# Tag Info

### 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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### 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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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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### 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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### 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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### 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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### 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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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 ...
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### 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....

### 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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### 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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### 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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### 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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### 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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### 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 ...

### 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 ...
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### 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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### 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. ...
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### 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 \$\...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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