Inspiring Maths

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Inspiring Maths

#1  Postby Spearthrower » Jan 05, 2021 9:38 am

To the mathematically proficient out there: how do I, as an acknowledged mathematical philistine, hope to inspire in my kid an appreciation of the beauty and utility of mathematics?

Most subjects I've got covered, and we're having a great time touring the precincts of human knowledge. But maths? I keep avoiding it because I'm just plain afraid of making it seem boring and complicated. Thai government school education is all about rote learning & repetition, but I believe that inspiring fascination and wonder results in a much deeper and more fulfilling relationship with learning and knowledge.

I can conceive in the abstract of equations being sublime, of mathematics being the key and the door to physical knowledge, and offering a language with which to directly interrogate the universe but sadly I can't actually do those things in the concrete - poetry in a language I don't understand, and can barely even hear. At best, I can think of the things I find mind-blowing, like fractals for example... but pointing at fractals and saying 'Math! See?' isn't really very inspiring.

So does anyone have any ideas, or sources suitable for a 7 year old that might fit the bill here?
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Re: Inspiring Maths

#2  Postby Mike_L » Jan 05, 2021 10:27 am

Kids love games, so...

Math games!

Just one link, taken at random from the results of a Google search (there are many more)...

https://www.mathplayground.com/
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Re: Inspiring Maths

#3  Postby Cito di Pense » Jan 05, 2021 10:29 am

Spearthrower wrote:To the mathematically proficient out there: how do I, as an acknowledged mathematical philistine, hope to inspire in my kid an appreciation of the beauty and utility of mathematics?

Most subjects I've got covered, and we're having a great time touring the precincts of human knowledge. But maths? I keep avoiding it because I'm just plain afraid of making it seem boring and complicated. Thai government school education is all about rote learning & repetition, but I believe that inspiring fascination and wonder results in a much deeper and more fulfilling relationship with learning and knowledge.

I can conceive in the abstract of equations being sublime, of mathematics being the key and the door to physical knowledge, and offering a language with which to directly interrogate the universe but sadly I can't actually do those things in the concrete - poetry in a language I don't understand, and can barely even hear. At best, I can think of the things I find mind-blowing, like fractals for example... but pointing at fractals and saying 'Math! See?' isn't really very inspiring.

So does anyone have any ideas, or sources suitable for a 7 year old that might fit the bill here?


Why don't you watch some of Grant Sanderson's (3blue 1brown) videos on youtube? They focus on calculus and above, but don't worry about that. Look for his techniques in suggesting how you might have discovered something for yourself, if you were curious enough. Geometry is a great place to start, or even with the concept of algebraic equality or identity, which is not that abstruse, but oh, so central. Trying to educate your kid without educating yourself first is likely to fail.

Watch some Feynman, "The pleasure of finding things out". Yeah, it's intimidating to find the good stuff. As Theodore Sturgeon declared when somebody told him 99% of SF is crap, he said, "99% of everything is crap". You have to have an outlook like Gwyneth Paltrow's to insist otherwise.
Хлопнут без некролога. -- Серге́й Па́влович Королёв

Translation by Elbert Hubbard: Do not take life too seriously. You're not going to get out of it alive.
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Re: Inspiring Maths

#4  Postby Spearthrower » Jan 05, 2021 11:15 am

Cito di Pense wrote:Trying to educate your kid without educating yourself first is likely to fail.


All too sadly aware of that, which is at least part of why I'm going more for the inspire part. If he's interested, he'll learn. Sadly, back in the day, my math teachers were not inspirational, so I learned what they taught rote, and never realized what I was missing until later in life.


Cito di Pense wrote:Watch some Feynman, "The pleasure of finding things out".


I'm not sure if he's capable yet of understanding that just yet - English is his 2nd language.
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Re: Inspiring Maths

#5  Postby Spearthrower » Jan 05, 2021 11:15 am

Mike_L wrote:Kids love games, so...

Math games!

Just one link, taken at random from the results of a Google search (there are many more)...

https://www.mathplayground.com/



Thanks, but not what I am asking for. That kind of thing he can get at school.
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Re: Inspiring Maths

#6  Postby Cito di Pense » Jan 05, 2021 11:43 am

Spearthrower wrote:
Cito di Pense wrote:Trying to educate your kid without educating yourself first is likely to fail.


All too sadly aware of that, which is at least part of why I'm going more for the inspire part. If he's interested, he'll learn. Sadly, back in the day, my math teachers were not inspirational, so I learned what they taught rote, and never realized what I was missing until later in life.


Cito di Pense wrote:Watch some Feynman, "The pleasure of finding things out".


