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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 ...
Matthew Daly's user avatar
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41 votes

What can (and should) an educator do about ambiguous terms like "triangle", "square", etc?

I'm confused. Are you really going to try to make this sort of distinction when teaching geometric figures to students "around 9-13 years old"? Students that age (and engineers my age -- ...
Flydog57's user avatar
  • 595
37 votes
Accepted

Why should or shouldn't we teach functions to 15 year olds?

In the U.S. Common Core standards, functions are supposed to be introduced in the 8th grade, i.e., around age 13-14. So arguably age 15 is a year or two behind where they ought to be. The standard for ...
Daniel R. Collins's user avatar
36 votes

In what curricula are "rectangles" defined so as to exclude squares?

I cannot answer the OP's question about cross-cultural/international perspectives, but here is a historical perspective that may be helpful. The issue here (whether the category "rectangles"...
mweiss's user avatar
  • 17.4k
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
  • 7,607
28 votes

Getting students to actually read definitions

In my experience, students are often predisposed to "learn" by memorizing facts; that's how much of their early education worked, so that's what they're used to. When you give them a ...
Reese Johnston's user avatar
26 votes

What can (and should) an educator do about ambiguous terms like "triangle", "square", etc?

One encounters exactly the same issue teaching multivariable calculus when one treats integrals over three-dimensional regions and integrals over the surfaces that are their boundaries. In particular ...
Dan Fox's user avatar
  • 5,869
23 votes

What can (and should) an educator do about ambiguous terms like "triangle", "square", etc?

I think the distinction you are raising is not natural to students at this age. I teach undergraduates and graduate students, not elementary schoolers, but I find that it is not natural for ...
David E Speyer's user avatar
21 votes
Accepted

Can this be a better way of defining subsets?

In both formal and informal treatments of set theory, we need to specify which operations and relations are allowable and build from there. Usually we take sets and set membership as primitives. We ...
Steven Gubkin's user avatar
18 votes

What is an intercept?

This is a case where you might be looking for a distinction that's pretty subtle. By definition, the y-intercept occurs at x=0. In one notation, it's literally f(0), where the x is explicitly offered....
JTP - Apologise to Monica's user avatar
17 votes

Getting students to actually read definitions

First of all, you should test them on remembering the definitions. Second, there are probably a significant number of your students who do not understand the definitions. Suppose you gave them an ...
Alexander Woo's user avatar
16 votes

Different Kinds of Variables

$2x$ is an expression, a phrase. Compare it to "two ducks". This is neither true nor false. It doesn't have a 'truth value'. $2x = 4$ is an equation, a statement. Compare it to "two ducks have four ...
Sue VanHattum's user avatar
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16 votes

Why should or shouldn't we teach functions to 15 year olds?

Functions are far broader and more applicable than you give them credit for. Consider the following: Country or state Capital Elevation (in meters) Bolivia Sucre 2783 Ecuador Quito 2763 Colombia ...
Matthew Daly's user avatar
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15 votes

For purposes of teaching, should constant functions be considered "linear functions"?

A linear function is not necessarily a first degree polynomial function: zero function is also linear. In France the terminology is more appropriate than the traditional English one: a linear ...
Alexey's user avatar
  • 341
14 votes

Are degrees of polynomials illogically defined in elementary algebra, intermediate algebra and college algebra courses?

I'm going to rewrite this answer to clarify what I think the issue is. I think the OP is imagining a different definition of the ring $k[x]$ than most answerers are. Here are two reasonable ...
David E Speyer's user avatar
14 votes

What can (and should) an educator do about ambiguous terms like "triangle", "square", etc?

Educator here who has worked with many students in the aforementioned age range (9-13) on triangles and squares. In my experience, it has never come up that a confusion between the boundary and the ...
Rivers McForge's user avatar
13 votes

Can this be a better way of defining subsets?

In addition to the very good answers that you've already received, it's probably worthwhile to also mention the following point: The alternative that you suggest might lead to a similar type of ...
Jochen Glueck's user avatar
12 votes

For purposes of teaching, should constant functions be considered "linear functions"?

While I haven't done a systematic survey, my impression is that the overwhelming majority of pre-calculus and calculus texts define a linear function to be one of the form $f(x) = mx + b$ with no ...
John Coleman's user avatar
  • 1,536
11 votes

In Polynomial Form, After Simplification (But Not Before!)

I think this is a great question because it is really about the ambiguous/confusing way we use the word polynomial. So I want to use the terms "polynomial" and "polynomial function" separately here. ...
Amanda's user avatar
  • 553
11 votes

How is $\frac{a}{b}$ interpreted?

A US specific answer: The Common Core State Standards define $\frac{1}{b}$ by saying it is one of $b$ equal parts making up a whole $1$. $\frac{a}{b}$ is then defined as $a$ of these. Connecting $\...
Steven Gubkin's user avatar
10 votes

Against introducing precise definitions first

At the secondary level, students have not yet mastered formal mathematics and most will need to continue learning concepts before definitions in many cases. The van Hieles (the Dutch educators who ...
Scott Eberle's user avatar
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

What can (and should) an educator do about ambiguous terms like "triangle", "square", etc?

I confess, when you start talking about 1-D triangles, my own first thought is "how can you have non-colinear points in 1-D?". So, I imagine most students that age will have a far more ...
Brian's user avatar
  • 101
10 votes

Importance of standardization of definitions of mathematical terms

IMHO one can tolerate such small differences in the mathematical language pretty easily (much easier than to tolerate the ambiguity of the natural language that lacks specifiers for almost any ...
fedja's user avatar
  • 3,909
9 votes

In what curricula are "rectangles" defined so as to exclude squares?

There is a model of how people progress towards abstract reasoning through the subject of geometry called the Van Hiele model. The model describes five levels: visualization, analysis, abstraction, ...
user52817's user avatar
  • 11k
9 votes
Accepted

Why is a translated exponential function considered an exponential function?

To start with an opinion, I think that this classification exercise is kind of silly. The student is being asked to put functions into some categories without having a clear idea about what those ...
Xander Henderson's user avatar
  • 8,225
9 votes

Why should or shouldn't we teach functions to 15 year olds?

Some abstraction is perfect for children. Mitsumasa Anno was a Japanese writer of children's books and he had many innovative approaches to introduction functions. This is from Anno's Math Games II. ...
user52817's user avatar
  • 11k
8 votes

Against introducing precise definitions first

One of my most vivid memories of graduate school was working on a problem concerning the Gelfand-Kirillov dimension of certain rings. I had been puzzling over the definition (which was rather new to ...
mweiss's user avatar
  • 17.4k

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