What distinguishes humans from other animals on our planet? Humans are surrounded by hardware stores, drugstores, furniture stores, and haberdashery departments! More abstractly expressed: humans invent and use tools, and stores are full of them, ranging from sewing needles and cabinets to toothpaste and lawnmowers. Tools allow us, on a small scale, to manipulate individual cells or, on a large scale, to explore the solar system.
Tools can be understood as materializations of rules. For example, crowbars and door handles implement the principles of leverage, indirectly revealing how humans operate: through observation, we initially identify regularities in the form of if-then relationships. We isolate and reproduce these effects through experiments and ultimately preserve them as devices or tools, which enable these discovered regularities to be put into effect in specific situations.
The foundation of all this activity is the ability to handle rules—that is, to recognize, apply, and purposefully combine them —a capacity summarized by terms like rationality or understanding.
In this context, mathematics can be viewed as a particularly distilled form of human rationality because it allows for precise formulation of regularities and their combination into new rules. Mathematics, as the science of the regulated, is thus an abstract version of tool-making and tool-use, becoming itself an important and powerful tool for humans.
If the goal of education is to develop and train the innate human abilities for movement, communication, empathy, emotion, and rationality, then the question of whether mathematics plays an important role here is not really in doubt. However, whether our current approach to mathematics education is genuinely aimed at training these rational skills should always be re-examined.
It is clear that mathematics cannot be conveyed without mathematical content. On the other hand, these contents should not be so prominent that they become the primary focus of training and assessment. The aim should be mastering rationality, not mastering fractions: someone who approaches problems rationally can find the rules for fractions on their own and repeatedly if they forget them. Those who have less training in systematic, rational approaches may struggle more and cling to ready-made formulas and recipes, which can quickly become jumbled in their minds. When, at the end, simplifications are made from sums, it is not always clear whether the student or the education system is at fault.
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