I'm not sure if he's capable yet of understanding that just yet - English is his 2nd language.


Well, then you are amid exceptional circumstances. I'm sure not going to try to inspire someone in a language in which I am not utterly fluent. When I was seven, people did try to engage my curiosity; however ineptly they did it, a spark was struck. That spark came from me, not from them.

I recently participated in a comment thread at the NY Times on an article about how educators in India were contemplating how to teach pupils to write computer code from an early age. Imagine how that went.
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Re: Inspiring Maths

#7  Postby newolder » Jan 05, 2021 12:00 pm

Have you found parallel by Simon Singh yet? It's aimed at 11 to 15 year olds but everyone, including parents and teachers, is welcome.

...
Dr Simon Singh, author of the No. 1 bestseller Fermat’s Last Theorem and The Simpsons and Their Mathematical Secrets has created a set of weekly maths challenges – just 15 minutes of interesting, fun and challenging material that goes beyond school maths: mystery and history, activities and oddities, puzzles and problems. (After Christmas, the challenges will take a bit longer.)

Sign up and each week on Thursday you will receive a Parallelogram, a weekly set of maths challenges.
It’s FREE to sign up and all the materials we offer are FREE.

...
I am, somehow, less interested in the weight and convolutions of Einstein’s brain than in the near certainty that people of equal talent have lived and died in cotton fields and sweatshops. - Stephen J. Gould
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Re: Inspiring Maths

#8  Postby Mike_L » Jan 05, 2021 12:51 pm

Spearthrower wrote:
Mike_L wrote:Kids love games, so...

Math games!

Just one link, taken at random from the results of a Google search (there are many more)...

https://www.mathplayground.com/



Thanks, but not what I am asking for. That kind of thing he can get at school.

:scratch:

Spearthrower wrote:Thai government school education is all about rote learning & repetition...


:dunno:
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Re: Inspiring Maths

#9  Postby Spearthrower » Jan 05, 2021 2:10 pm

Mike_L wrote:
Spearthrower wrote:
Mike_L wrote:Kids love games, so...

Math games!

Just one link, taken at random from the results of a Google search (there are many more)...

https://www.mathplayground.com/



Thanks, but not what I am asking for. That kind of thing he can get at school.

:scratch:

Spearthrower wrote:Thai government school education is all about rote learning & repetition...


:dunno:



:scratch:

Spearthrower wrote:I... hope to inspire in my kid an appreciation of the beauty and utility of mathematics



Mike_L wrote:Math games!


:dunno:
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Re: Inspiring Maths

#10  Postby Mike_L » Jan 05, 2021 3:12 pm

I took "an appreciation of the beauty and utility of mathematics" to be a goal, with math games as a possible step toward getting there.

If it's to an immediate appreciation of the beauty and utility of mathematics, then I suppose something like...

* The Mathematics of Origami (paper folding)
https://en.wikipedia.org/wiki/Mathematics_of_paper_folding

* The Mathematics of Perspective Drawing
http://www.sfu.ca/~rpyke/perspective.pdf

* Pop-Up Cards as a Vehicle to Teach Mathematics
http://www.science.smith.edu/~jorourke/DTS/Sum05/WebPages/Faith/weblog.htm
and...

* Geometry of Pop-Up Books
https://www.youtube.com/watch?v=T3JM_5_ARjo

(2 minutes, 22 seconds)
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Re: Inspiring Maths

#11  Postby Spearthrower » Jan 05, 2021 4:58 pm

Not sure if you recall, but I've got more than 15 years experience as an educator, so of course I can find exercises on any topic for a 7 year old. And games like those, regardless of their ability to keep attention levels a bit higher, are just a repetition exercise.

I don't want to practice learned concepts, particularly low level arithmetic; I want to inspire curiosity.

To that end, some of those links above are very interesting, thanks!

I do think that, as Cito also mentioned, geometry is a good place to start. First, you can do stuff with it that stretches credibility, so it satisfies the utility and wonder elements, plus it's got a host of simple rules which give a new way of accessing aspects of the world, so I think it can show why mathematics is a potent tool worth learning to use.

Guess it's time to brush up on some geometry.
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Re: Inspiring Maths

#12  Postby scott1328 » Jan 06, 2021 4:27 am

Cito di Pense wrote:
Spearthrower wrote:To the mathematically proficient out there: how do I, as an acknowledged mathematical philistine, hope to inspire in my kid an appreciation of the beauty and utility of mathematics?

Most subjects I've got covered, and we're having a great time touring the precincts of human knowledge. But maths? I keep avoiding it because I'm just plain afraid of making it seem boring and complicated. Thai government school education is all about rote learning & repetition, but I believe that inspiring fascination and wonder results in a much deeper and more fulfilling relationship with learning and knowledge.

