This was a surprisingly interesting and well-written article. I hope people read it and not just the comments here :-)
I'm unsure if it implies that "lame school exercises" are unnecessary or just not sufficient (I've recently read articles about how teaching "insight" without exercises is detrimental, though perhaps doing problems implies getting that repetition-work).
Does anyone have good experiences with keeping kids math-interested as they get into their teens? My kid used to enjoy math in school, and love talking about math problems ("can you help me set up that triangle pyramid thing with the sums again"), but now is seemingly disillusioned and finds the school exercises boring. Combine that with, well, teen-age, and I fear it's going to be hard to get back the spark. Not that it has to come back, but I'd hate for the interest to turn into dislike due to lack of opportunities.
Can't speak for my own kids (yet), but personally I was able to hold my interest in math through my teens for 2 reasons:
1. I had some good teachers who showed (glimpses) of the how and why, not just the "what", so it helped math feel like it made sense, rather than being just facts and calculating algorithms to memorize. One demo that left a particular impression on me was the teacher asking us to go around the unit circle in increments of 10 degrees and plot the ratio of the opposite side and hypotenuse of the inscribed right triangle. Watching the sine function -- until then some mysterious thing that just existed with no explanation or context -- materialize in front of me on graph paper was magical.
2. I was shown that math is useful. In another great high school demo, the teacher assigned every student a length, width, and height, to be cut from construction paper and taped together in a box. After we were done, we laid out all the boxes together and computed their volumes. Then the teacher worked through the calculus on the board to figure out the dimensions of the box with the highest possible volume, given that fixed amount of construction paper. That was a really big moment for me, because until then I "hated" math, being a silly waste of time messing around with numbers and shapes just for the sake of doing it.
I think the amount of approachable math content online is incredible. Some comments mention 3Blue1Brown on YouTube, which is good but can go on a bit and maybe be a bit advanced. I think Numberphile is great for having a lot of videos, each of which is pretty short, and which still show some small proof of some problem. There are also books of interesting mathematical tidbits or puzzles, eg the Martin Gardener or Ian Stewart books. I guess one other thing to say is that it probably helps for you to appear to be interested in the math things instead of you being interested in your kid doing what you want (ie looking at the math things)
There are a good amount of entertaining maths communicators on the internet. I'd like to recommend standupmaths/matt parker, as well as looking at the SoME playlists which contains some interesting content too
I took my kids out of school before they were teens. The younger one who did not like maths at school does like it now and is doing maths A level (UK exam, similar level to APs in the US AFAIK) at school (well, "sixth form college" school for 16+) and got a 9 (top grade) in her maths IGCSE (UK exams taken at 16 in schools, but mine took some of them younger).
I think the problem is that schools here (and I think most countries have similar problems) is that they focus on grades rather than keeping kids interested. Too much pressure takes away the fun. The article has some other clues to things that can go wrong with bad teachers: for example
> "Worst of all, the teacher docks points when the kids use techniques that they “aren’t supposed to know yet.”"
That is really terrible (and not typical, I hope - would not have happened in my school, I think) but it does happen.
I think there maybe a problem with insight without repetition, but it is definitely possible to keep kids interested while doing repetition as long as they feel they are getting better. My kids did do a lot of practice with minimal supervision.
I also think you absolutely have to provide enough interesting stuff to make kids feel the subject is interesting, even if there is some grind. Having a parent who is interested and will do things like answer questions is a huge help.
Here's one possibility it looks like no one has suggested yet: it's boring because it's too slow.
I was bored with math up until 8th grade (age 13-14) but didn't realize why until that year. Up until that point I got straight As pretty effortlessly, but due to an administrative error I skipped a year of math. I was supposed to have a year of pre-algebra, but got placed into algebra instead. Luckily that teacher decided to do a month of review before starting new topics, which effectively meant I was taking a year long class in just a month. I actually had to put in effort for once and averaged a C.
It was during that year when the pace slowed back down that I realized I did like math in general, it was how slow classes had to go to accommodate all the students that I didn't care for.
>it was how slow classes had to go to accommodate all the students that I didn't care for.
This is a huge problem across all classes, not just math. It's not as bad if the lowest portion of the bell curve is shuffled off to remedial learning classes, but really you need 3 tracks or more to keep the highest performing students challenged enough.
> Does anyone have good experiences with keeping kids math-interested as they get into their teens?
Not a parent, but what kept me engaged at that time was programming simple games or interesting visualizations and animations. I "discovered" quite a bit of useful trigonometry, linear algebra and statistics by just fooling around and following my curiosity. And the intuition I gained definitely helped later on with university math
> Does anyone have good experiences with keeping kids math-interested as they get into their teens?
The first step would be to get them into a math circle. There are 100s of them https://mathcircles.org/map/. I run a math circle as well, and work with a bunch of teens/preteens. I've had a lot of success with them. AMA.
For example UCLA math circle is very exclusive and there’s effectively no admission in elementary school if you miss enrollment or don’t do well on their kindergarten test.
