Communicating Advanced Mathematics to Kids
blogs.ams.org
blogs.ams.org
The most popular seems to be "Wurzel" [1] (translated: "root").
The target audience of the magazines are students (pupils) at school, as well as young students at university. The articles explain advanced math topics. Not too advanced, of course, but still far outside what they see in school. The articles use formalisms only where they make things more clear. They build on common school knowledge. Advanced formalisms are only okay if they are introduced in the article. And the articles aren't too long. The goal is that everything can be understood by students, if they are interested.
Also, the magazines usually contain reports about math olympiads and similar activities, usually written by participants (i.e. by students, not teachers).
Finally, these contain math exercises. But don't think of school exercises. Think of math olympiad, just that you have more time to solve them. Students send their solutions in and it is published in the magazine which students solved which exercise successfully.
The latter one should not be underestimated - it is more or less how all these magazines started: You have math competition, the students go back to school, and you want to keep in touch with them, so you offer them so solche math puzzles in their free time, give them feedback to their solutions, and do everything they don't feel alone with their mathematical interests. You can do that individually only with so many students, so you start a magazine for a more efficient communication.
Did they just add it?
I'm using the TLS version, though:
How were you previously trying to find the RSS feed? It's still not obvious to me from the front page that it exists.
There's so much cool mathematics out there that school kids can understand given a good enough teacher. It's really disappointing to me the crud that gets taught in school, but then i remember that a big part of this is the teachers just aren't that inspired. All it takes is one or two good teachers to really change a kids outlook on mathematics.
I might argue that kids should get a double period for math, especially earlier in their school careers — one class to teach the fundamentals, another class to simply explore. The benefit being that critical thinking is enhanced in all academic areas and not just the obvious in mathematics.
My first real mathematical experience was in second grade. My teacher had some hoops on the floor, and was putting coloured shapes inside. Red shapes in one hoop, and triangles in another hoop. But then the problem arose: what to do with the red triangles? It was total magic when she dragged one hoop to overlap with another hoop and placed the red triangle in the overlap. Venn diagrams for six year olds.
And I still can't multiply six times eight! I have to work it out every time...
Because there aren't any. The prime numbers would be gaps if you pivoted the table into a division table but, alas, that's not how it's presented (even in the context of teaching division.)
No offense intended, but if that still bothers you, it's time to learn it.
Either by accepting the offer you got with 5 (that is, memorizing the multiplication table - BTW, it is more fun to do this with 15x15 or 20x20 rather than 10x10).
Or by playing mental calculation games and trying to get high scores in speed.
(Essentially, this is the same as with learning vocabularies for a foreign language. The word "4x3" is translated to the other word "12".)
Maybe your teachers failed to make it interesting to your 5-year-old self. But your current self does seem to be interested, so that can't be the issue anymore.
You can critize your teachers that they didn't make your learn it at 5, but you can't critize them for still not having it learned later as an adult.
It was by solving one of these problems (using math that was above the course I was currently taking) that I started to interact more with the math department and decided to take more advanced classes!
The other day I ran across an issue of Scientific American from the late 70s (http://flowcytometry.sysbio.med.harvard.edu/files/flowcytome...) and I was super impressed by the quality and educational value of the content. Much superior to its current version. They have a ton of super interesting "mini" papers about all sorts of topics. In that issue alone I learned about:
- The metabolism of alcohol
- The meteorology of Jupiter
- Simpson's paradox
Take a look at it. I think you might be impressed too.
The other site I have fallen in love with recently is Fermat's Library (https://fermatslibrary.com). They essentially publish an annotated paper every week (usually physics, cs, math). Reading their papers is now a part of my weekly routine.
A friend showed me a magazine at the library; it was Scientific American and it completely hooked me. This was 1963 and SA was then a great magazine. I read every current issue and then read the bound volumes of past issues. My favorite monthly columns were Martin Gardner's Mathematical Games and the Amateur Scientist. I read every one of those two columns ever published.
The schools I attended were not very good, but I learned a lot of math and science during those years at the library. I went on to score very high in math and science, got into great colleges and ended up with a very successful career.
Martin was a huge influence on my life, and I treasure the afternoon I spent with him in person, and the thousands of hours I spent with him via his writings.
