Significance attached to the ‘what’ of mathematics education

The lack of clarity on how decisions about the what of mathematics education can be made, contrasts sharply with the importance that is attached to it. A broad audience, ranging from parents, caregivers, business leaders and politicians to professional educators, cares deeply about what children learn in school (see Senk & Thompson, 2003). The many countries that formulated standards in the last decade and the interest in international comparative studies such as TIMSS and PISA also reflect this engagement. Although it looks as if this awareness is typical of our present time, this is not the case. Even the question of whether school was teaching how to calculate the amount of tax that citizens have to pay (see Kilpatrick, 1992).

Just as in Horace Mann’s time, the question about the what comes up most significantly within the context of evaluation and assessment. We have to ask very explicitly what knowledge we want students to have (Romberg, 1993). Moreover, we have to ask this question continually. As Romberg and Kaput (1999) make clear, we can no longer assume that mathematics is a fixed body of concepts and skills to be mastered. Mathematics is a living, dynamic discipline, and therefore further changes in school mathematics are inevitable.

Despite the great concern about what schools are teaching, asking for content specification is often—in one way or another—viewed with some suspicion. For this reason, what-questions are often followed with an explanation that tones down the question and with warnings about the danger of focusing on content. According to Schoenfeld (1994) the danger of the ‘content inventory’ point of view comes from what it leaves out: the critically important point that mathematical thinking consists of a lot more than knowing facts, theorems, techniques and so forth. In line with him, Burton (2002a) mentions the danger of fragmentation. This danger is also often linked up with the traditional view on subject matter which still holds sway, especially in state guidelines and in textbook specifications (see also Bereiter, 2002). In contrast to the conventional approach of itemizing what is to be learned, the more modern view, associated with labels such as constructivism and conceptual change teaching, looks at subject matter somewhat differently (ibid.). The focus is more on mathematics as a whole and the relationships between the different parts of content. As is shown in the first part of this paper, this focus does not necessarily imply similar content choices—with a result of different learning outcomes.

There is a wide range of studies showing that ‘students learn what they have an opportunity to learn’—as Hiebert (1999, p. 12) says. If extra attention is paid to particular content, then, on average, students learn this content. On the other hand, if a mathematical topic is not taught, then it is often not learned by the students. In other words, different goals and curricula lead to different patterns of achievement (Hiebert, 2003; Kilpatrick, 2003). Again this is not a new finding. The study by Walker and Schafferzick (1974) in which they reviewed a number of curriculum evaluation studies is well known. Similar results were also found in a study carried out in the Netherlands aimed at comparing the achievements of students that had been taught using two different textbooks (Gravemeijer et al., 1993). These findings have important consequences for decisions about the what of mathematics education.

The problem is that the what is not a unified thing: ‘In different countries across the world and within countries themselves, school mathematics looks different’ (Lerman, 2004, p. 340). The substantial differences can be found in grade placement of mathematics topics (Senk & Thompson, 2003) and there is little agreement across the nation on the most appropriate content for any grade level (Lambdin et al., 2004). On an international scale the situation is similar. This is clearly shown by the variation in topics in textbooks found in the TIMSS related study ‘According to the book’ (Valverde et al., 2002).

The reason for this content difference has to do with making different choices. As stated by Lerman (2004, p. 341), ‘to teach in school is always a selection from what we (or whoever decides) perceive to be Mathematics (academic, in business/ industry, etc). Values are always associated with that choice, values as to what education should be all about and in particular what mathematics education should be all about. … These are political battles …’ In other words, it might be no surprise that the NCTM Standards sparked a nationwide ‘math war’ in which positions were taken by mathematicians, mathematics educators, teachers, administrators, parents and politicians (see Becker & Jacob, 1998; Lambdin et al., 2004). Senk and Thompson (2003) stress that this debate emerged because there is much disagreement about what skills are

needed for productive citizenship as well as on whether students can apply their knowledge in everyday life, the workplace, or higher education. According to Hiebert (1999) this lack of consensus is understandable given rapid changes in mathematical competencies that are important in the workplace and the increasing availability of computational technologies.

However, in addition to this, we should not forget that the different agencies that are struggling for what should constitute school mathematics are unlikely to accept that one group dictates what should be ‘worthwhile’ knowledge (Burton, 2004). Moreover, at the same time it is not very clear who has to decide upon the what. In this respect the April 1999 issue of the NCTM discussion journal Mathematics Education Dialogues was very revealing. This issue was completely dedicated to the question ‘Who should determine what you teach?’ In the editorial introduction it stated that there are different ideas about the level on which decisions about the what should be made. Some people believe that teachers, who are the closest to their students and who know their local situation better than outsiders, should make these choices. Others think that they should be made more centrally, at the school level, the school district level, state or provincial level, national level, or even internationally. By discussing the who-question in this way it looks as if the what of education is a kind of optional up-to-you thing. But is this really the case? This question can be asked in particular when a whole curriculum has to be designed. The macro-decisions that are required for such a design go far beyond the level of the individual teacher—which however does not mean that teachers cannot be involved in making choices of what content is most worthy of being taught.

Because ‘no teaching is possible without choices concerning goals, content and methods … it doesn’t seem

plausible to leave the questions of norms and values outside the discipline of didactics’ (Bengtsson, 1997, p. 2). As discussed earlier, this is in contrast with the perception that many researchers of mathematics education have that including values imposes a threat to the scientific quality of their work. This narrow interpretation of what it means to be ‘scientific’ implies that questions about the what can only be answered outside the ‘scientific’ discipline and that ‘scientific’ activity is restricted to monitoring the process of negotiation with all interest groups by ‘scientific’ survey methods. I believe that such an a posteriori approach does not offer us enough possibilities for further development of the practice and the theory of mathematics education. Instead of putting decision-making about the content and goals outside of mathematics education research, I believe that it should be at its heart.