Wau: The Most Amazing, Ancient, and Singular Number
In this video Vi Hart talks about a number that was discovered in East Asia and in many other ancient societies. She describes the different mathematical relationships, usually infinite fractions, that included Wau (some that just included Wau and others were equal to Wau). She related the special number to nature and other fields of study like physics.
https://www.youtube.com/watch?v=wsOXvQn3JuE
Why Every Proof that .999… = 1 is Wrong
In this video Vi Hart goes through some of the many proofs that are used to show that .9 repeating is equal to one. She goes on to prove or attempt to prove these wrong to say that the two are different numbers that are independent of each other. She invalidates these proofs and then briefly talks about why mathematicians believe it is so.
https://www.youtube.com/watch?v=CfJzrmS9UfY
Doodling in Math Class: Stars
In this video Vi Hart talks about drawing stars and connects this activity to math. She talks about stars with different numbers of points and how the number of points relates to the number of sides in each of the polygons that make up the star. She then makes this star drawing into a “game”.
I was a little confused, especially during Hart’s “Doodling in Math Class: Stars” video not because I did not understand what she was saying just because I felt that she was going a little to quick while speaking and the video recording was also sped up so while trying to process what she had just said she was already on to the next part of her speaking and all throughout her hand was moving very quickly on the paper. She did show what she was saying at the same time that she was saying it. For this reason I had to pause the video multiple times to reassess what she said. There was not anything in the video that particularly bewildered me or made me say wow except for when Hart started drawing the much larger stars with many points (when she began using the protractor): I thought that the stars that she made were beautiful and they also require a lot of talent to make. I learned that a star with p points can be drawn with 2 p/2- gons; this was a mathematical relationship that I had not been aware of. In fact I have not ever looked into the math behind stars before. I also learned that even with a set number of points in the star and a set number of polygons formed within the star, there are still multiple ways of drawing the star even with the restrictions. Although I did have to pause the video ever so often so that I did not fall behind on what Hart was saying, I for the most part understand everything that she said. So the only question I have after watching the video, is not directly about the math or the drawing in the video, it is: how does Hart discover the relationships between things like the sides of the stars and the number of polygons in the star; or rather: how is she so successful at finding things like the “game” in this video, that still relate to math, but that a majority other people would not be able to discover?
I think the video could really help me with my own math teaching, even though I did not really enjoy the fast pace of it. What I can learn from Hart is that I need to implement creativity in the classroom at least every once in a while especially in order to keep the students engaged. As Hart emphasizes in the beginning of the video while talking about how she was bored during her math class while learning about factoring, classes that are only lecture can get boring and can cause students to become distracted. So rather than have my students lose focus like Hart did from what the class was actually about: I will give my students an activity to do in class that directly relates to the topic of the day. This way they have something fun to look forward to and are then more inclined to pay attention during class so that they can preform well during the activity. This way they get the fun that they want in class, without losing focus from the actual class and making fun for themselves. Hart’s video really helped me come up with this idea for my own classroom because she expresses that class can get boring, which is true for any student at one point or another and that there are fun and thought-provoking activities to do during class that can still be mathematical and related to class. Kids have a hard time focusing on school work especially for note-taking and problem solving but they are often all for interactive activities, and so I have come to realize after watching this video that combining the two ways of learning can help to take away some of this lack of focus.
Yes I certainly agree with you. Hart is able to bring creativity into Math which is somewhat lacking these days. This creativity is able to inspire others since it shows how something simple can be made fun and entertaining. Teachings of this sort would make any math class more entertaining than boring. This perhaps would make students outside of class to begin to delve into math since we mostly just use it then continue on with our day as if it never happened.
I also find myself watching these videos in a different way than I watch other youtube videos – I pause a lot, too! – so in a sense I’ve had to learn a new way to watch in order to really appreciate them.
I think the way you’ve channelled this into your teaching ideas is wonderful!
I'm currently reading these: https://www.mcmp.philosophie.uni-muenchen.de/students/math/math_logic_munich.pdf lecture notes and am kind of stuck on the proof on page 58 line 4 where $$\Gamma \lnot \varphi \lnot \varphi$$ is derived using (Ass.). I don't understand how, none of the rule thus far introduced allow me to manipulated the consequent except contradiction which I can't use in this […]
I think calling a proposition an "implicature" means something like this: If there is a proposition "p->q" and this proposition is true, then we think that p "implies" q. So if p is false and q is true, does “false” imply “true”? My own thoughts are, Premise: ((I am a cup) and (the cup is […]
The problem shows up in Troelstra's Basic Proof Theory, exercise 2.3.6A. Show that in [classical logic's Natural Deduction], for the languages without ∨, ∃, all instances of ⊥c derivable from instances ⊥c with atomic conclusions. I have a solution to the problem, but I'm not sure if it's correct as it seems trivials, perhaps because […]
I have always been confused about a logic problem, which I can not form it into a general statement. Here is an example that may express my ideas. We know given a set $A$ and an equivalent relation $\sim$ on it. We can define equivalence class by $\forall x\in A, [x]=\{y,|y \sim x\}$. It seems […]
Problem: Prove the sentence $\forall x \forall y(xEy \rightarrow \neg x=y)$ in the vocabulary of graphs using the axioms of graph theory. So far: The axioms of graph theory given are antireflexivity and symmetry. Here $xEy$ means there's an edge between the vertices $x$ and $y$. We were told we'd need identity axioms for this […]
Problem: Using identity axioms, prove $\exists x P(x) \rightarrow \forall x P(x)$ from $\forall x\forall y \, x=y$. So far: I'm quite stuck on where to even begin. Working backward, I know we want $P(x)$ and $\exists xP(x)$, but I can't quite see how to write it down. I'm used to proof trees over Fitch-style […]
Problem: Show that the formula $$\exists x \forall y R(x,y) \rightarrow \forall x \exists y R(x,y)$$ is not provable by natural deduction. So far: First thought that came to mind was simply considering the natural numbers with an order relation to give a counter-example (considering an assignment such that $s(x)=0$). However now $R(x,x)$ is not […]
The following deduction seems correct: $(x+1)^2=x^2+2x+1$ $\forall x (x+1)^2=x^2+2x+1$ But the following deduction is obviously incorrect: $x+1=2$ $\forall x (x+1)=2$ Did I use the universal generalization rule in a wrong way?
I already know a few things about theorem proving however I was wondering how to use this when we are dealing with a non finite amount of axioms. In particular I was hoping to understand how to use automated theorem proving for set theory like ZFC for example. The method I know is using sequent […]
Convention: when a type depends on a variable from the context, this variable will be omitted from the type. For example, I will write $\Gamma, x:A \vdash B \text{ type}$ as opposed to $\Gamma, x:A \vdash B(x) \text{ type}$. The notation $B(a)$ will be reserved for the cases when $a$ appears in the premise; e.g. […]
Yes I certainly agree with you. Hart is able to bring creativity into Math which is somewhat lacking these days. This creativity is able to inspire others since it shows how something simple can be made fun and entertaining. Teachings of this sort would make any math class more entertaining than boring. This perhaps would make students outside of class to begin to delve into math since we mostly just use it then continue on with our day as if it never happened.
I also find myself watching these videos in a different way than I watch other youtube videos – I pause a lot, too! – so in a sense I’ve had to learn a new way to watch in order to really appreciate them.
I think the way you’ve channelled this into your teaching ideas is wonderful!