58 votes

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

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 and parsing word problems, as well as the ...
ryang's user avatar
  • 1,822
57 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 ...
Nuclear Hoagie's user avatar
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 ...
Matthew Daly's user avatar
  • 5,619
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\\ ...
Gerald Edgar's user avatar
  • 7,489
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 ...
Rivers McForge's user avatar
27 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 ...
Ross Millikan's user avatar
25 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 ...
ljrk's user avatar
  • 449
21 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 ...
Amy B's user avatar
  • 8,017
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 ...
Tommi's user avatar
  • 7,025
17 votes

What value is there in requiring students to declare the dimensions of an answer when it is already clear from context?

In my experience with remedial-level community-college students (USA), it is simply never the case that the units are trivially "clear from context". I can easily see some of my former ...
Daniel R. Collins's user avatar
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 ...
Nick C's user avatar
  • 9,436
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 ...
Milo P's user avatar
  • 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 ...
Thinkeye's user avatar
  • 741
13 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 ...
Ferenc Beleznay's user avatar
13 votes

Do any middle-school texts indicate that irrationality requires proof?

Here's a quote from the syllabus for the 9th grade in the school type "Gymnasium" in the federal state of Bavaria in Germany: Kompetenzerwartungen und Inhalte Die Schülerinnen und Schüler [....
Jochen Glueck's user avatar
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 ...
Steven Gubkin's user avatar
11 votes

What value is there in requiring students to declare the dimensions of an answer when it is already clear from context?

The legitimate purpose of this is trying to get the student to actually read the question and take note of the fact that there is indeed a context. Many students approach mathematical exercises as &...
Arno's user avatar
  • 966
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 ...
idmercer's user avatar
  • 345
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 ...
JRN's user avatar
  • 10.8k
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 ...
guest's user avatar
  • 109
10 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 ...
Geoffrey's user avatar
  • 898
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 ...
Federico Poloni's user avatar
10 votes
Accepted

Do any middle-school texts indicate that irrationality requires proof?

Here are two typical examples in print and digital educational content of how this is done: From Open-Up Resources: In your future studies, you may have opportunities to understand or write a proof ...
Marc Shelikoff's user avatar
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 ...
Sue VanHattum's user avatar
  • 20.3k
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 $n$ ...
Brendan W. Sullivan's user avatar
9 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 ...
Joseph O'Rourke's user avatar
9 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 ...
David Mulder's user avatar
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 ...
Silverfish's user avatar
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 ...
mweiss's user avatar
  • 17.4k

Only top scored, non community-wiki answers of a minimum length are eligible