.

Explain the differences in the graphs: The student is given five rational functions to graph, each function looks only slightly different mathematically but produces very different results.

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Interdisciplinary Courseware to the Rescue?

In the midst of all the bling of media-rich, adaptive, personalized, [insert-buzzword-here] digital products, there is a lurking underlying problem:

The general education curriculum in higher education has barely changed. Today’s world is cross-disciplinary, culturally diverse, and team-oriented. There is almost no problem that can be solved in a silo content area with a team of one.

Map showing the interconnected nodes between a variety of subject areas in research.

Interdisciplinary Thinking, from New Scientist’s article “Open your Mind to Interdisciplinary Research”

We need new cross-disciplinary curriculum. We need courses that are more engaging and reflective of today’s real issues. We need courses like these (referenced from my 2009 post on Hacking Higher Education):

  • Trend Analysis (Math + History)
  • Biology and Human Enhancement (Biology + Philosophy)
  • Science of Exercise (Science + Health & PE)
  • Exploring Water Issues (Science + Politics)
  • Design and Digital Presentations (Graphic Design + Communication)
  • Data Analysis and Information Presentation (Statistics, Graphic Design, and Communication)
  • Exploring Recycling and Refuse (Science, Government, and Humanities)
  • Chemistry of Nutrition (Chemistry + Health & PE)
  • Poverty and World Culture (Humanities, Government, and Sociology)
  • Sociology and Psychology of the Web (Sociology + Psychology)
  • How Computers Think (CIS + Philosophy)
  • Art, Media, and Copyright (Fine Arts + Law)
  • Writing for the Digital Age (CIS + Communication + English)
  • Energy (Physics, Chemistry, and Government)
  • Information, Query, and Synthesis (Literacy, Logic, English)

The problem is that very few faculty can teach courses like this without extensive learning or teamwork, and very few authors that could write such a curriculum from scratch.

This is exactly the moment when “digital courseware” should rise to the occasion. Digital courseware could be built to support these kinds of inter-disciplinary courses with a well-designed learning experience (not just text, but formative assessment and designed interactions with students and faculty). It could be multimedia rich, adaptive, personalized, and all that good buzzword stuff.

With a solid digital courseware backbone to support the learning, faculty could be tapped from different disciplines to evaluate work, answer questions, and coach students in their learning. No one faculty member would have to learn all the nuances of the course immediately.

So why aren’t we getting that? Why are we just getting more Algebra, English Comp, and Freshman Biology courses? Because that’s what we keep asking for. We keep saying, “give us better pass rates for these courses we currently teach.” We keep funding the rebuild (and rebuild) of those courses that create retention and graduation pressure in higher education. What if the problem is not the delivery of the course, but in the course itself? What if students are never going to do better in these courses because deep at the heart of the issue, the student knows the course isn’t applicable to the world they live in?

The Big History course (funded by Bill Gates) is an admirable step towards creating a more modern and more interdisciplinary curriculum. MOOCs do not have to pay attention to credit counts, what “department” the course lives in, or how it will or will not count as an elective towards multiple degrees. Consequently, MOOC providers have the freedom to build interesting, modern, and cross-disciplinary courses like The Science of Everyday Thinking (from EdX) or Politics and Economics of International Energy (from Coursera).

But why is it outsiders to education that have to lead these efforts? Educators should begin asking for the “right” curriculum from courseware providers (looking at traditional publishers, digital platforms, and MOOCs). We need to ask for the curriculum we want to teach instead of that which we have always taught.

Of course, courseware providers aren’t going to build something they don’t think has a market yet – and so we have a classic “chicken and egg” problem. This seems like exactly the kind of problem that needs a funding push. If a beautiful digital course on “How Computers Think” or “Poverty and World Culture” became available nationally at a low cost, I’d like to think that institutions and faculty would be able to step up to the challenge of figuring out the rest of the logistics to offer these courses.

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