Absolutely. Key for me was to invite children's friends (and family) along, host it in our house and make it a recurring weekly thing. The books (presumably you also have these from the MSRI's 'Mathematical Circles Library') are great, but week-to-week I've found the free online NRICH resources much more directly useful: https://nrich.maths.org/about-nrich
Practically speaking (running a little maths circle in the UK for my children, and some of their friends from nursery and primary school), I have found the Nrich website to be the single best source of resources:
https://nrich.maths.org/about-nrich
The book by Zvonkin described in the article is a very good motivator, particularly for the honest descriptions of lessons gone badly wrong, and staying up late cutting out pieces of cardboard! But it's quite difficult to use as a teaching resource.
One of the joys of high school was discussing these with a friend (after submission---but one could forego the contest and just do the circle thing with them for fun).
I would love to hear more about your experiences running a maths circle here in the UK. My two daughters are a little young (3.5 and 0.5), but this article has inspired me to get the ball rolling.
It's been really rewarding. I definitely recommend jumping in. I started with my reception-age child (+ school friends), and have just extended it to their younger sibling (+ friends from nursery). Your 3.5 year old will have started the EYFS (Early Years Foundational Stage) syllabus at nursery if they attend (which is also what they do in the 'Reception' year at primary school, before starting the national curriculum in the first year), so they will now be exposed to counting and comparisons. The perfect time to get started, in other words!
There's some NRICH funded research that showed that exposure to symmetry and reasoning at this level was much more predictive of future abilities than numbers and counting. I think when parents try and help at the early stages, they often try to e.g. get their kids to count to 100, which is conceptually identical to counting to 10.
For number fluency there is the free White Rose '1 minute maths' app, which does a very nice job of gamifying subitising & etc. A lot of primary schools in London seem to have adopted the White Rose teaching resources.
https://whiteroseeducation.com/1-minute-maths
What do you not like about Feynman's "little arrows" / rotating clock hands in the QED book? I can't think of a more simple metaphor for the exponential of a complex phase, exp(i omega t). I suppose you could try and do it with more commonplace trigonometric functions, but then you lose the simple vector interpretation of adding the contributions. Or are you arguing that you should always try and teach complex numbers and the Euler identity to avoid strained analogies?
> What do you not like about Feynman's "little arrows" / rotating clock hands in the QED book?
It’s difficult to articulate, but two aspects are:
The amount of times I have only confused people more by trying to explain even modular arithmetic by calling on the clock analogy.
And the fact that the little “clock hands” are a complete abstraction from both the physics being described and the mathematical models that describe that physics. ~“Quantum physics is just about adding clocks?”
> I can't think of a more simple metaphor for the exponential of a complex phase, exp(i omega t).
As I noted in the gp I think code implementations or numerical methods should be the goal.
The solution to the confusion about referencing clocks when talking about modular arithmetic was just to write down a complete numerical example, ie all natural numbers mod 6 up to 10, and use that as the abstraction for further discussion: negatives, reals, periodicity, infinities, applications, et al.
> As I noted in the gp I think code implementations or numerical methods should be the goal.
I’m 100% with Feynman on this one. I loved the book because of the intuition it gave me about quantum physics. He even has this amazing analogy for how to teach arithmetic without numbers. Now, you could absolutely claim that he fails in his analogies (I’m not among the .1% of people if not less who can debate that), but I can still say claim confidently that math is not the goal. Abstraction is not intuition.
In mathematics, geometric and algebraic explanations are complementary.
If you plot a function, you can observe many properties easily, for instance where does it cross the axes? Is it symmetrical? How quickly does it grow?
However there are also many properties that are easier to observe algebraically. For example if you plot x^n you can see if n is odd or even, but you can’t see what value n has because x^10 looks very much like x^12. But if you have the algebraic representation you can read it off.
The issue with Feynman’s clocks is that he only provides the geometric explanation (what physicists would call “intuition”), and not the algebraic explanation.
This only helps two kinds of people: 1) people not capable of understanding the algebra, 2) people who already know the algebra and want to develop intuition.
