Background: doing a freehand macrame project with a craft circle. Kids kept asking whether or not an object could be tied in. When I started to explain the general principles of how one can decide this… Whoohoo!
In an overview of the Natural Math project, an audience member asked why we are doing such diverse things. Why work, all at once, on mathematical art, programming-based math, algebra for toddlers, meta-cognition and quite a few other directions we pursue at the same time? These are necessary tools for a gradual, gentle cultural change we are making.
The future is already here, it’s just not evenly distributed. The future where toddlers and their parents play with algebraic ideas, where kids contribute to real work as apprentices, where everybody is able to create or improve mathematical conjectures, definitions and metaphors. We are working on inviting more and more people to work and play in this future, now. This mind map shows some of our tools.
Algebra is the study of patterns, structures and changes in them.
The language of algebra includes representations such as symbols, graphs, tables, and words that describe mathematical models of situations.
You can edit this map freely: it’s a wiki-map. Click on the MindMeister icon to edit. I have no idea what people will do with it.
Parents and other people working with beginners in geometry often find that beginners don’t understand any reason to prove their work. Children in particular can become very negative, sad or hostile about the whole idea of proof. And for very good reasons! This is what Lockhart has to say in his “Lament”:
“The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.”
This short paragraph is wonderful, because it lists all the activities you can arrange for beginners to help them find beauty and meaning in proofs:
What do we have in typical geometry courses? Kids work with pretty obvious facts from classic geometry that some long-dead ancients, not kids themselves, created literally thousands of years ago. The proofs are not made for the purpose of solving a problem or convincing someone who thinks you are wrong. The problems have been solved before the course began, and nobody doubts the rightness of the theorems and lemmas presented for “proving.” Actually, the proofs are not MADE at all! Not by the learners. They are received, from the same ancients, and rehearsed.
Yet the very meaning of proof is in the PROCESS of proving. It is said you can’t really appreciate romance, like “Romeo and Juliette,” unless you’ve had some love-related experiences and feelings. You can’t appreciate other people’s proofs unless you created some of your own, from scratch, and went through situations where proving was vital: defending your creations against the opposition, or else using proofs to convince yourself your solution to a tricky problem is not a mistake or a glitch.
Phoenix Wright, a cartoon lawyer famous for his “Objection!” yell, often quoted in our Math Clubs.
So, you need to invite kids into situations where they will be creating their own statements and proving and disproving them. The situations have to provide “natural environments” for proofs and proving. One easy way is to replicate the natural situations existing in our culture. You know how people “defend” their dissertations? That’s because they are attacked! Typically, if you work in math or science research, you will find yourself in somewhat warlike situations where people actually attack your creations. A good model of that situation, for kids, is a debate or rhetoric club. A game of examples/counterexamples works well: some club members become your opponents, attacking your statements and proofs and creating examples and counterexamples that poke holes in them. “Definition Wars” we sometimes play at the Math Clubs is a good example. As a result of being contested and argued, the proof becomes tight and rigorous. The irony is that most people who learn geometry never find themselves in these situations of “scientific attacks and defenses”, so the rigorous prophylactics against attacks is totally lost on them. Never having experienced the need to defend their arguments, they just don’t feel any need for proofs. You can create the need by providing appropriate activities. Make sure you don’t become too hostile, like some conferences I know 
Non-Euclidian geometries, by Dirk Laureyssens
Another good source of situations that require proofs is mathematical weirdness. Sometimes our visual intuitions are only “almost right” - and some weird marginal cases escape them. A proof, again, provides prophylactics against such weirdness and counter-intuitive examples. Ironically, again, most geometry students never meet any weirdness, instead doomed to prove things that look and feel (and are) quite obvious and straightforward. Good sources of mathematical weirdness for your kids are paradoxes, and also optical illusions if you talk about geometry specifically. You can also go into non-Euclidian spaces where our intuitions don’t work as well, for example, studying all your theorems on a ball instead of the plane. This way, the proof becomes a daily tool of exploration and problem solving, rather than the spare fifth wheel you only use once in a blue moon, if ever.
Yet another study reports that gamers are more fit and active than the average. A meaningless piece of statistics, in itself, but with an interesting explanation by Dmitri Williams, the researcher. To quote the article:
Williams pointed out that TV watchers get bombarded by messages about “buying, consuming and eating,” while video gamers get messages about “taking action” within the game. “I think a part of it is that the culture of video games is not necessarily a culture of consumption, whereas the culture of television clearly is,” Williams noted.
