My trainee maths teachers love setting students challenging, interesting problems. I see students scribbling away trying to be first to solve it, or staring at the table, or scribbling away pretending not to stare at the table.
The feeling that the trainees (and I) get when the first couple of students solve it is great. That’s the seduction, but it’s important to take a wider view.
First, the students who solve it may benefit. I am uncertain. Cognitive Load Theory shows that working with a high cognitive load inhibits learning.
Second, teachers typically wait until less that ¼ of the class have solved the problem before going over it. The time spend trying to solve it is possibly wasted for 3/4 of the class.
Third, if the successful students learn something, the gap between those who solved it and those who didn’t is now wider.
An evidence-based strategy to teach problem solving is to teach the method. Cognitive Load theory shows that worked examples and completion problems are effective teaching strategies. We are using Paul Bambrick’s Get Better Faster as a training handbook, which recommends (as does Lemov in TLaC 2.0) I do/We Do/You Do. This is a structured method for using worked examples and completion problems. You will need several versions of your problem.
I Do: The teacher models the solution to problem version #1. The important point is that you are teaching a problem solving strategy, not finding the specific answer to a specific problem (who cares how many jam-jars John needs?)
We Do: The teacher shows problem #2. The problem is solved as a class. You could use Q+A, cold call, hands up, TTYP (turn to your partner). The important thing is that the two problems are very similar.
You Do: Students solve a very similar problem using the same method.
Deep Structure
The point of teaching this problem is not to find the answer (no one cares how many jam jars John needs). The point is to teach a problem solving strategy for students to commit to their long-term memories. The first step is learning the method (it is knowledge, just like any other fact – all of the tricks of cognitive psychology are relevant – retrieval practice, spaced practice, interleaving).
The second point is about deep structure. Deep structure is the underlying maths of the problem. Deep structure won’t be obvious to your students because the surface features hide it (the jam jars). The solution you taught has many applications, but at the moment your students only know how it applies to questions about the context of your model problem (jam jars). The next step is to show them how the same strategy can be used to solve hundreds of different problems, many of which don’t look similar.
The way to teach deep structure is to repeat the I Do/We Do/You Do cycle using dissimilar problems with the same deep structure. The more of these you can teach, the more likely your students will identify the appropriate solution to a novel problem.
Example from NRICH (https://nrich.maths.org/34)
There’s a room in your school that has three tables in it with plenty of space for chairs to go round. Table 1 has one block of chocolate on it, table 2 has two blocks of chocolate on it and, guess what, table 3 has three blocks of chocolate on it.
Now … outside the room is a class of children. Thirty of them all lined up ready to go in and eat the chocolate. These children are allowed to come in one at a time and can enter when the person in front of them has sat down. When a child enters the room they ask themself this question:
“If the chocolate on the table I sit at is to be shared out equally when I sit down, which would be the best table to sit at?”
First I adapted the problem so that I had at least three versions, all more or less the same difficulty.
| Problem version | Table 1 | Table 2 | Table 3 | Number of children |
| 1 | 2 | 1 | – | 6 |
| 2 | 3 | 2 | 1 | 6 |
| 3 | 2 | 3 | – | 10 |
| 4 | 3 | 2 | 1 | 30 |
Demonstrate problem #1, solve problem #2 as a class and set problems #3 and #4 as independent tasks.
Deep structure: to begin to get students to get the deep structure of the problem, I change the context a little: herons and fish ponds, sofas and armchairs at a party, bottles of lemonade (500ml, 1l and 1.5l). These are used on subsequent days and weeks.
So if you want your students to learn from the tree of maths knowledge, don’t be tempted to set tricky problems and let them get on with it – teach the solutions first.
Reading for Learning 