Let us learn from maths mistakes
Writing in the Mail & Guardian, Professor Tamsin Meaney highlighted common assumptions about mathematics education that she found to have no substantive supporting evidence or whose basic suppositions were in some sense faulty (“Myths of maths and education”, M&G, March 28).
She argued that these “myths” (as she called them), if believed and acted upon, would have detrimental effects on young children’s mathematics education. Her article then proceeded to describe the myths and present the counterevidence, or the varied evidence, which in fact illustrate that these common assumptions remain unvalidated.
A letter by Professor Johann Maree, published in the following week, responded to the article and accused Meaney of propagating dangerous myths herself (“Maths myths don’t add up”, M&G, April 4). It is my contention that Meaney was not given a fair reading here. In particular, exposing “common sense” for not having substantial empirical support (as Meaney argued) is not the same as propagating a countermyth (as the letter suggested).
Meaney’s article was her summarised version of a plenary presentation she made at the South African Association for Research in Mathematics Science and Technology Education conference in January. As I see it, Meaney opened a much-needed debate, because the attempts to improve mathematics proficiency in this country over the past 20 years have yielded little confirmable result, despite the repeated systemic and international testing.
The challenges Meaney presented are not without support from esteemed organisations around the world. Regarding the vague belief that “regular testing will contribute to raising standards”, the research and development division of the United States Educational Testing Service is now proposing alternative forms of systemic testing. Its rationale is that the current and recent approaches to systemic testing in the US are feared to be doing more harm than good for the very communities they were supposed to benefit most.
Critical points in the learning cycle
This division, under the leadership of Professor Randy Bennett, is piloting assessment programmes that are more closely aligned with teaching and learning. These tests are administered regularly during the year so that teachers can engage with the results at critical points in the learning cycle, rather than only at the end of the year. The current year-end systemic assessment in this country, as Maree asserted, may be giving the system some information that can be acted on to improve the schools’ performance.
But it is important to note that improving performance is not necessarily the same as developing mathematical proficiency. “Improving performance”, especially if tied to performance management, may give rise to “coaching” and “teaching to the test”, which, in many cases, as Maree asserted, result in only some surface-level knowledge, great anxiety and a dislike of anything mathematical.
The proponents of reform are not against assessment. In fact, we firmly believe that regular diagnostic and supportive assessment is critical to learning, and should be aligned with teaching and learning in each specific classroom. There needs to be repeated mutual feedback between teacher and learners, using well-designed tests with suitable diagnostic marking memos.
An interesting idea from Australian researchers is “front-ending” assessment. Here the teacher decides, concurrently with planning her teaching and pupil learning, about the concepts that must be learnt and therefore assessed.
But, even here, in arguing for finely tuned assessment, we need to be sensitive to the learning of complex mathematical concepts, especially if we want to encourage deep learning, an ability to solve problems and a cycle of increasing abstraction.
Complexity in proficiency
Mathematical concepts are not learnt singly, in isolation, but in relation to other concepts: for example, the concepts of fraction, ratio and proportional reasoning will and must develop concurrently. The fact that a particular mathematical concept may apply to numerous problems, and that solving a single problem may require using many different mathematical concepts, points to the complexity in developing mathematical proficiency.
The complex relationships between mathematics teaching and learning require a programme of professional development that builds on teachers’ existing knowledge and extends this knowledge to include a network of related concepts and ideas.
South Africa’s particular challenge is empowering teachers to be innovative and responsive to their particular classrooms. This is critical, because the most fundamental locus of learning is the learner and the most proximate resource to a class of learners is the teacher. Where there are persistent teacher absenteeism and time-on-task problems, these need to be addressed at the individual, school and larger structural levels.
Meaney’s article also challenged the idea that “doing well in mathematics is likely to support the country’s economic progress”.
Maree conceded in his letter that the relationship between mathematics performance and the economy is complex, and indeed it is.
Zalman Usiskin, emeritus professor at the University of Chicago, has investigated Singapore’s exceptional performance in an attempt to assuage the US panic about its “underperformance” in successive reports of Trends in International Mathematics and Science Study (Timss). He found little direct correlation over time between the US economy and Timss results, and concluded, like Maree, that the relationship of school mathematics performance and socioeconomic factors is complex.
In order to improve innovation and employability, it must be our imperative to focus on mathematical proficiency of the kind where the learner enjoys learning, welcomes the challenge of solving problems not previously encountered and regards mistakes as opportunities to improve understanding.
Proficiency that is ingrained through trial and error and re-trial is more likely to contribute value to the economy than a syllabus that is subjected to rote learning and coached in methods that may improve performance but are unlikely to prepare learners mathematically for the world of work or tertiary education.
It is at tertiary level that real mathematics proficiency is likely to have an impact on the economy. What could we be doing at school-level mathematics to maintain a natural inquisitiveness?
Bonginkosi Mnisi, a University of Cape Town astrophysics student who came from a school in Mpumalanga that is poorly resourced materially but richly resourced in human terms, has cited energy, enthusiasm and passion for the subject as the key factors in learning mathematics and enabling learners to achieve their potential.
Teacher is the critical factor
It is evident from this that even in the 21st century – with open-source materials galore to be found on the internet – the teacher is the critical factor in the system.
Teaching is a vocation and usually taken up by people who care for others and who want to make a difference. Mnisi’s account makes it clear that the teachers and principal at a remote Mpumalanga school, and other interested support organisations, provided the vision for a rural child to make his dream of studying astrophysics come true.
In the case of this student, education will “lift him out of poverty”. But what of his entire community? Surely, as Meaney warned, our goal is not to clone the “middle-class” child but to change our society so that all other children in that Mpumalanga classroom can also succeed, whether at farming, teaching, nursing or creating small businesses for the entire region, as well as one pursuing a career in astrophysics.
Mncedisi Jordan’s recent article (“Making education more homely”, M&G, April 11) is relevant here: it argued the case for a curriculum that connects the science rural children encounter every day with the formally constituted science and mathematics they learn at school.
Professor Poonam Batra in Delhi is one of many who advocate the goal of social transformation through education. She makes the bold claim that, through an empowering professional development programme for teachers, which includes the dynamics of politics, social change and education, transformation can be achieved in a single generation.
Merely challenging the myths Meaney identified was not her main aim. As her article said, her concern is that imposing a narrow formal mathematics curriculum on preschool children may be counterproductive.
And her challenge to the mathematics education community is to think about the kind of mathematics education that will contribute to an economy in the 21st century.
Meaney’s question still remains valid: Why are so many people, even scientific researchers, so quick to reject alternative hypotheses about the inefficiency of our education system?
Caroline Long is a senior lecturer in the faculty of education, University of Pretoria. She writes here in her personal capacityLet us learn from maths mistakes