25

Perhaps emphasize to students the spirit of equivalence relations. They partitions sets into equivalence classes--cutting down the amount of cases necessary to prove something. To illustrate this, take a geometry example first. "Is similar to" is an equivalence relation on the set of regular polygons (I'll omit a proof here). Now say I need to ...


16

In general, all applications are going to be more of the "here is what we have to check to make sure that our algorithm/theory/definitions work". We don't usually encounter practical problems where we're given a completely arbitrary relation and have to check if it's an equivalence relation. Here are some of the more common equivalence relations ...


13

"A well-chosen example illustrates ... and is entirely convincing." For me, all of what is usually called "generic proof" satisfy your criterion. Consider the Euclidean algorithm for finding the greatest common divisor of two numbers. A well-chosen example tells your students all they need including why the algorithm gives the greatest common divisor, how ...


11

Equality vs. Identity Alexei touched on this topic by mentioning hash tables, but I would like to spell it out more explicitly, because this is a critical and fundamental topic in software engineering, and essential for every programmer to know and understand. Every high level programming language has a mechanism for comparing two values for "equality&...


10

Analyzing data (probably by pushing a button in a statistics programme) instead of learning mathematics is not the same thing as "emphasizing the discrete" as in discrete mathematics. As far as the discrete vs. continuous goes: I do think that it is a good idea to teach more discrete mathematics early in the curriculum because it is kind of strange if ...


10

Society always benefits when new mathematical tools are developed to get insights into the world around us. There is little doubt that Calculus has helped change the quality of the lives we can lead over the hundreds of years since it was born and Calculus has evolved in many unexpected ways. The real question should be to what extent it is worth ...


8

There are many situations in which we have a clear collective understanding of intention, or goals, and examples which persuade us that these goals are plausible, as well as illustrating apparent causal mechanisms identifiable as "reasons" for things being the way they are. For many reasons, the contemporary style of proof-writing, especially in intensely ...


7

Obviously, one place to look is in the huge amount of “new math” curriculum material that was written during the late 1950s to early 1970s, but I’ll leave that for you or someone else to search through. Regarding the papers listed below, I looked through a lot of papers that I've collected over the years on elementary logic issues (far more than what’s ...


7

The list of topics you want to study corresponds rather to abstract algebra than group theory. You did not say why you are interested specifically in group theory, but I believe that acquaintance with various algebraic structures in addition to groups would be beneficial for any person learning mathematics beyond school mathematics. Therefore, abstract ...


7

The greedy algorithm fails quite spectacularly for the Traveling Salesman Problem (TSP): Bang-Jensen, Jørgen, Gregory Gutin, and Anders Yeo. "When the greedy algorithm fails." Discrete Optimization 1.2 (2004): 121-127. (PDF download.) Abstract. We provide a characterization of the cases when the greedy algorithm may produce the unique worst possible ...


6

If you have looked at modular arithmetic, then one possibility is: Give/recall some example of an algebraic argument in modular arithmetic; then point out that the argument is implicitly relying on the fact that congruence is an equivalence relation. So then you can explain: equivalence relations are designed to axiomatise what’s needed for these kinds of ...


6

The notion of equivalence relation is one of the basic building blocks out of which all mathematical thought is constructed. (Paul Halmos) What's the point of learning equivalence relations? The concept of equivalence relation is a generalization of the concept of equality. Why is it good to know that $a$ is equal to $b$? Because, in this case, all we ...


5

I have found the best thing to do in tutoring a proof based class is to constantly ask the questions "why is that true?" or "what is the definition of that?". These questions help them build relations between definitions and highlight the connections to proofs as filling in the details relating things. Yes, on occasion you might have to do an example of a ...


5

There are tons of tasks where greedy algorithms fail, but the best in my opinion is the change-making problem. It is great, because whether the obvious greedy algorithm works depends on the input (i.e. the denominations). For example, if you have coins $1,6,8$, then $12=6+6$ is better than $12=8+1+1+1+1$. Some other tasks: Shortest paths where the edge ...


5

To repeat my comments: I can recommend Fraleigh's classic introduction. It is easy to read for beginners, with many exercises, from easy to difficult, on which self-learners could check themselves. John B. Fraleigh A First Course in Abstract Algebra, 7th Edition. Pearson, 2002.           Because it's been around so ...


