Joel Reyes Noche
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Why is it possible to teach real numbers before even rigorously defining them?
43 votes

It is possible to teach real numbers in elementary school before even rigorously defining them by using what H. Wu ("The Mis-Education of Mathematics Teachers," Notices of the AMS, vol. 58, no. 3, p. ...

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What's a replacement for "married couples" in combinatorics problems?
34 votes

Try objects that often occur in pairs but are distinct from each other: forks and spoons (or forks and knives), left and right shoes, salt and pepper shakers, and so on (where each fork has an ...

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Natural occurrences of a to the (b to the c)?
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27 votes

If your students have learned some statistics, then you could point out that the normal distribution's probability density function uses this double exponential. $$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\...

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Why is absolute value difficult?
25 votes

(My answer is just a guess and not based on any formal research.) I suspect the absolute value function may be difficult to understand because it involves "negative numbers that aren't negative." ...

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An introductory example for Taylor series (12th grade)
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24 votes

One practical reason for choosing a Taylor Series approximation of a function over the function itself is if you are able to compute using only the four arithmetic operations. For example, if you are ...

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How to convince students of the implication truth values?
24 votes

I make the statement "If it is raining, then I have an umbrella." Did I lie? If it is raining and I do not have an umbrella, then I lied. If it is raining and I do have an umbrella, then I didn't ...

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What female mathematician can I introduce to my High School students?
23 votes

If your main interest is to provide a role model that students can identify with, you might want to look at Danica McKellar. According to her Wikipedia entry: McKellar studied mathematics at UCLA, ...

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How does a teacher come up with plausible wrong answers for multiple choice tests?
22 votes

If a teacher has taught the course before, and has asked questions that are free-response (not multiple-choice), then the teacher can look at the incorrect answers previously given by the students. ...

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Content for a 40-minute lecture on graph theory for high schoolers
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20 votes

I suggest you discuss the Seven bridges of Königsberg problem (the problem that essentially started the field of graph theory), then discuss the Three utilities problem. For each, discuss the problem ...

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What is a good handwriting font for mathematics?
20 votes

$\times\ \ +\ \ 4\quad 2\quad \omega\quad 9\quad a\quad 1\quad i\\ x\quad t\quad y\quad z\quad w\quad g\quad q\quad l\quad j$

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Examples of vocabulary that have different meanings in Mathematics compared to "everyday" English
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18 votes

This is a problem for some English language learners: The triangle on the left is also a right triangle.

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How to explain that a negative number multiplied by a negative number is a positive number, and that $-(-x)=x$?
17 votes

I would use the (real) number line. First introduce the concepts of positive numbers (distance to the right of zero) and negative numbers (distance to the left of zero). Then introduce the concept ...

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A PEMDAS issue request for explanation
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16 votes

If the expression were, say, $48\div 4\times 12$, there would not be much disagreement (multiplication and division are performed from left to right). But an expression such as $48\div 4(12)$ results ...

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What was the problem with New Math? Why did it end?
16 votes

You might want to read Kline (1973). I haven't read the book, but according to Wikipedia, In 1973, Morris Kline published his critical book Why Johnny Can't Add: the Failure of the New Math. It ...

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Response to Students Who Say "This Is Not Important"
15 votes

The American Mathematical Society provides posters promoting awareness of mathematics, its beauty, and applications. Here's a quote from the AMS Posters website: "Students frequently ask when ...

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What is fairly new theorem one can teach (and prove) to an undergraduate student?
15 votes

Chvátal's art gallery theorem, a visibility problem in computational geometry, was proposed and proved in the 1970s. $\left\lfloor n/3 \right\rfloor$ guards are always sufficient and sometimes ...

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Vocabulary for giving just numbers, not a full answer
14 votes

The problem with the word "solution" is that it could mean the final answer or it could mean how the final answer was obtained, so I suggest that you don't use it. How about the following? Clearly ...

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Multiplying the square roots of negative numbers before we calculate a result using $i$
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13 votes

From Wikipedia: The identities $(bc)^x=b^xc^x$ and $(b/c)^x=b^x/c^x$ are valid when $b$ and $c$ are positive real numbers and $x$ is a real number. But a calculation using principal branches shows ...

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Why is learning mathematics compulsory?
12 votes

Underwood Dudley answers the question "What is mathematics education for?" in this article from 2010 (Notices of the American Mathematical Society, vol. 57, no. 5, pp. 608-613) (even though ...

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In teaching mathematics, should one always follow some international standards such as ISO 80000-2?
11 votes

The standard that you link to (ISO 80000-2:2009) seems to be not available for free. That is, in order for me to follow the standard, I have to be able to read it, and in order for me to read it, I ...

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How to correct visualization of mathematical expressions?
11 votes

One way to see if the student understands the commutative property of addition is to have "fill-in-the-blank" questions such as $$2+3=3+\_\_$$ $$2+3=\_\_+2$$ $$2+\_\_=3+2$$ $$\_\_+3=3+2$$

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Why don’t all professors let students use notes, books, etc. on exams?
11 votes

I allow students to have one index card of notes during their exams. I believe this allows me to test for understanding and skills that are more important than the ability to memorize. I also ...

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What reasons are there preventing me from using an old edition text?
11 votes

Some universities (like mine) are accredited by a third party agency. One of their requirements for a university to be accredited is for it to have an "updated set of reference books" where "updated" ...

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How many problems do we have to do as undergraduate mathematicians in order to learn a subject?
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11 votes

I think your question is good, but a little vague. When you say "how many problems," would ten easy problems be counted the same as ten difficult problems? If exactly the same problem was ...

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How to introduce group theory to a general audience in 15 minutes?
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10 votes

One way to introduce group theory to a general audience is to talk about how to flip a mattress so that it is in "a different position each time, so as to pound down the lumps and fill in the sags on ...

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What is it called when terms disappear when reducing fractions?
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10 votes

If you continue the operations until $$\require{cancel}\frac{x}{b}=\frac{c}{b},\qquad\left(\frac{x}{b}\right)b=\left(\frac{c}{b}\right)b,\qquad x\left(\frac{b}{b}\right)=c\left(\frac{b}{b}\right),\...

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How to Write Steps of Solving Equations?
10 votes

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 ...

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Software for randomizing multiple choice tests
10 votes

If you use LaTeX to prepare your documents, there are many packages that automatically randomize the order of choices in multiple-choice questions. One recently updated package is esami. (This ...

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Literature on learning from errors in mathematics
10 votes

You might be interested in systematic errors. From page 14 of Kurt VanLehn's "Mind Bugs: The Origins of Procedural Misconceptions" (MIT Press, 1990): Systematic errors appear to stem from ...

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Real-world examples of more "obscure" geometric figures
10 votes

For a super-ellipse, one example would be the fountain at Sergels torg, in Stockholm, Sweden. For a circular segment, one example would be the cross-section of liquid in a horizontal-axis circular ...

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