Episode #125 of the Stack Overflow podcast is here. We talk Tilde Club and mechanical keyboards. Listen now
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. ...


25

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 ...


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 ...


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

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 ...


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, ...


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

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 ...


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

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

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 ...


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 ...


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

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: $(n-2)\cdot180^\circ$....


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-...


12

I would discuss with these students the rather healthy point of view of Terry Tao on the subject. Summary One can roughly divide mathematical education into three stages: The "pre-rigorous" stage, in which mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and hand-waving. The "rigorous" stage, in which one ...


12

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-...


12

I wonder if we (or some formal international body) should rethink the entire curriculum? Aeryk's very logical and reasonable curriculum is focused totally on content, as opposed to what we want the students to learn along the way. In other words, it is not only sequence that matters! I am particularly concerned about three aspects that disappear in content ...


12

In reality, I think this is not the most important topic, and if I was designing a curriculum from scratch, I would probably omit it. We rarely have that luxury however. In my state, for example, all state colleges must meet certain common standards to be able to transfer credit: one of these standards is teaching trig sub. If I had to include it, I ...


12

Not a complete answer (could there be one?), but too long for a comment. why is it [Set Theory] not being taught at the very outset of math education? It has been tried, most likely still is in certain places at different degrees. For some history and background, lookup the New Math of the '60s, possible keywords Belgium, Willy Servais, Georges Papy. ...


11

The textbooks we used for Algebra 2 at Miramonte High School, Orinda, California, in 1967 - 1968 must have been SMSG Units 17 - 18, entitled Intermediate Mathematics, Part I and II. I recognize the content as being that of the PDF Documents (scan) available at ERIC corresponding to the same titles. Ref.: http://eric.ed.gov/?id=ED135625 http://eric.ed.gov/?q=%...


11

I was a bit like that in my first three years if study, then I read the first volume of Schwartz' Analysis, where he introduces the ZFC axioms. This is the point when I understood that one simply cannot make sense out of set theoretical axioms before having manipulated higher-level mathematics (here "higher-level" is to be understood in the sense of higher-...


11

I suggest having a look at the following book by John Stillwell: Naive Lie Theory. A fast-paced week or (if you're lucky enough... or believe it's worth it as I would) two weeks could be created by pulling the big ideas from chapters 1, 2, and 4. I would argue that this shouldn't be the first time the students are seeing groups like SO, SU, etc. Those ...


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