Week 14
Tuesday
I'm reading your proof portfolios, so let's take another day off while I try to find something useful, proof-wise, for us to do this semester. First, I have another problem to think about: Newcomb's Problem. Let's see where we fall in the reasoning. The next task is learning how to divine the day of the week corresponding to any date, the Doomsday problem. I want us to also consider the ways this discussion is written. I've given you 2 handouts: Conway's notes and a collection I put together from others. Here's a nice online calculator to check our work.
Thursday
Someone here said "who's Archimedes", so let's address that first today: the palimsest. Perhaps some Vedic math next? or a birthday magic square?
Week 13
Tuesday
One of my favorite modular projects is business card origami, although it's not a project to start and then move. You might try at home after watching a couple of videos. Unit cube with business cards (see a Sponge by Dr. Jeannine Mosley at the Institute for Figuring)
Instead, we'll try a gyroscope while we watch some more sophisticated videos about folding.
Look at what Robert Lang has to say about the science of origami. You may also wish to read about Erik Demaine on the deep mathematical roots of folding, or look into his grad class at MIT. I think the video of the 1st class is quite accessible. Another fascinating place, The Institute for Figuring.
Week 12
Tuesday
Thursday Today we watched a video about Robert Wiles made by the BBC. Everyone received a copy of the booklet that accompanied the first US celebration and video to announce the proof of Fermat's Last Theorem. The film provides many insights, including why the name. Of the many claims scribbled by Fermat in the margins of his copy of Diophantus's book, this theorem is the last to be proven.
Week 11
Tuesday Induction. Today we'll start with a worksheet where the arguments aren't quite proofs as some correction is needed. Then, time permitting, we'll look at a set of declarations, all of which lend themselves to induction.
Thursday
Week 10
Tuesday No class --- Lynne ill.
Thursday Last week we resolved epsilon-delta pairings to confirm continuity. I created a handout that includes that work, and additional strategies. I hope you find them useful. Another loose end is the promised mathematical card tricks.
Week 9
Tuesday
Thursday
Thursday
Week 8
Tuesday
Thursday
Week 7
Tuesday
Thursday
Thursday
Week 6
Tuesday
Thursday
Thursday
Week 5
Tuesday Today I shared some student work on the Graham sequence (part 1, part 2), the proof that a student submitted for Fido Puzzle. Has anyone made progress using the online app for the Towers of Hanoi? Solving this puzzle some employ a visible deductive process. We'll discuss this a bit. Time permitting, we'll return to our text. You might want to try your hand at Devlin's 3rd HW.
Also, now might be a very good time to consider statements related to the implication p=>q: ~q=>~p (contrapositive) and q=>p (converse), and the contrapositive of the converse (inverse of p=>q ): ~p=>~q. Proof by contrapositive appears on page 54, and the Contrapositive and the Contradiction methods differ in a very subtle way. t
- Method of Contradiction: Assume P and Not Q and prove some sort of contradiction.
- Method of Contrapositive: Assume Not Q and prove Not P.
The method of Contrapositive has the advantage that your goal is clear: Prove Not P. In the method of Contradiction, your goal is to prove a contradiction, but it is not always clear what the contradiction is going to be at the start.
Week 4
Tuesday I asked that you make comments on the rewrite of a document that compares HS math to College, and then we looked at (yet another) puzzler: Fido Puzzle. I related a story, celebrating the incremental solution from a previous semester: 1) the email that a student wrote immediately after first seeing Fido, and (2) a second email from another who had spend quality time on the problem. We also realized how to play using the 9s property, but don't yet know WHY 9 plays this role.
Thursday Today I'll ask for any comments on the HS/College comparison document, comments about Fido, and then we'll watch an 8 minute video of Devlin taking about truth and proof. Has anyone done more work on the Graham Sequence? Reactions? Just to balance the misconception that mathematical thinking is an adult activity, let's also watch a maybe 20 minutes of video (from a longitudinal study done in NJ) discussing mathematical learning and thinking. In case you want to get more involved, here's an online app for the Towers of Hanoi.
Week 3
Tuesday We return to STEM 215 for a second day with Latex. One new document to help is a collection of links and rules used in mathematics for formal writing. Tutorials abound. Here are two rather good videos on just mathematical expressions: video 1, video 2, .
Thursday Today we looked at the some of the Latex docs you have produced.
Week 2
Tuesday I mentioned Thursday math societies: AMS, MAA, SIAM. Each provides very useful resources and publications. For example: (1) The American Math Society as a large list on interest areas called Math Subject Classification: MSC2000; (2) the Math Genealogy site. . Graham Sequence paper also relates to more math culture: On-line Encyclopedia of Integer Sequences (OEIS) site. Before we begin with Graham sequences themselves, let me give you another problem to think about: You have 12 balls, and one has a different weight. How would you identify this ball using a 2-pan balance only 3 times?
