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
38 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 -- ...
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  • 525
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 ...
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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"...
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  • 16.1k
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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  • 6,171
28 votes
Accepted

Why do students have problems with showing that something is well-defined? How can this be improved?

Maybe, your students have a belief problem. They will rarely (maybe never) have encountered problems where something was not well-defined. If you have never been in trouble since everything you were ...
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  • 4,733
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 ...
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24 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 ...
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  • 4,948
21 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 ...
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19 votes
Accepted

The definition of natural log and e

I think it is a good thing to talk about how there are some concepts where there are choices for where you start when definining them. It happens in linear algebra too, with the definition of linear ...
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18 votes

Are there disadvantages to teaching complex numbers as purely geometrical objects?

I think there are serious pedagogical problems with such an approach. Here is a good general rule for explaining any kind of math: Skipping over the motivation doesn't make something easier to ...
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  • 7,930
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....
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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 ...
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16 votes

Are there disadvantages to teaching complex numbers as purely geometrical objects?

It's funny you should ask this now, because I just taught a Math Ed graduate class on this topic the other day, so a lot of these thoughts are fresh in my mind. The pedagogical sequence we ...
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  • 16.1k
16 votes

Can students tell the difference between the "definition if" and the "theorem if"?

Not formal research, but some decades of experience teaching both undergrad and graduate level courses, and "editing" PhD theses and such: It appears that even many serious professional ...
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  • 13.4k
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 ...
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  • 17k
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 ...
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  • 5,539
15 votes

Rigorously defining the concept of an angle for high school students

Many high school geometry textbooks define an angle as simply the union of two rays with a common endpoint The advantage of this definition is its simplicity. Among its disadvantages: It does ...
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  • 16.1k
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 ...
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  • 271
14 votes

The definition of natural log and e

I think a lot depends on context -- what course you are teaching and what the characteristics of that course are. If by "Freshman college calculus" you mean what I think you mean -- namely, a non-...
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  • 16.1k
14 votes

Why do students have problems with showing that something is well-defined? How can this be improved?

I've never tried this in a classroom, but I suspect a lot of the trouble with functions is that students haven't been taught the vocabulary to deal with things that are weaker than functions. For ...
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14 votes

What is a variable?

This is a very difficult question to answer; I recommend as a first place to look: Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. The ideas of algebra, K-12, 8, 19. Link (no ...
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13 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 ...
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12 votes

how is volume different than capacity

The situation does indeed change your interpretation of the terminology. Volume is a general concept of the amount of 3D space something takes up. For a solid object like a brick, or for a liquid ...
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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 ...
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  • 1,426
11 votes

Why do students have problems with showing that something is well-defined? How can this be improved?

I think that you hit the nail on the head, when you said some are not even aware of what well-defined means. As Anschewski suggests the problem may be that students have not encountered enough non-...
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11 votes
Accepted

Is multiplication by zero clear for and understood by K-3 students?

Turning my comment into an answer as per request: I think the graphical approach gets the idea across. You could represent $3\times 1$ by a row of three dots; likewise, $3\times 2$ is two rows of ...
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11 votes
Accepted

How can I motivate the formal definition of continuity?

Have a look at the paper written by Nunez et all: EMBODIED COGNITION AS GROUNDING FOR SITUATEDNESS AND CONTEXT IN MATHEMATICS EDUCATION. In essence, they argue that it is better to be causious if ...

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