Imbuing a six year old with a sense of mathematical wonder

35

10

My six year old started school a few months back and he's loving it. This first year is more about social skills than anything academic and I like that approach. But we're spending some time at home with letters and numbers. In the numbers department there's a sense of excitement on his part and I want to kindle it. Here are some things he finds amazing:

  • The numbers never end. Not even after a 1000! You can go on forever.

  • You can get to a number in many different ways. For example you can get to 6 by adding 4 and 2, but you could also do it by adding 3 and 3.

  • The number zero.

  • Numbers feel different. They're like personalities.

The other day I told him about odd and even numbers. We talked about the fact that there's an infinite number of even numbers but that if you add in the odd numbers you'll get twice as many numbers. That's crazy.

Now I'm just looking for more stuff to keep him going. I'm not looking for puzzles to solve as much as things to marvel at.

Mathdad

Posted 2015-12-09T12:05:43.037

Reputation: 325

2Perhaps keep on experimenting with (positive) whole numbers. For example, can he see how the sum of two odd numbers is always even?Joel Reyes Noche 2015-12-09T13:10:56.680

1One thing I could recommend, from the way I was taught by my parents, is to show that an "advanced" concept exists (ie. multiplication) and explain a possible derivation. My parents explained multiplication when I was about 6 (amazing I remember!), and then I remember writing tons of numbers down in a notebook and discovered simple "rules" of multiplication. Try powers of 2 to be simple maybe. But I was also an odd child who wanted to prove everything to myself.aidan.plenert.macdonald 2015-12-09T17:04:38.963

2I remember being interested in the problem of making two lines of equal length using blocks of unequal length (i.e. least common multiple).Rhymoid 2015-12-09T17:47:20.887

6

"We talked about the fact that there's an infinite number of even numbers but that if you add in the odd numbers you'll get twice as many numbers. That's crazy." The real crazy part is that you can match up the elements of the combined infinite set of odds/evens with the elements of just one. https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel (It might be too early to teach him about the cardinalities of infinity, though.)

JAB 2015-12-09T19:04:32.737

3As JAB almost said, I don't think it's quite true to say you get twice as many numbers when you add in the odd numbers. You get the same number of numbers. Infinity + infinity = infinity.bdsl 2015-12-10T00:39:07.773

1I read the title as if you were trying to create some sort of mathematical super power in a childAequitas 2015-12-10T02:25:53.693

2What does it mean if I still find 3 of those 4 things awesome?Cort Ammon 2015-12-11T03:43:31.847

Thanks for all the great feedback! There's a lot here to keep us busy.Mathdad 2015-12-11T09:55:16.267

A bit early for a 6yo, but I really enjoyed reading The Number Devil at about age 10. And it helped me understand a large part of high school math.

Kevin 2015-12-11T14:49:05.437

I'm still on the lookout for a good prime-number-decomposition game (preferably a physical one rather than a tablet one, both are good though, maybe the tablet one for further ages?).Jack Maddington 2015-12-11T19:08:11.317

Just an anecdote: I once got an eight year old to think math was cool enough to talk to his friends about it by explaining to him that it is impossible to write down the number googolplex because it has more zeros ($10^{100}$) than there are atoms in the universe (about $10^{35}$).Todd Wilcox 2015-12-11T22:23:20.317

" there's an infinite number of even numbers but that if you add in the odd numbers you'll get twice as many numbers. That's crazy." -- No, you get the same number of numbers, which some think is crazier. You can see that there are as many whole numbers as even whole numbers because they can be matched up -- each whole number to its double, each even number to its half. If you want to see how much fun and fascinating math can be, check out Vi Hart's youtube videos.Jim Balter 2015-12-12T11:02:02.797

I know my dad taught me negative numbers sometime around grade 0. Got in trouble with the teacher at school who gave me 2 pencils and asked how it is possible to take 3 away. Shame on you, teacher!Richard Le Mesurier 2015-12-13T12:43:48.737

Answers

18

I remember being excited about the following at a young age.

