Learning through problem solving
Owning the problem
How can maths teachers teach basic questions of volume? How can they make learners want to understand?
Dan Meyer does it this way: he asks his students how long it will take to fill a large water tank using a garden hose connected to a tap. To make the problem real, he even starts playing a video of somebody doing it, then sits back and waits.
“It’s kinda agonisingly slow” he says. “It’s tedious… students are looking at their watches and rolling their eyes, and they’re all wondering ‘man, how long is it going to take to fill up?”
“And that”, he says with a wicked grin, “is when you know you’ve baited the hook”.
Filling a water tank - or a bath - is a classic problem in maths textbooks, and mentioning it may have given you painful flashbacks to school maths lessons. But some sort of magic seems to happen when it’s transferred from the page to Dan Meyer’s classroom. Because after a while, the conversation begins: how big is the tank? How could we measure its volume? What’s the rate of flow? How could we work it out? Suddenly, the students are engaged, curious, throwing out tentative theories and testing possible answers.
In fact, there’s no magic involved. The engagement happens because they start owning the problem.
Dan asks the shortest question he can, avoiding distracting conceptual clutter. The students haven’t been channeled into the textbook procedure for measuring volume – they’re free to solve it any way they think best (though in this case it’s likely they’ll come round to the classic method eventually). Indeed, Dan’s point is that his students formulate the problem themselves.
This is a great way of learning, but needs to be approached carefully. The utility of classic word problems, such as “you walked for 20 minutes at 5km/h, how far did you walk”, is questionable, perhaps because they’re given as a “bitter medicine” rather than an intriguing puzzle. It’s hard to ‘own’ a problem which begins with a lie!
On the other hand, an article explaining the digital currency Bitcoin by Michael Nielsen seems to work rather well as a series of problems. “So how can we design a digital currency?”, he begins. In case you think it’s easy, he offers a couple of obstacles straight off the bat – that if each coin is just a string of numbers, you can immediately start forging, and that it’s difficult to prove that you’ve paid somebody.
Michael then starts explaining how some of the problems can be solved, using an imaginary currency called Infocoin. “Our first version of Infocoin will have many deficiencies, and so we’ll go through several iterations … with each iteration introducing just one or two simple new ideas. After several such iterations, we’ll arrive at the full Bitcoin protocol. We will have reinvented Bitcoin!”
Learning as problem-solving isn’t new – Socrates guided Meno’s slave boy to discovering the area of a square via a series of puzzles - it’s just that more people are studying how it works and figuring out how best to construct and present the problems. Nor does it necessarily work in every domain: you could hardly claim that a concert pianist with ten thousand hours of practice under her belt has been solving problems except in the vaguest sense.
But when it’s constructed well, it’s an effective way of provoking engagement.
Many adults today are partly self-directed learners, deciding for themselves what they should study and devising their own learning journeys and timetables. That’s hard! They need to find a lot of their motivation within themselves. Presenting them with problem scenarios they can relate to and want to solve is one way to make it easier.