The Law of Unintended Patterns

For any matched pair of non-trivial examples
there exists (n == 1) pattern that the creator of the examples intended to highlight
but there also exist (1 < n <= infinity) unintended patterns that students will find.

It’s difficult to live-code programming examples… the conventions we use by habit often invite students to find the unintended patterns.

As an instructor, how do I get students to see the single pattern in which I’m interested, rather than the possibly infinite patterns that exist? Or, is that even the best goal? Should I, instead, be encouraging students to look beyond the first pattern they detect in order for them to appreciate the inherent complexity of interpretation?

"Will this be on the test?"

Students reasonably need to understand what is expected of them in a course. Educators need to make clear what is acceptable and unacceptable student engagement with a course. The syllabus is the natural place for this to happen, as long as both students and educators recognize it for what it is.

Students shouldn’t approach the syllabus as the maximum they’ll do… education is about expanding your horizon! The syllabus is the absolute minimum you should expect to do; the engaged and interested student will use it as a lower bound, not an upper bound.

“It’s the model that matters!” — Eric Mazur

At the ICER 2011 keynote, Eric Mazur reported that when students see a demonstration and either do or do not engage in a discussion of the demonstration, they adjust their memory to fit their model1. In other words, they retain their prior (possibly non-cononical) mental model and mis-remember the facts of the event to fit that model, rather than updating their mental model to account for the new facts2.

In physics education, given the following modes of instruction

  • No demonstration
  • Demonstration to students
  • Student Prediction without discussion
  • Student-to-student Discussion (similar to peer instruction)

students do equally poorly on a standard instrument intended to assess students’ understanding of Newtonian mechanics.

So, if we assume that we can’t skip demonstration altogether, and if we can’t just demonstrate, and if demonstration followed by discussion all suffer this fate, then what can be done? Engage students directly.

It’s not the act of predicting or discussing a prediction that triggers changes in student mental models, but rather confrontation with confusing experiences: staking ones intellectual ground so that one knows what one believes, then being confronted by a confounding example, and finally needing to substantially defend and explain the new experience.

Confusion seems to be an essential part of the learning process, or at least the ability of students to reflect and express their confusion3. In a physics class where students were asked to report on what they were most confused about each week, those who expressed confusion did much better than students who claimed no confusion. Willingness to express confusion positively correlates with understanding4.

So, in a peer instruction environment, we teach by questioning, not by telling or showing. We facilitate students’ engagement with the material rather than their obedience while in our classroom.

There is work on students’ use of mechanistic reasoning (i.e., trying to articulate the underlying entities, entity properties, activities in which entities engage, and the mechanism by which those activities give rise the the phenomena of interest) in physics and math education by David Hammer (now at Tufts), Rosemary S. Russ5 (now at Northwestern), Andrew Elby, Ayush Gupta, and Brian Danielak that relates to this… how students express their understandings of and reasoning about mechanisms underlying physical phenomena.

In short, if we’re not changing students’ mental models, than any learning that may occur is shallow and fragile. Some modes of instruction have a better chance of engaging students and changing their models, but unfortunately not the most popular modes of instruction, currently.

  1. Mazur, E. (2010) International Computing Education Research Conference (ICER) Keynote. Providence, RI. Slides available from http://mazur.harvard.edu/talks.php
  2. The keynote is also discussed by Mark Guzdial on his blog at http://computinged.wordpress.com/2011/08/17/eric-mazurs-keynote-at-icer-2011-observing-demos-hurts-learning-and-confusion-is-a-sign-of-understanding/
  3. see the Dunning–Kruger effect http://en.wikipedia.org/wiki/Dunning–Kruger_effect
  4. forthcoming from Mazur, E., et al
  5. Russ, R. S. (2005) A Framework for Recognizing Students’ Mechanistic Reasoning. A dissertation available from http://drum.lib.umd.edu/handle/1903/4146

Making Values and Culture Manifest and Manifold

Over at his blog, Mark Guzdial has raised questions about the ability of (a) curricula and (b) instruction to be value-/culture-neutral. I wonder whether it isn’t more important that they be manifest and manifold in education.

In other words, we need value transparency, to express the values and cultural biases in our designs clearly and publicly. When we choose what learning outcomes to include in a curriculum and when we create instructional plans intended to help learners attain those outcomes, we make value choices based on our own prior experience, and often do so unconsciously. Probability examples that rely on a 52-card deck and programming exercises that remake western-style games are necessarily rooted in our past experience. That implies that some learners– those who don’t share our experiences– will have a higher cognitive load when faced with these tasks, working to attain not only our intended learning outcomes, but also to build knowledge and skills related to the new (to them) problem context.

We need to be sensitive to this and provide the supports necessary to promote success. One way to do this is to represent core ideas in multiple ways, creating banks of culturally diverse, parallel examples of instruction that speak to the same set of intended learning outcomes. For example, do we need probability examples to rely on dice, cards, and coins? How else might one think about probability, assuming that those objects aren’t part of your daily life?

I can imagine a rich collection of activities, presentations, etc. that could be used not only as teaching aids, but also as tools to train teachers about diverse ways to represent ideas. Even within my own cultural context, I find myself often looking for new ways to introduce learners to a topic (nifty assignments, anyone?).