For the third group of 3) people are capable of understanding the algebra but haven’t learned it yet, only talking about clocks is a bit dizzying.
I strongly disagree. The geometric explanation lets you understand the main concept without the hassle of algebra. The algebra isn't needed for these fundamental topics.
There's no requirement to do opaque algebra before approaching intuition.
Feynman invented Feynman Diagrams, which are a major contribution to Physics because they avoid algebra, and physicists are certainly capable of algebra.
And after you'll learn negatives, reals, periodicity, etc., you'll find that a rotating clock hand is a completely fine analogy. So, maybe it's not that bad to have this analogy from the beginning to not lose the forest behind the trees.
> Or are you arguing that you should always try and teach complex numbers and the Euler identity to avoid strained analogies?
I think it’s okay to be explain complex numbers. I think it’s just best to additionally explain why. That is, show why (real, imaginary) is a better numerical system than the more broadly taught (x,y) of the 2 dimensional space being explored.
As for the Euler identity I suppose you could include that when explaining why we use the exp() function, which is because it plays nicer with integration and derivation than other numerical representations.
I want the analogies to be representative of the work rather than my own mental model of it.
That's great! Their motivation & updates seem very sensible.
Calculus Made Easy is a very nice compact book for university students who have started to forgot their high school training, and I will certainly point people in the direction of this project.
If you want the opposite of that: University students that come into college with strong calculus fundamentals, then I suggest "Calculus" by Michael Spivak.
I haven't used calculus in a decade, but I use things I learned from working through that book pretty much every day I write code.
This is super useful to know about!
The sprite designer & waveform editor / tracker is a really good creative introduction to computers for small children. And you can jump straight in to doing this with the above web link.
(For those new to Pico-8, hit 'esc' from the Lua console to bring up the editor tools, then click on the icon in the top right.)
But it is theatre.
A single £0.01 Zener diode can generate vastly more guaranteed (by quantum mechanics!) randomness, without all the possibility of failure, entropy leaking etc.
Yes it is much more cool to look at than a zener diode, but it is still a source of physical entropy. And one could argue that a whole bunch of chaotic pendulums are more resilient to failure than a single diode.
The speed of sound does not depend on pressure, only temperature.
I suspect this is occurring because the (sonic) flow through the nozzle cools as it expands, therefore the speed of sound drops, making the same flow now supersonic in the cooled gas.
>For a given ideal gas the molecular composition is fixed, and thus the speed of sound depends only on its temperature. At a constant temperature, the gas pressure has no effect on the speed of sound, since the density will increase, and since pressure and density (also proportional to pressure) have equal but opposite effects on the speed of sound, and the two contributions cancel out exactly.
I never knew this either. This makes sense as when you tap on a propane tank they always sound empty, different than a metal tank of water. Even though it's a liquid and under extreme pressure, it's only the temperature that matters.
Not saying it is, but it's very volatile. It really doesn't want to be a liquid at room temperature. I can't tell if it's empty or full when a tap on it, but definitely can with with water. It's an impedance thing.
That's got nothing to do with some fluid behaving like an ideal gas or not. Also, except for very low pressures, vapor does not behave like an ideal gas.
Sure it does, it's riding the border of a phase change between gas and liquid. It's in a super compressed equilibrium so it's going to behave similarly to an ideal gas.
That's - not how that works. I suggest you look at any fluid data `riding' that border. "Super compressed" and "ideal gas" are mutually exclusive. "Equilibrium" has even less to do with any of that.
Whoa that's something I didn't realize. I always had the intuition the speed was faster in denser materials but you're right that pressure doesn't matter.
The temperature difference does make sense for the same reasons here though!
I think it's just relativity. After all, the gas in the Earth's atmosphere is traveling 29.78km/s relative to the sun (~Mach 85). Inside of the nozzle it doesn't really matter that it's traveling supersonic relative to the air outside the nozzle until it leaves the nozzle.
I don't think the difference in speed of sound will have much effect in any case.
