I teach university courses in mathematical problem solving at St. Francis Xavier University during my Summer break. The classes involve initiating numerous problem solving episodes and then interrogating and filtering our collective experience through the lens of current theory in the field. This structure provides plenty of opportunity to workshop new ways to launch tasks, and this year, I began experimenting with a new sort of launch routine that had pleasant results. This post is an attempt to reflect on why that may have been the case.
First, however, you must indulge me by responding to a prompt in five seconds or less.
Which number from one to one-hundred has the most factors?
Write down your gut feeling and then a sentence or two describing why you think you went with the number you did.
I hope you felt rushed; that was the point. My goal was to pressure you into making a quick, intuitive decision–one based on your history of mathematical experience. I don’t think there are enough of these opportunities in math class. Let me explain.
Many years ago now, I was introduced to the notion of System 1 and System 2 thinking theorized by Daniel Kahneman and Amos Tversky (and then later popularized by Kahneman in the book, Thinking, Fast and Slow). In a nutshell, the theory posits that human cognition exists in two separate “systems”, and together the two systems compose the gamut of human responses to stimuli in the environment.
- System 1: An automatic, unconscious process responsible for actions/decisions that occur near-instantaneously
- System 2: A rational, conscious process responsible for actions/decisions that take deliberate concentration.
The influence of Kahneman and Tversky can be found across many disciplines–including probability education (which is where I got my introduction). A humble attempt to summarize the general trend across the paperbacks written to popularize the psychological ideas might go as follows:
- We want to harness the elusive, intuitive power of System 1, but it is to be cautiously trusted.
- The reliability of System 1 increases alongside the expertise of the agent.
Some researchers are overly-skeptical of System 1 thinking, painting it as brash and easily-manipulated (Kahneman and Tversky can be put in this camp), while others move the study outside of the laboratory and into the field to demonstrate that power of System 1 as compared to the clunky and indecisive System 2 (My personal favourite, Gary Klein, fits here).1
Ultimately, there seems to be tentative agreement that System 1 is more stable as an agent’s expertise increases. This got me thinking (well, got my System 2 thinking) that engaging a student’s intuitive response to a mathematical scenario might be an interesting entry to their expertise. After all, most of the studies supporting this two-systems thinking theory are designed to sponsor an intuitive response and then analyze that response afterwards. (Interestingly enough, many of the studies involve mathematics–largely arithmetic and logic). Why can’t we borrow this framework to launch tasks in mathematics class?
1. Safety. Sometimes you can amplify one thing by constraining another. In this case, restricting time amplifies academic safety. There is no room for polished mathematical argument yet, only gut feeling. Gimmie what you got.
2. Starting points. Whether or not they are reliable, our intuitive responses are rooted in something. These seeds of argument then become the starting points for discussion and interrogation. Then, as our experience and expertise develop they become more refined and reliable. Or as Gladwell puts it, an increase in “knowledge gives their first impressions resiliency” (Blink, p. 186).
3. Reflection. Starting intuitively almost demands that we fold back after our expertise develops–under the guidance of System 2. After some deliberate interaction with the topic, it feels natural to return to our original gut feeling armed with a new sophistication and precision. Ultimately, it normalizes the making and updating of conjectures–two actions that can feel pretty risky when the weight of the red pen implies that all math be done entirely with System 2 thinking.
I’ve found that students are sometimes surprised and sometimes validated by their intuitions. When given the same prompt I gave you at the start of this post, I had one student choose 50 because they felt that multiples of 10 seemed to have lots of factors.
That isn’t wrong; that’s interesting. What about the multiples of 10 gives that impression? When can we trust it? Exploring the variety of intuitive responses is (partially) what gives this launch technique intrigue
I’ve found this sort of launch to work best when a response can be stated relatively simply. So in the example above, the response is a single number between one and one-hundred. Another example might be: “How many 7s are there between 1 and 1000?”
Or: Some animals are playing Tug of War. These two matchups were both ties.
Who will win this matchup?
In five seconds or less, Write down your gut feeling and then a sentence or two describing why you think you went with the response you did.
I’ve found the simplicity of this sort of task launch to be beautifully generative. Give it a try, if your System 1 feels it’s worth a shot
- If you are interested in diving into some of the ideas deeper, Malcolm Gladwell’s, Blink explores some of the ideas in a light, journalistic style.