Here I set out the start of a manifesto for expanding learners’ experiences of what mathematics “is”. My intention is not to decry what we already have (although I do think that it should be open to critique), rather to suggest that as it stands, for many students, our school mathematics curriculum is particularly ineffectual in preparing students for life beyond school. My concerns in this regard have been heightened by our recent experience of maths being centre stage as we, and politicians, drew on “the science” to understand the spread of covid-19. It is likely that we all have heard terms such as “flattening the curve” and “keeping the R value below 1”. We’ve all seen graphs with log scales, but what is everyone making of them? What do they understand?
Here are two graphs of the type which the government at it’s daily briefings used in early stages of the pandemic in the U.K (although they ceased to be shown to the public when the U.K could be seen to be performing badly by the particular measures they are based on). One has a linear scale and the other a log scale.
It is one or both of these graphs that government ministers referred to when they spoke of “flattening the curve”. But what might the majority with an experience of school mathematics make of these graphs? Although not everyone comes across logarithms at school, of those who do, what do they understand of a graph with a logarithmic scale? What is an exponential function? What is fundamental about systems that exhibit exponential growth?
Unfortunately, for too many, their experience is more about learning the “rules and procedures” than “relational understanding” in the sense of Skemp (1976).
Curriculum dimensions and mathematical practices
Many countries around the world prioritise mathematical content in their curriculum specifications and these have possibly the most convergence of any curriculum domain of knowledge. It seems likely that students in schools around the world study much the same “mathematics” in terms of content. However, although it is mathematical content that dominates, and I might add primarily a western European view of what counts, this is not all there is to mathematics as a discipline.
Particularly important in narrowing, but also potentially broadening thinking around the world is the PISA (Programme for International Student Assessment) which publishes a framework for mathematics which,
“defines the theoretical underpinnings of the PISA mathematics assessment based on the fundamental concept of mathematical literacy, relating mathematical reasoning and three processes of the problem-solving (mathematical modelling) cycle.”
Mathematical content knowledge is
- four content categories: quantity, number systems and their algebraic properties,
- four categories of contexts: personal, occupational, societal, scientific,
and seeks to provide challenge in real world contexts. (see, https://pisa2021-maths.oecd.org/files/PISA%202021%20Mathematics%20Framework%20Draft.pdf )
In many countries ideas such as these underpin attempts to prioritise what has come to be thought of as mathematical literacy or numeracy.
As Usiskin speaking at ICME-13 in 2016 identified,
“five kinds of curriculum: ideal (intended), textbook (materials), implemented (taught), tested, and learned. The first four of these are created by people called policy makers, curriculum developers, teachers, and item writers, respectively. He offered three broad paradigms of curriculum development: traditional, test-influenced, and innovative. In the traditional paradigm, these five kinds are developed in the order shown above. In the test-influenced paradigm, tested curriculum specifications occur before materials are written. Discussions during TSG-37 sessions indicated that test-influenced curricula are exceedingly common throughout the world, with PISA and TIMSS results having particular impact in some countries.” (Rampal et. al. 2017)
However, more broadly than perhaps test-led curricula might suggest, many curricula recognise other mathematical practices or ways of “being mathematical”. For example, the United States Common Core State Standards identify eight mathematical practices:
- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning.
In a seminal article Cuoco and colleagues (1996) identify eight different “habits of mind” for students as learners of mathematics, arguing that they should be:
- pattern sniffers
Here, I wish to support attempts such as these to broaden students’ experiences of mathematics and suggest further that to equip them to be prepared as critical citizens and ready for all aspects of adult life they additionally need to have experience of working with mathematical ideas that support mathematical:
- modelling, and
It is my intention to write three further pieces in which I set out in greater detail each of these separately, but briefly:
Mathematical comprehension, is perhaps best thought of as, making sense of the mathematics of others. Importantly, this is the mathematics of others both in the world of mathematics itself and also in a wide range of applied contexts.
Mathematical modelling, has an important role to play in applying mathematics to make sense of the world. It involves simplifying the complexities of reality so that we can use mathematical ideas and techniques to solve problems and understand situations.
Mathematical communication, involves using mathematical writing and representations (including diagrams, charts, graphs and formulae to communicate) mathematical thinking and the outcomes of mathematical work.
My concern is that, as currently is specified and played out, school mathematics inadequately prepares citizens for making sense of the world in which they live, work and play. The question I address is how we might ensure that school mathematics does a better job.
Warrant: Here I declare my interest and background (and bias). My agenda for mathematics curriculum reform is precipitated by a longstanding concern about the lack of focus of applying mathematics in our curriculum. This has been informed by many years of working in the field of mathematics education research and curriculum design. In particular, in this area, I have been privileged to have led work in developing in the late 1990s and 2000s the Freestanding Mathematics Qualifications, AS Level and A Level Use of Maths, being part of the Expert Panel developing the design principles for Core Maths qualifications in the 2010s and most recently the principles of Maths in T-Levels (https://royalsociety.org/~/media/policy/topics/education-skills/Maths/Mathematics%20for%20the%20T%20Level%20Qualifications%20-%20a%20rationale%20for%20GMCs.pdf ).
Cuoco, A; Goldenberg, E. P., and J. Mark. (1997). Habits of Mind: an organizing principle for mathematics curriculum. Journal of Mathematical Behavior, 15(4), 375–402
Rampal, A., Usiskin, Z., Buchter, A., Hodgen, J. and Osta, I. Topic Study Group №37: Mathematics Curriculum Development in G. Kaiser (ed.), Proceedings of the 13th International Congress on Mathematical Education, ICME-13 Monographs, DOI 10.1007/978–3–319–62597–3_64
Skemp, R. R. (1976). Relational understanding and instrumental understanding, Mathematics Teacher 77, 20–26.