# Teaching the History of Mathematics in High School

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?

• The answer to "Is any time being spent on the history of mathematics in high school classes today?" depends on the geographical area of interest. If you want to know the situation in different places, bear in mind that the concept of a high school may not be globally well defined. May 24, 2015 at 16:46
• Hungarian schoolchildren may learn all about how Bolyai discovered noneuclidean geometry. Whereas Russians may learn that Lobachevsky did it. May 24, 2015 at 18:46
• Long ago my math teacher gave me a copy of Men of Mathematics by E. T. Bell. Most of my classmates, though, were probably glad they didn't have to learn that historical stuff. May 24, 2015 at 18:50
• @JoonasIlmavirta - Point taken, but I am not very particular about "high school". Are students of mathematics ever exposed to the lives of mathematicians by teachers? I wonder how many students would come to know of John Nash through the news of his death! May 25, 2015 at 16:41

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.

• It is embarassing the fact that you wholeheartedly disagree with me and I almost completely agree with you. But English is not my 1st language and maybe it is difficult for me to translate idea into words correctly. I did not mean to say that you shouldn't mention people. Just that one should not forget his aim. Example: there is the wonderful story of Cardano, Ferrari, Del Ferro and their public duels in solving algebraic equations, with people hiding known formulas, and basically betting their positions in such occasions (in Bologna University you couldn't refuse such duels as a rule)... May 25, 2015 at 20:17
• It is a fascinating history, it contains infos on how was university and society related in XVI century, it contains infos on the (terrible) math notation of the time, it says a lot about negative numbers still not being considered numbers but only math devices (which in few years with repeat for $\imath$). You can make a burlesque out of it that all your students will remember forever but without getting out anything about math. Seen it. Or you can use the duels and insults between Cardano and Tartaglia to have them fun and at the same time get the best math out of it. And I favor this last. May 25, 2015 at 20:22
• I quite agree with: "In math, we might learn the names of some mathematicians and very little of the history, with most focus on the ideas only." I tried to write a piece some time ago about Paul Cohen's technique of forcing (admittedly not a high school math topic!) that shifted the focus away from the pure mathematics and towards his own creative development. I do not know of an MESE policy on self-advertisement, so I point to the paper here (no pay-wall). May 25, 2015 at 22:26
• I disagree. I studied Engineering before ending up in Computer Science. No history to be seen anywhere. Neither much (except for "this and that paper by XYZ solved a longstanding problem/is a breakthrough", sometimes when discussing very recent results) history discussion there. And that is understandable: Have barely enough time to cover the technical contents, leave history/philosophy for out-of-class discussions, if at all. Feb 14, 2020 at 16:51
• This is an old thread, but I wonder if the situation in USA has changed. In the high school science material that I am currently reviewing, there are almost no detail on Curie, Feynman, and even Einstein. The only required discussion of Curie is quite insulting: "Hey look! A woman". Tesla is discussed in detail, but not Tesla the man. So while I agree with the main criticism of teaching math in vacuum, the counterexamples given do not agree with what I'm seeing right now.
– user13395
Mar 15, 2021 at 14:04

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

I taught a 9th-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.

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.

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).

• It is not clear to me why you think discussing the lives of mathematicians would make the subject more attractive and be ok for entertaining a general audience yet not fit well into actual class time. Is the assertion that stories about people and anecdotes are, in general, pedagogically harmful? A waste of time? I see your opinion, but I do not know on what it is based. May 25, 2015 at 17:18
• I have not said that they are harnful, but that I do not see a pedagogical content in it. So in a way a waste of time. This is possibly biased by the fact that nuch of the general audience math I see around forgets to discuss math and concentrate on anedoctes abouth mathematicians. I think school should avoid this, in general. (Of course we can sit down and produce plenty of counterexamples, where personal info would also carry on significant informations, just as a general rule). May 25, 2015 at 17:42
• Similar to @BenjaminDickman's comment: The pedagogical content, in effect, might be that students pay attention? De-humanizing a human activity (e.g., mathematics) is not universally perceived as a help, or as making a scientific topic clearer. Arguably, there is a range of pschological "types" with corresponding reaction to people and chronology. I know that some people "like mathematics" because (purportedly) it disconnects from the sordid, mundane world. :) But for more general teaching, "even" of undergrad math majors and grad students, I think some connection makes it more real to many. Aug 3, 2015 at 19:12

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.

