MariaD’s blog

This is a sketch for an Early Algebra activity from Math Clubs. Kids can draw their own family trees, use photographs, or clipart of their favorite characters.

Once the tree is built, it can be used for several activities. Start from common words, gradually moving to mathematical terms:

• How many grandparents are there? (point to the “grandparent” level on the tree). What about great-grandparents?
• We have one child, and we have two parents in the first generation from the child, and four grandparents in the second generation, and eight great-grandparents in the third generation… How many people are in the fourth generation? Fifth? How do you know?
• Mathematicians use the term “power” here. For example, we can say “grandparents” or “the second generation from the child” or “two to the second power.” Two to the first power (parents) is two. Two to the second power (grandparents) is eight. What is two to the third power? There is a symbol for it:

  2³=8

• What generation has eight people? What power of two makes sixteen?
  Figuring out which generation each quantity means is a lot like logarithms.
  We can say, “What generation has sixteen people?” or we can write:
  log₂16=4
• Add up all generations up to a certain level, say, “grandparents”. Compare to the number in the next level. What do you observe? Is it always the case?

This parking meter is a contribution to What Can You Do With This, or WCYDWT, project. Dor Abrahamson sent it to me, together with this story for WCYDWT people:

“I’m co-teaching w/ Prof. Alan Schoenfeld an undergraduate course for future teachers, at UC Berkeley. It’s a problem-solving based course, and the problems are often based on mundane situations. We used this authentic document - a Berkeley parking meter - to explore multiplicative structure and rational numbers. Rate, I guess. Basically, we show this picture and ask what the best bang for the buck is. To our great surprise there was much confusion in terms of concepts, notation, vocabulary. Diverse approaches, some we couldn’t make sense of.

Judging by your blog, I need not demonstrate what a talented math teacher could do with this material. The mind reels, right? One of the interesting angles here is that some kids calculate min/cent, and others go for cent/min. We all wish to equalize the “denominator” so that we can compare the “numerators” directly. However, for me the min/cent feels intuitive, b/c I want to know how much bang I get for the buck. And yet, think of those ‘value’ numbers we see in the supermarket, which help us choose between comparable products that come in different volumes — they tell you, e.g,. 39c/ounce. So the equivalent here would be cent/min, right?

And so on. We played with polynomials so as to figure out what exact minute totals we could produce and how much they would cost us.

Time is money.

Oh, and the person whose image we can just discern in the display area of the parking meter is Becky Blessing — this is her, errr, reflection piece.”

UPDATE: Becky already submitted this picture in February, and there is a very nice discussion there at dy/dan blog.

I would like to invite you to participate in the second round of a research and development effort called Family Multiplication Study. Every week, participants choose an activity, helping their children write, design, draw, fold, program, define, conjecture and otherwise author some mathematics of their own, related to the broadly defined multiplicative conceptual field. Participants can also request activities to be developed for them, and develop their own. This study group is perfect for people who like to try new things, actively communicate with other families, help their children understand deep ideas based on multiplication, and help the world by building an innovative and open learning resource. The first round was energetic and active, with about thirty families participating.

Participants from the first round said it helped them have more math in their lives, provided accountability for regular math activities, allowed to meet other families, answered particular questions, and supported seeing math as beautiful, useful and fun. I started this research because there is a glaring lack of materials out there that even attempt to help children become authors of mathematics, or help families customize their mathematical experiences on the activity-to-activity basis, rather than choosing whole curricula.

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This is a beginning of an “extensional definition,” that is, definition by naming many examples, of multiplication and multiplicative reasoning. I would like to thank members of Math-Teach forum (1, 2, 3) for a productive ongoing discussion!

See also: Family Multiplication Study.

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.

This is an amazing video of research about a blind artist:

The scientists are attempting to understand how Esref Armagan can paint perfect perspective and other purely visual qualities (and quantities!) without ever having actually seen anything in his life. They found that the same visual centers of the brain “light up” for him on scans as for a sighted person, processing visual ideas without vision. These areas also light up when people dream, though the eyes are disconnected from the brain during sleep. This, to me, confirms one of my strongest beliefs that our perceptions of “the reality” are firmly based on our UNDERSTANDING. Armagan can understand perspective, so he can “see” it in his mind. We don’t see with our eyes, we see with our minds!

by Esref Armagan

I bet each of us can recall examples of our kids or students or ourselves not seeing something glaringly obvious until the conceptual understanding developed! Linguists who are also neurologists claim, for example, that the ability to distinguish colors is based on having names for them, and varies from language to language. Probably the most famous examples from mathematics education are conservation experiments of Piaget, where young kids, for example, claimed that stretching a piece of modeling clay makes more clay.