If you add consecutive numbers you get triangle numbers. Triangle numbers are fun.
triangle numbers

If you put two consecutive triangle numbers together you get a square number. enter image description here

You can also make a square number by adding the next odd number. enter image description here

Amy B

Posted 2015-12-09T12:05:43.037

Reputation: 3 622

That last point is super cool! Never seen that before.Richard Le Mesurier 2015-12-13T12:35:56.363

1Actually, these two facts are related ... the red circles from two consecutive triangles are just the odd number of orange squares.Paŭlo Ebermann 2015-12-15T23:46:44.297

@PaŭloEbermann The pictures make it very clear which is why I included them. Thanks for making it even clearer.Amy B 2015-12-16T01:14:21.077

16

How about:

  • Numbers go the other way, too (negative)
  • You can cut numbers in half, forever
    • What if you cut a number into three pieces?
  • 1 million is a thousand thousands (100 is ten tens)
  • If you don't know the number, call it a letter (or name it :) )
    • You can add letters together too
  • What if you cut triangles in half?
    • Rectangles?
    • Can you make a box from squares?
  • Start at 1. Every day, double it.

Ben

Posted 2015-12-09T12:05:43.037

Reputation: 261

13that last one is pure evil, soon they'll be drowning in rice!Aequitas 2015-12-10T02:28:15.920

8I would not encourage a 6 year old to confuse letters and numbers. It's too early in the development of their understanding of language. Once they've mastered the notion of letters are letters and numbers are numbers, only then should they be exposed to "numbers are just symbols like letters!"corsiKa 2015-12-10T23:00:32.930

1@corsiKa I wonder if you could resolve that by teaching them base 16 as a "secret way of counting" and let them invent the names and symbols of the last 6 digits, instead of using letters. That could be an amusing question to ask on Parenting.SECort Ammon 2015-12-11T03:45:05.327

1@Aequitas Actually, that last one should be fun. Multiplication by two can be done easily on paper, and if you keep it up for a month or so, you can see that while the number grows very big, you only need to add a digit every few days.Sanchises 2015-12-11T07:40:21.260

@sanchises after a month you'd be up to 537 million, in two months; 600 quadrillion, in a year 3.7*10^109, I know they'll be 7 by that time but that's still probably pushing itAequitas 2015-12-11T08:58:15.167

But take the log10 of that and behold, the length of the number grows constantly in time. And I would certainly stop after a month. XDSanchises 2015-12-11T09:00:21.140

2@corsiKa I'm not entirely sure I agree. I know I was around that age (at most a year or two older) when I started getting problems in the form of "6 + _ = 8" and being told to fill in the blank. When I started learning basic algebra my mental reaction was to wonder why it was a big deal, it was just kids stuff I'd learned years ago.Dan Neely 2015-12-11T16:28:48.910

@DanNeely my thoughts exactly. I remember being nonplussed about algebra because it was just letters for numbers. Wish somebody had told me that earlier - rather than telling me that I shouldn't be learning that yet.Ben 2016-03-25T07:26:17.073

7

There is a great old Disney short called Donald Duck in Mathmagic Land.

As well as being delightfully drawn in the traditional Disney style, it contains lots of useful and occasionally surprising information about where math can be found in everyday life.

Personally I found it informative and inspiring, I imagine there would be plenty of conversations to had from watching it. YouTube

just passing

Posted 2015-12-09T12:05:43.037

Reputation: 71

3Yes, although it's also full of a bunch of bunk about the golden ratio and art and architecture.mattdm 2015-12-11T21:26:10.890

6

The earlier question, "Teaching a very enthusiastic and bright 5 year old" could help.

Building and manipulating shapes enhances geometric imagination. Consider polydrons, or snap-cubes:


      SnapCube
      Learning Resources Snap Cubes

Joseph O'Rourke

Posted 2015-12-09T12:05:43.037

Reputation: 13 356

6

Below are some great and inspiring books by an excellent mathematician. (In the Really Big Numbers book, on the page where counting by tens is discussed there is an inspiring error (?)…Big Bird is right, everyone makes mistakes!)

Really Big Numbers and You Can Count on Monsters by Richard Evan Schwartz.

Let me also add this wonderful activity JDH did with his child's class: http://jdh.hamkins.org/tag/graph-coloring/. Graph coloring is an excellent way to engage small children with the feel of mathematical thought. The idea of a coloring book incorporating this was just fantastic! For 4 year olds it is a bit early, but the activity is just wonderful.

