## Higher Mathematics for Children

The exchange below contains many statements that start in common experience, but end up outside the general experience of non-mathematicians. None of the ideas is too complex for us to discuss, even though the full mathematical apparatus of notations, axiom systems, calculations, and proofs is not for everyone, any more than, say, the elaborate and ever-changing sign languages used by professional baseball players. However, I will have to leave most of the explanations for other times. I have added links for these topics, sometimes to discussions, and sometimes to images, so that you can get a better idea of any that interest you.

The issue is that these ideas lie within the experience of preschool children, but we don’t know how to talk to children about them. We may not even know that these are important mathematical ideas. How much can we put in terms that children understand, starting from actual experience? (How much can we understand, too?) Nobody has done the research. I will come back to various topics on this list and explore these possibilities.

Another issue that begins in common experience and can rapidly exceed anybody’s ability to follow is that we understand anything in our experience in part, and are ignorant in part. Math and science give us reasonably-well-understood methods not only for learning more, but for managing our ignorance where we cannot answer the current questions. One of those methods is that of conjecture and hypothesis. The famous solution to Fermat’s Last Theorem came from Andrew Wiles and others turning the Taniyama-Shimura Conjecture into the Taniyama-Shimura Theorem. We do not yet have a resolution of the Riemann Hypothesis, but we know how to explore its consequences in number theory, cryptography, and other areas while we try to find a proof or disproof.

So when you run into something that you don’t understand here, remember that that is perfectly normal. There will not be a quiz. We are not in a school where you have to know the Right Answer to everything being discussed, or be labeled a failure. We are doing what real mathematicians do, not what schools have pretended they do, and I am welcoming you and millions of children into our community.

On Thu, July 21, 2011 11:31 am, David Corking wrote:
> You wrote:
>> The curious thing is that life exposes them (pre-schoolers)
>> to higher
math, but we pretend otherwise.
>
> Have you started listing examples? I tackled about 60 credits of math
> at uni, which fascinated me, but only scratched the surface of higher
> math, and omitted all of pure math, so I have trouble spotting it in
> everyday life.

You give examples below yourself.

• Tesselations demonstrate symmetries of the plane, which leads directly to the theory of groups, an essential structure in modern algebra, art, crystallography, physics more generally, and other branches of math. I am doing a tutorial on symmetry groups in Turtle Art, starting with the idea of a toddlers’ block sorting toy. In how many ways can you put a particular block through a hole of a particular shape? In how many ways can you turn a block from one of those orientations to another? What happens when you combine turns, or undo a turn?
• You can tesselate a sphere, and it is done sometimes with toys, into eight spherical triangles of different colors, each with three 90 degree angles. This is an example of  Riemannian geometry [aka elliptic geometry], and also of central projection of an octahedron onto the sphere. Each of the other Platonic solids gives another tesselation by projection. Similarly the common pattern for soccer balls, a tesselation into 20 hexagons and 12 pentagons, based on the icosahedron and dodecahedron, is another example, and there are many more.
• Escher has several tesselations in the Heaven and Hell series based on hyperbolic geometry.
• Escher uses reflection in a sphere, which is an important mathematical inversion. Inversions in circles are an important part of geometry and of complex arithmetic.
• Dice for Dungeons and Dragons come in the shapes of each of the Platonic regular polyhedra, supporting dice throws in ranges 1-4, 1-6, 1-8, 1-12, and 1-20, and in shapes of some other semiregular polyhedra, including one with 30 faces. It is of course possible to number dice starting at 0 or any other number.
• Shuffling a deck of cards is a permutation. Dealing cards, along with throwing dice and tossing coins, lead directly into probability theory.
• Clock time is an algebraic ring, a second essential algebraic structure. So is integer arithmetic.
• Rational numbers, that is fractions, form a field, the third of the most important algebraic structures.
• Motion in space is vectorial. We add vectors by putting the tail of one on the head of the one before. We can also walk from A to B, and then from B to C, in exactly the same way. Vector spaces are the fourth of the most important algebraic structures.
• The rotations of a circle or a sphere form a Lie group, that is, a group that is also a differential manifold, with a connected, compact differential topology. They and the Lie algebras that come out of them are fundamental in modern physics.
• Fractals appear everywhere in Nature.
• The apparent convergence of railroad tracks is an instance of perspective, which leads directly into projective geometry.
• A flashlight shone on a wall in a dark room makes a conic section, that is, a circle, ellipse, parabola, or hyperbola. No matter how you look at a conic section, you see a conic section. (projective geometry, perspective)
• A water fountain produces a parabolic shape. Galileo proved that the parabolic paths of projectiles and streams flowing in air are equivalent to a law of constant gravitational acceleration. (Leonardo da Vinci discovered this fact more than a century before, but was not in a position to publish his discovery.)
• Children can catch balls using Galilean relativity. Once you learn to catch a ball coming on a parabola right to you, you can learn to run so that it looks like it is coming right to you, putting yourself on exactly the course to reach the ball when it comes down to eye height, the right height for catching.
• Bouncing a ball or throwing it straight up and catching it without moving your hand in a moving car, bus, train, or airplane also demonstrate Galilean relativity.
• If you drop two reasonably dense objects at the same time from the same height, they hit the floor or the ground at the same time. This proves that all materials fall with the same acceleration. This in turn shows that neutrons and protons fall with the same acceleration, because different elements and isotopes have different ratios of neutrons to protons, from 0 in ordinary hydrogen up to 2 in tritium, generally close to 1 for other light elements, and somewhat more than 1 for heavy elements. This in turn shows that up and down quarks fall with the same acceleration, since neutrons and protons have the compositions UDD and UUD. A slight variation of the experiment shows that different weights fall at the same rate.
• Placing a ruler tangent to any curved object shows the direction of the curve at that point, leading to the derivative function. It is easy to see that the ruler is level at the top or bottom of a curve.
• Cutting any object, such as a potato, into thin slices demonstrates the technique of the Riemann integral and the Fundamental Theorem of the Calculus, that the derivative is the inverse of the integral. This is because the volume of one slice is the approximate rate of change (that is, the derivative) of the indefinite integral at that point along the length of the potato.
• There is a joke that a topologist is a person who cannot tell the difference between a coffee cup and a donut, because each has just one hole. There are many other instances of rubber-sheet geometry in daily life. It is trivially easy to make a one-sided Möbius strip, by joining the ends of a narrow strip of paper with a half twist.
• Knots and braids make up an important branch of topology, with many practical applications. The Pacific hagfish ties itself into an overhand knot at intervals to scrape its skin clean of the mucus it secretes to prevent predation. There is an enzyme which twists and untwists DNA loops to regulate gene expression.
• The arrangement of ten bowling pins and other such arrangements in squares, pyramids, and so on give demonstrations of figurate numbers and various packing geometries for circles, spheres, and so on.
• There are many children’s games that have important mathematical theories.
• Children love secret codes, which lead to mathematical cryptography and computer security.
• The concept of duality (Do A, then do B, then undo A) is completely familiar in daily life. Open your toybox, take out a toy, close the box. This is an enormously powerful idea at the heart of logarithms, Fourier transforms, category theory, recent applications of the Erlangen Program, and, it has recently turned out, string theory in physics. It is also essential in computer programming, though I know of only one language,  Ken Iverson’s J, in which this is recognized and implemented.
• Computer searches involve elementary set theory and logic. When you say to match any keyword, you are asking for a union of sets, or an oring of properties. When you say to match all keywords, you are asking for their intersection, or an anding of properties. The more elaborate forms of query lead to relational algebra, SQL, and Query By Example (which is more powerful than SQL).
• We can resolve multiple pitches in music because we have a mechanical system for Fourier analysis in our ears and brains. The Measure activity in Sugar can display waveforms of sound and also their frequency analysis.

