MariaD’s blog

I inadvertently got into a flame war at a forum where I tried to talk about multiplication. Somehow, the discussion wandered into our past graduate school experiences. My last graduate school was pretty meaningful. Many people contributed to that fact, including Laurie Cavey. As graduate students, Laurie and I spent several years working with our advisor, Sally Berenson, and conducting some of our research together. Doctoral dissertations can be horribly lonely affairs, and only about 40% of people who start doctoral programs get their degrees. I think having ties with other people, beyond your advisor, makes or breaks your graduate school experience and success. I’ve been coaching people through their dissertation work, and they claim I am doing a wonderful job as a coach. However, I think the mere fact of a tight connection with an interested human being is doing some 90% of the coaching job.

Here is a snowflake for Laurie, who made a difference. And for everybody, I wish you to find good fellow travelers in your research journeys.

What is my job title? Once I found a directory web site that had my personal page. I never registered there. I assumed the site’s robots found my information elsewhere. The robots thought my job title is “creator.” I thought it was a bit too kind of them.

Today, my husband Dmitri found a perfect job title for me: Snowflake Moderator. We started beta testing SpecialSnowflake a couple of days ago. The Natural Math beta testers are a wonderful team. In these two days they submitted about twenty suggestions, little bug reports and improvement ideas. And of course, I had the honor to moderate all their beautiful snowflakes!

In a move that surprised even myself a little, I sent out this invitation today. I’ve been planning and dreaming about this for years. No time like now!

My name is MariaD, and I love multiplication. Natural Math is starting a research and development family group about this topic. You are cordially invited! Please forward this invitation to other families who may want to join.

There are three main benefits. You receive individual family math coaching. You access a community of other parents sharing questions and ideas. And you contribute to a beautiful and much needed web resource for the future. There are two main responsibilities. At least weekly, you will run custom family math activities you select. As needed, you will talk with me or other group members about your activities. We can talk by email, chat, voice, or face-to-face in Cary, North Carolina, USA. At this early stage, we need active group members. If you plan to be a quiet fly on the wall, please wait until the next round of development. Time estimate is that the group will provide your family at least an hour a week of math and community activities.

Multiplicative reasoning is the capstone of arithmetic: it ties all the parts together. It is the cornerstone of algebra and the basis of pattern thinking. It is also one of the most badly taught areas of math. People spend a lot of effort and many years on times tables, division, fractions, and proportions. Still, many struggle with these multiplication topics for the rest of their lives. I am a strong believer in multiplication. A kid who “gets” multiplicative reasoning will probably be just fine with algebra and math in general. Based on this faith, I’ve spent more than twelve years collecting, researching and creating multiplication-related lore.

My collection includes psychology of multiplication. It explains why 7*8 and 6*7 are hard to memorize without gimmicks, or how doubles relate to our innate sense of health, beauty and order. There are tidbits about multiplication from histories of many cultures: Ancient Greek music of the spheres, and medieval Chinese secret finger codes for trades. The collection has a lot of modern children folklore. It includes rhymes, finger tricks for times nine and all times tables beyond five, silly pictures and jokes. There are all kinds of contraptions: abacuses, mirror books, bead strings, and Napier bones. There is software: powerful Excel, or small applets for a kaleidoscope, a snowflake creator, or a base two calculator. There is cutting-edge as well as classic research: hundreds of articles, conference presentations and books. Some of these are actually useful, but most are ever read by just a handful of academia people. Speaking of which, there are also people in my collection! Among our contemporaries, there are parents, researchers, designers, and writers who love multiplication, too. This collection of multiplication stuff, and people, can help us start.

I envision a “multiplication planet” map, connected by a web of many paths. Each family can start at a different entry point, depending on their goal. If you want to memorize times tables in three hours, your will probably trek through algebraic shortcuts, memory tools and work with patterns. If you want to have rich, deep experiences connecting many human endeavors, you will also visit algebraic shortcuts. But then you will travel to geometric explorations, history-centered projects, or psychological experiments. If you want arts and crafts, you’ll head for drawing, cutting, or computer animation activities. This first stage of research has five main goals for the map.

1. Develop and find major multiplication activities to put on the map. As all Natural Math activities, they will be centered on creating something.
2. Develop paths between activities, following each family’s travels.
3. Find out what kinds of families use each path, and for what. Use this knowledge to start a guide for new families joining us.
4. Find out what support people need in their journeys.
5. As we do all of the above, plan web tools that can help us do it better.

Update: The study’s discussion group is up and running.

I don’t know how many times I got, “But it’s not for everybody!” in response to my math or social ideas. It’s supposed to be a killer argument totally dismissing the idea. As if every education-related concept has to work for everybody.

I am reading “Tribes” by Seth Godin The book makes me cry at least every ten smart, tiny pages.

Here is what Godin has to say on the subject:
“All you need to do is motivate people who choose to follow you. The rest of the population is free to ignore you or disagree with you or move on. Starbucks doesn’t serve coffee to the majority of people in the United States. The New York City Crochet Guild appeals to just a small percentage of the people who encounter it. That’s okay. You don’t need a plurality or even a majority. In fact, in nearly every case, trying to lead everyone results in leading no one in particular.”

Being easily amused, I like this:

This is a short collection of decent mathematical novels. I started to think about mathematical novels because I want to play NaNoWriMo this year. If you are playing too, let’s be writing buddies. Wish me luck!

I think it’s pretty, but maybe it’s my imagination. You can help Natural Math a lot by being a beta tester.

People who run classes both for school kids and for homeschoolers often comment that homeschool classes are “lively.” What does this vague feeling mean? There are several slightly less vague points I noticed, for example:

• homeschoolers are more eager to answer questions
• transitions from activity to activity happen faster with homeschoolers
• homeschoolers will ask “wait, what?” if they don’t understand directions, rather than doing nothing
• homeschoolers volunteer helpful explanations, examples, and anecdotes

In practice, it means that homeschool classes run faster. Today, driving around in the late afternoon and looking at all the tired school kids leaving buses, I was thinking of the sheer length of their workdays. Over the years, kids learn tactics allowing them to survive these daily school marathons. They grab pauses whenever they can, don’t invest too much energy in any one activity, derail, delay and slow down teachers trying to drive lessons too fast, and in general conserve strength. The slow pace often frustrating me in school-oriented textbooks and other teacher materials is there for a purpose. Schoolchildren need to pace themselves for long hours.

Marathon runners, by Infomatique

There are studies showing that most adults are capable of focused productivity for about three to four hours a day. Most homeschoolers report having one to four hours of “school work” a day. No wonder homeschoolers are used to a fast, lively pace while working. They know they can sprint during the class and rest later.

Sprinter, by Felice de Sena Micheli

This is a prime example of an ancient maxim, “Never send a human to do a machine’s job.” I have little patience with tasks that simple machines can do better than humans, except for meditation and relaxation. Specifically, I think it’s better to avoid such tasks as learning experiences, even for young people. “And when you grow up, Jonnie, you can become a calculator!” - ugh, no.

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:

• Pose your own problems and solve them. The captured process of posing and solving problems is an informal proof.
• Explain solutions to your own problems to others. An explanation to another person is an informal proof.
• Engage others in arguments, discussions, “court sessions,” thesis defenses and other contest-oriented, competitive rhetoric experiences. Competition leads to proofs becoming more rigorous as you defend them against the opposition.
• Work at a level where you will often be wrong and creatively frustrated. If you are not sure if you are right, you will want to prove the truth of your solutions to yourself

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.