Paper presented at the British Educational Research Association Conference, Cardiff University, September 7-10 2000

A rational model for the teaching of mathematics:

the impact of comparative studies

Paul Andrews and Anne Sinkinson

University of Cambridge

Faculty of Education

17 Trumpington Street

Cambridge

CB2 1QA

+44 (0) 1223 336290

Comparative studies: a rational approach to the teaching of mathematics

Abstract

Recent international tests indicate that English mathematics teaching may be less effective than that found in economically comparable countries. In this paper we review some of the literature pertaining to mathematics teaching overseas and hypothesise the existence of practices and perspectives common to successful systems which, we argue, could be transferable to the English context. We present our findings in two parts. The first addresses class management and lesson structures whilst the second focuses on the conceptualisation of mathematics for teaching. Among the former are the whole class teaching of mixed-ability teaching groups, the operation of learners in the public domain and lessons dominated by teachers talking, or managing the talk of others, for the majority of the lesson. Among the second are teachers acknowledging the complexity of mathematics and its commensurate language; mathematics as problem-based with problems exemplifying generality, teachers encouraging an engagement with proof, forcing learners to think and revisiting material covered previously; mathematical applications being subordinated to the subject; and little emphasis on the practice of routine procedures. We argue that much of the British educational tradition and policy is falsely premised and propose that attainment will not be raised until these are addressed. Some examples of the sorts of problems we regard as exemplifying the rational approach to mathematics teaching are offered.

Comparative studies: a rational approach to the teaching of mathematics

Introduction

The last few years, largely as a consequence of the third international mathematics and science study (TIMSS), has found the teaching of mathematics under scrutiny from politicians, educationalists and the media. The most extensive piece of comparative research ever undertaken, TIMSS surveyed the attainment of "more than half a million students at five grade levels in 15,000 schools and more than 40 countries around the world" (Beaton et al 1996, p7). Most of the interest lies in the study's having identified discrepancies in the educational attainment of children from nations with largely comparable economic, industrial, and, frequently, cultural and social traditions.

Inevitably the study and its processes have not been without criticism. Wiliam (1998), for example, has argued that inconsistencies in sampling procedures, United States domination of the design process, problems of question translation and irregular test implementation have colluded in producing results of dubious validity. Keitel and Kilpatrick (1999) criticise the organisation, scale and accountability of the study and argue that supplementary studies designed to inform the surveys were too large and too biased to be of use. Stronach (1999) suggests that projects such as TIMSS have (inadvertently) become comparisons of global capitalist competitiveness and lost sight of more fundamental educational objectives.

Keitel and Kilpatrick (1999) challenge the study in respect of its management, funding, dissemination and the relationships between them. Indeed, they cite examples of individuals who framed national findings in ways which persuaded respective governments to fund their own research interests. They express concern about the representativeness of samples and the generalisability of findings and argue that many analyses were too flawed to be of value. Indeed, they write that those responsible frequently failed to re-define objectives in the light of evidence showing meaningless results - they were too wedded to the use of quantitative techniques in their interpretation of qualitative data. Looking beyond the surveys, and acknowledging the ambition of an attempt to explain, they criticise the video study - dominated by the US in respect of management, analysis and dissemination - and the case studies - too large and generating too much data to be of use.

Stronach (1999) attacks the international competitiveness which, he believes, underpins much recent educational policy. He writes that the school effectiveness movement, and with it studies like TIMSS, are mappings onto economic effectiveness where the "connections between the economic and the educational are far from self-evident, but are construed as a kind of universal common-sense magic" (p4). Sadly, Keitel and Kilpatrick (1999) argue that politicians' interest in projects like TIMSS lies more with ensuring their nation compares favourably with those perceived to be successful than any genuine sense of educational improvement - whether selfishly at home or altruistically abroad.

There is validity to many of these criticisms although they are inevitably framed by interests which are not beyond challenge. For example, Wiliam (1998) argues that the TIMSS sampling procedures militated against English participants - not least because of the practice in some countries of repeat years and the lower levels of pupil absenteeism reported by English teachers. Alternatively, of course, it would not be unreasonable, by virtue of their having received an extra years' schooling, to expect the performance of English children to exceed that of children from overseas.

Both Wiliam (1998) and Keitel and Kilpatrick (1999) suggest that the United States' dominance of the processes of the study militated against the interests of other participants. Wiliam writes of a process of "'horse-trading' whereby representatives of each country try to get items that reflect their own distinctive approaches to topics included in the tests" (p34) and argues that such a process inevitably favour the strong. Keitel and Kilpatrick believe, despite claims to the contrary, that the use of English as the official language of the project may have militated against equity. An alternative perspective might be that the relatively poor performance of both England and the United States was indicative of little such bias.

Keitel and Kilpatrick also question the representativeness of the lessons used in the video study and suggest that they had the appearance of model lessons of the form used in Open University training videos. However Matthews (Bloomfield and Matthews 1999) has stated that he regards the lesson shown in the TIMSS publicity video as typical of lessons to be found in the United States - teachers simplifying mathematics to meaningless levels and then answering many of their own questions.

