Mindful Mathematics

Mindful Mathematics by BEE

Mathematics. You either love it or you hate it- I don’t know of anyone who doesn’t have strong feelings towards it one way or the other. The irony though is that contrary to popular conception, humans are naturally mathematical animals; “We are as a species particularly well oriented for perceptual and physiological reasons to see the world in certain terms and create symbol systems that reflect our insights,” divulges Dr. Bill Higginson, a professor of mathematics education at Queen’s University in Ontario Canada. “And to the extent that we deny and frustrate that we are doing something that is quite unhealthy.”

That mathematics is a source of joy for only a very small percentage of the population might be because most of us have bad experiences with it in school; “it’s pretty rare for a child to go through public education for an extended period of time one year after another and have a positive experience with the subject,” Higginson observes.

No wonder so many people are math phobic! No wonder parents are fearful of transmitting their anxieties over to their kids. But can we change those numbers a bit? Can we teach our children and ourselves to get a better understanding of mathematics and even learn to appreciate it?

Necessary Mathematics

According to Higginson, if there is one thing that we can do to change the situation is to have people who present math to children be positive about it;”this is interesting. This has potential.” Even with some of the children that do very well “we haven’t given them an authentic picture of what math is,” says Higginson. And so they gain a false impression of what mathematics is, unaware that a lot of what they are taught, “is not nearly that important.”

To an environmentally conscious man like Higginson, “moronic” persuasions that “you need to learn math because you don’t want to get ripped of at the supermarket when you go in to buy those doughnuts,” really get’s his goose. Such a response to mathematics is reflective of our consumer, commercial-oriented society and the way we view our world.

But mathematics according to Higginson is essentially about sensitivity to proportion; “the ethical version of that is about justice, balance, understanding the way systems work well enough.”

People who have been well educated in math, Higginson believes, have comprehended that as with it’s foundations and going back to the Greeks, “it’s a sensitivity to balance in it’s many, many manifestations. Fundamentally it’s about rhythms, and understanding connections.”

For Higginson, approaching mathematics education of children is possible from an aesthetic perspective- mathematics as an art; “the idea of fit.” He proposes,”that the world is a set of connections built around relationships of certain sorts.”

Math is one way of understanding relationships. And of course, the environment. Mathematics is a way to understand the limits of the world your in. “You understand the dangers you run in to when you transgress the limits,” reflects Higginson.

Mathematics is everywhere

It was only when I was exploring mathematics with my own daughter that I learned about the Fibonacci number series (and I’ve even taken university math!). I was completely amazed that this number series 1, 1, 2, 3, 5, 8,13, 21,34....is found in nature; from the petals of a flower, to the swirl of a pine cone or a pineapple, or... a galaxy. The more we hunted in the garden amongst the roses, the black-eyed Susans and the sunflowers, the more we were impressed by this series appearing in nature’s patterns. We immediately wanted to know more about number series.

The ‘doors of perception’ had swung open and what I’d always had an inkling of, became obvious and as clear as 1, 2, 3. Mathematics is everywhere.

Studying art I learned the use of the Golden Section used in Renaissance painting, a sort of ‘divine proportion’ responsible for how, right across cultural boundaries, people define beauty.

Playing the piano, my daughter told me she was doing math; she was encountering fractions; half notes, quarter notes, eighth notes, sixteenth notes all in musical notation as well as patterns and rhythm.

We started to notice that maths is every where for us to use and discover; estimating, symmetry, proportions, balance, shape, space, logic. From cooking and baking (fractions), quilting and knitting (geometry), building a bookshelf (trigonometry), to balancing your account book (algebra), calculating interests on loans the kids have given me, keeping tabs on basketball scores, and using tessellations (geometry again) in art making.

We pay attention to how our own mathematical problems arise from questions we ask ourselves about aspects of our world. My youngest asks “how many dots do you think are on the ceiling tiles?” She tries to figure out how many turns of her bicycle wheels will cover the area from the front door to the back yard.

