Hello Mr. Bard, Thank you so much for all of these resources - I especially like the idea about the 4-variable Venn diagram (as there is a lot of gateways to other topics in math and CS that start from Venn diagrams that I think high schoolers would find interesting and moderately challenging as well)! I will definitely look at *Discrete Structures in Math *as well as *Sage for Undergraduates*.
I really appreciate the detail and explanation for each of these links - I will have to take some time to examine all of them! Sincerely, Tanmay On Friday, June 17, 2022 at 11:46:50 AM UTC-7 [email protected] wrote: > Hi there! I'm sorry for the delayed response. > > I also love Venn Diagrams and discrete mathematics. I had been working on > a textbook for discrete mathematics, *Discrete Structures in Math---A > Problem Solving Approach*, for several years, but I had to pause it > because of the urgent need to update *Sage for Undergraduates*. Though my > discrete mathematics coverage has nothing to do with Sage, I think you > might find that motivated high-school students are fully able to do most of > the problems. (See Ch 1 and Ch 2 here: > http://www.discrete-math-hub.com/textbook-in-progress.html .) At some > point in the future, I do plan on finishing that book, but the drafts are > rather clean. By the way, I think Venn Diagrams are best done with > paper-and-pencil... but you could teach some very elementary SQL alongside > the set theory, to combine computer science and mathematics. > > If you've been studying group theory and discrete mathematics, I think > you'll find the 2nd edition of *Sage for Undergraduates* very easy to > read. It should be released in August. In Chapter 1, which introduces Sage > to the beginner, I deliberately quarantined the calculus until near the end > to provide for the possibility of use in high schools. Prof Beezer's book > on linear algebra (http://linear.ups.edu/html/fcla.html) is also easy to > read. > > In Ch 5 of *Sage for Undergraduates*, I teach the reader to program in > Python by using Newton's Method ( > https://en.wikipedia.org/wiki/Newton%27s_method > <https://en.wikipedia.org/wiki/Newton's_method>), which requires > calculus, as the running example. This could have easily have been instead > the secant method (https://en.wikipedia.org/wiki/Secant_method), which > despite its name has nothing to do with trigonometry. This could give a > high-school student exposure to the concept of an iterative algorithm, root > finding, and successive approximations. It also lends itself well to > graphical explanations, and even animation. > > If you like random walks, then you might want to do something with the > binomial model of stock prices and pricing stock options. You could combine > a bit of finance, a bit of computing, and a bit of mathematics that way. > You don't need to know very much at all about stocks to study this topic. > There's already an interface where you can get stock data from inside of > Sage. > > For younger students, I think it would be very good for their critical > thinking and even their reading ability, as well as their math, to solve > mathematical word problems and puzzles that result in systems of linear > equations. The student must analyze the word problem, and come up with a > system of equations. Let the computer solve the equations, but then let the > student interpret (and verify) the result. Some superb but challenging > examples can be found in *Problem Solving through Recreational > Mathematics*, by Bonnie Averbach and Orin Chein, re-published by Dover > Publications but originally from 1980. Some easier examples can be found in > *The > Complete Idiot's Guide to Algebra Word Problems Paperback*, by Izolda > Fotiyeva > <https://www.amazon.com/Izolda-Fotiyeva/e/B005GS7D3I/ref=dp_byline_cont_book_1>, > published > by Alpha Books in 2010. Actually, for 2 variables and 2 equations, I'd say > that the student should solve it by hand, and for 4 equations and 4 > variables (or larger) let Sage do it. The case of 3 and 3 to be left to > those instructors who are more familiar with this age group. > > The artificial intelligence method called "model checking," or sometimes > "entailment," would combine a bit of combinatorics and a lot of logic. I > always wanted to code up something in Sage for problems of roughly 24 or > fewer variables. In contrast, a 4-variable Venn Diagram is already hard to > draw ( > https://en.wikipedia.org/wiki/File:Venn%27s_four_ellipse_construction.svg > <https://en.wikipedia.org/wiki/File:Venn's_four_ellipse_construction.svg>). > In any case, model checking---when done rigorously---uses "certainly > false," "certainly true" and "not certain," a way of looking at the world > that I have found useful in ordinary life, in stark contrast to boolean > logic with only "true" and "false." As algorithms in artificial > intelligence go, it's one of the easiest. > > As you can see, it took a lot of research to dig up all these resources > for you, so I hope you will forgive the delay in responding. > > Enjoy your summer! > ---Gregory V. Bard > On Saturday, June 11, 2022 at 3:20:22 AM UTC-4 [email protected] wrote: > >> Hello all, >> >> My name is Tanmay Kulkarni and I am a rising sophomore. I have also been >> taking several extracurricular math classes with Squares & Cubes >> <https://www.squaresandcubes.com/> on things like number theory, group >> theory, discrete math, and linear algebra. In these classes we have >> utilized Sage to explore mathematical patterns. For instance, in my >> discrete math class, I used Sage's graph functionality to take a stab at >> graph isomorphism, which eventually lead to a magazine article >> <https://chalkdustmagazine.com/features/a-walk-on-the-random-side/> on >> using random walks on graphs to solve graph *connectivity*. >> >> During these various explorations, I realized that Sage was a very >> powerful tool to explain and provide intuition for complex mathematical >> concepts, however, (a) it is mainly used by those working in higher math, >> and (b) there is a high barrier of entry to implement concepts (even ones >> in lower math) in Sage. >> >> Thus, I wanted to contribute to Sage and* implement specific concepts >> which I felt high school students like myself would find interesting,* >> and use them for educational purposes (e.g. at my school). Two basic ideas >> I thought of were: >> >> 1. Random walks. I think mathematics is often far more engaging with >> a visual component (for instance, teaching graphing skills and different >> types of equations through a Desmos art project), and I think when >> talking >> about probabilities and randomness, an excellent visual representation of >> stochastic processes is random walks, which are currently not implemented >> in Sage. The other advantage of this is that random walks are often >> present >> in other places such as physics (in Brownian motion). This could expand >> into >> 2. Venn diagrams. Venn diagrams are incredibly important; however, I >> could not find any Sage implementations of Venn diagrams beyond simply >> plotting intersecting circles. Having a more solid implementation could >> provide a strong, visual intuition for a variety of concepts, like basic >> set theory, logical operators, probability, and even open the door for >> Edwards-Venn diagrams! Such an implementation would utilize Sage's 2D >> graphics (specifically the circle and text functions) as well as the >> detailed Set implementation. >> >> >> Several people who I contacted referred me to this group, and thus I am >> wondering if anybody would be generous enough to (a) *provide thoughts >> on the feasibility and usefulness* of such an endeavor, (b) *provide >> some direction or guidance* as to where to begin, and (c) offer *potential >> avenues *where this could be used. >> >> Until then, I will be beginning to work on any very simple bug fix I can >> find to familiarize myself with developing in Sage. >> >> Thank you so much! >> >> Sincerely, >> Tanmay Kulkarni >> > -- You received this message because you are subscribed to the Google Groups "sage-edu" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. 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