79

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 ...


64

Why is Mathematics a compulsory subject for high school students, especially those who are clearly studying in Humanities streams? A kid at age 14 is not ready to make irrevocable decisions that will affect them for the rest of their life. That's why we don't let them get married. I have a friend who, at age 30, decided to apply to grad school in sociology, ...


46

TL;DR 2-semester course is not enough. Disclaimer: I write this as a computer-scientist that uses math a lot in his work (I'm a research assistant at a university). Introduction: There are three (overlapping) aspects of math in computer science: Math that is actually useful. Math that you can run into, and is generally good to know. Math that lets you ...


37

I'm really wondering if this is the end of the world or the beginning of an improvement in the way we teach math. I'll answer this first, and then talk a little bit about the rest of the question. Please excuse my bluntness. I will try to give you the opinion you ask for by weighing what I think may be most important relative to quality of math education. ...


30

I just start with constant rates of change, where it's pretty blazingly obvious that the chain rule works. E.g., Jane hikes 3 kilometers in an hour, and hiking burns 70 calories per kilometer. At what rate does she burn calories? Our class is generally much more comfortable with the $f'(x)$ notation, and as a result I stayed away from the $\frac{dy}{dx}=\...


29

I cannot answer the OP's question about cross-cultural/international perspectives, but here is a historical perspective that may be helpful. The issue here (whether the category "rectangles" includes or excludes the category "squares") is one aspect of a larger question having to do with whether the classification of quadrilaterals should be partitional or ...


27

The OP may be interested in the controversial 2012 article by Andrew Hacker in the NYTimes: Is Algebra Necessary? Here is one reply, by Peter Flom: A reply to Andrew Hacker. His closing remarks: Is algebra necessary? In the strict sense, no. You can live without it. You can also live without art, music, literature or sports. Would you want to? And here ...


24

It is possible to usefully mention "Lie groups (and Lie algebras)" in an introductory course, if one does not give formal definitions, but, rather, examples. It is not necessary (or advisable) to "define" smooth manifolds, which seems to have considerable baggage-of-abstraction of its own. Just give important examples, noting that they do seem to have a lot ...


23

In my experience, students in sophomore or junior level math courses usually have very little trouble picking up LaTeX on their own. They typically require the following assistance: Some guidance in downloading and installing it, e.g. links to user-friendly distributions for both the Mac and the PC. I post links to my course webpage. A sample LaTeX source ...


22

Edit 9/5/14: It has recently come to my attention that another helpful paper is: Confrey, J. (1990). A review of the research on student conceptions in mathematics, science, and programming. Review of research in education, 3-56. Link. Though there are sure to be more technology-related errors today, you can find a late 70s article on this subject from ...


20

TL;DR: It's not the triangles that are interesting; it's the mathematical concepts that can best be explained by using one of the most primitive geometrical shapes. The reason for intensive use of triangles goes beyond knowledge about triangles per se. It's the act of and steps in proving a theorem that's important to learn at this stage - start with ...


19

To properly understand and appreciate the different rounding rules, one really needs to have some grounding in statistics and probability theory. It is a little hard to describe the full theory in detail (the least being that I don't have a entirely coherent formulation of the theory off the top of my head), but here are some examples: Naively if we ...


19

As a first guess, I'd give the following categories: Overgeneralization of known theorems or techniques. Application of these theorems or techniques in unsuited situations. Overconfidence in intuition Making intuition into false theorems. Applying intuition where it contradicts known theorems. Nonapplication of known theorems or techniques. (Mistakes ...


19

One angle you could look at is molecular geometry. Not really my subject area but a couple of examples: Organic molecules can have different chiralities. That means that while one is the mirror image of another you cant rotate one molecule to the other. The reasons for this are pretty deep mathematically, but chemically give rise to interesting things as ...


18

I've run into a few students like this. I usually try to convey a few messages. It is great that you are so interested in foundations and there is absolutely a place in math for people with this perspective. Followed by a recommendation of books suited to their interests: At a variety of levels, I might include Spivak's Calculus, an intro set theory book, ...


