At the end of Term 1, we ran our student-parent-teacher interviews. Instead of having a parents’ evening for a each year group, once a week for several weeks, we complete all our interviews in one go and we suggest the students attend. This is not the only interesting aspect of my school. As you can probably imagine, we have adopted explicit teaching and a knowledge-rich curriculum and in one of my upcoming posts, I am going to explain how we teach mathematics. We also have an unusual method of curriculum improvement and staff-student relationships are, in my view, extremely positive. If any of this sounds interesting, you should check-out the series of positions we have recently advertised.
As I was conducting these student-parent-teacher interviews, I found myself making the same comment repeatedly. I would ask students to not spend more than five minutes stuck on a maths problem. Instead, I would suggest they circle the problem and raise it individually with me or in class. I have a box on the whiteboard where students can write-up homework queries at the start of the lesson or add a tick to a query someone else has raised. Usually, the same question causes problems for several students and so I can tackle them with part of the class as the other students work independently.
Instead of being stuck a long time, I ask students to move on and attempt other problems. I tell them that given enough time, I would back them to solve the problem but this is not efficient when they could spend that time in additional practice. I claim there is nothing to be gained from prolonged periods of being stuck, there is no medal available and no cash prize.
You may agree with me. If not, you probably believe one or both of the following:
Working through a problem and eventually figuring out how to do it leads to better learning.
Struggle builds resilience or some other non-cognitive ability.
As far as I am aware, there is no strong evidence to support the idea that figuring something out for yourself delivers learning gains.
One classic 2004 study by David Klahr and Milena Nigam investigated this hypothesis in the context of third and fourth grade students learning about the need to control variables in science experiments. Half the students were taught the principle directly and half were given an exploration task that facilitated their discovery of this principle.
As we might expect, fewer students learnt the principle in the discovery learning condition, but this is not the most interesting finding. Those student who learnt the principle were then given the task of evaluating science fair posters. These described two science projects other notional students had conducted and that did not properly control variables. It was an example of a transfer task because both groups had learnt about variables in the context of balls rolling down ramps whereas the science fair posters were about memory and how holes made in a ping-pong ball would affect its flight.
Critically, the students who had managed to discover the principle of controlling variables for themselves were no better at evaluating the science fair posters than those who had learnt it by direct instruction.
OK, so this research is not about maths problems. However, the burden really lies with those who argue an advantage to struggling with a problem for prolonged periods to demonstrate it. I am aware of little. By contrast, the benefits of further practice — retrieval practice — that students could complete by practicing other problems instead of being stuck are well documented.
So, what of the idea of building resilience? Again, it is an hypothesis in need of some supporting evidence. I guess that if a student does eventually solve the problem, this could be a positive experience. Or, it might not be — they may feel silly for not seeing the solution earlier. And what if a student spends 30 minutes on a problem and still does not solve it? May they perhaps grow frustrated? Maybe this will even feed into self-talk about not being any good at maths? I remember feeling this way at university after spending an hour trying to solve a physics problem.
As teachers, once we stray from the objective of teaching our subject and start pursuing psychological objectives which are out of our domain of expertise and are also, well, speculative, I think we are in trouble.
And maybe even five minutes of struggle is too much? I don’t know. Five minutes is a long time.
This reminds of a recent tweet I saw from a US parent whose 5th-grade son got a math puzzle for homework that no one could figure out (except those who searched the internet for the solution--or whose parents did). Kids were given no strategies or hints. The parent said "the idea seemed to be: stick with it. Own your frustration, etc. The kids were all having fits; parents were emailing each other." Obviously a big waste of everyone's time.
On the other hand, I've seen lessons (at least, non-math lessons) that clearly demand too LITTLE struggle on the part of students, leading to boredom and disengagement. E.g., a 15-minute lesson on "figuring out the topic" of a book chapter where the topic is stated right in the chapter title. Also a waste of time.
I always found and still find math puzzle problems to be a lot of fun. I had a knack for them in high school and got a lot of pride from it. Looking back at my k-12 math education though, I would have learned a lot more if someone told me to read more interesting math books instead of trying to figure everything out myself. Then I could have puzzled over more interesting problems at a younger age.