Yesterday I came across this quote from a podcast interview between Professor Anna Stokke from the University of Winnipeg and Professor Emeritus John Sweller, best known for formulating Cognitive Load Theory.
This seemed too binary to me. That a problem can either be impossible or simple depending on the schema that a person has seems to belie the complexity of how learners develop mathematical knowledge. Surely, there must be points where the solving is difficult, but achievable, and this difficulty lessens over time.
One example given in the podcast is a pair of simultaneous equations: x + y = 5 and 2x - y = 8. Now, of course, to someone who knows lots about simultaneous equations, the path to finding the values of x and y here is relatively clear. As Anna said in the podcast, you would add the equations together to eliminate y, find the value of x and then substitute to find y.
Clearly one needs enough knowledge of algebra to even interpret the question. If I don't have some knowledge of the concept of x and y as unknowns here, I won't even understand what the equations themselves mean, never mind what asking me to 'solve the pair of equation' means.
However, there are many other ways to solve this pair of equations. The podcast mentioned trial and error. Although not efficient, trial and improvement is valid. Other numerical methods such as the Gauss-Seidel method are also valid. Alternatively, we could plot the two linear graphs and look for their point of intersection. We could employ matrix approaches involving the inverse matrix or reduction to row echelon form. If I know anything about any of these approaches, the problem is not impossible even though my schema may not contain any knowledge of solving pairs of equations using elimination.
What I think Professor's Stokke and Sweller mean by 'impossible' in this case is actually 'unreasonable to expect learners to do using the approach intended by the teacher'. This I have more sympathy with. If I offered that pair of equations to pupils with the intention of them 'discovering' elimination as an approach without having ever manipulated pairs of equations, I don't think many (if any) of them would work out the approach for themselves. I am not, nonetheless, in 100% agreement with Sweller and Stokke's point of view. I think that, if I did teach pupils more generally manipulating systems of equations - showing them how to add, subtract, multiply and the like single equations or pairs of equations - without the goal of 'solving' the pair, and then explained what it meant to solve a pair without modelling or exemplifying the approach, I think it much more likely that some pupils would then 'discover' the elimination approach.
To be fair, having listened to the podcast, I don't think Sweller or Stokke really think it is impossible anyway. The point they seem to be making is that it is simply not a good approach to ask learners to employ if the goal is them learning to recognise and appropriately deploy that strategy. The much more useful approach that will support more learners in achieving this goal will be to have an expert exemplify and model the approach, and then have learners practice application of what they have seen/studied to an increasingly complex array of carefully chosen and structured problems to support development of increasing fluency. This I do 100% agree with, given that the goal is the learner getting to the point where they recognise and can effectively deploy the strategy.
The bit that I have more of an issue with is that this should not be the only goal of a mathematics education - to be taught lots of strategies, how to recognise when to deploy said strategies, and then to deploy them automatically. In my opinion, there does need to be space created for learners at all levels to be able to grapple with uncertainty, deal with competing constraints, and examine the pros and cons of different approaches. There is a phrase that I first encountered in Colin Foster's
MT article that comes from Japanese "problem solving" lessons; 'the lesson begins when the problem is solved'. As teachers, part of our goals for a mathematics education must include opportunities for engaging with authentic problems, not simply questions which are very closely related to a single mathematical approach or result that has been recently taught. One can argue that questions like solving the pair of simultaneous equations given above, once pupils have been taught elimination as an approach, cease to be a 'problem' in a mathematical sense rather than simply questions that should cue a particular recognition and deployment. Indeed, at GCSE, such a question would be considered an AO1, "use and apply standard techniques" question, rather than an AO3, "solve problems within mathematics and in other contexts" question.
Contrast that with this question taken from the Corbett Maths website:
This problem has multiple possible approaches. Yes they all revolve around having equal amounts to compare - either equal volumes of Cola or equal values of money. However, there are a number of different volumes or amounts of money that could be in consideration here, 6 litres, 100 ml, 1 ml, 1p, 10p, £1 or more are all feasible. This is entirely the sort of problem I can see featuring in a Japanese style lesson, with the teacher introducing the problem and providing any necessary input around scaling of volumes and pricing or the like, before allowing pupils to approach the decision in their own way and generating meaningful discussion about how different approaches that pupils might take compare to each other.
Another issue to consider is the role that engaging with problems prior to learning an approach specifically tailored to the problem type might have on motivation. This is a complex issue. It might be that, in certain circumstances, having to consider problems for which the solution isn't obvious provides a motivation to learn the techniques that will make the problem easier to solve. Conversely, it may be that this negatively impacts pupil motivation if they feel the concept is too difficult to grasp due to early exposure with challenging problems. I also recall a phrase from Skemp here, 'well is the enemy of better'. If pupils are able to solve the initial problem using an inefficient but adequate strategy, they may be less motivated to move away from that strategy even if it is more efficient.
I think a lot of this is likely to do with how invested pupils are in the initial problem - either because of a positive attitude to maths in general or due to some 'hook' in the problem itself that piques pupil interest. I do believe, however, that the use of problems to motivate a need (or at least usefulness) to engage with new mathematical learning is one that is worth examining more fully.
In my response to the first Maths Horizons report I shared this image which I think contributes to the role of problems in learning maths:
Rather than problems being something that learners only attempt once fluency has been achieved, which is the view in some places, I strongly believe that teachers of mathematics need to recognise the cyclical relationship between developing fluency, reasoning and problem solving, This was based on a sesson I had done at a LaSalle MathsConf a couple of months earlier, where I shared this anecdote:
A personal anecdote:
It may just be me, but it seems that the pupils I am teaching are less prepared to try and think about an idea.
They seem to be expecting me to do all of the hard work, and show them every little aspect of everything.
I wonder if they have formed an expectation about maths lessons that all they have to do is sit and listen to the teacher and then try and do what they (we) do.
I wonder if this is because of their experience of maths lessons to this point.
Learning is effortful, and mathematical learning should require the employment of reasoning, deduction, conjecture and the like. I think that exposure to problems where the solution isn't obvious may have a role to play in impressing upon pupils that they are expected to think in maths lessons, expected to bring their reasoning skills alongside their prior knowledge to the table, and that mathematics learning is not simply watching the teacher do something and then try and regurgitate it.
So, what is the role of 'problems' in learning mathematics. I don't think there is a simple answer. They both build and test fluency. They affect motivation and provide points for discussion. They communicate something of what we value in learning mathematics and in seeking a mathematical education. This is probably what makes their role so debated and difficult - 'problems' and how they are used have many roles in the teaching and learning of mathematics and how and when to use different problems and problem types will depend a lot on the pupils you have in front of you.