Thursday Meet in STEM 215 computer classroom to start learning LaTex. To get started, you may wish to follow a step-by-step approach like that offered by the AoPS site. Your assignment for next Thursday is to have some piece of mathematics Latexed. It can be something for a book, another class, or something from the list of 29 theorems on this site.
Week 1
Week 5
Tuesday Today I shared some student work on the Graham sequence (part 1, part 2), the proof that a student submitted for Fido Puzzle. Has anyone made progress using the online app for the Towers of Hanoi? Solving this puzzle some employ a visible deductive process. We'll discuss this a bit. Time permitting, we'll return to our text. You might want to try your hand at Devlin's 3rd HW.
Also, now might be a very good time to consider statements related to the implication p=>q: ~q=>~p (contrapositive) and q=>p (converse), and the contrapositive of the converse (inverse of p=>q ): ~p=>~q. Proof by contrapositive appears on page 54, and the Contrapositive and the Contradiction methods differ in a very subtle way. t
Also, now might be a very good time to consider statements related to the implication p=>q: ~q=>~p (contrapositive) and q=>p (converse), and the contrapositive of the converse (inverse of p=>q ): ~p=>~q. Proof by contrapositive appears on page 54, and the Contrapositive and the Contradiction methods differ in a very subtle way. t
- Method of Contradiction: Assume P and Not Q and prove some sort of contradiction.
- Method of Contrapositive: Assume Not Q and prove Not P.
The method of Contrapositive has the advantage that your goal is clear: Prove Not P. In the method of Contradiction, your goal is to prove a contradiction, but it is not always clear what the contradiction is going to be at the start.
Week 4
Tuesday I asked that you make comments on the rewrite of a document that compares HS math to College, and then we looked at (yet another) puzzler: Fido Puzzle. I related a story, celebrating the incremental solution from a previous semester: 1) the email that a student wrote immediately after first seeing Fido, and (2) a second email from another who had spend quality time on the problem. We also realized how to play using the 9s property, but don't yet know WHY 9 plays this role.
Thursday Today I'll ask for any comments on the HS/College comparison document, comments about Fido, and then we'll watch an 8 minute video of Devlin taking about truth and proof. Has anyone done more work on the Graham Sequence? Reactions? Just to balance the misconception that mathematical thinking is an adult activity, let's also watch a maybe 20 minutes of video (from a longitudinal study done in NJ) discussing mathematical learning and thinking. In case you want to get more involved, here's an online app for the Towers of Hanoi.
Week 3
Tuesday We return to STEM 215 for a second day with Latex. One new document to help is a collection of links and rules used in mathematics for formal writing. Tutorials abound. Here are two rather good videos on just mathematical expressions: video 1, video 2, .
Thursday Today we looked at the some of the Latex docs you have produced.
Week 2
Tuesday I mentioned Thursday math societies: AMS, MAA, SIAM. Each provides very useful resources and publications. For example: (1) The American Math Society as a large list on interest areas called Math Subject Classification: MSC2000; (2) the Math Genealogy site. . Graham Sequence paper also relates to more math culture: On-line Encyclopedia of Integer Sequences (OEIS) site. Before we begin with Graham sequences themselves, let me give you another problem to think about: You have 12 balls, and one has a different weight. How would you identify this ball using a 2-pan balance only 3 times?
Thursday Meet in STEM 215 computer classroom to start learning LaTex. To get started, you may wish to follow a step-by-step approach like that offered by the AoPS site. Your assignment for next Thursday is to have some piece of mathematics Latexed. It can be something for a book, another class, or something from the list of 29 theorems on this site.
Week 1
Tuesday Today's "take-away" should include that "doing" math and "solving problems" are overlapping but distinct skills. We'll took a look at the course syllabus, and then moved onto introductions. I introduced Paul Zeitz, and his views of problem solving. We watched a few minutes of Paul talking about problems from the Great Courses site. What luck! A pdf of Paul's book is online. We quickly resolved the pill problem by cutting up the pills; we ended with the camel problem.
Thursday We started with a two-page handout containing a description of another Zeitz video (sound is too soft to see together), and a couple of problems to start our semester. We looked at the lightbulb problem, and some made a lot of progress on the camel's problem. Noah shared his write-up. We looked at another popular mathematician who thinks about thinking, Steve Strogatz.
If you wish to get a head start on next week, look at these documents about the Graham Sequence, and documents in Latex: source file,pdf file, article on Graham Sequence.
If you wish to get a head start on next week, look at these documents about the Graham Sequence, and documents in Latex: source file,pdf file, article on Graham Sequence.
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