Jon Bannon

Posted 2015-12-09T12:05:43.037

Reputation: 3 411

5

Another book, to add to @Jon Bannon's list, that my 4 year old daughter and I can recommend is Introductory Calculus For Infants. We also seem to discuss Graham's number a bit after watching the Numberphile videos

David Pell

Posted 2015-12-09T12:05:43.037

Reputation: 151

5

Arranging counters into groups (multiplication and division), so arranging 12 counters into 6x2, 3x4 etc, and realising that there are some numbers that cant' be arranged, no matter how hard you try (prime numbers). Then, how many prime numbers are there, can you work out which number is going to be prime?

Joe Miller

Posted 2015-12-09T12:05:43.037

Reputation: 151

1I think this is great, not only because primes are fascinating, but also because it probably requires a mix of theoretical thinking and physical manipulations. Assuming that the child can't do long division and doesn't come up with the sieve of Eratosthenes, the best approach would be to take a certain number of things (legos, matchsticks, whatever) and try to arrange them into same-sized groups. Maybe the child will come up with ways of doing that more efficiently; boom, they've invented engineering!yshavit 2015-12-11T08:55:08.840

4

Just to complete the list of books. Some time ago I read The Number Devil and liked very much. I think it is the proper book for your needs as the argument shows how a "math devil" shows math concepts to a child in a way that makes the child more interested in new concepts.

Juan Carlos González

Posted 2015-12-09T12:05:43.037

Reputation: 141

the book that got me interested in math as a kid, i read it about 50 times in a rowceleriko 2015-12-15T01:49:11.100

3

Try folding a piece of paper in half, then in half again and see how many times you can do it. start with A4 then find larger pieces-newspaper then wallpaper perhaps.

grimbold

Posted 2015-12-09T12:05:43.037

Reputation: 31

3

Base 2 counting on fingers - gets you up to 1023.

Base 12 counting - need two more numbers flip & flap. 1, 2, 3, 4, 5, 6, 7, 8, 9, flip, flap, flap-one, flap-2, flap-3, flap-4, flap-5, flap-6, flap-7, flap-8, flap-9, flap-flip, and flap-flap !!!!

Review of "Mathsemantics" by Edward MacNeal:

The language of mathematics seeks precision--only one interpretation. Everyday language such as English, colorful, emotive, and artistic, allows many interpretations.

Edward MacNeal's Mathsemantics presents a whole new way of looking at math that liberates mathphobes from math anxiety, enables businesspeople to do their jobsmore effectively, entertains and informs math buffs, and provides educators with the tools to teach match without instilling fear.

This divorce between numbers and what they mean begins in childhood and pervades every aspect of life from then on. One result is innumeracy. LIke John Allen Paulos's bestseller of that name, this book describes the symptoms of the problem, but unlike Innumeracy, Mathsemantics offers a solution: a revolutionary way of looking at math as a language, something we've all heard but which never made sense until now.

Mathsemantics takes off from a quiz that was given to job applicants for the author's consulting firm who described themselves as "good at numbers." Most of them, it turned out, weren't in fact good at numbers, because they couldn't draw conclusions about what the numbers meant. The good news is that many people who think they're terrible at numbers will find after reading this book that they aren't so bad after all. They'll learn how to one-up the number crunchers.

Mathsemantics is that rare book that will change the way readers look at the world. It provides the most promising and entertaining answer yet to the problem of American innumeracy.

Might be little advanced, but with some parental guidance it should do the trick. (Flip-flap example drawn from the book)

Rick Grant

Posted 2015-12-09T12:05:43.037

Reputation: 39

The base 2 counting reminds me of Rick Garlikov's use of Socratic method to teach binary arithmetic to third graders. It's an interesting read. http://www.garlikov.com/Soc_Meth.html

Travelling Man 2015-12-11T17:49:40.420

Base six counting is another good one — ones on one hand, sixes on the other. Or skip the thumbs and do base five.mattdm 2015-12-11T21:27:44.613

2

While this is going to above the "age: 6" mark, I ran across this Youtube video just the other day that really fits the topic of this question, although for a different age group.

For those of you old enough to understand infinite sums, this is What it Feels Like to Invent Math.

