Teaching the History of Mathematics in High School

Question

Is any time being spent on the history of mathematics in high school classes today?

Few observations as a student -

1. I had to discover Cantor many years after I was introduced to set theory.
2. I had to discover Fermat's last theorem and Andrew Wiles five years after I learnt the Pythagoras theorem.
3. I had no idea about the Konigsberg problem or Euler. That had to wait for several more years.

I am not suggesting a detailed study of Fermat's last theorem, I have no idea about Wiles' proof of the same. But I would have loved to hear these stories from my teachers.

Why do mathematics classes ignore mathematicians and exclusively focus only on the mathematics that they have handed over to us?

Answer 1

I disagree wholeheartedly with Nicola. I can only speak to the US' and Canada's mathematics systems so it may not be broadly true. Mathematics is largely taught in a vacuum. In every other field of study (physics, chemistry, economics, literature, etc.), we learn about the people, the history, and the ideas. In math, we might learn the names of some mathematicians and very little of the history, with most focus on the ideas only. This is a huge detriment. It makes mathematics extremely dry in my opinion. People love talking about Marie Curie because she was one of the first big shot women in the sciences; people love talking about Feynman because he was kind of crazy in a fun way; people love talking about Einstein because he had such an unlikely story; people love talking about how much of a visionary Tesla was and how completely crazy he was; people love talking about Thoreau because he was extremely misanthropic; people love talking about Rockefeller because he was a ruthless capitalist; people love talking about J. Edgar Hoover because he extremely flawed and polarizing. Having these figures attracts people to a topic. Talking about history without getting into the people behind it is extremely boring. It would be like reading the line of kings in Lord of the Rings or the family trees in the Bible - just a series of words with not much to connect to at a personal level.

As for why this happens.. It's because of a really vicious cycle. Our educators don't know anything about the mathematicians or history behind the mathematics they teach because it was never taught to them. Thus none of the students know because they weren't taught and when a small fraction of those students become math teachers, they can't impart this knowledge because they never had it. This probably has its roots in the ivory tower aspect of mathematics. Mathematicians historically shied away from the spotlight, did math and didn't worry about anything else. Other disciplines were about understanding tangible things about the universe around us which is something we can latch on to and feel passionate about like the fate of the universe or the birth of the universe or abiogenesis or space travel. It's harder to feel passionate about abstract things like a scheme or moduli space. This is made worse by the completely abysmal state of math journalism. Other fields get fairly regular press, math gets incredibly infrequent press.

I take this topic pretty seriously and it's one that I am very passionate about. After reading Dunham's Journey Through Genius, math leapt off the page and I felt like I was watching it unfold before my very eyes. I could see the arguments and spats between mathematicians. Seeing something you love come to life like that is astounding. Unfortunately, the education system in the US is so entrenched that there's not much hope for change when it comes to math education.

Answer 2

Caveat: This is merely an anecdote, from a college teacher, not a high-school teacher.

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I taught a 9^th-grade (~14 yr olds) geometry class at the invitation of a high-school teacher last year. Tangentially & interactively—in the context of "What is a proof?"—the $4$-color theorem arose, and I started to explain the theorem and the tangled history of its proof, from Kempe, to Appel & Haken's proof in 1976, down to the recent developments (e.g., the Robertson, Sanders, Seymour, Thomas proof). They were fascinated, and peppered me with questions.

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This experience led me to believe there is a hunger for this type of personalization of mathematics, which is not currently being satisfied in the highly constrained US curriculum.

Answer 3

I would keep separate "life of mathematicians" from "history of math".

While the first certainly contains much narrative material, which would make the subject more "attractive", its pedagogical content is, to me, doubtful.

As for the second, on the other hand, I am inclined to say that everybody would get something from presenting math as a lively subject; and lively means also that it has a history, containing evolution, wrong turns, errors, repetitions, trends and all the like.

Unfortuntaley, at least for what concerns my country, Italy, very little emphasis is usually, at high school level, put on such a presentation of math. The reason being that it would take time away from "real math". I think this is an error in perspective, since it mantains a status quo in which a large majority of students exit high school convinced that math ended somewhat in the 19th century with nothing new going on after that. As a research mathematician one of the most common questions I am asked is: "is there really something new to say about math"? (Guess this is common)

You mention Koningsberg problem. I devoted some time to think abouth Euler charachteristic in the context of polyhedra. If you browse the book "Proofs and Refutation" by Lakatos you may well find an enormous amount of informations, mathematical and historical, that could be presented in a class to show how that subject, starting from an apparantely simple remark open a whole new field of research for years.

I referred to the fact that I'd rather keep separate this from "life of mathematicians" since I fear that encouraging history of math in high school could result in simply adding to math some spicy story on people, anedoctes, dubious interpretations (like Cantor being driven crazy by infinity) which are ok for entertaining a general audience but not for class time (imho, of course).

Answer 4

I believe the history of math is part of the curriculum, almost everywhere (at least it has been in the various school systems I attended as a student, and have taught in as a teacher), however it is rarely covered in depth.

One reason has already been brought up by @Cameron Williams regarding the cycle of ignorance among math educators (myself included, I do not know nearly as much about the history of mathematics as I would like).

Another reason is standardized testing. Unfortunately, the history of mathematics is rarely, if ever, tested. In systems where student performance is measured by a standardized test, subjects like the history of mathematics will fall to the wayside.

Answer 5

In teaching a course on geometry to sophomore's in the previous year, I definitely included as much history as possible. There are numerous ways to do this.