Totally agree with you. That was the case in my area in the midwest as well. That's why I started my own mathcircle. Its not as hard as you might think - you just need a few interested students, a few textbooks, and plenty of time.
We focus entirely on competition math - so mathcounts, amc 8/10/12, aime/jmo/imo. The material gets real hard real fast, so kids will drop, new kids will join etc. The ones who stick around benefit immensely. I've had 11-12 year olds in my group qualify both for the AIME (one of 6000 kids in the usa) as well as mathcount nationals (one of 200 kids in the usa).
Which textbooks do you recommend for 6 year olds? I have a few Russian math circle ones and the UCLA one. But it’s kind of daunting because the material is not sequenced into lesson plans.
As a point of reference, reading curriculum is very easy to teach because there are scripts to follow in the lesson plans.
For the "exercises are important and also build conceptual understanding", see "BASIC SKILLS VERSUS CONCEPTUAL UNDERSTANDING, A Bogus Dichotomy in Mathematics education" by H. Wu.
The big challenge in teaching is not to make the exercises seem lame.
I haven't read this whole article yet (PDF here https://math.berkeley.edu/~wu/wu1999.pdf), but this idea is my #1 problem with Tom Lehrer's "New Math" song. His big complaint at the beginning is that the emphasis is too much on understanding and not on efficiently/correctly calculating an answer. As funny as the song is, I don't think that complaint has aged well at all. Also the old algorithm was just as complicated as the new one, but at least the new one makes it easier to see what's going on, so the whole joke kind of falls apart.
As far as I understand the article, the author (and myself) absolutely wants people to practice longhand addition etc., and is pushing back on the idea that you can teach how to calculate XOR how to understand place value.
If some of the kids go on to study CS, they can then think about the similarities and difference between decimal, hex and binary addition, how half and full and ripple-carry adders work, and how you add bigints. At that point you need both a conceptual and a procedural understanding of digit-wise addition.
If you want to do say 67 + 24, perhaps even in your head, there are more efficient ways to add than the standard algorithm, and I think that's what new math was trying to get at. But at some point you might want to add 25137 + 1486 and then your neat tricks no longer work and you need something that scales.
New math or common core or any approach that tries to center ‘understanding’ over rote procedure is definitely pushing in the right direction. I think that people who have the closed understanding of addition, that there is one true algorithm for doing it and who will start your 25137+1486 problem off by adding six and seven to get three and carry the one… are missing out on a deeper intuition about numbers, because they only think of those numbers as sequences of digits.
But someone who sees that as ‘add fifteen hundred and take away fourteen’ is much closer to understanding what that expression actually represents, as well as being able to produce 26623 almost immediately without writing anything down.
It’s not about ‘neat tricks’, it’s about numbers having shape and feel and flavor.
> 25137+1486 problem off by adding six and seven to get three and carry the one… are missing out on a deeper intuition about numbers, because they only think of those numbers as sequences of digits.
This is precisely the dichotomy that is bogus according to the article.
25137 = 20000 + 5000 + 100 + 30 + 7 and 1486 = 1000 + 400 + 80 + 6, then you add (7 + 6) + (30 + 80) + (100 + 400) + (5000 + 1000) + (20000 + 0) to get the result. The fact that we can do that and combine it all tightly into columns is IMO a very deep insight into what a "number" really is, while also providing a general pen-and-paper algorithm for adding any two numbers. The insight provides an algorithm, and the algorithm leads us to an insight.
Discovering that 1486 = 1500 - 14 isn't a particularly deep insight into numbers either. It's a useful technique and I think it's fine that we teach it, but I don't think it has any particular conceptual merit that the standard algorithm lacks. I certainly don't see how it puts a child any closer to understanding what addition really means.
No but that’s actually exactly what ‘new math’ was about. The thing Tom Lehrer was lampooning was all this talk of the ‘tens place’ and the ‘hundreds place’ rather than just plugging and chugging the digits, you know;
Seven plus six is thirteen carry the one leaves three, four plus eight is twelve carry the one leaves two, two plus four is six, five plus one is six, two six six two three…
Seeing that as a decomposition of multiples of powers of ten and how that makes ‘carrying’ happen is exactly a result of having a deeper understanding of the way the numbers work.
For the student who doesn't understand, one rote algorithm is as boring and stupid as any other. That student is plugging and chugging all the same, whether or not they have heard of a "tens place".
For the student who does understand, the "new" algorithm at least is elucidating and actually makes sense as a direct application of the basic principles of our number system. The "tens place" is in fact a real thing, regardless of what you call it.
I’m not remotely arguing against encouraging kids to discover the mathematical principles themselves. I’m also not advocating teaching swapping + (1500 - 14) into -14 + 1500 by a careful application of the laws of commutativity. I’m saying that having a comfort and confidence with what summation is is way more valuable than learning that addition is a procedure applied to digits.
School is where you kid spends the majority of the time. If they find it boring, and the school is unable/unwilling to provide enrichment, then it is an uphill battle to resist that. The ideal solution is to make the work they do at school engaging, then they'll seek out enrichment at home themselves.
