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 ...
- 5,609
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 -- ...
- 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 ...
- 24.4k
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"...
- 17.3k
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\\
...
- 7,369
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 ...
- 1,984
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 ...
- 5,728
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 ...
- 5,243
20
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 ...
- 24.4k
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....
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 ...
- 2,484
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 ...
- 14.2k
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 ...
- 20.1k
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
...
- 5,609
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 ...
- 17.3k
15
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 ...
- 18.4k
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 ...
- 331
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 ...
- 5,243
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 ...
- 540
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 ...
- 1,695
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 ...
- 1,506
11
votes
Accepted
What is the intuition behind the limit superior?
I have two intuitions to offer:
A sequence $(a_n)$ may have cluster points (these are points such that every neighborhood contains infinitely many elements of the sequence, or, more precisely, for ...
- 2,992
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 ...
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.
...
- 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 $\...
- 24.4k
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 ...
- 421
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 ...
- 109
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 ...
- 3,599
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