Another daughter becomes preoccupied with the shape of black holes, infinity, motion, light, string theory. I can only wonder at where these interests will lead them to.

“The secret to math is finding problems that you want to think about that you find interesting, that you fester on,” approves Higginson, unlike the conventional schooling approach where you try and close them down. “You try in fact to open them up- what would happen if we didn’t look at triangles we looked at squares? What would happen if we didn’t consider it in 2 d but 3 or 4 d?”

As the 16^th century polymath Leonardo da Vinci is known to have said, mathematics really is “the language of the universe.”

The Basics

While it is true that "mathematics is no more computation than typing is literature,"(John Allen Paulos) still we know that basics are important to function in the world but we don’t need to sweat it to learn them; there’s a reason why the basics are called basics; because that’s what they are! Most people given time will grasp the concepts with little trouble.

Happily, basics can be learned in ways that don’t involve pencil paper and a work book. And the best part is that parents too can overcome their fears as they join with their kids to explore non traditional ways of gaining the language of mathematics.

Read it!

There’s all sorts of possibilities around mathematical lit that merges language and mathematics; right now there is an explosion of math books that parents can use as a basis for discussion with their children.

In San Diego, one home educating mother of 4 has set up a web site gathering into a compilation an extensive listing of math literature. She’s calls her project ‘Living Math,’ approaching the subject of mathematics from a “human angle,” and from real life situations.

A former CPA, Julie Brennan was searching for ways to “close the gap between math and the various disciplines, the fact that math is taught in an isolated fashion and with no connection to the real world or the human drama,” i.e. with very little context.

She fell on the idea of math in literature, because her oldest love to read so much.

“It’s a jumping off point that might spark of interest- which makes it more exciting. In this way what is happening is that they are ‘grazing on math concepts,” Brennan comments.

As a family Brennan and her children read about Pascal and his famous Triangle and then went off and researched ‘how does this thing work?’ “ It’s much more meaningful and easier to remember.”

Take the Greg Tang books- these are interesting rhymes with matching illustrations to go with them; not necessarily to master the concepts but to prepare the ground work.

When you’re a bug you must beware

Of danger lurking everywhere

A sticky tongue right on your back,

soon you’re just a tasty snack!

How many beetles do you see?

Count them fast before they flee!

Here’s a little helpful fact:

Adding’s quick when you subtract!

From The Grapes of Math by Greg Tang

“Riddles help you learn to look at objects and find a way from going rote counting to 2, 4 groupings. So it teaches you mathematical thinking and for parents it helps them think mathematically too!” Brennan enthuses.

She believes that the best books are books that they whole family can benefit from. Math curse by Jon Szeska is one of my very favourites Brennan says. No matter their ages each on of her children get something out of that book.”

For older children, the I hate Math and Math for Smarty pants by Marilyn Burns, Theoni Pappas books and The Number Devil by H. M. Enzensberger are more amusing and thoughtful books to sink into.

Learning mathematics from a 'human angle,' such as through history like the Mathematicians are people too series is also helpful. The murderous maths series by Kjartan Poskitt is another great series, that include comics and can be enjoyed by adults too.

Mr. God This is Anna by Fynn, a beautiful tender tale of a child’s mathematical observations along side her adult mentor and friend are also recommended.

In your Face

Brennan has a concept of 'strewing.' Strewing is the practice of leaving math tools and materials such as rulers, compasses, calculators, abacuses within children's reach and not locked up somewhere out of their view. “I use clear plastic containers, rotate them put books on the coffee table.” It’s suggesting without being pushy; “They may or they may not use it and that’s okay. The idea is to find ways to engage them.”

Brennan who has invested in a graphing calculator for her children finds that it gets a lot of mileage for the money, “it has a nice display and they can see the trial of what they’ve input and what comes out.”

Almost all games have a mathematical component;“games are to math what reading is to language,” is Brennan’s opinion so building a games library for the family to use is a good idea.

