Below is the writing I undertook in preparation for my presentation at ResearchEd Rugby 2017. I have to say that despite the incredible amount of nerves, both in the initial writing of my talk and during the course of the day leading up to speaking, it was a very rewarding experience that I might very well consider repeating in the future! I hope that anyone who came along to hear the talk took something away from it, and that if you have found your way here you might take something away from it too.
The slides from the talk can be found here.
I used a video of Dylan Wiliam speaking on this same subject at the Wisconsin Mathematics Council Annual Conference, which can be viewed here.
I completely welcome any discussion, comments and questions below.
I’d like to start by just giving you a chance to have a look at this initial problem.
Take a couple of minutes to have a look through it, and begin to consider your strategies for solving it – but it will return later.
Unfortunately David Didau about 6 weeks ago decided to use the exact title I was going to use for this session as a title for a blog-post, so I had to adjust it slightly!
Developing critical problem-solvers of the young people we teach Mathematics to has for a long time been seen as the holy grail of our Mathematics teaching – students who can be fed information, sometimes in an unfamiliar situation, organise it, and solve a presented problem. We have our GCSE objectives;
AO3: Solve problems within mathematics and in other contexts
- Students should be able to:
- Translate problems in mathematical or non-mathematical contexts into a process or a series of mathematical processes
- Make and use connections between different parts of mathematics
- Interpret results in the context of the given problem
As I embarked on my teacher training, I was struck by how much we were expected to teach students using a constructivist approach – allowing students to construct their own meaning from the problems that we posed them. Teaching through problems and inquiry would not only provide contexts for students to understanding the Mathematics they were learning, but create learners with ‘grit and resilience’ as they tried to navigate their way through problems, unwrapping the Mathematics that they needed to use and applying their understanding to these situations. “People learn best in an unguided or minimally guided environment, generally defined as one in which learners, rather than being presented with essential information, must discover or construct essential information for themselves.” (e.g., Bruner, 1961; Papert, 1980; Steffe & Gale, 1995) (Kirschner and Sweller, 2006)
Except none of this felt right to me. What understanding were they applying? They were being left to look for solutions that they had no real understandings of the Mathematical structures that underpinned them, and as such the learning experiences happening from the problem-solving approach to teaching. Part of the issue came from, and continues in some quarters, to come from the conflation of those novices in the learning of Mathematics with those who apply and are experts within Mathematics, as Kirschner, Sweller and Clarke have alluded to – the idea being that learners in our classrooms can have the same learning experiences as those who are experts in their fields by doing what those experts do.
Due in part to the work of researchers, prominent bloggers, some of whom are speaking here today and ResearchEd itself, today teachers are empowered to really look further into and challenge what in some cases has become a prevailing orthodoxy within the fields in which we teach. And it is thanks to this growing movement that I feel I have been driven and motivated to do exactly that in developing my teaching through my short career so far.
The main issue I had when being advised that teaching through application was the most effective way for students was the fact that I couldn’t work out exactly what it was that these students were supposedly applying. With ‘teacher-talk’ the work of the education-devil, students were being allowed to develop understanding that may have been riddled with misconception, not necessarily on a superficial level, but deeply and damagingly.
Hand-in-hand with the constructivist inquiry-based learning approaches comes the idea that direct instruction of mathematical concepts “impairs the ability later to retrieve correct responses from memory on their own.” (Wickens, 1992) Instead, students should be allowed to draw on unique prior experience… to construct new situated knowledge that will achieve their goals.”
I therefore want to initially start by looking first of all at the failings of the problem-solving based approach to teaching and learning, before moving to consider how we do provide students with opportunities to develop skills in applying their understanding.
I’m part of a very strong Maths department, and we’re currently going through the process of designing a 5 year programme that places real importance on students understanding of the Mathematics we teach in a way which explicitly builds in links and provides the opportunity for teachers to give students chances to engage with the material being taught not just superficially but with real depth. This has been borne out of a belief that in order to know how best to approach a Mathematical problem, a student needs to have an understanding of the structure of the problem being posed if they are to solve it in the most effective and efficient way.
The role of memory in the solving of problems has been researched for a number of years, as far back as the 1920s, and it has been found that long-term memory stores the necessary information for solving problems, and those most effective and dealing with unfamiliar problems are those who are able to quickly select and apply the best procedures to apply to the problem faced. Those who have the most skill in an area are those who have the most information concerning the area stored, waiting to be recalled and applied.
Working memory and long-term memory are large factors of cognitive load theory, which analyses the way in which the brain is able to incorporate new understanding into existing schemas, and therefore create new understanding.
In the cognitive load model, information enters the sensory memory, and then is either forgotten, or works its way into the working memory. This is where the information being presented can either simply be forgotten, or through rehearsal and use can be encoded into long-term memory. This is when, as Sweller describes it, learning has actually taken place. In this model, Long-term memory has essentially infinite storage space, and information stored here can be retrieved in order to apply or combine with new information to form understanding. Working memory, however, is quite limited, and we should therefore consider carefully how we wish to place burden on those limited resources in order to maximise learning.
