# Imbuing a six year old with a sense of mathematical wonder

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.

• Perhaps keep on experimenting with (positive) whole numbers. For example, can he see how the sum of two odd numbers is always even? Dec 9 '15 at 13:10
• I remember being interested in the problem of making two lines of equal length using blocks of unequal length (i.e. least common multiple). Dec 9 '15 at 17:47
• "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. 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
Dec 9 '15 at 19:04
• As 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
Dec 10 '15 at 0:39
• What does it mean if I still find 3 of those 4 things awesome? Dec 11 '15 at 3:43

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

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

If you put two consecutive triangle numbers together you get a square number.

You can also make a square number by adding the next odd number.

• That last point is super cool! Never seen that before. Dec 13 '15 at 12:35
• Actually, these two facts are related ... the red circles from two consecutive triangles are just the odd number of orange squares. Dec 15 '15 at 23:46
• @PaŭloEbermann The pictures make it very clear which is why I included them. Thanks for making it even clearer. Dec 16 '15 at 1:14

• 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.
• that last one is pure evil, soon they'll be drowning in rice! Dec 10 '15 at 2:28
• I 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!" Dec 10 '15 at 23:00
• @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.SE Dec 11 '15 at 3:45
• @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. Dec 11 '15 at 7:40
• @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. Dec 11 '15 at 16:28

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

• Yes, although it's also full of a bunch of bunk about the golden ratio and art and architecture. Dec 11 '15 at 21:26

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:

Learning Resources Snap Cubes

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.

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

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?

• I 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! Dec 11 '15 at 8:55

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.

• the book that got me interested in math as a kid, i read it about 50 times in a row Dec 15 '15 at 1:49

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.

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)

• 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. garlikov.com/Soc_Meth.html Dec 11 '15 at 17:49
• Base six counting is another good one — ones on one hand, sixes on the other. Or skip the thumbs and do base five. Dec 11 '15 at 21:27

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.

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!

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.

• Is there a webpage that describes the method? A link to such would help those who want more details. Dec 12 '15 at 13:32

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.

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. :)

• As 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. Dec 10 '15 at 22:29
• 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. Dec 11 '15 at 16:48
• @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. Dec 11 '15 at 22:28

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

• Is your answer supposed to contain a link to a video? It seems to be missing. Dec 11 '15 at 15:14

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.

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.

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.

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.

• Your two links at the end seem to be identical. Is this intentional? Dec 13 '15 at 10:43
• @JoonasIlmavirta Nope! Thanks, I fixed it. Dec 13 '15 at 11:05

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.

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...