The Dark Side of Pascal's Triangle
Audience: high-schoolundergraduate
Tags: combinatoricstaylor-seriescombinatorial-reciprocitystirling-numbersfalling-and-rising-factorialscombination-numbersbinomial-coefficientsgregory-newton-formulaforward-differencesbackward-differencesdiscrete-calculusumbral-calculusforward-newton-seriesbackward-newton-seriescombinatorial-identitiespascals-trianglefinite-differencesinterpolationpolynomial-interpolationnegative-binomial-coefficientsdiscrete-sequencespascals-identitysymmetric-identityhockey-stick-identityvandermondes-identitylinear-operatorsanalytic-functionssummationshift-operatordiscrete-mathanalytic-continuationfunctions-as-vectorsoperators-as-matrices
An informal introduction to the negative rows of Pascal's triangle, discussing the motivation and intuition behind some of its basic applications, before diving into discrete calculus, the forward and backward Newton series, umbral calculus, and as a consequence, a quick overview of Stirling numbers, for the purpose of highlighting some deeper, underappreciated treatments of discrete math. The target audience is meant to be a wider net, showcasing fun recreational ideas and basic combinatorics in the early chapters, slowly introducing connections to calculus and linear algebra as the video progressively gets more advanced, but hopefully the ideas throughout the entire video are novel enough that almost everyone will find something interesting within it. The main message conveyed here is that discrete math and combinatorics especially is often underappreciated, in my opinion, and this is just one particular direction that combinatorics can be pushed into, an outline of how its ideas can help shape intuition in another field. The value of pedagogy here for teachers is admittedly not explicitly high, more about the high-level message than about the particular ideas in this video.
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Wow this is a tour de force! This is an amazing YouTube video, but for a teaching tool there are some downsides. Covers a huge amount of ground in a short time, which in my experience would need to be gone over a lot slower for students to understand what was going on. The huge range of things covered at different levels makes it a bit less useful as a teaching tool. It could be a nice overview video for a multi-week long chapter in a combinatorics course or something, where after the students have learned the material serves as a summary. Some of the concrete examples, like the 2^x example are more useful for in class as it gives nice visuals to go with the formulas; these snippets would be the main teaching value in my view. I also don’t think the tag of “high-school” is appropriate for 90% of the video; especially towards the end of the video with the infinite matrices etc is very advanced undergraduate content (like 4th year undergrad) or early graduate content. Overall score as a Youtube video: 7.5 Overall score as a teaching tool: 6
The addition rule of Pascal’s Triangle is just a flipped version of the finite difference formula!
The explanations and animations are literally perfect! This deserves to win!
Too long, but very interesting. Good voice. The scheme at the end is overcrowded and way too big.
The visualization is awesome. I love that it answers the questions I have in mind as the video goes on as if I’m having a conversation with the author. It also goes surprisingly deep and connects a bunch of seemingly unrelated fields together. I think this is exactly the new perspective that is lacking in the modern day classroom system. We follow the curriculum structure way too carefully and overlook the important connections that students ought to make with other topics / subjects. In my opinion this is the exact type of the supplimentary educational tool / video that I wish I had outside the class.
Magnificent, exceptional work on the deep connection between finite differences and differentiation, showing that tools from one side may solve problems from the other, and the many connections of the mathematical web between analysis, algebra and combinatorials. Hats off!
Beautiful video, very interesting for mathematicians. It remains largely out of reach for high-school students, but that was stated from the beginning. I cannot give higher marks because the aim of SoME4 was mainly to support students in difficulty (“How useful would this entry be in supporting student understanding?”).
This was a cool video with lots of deep content! I really liked the connections that you drew between discrete and continuous areas of math.
The video covered a lot of content very quickly. There were points where I was trying to keep up — perhaps a second viewing will make it more clear. I also thought the video lacked some direction at the beginning (especially when discussing facts about Pascal’s triangle).
Actually watched this earlier already :) quintessential SoME goodness
WOW! Insanely interesting video. I had heard a little about umbral calculus before but this just blew my mind! I think this video would be a lot for highschoolers but for math majors and grads this video is very enlightening! And it shows something that I think is very lacking in most math education - the deep connection between mathematics fields. Most math classes I think very much stick to the field they are in but there is a wider deeper world out there that even a lot of graduate students and mathematicians do not see. Thank you for giving me another glimpse into this world!