It has been found that the “problem-solving” search that novice learners undergo in interpreting and accessing the problem at hand is not designed to alter long-term memory, simply to provide a solution to the problem at hand. As Sweller (1988) tells us, problem solving…places a huge burden on “working memory”, and if these cognitive resources are being spend on the problem-solving search, they are not involved in the process of actually learning anything. A large part of cognitive load theory in relation to problem solving also places emphasis on the fact that whilst problem-solving, the load placed on working memory is affected by the fashion in which problems are approached, particularly the difference between how a novice may use a means-end approach to a problem, and how an expert in a field approaches it by categorising it into a certain category of problem. This is alluded to in the recent blog by David Didau, which I made reference to at the beginning, “the problem with problem-solving,” in relation to remembering how to solve the clock on his car twice a year, and the fact that he can never remember how to, despite the fact that after looking it up, he is instantly familiar with the method again. As mentioned earlier, the reason that problem-solving does not serve as a useful way of learning for novices, is that novices will use a means-end analysis to approach a problem, a method which simply looks at the stage that needs to be reached, and attempts to work backwards. This will place an extraneous load on working memory, as students spend their limited processing power in working their way backwards through the problem. An expert, who has developed the necessary schemas, will be able to categorise the problem as belonging to a group of which has been previously solved, thereby limiting the problem-search aspect of the process within working memory, and apply this previous understanding to the problem to hand.
If we are to develop effective problem-solvers of our students, we as Maths teachers need to ensure that the working memory is not overburdened when it comes to expecting our students to learn. It is for this reason that we want our students to be fluent in their times tables for example – consider the incredible things that knowing your multiplication tables allows you to do without a second thought – find a common denominator for fractions. Cancel down a fraction. Factorise a linear expression, a quadratic expression, a cubic expression. If the foundations of doing these things are effectively embedded in long-term memory, then their application, for example learning to simplify fractions by finding a common factor, can be understood, learned and itself committed to long term memory far more effectively than if the working memory is pre-occupied with trying to work out the individual factors of the numerator and denominator.
Going back to the initial problem I posed – What different areas of Mathematics did we identify in there?
I know that it is not possible to answer the actual question; what is the area of the triangle formed between the axes and the straight line graph?; without having a strong understanding of how co-ordinates work, the equation of a circle graph, calculating gradients between points, gradients of perpendicular lines, finding the equation of a straight line, and finally, the area of a triangle. Consider how much load is going to placed upon each individual aspect of this problem if it is going to be solved without a thorough understanding of they each work and interact with each other, no matter how skilled a student may be when it comes to “problem-solving.”
This means, as Mathematics teachers, our most effective way to enhance students problem-solving ability is to teach them stuff. Teach them topics well and in depth. Make links between topics explicit. Demonstrate, work through and model how problems should be approached. It is only through students having a well-developed set of schemas in place that they are able to assimilate new information into these.
Learning in students involves the development of schemas that allow them to analyse a problem posed, and determine which category of problem it belongs to, and therefore which moves are going to be necessary to move through them (Sweller, 1988). This involves students being exposed to approaches with which to tackle a wide range of problems.
Which brings us to how best to develop problem-solvers of our students? As I mentioned earlier, I am currently in the process, in collaboration with colleagues, of working out how best to arrange our schemes of learning and content in order to facilitate the development of the understanding required to problem-solve in Mathematics effectively. We’ve taken a far bit of guidance from how schemes of learning are being created by the White Rose Maths hubs in Yorkshire, and started to put together our teaching calendar and resources for next year for our Year 7s and 8s. What we want to do is give time to topics, enough time to teach in such a way that instead of rattling through a set of objectives, teachers are able to explore strands of the curriculum in depth, building in links so students can begin to appreciate how interconnected Mathematics is. This appreciation is the first step in being able to solve problems, as the understanding of how different areas of Mathematics overlap and interact is crucial if various areas of Mathematics are to be pulled together. It is through the in-depth of understanding of the structure of the aspects of a problem, that the structure of the problem itself can be understood, and therefore solved.
Sweller and Mwangi undertook research into the solving of Word Problems which required students to make comparisons, and contrasted the affect of what they refer to as “split-attention” examples, which I will elaborate on shortly, and integrated worked-examples, and looked at the effect this had on student’s ability to explain their own reasoning. It was found in their initial experiment that in a comparison between students who had studied worked example problems and those who had a more general problem-solving approach, that those whose received instruction that involved worked-examples made far fewer incorrect attempts at problems within a given time period. This can be explained by the importance that cognitive load theory places on reducing the load on working memory. The students had generated problem-solving schemas that then allowed them to identify the type of problem they were looking to solve, and compare it with the examples they had studied. The cognitive load was not being placed on the problem-solving search, as students were able to retrieve comparable approaches from their long-term memory.
A second experiment within this research involved “split attention,” and the extent to which poorly-designed examples can reduce the effectiveness that they have in reducing cognitive load.
