The Problem Solving Methods Learners Use – and How to Develop Them

Solving problems, especially in physics, has been well studied (e.g. Larkin, McDermott, Simon and Simon 1980: here and Chi, Glaser and Rees, 1982: here). There are a number of strategies typically employed by novices and experts. In this post I am describing two approaches, one typically used by novices and the other by experts. I then describe three proven strategies to help develop expert problem solvers.

I have used an example problem to illustrate the two problem solving approaches and the training techniques. 

Example Problem

A block of niobium, a metal with density 8570 kg/m3 , has sides of length 3 cm, 4 cm and 5 cm. What is the maximum pressure that can be exerted by this block when it is stood upright on one of its faces? 

1. 4.2 kPa 
2. 430 Pa 
3. 2.6 kPa 
4. 510 kPa

Taken from the Oxford University physics aptitude test (2006). Physics Aptitude Test

Means-Ends: Backwards Reasoning

The means-end strategy is an effective problem solving strategy when you don’t know how to get to the answer. You start with the end and move backwards. Below is an example of means ends backward reasoning to find the solution of the niobium problem.

1. You know you need to find the pressure of the block on the ground, so you’ll probably need P=F/A. But you don’t know F and you don’t know A.
2. You decide to start with F. You know F is probably the weight and you know weight=mg, so you’ll need to find m.
3. You know from the question you’ve got the density and you know the formula ρ=m/V.
4. You know you can work out V from the dimensions (lbh) and so you can rearrange the density equation and find the mass. You’ve come to the end of this trail. You can now go forwards again, plugging in the values until you’ve got F.
5. Next you need to work out A. If you know that for the highest pressure, you need the smallest area, you’ll be able to short cut and go straight to 3×4. Otherwise, you might need to work out all of the areas and then divide F by each until you find the greatest P and you have your answer.

If you know about cognitive Load Theory, you’ll know that holding these steps in your head is hard, if not impossible, even if you can do each step easily. So you’ll probably reduce the cognitive load by scribbling down notes on paper. If this problem is at a learner’s capacity, she’ll have insufficient cognitive resource left to put the problem into her long term memory. She may solve the problem but learn nothing. This is really important. Giving pupils lots of problems, at or near their ability is an inefficient way to teach.

Expert Problem Solving

Experts solve this problem quite differently: they use memory. They have solved similar questions before and can remember all or most of the sequence as chunks:

1. Find the mass using the volume and density.
2. Find the weight of the block.
3. Divide the weight by the area of the smallest face.

That’s it. If the expert can’t remember all of it, she’ll have chunks of it in her long term memory, reducing the cognitive load.

We glorify IQ and ‘intelligence’ but these only apply to means-end problem solving. A learner who is trained in problem solving will out-perform a novice learner with higher ‘intelligence’. It is our job as science teachers to support learners build chunks of problem solutions in their long term memories. Problem solving for experts is memory.

Just so you know the scale of the problem, it’s been estimated that chess masters can recognise tens of thousands of board arrangements (here) – the same order of magnitude as the number of words in a graduate’s vocabulary. Perhaps we should be treating problem solving ‘chunks’ as seriously as we treat vocabulary.

Improving Student Problem Solving – Three Applications of Cognitive Load Theory

#1 Worked Examples

The teacher models the process, taking away the cognitive load of navigating the problem and leaving mental resources for the learner to commit the problem to long term memory.

1. I know the greatest pressure will be when the area is smallest, so the block will need to be on its end. Therefore the area will be 3×4 = 12 cm² = 0.0012m²
2. The force is equal to the weight of the block. I can work out the weight by first finding the mass and then multiplying that by g to get the weight. So…
   1. m=ρxV = 8570 x 0.03 x 0.04 x 0.05 = 0.514kg
   2. F=mg = 0.514 x 9.81 = 5.04N
3. I can now calculate the pressure: P=F/A = 5.04/0.0012 = 4.2KPa

(The MCQ answer at the start is answer 1.)

Completion Problems

The next step after modeling is to set scaffolded problems ‘minimally different to the fully modeled problem. For example , you might change niobium to molybdenum (ρ=10200 kgm^-3) and provide the same script:

1. The greatest pressure will be when the area is smallest, so the block will need to be on its end. Therefore the area will be ______ = __ cm² = ______m²
2. The force is equal to the weight of the block. I can work out the weight by first finding the mass and then multiplying that by g to get the weight. So…
   1. m=ρxV =________ x _______ x_______x______ = ________kg
   2. F=mg = ______x ______= _____
3. I can now calculate the pressure: P=F/A = ______ / ______= _____Pa

Do this several time over several lessons – changing the material, the dimensions and the shape of the block.

Goal Free

A method of exploring the problem space (the possible problems and solutions associated with a situation) is by removing the goal of the question. The learner then simply writes as much relevant information about the situation as she can.

A block of Niobium, a metal with density 8570 kg/m3 , has sides of length 3cm, 4cm and 5cm.

You can find…

• the volume
• the mass from the density and volume
• surface area
• it would sink in water
• is niobium an element? Can I find it on the periodic table?
• It’s a metal so it would conduct electricity and heat.

etc

The point is not that the learner would necessarily find the original question and solve it accidentally. The point is that without the load of finding a solution to a problem, the learner is free to develop associations and schema which may be helpful in solving problems in the future.

This may look like daydreaming, but Sweller showed that this activity lead to improved problem solving (here).

(I first wrote about this 3 years ago here – I still think it’s really important!)

A Final Note: Expertise Reversal Effect

Once your learners reach a level of expertise, scaffolding can actually increase cognitive load. Your learners already understand the problem but have to interpret and implement your method instead of theirs. This introduces additional unhelpful cognitive load.