The TIMSS data suggests that the mathematical attainment of children in both England and the United States is poorer than either would like. Indeed, it indicated that, at age 13, English and American children were less successful on every one of the six reported topic areas than children from a range of Asian and European countries (Beaton et al, 1996). In many ways these countries are our economic, and in some cases cultural, peers and their apparent higher achievement in mathematics education is a cause for concern. To deny the crisis and criticise the means by which the concern was raised is to miss an opportunity to move forward.

However, in order to make progress we need to understand more fully those pedagogic practices and traditions which contribute to pupil achievement. There have been appeals from some quarters to look to the Pacific Rim for solutions. Such appeals may be unhelpful and we sketch below our reasons for believing this to be the case. Korea, for example, is clearly a successful nation in respect of TIMSS attainment. However, there are voices from within which question its practices. It operates an "inflexible skill- and fact-oriented curriculum" (Lew 1999, p220) which, he argues, is harming the nation's children. He describes how learners, despite mechanical successes, fail to derive meaning from their experiences of mathematics with intended curricular aims being poorly reflected in pupils' attainment. Lin and Tsao (1999) describe the situation in Taiwan, another successful nation in respect of mathematical attainment, where understanding and the intended curriculum are subordinated to the need for students to pass entrance examinations to, first, the best secondary schools and, second, the best universities. Their description of Mr Chen, a highly respected teacher, is enlightening.

"Mr Chen,..., bases his teaching on a meticulous and fine-grained classification of mathematics content. He has classified simultaneous linear equations into 20 categories, some based on coefficients such as integral, decimal, fractional, with or without symmetric pattern, some on types of substitutions, some on the form of expressions, etc. He has devised a solution for each type. Experience shows that good performance is achieved in any test if a student is familiar with all these types and the corresponding solutions" (228).

The initial reaction of a Western mathematics educator is to reject such teaching as anachronistic in a discourse dominated by different species of constructivism. However, summary dismissals may be premature. Lin (1988), for example, replicated the mathematics component of the Concepts in Secondary Mathematics and Science (CSMS) study (Hart 1981) with more than two thousand Taiwanese students and obtained some unexpected and informative results concerning students in the middle years of secondary education. Firstly, and despite an emphasis on algorithmic teaching, a significantly greater proportion of Taiwanese children attained higher levels of mathematical understanding than children in England. Secondly, a higher proportion of Taiwanese than English children failed to attain the lowest level of mathematical achievement as defined by the CSMS study. It was conjectured that the predominance of Taiwanese children at the lower end of the attainment spectrum was a consequence of their never having developed their own strategies to solve simple problems - strategies which English children had been encouraged to develop and which, it is argued, contributed to their inability to handle more sophisticated problems in the way that Taiwanese children, with their repertoire of algorithms, could. Such research serves to remind us that effective teaching and learning are complex activities which are not easily reduced to lists of observable behaviours.

However, complexity should not deter us from seeking productive ways forward. There is evidence of the existence of national pedagogic traditions (Schmidt et al 1996) which should prove informative to a review concerned with identifying effective practice. The Survey of Mathematics and Science Opportunities identified, at least tentatively, the "characteristic pedagogical flow" (Cogan and Schmidt 1999 p74) of each of the countries they researched - characteristic because they appeared to stem from shared perspectives and experiences and flow because the practices seemed unconscious and familiar in their manifestation. That is,

"teaching and learning are cultural activities (which)... often have a routineness about them that ensures a degree of consistency and predictability. Lessons are the daily routine of teaching and learning and are often organized in a certain way that is commonly accepted in each culture" (Kawanaka et al, 1999, p. 91).

This paper is an attempt to identify, from the available literature, some of the characteristics of, or factors which are believed to contribute to, effective teaching and learning of mathematics. The criteria for success are the considered responses, evaluations and judgements of professional observers looking in the classrooms of teachers. Indeed, we argue that despite the criticisms of, say, Keitel and Kilpatrick, the TIMSS video study has proved informative through its exploration of three keys areas of mathematics education: the nature of the work environment, the nature of the work in which students are engaged and the methods teachers use for engaging students in their work. (Kawanaka et al 1999).

Most comparative work has been undertaken and reported by foreign observers. This means that whatever has been observed will have been interpreted in various ways before reaching the page - not least because it is difficult for foreign observers to suspend their own cultural perspectives when observing another. Also, accounts of lessons, no matter how well constructed, rarely account for why teachers behave as they do. Consequently, inferred generalities are necessarily tentative. In general our review has been undertaken within the following loose framework - loose in the sense that informative studies not fitting the criteria have not necessarily been excluded. Thus, in general, we have focused on

•Studies from countries which formed part of the TIMSS sample in order to provide a contextual, or mathematical attainment, base-line. In reality, and acknowledging the availability of published research, this has meant studies undertaken in France, Hungary, Japan and the United States. The first three because they represent a cross section of countries with high mathematical attainment and the latter because, like England, the US appears to have problems of underachievement.