• Where is history part of the curriculum? I found curricula for the US (see my answer), Britain and Australia, and found history in none of them. In the South African curriculum, history is mentioned as a way of exploring mathematics in culture, but with no details. An example of a curriculum with history would help.
– user173
Aug 2, 2015 at 14:21
• Ontario: edu.gov.on.ca/eng/curriculum/elementary/math18curr.pdf p. 28 mentions the importance of integrating real-life mathematics and the history of mathematics, this can be interpreted in different ways, but when I was in school we had to do projects on specific mathematicians and their impact historically, and when in teachers' college/university, I had to take a class on the history of mathematics to help with teaching Aug 2, 2015 at 18:38
• Perhaps I misunderstood the question, but I was under the impression that the question was about whether it is being taught/are students being exposed to the history behind mathematics, and not is there a specific unit or course dedicated to it Aug 2, 2015 at 18:39
• In school, in the American, and Canadian systems, I had to learn about various mathematicians and their part in history. In the IB program I was required to teach it (assets.cdnma.com/9136/assets/Courses/math_hl_fact_sheet.pdf) (pamojaeducation.com/IB-online-courses/…) Again, students are being exposed to it/taught a little about it, but it may not be a whole unit, or a whole course Aug 2, 2015 at 18:43
• SORRY! I gave the wrong curriculum documents for Ontario, here is the highschool curriculum: edu.gov.on.ca/eng/curriculum/secondary/math910curr.pdf page 3 and 26 and edu.gov.on.ca/eng/curriculum/secondary/math1112currb.pdf page 4 and 35 Aug 2, 2015 at 19:05

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!)

• Alas, they aren't. I obtained them via other methods, so I only have pdf's that I'd prefer sharing privately. But! I've edited my post to include an extra resource that I rather appreciate. Aug 10, 2015 at 2:15

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.

• Indeed! I'll be amazed, though, if any such change, even incremental, occurs anytime, anywhere soon in the U.S., given the partisans in favor of "traditional" mathematics and its accoutrements. That is, the cultural role of mathematics has terrific inertia (as we here know), even, or perhaps especially, among people who know very little about it or what mathematics gets used for outside k12 or undergrad classrooms. Much social politics and even ideology involved... But, yes, drop some of those antique (and never seen again) little topics, and there'd be a few minutes left over... :) Aug 3, 2015 at 19:17
• I think the curriculum will get streamlined, not incrementally but disruptively. E.g.: As more companies use commercial tests of quantitative skills, those tests may become more important than the curricula, and schools and governments may ignore the parts of the curricula outside of what the companies want.
– user173
Aug 5, 2015 at 13:04
• Indeed, that sort of dystopian "trained workers" (rather than "educated citizens") goal is far more plausible by this year than it would have been (in my perception, anyway) 20-30 years ago. The goofy antique-ness of the k12 math curriculum seems ever-more a waste of time, especially for the vast majority. And even for undergrad courses, my dept still has arguments about "allowing calculators"... :) Aug 5, 2015 at 13:14

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

• Don't restrict yourself to high school, most college classes here are also centered on "getting them to compute ...", little (if any) is discussed on "how it all hangs together", and much less on "how came our current (understanding, notation, technique, ...) to be?". And not just math. Aug 3, 2015 at 19:00

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.

• There's plenty of history to teach without worrying about proof techniques. E.g. in an Algebra II class, you could do several sessions on ellipses in history. Appolonius defined them as conic sections. Euclid used them as loci of constant distance from foci, for reflectors. Kepler used their polar equation for orbits. Galton used their rotated Cartesian equation for correlation and confidence intervals. Maybe old proofs cause difficulties, but old applications tend to stay relevant, and provide human context that people find interesting.
– user173
Aug 11, 2015 at 1:55