Further

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.

Hung-Hsi Wu: “Fundamental Principles of Mathematics“
What is mathematics, anyway? Whatever you think it means, it sounds the same and is spelled the same. Is it—as many people think—remembering what to do on the way to getting an answer? (No, although when you know what you’re doing, it’s quite natural to remember what paths to take and tools to use on your way to a reasonable answer.) For me, mathematics is what conforms to Wu’s Fundamental Principles of Mathematics https://math.berkeley.edu/~wu/Stony_Brook_2014.pdf 
  • 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
For a more recent (and fuller) discussion, see Wu’s book “Teaching School Mathematics: Pre-Algebra”. Also “The Mathematics Early-Grade Teachers Need to Know—and What It Means to Know It” For many more articles and essays on the mathematics of K-12, see Hung-Hsi Wu’s homepage at U.C. Berkeley. For available books, see Wu’s Author page at Amazon or go directly to American Mathematical Society. With the publishing this fall of the final three volumes, now all six of Wu’s books on K-12 mathematics are available, in print and electronically.  (AMS is also where you can get these books in searchable electronic form.)
In an address to middle school teachers, Daro contrasts two goals (1) “How can I teach my kids to get the answer to this problem?” vs (2) “What’s the mathematics they’re supposed to learn from working on this problem? How can I get them to learn that mathematics?” Great quote from his daughter when she was in high school: “Daddy, I don’t have time to understand it; I just want to get it right on the test.” Our mile-wide-inch-deep curriculum comes from our promulgating answer-getting techniques that hide or avoid the math.

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.)

Babies’ and young children’s consciousness is like a lantern; adults’ consciousness more like a spotlight.

Underwood Dudley: “What is Mathematics For?” 

“Reasoning needs to be learned, and mathematics is the best way to learn it.”

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.

Frank Quinn: A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today”

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.”
Professor Kenschaft did a small survey in the 1980’s of black people in New Jersey with at least one degree in mathematics, asking “What can be done to bring more blacks into mathematics?” The most common answer was “Teach mathematics better to all American children. The way it is now, if children don’t learn mathematics at home, they don’t learn it at all, so any ethnic group that is underrepresented in mathematics will remain so until children are taught mathematics better in elementary school.”
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.
A good illustration of this might be a pair of slides in my PowerPoint “Picturing Multiplication”, showing how 12×23 is like (x+2)•(2x+3).
 
 
“Teacher education does not exist in the United States. There is so much variation among all programs in visions of good teaching, standards for admission, rigor of subject matter preparation, what is taught and learned, character of supervised clinical experience, and quality of evaluation that compared to any other academic profession, the sense of chaos is inescapable. The claim that there are “traditional programs” that can be contrasted with “alternative routes” is a myth. We have only alternative routes into teaching.”
 
Joseph D. Becker: “The Phrasal Lexicon”
“…an understanding of the use of phrases is basic to the understanding of language as a whole.” [All the more reason, I would say, why precise definitions are useful, even crucial for learning math because of their meaningful but restricted lexicon.]
 
combine/compare these with Alison Gopnik’s Scientist in the Crib
George Miller: “The Magical Number 7, Plus or Minus Two” (1956)
Start with an intro and overview. Then, if you like, go to the article itself.
For more on Miller, including more on chunking, see “The Miller’s Tale: A Genealogy of the Father of the Cognitive Revolution” by Richard Hébert. And from 2020, an update “Overtaxed Working Memory Knocks the Brain Out of Sync”.
 
David Foster Wallace: “This Is Water” (video)
Here’s a link to his Kenyon College Commencement Address to the class of 2005, followed by a link to an article that tells a bit of the backstory, “David Foster Wallace’s Famous Commencement Speech Almost Didn’t Happen”.
This account of the Usage Wars seems to me quite similar to the Math Wars. I think DFW and Bryan Garner’s approach may offer a way out.

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…”

Errol Morris wrote a very good introduction to the Dunning-Kruger effect, “The Anosognosic’s Dilemma”. We found out about this whole thing as we listened to an episode of This American Life “In Defense of Ignorance: Act 2” one Saturday morning.