79
Emmy Noether comes first to mind, as one of the most influential mathematicians in abstract algebra, specifically in the development of Noetherian rings (along with many properties of ideals).
One aspect of her work that high school students might like is from another area, analysis. Noether's theorem says that every symmetry of the laws of nature (or the ...
54
Anscombe's quartet is pretty good:
All four of these sets have almost identical mean and variance for both x and y coordinates, correlation, and best-fit linear regression. But they're obviously very different!
52
Here are two well known examples:
If someone tests positive for a rare disease (say its prevalence is 1 out of 100,000) with a test that has a 1% false positive rate, it is tempting to say that we are 99% sure they have that disease. This isn't true if you go through the numbers; they probably don't have that disease and are a false positive. (Bayes)
If you ...
51
Julia Robinson! I recommend her for a high school audience for a few reasons:
Mathematical reasons: She is best known for her work towards the solution of Hilbert's 10th Problem, regarding an algorithm for solving Diophantine Equations. High school students can absolutely recognize and solve particular Diophantine Equations. Furthermore, and more relevant ...
49
Perhaps not strictly a mathematician in the traditional sense, but I think Ada Lovelace might be a great woman to start with in today's digital world. She even has an important programming language named after her: Ada.
Augusta Ada King, Countess of Lovelace (10 December 1815 – 27 November
1852), born Augusta Ada Byron and now commonly known as Ada ...
48
All four of your options lead with "They are told..." Consider asking the student questions instead. At the very least, this shows interest, and they may end up catching their own mistakes as they try to explain to you what they had glossed over in their own heads.
When I have the opportunity, I like to challenge my students to explain EVERY step of their ...
40
A book I remember has the title "the egg-laying dog". The titular dog enters a room where we placed 10 sausages and 10 eggs. After a while the dog leaves the room, and we observe, that the percentage of eggs relative to the sausages increased, so we conclude that the dog must have produced eggs.
It's easy to spot the mistake in the above example, because ...
39
Sophie Germain and her work on Fermat's Last Theorem.
38
(Disclaimer: I peronally really don't care if one uses $\tau$ or $\pi$, both are just numbers for me.)
But I would strongly recommend to use $\pi$. Why?
In every technical literature, in many popular literature, the people always use $\pi$ (even worse: $\tau$ is used for different things than $\tau=2\pi$ which would confuse when reading those literature). ...
38
Sally Clark (http://en.wikipedia.org/wiki/Sally_Clark) was convicted in the UK of murdering both her infant sons, when in fact it is much more likely that they died of natural causes. The case against her was largely based on invalid statistical reasoning. The Royal Statistical Society made a statement about at at the time, which begins as follows:
In ...
37
As a personal tutor, I’ve been teaching algebra to kids from ages 8 to 16 for many years. Mostly I find myself in the position of picking up the pieces when the kids are failing and fearing more failure.
The root of the problem, in my experience, is the way algebra is taught as something alien, and in particular, different from arithmetic, which it really ...
36
One thing you might do is contrast reading mathematics textbooks with reading novels. I have seen this done at the start of a textbook draft for a course on the Kuratowski closure operators (MESE sketch).
The material is not generally for distribution, but here is a brief excerpt from the start of the book:
I see that Topology was mentioned in a comment ...
35
Trapezoid
Native Peruvian architecture makes heavy use of the trapezoid for stability in earthquakes. (The Spaniards thought they were primitive as they didn't use arches ... but most of the Spanish buildings have collapsed or had to be rebuilt).
It's especially apparent in their doorways and windows.
(hi res)
Other examples with licensing such that I ...
34
I do not know if there is an accepted definition of what a mathematician is. There are teachers of mathematics and professors of mathematics, for example, and most people agree that people of the latter category are mathematicians while I am not sure if there is any consensus about the former category.
So my suggestion would be to not got into the trouble ...
33
Simpson's paradox: see http://en.wikipedia.org/wiki/Simpson%27s_paradox.
To summarize the Berkeley Admissions example: in 1973, 43% of men applying to graduate school at Berkeley were admitted, but only 35% of women. But, broken down across the six departments, women either did better than men, or the difference was not significant. The paradoxical result ...
