International comparative research in mathematics education

SIMILARITY AND DIFFERENCE IN INTERNATIONAL COMPARATIVE RESEARCH

International comparative research in mathematics education is a growing field. Experiences from recent and ongoing studies seem to have huge impact on both the field of research and the field of practice. The very idea of both grasping and making use of diversity lies in the heart of all comparative approaches. However there is an ongoing need for enlightened discussion on how the character of these results relate to the research methods and techniques used and the theoretical and analytical perspectives enacted in the research. The main focus of the forum is how these different comparative approaches, and the consequent and profound differences in project outcomes, can inform our individual and collective ways of understanding

learning and teaching in mathematics.

The idea is to contrast and discuss different approaches, and to discuss both differences and similarities, especially in the character of what we can learn about the learning and teaching of mathematics in classrooms from these studies. What are the possibilities and limitations associated with different approaches? The different types of comparative research that are represented in this forum are:

1.OECD-PISA, Organisations doing large scale studies with questionnaires and tests

2.IEA-TIMSS, Organisations doing large scale video studies

3.LPS, Researchers doing large studies on their own initiatives

4.Small scale comparative studies

Schmidt, McKnight, Valverde, Houang and Wiley (1997) investigated the mathematics curricula of the “almost 50” countries participating in the Third International Mathematics and Science Study (TIMSS). The documented differences in curricular organisation were extensive. Even within a single country differentiated curricular catered to communities perceived as having different needs. Countries differed in the extent of such differentiation, in the complexity or uniformity of their school systems, and in the distribution of educational decision-making responsibility within those school systems. Given such diversity, the identification of any curricular similarity with regard to mathematics should be seen as significant. And there were significant similarities. There were similarities of topic, if not of curricular location; broad correspondences of grade level and content that became differences if you looked more closely; differences in the range of content addressed at a particular grade level, but which repeated particular developmental sequences where common content was addressed over several grade levels. In another international study of mathematics curricula, the OECD study of thirteen countries’ innovative programs in mathematics, science and technology found that, “Virtually everywhere, the curriculum is becoming more practical” (Atkin & Black, 1997, p. 24). Yet, despite this common trend, the same study found significant differences in the reasons that prompted the new curricula (Atkin & Black, 1996). These interwoven similarities and differences are the signature of international comparative research in mathematics education (Clarke, 2003).

Schmidt, McKnight, Valverde, Houang, and Wiley (1997) reported that differences in the characterization of mathematical activity were extreme at the Middle School level; from ‘representing’ situations mathematically, ‘generalizing’ and ‘justifying’ to ‘recalling mathematical objects and properties’ and ‘performing routine procedures.’ Despite the apparent diversity, it was the latter two expectations that were emphasised in the curricula studied. Given the documented diversity, it is the occurrence of similarity that requires explanation. Some curricular similarities may be the heritage of a colonial past. Others may be the result of more recent cultural imperialism or simply good international marketing.

In attempting to tease out the patterns of institutional structure and policy evident in international comparative research (particularly in the work of LeTendre, Baker, Akiba, Goesling, and Wiseman, 2001), Anderson-Levitt (2002) noted the “significant national differences in teacher gender, degree of specialization in math, amount of planning time, and duties outside class” (p. 19). But these differences co-exist with similarities in school organization, classroom organization, and curriculum content. Anderson-Levitt (2002, p. 20) juxtaposed the statement by LeTendre et al. that “Japanese, German and U.S. teachers all appear to be working from a very similar ‘cultural script’” (2001, p. 9) with the conclusions of Stigler and Hiebert (1999) that U.S. and Japanese teachers use different cultural scripts for running lessons. The apparent conflict is usefully (if partially) resolved by noting with Anderson, Ryan and Shapiro (1989) that both U.S. and Japanese teachers draw on the same small repertoire of “whole-class, lecture-recitation and seatwork lessons conducted by one teacher with a group of children isolated in a classroom” (Anderson-Levitt, 2002,

p.21), but they utilise their options within this repertoire differently.

LeTendre, Baker, Akiba, Goesling and Wiseman (2001) claim that “Policy debates in the U.S. are increasingly informed by use of internationally generated, comparative data” (p.3). LeTendre and his colleagues go on to argue that criticisms of international comparative research on the basis of “culture clash” ignore international isomorphisms at the level of institutions (particularly schools). LeTendre et al. report yet another interweaving of similarity and difference.