1. The first thing is that -- like @Cameron Williams said -- you have to know this material yourself before you can teach it. There are many resources out there in terms of texts and online videos. Norman Wildberger's History of Math lectures can be found on youtube at that link. They are quite good and would be great for any high school math teacher to incorporate parts of this into class. They are based on John Stillwell's Math and Its History. Journey Through Genius has already been mentioned.

2. I started class each day with a Historical Date in Mathematics [I might change that to 3 days a week or 2 depending on time]. I used the Math Timeline as a base to help me organize myself and also provide other context for the students that they might be familiar with (e.g. pyramids were built around the same time kind of thing). They students were required to have their own timeline for the dates we talked about in class. I didn't give an assessment of their recall of the dates, but they were required to do a few research projects. In the future, I am considering having them do independent MATH research based on some historical problem. These do not have to be problems just from ancient Greece. Students can handle investigations into number theoretic problems too. Symmetry study? Sure! Methods for solving systems of linear (and non-linear, oh my) equations? Definitely.

3. When I introduce a topic, I try very hard to motivate it and this includes talking about why anyone considered it in the past. The reason for (more advanced example) the current state of knowledge around unique factorization domains is due to Gauss's interest in seeing systems similar to but different from the integers. This alone begins to demonstrate to students how math even happens. What is math research? -- Start with something you already know and tweak!

4. (really 3a) The historical motivation is sometimes the best one in my opinion. This was particularly true for logarithms. They are usually taught as inverses of exponentials and then properties are listed, some calculations and graphing are done... moving on. That's doable (and how I first learned) but often this leaves one with a feeling of... why did I just learn that... or even more so, what are logarithms REALLY? However, if one talks about their original purpose and how to arrive at them naturally, it's rather wonderful. Then suddenly their uses for things like linear regression make since -- it's their original purpose - LINEARIZE!

I have some history modules that teach concepts via historical routes that are rather good (admittedly, I change them some) if anyone is interested. Some material goes from middle school up through first year college.

Another great resource is: Teaching Math with Original Sources -- This is done with an eye towards undergrads, but there are many links and resources on the page that are for high school or high school content. The site is that of authors of two sets of books that are great (again, some advanced stuff -- do some digging). I think it would be great to incorporate some original source material at least once or twice a year in a math course at any high school in the US! Talk with your history colleagues and get them in on this! (especially those AP Euro teachers, Document-Based Questions are a big essay component!)

Answer 6

Mathematicians and history of mathematics are not in the standards for high school math or the mathematics tests. I think that answers why they aren't taught.

Instead, the standards include things like: factoring, completing the square, directrix of a parabola, quadrilaterals inscribed in circles, medians of a triangle meeting at a point, trigonometric sums and differences. In my view, those are all poor choices.

Take some topics out of the curriculum, and there'd be more time for history and biography.

Answer 7

The question and some of the other answers make me wonder at the purpose of education.

I can imagine situations where you are supposed to learn "something" ( a body of knowledge, a set of skills, a compendium of facts; I am currently challenged to provide an accurate description of "something"), that this learning takes some time, and that incidental information may get in the way. In such situations I can see the history of mathematics getting in the way of the material; no pedagogical purpose is satisfied, just an ancillary purpose of entertainment (which in practical settings may be needed to get a student through the semester lesson plan, but I'm looking at things a little more abstractly for the present) . I imagine Nicola Ciccoli is making this kind of point in (gender presumption) his answer.

If the idea of the school system is not just education, but edification and entertainment ( so that some practical and political purpose is served by "keeping kids in school and out of trouble" ), then I can see where it may be useful to inject some additional material "that doesn't have to be on the exam" to make the lesson plan more bearable and enticing. Doing such in an educationally useful way is the craft of teaching, and such craft is what I and others hope to learn with the help of a forum like this.

If we see high school (or whatever the non U.S. equivalent is; forgive my ignorance) as a place to continue awakening the interests and develop the capability of the next generation, then I think a major societal reform is in order. There are some schools and programs in the United States that help this along, but my impression is more schools are doing without it. Technology (and more importantly, information storage and retrieval) has advanced to the point where most physical and social barriers to education are being removed, the economic barriers are being overcome, and the major barrier becomes personal: do you have the interest and personal resources to learn? Lectures and tutoring and books will not immediately become things of the past, but videos, visual and auditory guided exercises, and the ability to bring people of like interests together to discuss and learn regardless of their physical location will only increase. The one thing that will remain are the questions to be asked. Many of those will be "What was math like in the 21st century?"

Happy Memorial Day, here and abroad.

Gerhard "Better Write It Down Now" Paseman, 2015.05.25

Answer 8

Personally, the history of mathematics (and science in general) interests me quite a bit. Just biographies can be entertaining, but they don't add much to understanding the subject matter, you need to explain the development of the ideas and techniques. But there you soon run into a snag, the notation and methods used e.g. by Euler are often hairraising. To explain them, and explain why they aren't acceptable nowadays, and how to fix the "proofs", requires quite a bit of explanation. Going further back, appreciating Archimedes' work on solids is hard work, as he uses methods that are quite alien to us. William Dunham has written several books that do an outstanding job at translating historical masterpieces to something a bright student can understand, but that is clearly a complex task.

Given the pressure to "cover the required subjects" I'd leave that out, with regrets. Besides, sidetracking through strange (or in retrospect downright wrong) developments presents the risk that your students, with not-too-firm grasp of the subject in the first place, get confused or end up using outdated techniques.