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12

If you solve a mathematical problem the wrong way, you get a wrong result, which can be checked. If you "solve" a philosophical problem there is no way to check the result in any decent timeframe. May be you made no sense, may be you invented a new branch of philosophy. In the hard (read: checkable) parts of philosophic thought, the tools are mathematical ...


12

I think the level of the student is very important to this question. If the student has never had an abstract math course (like my students), then the lack of a definition of "number" is a great way to introduce the idea of abstract algebra. They are very happy to initially believe a definition like A number is anything that you can add and multiply, such ...


12

What do we understand as mental processes? All of us (Math teachers) dream of entering the brain of our students, see what's happening and adjusting some connections... However, the thing is that their brains are a kind of black box for us. So the only way we actually have to be sure they have catched some mathematical concept (or argument, or property, etc....


8

This grew a bit long for a comment. (My first note is similar to Alexander Woo's remark about factoring polynomials; perhaps he intended "polynomials" to subsume the case here, in which we add constant functions...) Given $413 + 91$, it may not be clear that this can be re-written as $7 \times 59 + 7 \times 13 = 7(59+13)$. (Plenty of people seem to ...


7

For all of the community college algebra classes I teach, I certainly make proper mathematical writing the number one priority; which is not to say that I have students compose everything in English writing. It's already an overwhelming challenge for students to get the algebraic grammar and syntax right, such that I already feel like there's not enough time ...


7

My take for what it is worth. I see at least two very strong statements you make. I don't see strong statements as necessarily problematic; sometimes it is very important to take a stand. But stands often have to be defended. The first very strong statement essentially summarized in your sentence: Math never cares what your social background is, it does ...


7

First, a direct answer: Yes, there is work in mathematics education connected to Gadamer. One soure for such studies is articles by Brent Davis; more generally, google scholar yields such connections readily. Here are some examples of BA Davis' work in which Gadamer is cited: Davis, B. (1997). Listening for Differences: An Evolving Conception of ...


6

Modulo is not a well-behaved operation when thought of as being solely defined in terms of integers, in the context of how it interacts with the other basic operations. Taking as the definition that $mod(m,n)$ is the unique number $r$ with $0 \leq r < n$ and $m - r$ is divisible by $n$, then you run into unpleasantries such as $$mod(m_1 + m_2, n) \neq ...


5

Whatever teachers may think about the nature of numbers, the foundations of "arithmetic" and the nature and concept of number in particular are very subtle. For a recent and sophisticated look at the issues, see: John Horton Conway, On Numbers and Games, second edition, A.K. Peters, 2001. Conway gives his approach to the surreal numbers, relates these to ...


5

Here's a few things you might consider that distinguish the modulo operation (as referred to in computing, that is, remainder after division) from the elementary operations (add, subtract, multiply, divide): The graph of modulo is not continuous. Modulo does not have a derivative everywhere (not analytic). The function x ◦ 3 is one-to-one for the elementary ...


4

Another thought occurred to me regarding this, after having read Reuben Hersh's collection of essays. There is a quote of Bill Thurston, which I paraphrase as "thinking is the same as seeing". In a sense, having the logic of one's proof "at the tip of the tongue" produces a sense of unity in the proof that is analogous to an object one can mentally ...


4

I did a search on this question as well as the three existing answers for the term "applications" and did not find any occurrences. The natural numbers $\mathbb N$ do not cause students any particular difficulties of motivation. Arguably the main point to emphasize in introducing the successive enlargements $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\...


4

I think this question is important. I'd love to see an actual answer to it and cannot upvote it enough. I don't have an answer, but would like to share some intuitions/speculation. I think the following is what teachers have absorbed as the notion of number. I do not think most teachers could articulate this. Our early exposure to numbers comes about ...


4

This is too long for a comment. This question reminds me of a story that Feynman told in Surely You're Joking, Mr. Feynman: There was a Princess Somebody of Denmark sitting at a table with a number of people around her, and I saw an empty chair at their table and sat down. She turned to me and said, "Oh! You're one of the Nobel Prize winners. ...


4

I'm apologizing in advance for the subjective answer but I think a short story is the best way I can organize my thoughts around this. I started out in Medicine in the mid 1990's. But I was unhappy. Medicine was too concrete. While we learned plenty theory in medical school, that theory wasn't very helpful in making accurate predictions. You never knew ...


3

Based on a quick Google Scholar search, it seems like there was a flurry of activity in this area ("education literature on writing to learn mathematics") circa 1990, with at least 3 book-sized collections on the subject: Connolly, Paul, and Teresa Vilardi. Writing To Learn Mathematics and Science. Teachers College Press, 1234 Amsterdam Ave., New York, ...


3

The other (correct) answers explain why the modulus operation doesn't fit with the other ordinary operations. Nonetheless there are very good reasons for introducing it at many places in the K-8 curriculum, with an appropriate degree of rigor at each place. I've had a lot of fun with "clock arithmetic" on various sizes of clock. The first hurdle (and kids ...


3

How about: The Book of Numbers by John H. Conway, Richard Guy


3

'Number' is just a word. It's meaning strongly depends on the audience and the information you want to give. For a small child a number is something she can show on her fingers. Growing up with number, it is something used to count things (N), measure dimensions (R), describe positions (C) and so on. You decide where to stop. For me, the number is the ...


3

As the above comments explain, that it is called "distributive law" is just a name. Distributing and combining (factoring) are logically equivalent. The identity can be applied in both directions. A practical difference is that distributing (multiplying out) can be done mechanically, factoring (combining) often has several potential common factors among ...


3

My apologies for my small sample size... but, still, 40+ years of teaching almost gets me to statistical significance? :) Yes, early in my career-arc, I was essentially conformist, and would aver that the course objectives officially espoused were truly what they were. Students did not "like" this, but could easily endorse it. (Yikes...) This is not about ...


2

Having an inquisitive, but impatient 3rd grader can help you to get the answer pretty quickly: Distribution can be explained and applied directly out of the box, because a, b and c are all known when starting on the left. In order to work from the right hand side, the combination requires finding (isolating) the common part (i.e. denominator), which might ...


2

This leads me to wonder if it is possible to consider a terrible heresy: Is it possible to imagine mathematics without problems?... What if "conversation" replaced "problem" in the mathematics classroom? I'm going to focus on this other question asked here. Personally, in the math courses that I teach (community college remedial algebra, college ...


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