Above is an example which demonstrates the potential effectiveness of two worked examples to the same geometry problem. The first involves the diagram with the worked through steps below. The Integrated example on the right has every step of working within the diagram. Students attempting to study the split-attention example are going to suffer from the fact that they are having to use more cognitive resources in an attempt to reconcile the explanation below with the diagram above. Conversely, with an integrated example, the steps for solving the problem are contained within the diagram. It should be noted however that whilst we eventually want out students to be able to articulate their steps as seen in the split-attention example, the integrated approach works on the basis that students have strong enough domain-specific knowledge to understand the steps being taken, thus negating the need for explanations within. Learner’s attention is not split between the two, and hence extraneous cognitive load is avoided. Research into this led Sweller and Mwangi to the conclusion that instructional materials that involved students spending less time having to process materials improved understanding over those who studied worked examples that involved split attention. The work carried out by Mwangi and Sweller problems involved worded problems are seen here:
The way in which each question is being worked through is identical, but the placement of the working within the text with the integrated allows the student to see precisely how the worked problem is being translated into diagrammatic form at each stage, and not having to place load on their working memory with trying to link each stage of the problem to its working step. Again, Sweller and Mwangi found that those who has studied the integrated format examples performed better when presented with similar problems to solve, and concluded that the integrated examples placed a lesser load on working memory, and therefore facilitated learning.
I decided to attempt my own worked example of the initial problem I have referred to throughout the presentation. I hope it is clear how I have worked through each step, and provided a route through the problem. Whilst this is a relatively complex problem, it has been broken into 5 clear steps, focussed on establishing the equation of the straight line graph RQ, and therefore allowing us to find the area of triangle OQR.
This research therefore points us to a situation where problem-solving does not simply make students better at problem-solving. Initial instruction will involve building up the domain-specific knowledge to provide students with the understanding of the mathematical structures that underlie the problems that they will face. When it comes to introducing problems that then involve an application of this knowledge and understanding, the most effective practice will involve integrated worked examples for students to study and become familiar with, which in turn will provide them with problem-solving schemas that are built on an understanding of how to solve particular problems. Once students have these schema in place, they will be able to organise, categorise and apply knowledge in order to become far more effective problem-solvers than those who are repeatedly presented with problems and will persistently use means-end analysis methods that, whilst may eventually lead to a correct attempt, will provide little in terms of learning, and little in terms of developing the schemas that allow them to approach problems in a fashion that free up working memory.
My own personal challenge, and our challenge collectively as Mathematics teachers is to use this understanding and the implications of cognitive load theory and its relationship with problem-solving to develop teaching practice that does as we’ve discussed. As I said earlier on, we are looking within my department at how we organise the teaching of topics in a less discrete, boxed-in way, and I feel that this will allow us to demonstrate and model the applications of different areas of Mathematics across the subject. The intention is that not only are we able to spend the time building up the domain-specific knowledge and providing firm foundations of understanding and of structure, but we have the opportunities to model problems that involve application and allow students the opportunity to develop their understanding of different forms of problems.
What has become very clear through this review of the research is the care with which we should approach the creation and use of our resources when it comes to providing worked examples. Worked examples need to ensure that they maximise their potential for developing student understanding by integrating the working steps and not creating a split-attention effect. This is certainly an area in which I will be looking to develop my own practice going forward, and doing some of my own classroom-based research into what the best modelled examples look like, and their potential for improving understanding.
I really hope that this has given everyone here something interesting to take away with you from today, and an opportunity to consider how this fits in with your own practice. I’d like to now give an opportunity to ask me any questions, ask me to clarify anything I’ve mentioned in the session, or generally to open things up to a wider discussion for a few minutes around the areas I’ve talked about.
Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12, pp257–285.
Kirschner, P, Sweller, J, and Clarke, R. (2006) Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching , Educational Psychologist, 41(2), pp75–86
Didau, D. (2017) The problem with problem solving (or, why I struggle to reset my clock), http://www.learningspy.co.uk/psychology/problem-problem-solving-struggle-reset-clock/ [accessed June 20th 2017]
Mwangi, M and Sweller, J. (1998) Learning to Solve Compare Word Problems: The Effect of Example Format and Generating Self-Explanations, Cognition and Instruction, Vol. 16, No. 2 pp. 173-199
Mark Ward & John Sweller (1990) Structuring Effective Worked Examples, Cognition and Instruction, 7 (1), pp 1-39
Bernstein, D. A., Penner, L. A., Clarke-Stewart, A., Roy, E. J., & Wickens, C. D. (2003). Psychology (6th ed.). Boston: Houghton-Mifflin.
Cognitive Load Model Diagram: Adapted from Atkinson, R.C. and Shiffrin, R.M. (1968). ‘Human memory: A Proposed System and its Control Processes’. In Spence, K.W. and Spence, J.T. The psychology of learning and motivation, (Volume 2). New York: Academic Press. pp. 89–195.
Tarmizi, R.A.; Sweller, J. (1988). “Guidance during mathematical problem solving”. Journal of Educational Psychology. 80 (4): 424–436.