Draco18s

Posted 2015-12-09T12:05:43.037

Reputation: 121

2

I have been teaching my daughter mathematics now for nearly a decade. She is 13. I have been telling her that mathematics is a language- with symbols for imagining big numbers and patterns. Very much like spoken languages in which we can spell her name. The one that I remember vividly was that I showed her that fractions are very special types of numbers, different from the other numbers she had used (integers). I showed her that 1/2, 2/4, 3/6,...are all symbols to represent the same concept - half. It was based on a story of she and her friend sharing a soup on a day when she had money only to buy a soup. I asked her how she would share it equally and she had a method - get another bowl and pour out some into it so that the level are the same. I told her she could write the experience as 1/2, read out as One-by-Two. Then, the next day she had money to buy two soups, but 4 friends (obviously, the word her spread that she was buying soups). How would she split the two equally among the four- sure, she had a clear answer. But, did she have more on the second day or on the first day? Same on both days. How did she write it? Two-by-Four. So, one-by-two is same as two-by-four. But English is a funny language- we have agreed on this often. Mathematics is precise so we write 1/2 = 2/4. Then, we discussed the value of this crazy possibility. I believe I made rational number arithmetic very intuitive. Today we are reading "Challenge and Thrill of Pre-College Mathematics" by Pranesachar et al, "Mathematical Circles" by Fomin, Iternberg, Genkin. Most of my teaching has been when I was jogging and she was cycling with me, and these days she reads mathematics herself. I only need to help her read carefully once in a while - especially when it comes to a new concept.

Nari

Posted 2015-12-09T12:05:43.037

Reputation: 21

2

I would try Pascal's triangle. It's easy to set up (requires only addition), and it has a bunch of cool patterns that your son can find.

One of those patterns is the Fibonacci sequence, which may be worth diving into in its own right. Like Pascal's triangle, it only requires addition to set up. It's fun, and it'll come up again and again in his education -- starting with the fact that it shows up in Pascal's triangle! I always liked "themes" like that in math, those things that just seem to crop up again and again unexpectedly. And it's always neat to come across a "new" concept in class that you already know from home -- it'll make your son feel smart (I'm not implying he's not, of course! He sounds like he is).

And while we're on recursion, you can venture out a bit from the world of numbers and take a look at the Tower of Hanoi. You may be able to help him come up with the recursive solution to it, and you can tie it back to the world of numbers by seeing if there's a pattern in how quickly the number of steps grows as the number of levels grows. This also shows the connection between numbers math, puzzles, and the physical world -- which is a neat connection at any age!

yshavit

Posted 2015-12-09T12:05:43.037

Reputation: 121

2

You could introduce your child to CircleMath, invented by Dr Stephen Taylor. It is a very simple process for using 2 bases at once, eg 9 & 10, 10 & 11, 9 & 11 is best because it has many more patterns and involves 'Greek 9s'. The method relies on estimation, which I suggest is the starting point for all calculations. This works in any base... and, until metrics and decimals and calculators took over, everyone once used multiple bases every day... pints/gallons, inches/feet, days/weeks, etc.

When I first met Dr Taylor in 1985 he was tutoring a half dozen 5-year olds in his lounge using a whiteboard. The children could do 'sums' like 37x58 in their heads... in under 10 seconds... and tell me, a complete stranger - how they did it. As I was at Teachers' College at the time, preparing to teach high school math, I realised I knew very little... So, Dr Taylor and I became friends, I have most of his books, and still teach CircleMath when I get a chance. I recently wrote a PowerPoint intro - for adults - if anyone would like to see it.

I frequently get amazed looks as people realise how intuitively obvious it is, but it has been overgrown by dull modern mono-base thinking... and there is always a real buzz about this 'new thing'. It's the buzz all teachers and parents love...

As an HoD Math I put this to the test at one of our regional math assoc meetings, and talked about 15 capable, qualified math leaders through the basics, building up to this one: 114622/514. Following the very simple rules they got the answer correct in about 20 seconds... then looked at me and each other in silence... then said 'how the hell does that work?' Tada! CircleMath :)

It had taken only 45 mins, and they all got it correct, but few really got it at all.

Sad to say, for years afterwards some of them joked about that session as the workings of a crazed mind... not at all open for people whose job it was to open students' minds.

Edit for clarification: there was a link but has lapsed since Dr Taylor died in the last couple of years. I'll contact his son who was also brilliant at this stuff.