It’s not just the content but the ads and the copywriting is just incredible. In this age of 140 chars You don’t see such long form copy anymore...
I don't have the opportunity to read the linked magazine right now, but I wouldn't be surprised if much of the research cited in it has been disproven.
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A lot of stuff is intentionally overcomplicated. My oldest was furious when I finally explained to him that algebra is basically renaming the blank space in an equation as X so you can more easily move the blank space around.
He had been doing algebra in his head for years to infer stats in games while simultaneously feeling baffled and intimidated by formal math, like the concept of algebra. He had assumed all variables were like E=mc^2 where letters stood for specific concepts, not for floating "=___".
For some reason, one of my earliest memories I have is from when I would try to show off to older kids that I understood algebra.
Like "We are just going to call it X or Y instead of the unknown number we are trying to figure out."
I find that visual approaches work well, but when you write an English letter which has already been used for other purposes, it's a little less clear.
A great resource for "mathy" kids is Art of Problem Solving. https://artofproblemsolving.com There are free resources, their math books are outstaningly excellent (well, they could use more excercizes, but the exposition is superb), and they have online classes. The online classes are run with a live chat room that typesets LaTeX. The pace of the classes is blistering, so not for everybody. But you can buy the same books for cheap and use them at home at a pace that is right for your kid.
We used AoPS as the main spine of our child's math curriculum. At the time, it topped out at about AP Calculus, I'm not sure if they go beyond single variable calculus yet. Anyway, after that a local university let our kid take multivariable calculus along side the freshman engineering students. We were lucky to have that resource. Next problem: when your kid completes multi-variable calculus at age 13, what next? We did discrete math and freshman engineering physics, and counted them toward the high shool transcript.
Which is another problem homeschooler's face -- the so-called "mommy transcript". When it comes to college applications, it's good to have somebody in addition to mom posting grades to your transcript. The Uni grades in math and physics and recommendation letters from university profs helped the college application package a lot. For that matter, AoPS is well ennough known now that it is good on a transcript.
While I'm on math resources, http://www.mathpath.org is a truly wonderful summer math camp for kids in the 11-14 age range. Great faculty, great staff. Now would be a good time to think about applying for 2018. Just so you understand the scale here: My kid spent all of their spare time for two weeks on the quailifying quiz, attempting 5 of the 7 questions, a couple of those attempts were a bit weak. (A good solid attempt at 2 or 3 will get you in.) If you think your child might like MathPath, I'd encourage you to download the quiz now and let your kid start pondering the questions.
I bought math books full of words off the bargain table for as little as a dollar and, after sorting things out with his father, just gave my son the stack of books to read. No tests. No written work. No problem solving. Just read these books. That finally worked and he has a good grounding in math, though he can't crunch numbers. He is, as they say, calculator dependent. But he absolutely understands important concepts. To my mind, this is vastly better than a kid who can estimate to 8 decimal places in their head, but can't grasp GIGO (garbage in, garbage out).
I grew up in rural fly-over land, and my little farm country high school with a very high proportion of kids on the lunch subsidy didn't prepare me very well for engineering school. Maybe one kid every other year would attempt engineering school. My pig farmer father died when I was 5, but my mother managed to stuff enough money away even in the lean years that I had money for a cheap land-grant university.
As I mentioned, I wasn't all that well prepared. I remember one night when DiffEQ homework was kicking my butt, and said to my self: "Ether get this stuff under your belt, or go home an clean hog barns for the rest of your life." I chose option A. My wife's parents were dairy farmers. She was the guru of scholarship application essays... that paid for her undergrad degree.
So, I get what you are saying. I've lived it. My wife lived it. But between us we've accumulated 5 degrees, made some good investments, been in a couple of situations where stock options were pleasantly large, and yes, work is optional right now, even living at Silicon Valley prices. Pardon me if I don't appologize for accompishing what YC and HN are all about.
So your post is a little short on actionable solutions to the problems we both see as clear as day. Here is a suggestion: MathPath has a need-tested scholarship fund so that kids that can't afford the camp can go. Tax deductable donations gladly accepted. Join me in donating. Then spread the word about MathPath -- anybody can apply.
I don't see any reason to restrict this to those homeschooling for nonreligious reasons. There are plenty of bright homeschooled kids interested in advanced subjects regardless of their parents' primary motivation for homeschooling.