I can conceive in the abstract of equations being sublime, of mathematics being the key and the door to physical knowledge, and offering a language with which to directly interrogate the universe but sadly I can't actually do those things in the concrete - poetry in a language I don't understand, and can barely even hear. At best, I can think of the things I find mind-blowing, like fractals for example... but pointing at fractals and saying 'Math! See?' isn't really very inspiring.

So does anyone have any ideas, or sources suitable for a 7 year old that might fit the bill here?


Why don't you watch some of Grant Sanderson's (3blue 1brown) videos on youtube? They focus on calculus and above, but don't worry about that. Look for his techniques in suggesting how you might have discovered something for yourself, if you were curious enough. Geometry is a great place to start, or even with the concept of algebraic equality or identity, which is not that abstruse, but oh, so central. Trying to educate your kid without educating yourself first is likely to fail.

Watch some Feynman, "The pleasure of finding things out". Yeah, it's intimidating to find the good stuff. As Theodore Sturgeon declared when somebody told him 99% of SF is crap, he said, "99% of everything is crap". You have to have an outlook like Gwyneth Paltrow's to insist otherwise.

3blue 1brown is inspiring. i would recommend that channel for anyone who made it past middle school algebra. i was going to recommend thus, but Cito beat me to it.
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Re: Inspiring Maths

#13  Postby i have no avatar » Jan 06, 2021 7:45 am

Hi ST,

One thing that you and your kid may find interesting involves simple arithmetic, especially since he is 7. :)

I'm not sure when I learned my math multiplication tables but I assume that he can divide by 7, with a known remainder (knowing the multiplication table for 7 is very helpful, of course).

I do not mean to under-estimate the abilities of your son, but for now, let's let him tell a friend (perhaps an older friend who can "appreciate" what he is doing) that he can multiply two 3 digit numbers in his head and write down the answer, left-to-right. Have the friend suggest one 3 digit number and your son will "arbitrarily" fill in the other number for multiplication as being 143.

Since 143 = 1001/7, then abc*143 is simply abcabc/7. So all he has to do is concatenate the number upon itself and divide by 7.

Example: The friend gives the number 432. So 432*143 = 61776. And 432432/7 = 61776. With a little practice, this may become almost second nature, and it may amaze his friends.

When he is older, he may want to tell people that he can multiply two 9 digit numbers in his head and write the answer from left to right. If there is a group of people, I find that asking members of the group to provide a random digit (9 times) tends to suppress any pretense. Then he will multiply this number by 142857143 (as "made up" by him).

Since 142857143 = 1000000001/7, then abcdefghi*142857143 = abcdefghiabcdefghi/7. Again, all he has to do is concatenate the number upon itself and divide by 7.

Example: The group gives the number 361214813. So 361214813*142857143 = 361214813361214813/7 = 51,602,116,194,459,259.

Of course, about 2% of the fun is having people briefly think that you are a human calculator, and 98% of the fun is telling them how it works a short time later (which I always do). I also use this to try to stay mentally sharp (yeah right, impossible for me) by, on occasion, making up a few 9 digit numbers and doing the calculations.

Some things to consider:
  • If the number that the person(s) gives is divisible by 7, then there will be repeating digits in the result. Example: In the 3 digit case, the person suggests 847. 847*143 = 121121 so you say something like "Wow, isn't that interesting" (before you tell them your method).
  • If you make a mistake, you can usually discover it since the final two digits will most likely not be divisible by 7. In which case, you say something like "Wow, this is a hard one, let me re-calculate".
  • Of course, you can only do this once with 143 and 142857143 as the given second multiplicand, which helps lead to the fun of telling your audience how it works if they ask you to do it again.
  • Verifying the results of the 9 digit calculation became a lot easier after the calculators that are now common on computers became able to provide the requisite number of digits.

Note: I learned this from a book by Martin Gardner, which my dad gave me when I was 12 or so. Gardner also gives the problem of 1443 times a 2 digit number, since 1443 = 10101/7, but I never use this (I simply don't like it as much as the 3 and 9 digit problems). Ex: 1443*96 = 969696/7 = 138528.
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Re: Inspiring Maths

#14  Postby Spearthrower » Jan 06, 2021 9:10 am

That's wizardry! I want to teach him math, not dark magic!

:grin:

Seriously though, that's kind of fun... not sure if he's quite up to that yet, but I'll definitely keep that in mind.