Edit2 for clarification: Dr Taylor wrote a book Theory of Mind, in which he identified and promoted the human brain's 'intrinsic base', meaning if things are couched in the right way then they resonate with the brain's OS, as it were, thus obviating the need for us to try to engender or pander to the child's interest. Things put in the right way will be interesting to the child. It is a provocative book about what is intrinsic and extrinsic to mind, learning and understanding.

Stephen

Posted 2015-12-09T12:05:43.037

Reputation: 21

2Is there a webpage that describes the method? A link to such would help those who want more details.Joonas Ilmavirta 2015-12-12T13:32:47.233

1

Do an exercise a couple times per day with a hidden number. For example:

I just wrote a secret number down on this little piece of paper. I can't tell you what it is, but I'll tell you a secret: If you add 1 to this number, then you get your own age! Can you guess what it is?

It may take a couple examples, but he will eventually understand this concept. And this will give him an advantage later on in math.

Count by 2's to 100. Now count by 5's to 100. Now 10's.

Once you get into any multiplication (even very simple numbers), teach him how to do 11. The trick is to split your number then add into the middle. For example take 21. Split it and get 2_1. Now stick 2 + 1 into the middle and get 231. Never give him two numbers that add to 10 or higher and he'll be able to impress family and have fun doing it.

Paul

Posted 2015-12-09T12:05:43.037

Reputation: 133

1

At some point you need to remember why we should even educate at all -- because there are useful things to know so we can do things in life. The more practical and relevant things you can demonstrate and integrate into one's life, the more one is going appreciate it and hang on to it.

So, there is a lot of art in all this, but the kiddo's going to either appreciate it or not on the level of the mechanics and novelty of various facets... that'll be the child's preference. What I'm saying is focus on "why", on how useful mathematics is, because it truly is amazing what you can do with it, especially the deeper you go when you have a problem to solve. You can at least foster some long-lived appreciation and life skills. That goes for the future engineer, scientist, and everyday Joe.

With that all said, here's a fun starter: How to use algebra to derive a way to calculate post-sales tax prices in one step... maybe a little early for the 6 years old, but hey. :)

jeremy simon

Posted 2015-12-09T12:05:43.037

Reputation: 111

6As a kid, I was never interested in applying mathematics. I just liked math. Focusing on applications might have actually killed my interest. Nowadays I hold a PhD in mathematics and I work on problems closely related to practical applications, but the drive still comes from the desire to understand cool abstract things.Joonas Ilmavirta 2015-12-10T22:29:05.517

Ya. Each person is different. So some may have a distaste no matter what. Others may just have an affinity "just because". With the application and stuff, I keep reflecting on how much stuff I learned in school and college just to get a good grade. I actually found calculus fun "just because", but in college for as much as I paid, I wish they would've spent some time on all the great things it's good for...after all, I was spending my time and money to qualify to work in some desired fields. For kids, small things like allowance goals, making good decisions, and fun games are great.jeremy simon 2015-12-11T16:48:11.063

1@JoonasIlmavirta I compare abstract math to weight lifting. Say you do curls to improve the strength of your biceps. You might use the fact that you have strong biceps in some real situation, but it's unlikely that when you use your biceps, it will be to actually curl something heavy up to your shoulder and release it. Likewise, you strengthen certain parts of your brain when doing abstract math. We assume that sooner or later you will use that stronger part of your brain, even if you're not actually using it to solve for $x$ or anything like that. I agree applications can be boring.Todd Wilcox 2015-12-11T22:28:24.250

1

I find this video by VSauce on logarithmic thinking really interesting. I dont know if its appropriate for a 6 year old, but in the video he mentions this is the way we naturally understand the world, and that children before 4 years old think in this way.

Also the story of Gauss figuring out how to efficiently find the sum of the first 100 whole numbers when he was 6(?) is really interesting. This link is one I found with a quick google search: http://mathcentral.uregina.ca/QQ/database/QQ.02.06/jo1.html

Nick Steffen

Posted 2015-12-09T12:05:43.037

Reputation: 11

Is your answer supposed to contain a link to a video? It seems to be missing.Joonas Ilmavirta 2015-12-11T15:14:45.967

1

Inspiring question..

Try explaining bisecting: Get a number between 1 and 10(00) and let him guess by cutting the range in half.