33
A really nice example is factorization of the Hilbert numbers - that is the numbers
$$1,\,5,\,9,\,13,\,17,\,\ldots,4n+1,\,\ldots.$$
Now, we can talk about factoring in this domain too - for instance, $45=5\times 9$ is a factorization of $45$. Moreover, $5$ and $9$ are both "Hilbert Primes" - they are not the product of two other Hilbert numbers - meaning ...
32
As someone who teaches calculus to college students, I expect my students to have seen point-slope form. We just start using it (because it's the right way to talk about tangent lines and linearization) without teaching it, because we consider it part of the standard algebra curriculum, so students who haven't seen it are at a disadvantage.
Further, ...
31
I agree with vonbrand that it is important to stress that this is a convention that is used sometimes but not others. But I would add the emphasis that all conventions are local. There are places where it is helpful to adopt this convention but other settings in which it would be a disaster. My preference would be to make sure students understand that the ...
31
I feel that it is perhaps a little irresponsible to teach $\tau$ instead of $\pi$. As a first introduction, it is the norm which should be taught: teaching a rare alternative to $\pi$ only serves to confuse students, especially when almost all available resources use $\pi$ instead of $\tau$. Imagine a student's confusion when they see $\tau$ in class and $\...
31
Assuming we're talking about mostly US students, most American high schools teach calculus in a way that's very focused on the AP test. The pressure to get students through that with an adequate success rate seems to have lead to some streamlining and cutting corners in ways that mean that what gets taught doesn't really stick, and also doesn't set students ...
30
Maryam Mirzakhani, who just won the Fields Medal, and also was the first Iranian student to win a gold medal in the IMO in 1995 with a perfect score.
My colleague Mohammad Javaheri was on Iran's IMO team with her in 1995. He told us the other day that after Maryam won the gold, when the rest of the team went up to congratulate her she said "next, the Fields ...
29
The most intuitive reason I know for $0^0 = 1$ comes from interpretation in terms of functions, namely
$$\text{There are } |B|^{|A|} \text{ functions } A \to B \text{ for any finite $A$ and $B$}.$$
Now, there are no functions $\{\spadesuit\} \to \varnothing$, so $0^1 = 0$, but there exists exactly one function $\varnothing \to \varnothing$ (its set of ...
29
This is what really happens during education (not in the mind of a mathematician): The teacher introduces $4^9$ as just a way to abbreviate $4\times 4\times 4\times 4\times 4\times 4\times 4\times 4\times 4$. So, the $a^n \times a^m=a^{m+n}$ and othere rules can be shown to be true, simply via thinking to the meaning of this abbreviation.
Then, someday, $a^...
29
An identity always holds for some values of the free variables. Sometimes the allowed values are all real numbers, sometimes something else. In this case the identity holds whenever $\sec x$ and $\tan x$ are defined (they are defined in the same set). In a certain limiting sense the identity is also true at $x=\pi/2$, but you probably don't want to go into ...
28
One of the reasons your students are putting the $\pm$ on the 10 is probably because someone told them that when you remove a pair of absolute value signs that the $\pm$ goes on the number after the $=$. They are simply doing what they thought they were told to do.
So I try to avoid telling them this sort of thing. I like to tell them is that $|x|$ is equal ...
28
The National Library of Belarus,
a rhombicuboctahedron:
27
Any Living One who is friendly enough to come talk with them.
Seriously, learning about "people in books" can sometimes be inspiring. But actual live role models are best. Write a local college, university, or business to find a woman who self-identifies as a mathematician. Invite her to your school to spend some time with your students. You want a real ...
27
Foremost: It depends on what the lead-in lesson/topic/direction was. If this was the essential point being exercised, then I would interrupt ASAP and refocus them on the lesson/direction that just occurred. Otherwise I would let them complete their process (unless obviously totally infeasible) and then compare to a faster way afterward.
26
I have never quite understood why it was impressive and/or beautiful, and it always frustrates me when people claim that it is. Therefore, I would say "no, it is not good motivation", because beauty is subjective.
On the other hand, if you explain to students that the formula is based on $e^{i\theta}=\cos\theta+i\sin\theta$ and this essentially allows you ...
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