I've been running through basic geometry today in preparation - I think it's a great place to start as it has so much utility in the world... you can just do so much stuff with it and see familiar things from a new perspective. I think running through the basics then just touching on trigonometry will cause some wonder when we can calculate the distance to a far-off point just using known angles
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Re: Inspiring Maths

#15  Postby Spearthrower » Jan 06, 2021 9:19 am

I'm not sure when I learned my math multiplication tables but I assume that he can divide by 7, with a known remainder (knowing the multiplication table for 7 is very helpful, of course).


Incidentally, I am not sure when this becomes normal in most countries, but he hasn't been taught division at school yet (he can readily regurgitate times tables in Thai, but he actually has to calculate mentally in English). I've given him the basics in the past and presented it in terms of interchangeability: the easiest way to get a grip on division as a concept, I think, is just to deal with it as the 'reverse' of multiplication, so if he can do 7 x 8 = 56 then he can easily do 56/8 - he was mightily impressed with himself when he clicked on this.

So far, showing him such 'tricks' has seemed to give him the most pleasure.



* just discovered there's no ÷ key on my Thai keyboard - that unfortunately shows a lot about the parlous state of mathematics in Thailand. By and large, people cannot do simple mental arithmetic, such as working out how much change to give. Amusingly, even with my generally poor mathematical skills, I am sometimes treated like I'm doing dark magic when I can tell them how much change before they've had the chance to type the numbers into the omnipresent calculators.
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Re: Inspiring Maths

#16  Postby i have no avatar » Jan 19, 2021 2:53 am

Hi ST,

Thanks for the feedback, and I apologize for practicing dark magic, especially on a rational forum. :grin: I also apologize for my delayed response as work has been crazy the last dozen days or so.

then just touching on trigonometry will cause some wonder when we can calculate the distance to a far-off point just using known angles


Have you taught your son how to calculate (using trigonometry) the distance to the horizon for any height above the ground? It is fairly easy to look up how to do this but I include a relevant diagram below (I think I copied it from one of Cali's posts from the distant past :) but I cannot provide a link):

earth_curvature_distance_height_formula.JPG
earth_curvature_distance_height_formula.JPG (53.12 KiB) Viewed 159 times


Of course, there are complicating factors such as refraction, etc., and maybe this is a bit boring, but hopefully this generates enough of your son's interest to calculate the circumference of the Earth (see next post).
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Re: Inspiring Maths

#17  Postby i have no avatar » Jan 19, 2021 2:56 am

As I stated in the previous post, how about calculating the circumference of the Earth? You could use a method similar to that of Eratosthenes but this would involve driving a fairly long distance for you and your son.

So why not measure the radius (and hence, circumference) of the Earth using a stopwatch from a single location (almost instantly)?

If you click on the first diagram of the link, it gives you a "movie" of the principle involved. Since links tend to disappear, I have snipped 3 images to capture content (other than the movie) for future reference (see below).

At the bottom of the last picture, the experimental data given is, and calculates as:
Height: 165 cm
Time to re-appear: 10 seconds
Calculated radius: 6200 km

My calculations are:

theta = measured_time/86400 * 360 (degrees) = 10/86400*360 = 0.0416666667 degrees.

a = (a variable I made up for compactness) = cos(theta) = R/(R + H) = cos(0.0416666667 degrees) = 0.9999997355751681

So:

a = R/(R + H) and we can solve for R:

R = (a * H)/(1 - a) = (0.9999997355751681 * 1.65)/(1 - 0.9999997355751681) = 6,239,958 meters = 6240 Km.

(There are complications: We really want the length of the sidereal day (86164 seconds), not the solar day (86400 seconds), but just using the solar day is easier (so you don't have to explain the difference), latitude, etc.

Also, it seems like sunset would be easier, and you may want to run up a tower of known height, etc., for better accuracy although differences in refraction may come into play.

earth_circ_1.JPG
earth_circ_1.JPG (130.65 KiB) Viewed 157 times


earth_circ_2.JPG
earth_circ_2.JPG (88.51 KiB) Viewed 157 times


earth_circ_3.JPG
earth_circ_3.JPG (64.75 KiB) Viewed 157 times
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Re: Inspiring Maths

#18  Postby i have no avatar » Jan 19, 2021 5:53 am

Sorry, I'm not finished yet. :grin:

Maybe your son can imagine very large numbers (and very small numbers and probabilities). And maybe he is familiar with a standard card deck. Where I am going with this is that with a well-shuffled* deck of cards, he can create a deck ordering that has never been "seen" before in all of history.

So let's get to big numbers:

Example: A trillion (1E12) is a 1 followed by 12 zeros: 1000000000000.

But the number of possible orderings of the cards in a standard deck is 52! = 52*51*50* ...*3*2*1 = 80658175170943878571660636856403766975289505440883277824000000000000 =~ 8E67, which is a LOT larger.

