MariaD’s blog

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:

• Excitement: creating your own answer is more interesting than “guessing” (or finding) that single right answer that already exists. There is a sense of danger and adventure and exploration there.
• Therapy: if you are making many different answers anyway, the stakes aren’t as high. If one of your many answers is botched, it’s not a big deal, since you can make many more good answers. On the other hand, trying to find that one good answer and getting it wrong is nerve-wrecking.
• Natural ordering and abstraction: many answers to the same question create a menagerie that simply begs to be sorted and ordered. Humans have a very strong instinct for organizing collections into categories. That’s why closet organizers are such a blooming business. When you sort examples of answers from the same topic, you will create some categories, notice patterns, make more abstract generalizations. Congratulations! This is how mathematicians work, or how best students learn math.
• Collaboration: if a question has many answers, your group can work on it together. People can discuss and compare their answers, make answer collections, and then order and sort the collections together. If there are different levels of abilities or interests in the group, some members can create many answers and some can make few; some can make complicated and difficult answers and some can make easy answers.

In an article “Bloom’s Taxonomy Blooms Digitally” you can see an ordering of thinking skills:

This diagram is handy when you need to analyze learning activities. For learning to be meaningful and satisfying, one needs a healthy balance among all levels. For example, if there are too many low-level activities, learning turns into dull drudgery. If there are too many high-level activities, there is a risk of burnouts. This balance is highly individual and depends on personality and experience in each field. But does it depend on the field? How come even the newest newbies are encouraged to create and evaluate in many subjects, from participating in science fairs to writing little poems and essays, but rarely in mathematics? The situation has been slowly changing for the last fifteen years or so, but it still has ways to go. I am going to quote from the article, British spelling and all, the list of verbs describing thinking levels, and invite people to design math activities supporting higher order thinking.

* Remembering - Recognising, listing, describing, identifying, retrieving, naming, locating, finding
* Understanding - Interpreting, summarising, inferring, paraphrasing, classifying, comparing, explaining, exemplifying
* Applying - Implementing, carrying out, using, executing
* Analysing - Comparing, organising, deconstructing, attributing, outlining, finding, structuring, integrating
* Evaluating - Checking, hypothesising, critiquing, experimenting, judging, testing, detecting, monitoring
* Creating - Designing, constructing, planning, producing, inventing, devising, making

It is much harder to design and program software for higher order activities, compared to drill software. Likewise, it takes significant thought and knowledge to design face to face activities, especially group activities, aimed at higher orders. Creating always takes significant human support, but it’s worth it. Imbalances toward lower order thinking lead to immediate unhappiness among learners, and to the subject becoming hateful. In the long run, it can lead to general cultural stagnation in an field, if not enough people are capable of contributing original thinking.

Math is a curious field. A small minority of “math nerds” engage in higher order math thinking, not accessible to the majority. It’s been like that at the time of Pythagoras and his secret, sacred, exclusive circles, and it ironically remains so to this day, for pretty much all math topics beyond the basest arithmetic. Even with the current explosion of casual creativity on the Web, the (relatively few) social math sites and groups aren’t accessible to novices or casual participants. Ironically, math is supposed to develop higher order thinking - and of course it does, for the select few who get to participate in higher order math activity circles. I wonder what will happen over the next ten or fifty years.