Summing a sequence of numbers (1 + 2 + 3 + 4 + 5 + 6) = 1+6 + 2+5 + 3+4

Mounty Hall (and increase the number of doors to 10 and leave one closed).

induction puzzles: Like The king's wise man

Oh yeah I loved it when my dad let me discover PI by cutting out cardboard wheels and rolling and measuring distance. But i think I was 10 at that age.

dr jerry

Posted 2015-12-09T12:05:43.037

Reputation: 111

1

As a parenting question (rather than a formal teacher question), we found it good to put the number in context, or rather, extract the numbers from our local context.

For example, when driving home from a long journey you can ask what are the distances to places when the sign boards come by.

You can as how much will be left to get to the further away place when you get to the next place.

You say how far 'home' is from some place listed and ask how far is it to home (really good for subtraction if home is before the listed place). You can ask how many times you have to do the little bit you did before you get home (5 miles from last village, 20 miles to go, -. we'll have to do that same distance four times'...).

You can show them tricks you know and use, such as rounding to get a handle on things. Here we have petrol/gas/deisel in litres but everyone talks in miles per gallon, so try some approximations - we got 300 miles on that fuel fill, we filled with 22.5 litres, how many miles can we go for every gallon? If the tank has 60 litres can we drive all the way to grannies - she's 800 miles away.

The same can be done when shopping (and it's good budgetting).

Estimate the beans in the jar...

The key here is to use the context you have, so it's normal and natural.

Philip Oakley

Posted 2015-12-09T12:05:43.037

Reputation: 111

1

There are many different ways to write numbers - base-10, binary, hexadecimal. You can have any number of digits 2 or greater, as long as one of those digits is 0.

Numbers that have an exact decimal representation in one system may not in another. For example 1/3 in base-10 repeats forever, but in base-3 it is just 0.1.

Also, Roman numerals, tally marks, other ancient ways of writing numbers.

Jayd

Posted 2015-12-09T12:05:43.037

Reputation: 11

1

Pull out a couple of Möbius strips at the dinner table. There is nothing like a "magic trick" to inspire wonder.

Know all of the fun stuff you can do cutting them, double looping them etc. If you need a refresher, look here and here.

Jolenealaska

Posted 2015-12-09T12:05:43.037

Reputation: 111

Your two links at the end seem to be identical. Is this intentional?Joonas Ilmavirta 2015-12-13T10:43:05.463

@JoonasIlmavirta Nope! Thanks, I fixed it.Jolenealaska 2015-12-13T11:05:22.713

1

I recently pulled out my old polyhedral dice from my role-playing days and gamified multiplication tables with my seven-year-old twins: roll two dice and multiply them. How many can you do correctly in five minutes? We are up to two ten-sided dice (2d10) now, but I still have a couple d12 and d20. And you can always use more than two dice.

Well, there isn't really all that much mathematical wonder in multiplication tables, I'll admit that.

So I took the d4 and had my daughter (the son had gone off) count the number of vertices $V$, the number of edges $E$ and the number of faces $F$, then had her calculate $V-E+F$. Then we did the same for the d6, the d8 and so forth. (The little tricks I explained to calculate $V$ and $E$ for the d12 and the d20, by noticing the regularity of the faces and accounting for double-counting, went a bit over her head.) She was fascinated by the fact that the result of $V-E+F$ was the same for all dice, got her brother and explained this to him. (With a little help from daddy.)

My kids now know the Euler formula for convex polyhedra. And they seem to like it; especially my daughter brings it up every other day or so.

No, I haven't gone into proofs or CW complexes yet. Time enough for that when they get into high school.

Stephan Kolassa

Posted 2015-12-09T12:05:43.037

Reputation: 111

-3

Here are a list of books and articles he might enjoy reading.

Mean Field Theories and Dual Variation - Mathematical Structures of the Mesoscopic Model

Hardy Spaces on Ahlfors-Regular Quasi Metric Spaces

Hamiltonian Partial Differential Equations and Applications

Approximation of Stochastic Invariant Manifolds

I'd suggest that you replace bedtime stories with these articles as they are guaranteed to instill a sense of mathematical wonder in your child, and put him right to sleep...

naughty boy

Posted 2015-12-09T12:05:43.037

Reputation: 101