To help show how large it is, let's imagine a clock set to 80658175170943878571660636856403766975289505440883277824000000000000 ticks, and let it count down at a rate of 1 tick per second. Before the clock reaches 0, we can (theoretically):

  • Starting at a known point on the Earth's equator, take a step.
  • Wait a billion years, then take another step. When we reach our starting point (completely around the Earth):
  • Take a teaspoonful of water out of the Pacific Ocean (and put it, um, somewhere :scratch: ), and start the "stepping" process all over again.
  • Continue until the Pacific Ocean is empty. Then:
  • Put down a sheet of paper, refill the Pacific Ocean (instantly), and repeat the process of "stepping" and "draining".
  • When your stack of paper reaches approximately to the nearest star, the clock will reach zero.

Note: I did not originally come up with the idea of presenting 52! in this way, and I cannot provide a reference, but I did change the parameters of the original idea to give a closer approximation (if I remember correctly, the original countdown time was only about 0.1% of the initial clock time).

However, having the same order of cards in a deck of cards is analogous to The Birthday Problem wherein only about 23 people need to be in a room in order to have a 50% chance of at least two people having the same birthday. Does a similar phenomenon happen for the card deck, and if so, does it show that a well-shuffled deck is likely to have been repeated?

We can answer that by looking at the square approximation (scroll down a little in the link, and I pick this approximation because I can derive it using two different methods).

So if we take n as the number of orderings in all of history for a deck of cards, and m as 52!, then:

p(n) =~ (n^2)/(2 * 52!), where p(n) is the probability of any two orderings being the same.

I will let the reader come up with their own values of n (and ST, maybe you and your son can make an estimate), but as an extreme value on the high side (lowering the probability of a repeat), there have been approximately 37 billion people alive since the card deck was standardized and if each of them performed 1000 shuffles per day for 50 years, that is about 6.75E17 different ordering of cards. I think we can all agree that this is way too high.

Yet, using the formula given above:

p(n) =~ ((6.75E17)^2)/(2 * 8E67) =~ 2.85E-33 =~ 0.00000000000000000000000000000000285, which is incredibly small.

Perhaps, for a more reasonable prediction, assume that there have been a billion orderings a day for 1000 years. This is n =~ 3.7E14. Then the probability is only ~8.55E-40 (about 3.3 million times smaller).

___________________________

For those who wish to check my calculations, the following data was used (sorry, these are given w/o references, but these can be looked up easily, and it provides an additional check if looked up):

Circumference of Earth: 4.38E7 steps
Volume of Pacific Ocean: 710E6 Km^3
Thickness of paper: 0.1 mm
Distance to the nearest star: 4.24 light-years


* Being a rational forum, I should define "well-shuffled" but I will ignore that definition and let the reader research it, if desired (I'm trying to keep it simple for a 7 year old - or, I'm just being lazy :nod: ).
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Re: Inspiring Maths

#19  Postby i have no avatar » Jan 19, 2021 6:30 am

Ok, I will introduce one more "problem", involving probabilities that are more reasonable than in the preceding post.

Rather than describe all of it, I will provide a link to the Monty Hall Problem and make some suggestions and comments.

ST can actually play this out with his son by, for instance taking 3 playing cards (maybe two 2s and an ace). ST randomly mixes the cards out of sight of his son and then looks to see where the ace is and places the cards face down. His son then comes in and chooses a card. ST plays the role of Monty Hall and turns over a 2.

Maybe do this 20 times where his son always switches from the card he chose and then do it 20 times where his son always keeps the card he chose. There is a fairly high probability that in the first case, he will win (get the ace) about 2/3 of the time and in the latter case he will win about 1/3 of the time.

There are some assumptions given in the "Standard assumptions" near the top of the article given in the link. But I think that they left one out:
In the case that the contestant picks the car, the host randomly picks which goat to reveal. In other words, he does not always pick the goat with the lower numbered door, for instance. I'm pretty sure this is explained later in the article though.

I think that the key to this problem, as Marilyn vos Savant explains, is that the host opens ALL of the doors except the one that the contestant picked, and one other door. So in the case of 1,000,000 doors, if the host opens up 999,998 goat doors, you're going to switch, right?
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Re: Inspiring Maths

#20  Postby Spearthrower » Jan 19, 2021 6:58 pm

Thanks for the great suggestions IHNA. The latter one I was thinking about already, although the pathway to take him from arithmetic to probability is not an obvious one for me. I'm not even sure what kids of his age in other countries would be doing by now, but I can't help thinking it'd be more advanced that multiplication.
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