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).

 

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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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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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Why prototype a digital course?

Very few of us would buy an unbuilt home without at least viewing a model home that conveys the look and feel of the interior and exterior of the rest of the community. We should be unwilling to build (or buy) an entire course (a “row” of units, modules, chapters, or weeks of content) without seeing at least one “model unit” first.

craftsman-exterior

From http://www.houzz.com/photos/36213135

In the software world, a low-fidelity prototype is used to give the look and feel of a future product. With this prototype there is some hand-waving (mockups) to explain away missing functionality and potential users are asked how they would navigate and use the product. This happens long before the product build, and is iterative.

In the learning world, we should consider that course builds (especially large-scale digital courseware) need the same kind of prototype.  Before the time and money is invested to build the a full course, consider building one unit as completely as possible, and make sure your stakeholders (students, faculty, instructional designers, deans, customers) actually want to learn in this course.  Choose a prototype unit that is most representative of the majority of the learning in course; this is usually not the first or last unit.

When the model unit is being designed and built, this is the ideal time to collaborate iteratively with students, faculty, IT, assessment, and instructional designers. While it will take some time to change the model unit as opinions shift, it will not take as much time as remodeling every unit in the course.

After you’ve got stakeholder approval for the model unit design, make sure to carefully document what features this prototype contains, since your team will need to apply it consistently across the full development. Here are just a few of the learning features you might want to apply across your multi-unit build:

  • content: where did it come from? what quantity per learning objective?
  • examples: how often, how relevant?
  • interaction: how much, what kind, and how often?
  • assessment: what kind? how often? authentic? purely for practice? for learning scaffolding?
  • images: for what purpose, how often?
  • videos: how long are they, what stylistic elements are there, how often do they occur?
  • simulations or games: for what purpose? how often?

As digital learning becomes more accepted (thanks MOOCs) and blended learning becomes a more standard model at traditional institutions, I hope we’ll see much more collaborative prototyping, followed by intentional design, in these courses.

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Idea Sex

Illustration by Mat Moore, Muskegon MI, 2010

I’ve just re-read the first few chapters of Steven Johnson’s “Where good ideas come from: The Natural History of Innovation” (you can watch the TED Talk or watch the RSA Animates video, but I highly recommend the slow read through the actual book) on the plane ride to Doha, and all week it’s been fascinating to watch the liquid network of minds at this event.  Johnson defines this concept of a liquid network as “bringing together a diversely-focused group of creative people.” Normally, I have quite a few ideas on any given week, but in this network, the idea generation has been fast & furious (even for me!).

Johnson also says that “good ideas come from the collision of smaller hunches” and here at the TEDxSummit, with 650 participants from 90+ countries, my smaller hunches are colliding with a whole new set of smaller hunches, and in particular, a whole new set of problem spaces.

The conversations and critical thinking at this event is much like playing video game in my brain (imagine the Angry Birds of idea formation).  With every conversation with a participant, a new “level” (problem space) presents itself.  I throw a collection of well-designed missiles (ideas) at it.  Sometimes it’s a direct and perfect hit.  Sometimes it’s not quite right, the problem space shows me the flaws, and I go for another throw.  Finding the solutions to problems is a lot like playing a video game.  The constraints of a new system actually birth more creative solutions.

One of the TEDx organizers, Eiso Vaandrager, used the phrase “idea sex” to answer the question, “What is TEDx?” and I think that’s about right.  And while the brainchildren of idea sex are brilliant and exciting, it’s important to remember and give credit to the parents of these ideas as well, and remember the stories of where and how those fires were sparked.

An idea is something powerful, and while an idea cannot be copyrighted or patented (only the form of an idea can be patented), it should be credited.  Without this, the TEDx community (and the other liquid networks you are a part of) will grow wary of sharing openly, and that would be a shame.  As we all go home and share the wonderful things we’ve learned, try to remember to give credit where credit is due.  The intoxicating new ideas we now carry were not generated in situ.

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