Improving Mathematics engagement through problem solving
We need to move to a problem centered and interdisciplinary teaching philosophy to engage all learners in the joy of Maths
David Vaccaro Director of Learning & Innovation, Wycombe Abbey School
My daughter is four years old, and she is on that magical journey of learning to read. Like most parents we are trying to sugar the pill of time spent stuttering out phonics by still reading out her favourite story books. These two activities together encompass both meanings of “learning to read”: the epistemic goal of linking letters to sounds, and the eudaemonic goal of establishing a positive relationship with books.
A similar dichotomy exists within learning Mathematics. Again, there are necessary skills, but there is also the joy of solving problems(1). Relatively few people ever experience this flourishing goal, indeed many pupils find it bizarre when I talk about enjoying Mathematics. Most have a sense that Mathematical skills are (sadly) necessary, but the subject is more associated with emotions of confusion, frustration, and embarrassment rather than pleasure.
It is easy to see the reason. Look at the questions on any GCSE paper and ask yourself whether you want to do them. Do you care? There is clearly almost nothing which is intrinsically motivating. Most questions have no context, no relevance, and no scope for personal input. A few pupils, due to inspirational teaching or independent study, may develop an appreciation of Mathematics, but more will only strive to achieve a good grade because they recognise its extrinsic value. Sadly, a significant number (almost a third) will lose interest, switch off and fail; a disproportionate proportion of these coming from less affluent backgrounds.
At first glance it seems counterintuitive to suggest problem solving as the way to improve engagement. In the current qualification questions earmarked as such tend to be the most alienating. Here is an infamous example:
However, is this question really problem solving? The apparent context is a charade bearing no relation to reality. This question was found difficult due to the perverse obfuscation rather than to any genuine richness. Once deciphered, it has a single correct answer obtained by a known technique.
Look instead at a genuine example of a mathematical problem. Early in 2021 the government needed to decide the order of priority for covid vaccinations. Some advocated prioritising those most risk of the contracting the disease, others for those most likely to become seriously ill. Gaining insight into such a real-world question is no easy matter, requiring input from Data Science, Computer Modelling and Epidemiology. There is no single approach, it is unlikely to achieve a definitive conclusion, and any publication needs to be honest about its scope and limitations. This example reveals some essential features of problem solving:
- Problem solving requires creativity.
- Problem solving is collaborative.
- Problem solving makes uses of all available tools- including published literature and computers.
- Problems are genuinely interesting.
The contrast between this list and GCSE could not be starker. Real Mathematics emerges as a literate activity(2), requiring critical analysis of data to reach (partial) conclusions supported by quantitative reasons. It is also social and interdisciplinary. Only in schools is the benchmark of mathematical competence measured by what people can do on their own and in silence, separated from books and technology.
While GCSE prizes Victorian virtues of flawless, solitary reproduction of known techniques, other countries teach in a way suited to the modern world. In 2013, Estonia, in partnership with Wolfram Research, introduced a curriculum that allowed pupils to investigate rich contextual problems using modern computation(3). Instead of calculating the focus was on evaluation and exploration. Using technology, it became possible to investigate problems akin to covid vaccination policy, as even if pupils cannot independently build simulations, they can interpret the output.
This problem-centred teaching philosophy may explain Estonia’s subsequent dramatic rise in PISA rankings(4). Similarly, Phenomenal Based Learning in Finland (another PISA high-flyer) gives pupils the chance to use taught skills to investigate rich multi-disciplinary problems. This opportunity to put Mathematics into action, is not something that is typically on offer in Britain, where many pupils see the subject as irrelevant.
It is easy to understand this exasperation. I similarly hated Mathematics at Primary School, where I developed the habit of working as quickly as possible to try and make it go away. When my report dismissed me with a grade C and the comment, “some good work- spoiled by inaccuracies”, I did not care. There was nothing to indicate at that point that Mathematics might be in any way interesting. At 13 my life changed, when I was invited to a local RI Masterclass Series. This sparked a lifelong love Mathematics. The atmosphere was friendly and supportive, and the emphasis was on exploration rather than pressure to come to an answer(5). I became prepared to think deeply, and the sloppiness described by my teacher turned out not to be an inherent deficiency.
Problem-centred pedagogy was central to unlocking my potential, and it is possible to ignore the formulaic nature of GCSE examination and teach in this way. Nowadays many schools will use UKMT resources, and other high-quality materials, as a source of rich and engaging exercises. A growing number of pupils are capable of reasoning that is more sophisticated than required, and the GCSE is seen as limiting for high attainers. However, this is a minor problem compared to lack of engagement from the “forgotten third”.
The fact that some pupils do well should not obscure the fact that the content, as well as the style, of GCSE assessment is wrong. A few outliers may derive pleasure from Simplifying Surds or Circle Theorems, but many more would benefit from an explicitly relevant curriculum like Estonia. This shift is happening, and this year California(6) started offering Data Science as an alternative to Algebra II. This move is to be celebrated. Complicated algebraic manipulation is often to blame for turning children off Mathematics, and as well as being more engaging, data interpretation is far more widely used(7).
When the inspirational President of the IMA, Dr Nira Chamberlain(8), does Mathematics outreach, often in deprived areas, it is no surprise that he seeks problems that are meaningful to his audience(9). For example, when working with football obsessed children in Birmingham, he may lead a workshop on “Saving Aston Villa”. Like other examples of problem-solving this is done in groups and the output is not just a calculation but a presentation. This intertwining of Mathematics with literacy again emerges as essential, mimicking the real world, where a scientist will use quantitative reasoning within an academic paper or a grant proposal rather than a stand-alone calculation.
In summary, here are the key features that one would wish from any revised curriculum:
- Collaborative Problem Solving
- Investigative Mathematics
- Contextual Mathematics
- Interdisciplinary projects combine Literacy, Mathematics, Computing and Creative Subjects.
All of these would, I believe, improve Mathematical outcomes but they also have flourishing in mind. Francis Su, the former President of the Mathematical Association of America, advocates asking all students:
The GCSEs failure to consider this question makes it akin to testing a child’s ability to read letters without ever expecting them to see a book.
- For more on the eudaemonic nature of Mathematics: Mathematics for Human Flourishing by Francis Su.
- Mathematical Literacy is a key strands of the PISA Assessment see PISA 2021 Mathematics Framework
- For more on Computer Based Maths see The Maths Fix by Conrad Wolfram.
- Information about Estonia’s performance in PISA can be found here.
- For more on introduce problem-based learning in the classroom see Limitless Mind by Jo Boaler or www.youcubed.org
- For more about California see Bringing Math Class into the Data Age
- A report from the Royal Society advocates the Integration of Data Science in the Primary and Secondary Currciulum.
- More information about Nira Chamberlain and the IMA can be found here.
- More about the importance of meaningfulness in Mathematics education can be found here