Although our knowledge about the teaching and learning of mathematics is far from complete, it looks as though we are making more and more progress on the question of how students can learn mathematics (see Schoenfeld, 1994; Wiliam, 2003). Moreover it seems that the views on the how are growing closer. For instance, Putnam (2003) describes the common assumptions on how children learn mathematics that can be found in the newest reform curricula for primary school mathematics education. In contrast with the traditional approach , according to Putnam (ibid.), these curricula view the teacher as guiding the students through discussions and learning activities and conceptualize the learning of mathematics as building on the students’ intuitive understanding, providing the students with settings and problems that are meaningful for them, and proceeding from informal, grounded understanding to more formal knowledge of the symbol systems of mathematics.
These ideas about the didactics of teaching mathematics are also very similar to the principles of Realistic Mathematics Education (RME), which was the Dutch answer to the need to reform the teaching of mathematics. The roots of RME go back to the early 1970s when Freudenthal and his colleagues laid its foundations. Briefly, these principles include taking students’ initial understanding as a starting point, providing them with problem situations which they can imagine, scaffolding the learning process via models, and evoking reflection by offering the students opportunities to share their experiences (for a more elaborated overview see Van den Heuvel-Panhuizen, 2001a). In the Netherlands, as in the USA, this approach contrasts with the traditional mechanistic way of teaching mathematics in which the students are, for instance, offered fixed solving procedures in which they are to be trained through the use of exercises.
However, contrary to what might have been expected, this agreement on principles in no way assures similarity in mathematics programmes. I became explicitly aware of this once again when I read McClain’s (2002) chapter in Goodchild and English’s (2002) book on researching mathematics classrooms (see Van den Heuvel-Panhuizen, 2002). The chapter reports on a classroom-teaching experiment carried out by herself and Cobb and his colleagues. This focused on the development of a socio-constructivist instructional sequence for adding and subtracting three-digit numbers.
As McClain describes, the content of the instructional sequence that was investigated in the classroom-teaching experiment was, among other things, based on RME principles. The result was quite remarkable. Although the sequence reflects some important RME principles, I would not call it an RME teaching unit. The narrow focus on algorithms in conjunction with heavy emphasis on place value supported by Unifix cubes is quite different from an RME approach. An example of this approach can be found in the TAL learning–teaching trajectory for whole number calculation (Van den Heuvel-Panhuizen, 2001b; more about TAL is in the section in which the second experience is discussed). Characteristic of this trajectory is the integration of written and mental calculations, an orientation towards numbers instead of digits, and the use of the numberline to support the development of strategies to solve problems such as 265–194. Without judging either approach, and certainly not saying that the second approach is better than the approach that was chosen in the classroom teaching experiment, I am left wondering about the evidence base that motivated this different choice in content. An explanation for this might be that the socio-constructivist instructional sequence was inspired by the teaching tenets of RME. These tenets, however, do not completely encapsulate the domain-specific education theory of RME. In addition to these teaching principles, RME also implies choices for particular goals and content—in its broadest sense—resulting from ‘didactical phenomenological analyses’ (as suggested by Freudenthal, 1983). These include the analysis of mathematical concepts from a didactical perspective, while bearing in mind knowledge of the history of mathematics, evidence from students’ learning, and using experience from collaboration with teachers, teacher educators, teacher counsellors and textbook writers. Taking this into account it is understandable that the content of instructional sequences based only on the RME principles may differ from the previously described RME approach.
Another thing is that the classroom experiment did not bring new insights that led to a fundamental revision of the conjectured trajectory. The revisions were mainly related to the micro-didactical how-questions (How do the students learn? and How to teach or how to organize an instructional environment to optimize the students’ learning?), rather than to the macro-didactical what-questions (What should be the learning goals? What content should constitute the programme?). Although this may lead to the inference that questions about content cannot be covered within the scope of classroom teaching experiments, I cannot believe that this is a correct conclusion.
Because the answers to macro-didactical what-questions are significantly informed by didactical phenomenological analyses loaded with experiences and opinions of experts in and out of the school, evidence from classrooms can play an important role in providing answers to such questions. However, to achieve this, it is necessary to include research activities that can inform our macro decisions.
As an example, I want here to draw attention to Treffers’ (1987) plea to use problems that can be solved in a variety of ways. Students’ work on these problems can bring to light their levels of understanding and arithmetic skills at a particular moment. Apart from the fact that this information is important for taking micro decisions it also guides the macro-decisions. The cross-sectional view of the class (the different levels of understanding of the students in a class at one particular moment) that is produced in this way shows at the same time a longitudinal section of a learning–teaching trajectory or a part of it. The solution strategies of individual students reveal collectively essential elements of the long-term path that students need to travel. What is found in the classroom in the present anticipates what is on the horizon and beyond.
Returning to the purpose of McClain’s chapter I think that research activities by which we can expose students’ future learning must have a place within the methodology of classroom teaching experiments. They can bring us closer to the heart of the design process where instructional trajectories and sequences come into being.
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