Here is a collection of readings, all accessible online, as far as I know, that to me are foundational and provocative. Many will have an excerpt or a little blurbette to give you an idea of why I like them so much.
- Every concept has a definition.
- Every statement is precise about what is true and what is not true.
- Every statement is supported by reasoning.
- Mathematics is coherent: the concepts and skills are logically intertwined to form a whole tapestry.
- Mathematics is purposeful: there is a purpose to each skill and concept
Phil Daro: “Against Answer-Getting” (video)
You can also download the entire video (about 335 MB) and thus not have to depend on internet connections or changing urls when you want to see it or show it to a group. Keith Devlin has some insightful comments on it in his blog.
Daro has another important video on planning, “Planning Chapters, Not Lessons” in which he explains that the writers of the Common Core Standards found that “mathematics does not break down into lesson-sized pieces”. (It does break down into chapter-sized pieces, however, and Daro says that chapters, not lessons, are what teachers or departments should focus on in their planning.)
Alison Gopnik: “What do babies think?” (TED talk video)
Babies’ and young children’s consciousness is like a lantern; adults’ consciousness more like a spotlight.
Underwood Dudley: “What is Mathematics For?”
Richard Hamming: “The Unreasonable Effectiveness of Mathematics”
In the far distant prehistoric past, where we must look for the beginnings of mathematics, there were already four major faces of mathematics. First, there was the ability to carry on the long chains of close reasoning that to this day characterize much of mathematics. Second, there was geometry, leading through the concept of continuity to topology and beyond. Third, there was number, leading to arithmetic, algebra, and beyond. Finally there was artistic taste, which plays so large a role in modern mathematics. There are, of course, many different kinds of beauty in mathematics. In number theory it seems to be mainly the beauty of the almost infinite detail; in abstract algebra the beauty is mainly in the generality. Various areas of mathematics thus have various standards of aesthetics.
Children are attracted to rule-based reasoning (think games), and rich applications and success downstream should more than compensate for initial obscurity. I suspect that it is a bigger challenge for educators to think this way than it is for children. The starting point would be to acknowledge the significance of the mathematical revolution a century ago and to see the new methods—properly understood—as profoundly rich resources rather than alien threats.
Håkan Lennerstad, Lars Mouwitz: “Mathematish – a Tacit Knowledge of Mathematics”
- “Fractions are not counting numbers; they are measuring numbers.”
- “…algebra can be viewed both as a symbol system and as a way of thinking.”
- “…both mathematical prose and Mathematish are established vehicles crucial for problem solving and proof activities in both school mathematics and mathematics research, and both have a language character.”
Roger Howe: “Taking Place Value Seriously: Arithmetic, Estimation, and Algebra” (especially pp. 33-35)
In a certain sense base 10 notation exploits algebra in the service of arithmetic because decimal numbers can be usefully thought of as ‘polynomials in 10.’ Emphasizing this relationship can both shed light on arithmetic and make algebra more familiar and learnable.
- text of the lecture: https://prelectur.stanford.edu/lecturers/hofstadter/analogy.html
David Foster Wallace: “Tense Present: Democracy, English, and the Wars over Usage”
David Dunning: The Dunning-Kruger Effect cuts every which way: no one’s exempt. It’s hilarious and deep and well worth keeping in mind:
“As Dunning read through the article, a thought washed over him, an epiphany. If Wheeler was too stupid to be a bank robber, perhaps he was also too stupid to know that he was too stupid to be a bank robber…”