17

In my job, I evaluate university math courses for transfer equivalency on a regular basis. In the US, "Calculus 1" typically refers to single variable differential calculus up to the fundamental theorem of calculus. So the course includes limits, the definition of the derivative, techniques and applications of the derivative including trigonometric and ...


16

After reading the previous answers, I'd add logic and proofs. To reason about a program's (in)correctness is proofs, even if you don't go to the length of proving correctness formally. Many students struggle with the idea of recursion, recursively defined functions/data structures, and the related proofs by induction. More on the soft side, being able to ...


16

The appendix A in the Common Core State Standards for Mathematics seems to agree that traditional curricula put logarithms somewhere after Algebra I. CCSSM isn't a curriculum, and it doesn't suggest teaching logarithms later, it just offers "pathways" that are supposed to help districts using traditional curriculum find a way to meet the standards. I mention ...


15

I'd be careful about trying to identify students for whom a math/science field is not in their future. At the college I work at, about half of our math majors do not intend to major in math or science when they arrive, but discover during their first few semesters that math is actually much more interesting than they had assumed. In the last five years, I'...


15

Years ago, as an undergraduate student, I experienced something close to your description of a utopian mathematics undergraduate-level curriculum at Sharif University of Technology in Iran. We didn't mind winning Fields Medal. Indeed,one did: Maryam Mirzakhani. I remember, one of the courses we had was "geometric analysis", quite uncommon as an undergraduate ...


15

Many people think that the clientele of our public school system is the students. Others act as if the clientele were the parents of the students. Those people are wrong. The client is the citizenry of his state and nation, who need the electorate to be educated and informed, in order to secure the benefits of a well-ordered government. The question, "...


14

Through the repetitive drilling of pointless, boring, exercises many people develop a disconnect between the mathematical formalism and semantics. Many people have lost the ability to reason logically about simple mathematical entities since they don't understand them as entities. This is, in my view, a horrible situation and the biggest mistake, as in a ...


14

This is not an answer, but I share what I take to be your skepticism. I think polygons are a rich topic. For example, just understanding that every simple polygon can be triangulated would be an achievement. And it would lead to the often surprising conclusion that all $n$-gons, regardless of shape, have the same sum of internal angles: $(n-2)\cdot180^\circ$....


14

I'll make some observations from a Western perspective (which may or may resonate with a Chinese student, say). Consider: Mathematics has been at the forefront of all education forever -- since long before the idea of compulsory education itself was conceived. In Classical Athens (~420 BC): More focused fields of study included mathematics, astronomy, ...


13

Historical comments. Early on, the study of logarithms and logarithmic tables was incorporated into trigonometry. For more on this background from the perspective of the history of trigonometry education, search through the following dissertation for the word logarithm: Van Sickle, J. (2011). A History of Trigonometry Education in the United States: 1776-...


13

If this is of any interest here is the case of Romania -Bourbaki's heaven (or hell): A year-long course of Analysis: starting with the Peano axioms, construction of Integer, Rational and Real Numbers (as the completion of $\mathbb{Q}$) sequences and series, general topology, differentiation and integration (Riemann) of function of one variable A year-long ...


13

In A Mathematician's Lament, Paul Lockhart describes what is wrong with math education in schools. To get a good idea of how he would do things differently, you should read his book titled Measurement. The ideas in it are for slightly older students, but the approach is what you're looking for. I recommend that you try to work through Measurement yourself, ...


13

TL;DR Triangles helped me understand both unit circle and trig functions. Super cool and super useful. Didn't really use much of my other geometry. Don't drop triangles. If you do, you should have a good reason and better replacement. I am not a mathematics educator. But I did take junior high geometry that covered all those topics. Here's a student's ...


13

Yes. The primary reason in my mind is that you want to have multiple procedures with which you can double-check work later. Knowing only a single procedure makes for very fragile understanding; on the day that you make an error, you have no way of recognizing or fixing that error. So I would claim that you want at least two or three methods of sanity-...


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