This strongly reminds me of the various math magazines we have here in Germany. (I know that other countries have similar magazines, but I only grew up with the German ones, except for one Canadian one.)
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.
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.
You also have the problem (at least in public schools,) that teachers are obliged to “stick to the curriculum” and much of a teacher’s time is spent catering to the lowest common denominator lest the test scores fall to an unacceptable level.
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.
I hated learning the multiplication table when I was 5. Basically, I boycotted the whole thing. It occured to me recently, why don't they talk about the holes in the multiplication table? These are the prime numbers! So easy to mention this and may have inspired my 5 year old self to dig a bit deeper.
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...
> And I still can't multiply six times eight! I have to work it out every time...
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 occured to me recently, why don't they talk about the holes in the multiplication table?
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.)
This reminded me of a math professor during my undergrad. He would print out copies a math problem from some journal that published these kinds of interesting problems that are pretty hard and a little different from what you normally see in class. He would then put the copies in the halls of the math building. If you could bring him your solution and explain it to him he would give you $20 and mail in your answer to the journal, who would then send you back a little certificate and publish your name in the next edition.
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!
I was reminded of the book 'The number devil' (Zahlenteufel) which showed me you can reason with children about math if only you do away with intimidating language: https://en.m.wikipedia.org/wiki/The_Number_Devil
reddit.com/r/ELIF is also good for explaining any topic in a simple way so even a kid also can understand. Its Not targeted towards maths though but biology / physics/technology etc. Have a look.
My son is now 4 years old, he has been showing a great deal of interest in playing with numbers and puzzles, so I have been thinking quite a bit about this lately.
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.
I grew up in an inner city Detroit neighborhood. The junior high school that I attended was pretty rough so, to avoid the gangs that formed after school, I quickly ran over to the near by public library and stayed there until it was safe to walk home around dinner time.
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 Gardner was amazing. I was privileged to meet him shortly before he left us, and he was still spry, mentally alert, and interested in nearly everything. Much of his work has dated badly, some equally, much of it is timeless, and just as engaging now as it was then.
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.
Scientific American looks awesome. Love the ads too,
p21:
Atari's new remote
pre-programmed computer ping-pong
game can be used with any size TV set, black
and white or color.
p14:
> HP's 9825A: a powerful new
desk-top computing system.
Through its novel architecture and speed, the 9825A runs
programs, accepts keyboard operations, and controls
instruments-with apparent simultaneity.
...
> The 9825's storage capacity includes
8K bytes of internal read!
write memory, expandable to 32K
bytes; in addition, each bidirectional
tape cartridge can store 250K
bytes. Its built-in tape cartridge
drive can access data in an average
of six seconds, and transfer data at
2750 bytes per second. At these
speeds, and with memory load and
recall capabilities not previously
found in desk-top systems, it becomes
entirely practical to interrupt
a long program, transfer it to tape,
run a short program, then reload
and continue the first one.
Check out Knuth's book "Surreal Numbers: How two ex-students turned on to pure mathematics and found total happiness". It's a book about discovering new mathematics, in this case Conway's Surreal Numbers.
The style of writing may well be very attractive, but it's important to remember that you're reading scientific articles written about forty years ago, and above that, in fields that are both advancing very quickly.
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.
A good place to promote this would be with secular homeschoolers. Those homeschooling for nonreligious reasons are often dealing with a bright kid interested in advanced subjects. They often need lay explanations that make advanced subjects approachable without dumbing it down.
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 "=___".
I'll add to that. We fell into the secular homeschooler camp. At some point, it takes less effort to just homeschool than to spend endless time arguing with school administrators to challenge your child appropriately. "We can give them extra worksheets for homework." Oh wow, thanks.
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.
It sounds like you are an upper middle or upper class family. There are gifted children in the US that aren't identified as such that come from lower class families who are deprived of their education. I checked the AoPs site and if you get the pre-algebra -> calculus textbooks + solution guides it's over $200. This may not sound like a lot to you (as you suggested) because it is due to your class. This is something some poor families cannot do. Even if they want to buy those books. Sending your middle schooler or high schooler to take college level classes is another sign of class status. I believe we should have schools like AoPs and the Proof School targeted not at the wealthy gifted children, but poor gifted children. I also think these schools shouldn't filter applicants out by SAT scores (since mostly gifted children from well off families seem to do better than gifted children without access to education). It seems like most places ignorantly assume the gifted only come from well-off places and put cultural barriers (such as exams) to identify them. This class advantage carries over into university admissions when the well-off students have 30-60+ credit hours from colleges coming in, as opposed to the gifted kids from poor families with 0 college credits. The former are admitted to elite schools, while the latter are denied. If you look around, these elite prep schools and elite colleges tend to lack economic diversity. Teaching advance math to gifted children of the elite, while excluding the gifted children of the poor (due to barriers of entry) do a disservice to our society as a whole.
I'm not sure why you were downvoted, as you are absolutely correct. I've lived both sides of that.
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.
My son is actually math challenged and also a science nerd. This is a whole other mind-boggling challenge.
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've actually been trying to explain that exact concept (algebra variables) to some advanced third graders. It's been surprisingly hard to get them to grasp it.
When I was a child, the first thing that made sense to me was that "x is the number that makes the statement true". That is, 100 = 100, so if 25 * x = 100, you need to figure out what value of x multiplied by 25 gives you 100. I didn't even understand the manipulations or how you would really solve for x at first, but I did understand that you were trying to figure out the number that would do that. I would just phrase it like a puzzle, like "what plus 9 equals 14" or "3 times what equals 42"?
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.
Have you tried analogous approaches, such as showing kids an empty box, a question mark, multiple metaphors, etc?
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.
I think it helped my son for me to explain that letters were just short "names" in place of the phrase "mystery number." Or something along those lines.
Like "We are just going to call it X or Y instead of the unknown number we are trying to figure out."
> A good place to promote this would be with secular homeschoolers. Those homeschooling for nonreligious reasons are often dealing with a bright kid interested in advanced subjects.
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.
So many things I could say about this, but I"ll let Alan Kay speak for me -
https://youtu.be/p2LZLYcu_JY Alan talks about how ideas in Calculus could be manifested from an early age and a richness in understanding built up as they aged up:
- 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:
The great teachers of math to kids in the US are of course the Ross Program folks. (When last I checked in I thought Boston University's Promys was actually livelier than the original Ross Program at Ohio State.) They basically take very smart kids and help the rediscover much of abstract algebra and basic number theory.
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.)
I'm interested in insightful descriptions of the physical world for young kids.
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?
Unfortunately if you follow the link to the proposed site for publishing kid-friendly mathematics articles, then search with a filter for mathematics, you get zero results :(
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).
Honestly, this is a fantastic idea. As a dad of a toddler I am starting to look for more math-oriented materials, and this will definitely be on the reading list.
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.
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.
[1] http://wurzel.org/