5

Dictionary/hash table relies on equivalence to bucketize items. So knowing that one would never try to build a hashtable by a distance between cities (objects on a plane): distance is not transitive. In real programming there is another common way to violate equivalence which is sort of implied in pure math/CS: "a == b hence in 5 minutes a == b too"...


4

Math With Bad Drawing has some images that approach an info-graph (and in general is just a great website for math education), for example: https://mathwithbaddrawings.com/2015/07/01/infinity-plus-one-please-check-your-intuitions-at-the-front-desk/ There are some good geometry ones, especially around old compass and straight-edge constructions but that ...


4

I don't believe that there is much of debate about Calculus today. Major universities continue to push the overly rigorous vision embodied in texts like Stewart or Hughes Hallet. The AB/BC Calculus curricula have seen little to no change in decades. It seems lost on many arguing for Stats over Calc, that they are one and the same. Statistics developed ...


4

I taught a geometry course which counted both for math majors (sophomore level), and was required for the math education program secondary education teacher majors, and we used the text: Edwin Moise: Elementary Geometry from an Advanced Standpoint, 3rd Edition, which was amply challenging for both cohorts of students. For math majors completing this course,...


3

I liked this Discrete Mathematics: An Open Introduction, by Oscar Levin, for generating functions, so I'm guessing it will be good for recurrence relations.


3

"Kids usually struggle with every one of these concepts, let alone all of them together. It is difficult to get the whole picture and all the moving parts. So this place (proof) seems like a good place to show all of this in action." I'm afraid there is no "complete picture" of all the facets of logic and proof that you mention, interrelating in one ...


3

I'll echo @XanderHenderson in the comments that Hungerford's Abstract Algebra: An Introduction is a really nice textbook. It's what I had as an undergraduate, and I'm currently re-reading it for maybe the third time. It's one of my favorite, well-written, clear texts. Note that he's committed to a pedagogy that starts from the most familiar/concrete and ...


3

There is Grinstead & Snell's Introduction to Probability, Second Revised Edition. It is freely available under GNU Free Documentation License, as explained on the linked page. The level is undergraduate and the authors state in the preface: The text can also be used in a discrete probability course. The material has been organized in such a way that ...


3

I think that whether or not a proof makes a theorem more convincing depends on who exactly you are trying to convince. While teaching an introductory math course to non-mathematicians (like say, calculus I in the U.S.), the emphasis is often on using the results. I think it is rare that a proof helps a student use a result, so proofs might not help a student ...


3

The debate seems to be where it was 20 years ago. The reason is that most universities have a "generic" mathematics curriculum, so if the engineers need calculus as it is now, everyone will also get it. That said, the quotation rings true to me: "The world has moved from analog to digital, and it is time for our mathematics curriculum to change from analog ...


3

One of the more popular high-school proofs (Pappus's) that the base angles of an isosceles triangle are equal relies on an equivalence relation. They typically don't call it that in high-school, but that's what it is. Indeed much of high-school geometry and trigonometry relies on this equivalence relation (the SSS equivalence).


3

OP here: There's so many good answers here, this is possibly my favorite question that I've ever asked on SE! I'm so glad that I thought to post the question here. For comparison purposes, having considered all the great responses to date, I'll share how I've decided to revise my lecture notes. Observe that we have a crushing time constraint in the course, ...


2

I actually have some sympathy for this point of view in that there is an awful lot of useful insight that can be gathered from algebra, exponential curves, trig, stats, investment calculations, nomographs, maneuvering boards, etc. However, anyone heading into engineering, physics, or chemistry still needs standard (Granville) calculus.


2

A source of simple example is in basic set theory. Let $A,B,C$ be sets and $f:A\to B$, $g:B\to C$ maps. Consider assertions of the kind "if $f$ and $g$ are one-to-one, then $g\circ f$ is one-to-one". You can prove it directly using this classical definition "$f$ is one-to-one when for all $x,y\in A, f(x)=f(y) \implies x=y$". Assume $f$ and $g$ are one-to-...


2

Discrete mathematics (some combinatorics, at least) is a requisite for probability and statistics. So it should be covered. For computer science, combinatorics, techniques to evaluate sums, recurrences, and some graph theory are a must, and should be covered early in some detail.


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