•Studies undertaken at the TIMSS population two level - children at age thirteen. This decision was informed by two important factors. Firstly, depending on the system under examination, children at this age are either in the last years of primary or the first years of secondary education which we felt this would be an informative positioning device for comparison. Secondly, the availability of studies conducted at this age meant that generalities might be accessible.

What is seen in classrooms where mathematics is taught successfully

If, as Schmidt et al (1996) suggest, national pedagogic traditions can be identified then it would seem reasonable not only to expect them to be evidenced in the ways pupils are grouped and lessons structured but also that distinctive structural devices would be related to mathematical attainment. We consider, successively, research into the pedagogic traditions of Japan, France, Hungary and the United States in order to justify our conjecture that among the more successful practices can be found the following:

•Learners are generally taught in mixed ability classes

•Teachers expect to teach each class as a unit

•Teachers dominate the lesson's proceedings with their talking, or managing the talk of others, for the majority of the lesson

•Learners are expected to operate in a public domain

•Learners spend little time working alone from textbooks

•Routine exercises are few and involve a small number of problems

In Japan pupils are taught in mixed ability groups because "grouping by ability within classrooms is discriminatory and may hurt students emotionally" (Stevenson 1999, p117). A "typical Japanese lesson revolves around one topic, which is introduced to the students as a problem to solve" (Jacobs et al 1997, p9). Lessons tend to follow a similar format in which the teacher conducts a discussion around the previous lesson's problem before posing the new one. Pupils work on the problem - individually or in pre-determined groups called hans - before a discussion in which their ideas are presented on the board. The teacher summarises the discussion, pupils work on several similar problems and homework is set (Whitman and Lai 1990, Stigler et al 1996, Jacobs et al 1997, Kawanaka et al 1999, Cogan and Schmidt 1999). The majority of a Japanese lesson is spent with pupils either attending to their teacher - in all its manifestations - or working collaboratively on the problems offered the class (Stigler et al 1987, Schmidt et al 1996) Relatively little time is given to the completion of text book exercises. A Japanese lesson moves slowly because, for Japanese teachers, thinking about a problem and focusing on, and sharing, processes is more important than the product (Stigler and Perry 1990).

French students are class taught the same curriculum in mixed-ability groups in all subjects up to the age of sixteen - a tradition premised on the notion that there must be equality of provision (Jennings and Dunne 1996, Pepin 1998). Teachers accept the premise of curriculum entitlement arguing "that it is not essential for a child to demonstrate a well-defined knowledge and understanding of a topic before moving to the next." (Jennings and Dunne, p50). The lesson itself comprises several phases and starts with homework being discussed and corrected. This is followed by the presentation of the new topic content before pupils begin a short period of individual working on problems set from a text. Lastly, the next homework is set (Schmidt et al 1996). At any time pupils may be called to the board to share solutions or to offer ideas because public sharing is seen as an integral part of the process (Schmidt et al 1996, Pepin 1996). During the middle, or teaching, phase pupils are frequently invited to work on tasks which verify that which has been introduced - proofs and confirmatory examples are discussed simultaneously (Jennings and Dunne 1996, Schmidt et al 1996).

Hungarian pupils are generally taught in mixed ability classes until they move to differentiated secondary schools at age fourteen (Báthory 1992, Andrews 1994, Hatch 1994). Lessons tend to follow a relatively well-defined pattern and begin with a review of the homework set the previous lesson. Homework rarely involves more than a handful of questions usually selected from the text that all children have with them at all times. Solutions are shared publicly with children being invited to the board (Andrews 1995, 1999, Hatch 1999). This may be followed by a brief period of revision or mental work with the teacher offering questions orally for immediate response before the day's work is introduced. The teaching phase of the lesson consists of, possibly, several episodes in which a question or problem is posed, children work on it for a few minutes on their own before solutions are publicly shared and the teacher reviews the outcomes from the task. At the end of the lesson homework is set in preparation for the next lesson. Pupils spend little time working through routine exercises or discussing their work in small groups (Hatch 1999) - teachers tend to view the whole class, rather than the individual, as the learner (Hatch 1999, Andrews 1999).

A different perspective emerges from the United States where a typical lesson can be divided into two main parts. In the first the teacher presents the topic for the day and lectures on it. In the second students work individually on practice problems in order to apply and consolidate the information they have acquired from their teacher (Schmidt et al 1996, Jacobs et al 1997. Described as the acquisition and application phases they may alternate several times within the one lesson (Kawanaka et al (1999). American children spend long periods of time working independently on practice problems and receive little instruction (Stigler et al 1987). They are given little opportunity to discuss their work or share perceptions because teachers see their role as encouraging individual work supported by one to one teacher-pupil conversations (Stigler et al 1996). Interestingly, there are indicators of divergent traditions within the United States which are probably a consequence of that country's geographical diversity - lessons in Hawaii are closer in structure to those of Japan than the mainland US (Whitman and Lai 1990).