We find some differences in how teachers’ work is organised, but similarities in teachers’ belief patterns. We find that core teaching practices and teacher beliefs show little national variation, but that other aspects of teachers’ work (e.g., non-instructional duties) do show variation (LeTendre, Baker, Akiba, Goesling & Wiseman, 2001, p. 3) These differences and the similarities are interconnected and interdependent and it is likely that policy and practice are best informed by research that examines the nature of the interconnection of specific similarities and differences, rather than simply the frequency of their occurrence. This Forum uses brief presentations relating to five different research projects, each representing a very different approach to international comparative research in mathematics education, as a catalyst for discussion of how such research might best inform theory and practice in mathematics teaching and learning.

In most international comparisons of Mathematics Education (ME) it is achievement in terms of test results that is compared. From such outcome comparisons we can conclude that students in some countries are doing better than students in other countries. Why this is the case is impossible to tell without further information. But we might also collect data about the prerequisites for learning mathematics, such as the size of per student investments in education in different countries, class size, number of hours in mathematics teaching etc. If the correlation between achievement and prerequisites variables, like those above, were high, we could possibly come up with conjecture, such as one country could boost achievement in mathematics by

increasing its investments in education, reducing class size, increasing the number of class hours etc. But such correlation evidence is extremely scarce. If outcome comparisons have such limitations the next move is in a way self-evident, since we need to know what is happening in the teaching process in order to understand the outcomes of this process. And this was exactly what the TIMSS-99 did in the most advanced attempt to produce plausible explanations of differences in Maths achievement between different countries. One hundred year 8 classes were selected by random sampling in seven countries. In each class one lesson was video-recorded.

When all the data were collected and analysed the results were published on the internet. We could compare the different countries with regard to, for instance: Length of lesson, Time devoted to mathematical work, Time devoted to problem segments, Percentage of time devoted to independent problems, Time per independent problem, Time devoted to practicing new content, Time devoted to public interaction, Number of problems assigned as homework, Number of outside interruption, Number of problems of moderate complexity, Number of problems that included proofs, Number of problems using real life connections, Number of problems requiring the students to make connections, Time devoted to repeating procedures, Number of words said by teacher, Number of words said by students, Number of lessons during which chalkboard was used, Number of lessons during which computational calculators was used. Now, it would not be unreasonable to expect several of such factors be correlated with differences in achievement between

countries, given that more or less the same Mathematical content has been covered in different countries. This means that not only factors like those presented above, referring to how Mathematics is taught varied between the countries but also that the content covered, i.e. what was taught in Mathematics varied between the countries as well. This in turn means that the characteristics of ME referred to different things in different ways. Small wonder that basically no correlations with achievement were found!

The close relationship between what the students have the opportunity to learn and what they actually learn is logically necessary and has also been empirically demonstrated (Marton & Morris, 2002; Marton, Tsui et al, 2004) and this relationship has been taken as a point of departure for improving learning in ME. By finding out the necessary conditions for a certain group of students to appropriate a certain object of learning and by bringing those necessary conditions about the likelihood of learning is most considerably enhanced (Lo, Marton, Pang & Pong, 2004). So, if we want to understand differences in achievement in ME between students in different countries we must explore to what extent the objects of learning reflected in the achievement test have been possible at all to appropriate. And in order to do so we have to look at how the same objects of learning have been handled in classes in different countries. Now, to us it seems that the international comparative studies such as TIMSS-R och TIMSS are not designed to be comparative in essence, since they show little interest in e.g. keeping the content invariant. So what are they then?

We put forwards two kinds of understanding. First, an important side-effect is making of what is important in Maths Education by means of the items that are used in order to measure knowledge in Mathematics. International comparisons such as the PISA or the TIMSS are not only producing data for comparisons, they also produce conceptions of what is important and of value in Maths Education. They are not only comparing, they are participating in the social construction of curricula in Maths Education. This thought is well developed in the work of Ian Hacking (1999). From this point of view, international comparisons are about homogenisation of Maths Education. Second, going back to the correlations between different variables that are sought in international comparisons we find another thought and that is that given a certain correlation it is predicted that some fact will have an impact on another fact. Given what we know about correlations on one side and explanations on the other side such conclusions are of course problematic. But we think that, on a pragmatic level, even the search for correlations between facts is problematic. What we find is an instrumentalistic system of reason, that construct technical directives (von Wright, 1972) based on abstract numerical relations instead of e.g. didactical arguments. Stated otherwise, what is compared in international comparisons of preconditions and outcomes are educational phenomena shrunk to fit an instrumentalistic system of reason. In a word, international comparisons carried out in this way are examples of intellectual thrift of content as well as of educational reason.