What messages bombard learners of math? Do our math activities promote “a culture of consumption”? The other week, I was looking at several wonderful math enrichment books for teens, talking about delicious topics like fractals, topology, or combinatorics. Lovely pictures, wonderful problems, engaging texts. The books made me quite sad. There wasn’t much the readers were invited to do, other than eat up the book content.
“They” say that to achieve happiness, you need to balance “giving” and “taking” in your life. Look at any math curriculum material you remember. Are activities about giving or taking? Here are some examples that come to my mind, and they are all about taking, about consuming knowledge, about eating up that math content:
- read some explanations (watch a movie, look at pictures)
- solve some exercises to better yourself
- do an investigation/exploration project to tie your knowledge together
People usually assume consuming the knowledge of how to solve quadratic equations is better for you than consuming an hour of soap opera. For sure, consuming math knowledge potentially allows you to give something back, to create, to contribute. But where and how do you learn to contribute, to create, to give, if you are only taking and consuming all along your learning process? You may learn quadratic equation, but will you have any idea how to create with them? How to apply them to something contributing to the community? How to make them a part of your life that gives to others?
As a parent, I used to pride myself on advanced knowledge of my daughter. But now I am at best lukewarm about all the feats of intellectual consumption, even if my own dear child performs them. How can we promote an active, community-oriented life position in our children without squishing their free exploration, or exploiting them for mundane labor? Specifically, how can we help kids to give, as well as take, in their mathematical endeavors?
Goats eating paper, by C&T
More questions than answers, surely. Even Google only brings about fifteen hundred results mentioning “intellectual consumption.” One of them a blog entry from about a month ago, asking similar questions, by Dave. ::waves::
I noticed if we plan our math activities and learning sequences too tightly, life gets boring. Randomness is an old and respectable tool of creativity, from opening Shakespeare’s sonnet book at random to get your business leadership seminar inspiration to casting sticks or cards to help you brainstorm about life events. I usually use several learning ideas specifically designed to introduce randomness into our choice of topics.
Brainstorm, by Unknown.
For example, I would ask kids to make up an equation. Any equation. I don’t know what they will make - one made a series of number equations, like 5=3+2=17-12=… and another an equation in two variables (a+b=12) and another something that wasn’t even an equation (3x+5 or some such). So with the first one we talked about infinity, equivalence and different operations, and with the second one we graphed a variety of two-variable equations and solved systems graphically, and with the third we worked on the definition of what an equation is, and created our own.
In another example, we open an SAT book (young kids can do it if there is a math-literate adult by them) and a kid selects any problem. Then we chase all prerequisites to the problem, use examples to quickly learn what’s needed, and, well, eventually solve the problem. It’s always an adventure with younger kids, and afterward I recap the story of that adventure - how examples helped, and where was a blind valley, and how we despaired at some point, and what topics came up. The story is an additional intrinsic reward, on top of solving the problem. We were just talking with my daughter Katya, turning ten tomorrow, comparing math SAT problems with verbal sections. Every math problem creates such a rich story! And a verbal problem, well, those are poorer for connections, links, prerequisites and so on, and so stories of their solutions are less dramatic. I never saw a verbal problem calling for a chain of prerequisites as long as math problems at that level. As Kaplans said in their excellent book “Out of the labyrinth: Setting mathematics free” - math is a tall structure! Randomly landing somewhere in the middle and climbing up and down it with a trusted sherpa is a great source of dramatic, occasionally dangerous, random adventure.
Math is tall. Photo by Centralasian.
I am going to write up my collection of tasks that support participants in creating their own mathematics.
Questions that have many right answers are probably the simplest tool for helping people make math. The beauty is that you can use pretty much any closed-ended little exercise with one right answer into a question with many right answers. Here are a few examples:
One answer: 2+2=?
Many answers: make up some numbers that add up to 4
One answer: 2/3=5/?
Many answers: create as many ratios as you want that are equivalent to 2:3
One answer: solve for x the equation 2x+5=7
Many answers: make up an equation with one variable
One answer: how many solutions does x2-4=0 have?
Many answers: make up an equation with two (one, none, infinitely many) solutions
Why would we want to do something like that?! There are several reasons:
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