55 votes

Why do we introduce the notion that triangles are "congruent" instead of just saying that they are "the same" or "equal"?

Colloquially, there's a lot of conceptual overlap between all of these terms, but "sameness" is not a well-defined mathematical property. Congruent shapes need not be "the same" or ...
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54 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

"Because we said so" is a bit of a conversation closer, I agree. But "Because some people agreed a long time ago to define it that way so we could have conversations where we all understood each ...
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  • 5,539
52 votes

Is there a virtue to learning how to compute by hand?

I couldn't agree more with @Steve's comment. The following response is written with elementary-to-high-school mathematics in mind. A lack of a decent number sense really does encumber making sense of ...
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  • 1,041
34 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

There was a multiplication table posted on the wall. Like this \begin{alignat}4 1 &\quad 2 &\quad 3 &\quad 4 &\quad\cdots\\ 2 &\quad 4 &\quad 6 &\quad 8 &\quad\cdots\\ ...
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  • 6,141
30 votes

Why do we introduce the notion that triangles are "congruent" instead of just saying that they are "the same" or "equal"?

The smart-aleck answer is that most congruent triangles, or congruent figures more generally, aren't actually "the same" or "equal". Usually when we say two things are "the ...
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24 votes

Is there a virtue to learning how to compute by hand?

I find the ability to estimate calculations quite useful and I think you need to be able do do calculations to estimate them. If you are keeping a grocery budget, I would suggest you should know what ...
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22 votes

Is there a virtue to learning how to compute by hand?

Yes! But the virtue doesn't lie in being able to do the calculation but in gaining a feel for numbers as well as algorithmic thinking. I teach Computer Science freshmen and one of the first things we ...
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  • 419
19 votes

What is the preferred way to denote the Pythagorean theorem equation?

Common knowledge The formula $a^2+b^2 = c^2$ is common knowledge and the words for hypotenuse and leg (is "cathetus" not used in English?) are basic mathematical vocabulary. Including these ...
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  • 4,544
17 votes

Is there a virtue to learning how to compute by hand?

I taught at the elementary and high school levels. At times we used calculators and at times we didn't. Students benefit from experience both ways. Students need to learn that calculators are only a ...
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  • 6,735
16 votes

Prisoner's dilemma formulation for children

Here's a silly example: Give students collections of the same type of thing, where each collection contains "good" objects and "bad" objects -- for example, a stack of Pokemon cards with both rare ...
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  • 7,686
14 votes

Prisoner's dilemma formulation for children

I found out about the Prisoners' Dilemma as a kid from a book about the Harry Potter phenomenon, which had a chapter about the problem, but presented as a story about Harry and Draco being accused of ...
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  • 241
13 votes

How to Write Steps of Solving Equations?

I wouldn't do that. The parenthesis in use are also used for legal expressions within equations. So you can end with one line containing the same parenthesis meaning different things, what looks like ...
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  • 731
11 votes

Is there an agreed upon difference between how we represent $\frac{a}{b}$ and $a \cdot \frac{1}{b}$?

The common core state standards definition of the fraction $\frac{N}{D}$ of a unit is to subdivide the unit into $D$ equal sized pieces. Each of these pieces is defined to be $\frac{1}{D}$ of the ...
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10 votes

How to Write Steps of Solving Equations?

The problem with $(4x+7=6x+2)-6x$ is that there is no subtraction operation that involves subtracting a term from an equation. Subtraction involves subtracting a term from a term. So the correct ...
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10 votes

How to Write Steps of Solving Equations?

Speaking as someone who has taught college precalculus several times, I have an intense dislike for the way that Geogebra writes this step. In my opinion, it is very important to emphasize to ...
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  • 345
10 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

FYI: here's some pro and con: http://primefan.tripod.com/Prime1ProCon.html One was originally considered prime. It is prime with the most convenient ("natural") definition. It got excluded from ...
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  • 109
10 votes

Is there a virtue to learning how to compute by hand?

Brian D. Rude, "The Case For Long Division." 2004. HTML link. This is a somewhat long (unpublished) article (which I haven't studied carefully), but maybe the excerpt below suffices to give ...
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10 votes

Is there a virtue to learning how to compute by hand?

I attended the Computer based math educational summit back in 2016 and found their ideas interesting. I agree with some of their points and disagree with others, but it is certainly interesting to ...
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10 votes

What is the preferred way to denote the Pythagorean theorem equation?

In Olympiad geometry, $a$, $b$, $c$ is the so-called standard notation for the sides of a triangle, so it makes sense to use it consistently when referring to a triangle (in isolation). However, in ...
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9 votes

Patterns that unexpectedly fall apart at large $n$

It might work better, with this age level, not to be concerned about how large n is when the apparent pattern falls apart. My favorite example is the problem of making all possible straight-line ...
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  • 16.9k
9 votes

Patterns that unexpectedly fall apart at large $n$

I second Sue Van Hattum's suggestion that you should not be so concerned with how large the $n$ is where the pattern eventually fails. I'll go one step further and recommend an example where that $n$ ...
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9 votes

Is there a virtue to learning how to compute by hand?

I have thought a lot about this question since posting it, and having read the other answers and the many comments, I want to add a perspective that no one else seems to have given. Most of the real ...
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  • 823
9 votes

What is the preferred way to denote the Pythagorean theorem equation?

Two alternatives I have seen used (am not necessarily recommending them, but will list some pros and cons) which don't seem to have been mentioned yet. Don't denote it algebraically at all! Draw a ...
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9 votes

Why do we introduce the notion that triangles are "congruent" instead of just saying that they are "the same" or "equal"?

Historically (and by historically I mean "in Euclid's Elements") the word "equal", when applied to geometric figures, meant "equal in magnitude". So for example: Euclid ...
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  • 16.1k
8 votes

Is the constant term a coefficient?

Your question is kind of two parts: one about a convention Is the constant term a "coefficient" and one about a philosophy, which I perhaps find to be a more important question to answer. Isn'...
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  • 3,477
8 votes

Prisoner's dilemma formulation for children

I like Nick C's idea more than modifying the typical formulation. The notion of snitching on a friend, regardless of the severity of the "crime", has real-world ramifications beyond the punishment ...
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  • 5,539
8 votes

Explaining why (or whether) zero and one are prime, composite or neither to younger children

How should one talk about the question of 1 or 0 being prime ... with primary or middle school children? Depending on what you did before you will have an easy or a hard task: If the children were ...
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8 votes

Is there a virtue to learning how to compute by hand?

Beyond having worked as a programming teacher I have no experience with math education, but this is a topic I have been fascinated with for years. Arguments in favor of mental/manual arithmetic can ...
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8 votes

Are these fraction problems different enough to warrant individual consideration?

i don’t think they are. In fractions there are (at least) 3 analogies: set (discrete objects), area (or volume), and length. Your 1st is set, 2nd maybe area (more like length) and 3rd is length. You ...
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