Mathematics holds an important place at the core of all curriculum models for good reason. The traditional focus on Literacy and Numeracy reinforces the special place that Mathematics holds in our educational thinking. The importance of Mathematical thinking to our daily lives is arguably increasing as we rely on computational models and large data sets. Industry, according to multiple reports requires more graduates with a STEM (Science, Technology, Engineering & Mathematics) background and the M in STEM is seen by many as providing the glue which holds the model together. Despite the importance of Mathematics and the high esteem it holds as a discipline too few students are pursuing it as a pathway beyond school and many people report a fear of Mathematics.
According to Stanford professor and author Jo Boaler mathematics requires a mindset makeover. “Mathematics, more than any other subject, has the power to crush students’ confidence’. (Boaler. 2009) What is needed is a deliberate shift away from perspectives that make it OK to believe that one is not good at mathematics. This perception of mathematical abilities as a fixed set of attributes which some possess and others do not is a significant hurdle to be overcome if we are to encourage all learners to achieve success in the subject. A first step towards this goal is for teachers and parents to adopt the language of a growth mindset for mathematics and ensure that their children and students are not exposed to negative attitudes of fear or failure that many adults carry forward from their experience of school mathematics.
A recent ‘Guardian’ article by Bradley Busch offers advice for teachers when responding to students about their mathematical achievements. He describes the ‘comfort strategy’ that is often applied when students underperform in mathematics. The comfort strategy tells students that their results, even when they are low, is not something to worry about, that mathematics is hard and that not everyone can be good at it, or that you are good at other things. The opposite approach offers strategies for improvement. The research cited by Busch shows that students offered the ‘comfort strategy’ tend to estimate that they will maintain consistent results while students offered a ‘strategy focused’ response believe they will improve.
How we learn mathematics is just as important as our attitude towards the subject. Boaler cites research she conducted as part of the last round of PISA (Programme for International Student Assessment) which looked not only at achievement scores but asked how students learned in mathematics. The clear finding was that countries with a focus on memorisation underperformed, countries which achieved positive results focused on problem solving, number-sense and challenge.(Boaler & Zoido. 2016) Boaler’s long-term research supports this finding with case studies that compare schools teaching through challenging problems linked to the real-world use of mathematics with traditional classrooms with a focus on mathematics as a set of rules and processes to be memorised. Classes which produced the best results, the students with the most positive attitudes towards mathematics and the greatest equity of learning where those where mathematics was taught in richly collaborative settings, where mathematical dialogue was the norm and where students embraced challenge. Beyond a focus on the repetitive manipulation of algorithms and manual calculations, successful mathematicians learn to use multiple representations (including pictures, diagrams, charts, tables, graphs and physical representations) of mathematical concepts that assist in the development of ‘number sense’ and particularly the understanding that numbers can be readily decomposed and recomposed.
The role of challenge in mathematics is important and the research shows that traditional approaches that seek to make mathematics easy by deploying rules, mnemonics and ‘simple’ processes are doing learners a disservice. Research by Manu Kapur (2014) shows that students are better able to develop a deep understanding of mathematical concepts when they are allowed to fail on their first attempt to learn new ideas. The experience of failure better activates prior knowledge and prepares students for subsequent instruction. This requires that students are presented with sufficiently challenging material and that it is presented in ways that allow for failure. Kapur (2015) also suggests that students are best served by opportunities to generate problems as a part of their mathematical learning and that doing so assist with conceptual understanding and transfer of learning to new situations. Boaler’s research supports this and shows that “our brains grow when we make mistakes because it is a time of struggle, and brains grow the most when we are challenged and engaging with difficult, conceptual questions”. (Boaler 2009) Steven Strogatz (2015) of Cornell University adds “This is not the way math should be taught, even at an elementary level. There really ought to be problem solving and imaginative thinking all the way through while kids master the basics. If you’ve never been asked to struggle with open-ended, non-cookbook problems, your command of math will always be shaky and shallow.”While we believe we showing care for students by making mathematical learning easy and by removing challenges from our students pathways (particularly those labeled as low ability) we are inhibiting their learning.
Boaler and others are concerned that the Mathematics is a diluted version of the real subject. This dilution of Mathematics to a set of rules to be mastered and applied robs the subject of its true beauty and real power. Conrad Wolfram as the founder of ‘Wolfram Research’ a company dedicated to mathematical applications is well placed to describe real world mathematics, the sort we should be teaching. "At its heart, math is the world’s most successful system of problem-solving. The point is to take real things we want to work out and apply, or invent, math to get the answer. The process involves four steps: define the question, translate it to mathematical formulation, calculate or compute the answer in math-speak and then translate it back to answer your original question, verifying that it really does so.” Conrad advocates for teaching that makes use of computers for much of the calculating that is required for problem solving just as is the case in the real world “In the real world we use computers for calculating, almost universally; in education we use people for calculating, almost universally”. Conrad has launched a site advocating for “Computer Based Math” as a solution to the crisis he perceives in mathematics education.
What becomes clear, as you dive further into the emerging research that connects what we know about learning, mindsets, dispositions for learning and the development of mathematical understandings, is that a new approach is required. We need to move away from memorisation and rule based simplifications of mathematics and embrace a model of learning that is challenging and exciting. We can and should be emerging all our students in the beauty and power of mathematics in learning environments full of multiple representations, rich dialogue and collaborative learning.
by Nigel Coutts
References & Resources:
- Bradley Busch (2017) Growth mindset: practical tips you may not have tried yet. The Guardian 10th January 2017 - https://www.theguardian.com/teacher-network/2017/jan/09/growth-mindset-practical-strategies-classroom
- Jo Boaler (2009) - The Elephant in the Classroom: Helping Children Learn and Love Maths. Souvenir Press, UK
- Boaler & Zoido (2016) Why Math Education in the U.S. Doesn't Add Up. Scientific American November 2016 - https://www.scientificamerican.com/article/why-math-education-in-the-u-s-doesn-t-add-up/
- Manu Kapur (2014) - Productive Failure in Learning Math. Cognitive Science 38 pp1008-1022
- Manu Kapur (2015) - The preparatory effects of problem solving versus problem posing onlearning from instruction. Learning and Instruction 39 pp23-31
- Steven Strogatz (2015) cited in The Problem with Math Problems: We're solving them wrong. by Jessica Lahey - https://parenting.blogs.nytimes.com/2015/04/02/the-problem-with-math-problems-were-solving-them-wrong
- Conrad Wolfram - School Math is the Wrong Subject via "Assorted Stuff” by Tim Stahmer http://www.assortedstuff.com/school-math-is-the-wrong-subject/
- Conrad Wolfram - Computer Based math - http://www.computerbasedmath.org/
YouCubed - https://www.youcubed.org/
A mathematics resource from Stanford developed by Jo Boaler author of Mathematical Mindsets
With + Math = I Can - https://www.amazon.com/gp/withmathican
Site promoting a growth mindset for mathematics. Support for a positive mindset and resources