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?

Fork me on GitHub

"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

My Classroom Rules

Recently, several students commented that I seemed to have a lot of classroom rules. This is an old refrain in my life, and, in a sense, it’s true. However, the rules I have are all just special cases of my basic three rules, which I share on my About Me page.

  1. If you are going to break the rules, don’t be obnoxious about it.
    • If you can’t be engaged, don’t distract others. It’s unfair to both you and them.
  2. Don’t disappoint me.
    • Don’t promise to focus, but fail to do so. Instead, acknowledge whatever is distracting you and address it.
  3. Be aware.
    • Know what questions your classmates are asking.
    • Recognize which questions are related to tweaking the solution and which are related to a different problem context.

Wil's Classroom Rules
A full sized version of my rules diagram

I think my biggest failing in the classroom is that I’m uneven in the application of the rules, which is perceived as me being arbitrary. Inconsistency and randomness seem very similar to the outside observer.

I sometimes let feature creep take over the problem statement, which can lead to unintentional complexity or student confusion as the problem changes. I need to spend more time up front specifying the problem completely with students so that it’s clear to them and me what the invariants are.

I also find it difficult to ask a student actually to leave the classroom. I’m forever optimistic that the unfocused student will find moments of clarity and engage with the course material. Often, they do, but unfortunately, while I’m waiting for that to happen, the class as a whole is affected and, generally, material isn’t covered as concisely, clearly, or completely as might have been the case otherwise, thereby disadvantaging the other students who could have gone further, faster. Such is the nature of a set of random people with diverse metacognitive skills and needs. Still, I’m certain that I could serve better both ends of the spectrum.