There might be math-oriented stuff online (3Blue1Brown is one off the top of my head) that keeps them wanting to understand more. That might anchor their school work a bit, or give them something extra to try. Books can help too.
Being a product of the US school system, the most important lesson it taught me is that teachers as well as students are caught up in a system that's bigger than they are, so in order to educate oneself, one need merely (a) organise one's own time effectively, but (b) avoid doing so indiscreetly, such that it could force other pawns to call one's lack of genuine participation to the attention of the Man.
90% of success in primary and secondary schooling is just showing up; as long as you keep your grades up, they won't demand mental attendance, only the physical.
denn meine Gedanken zerreißen die Schranken
und Mauern entzwei: die Gedanken sind frei.
I agree, but would like to add that the US public school system also has a bad habit of teaching to the test, namely standardized testing like the ACT or SAT. As it turns out, my time would have been better spent learning how to budget long term or navigating the US healthcare system as opposed to learning maths rarely seen in the wild or that the mitochondria is the powerhouse of the cell. The information imparted to me during the entire four years of high school has largely been forgotten, "replaced" by more practical knowledge that I wish I'd had even a tiny bit of upon entering adulthood.
The ideal solution is to make the work they do at school engaging, then they'll seek out enrichment at home themselves.
One thing I haven’t seen brought up in this discussion yet: technology. Seriously, what hope does a teacher have for getting students to engage when they’re competing against all the might of Silicon Valley? The industry is spending billions every year to discover and implement the best techniques for stealing teenagers’ attention and focus away from everything else in their lives.
There’s a growing chorus of people who want to get phones out of classrooms. That’s a strong first step but students’ struggles don’t end when they go home for the night. I volunteer as a tutor with high school students at an after-school homework club. We aren’t allowed to take their phones away! You can imagine how much of a Herculean struggle it is for these kids to put away their phone and actually get some work done.
In a small way, yes, and throughout "three to seven", actually up until she was nine, we had a lot of fun with what I think I can call problems (especially while home-schooling during the pandemic, where we had time to keep going back to these from different angles). But I'm not able drum up much interest these days, so I was wondering if people here had any insights on what if anything has worked with tweens, as opposed to how the younger ones learn.
I learned basic trig around ~10 because I wanted to make spacewar/asteroids-like games, which led naturally into matrix math, later on.
Parsing also interested me around ~12 (text games this time), but while I made some mechanical attempts, the theory never clicked until much later.
Sometime around that time I learned about recursion by reverse-engineering the display code for a tile based first-person maze crawler one of my father's colleagues had written. (yes, fib should've been simpler, but drawing those perspective walls was way more concrete)
[perspective was luckily something I'd been introduced to in second grade, so it was old hat at this point, and the scaling math was straightforward; the only jump I needed to make was grokking that having drawn the walls visible from this square, one could use the same routine, with fresh parameters, to draw the walls from all the still-visible neighbouring squares, etc. Unfortunately z-buffers make this entire approach obsolete; but maybe he'd take it as a challenge? this is trivial with z-buffering, but how might it even be possible without?]
Might Processing sketches (or whatever the new shiny might be) interest your kid?
Personally, that's how most of my math learning came. As a teen, I started to program and wanted to understand mathematics tools to solve specific problems, so I learned trig, Bezier curves, cryptography, number theory, etc like this.
Then later between my love of point and click adventure games and puzzles plus the fact that I had good foundations in maths, pure mathematics problems became increasingly fun.
I still have to make to that point, but answer from what I remember from my time there:
Abstract math, or "math per se" was utterly uninteresting for me. My drive was to solve actual problems I had or wanted to solve. For example, making something out of wood with complex shapes, or drawing with the computer. I would say you have to find an area of interest with which the kids get passionate, and needs math to solve the problems.
Try dropping him in one of these math discord groups (for eg: Summer of Math Exposition)? Teens care a lot about social approval and seeing so many people having fun with math might help.
when i was 12,13 i really enjoyed the algebraic manipulation parts of calculus without having to think too hard about setting up problems. just practicing integration, differentiation and limits was a good start
i didn’t even know i enjoyed math until i was fortunate enough to take multivariable calculus my senior year of HS, you don’t really get to do any cool shit in typical public schools so it’s hard to keep the interest
The article seems to imply though that the nurture/nature quotient is very low. As a one off example, I've always been interested in exact (applied) sciences, my kid much less.
I'm unsure if it implies that "lame school exercises" are unnecessary or just not sufficient (I've recently read articles about how teaching "insight" without exercises is detrimental, though perhaps doing problems implies getting that repetition-work).
Does anyone have good experiences with keeping kids math-interested as they get into their teens? My kid used to enjoy math in school, and love talking about math problems ("can you help me set up that triangle pyramid thing with the sums again"), but now is seemingly disillusioned and finds the school exercises boring. Combine that with, well, teen-age, and I fear it's going to be hard to get back the spark. Not that it has to come back, but I'd hate for the interest to turn into dislike due to lack of opportunities.