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Questions like this, or variants (from students, the notorious "when will I use this in real life") seem to be pretty common, and I'm always a little surprised, because the unstated premise - that high school is supposed to teach students things narrowly tailored to their future career - is so obviously false.
It's obviously false because almost no academic ...
83
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 ...
59
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!
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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 ...
58
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
Not very rigorously at all, but that doesn't (and shouldn't) mean just memorizing calculations. (I should add that I'm basing this on my experience teaching calculus to non-major college students, but I think the relevant issues are similar.)
Mathematicians have a bad habit of conflating rigor with conceptual understanding. A lot of this seems to come out ...
50
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 ...
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
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, ...
44
"Lately, my students keep telling me why what we are learning is not important. They ask me when will we use this in the real world?"
There's a quick reply to this that I think people won't like, and you would have to be very careful using, but actually makes a more serious than it first appears. One possible answer is: "You're quite right. ...
43
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 ...
43
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 ...
41
Sophie Germain and her work on Fermat's Last Theorem.
41
Imagine you are put in jail. You are forced to paint a painting every day for 10 years. You have no choice in the subject: one month you paint dogs, another month you paint horses, another month you paint lampposts. The prison guard verbally chastises you when your painting is not up to their standard. If you doodle something on your own, outside of the ...
40
The product of two numbers should be given with as many significant digits as the least precise of the numbers multiplied (see https://www.nku.edu/~intsci/sci110/worksheets/rules_for_significant_figures.html). 1.60934 km/mile has six significant digits (or, if a mile is defined to be an exact number of km, then the conversion factor has an infinite number of ...
39
Just before dividing, you can reason "Either $x=0$ or I can divide by $x$." This creates two separate cases to be analyzed.
This works for dividing by anything. You want to divide by $\sin(x)$? You need to make two cases: $\sin(x) \neq 0$ and $\sin(x)=0$. And then analyze each independently.
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
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 ...
37
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 ...
37
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 ...
37
To expand on my comment, I found that high school kids like watching YouTube videos. (I mean, they don't have to do any work right? Just sit and listen.) These are a few of my go to channels to pull mathematical ideas from. I try to show them short clips that might motivate them to think abuot math in a different way, not only just "plug it into the formulas....
37
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 the 8th grade says:
Understand that a function is a rule that assigns to each input
exactly one output.
So honestly that really doesn't seem like a hugely ...
37
Math is just as useless as almost any other subject
As a math tutor, I've thought about this a lot over the last 15 years or so. Aside from tutoring, I don't use my math education in "the real world". Here's a list of educational requirements that I have also never used in "the real world":
American literature (Hemingway, Flannery O'...
36
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, ...
35
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 ...
35
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 ...
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