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Teaching Discrete Mathematics via Primary Historical Sources

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This site offers written curricular materials, based on primary historical sources, for beginning and advanced undergraduate courses in discrete mathematics and computer science. Such courses, which often cover combinatorics, deductive reasoning (logic) and algorithmic thought, draw a variety of majors, ranging from computer science, mathematics, the physical sciences and engineering to secondary education. Traditional methods of instruction follow ``The Modern American Discrete Mathematics Text,'' which although thorough and mathematically precise, present the material as a fast-paced news reel of facts and formulae, often memorized by the students, with the text itself offering only passing mention of the motivating problems and original work which eventually found resolution in modern concepts such as induction, recursion, or algorithm.

This sort of apporach is very close to my heart, as LtU readers probably know. I wish them well.

By Ehud Lamm at 2006-12-18 20:46 | History | Teaching & Learning | other blogs | 9507 reads

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Providing context

I have to agree that providing context to help students see why particular mathematical theorems or insights are interesting is crucial. In fact I wrote about that very subject recently myself. I think this is an idea which is often missed in mathematics education. This then result in efforts to "make it relevant to the student" by trying to turn everything into practical applied examples that teachers assume relate to the stuents everyday life. Mathematics, however, doesn't always lend itself to such direct and immediate application - indeed a great deal of the mathematics that is truly interesting can only be applied to more mathematics. The aim should be to provide an idea of the context, the mathematical landscape into which the new material fits, so that new pure mathematics can be seen weaving itself into broad patterns that stretch across the entire vista of mathematics.

Obviously the method to achieve that here is through historical reconstruction - by building up the layers of context as they were developed historically we can reconstruct the reasoning that saw the more abstract developments as interesting in their own right. Following the exat historical road is, I would hope, not necessary however: history often followed a torturous path, and it might be nice to use the benefit of hindsight to carve a slightl more direct, but equally scenic road.

By Leland McInnes at Tue, 2006-12-19 00:04 | login or register to post comments

I find the tortuous path

I find the tortuous path interesting and informative. It's helpful to see how discoveries are made in reality, instead of as a neat and orderly progression. Maybe the best way is to present both paths, and let the student choose what works best (perhaps both).

By Jeff Nowakowski at Tue, 2006-12-19 02:15 | login or register to post comments

Self directed learning

Well the ideal is going to be self directed learning where the student picks the path that suits them. In practice that isn't always viable. A mix and match approach is going to probably be the best. For instance, with the material discussed in the article I think a historical approach would be fine, but to learn calculus via Descartes first pushes into analytic geometry, followed by both Newton and Leibniz's (different) rather half assed "appeal to intuition" approaches, a long and protracted battle over notation, followed by a long and heated debate over the meaning and validity of infinitesimals as we slowly start to build toward the concept of a limit and of continuous functions via epsilon-delta definitions before finally arriving at a rigorous formulation... Well that seems more effort than its worth. Honestly try reading some original early 18th century calculus: it's hairy, sketchy, hard to understand well, and generally unpleasant. In practice a little hindsight to put things in a more palatable order is worth it.

By Leland McInnes at Tue, 2006-12-19 02:27 | login or register to post comments

I agree, it shoudn't be

I agree, it shoudn't be mandatory to dig into that stuff. At the end of the day you should be tested on whether or not you can do the math. However, when I went to school there was very little of the history, so it wasn't even an option unless I wanted to do a lot of research on my own.

The stuff you mention in the development of calculus is fascinating to me, and really helps to put it into perspective. I wouldn't want to sift through all the classical works, but I would love to read "cliff notes" for the most significant ones.

By Jeff Nowakowski at Tue, 2006-12-19 06:38 | login or register to post comments

Suggested reading

Some suggestions for reading then - sort of a cliff notes of historical development of various fields:

The Equation that Couldn't be Solved by Mario Livio is really quite good, though far more historical than about Grup Theory.

Everything and More by David Foster Wallace is truly excellent - it provides both a precis of the history, and a healthy dose of reasonably hard-core detail for anyone who is interested.

Hopefully other people can add some more suggestions to the reading list in replies. I do enjoy good math history.

By Leland McInnes at Tue, 2006-12-19 08:00 | login or register to post comments

an historical approach to teaching real analysis

Understanding Analysis

A Radical Approach
A Radical Approach to Real Analysis
A Radical Approach to Lesbesque’s Theory of Integration

By Marlene Miller at Wed, 2006-12-20 04:04 | login or register to post comments

This book sounds very

This book sounds very interesting. Thanks!

By Ehud Lamm at Wed, 2006-12-20 08:43 | login or register to post comments

The review for

The review for Understanding Analysis sums up nicely the conversation we've had here:

The obvious solution to the dilemma is to present the problems and the counter-intuitive examples that informed and motivated the theory in the first place. One can be tempted, and I admit I was tempted, to attempt a historically accurate presentation of the subject. There exists a terrific book by David Bressoud for those brave (or foolhardy) enough to take this radical approach. I think this is a mistake for a first pass through the theory and, if a student is only going to see this material once, she should get the cleaned-up, elegant, modern version. Future mathematicians, high-school teachers and the historically inclined can then be exposed to a historical treatment and be in a much better position to understand the evolution of the ideas if they are not also simultaneously trying to assimilate the technical details.

By Jeff Nowakowski at Wed, 2006-12-20 20:03 | login or register to post comments

more real analysis

From the preface of Understanding Analysis mentioned above...

In a recent rereading of the completed text, I was struck by how frequently I resort to historical context to motivate an idea. This was not a conscious goal I set for myself. Instead, I feel it is a reflection of a very encouraging trend in mathematical pedagogy to humanize our subject with its history. From my own experience, a good deal of the credit for this movement in analysis should go to two books: A Radical Approach to Real Analysis, by David Bressoud, and Analysis by Its History, by E.Hairer and G.Wanner."

Analysis by its History
A review in The American Mathematical Monthly

By Marlene Miller at Thu, 2006-12-21 20:23 | login or register to post comments

Is it context that helps, or something else?

I've recently been thinking about teaching methods as well. As Leland suggests, following the exact history of a topic is probably unnecessary. In fact, I think the reason teaching via historical sources succeeds is because it gives a narrative to the topic at hand, and it is certainly not the only way to do so.

The reason to give a topic a consistent story (the historical one being a obvious example) is because then students can see why a particular formula, equation, or whatever was motivated. I think this gives far better understanding than the typical "news reel"-style presentation that was alluded to. Furthermore, when one has understanding of the motivations for something, it becomes easier to integrate into a conceptual framework and doesn't have to be memorized. Instead of using a formula 'cause that's what's on your cheat-sheet, one can instead use it because it's the obvious way to solve <insert problem here>, and it really just falls out of <property of problem>.

An analogy with programming is the writing of comments. Ideally, we try to write comments that explain why we do something, and not just what it is that we are doing.

Giving a student the formulas in a vacuum is kind of like giving the tranlation book to the person in the Chinese room. Sure they can talk about the topic at hand, but can they compose the results in new and interesting ways? Are they hampered by reliance on the translation book?

By Andy Kish at Tue, 2006-12-19 02:27 | login or register to post comments