I would love for people to share their examples of shifts in students perceptions based on shifts in understanding. Here is mine, and it’s quite fresh. Yesterday I was working with a student and a long-time friend Dasha, a bright 11yo girl. She claimed that a room 4.2 by 6.3 has the area of 264.6 From one perspective (pun intended), she just made a minor arithmetic error in multiplication, placing the point slightly wrong. But when her dad and I, working on the problem together, asked her to draw the problem on paper, we saw something fascinating. She drew it on graph paper, to scale, and when asked colored in the “whole part” (the 4 by 6 part, leaving correctly thin fractional strips along the side of the room) and correctly, and quickly, said its area is 24. But even upon seeing the tiny leftovers of the area left uncolored, Dasha insisted that everything together added up to 264.6 She understood the relative order of magnitude of 24 and 264.6 quite well: when her dad asked her to show their proportion in gestures, Dasha opened her arms as wide as they would go for 264.6 and then closed them down to, well, about a tenth of that size for 24. We adults had hard time believing one can look at a picture and see something like that, but for quite a while Dasha could not be convinced to see otherwise. It turns out she has not accommodated the work with decimals into the work with areas and had her own ideas, different from ours, of how decimal areas work. The big “Aha!” was in seeing how to compute the area of, say, the .2 by 6 rectangle in the way we adults were computing it - switching from tenth to wholes, several times, in a process mathematics education researchers call “reunitizing.” When that piece of the puzzle fell into place, the way Dasha, me and her dad saw the situation realigned dramatically.

Dasha was able to see what the adults are saying: that the thin strip .2 by 6 at the edge of her drawing of the room (the whole part separated) has a teeny tiny area that can’t be in hundreds. And we adults saw the nature of Dasha’s cognitive difficulty with the situation: not a minor mistake about decimals, and not any refusal to pay attention, but a powerful alternative picture involving areas and fractions. All told, that switch in perspective took us about an hour and a half to prepare. The adventure was definitely worth it. It just felt so good for all involved, not just me - I asked!

I do not know what would help parents more, though: working on learning all the fine details of every mathematical topic and what can possibly go wrong with it… Or believing that if a child is struggling, then the problem is hard. You may have to dedicate time and energy, develop the look and feel of very minor details, and grope in the dark for a while before you can see. Before the child can see the world your way, and you can share the alternative reality the child sees in her mind.

There is a lively discussion about division models going on in the Living Math mail list. A question came up: “Isn’t there a better written way to express what the actual relationship of 1 3/4 divided by 1/2 = 3 1/2?” I shared the funky idea Kelly, my student, used in a similar situation. It is all about extending the metaphor “division is sharing” into fractions, and the funk comes from using fractional people! Let’s consider an example.

Kyoma and Naruto, pictures by Harumi Uchiha, are sharing six pizzas. How much food will each one get?

The sharing is easy to set up, and the division problem directly follows the sharing metaphor:
6:2=3

Now let’s extend the examples into fractions! Half a person (insert giggles here) has a pizza and three-quarters. How much food will that one whole person get?

It is logical that if half the person got some food, the whole person will get twice as much food. Here are the formulas and a picture:

1 3/4 : 1/2 = 1 3/4 * 2 = 2 6/4 = 3 2/4 = 3 1/2

The fraction operations here follow the pizza slices nicely; if you have trouble, make a model out of paper and you will see what’s going on.

While lol-worthy, the fractional person sharing situation is not very realistic, even in the magical fighting world of Naruto where people sometimes do get sliced. But there are everyday situations involving this model. Here is an example.

Lucy had about a can and three-quarters of blue paint and she thought it would be enough to paint a room, but it only covered a half of it. How much paint would she need for the whole room?

The division problem in symbols: 1 3/4 : 1/2 = 3 1/2

Let’s try a similar problem with juicier numbers. Halves are too simple to manage, and this simplicity makes understanding of operations more difficult!

Chris was seeding his lawn. He had six little seed bags, which covered three-quarters of his lawn. For the next season, he wants to figure out how many bags he’ll need for the whole lawn.

Chris reasoned: if six bags covered three-quarters of the lawn, it comes to two bags per each quarter. Since there are four quarters altogether, it then comes to eight (2*4) bags for the whole lawn. If, for some strange reason, Chris wanted to write it down very formally (this is not a realistic desire, by the way), he could do this:

6 : 3/4 = (6 : 3) *4 = 8

This model shows, well enough, the “flipping of the fraction when you divide” procedure some textbooks stress. I’d like to note that division by fraction is NOT how most humans think of their day-to-day situations. It is more of an extension to make mathematics involved prettier and more streamlined. Division by fractions is rarely, maybe never done to make our day-to-day lives easier. It’s not practical, it’s beautiful, but appreciation of that beauty is inside the mathematics involved. Well, it is practical in the sense that it makes math theories easier and it makes advanced math go smoother. However, the examples about pizzas and paint would not involve division by fractions if real life humans set them up in their daily routines, or if the goal was just to do these computations. I still use such examples, though, but with a caveat that they are a part of a larger mathematical picture I have not even started to discuss. The beauty of the picture is there, but this post only shows a tiny little corner of the picture - not nearly enough to appreciate the beauty.