The Borrowers

We cannot raise standards in mathematics by cherry-picking ideas from Asia and Europe. Success comes from choosing an approach and sticking with it, says Mike Askew...

Although the popular press would have us believe that the two big international surveys of mathematics teaching and learning – TIMSS and PISA – are all about ranking countries in international league tables, both studies collect a wealth of data about schools, teaching, and learners. Data that, some hope, shed light on what effective teaching might look like.

With this in mind, colleagues and I were commissioned by the Nuffield Foundation to review the research into why countries that consistently rank high in these international studies maintain their position. One overall conclusion was that a nation’s attitude towards the importance of learning mathematics has as much, if not more, impact on standards than any particular style of teaching appears to have. Nevertheless, there are some hints as to what can make a difference, for example, an emphasis on problem solving and consistent teaching approaches.

Problem solving, making connections and challenge

One finding across various studies (not only TIMSS and PISA) is that if teaching focuses on learners making mathematical connections through working on challenging problem solving, this can lead to higher standards. For example, alongside the large-scale assessment and questionnaire data collected, TIMSS also conducts video studies of a small number of participating countries. The first of these concluded there were differences in the ‘lesson scripts’ of different nations (United States, Germany and Japan). These countries displayed broad patterns of teaching styles across the three countries – patterns that were greater than teaching differences within a country. Major differences between countries appeared to be in the extent of problem solving taking place in lessons and whether or not the emphasis was on conceptual understanding – more prevalent in Japan, a high performing nation – or on procedural fluency – more typical in the USA (which ranks around the same as England).

The studies also reveal a difference in the interpretation of what is meant by ‘problem solving’. A later TIMMS video study involved seven countries: Australia, Czech Republic, Hong Kong, Japan, the Netherlands, Switzerland and the United States. This looked more closely into the the types of problems pupils worked on in the videoed lessons. In five of the countries, the problems were described as ‘routine’ and only really engaged learners in more practice of certain procedures – the problems were really just ‘sums’ wrapped up in words. But where problem solving was richer and engaged learners in reasoning, looking for patterns and making generalisations (typical in Japan and the Netherlands) there was more evidence of problem solving helping pupils to connect different aspects of mathematics and to deepen their understanding.

Starting from the ‘real world’ or from ‘mathematics’?

One of the debates in mathematics education is whether learning is better promoted through starting from problems, or with formal mathematics. The former approach involves building on a learner’s ability to solve problems informally, with teaching that then focuses on helping him or her refine these informal solutions so they become more mathematically sophisticated. In contrast, the latter approach starts with teaching ‘pure’ mathematics and getting pupils to apply it later to problems.

Dutch research and development into ‘Realistic Mathematics Education’ (RME) advocates a problem-based approach1 many tables need to be put out for a meeting if each table seats six people?’ or ‘how many pots of coffee have to be made if each pot holds eight cups?’. The approach has been picked up and developed elsewhere, notably in Cathy Fosnot’s ‘Young mathematicians at work’ project in New York. The premise behind the approach is that mathematics originally developed through problem solving and so problem solving provides the starting point for learning.

The other approach rests on the assumption that there is a ‘body’ of mathematics – procedures, concepts and approaches – that pupils can learn and apply later. The mathematics teaching in places such as Singapore and China is more aligned to this philosophy. Closer to home, a major German project – ‘Mathe 2000’ – takes the approach of starting with quite abstract work in number and calculation and only applying it later.

So which approach is better? The answer seems to be that they both work! Researchers in, for example, Holland and Germany, provide evidence that both RME and Mathe 2000 can lead to good learning outcomes. A consistent approach seems to be the thing that matters.

Planning and practice

A study re-analysing the TIMSS 1999 video data compared the contexts and approaches to problem solving in the Dutch and Japanese classrooms. Although the Dutch lessons were originally coded as displaying more real-life connections, in examining the way teachers developed the problems within the actual lessons, the research concluded that the Japanese lessons were closer to the spirit of ‘Realistic Mathematics Education’ in the ways that the pupils were involved in ‘mathematising’ situations.

Some problems that looked as though they would challenge pupils and help them to make connections turned into routine tasks as lessons developed. Other problems that on the surface looked ‘routine’ actually developed into more challenging tasks. In other words, it not simply a case of planning lessons with a focus on problem solving, reasoning and connections; the way in which a teacher develops problems within a lesson is also important. These findings are similar to ones about effective teachers of numeracy in English primary schools that I studied around the same time.

Consistency of approach

Overall, the thing that seems to make the difference is the consistency of approach – a bit of realistic problem solving mixed with a bit of abstract mathematics looks like being less effective than choosing one approach and sticking with it. High attaining Asian countries such as Singapore, Japan, Korea and China all demonstrate a consistency to their mathematics curriculum and teaching that changes only incrementally over the years. (Although like most findings, there are also other countries with a long, consistent history that do not do so well.)

The briefer history of the England’s National Curriculum for mathematics presents a more fragmented and frequently changing picture, with no clear set of teaching principles having had time to emerge and bed down. The key to raising standards does not lie in adopting, wholesale, practices from elsewhere – the history and culture of those practices makes that impossible to do anyway. What we need to do is have a serious debate about the sorts of mathematical experiences our learners might best benefit from and develop these over timescales that go beyond periods of general elections.

* A more accurate translation of the Dutch phrase is ‘realisable mathematics education’ – the idea being that problems posed have the potential to be ‘realised’ into mathematics – they do not necessarily have to be realistic in the sense of ‘everyday’.

* You can download the research report ‘Effective Teachers of Numeracy’ from mikeaskew.net

The different emphases of TIMSS and PISA

TIMSS – Trends in international mathematics and science study

PISA – Programme for international student achievement

TIMSS and PISA are different surveys and, whilst complementary, each assesses different aspects of mathematics. PISA has a specific focus on mathematical literacy and places emphasis on application and realistic contexts – it is meant to assess the general outcomes of mathematics education, rather than whether any specific content has been taught. This benefits countries that have a similar emphasis in their curriculum, such as Australia and the Netherlands. TIMSS, on the other hand, is designed around what is agreed to be a core of commonly taught curriculum topics. So, not surprisingly, countries such as China or Singapore, whose national curricula closely match the TIMSS content list, perform better than nations with a less good match.