There is far more than I have time to find just at this moment. Thank you for the question. This will make an excellent blog post. Several, in fact, and a number of tutorials.

>> There are many such projects, and whenever I find one I invite
>> the people involved to join with us
>
> Please check these folks from my alma mater are on your list:

Thank you. I will write to them.

>> Presumably this is because introducing any idea into the formal
>> curriculum carries the implication that all of the teachers and parents
>> will have to learn this now, and they have been taught that they can’t.
>
> It is a problem both of motivation (why?) and of bootstrapping (who
> will teach the teachers? can they teach themselves?) but no different
> from any other curriculum change. The additional cultural fear of
> maths in some parts of the world is an extra barrier.

It is greatly complicated by the ongoing fights over New Math (still), Phonics vs. Whole Word, literary vs. vernacular language, Creationism, Global Warming, and other pointless controversies where people are talking past each other, like the blind men with the elephant, or regard each other as a threat to civilization itself.

> However, I am always delighted by the number and beauty of
> tesselations and Escher-inspired art on the walls of our local primary school.

Have a look at Tilings and Patterns, by Branko Grunbaum and G. C. Shephard. It starts with the easy part of the subject, the regular tilings, and winds up in several higher math topics, including the structure of asymmetric tilings and computability theory. It turns out that every Turing machine can be represented by a set of shapes that tiles the plane in exactly one way, thus demonstrating that there are unanswerable questions in the elementary theory of tilings.

> Symmetry is the first and only part of higher pure maths to
> have made it into the English national curriculum so far, but there is
> also a massive new emphasis on application of maths, which sounds a
> little dull when read from the curriculum documents,

Link, please? I would like to check whether, as I suspect, their notion of applications is bounded by complete lack of imagination ^_^ and nearly complete lack of information from the experts.

> but can be exciting in the hands of a good teacher.

It is possible to shoehorn all of these applications into an extended discussion of symmetries, because most significant branches of mathematics are equivalent to each other, like geometry and algebra in Cartesian analytic geometry, and much more so in Klein’s Erlangen program, which sought to classify all geometries (and thus all algebraic structures) in terms of the symmetries of their transformation groups.

http://en.wikipedia.org/wiki/Erlangen_program

> Similarly, I am delighted by the abstract thought of a young learner
> who, despite struggling with two-digit arithmetic, offered me soaring
> original descriptions of infinite numbers and negative numbers.

Exactly.

> I think English high school students study exponential growth, one of
> Alan Kay’s Powerful Ideas, but I would love to see most under-16s get

I am working on Kindergarten Calculus.

The exponential function is the solution to the differential equation

y’= y

From it we can define the trigonometric and hyperbolic functions and their inverses, among other things.

The hyperbolic functions sinh and cosh are the solutions of

y” = y

The trigonometric functions sine and cosine are the solutions of y” = -y.

The infinite series representations of all of these functions are closely related, being made from terms of the form xn/n! .

ei𝜃 = cos 𝜃 + i sin 𝜃

ei𝜋 + 1 = 0

> Keep up the good work (I’m sorry, I won’t be able to read any reply
> for a few days.)
>
> David

No problem. It is going to take me months to do all of the work that you and I have set for me here.