: Using worked solutions to explicitly teach non-routine problem solving strategies Using worked solutions to explicitly teach non-routine problem solving strategies
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Using worked solutions to explicitly teach non-routine problem solving strategies
Using worked solutions to explicitly teach non-routine problem solving strategies
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by Tudor Grange Research School
on the
Tom Pole
Tom is Research School Director and Associate Principal at Tudor Grange Academy Solihull
In my previous blog, I wrote about the possible links between the way non-routine problem solving strategies are explicitly taught (or not) in schools and the impact this might have on tackling the disadvantage gap. This quote from Colin Foster sums it up well:
"Students with more advantaged socio-economic backgrounds are more likely to have received enhanced opportunities for support... to develop dispositions that allow them to make good use of open-ended opportunities, which other students may simply struggle with and learn very little from."
The challenges around teaching non-routine problem solving
In this blog, I’m going to suggest one approach that might be helpful in addressing this challenge; but first, a caveat. I’m going to offer what, to some, might seem like a counter-intuitive suggestion. To give this some much-needed foregrounding, I’m going to begin with a discussion about the barriers to teaching non-routine problem solving that teachers have shared with me. I think this might help you better understand the motivation behind my suggestion later in this blog.
There are many reasons why non-routine problem solving might not get the time it deserves in classrooms, but in my experience, much of the nervousness mathematics teachers experience about teaching non-routine problem solving is tied up in the pedagogical complexity associated with the support we can offer to pupils. Too little help and the pupils might flounder, too much and there’s no problem left to solve.
To compound this, we need to think carefully about how the support we offer students directs their attention towards the specific problem or the more general problem-solving strategies that we want them to retain.
“What we need to be doing is scaffolding the problem solving…..When we scaffold the problem, we direct students’ attention to particular features of the specific problem that they are working on. When we scaffold the problem solving, we give more generic support, which is intended to help students when they meet other similar problems in the future.”
We also need to consider how the questions we pose and the interactions we have with students influence their perception of themselves as mathematicians.
“The actions teachers and students perform construct different identities and meanings for mathematics students and mathematics teachers. These are not just constructed at the classroom level, but at the level of an interactional context within a lesson. Whilst many researchers define problem solving … in seemingly objective ways, … the meaning of these actions is … related to the interactional context in which they occur. This has significant consequences for how students make sense of what it means to do mathematics and what it means to be a mathematics student.”
Even when we get all this right, we might find ourselves with a classroom full of students with wildly different but equally valid solutions. Orchestrating discussions where everyone seems to be in a slightly different place can be a real challenge for both the teacher and the pupils.
The Pupil Perspective
So let’s take a moment to also consider this from the pupil perspective. By its nature, non-routine problem solving involves having no ready-made solution to the problem and therefore being a bit stuck. Mathematicians quite like this feeling, but for some pupils, this is not such a fun experience.
Getting unstuck requires, amongst many other things, creativity, mathematical thinking to identify the underlying structures, metacognition and a high degree of self-regulation. This can be both cognitively and emotionally challenging. Add to this a class discussion where no-one in the class seems to be quite where you are in your thinking and some pupils might find themselves at the limits of their cognitive resources.
Is it therefore reasonable, with this level of cognitive overload, to expect students to then begin to consider what the most efficient problem-solving strategy might be?
Comparing worked solutions to problems
With the right PD, approaches and careful preparation, we can learn to address these concerns, but I want to offer an additional, perhaps indirect, approach that we might also use to support students to deal with these moments more successfully. As I mentioned before, this might feel counter-intuitive, as despite what I’ve said above, I’m going to suggest that there is a place for providing students with possible answers to problems.
Yes, this feels strange because we know how valuable and enjoyable it can be for pupils to experience solving problems for themselves. However, that doesn’t mean there isn’t also some benefit and room for pupils to compare the strategies used by others to solve problems.
Why might this be helpful?
In the section above, I highlighted the cognitive load for pupils associated with both solving the problem and considering the suitability of the strategy used. Worked examples, used as a scaffold for teaching non-routine problem solving strategies, might remove some of this cognitive load:
"Worked examples... reduce the cognitive load for pupils so that they can concentrate on the specific steps involved."
This mechanism is well supported by wider educational research:
“not only did worked examples, as expected, require considerably less time to process than conventional problems, but that subsequent problems similar to the initial ones also were solved more rapidly… accompanied by a decrease in the number of mathematical errors…..It was concluded that for novice problem solvers, general algebra rules are reflected in only a limited number of schemas. Abstraction of general rules from schemas may occur only with considerable practice and exposure to a wider range of schemas.”
By considering completed worked solutions, we can potentially also side-step some of the pedagogical challenges linked to managing discussions where students have all devised different solutions. Knowing in advance the strategies used in the worked examples, the classroom teacher can plan more carefully the questions they will use to support students.
Perhaps a further benefit of this approach is that a curriculum leader can feel more confident that students have already been exposed to particular problem-solving strategies when considering what rich problem-solving experiences they might have in store for pupils later in the term.
An Unexpected Solution
So, to return to where we started. You might find offering the worked solutions to non-routine problem solving tasks an unexpected solution. I’m certainly not advocating this as a replacement for students experiencing rich problem solving themselves. But perhaps explicitly discussing and comparing the strategies used in worked solutions might go some way to levelling an uneven playing field, supporting all students to be successful problem solvers.
Look out for my next blog I’ll explore what this approach might look like on the ground.
Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2(1), 59 – 89.
Teaching problem solving strategies
Using worked solutions as a structured framework to teach specific problem-solving skills.
