# Tag Info

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 ...

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 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 ...

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 ...

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 ...

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 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 ...

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 ...

2

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 ...

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-...

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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.

2

While not, strictly speaking, a textbook, Phillips Exeter Academy publishes their full problem set for their non-proof based discrete mathematics course (which they offer to advanced high school students). For more on how they use the problem set in their course, see my answer to this question. Re: your "Update," these problems do have the advantage of ...

2

The Vandermonde binomial identity. The Wikipedia page gives three proofs; an algebraic one (symbol pushing), a combinatorial one (counting in two different ways), and a geometric one (where a certain family of paths is decomposed recursively). All three proofs are short, simple, and insightful, and they complement each other.

1

Better yet, I very much liked generatingfunctionology by Herbert Wilf; it is the go-to text for what you are seeking. It addresses generating functions, and considerable help for understanding recurrence relations. (And it's free from the author, and downloadable as a pdf.!) In addition, you'll find extensive information at Ronald L. Graham, Donald E. ...

1

I've made some fairly nice graphics using Sage. I think that it uses matplotlib, at least partly, as a backend. It can render to a variety of image formats and supports LaTeX in labels. Edited to add what the code referenced in the comment produces.

1

Excel comes to mind. This took just a few seconds to make:

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I'd whip up something in LaTeX using TikZ (see for example Cremer's short tutorial, the full documentation is a real bear; or snoop around in TeX.stackexchange.com). For standalone, I'd probably use something like xfig (for hand drawing) or asymptote (for "programming" a more complex figure). Under Linux, there probably are Mac (and even Windows) versions ...

1

One important example is the equivalence of standard bottom-up induction and top-down induction. It is interesting that bottom-up induction does not extend immediately to ordinals (since successors and limits must be handled differently), while top-down induction extends without any change. On the other hand, top-down induction is usually easier to learn ...

1

Have you looked at Khan Academy? They seem to cover just about all of this in their video lectures and follow it up with exercises to mastery. Students can have you as their "coach" and you can assign those exercises that you think are helpful. I've used it with elementary - high school students and they have all found it useful. I assume you would ...

1

I hate answering my own question, but thought I should let others who find this know what I decided on: "Applied Discrete Structures" by Alan Doerr and Kenneth Levasseur Reasons for picking it: Light on proofs (but they're still there), good on computation problems, free online text.

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I may suggest Discrete Mathematics by Rosen, this book has a lot of computation problems, and the sections on proof are not really that necessary to continue with the advanced material on the book.

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