Abstract
Inspired by a certain reading of Wittgenstein , especially his notion of rule-following , this paper sketches an outline of an idea of elementary mathematics (arithmetic and basic geometry) as an intrinsic part of intellectual formation, or Bildung . In particular, I argue that we need to distinguish between mathematics as an activity and mathematics as a body of knowledge, where it is the first mentioned that is primary for the character of thinking. My claim is threefold: (i) The development of the capacity to think mathematically, together with the ability to read, write and speak one’s native tongue with clarity and precision, ought to be the primary aims of primary and secondary schooling; (ii) At this basic level, the capacity to think mathematically is inseparable from the capacity to reason in general and should be seen as an essential part of the latter; (iii) These two claims, if correct, have profound consequences for how we ought to think about the form and content of teaching .
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Notes
- 1.
The English phrase “in mufti” is here a colloquial translation for im Zivil, i.e., in civilian dress, especially for someone who normally wears formal attire such as military uniforms. The term derives from the dressing gowns favored by off-duty British officers in the early 19th century.
- 2.
Following convention, titles for Wittgenstein’s works are abbreviated (PI = Philosophical Investigations, RFM = Remarks on the Foundation of Mathematics, OC = On Certainty), with section (§) or page number (p.), with full citation and initials (e.g., RFM) in the References.
- 3.
Later in the essay, I will discuss in more detail an idea of Bildung , intellectual formation through education , which is not primarily concerned with erudition, eloquence or style, but which rather has to do with refined judgment, what Kant called, respectively, the maxims of “unprejudiced,” “enlarged” and “consecutive” thought in the Critique of Judgment. My reading of it is influenced by Hannah Arendt’s interpretation in Arendt (1992). See also Arendt (1961).
- 4.
For a comparison of the two methods of instruction, see Stevenson and Stigler (1992). In the book, the authors attribute the superiority of Japanese and Chinese student skills in mathematics over American students to this difference in pedagogical culture .
- 5.
The OECD’s Programme for International Student Assessment.
- 6.
Sabetai Ungaru’s forcefully criticizes what he considers the incoherent application of modern algebraic operations on the historical interpretation of Greek geometry, and the incommensurability of the two. In response to those who argue or assume that the Greeks dressed up algebraic formulas in cumbersome geometric outfits, he replies: “ Language is the immediate reality of Thought. The differences between the two ways of thinking are real differences […]. Different ways of thinking imply different ways of expression .” Or, as he writes in a footnote later on in the paper, “To what extent does one possess the method if he lacks the means to put it to use?” Similarly, my claim here is that there is nothing more to “scientific literacy ” than being able to recognize and knowing how to use certain terms in the “right way” in the appropriate context . The quotations are from Ungaru (1975), pp. 80 and 107, respectively.
- 7.
Benezet was Superintendent of Schools in Manchester, New Hampshire in the 1930’s. In 1935, under the heading, “The Teaching of Arithmetic : The Story of an Experiment,” he published a series of articles in the Journal of the National Education Association in which he offers a detailed and very enlightening account of his “experiment” and its results, from which I here borrow a few elements.
- 8.
Compare to Wittgenstein (OC §38): “Knowledge in mathematics . [One has to ask]: “Why should it be important? What does it matter to me?” What is interesting is how we use mathematical propositions .”
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Rider, S. (2017). Language and Mathematical Formation. In: Peters, M., Stickney, J. (eds) A Companion to Wittgenstein on Education. Springer, Singapore. https://doi.org/10.1007/978-981-10-3136-6_33
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