- at very young ages, kids really respond well through "doing" / the enactive channel. When asked to draw a circle, kids in Papert's group would first emulate what a LOGO turtle would do by rotating their body in a circle (making tiny increments in x and y).
- as they got a bit older, the visual / iconic channel was more developed and they could understand the abstraction of a circle on pencil/paper and how the concepts carried over there
- closer to early teens, symbols were much easier to grasp and relate to, etc.
With this context in mind, there have been some cool efforts to mix the second and third channels I just mentioned to communicate advanced math concepts. Vi Hart and Grant Sanderson's youtube channels come to mind. Here are my favorite videos by Grant:
- "What does it feel like to invent math": https://www.youtube.com/watch?v=XFDM1ip5HdU
- On the visual intuition behind a hard problem on the Putnam exam - "The hardest problem on the hardest test": https://www.youtube.com/watch?v=OkmNXy7er84
- From vectors to matrices to vector spaces to higher-level ideas like the link between linear algebra and calculus: "Essence of linear algebra" series: https://www.youtube.com/watch?v=kjBOesZCoqc&list=PLZHQObOWTQ...
What I especially like about Grant's videos, is that he often walks through what a mathematician would do, the questions she would ask, etc.
Decades later, I probably couldn't state quadratic reciprocity any more, let alone prove it. But proving the Two Squares Theorem is still one of the best ways to clear my mind of less pleasant thoughts, should I need such a device. It's just a whole lot of fun, even if you go in assuming next to nothing. (I start by proving that the Gaussian Integers are a Euclidean Domain and that all Euclidean Domains are Principal Ideal Domains, and proceed from there. It doesn't take long.)
https://quantum-journal.org/call-for-papers-good-popular-sci...
For illustration, what might it mean to teach friction well? Is it memorize-and-regurgitate of bogus definitions in late primary? Or plug-and-chug of Arrhenius's law of large objects sliding on pig fat in high-school? Or instead, could we talk about sock nubbies in K-3? Their similarity to cleats and crampons and klister. How to avoid slipping and falling. Sliding and slipping and sticking - pervasively surrounding us. Nanoscale origins connecting nicely with macroscale behavior. And for the numerate, a feel for reasonable order-of-magnitude values. So physics, but coming from sort of an engineering perspective - hands-on, pragmatic, rough quantitative.
Could we approach math similarly? What might it mean to teach category theory to young kids? Perhaps math as a vocabulary for describing similarities among everyday things? "Oh, you missed the square, but you can get it the next time you go around the board":"It's almost one o'clock now, so we'll wait for lunch-time tomorrow":"Missed the parking space, so we'll drive around the block". Movement around short loops is a theme of everyday experience. Could it be taught? Now, or with future AR/VR tech?
In preK-1, learning to describe physical properties is a thing. Rough, smooth, etc. But instead of drawing on well designed vocabularies from industrial design or material science, it's left to dysfunctional ad hoceries. Similarly, K-12 physics education leaves students having seen almost no physics. What about math?
Could math be taught not as a random fun thing to do, like crossword puzzles and fashion magazines, but as a deeply insightful and broadly illuminating vocabulary for describing everyday experiences?
fwiw I have made some attempts to get my own children interested in 'advanced' mathematics. I found the TV series "Story of Maths" by Marcus du Sautoy to be helpful. I've also had some positive results watching lectures from Harvard's E-222 class[2] with my older son (14).
[1] https://www.netflix.com/title/80093836 [2] http://matterhorn.dce.harvard.edu/engage/ui/index.html#/1999...
We need advanced concepts explained in simple clear language so that students have enough context to anchor the knowledge.
I especially love the "How to write for this audience" section. It says: be concise, convey the excitement, say exactly what is important and why, and draw the reader in.
This is how most concept should be explained, to adults or kids, doesn't matter. As someone who's gone through an engineering degree, so much of it was written/presented in convoluted "this is going to be difficult" format, instead of this playful "here, try something tasty" one. Wish more educators would adopt this kid-friendly way for older kids too :)
Edit: Do you know any dads who have been very successful teaching kids math at home, better than they are taught in schools? I'd love interview them for my podcast.