International Day of Mathematics
What makes learning mathematics difficult is not its abstract nature, but the common practice of avoiding abstraction in mathematics education. This stands in stark contrast to the observation of the Belgian mathematician Ann Dooms. She emphasizes that abstracting and dealing with abstraction are natural human traits. She points to how young children learn the concept of “cat.” We present children with countless examples—real cats, drawings of cats, toy cats, and so on—and always label them as “cat.” This is how the child forms the concept of “cat” by generalizing about and abstracting from concrete examples. As an early example in mathematics, she mentions the formation of numbers, which children abstract from the activity of counting. Of course, abstraction becomes more difficult when progressing in mathematics. Therefore, schools often revert to teach definitions, rules, and procedures—in order to avoid abstraction. The consequence of this, however, is that no conceptual foundation is provided. This causes problems when more flexibility is required and can lead to math anxiety.
There is, however, an alternative: helping students develop abstract mathematics. For this the renowned scholar Freudenthal recommends the method of “guided reinvention”. This involves drawing on our knowledge of the history of mathematics. Figuring out how concepts, notations, models, and procedures emerged, what unnecessary detours were made, and how and why breakthroughs occurred. We can use this knowledge to outline a smoother path by which a given part of mathematics could have been invented.
However, available classroom time is limited, and we may wonder whether there is enough time for reinvention processes. Until now, the trade-off between developing insight and pursuing procedural skills has favored the latter. Nowadays however, as machines perform more and more calculations for us, the emphasis can be shifted from striving for procedural mastery to abstraction and reinvention.
An important caveat is, that this is not easy. Reinvention requires a specific classroom culture. Students are expected to generate their own solutions, explain them, and justify them. Furthermore, they are expected to try to understand and, if necessary, challenge, the solutions and reasoning of their classmates.
Moreover, for each topic, a description of a reinvention route has to be available. Based on such a description, teachers may select daily teaching activities. More specifically, they are to choose assignments that connect with the students’ thinking and require mathematical reasoning that may lead to deeper mathematical understanding. Furthermore, teachers will have to ask appropriate follow-up questions and lead productive class discussions to further stimulate student thinking.
In conclusion, we may say that the time is ripe for making a transition to more conceptual mathematics education. However, we acknowledge that this requires a significant investment in designing teaching materials and equipping teachers with the necessary knowledge and skills.
Emeritus Professor at Eindhoven University of Technology in the Netherlands. He has made significant contributions to RME theory – a theory for designing mathematics education – and to the theoretical foundation of Design Research as a scientific research method.