Games like chutes and ladders, yahtzee, chess, tangrams, puzzles, lego, rubics cubes, backgammon, dominoes, checkers are fun for the whole family. Have a regular game day once a week; “when you sit down with your family to play a game of cards you are developing math skills,” says Brennan.

Again create an environment where a child can go in and choose a game.

Frequenting thrift shops and garage sells is a great way to add to the collection. Have a list of games and math readers handy and then you can recognize them when they come up.

In the world of computer games and dvds, you can pick up good programs such as well as free materials of the internet.

For Brennan, these ideas are akin to keeping a sort of ‘tool kit’ handy for what works and what doesn’t; “experiment with your child,” she suggests.

You can extend the idea of strewing to include talk; become aware of math words that we use such as “I’ll have a quarter of the pie,” or “this building is 100ft tall,” or “we are exactly 10 miles from home.” Don’t be afraid of using words and concepts that you think they won’t get. They’ll catch on soon enough.

Check out Brennan’s site at www.livingmath.net for more suggestions on ‘living math,’ math readers and games.

Scrap Math

“There’s an enormous amounts of profound math that can come out of very simple materials,” says Higginson, who is a self described “intellectual bag man.”

In our throw away culture there are just tonnes of materials you can pull out of recycling bins that you can use to teach math. For example, with origami, he uses recycled photocopied paper that holds a crease well. Paper folding, a fascinating cultural area “is great fun to do with kids and it raises all kinds of artistic and mathematical questions,” Higginson suggests, referring interested parents and teachers to the exciting work of Erik Demaine where he addresses “profoundly difficult math questions which he interprets through origami, an ancient Japanese art.” (http://theory.lcs.mit.edu/~edemaine/)

Other useful materials include decks of cards, “for all sorts of probability problems,” tooth picks and plasticine, nails, elastic bands and a piece of board to make a geo board and use it to create all sorts of geometric shapes. Soaked peas, marshmallows, that you can build attractive mathematical structures with bread bag tags, counters, buttons, soap bubbles, “build a structure and dip into a soap solution get a surface which because of the way math works happens to have properties of minimality,” Higginson points out.

With mathematics, a “no waste aspect” from an environmental point of view is pretty obvious and can readily be emphasized.

Higginson urges us to look at mathematics as an avenue to reflect deeply on issues of consumerism and commercialism, philosophical spiritual and political issues.

“Math is a gymnasium for the mind,” it is said. And it’s in the reach of all people. “There are some areas that due to socio- economic factors some people have a lot of advantage over others. Maths is not one of these areas,” concludes Higginson.

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recreational math

What shape is snowflake; Magical numbers in Nature by Ian Stewart

Entertaining Mathematical Puzzles, Martin Gardner

Giant Book of Math Fun, Raymond Blum, Glen Vecchione, Kurt Smith, Steve Ryan, and Adam Hart-Davis,

Creative Mathematics Higginson William; Phillips, Eileen; Upitis, Rena; Pimm, DavidMathematical origami David Mitchell

Dictionary of Geometry Curious and Interesting Numbers David Wells

Teaching mathematics; a source book or aid activities and strategies Max Sobel

Mathematics a human endeavor Herald Jacobs

math a science of patterns Keith Devlon

Each orange has 8 slices Paul Giganti

Murderous Maths by Kjartan Poskitt

Math trek 2; A Mathematical Space Odessey by Peterson Ivars

Life by the Numbers by Kevin Devlin

Africa Counts Number and pattern in African culture, Claudia Zaslavsky.

Tic tac toe; and other three in a row games from ancient Egypt to the modern computer, Claudia Zaslavsky.

The man who counted: A collection of mathematical adventures by Tahan, Malba

The Amazing Mathematical Amusement Arcade, Brian Bolt,

computer games

Road adventures by the learning company

Tetra game

Links;

www.mathpuzzle.com

www.mathcats.com

http://www.shout.net/~mathman/