Summer of Math Exposition

The use of examples and the clarity of explanation worked well. I especially like the final message about applying reasoning to all areas of life. Perhaps this message could have been more inherent throughout the whole video instead of just briefly mentioned at the end.

I believe the motivation is clear: making something daunting less intimidating. While the examples themselves are not novel, their selection to make this explainer is, and are quite convincing and clear. Few minor remarks:

• at 4:51 the voice should say “m+n is even” not “m is even”
• the triangle animations are a bit confusing. It’s nice they emphasize alpha, and beta like that, but I think making alpha disappear when presenting beta is not a good choice. Keeping it there, perhaps faded, would have been clearer in my opinion. I think memorability is the weakest point here, I don’t think I’ll remember this presentation after some time.

This is a great introduction to proofs. The large counterexample shows why intuition isn’t enough, and the example proofs are cleanly presented. Well done.

Beautiful explanation and visuals alongside!! I understood the topic beforehand, yet you somehow found a way to enlighten me more with it. The only feedback I’d give is to correct the “m is even” mistake at 4:52, but honestly, it happens to all of us. Great work!!

I like the personal touch! That is the author is telling of his journey. I wish that when he was talking about the Nile’s of a triangle adding up to 180 degrees that he had prefaced the statement with, “ In the plane…” The author might have also said that many times there is more than 1 way to prove a statement. I liked that the author showed that many examples did NOT a proof make and there are conjectures out there still to be proven. This gives the young mathematician a motivation and a goal and show that Mathematics is an ever growing field.

Very beginner friendly, well motivated from first principles. Unfortunately the content was neither memorable nor innovative. Examples were well-known and I didn’t see many new insights, even for middle school students.

This would definitely help my students! I would love to have my students watch this at the start of a geometry unit, or really any time proof comes up in conversation, or the general “why do we have to learn math” question is asked.. It really shows why math is worth studying and what types of things mathematicians really do, which is prove things.

The only feedback for improvement I have is that my attention waned a bit during the proof that all triangles’ angles sum to 180 degrees. I think choosing a different proof of that theorem would be better potentially? I think it might be hard for students to understand in the short amount of time this is given but it could be presented somewhat more intuitively. I bet that there are a bunch of proofs to this idea that are easier to understand? or maybe this proof could be reworked to make it easier to follow… not quite sure.

Awesome video and keep making more! This type of thing is great and the number theory stuff was SUPER interesting and would be interesting to students everywhere!

The video is pleasant to watch, clear and easy to follow.

If I had to criticize something I’d say that the proof of the sum of the angles of a triangle being 180° might be a bit confusing, because it relies on some other property (angles formed by one line that intersects two parallel lines being equal) whose proof might not be obvious for your target audience. Their next natural question would be “but how do we prove that? and where do we stop proving things?”. And then the discussion would naturally lead to axioms etc. But this would be a whole other topic. The “odd + odd = even” example was a safer choice for this, because I don’t think people would usually question the axioms of arithmetic.

The motivation, pacing, and clarity were all great, and I would definitely show this to someone at that level. The counterexample was a great start and plot twist as well. That said, some of the examples are pretty standard, it would have been interesting to find a topic that wouldn’t necessarily be covered in class for instance, but at that same level. You also seem to talk to judges more than the target audience (e.g. “but for many students, …” instead of “you might have seen …”).

You did a great job at explain you’re topic. In terms of the 4 principles, your explination did contain nothing special and your style also simply contains manim animations on a black background. While your message at the end is a good idea, I would suggest talking about it a bit more (instead of just showing it on the screen) and to make the connection to what you talked about in this video clearer or adjust your examples. Your example with natural medicine talked about accidentaly setting two things equal, in your video however, you talked more about, that you can not proof something by inductive reasoning. A conclusion, that would be more connected would thus be something like “Everytime my friends text messages are short, he’s made at me (like a I know from experience)” => “No it could also be, that he’s just busy this time” or something like that, I think you get what I mean. Anyways, these were simply the reasons why I didn’t gave 10 points, the rest was clear and understandable.

lovely video! I think it is insanely important to know why things are true, not just that they are. To me, that’s what mathematics is really about, the numbers don’t really matter, it just a logical exercise. (I say this as a highschool student btw :)

Some small things: The first example should probably address 1, as it’s a weird edge case that probably jumps to everyone’s mind right away, even if it isn’t a real counter example. “m is even” was a small verbal mistake.

Overall, really good, useful video. I would’ve loved to see a little more guidance for how a student approaches writing a proof (like what do you actually put on the page). The recap in Chapter 2 almost accomplishes that, but an additional example that wasn’t glossed over would be great.

Instead of working up to the answer, sometimes it is easier for people to fill in the building blocks from the answer rather than remember them while working toward it. I think the first one it took a little long to get to the counter example and might have been easier just to say there was one, then explain that you would have had to check hundreds before getting to it.

Going back to basics is fun. Also, I wondered if the presenter would get back around to defining inductive reason; no luck there.

Nice. In the proof of odd+odd=even, you initially showed odd numbers with balls, you could show a sort of visual proof also using balls, like showing half balls in pair, one extra, same with other odd number, then adding both creates a pair of extra balls of each, giving us an even number. But it was good nevertheless.

This is a really solid video. I think you explained the different kinds of reasoning and the basic idea of a proof really well. Thinking of my students, however, you may want to consider slowing down a little or maybe even cutting out an example to make the others longer. If this is meant for people going into a geometry class, any sort of advanced mathematical notation will only confuse (it doesn’t even have to be advanced). As a teacher, I totally get and need no further explanation about the proof of two odds is an even, but for my students (and I know because I did this with them; and they’re pretty smart kids too), the claim that 2l +2m +2 is even or that (l+m+1) is an integer will likely not be immediately clear to them. They might even need a reminder about what an integer is. I actually think that your example of odd+odd=even and the explanation of Goldbach’s conjecture might be sufficient (but this might be because this is the third SoME video that I’ve seen with the 180degrees in a triangle proof…). And one other note is that although your explanations and wording are very clear and succinct, your tone at times could be a little off-putting. Sometimes it sounds like you’re annoyed. This might just be how you normally talk, but I think your message would be better received if you tried to be a little more gentle/patient in tone. I also think that you don’t need to mention that you are a college mathematics student. This might actually diminish your authority, and it might be something that you look back on after a couple years and cringe at. You can just say that you’ve studied mathematics more formally, and this would be enough to establish you as a sort of authority on this subject. But overall, for your first video, I am very impressed! I do hope that you continue to make more! You seem to have a talent of explaining things in a very straightforward way.