Ms Catrin Huxtable, Director of Mathematics
It is 1995. A ten-tonne truck rattles through Zambia, destination Victoria Falls, Zimbabwe. On board, 22 passengers and crew bounce in their seats, as their driver assesses the track ahead, balancing the need for caution with that of a daylight arrival. The road is slow, a patchwork of dirt and neglected post-colonial tarmac weaving through the sugar cane landscape. Yet the company is making good time, dodging the washouts, until a deceptively large hole presents itself and the truck drops into it with some force. The usually sprightly engine loses power, the result of diesel spraying under high pressure past the failed injector pump gasket. The vehicle bounces to a halt. ‘No problem’, the driver states confidently to the concerned faces around her, ‘we carry a good set of spares, we can dismantle the pump, pop a new gasket on and be on our way’. Only, as luck would have it, the rather delicate, specialised part needed is not present among the set. The driver’s ingenuity, confidence and ability to problem-solve are the only tools at her disposal.
Fast forward to the present day and the truck driver, yours truly, now finds herself part of a bigger team facing quite a different situation, one even more challenging to address. The modern, and future, workplace demands that workers are able to solve non-routine problems in a range of professions and circumstances: complex problems, unfamiliar problems. Schools are charged with developing the skills that underpin successful problem-solving, across all subject areas. Mathematics is well-placed to contribute to this need, so as a Faculty we consider how we can best support the development of these skills in our students. However, there is another aspect that complicates the issue, particularly within Mathematics.
Students find problem-solving activities challenging (PISA 2012) and, in Mathematics, difficulties with problem-solving can lead to anxiety, especially when assessment is involved. Many students study hard only to find they struggle to apply their knowledge on the day of an exam. Occasionally, they run out of time, and sometimes they have difficulty understanding the question. On return of their marked paper, students may be frustrated when they realise they can now do what they could not under the pressure of an exam situation. Often, there are tears as their confidence suffers. Many profess to feel anxious beyond expected, normal (arguably helpful) levels and feel this impairs their ability to show what they can do. Many will comment that it was the problem-solving that defeated them.
Within the sporting arena, the phenomenon of ‘choking under pressure’ is relatively well-known, with research documenting that pressure situations can result in adverse effects on performance. Perhaps less well-known is that similar findings have been observed in mathematical contexts. In 2004, Beilock, Kulp, Holt and Karr supported the idea that anxiety could have a disruptive effect on mathematical problem-solving by consuming working memory capacity. Their research noted that while high levels of anxiety were detrimental to performance, with sufficient practice of similar tasks, anxiety and associated ‘choking’, were greatly reduced. This would seem to be logical, and indeed, for many students, additional practice and exposure to more difficult problems can be helpful. However, for others, this advice does not appear to work. Some students report that despite practising all of the problems available to them, they still have difficulties on the day of the assessment; ‘the questions were not like anything I’ve practised’ is a common refrain. Perhaps adding to the complexity of the issue is that by their very nature, non-routine questions must be different to those encountered before, so while practice is undoubtedly important, other strategies are also required.
Research from the the Organisation for Economic Co-operation and Development (OECD) has shown that girls, in particular, can struggle to understand and solve problems where the way forward is not immediately obvious. Translating a problem into a mathematical form in order to make progress is challenging, even for those considered to have a strong aptitude for the subject (OECD (2015), The ABC of Gender Equality in Education). In 1966, Webber labelled word problems requiring mathematisation as ‘demon’ problems. Anecdotal evidence would suggest many of our girls would find this title apt. So how do we help our girls overcome unhelpful levels of anxiety, improve their confidence, and learn to love the ‘demon’ problems?
Underpinning any application of skills must be proficiency in the basic skills themselves. PISA’s 2012 report on Creative Problem Solving notes ‘the highest intelligence, largest working memory capacity, or the most efficient brain cannot help to solve a problem if the person has no meaningful knowledge to process’ (PISA, 2012).
Yet, if we consider a complex, non-routine, task to be one where ‘the solution cannot be obtained by the automatic application of a ready-made algorithm’ (Mevarech and Kramarski, 2014), there is a clear need to push beyond procedural proficiency to a deeper conceptual understanding. Our teaching strategies at Girls Grammar are designed with this in mind. Yet on occasion, we will hear the question ‘So after this, I do this…?’ as students seek to clarify a process. In itself, this can be helpful, but if it is the only question a student asks as they are wrestling with new ideas, then our experience suggests knowledge development is likely to be shallow and short-lived, and opportunities to form critical links with prior knowledge could be lost.
The process of metacognition, thinking about our own thinking, has a history within mathematics as a useful tool in establishing a deeper understanding of situations and as an approach to solving difficult problems. In 1945, Polya outlined useful steps to consider when solving such problems in his book, How to Solve it (2014). Other models exist, which suggest a similar approach, and recent research has given support to their value. Kramarski et al. (2010) investigated the effects of metacognitive processes on anxiety and problem-solving for students across the ability range and found students confident with the process demonstrated greater gains in their performance and reported a reduction in anxiety.
Our aim is to help our students move toward becoming confident problem-solvers, by more overtly modelling these metacognitive processes. As we challenge the girls to tackle unfamiliar problems, we are encouraging them to slow down, to read and to think about what is in front of them. We are asking them to consider what they know, and what is familiar about the situation in front of them. What information is given and just as importantly, not given? If they could phone a friend, what is the one thing they would ask? (Other than the answer itself!). This can be a painful process, as regardless of experience or ability level, students’ pleas of ‘just tell me what to do’ are responded to with more questions. The pain will be worth it if over time a greater depth of understanding and confidence develops, and the anxiety associated with ‘getting it wrong’ decreases.
The reality for Mathematics teachers is that many of the concepts we are teaching may not be specifically utilised in later life. Undoubtedly, many past students regret they no longer find the need to apply the quadratic formula. However, few would argue that reasoning, perseverance, logic, creativity and flexibility—skills required to move successfully through the Mathematics curriculum—are not essential to life after secondary school. These skills stay with us for life. Perhaps unsurprisingly, my knowledge of algebraic axioms was not overly helpful in getting the show back on the road in Zambia, but years of thinking about what to do, when I did not know what to do (coupled with some rice bubble packets and silicone) definitely were.
Beilock, S.L., Kulp, C.A, Holt, L.E., Carr, T.H. (2004), “More on the fragility of performance: Choking under pressure in mathematical problem solving”, Journal of Experimental Psychology-General, Vol. 133/4, pp. 584-600
Mevarech, Z. and B. Kramarski (2014), Critical Maths for Innovative Societies: The Role of Metacognitive
Pedagogies, OECD Publishing. http://dx.doi.org/10.1787/9789264223561-en
OECD (2014), PISA 2012 Results: Creative Problem Solving (Volume V): Students’ Skills in Tackling Real-Life
Problems, PISA, OECD Publishing, Paris, http://dx.doi.org/10.1787/9789264208070-en
OECD (2015), The ABC of Gender Equality in Education: Aptitude, Behaviour, Confidence, PISA, OECD Publishing. http://dx.doi.org/10.1787/9789264229945-en