Some mathematical definitions seem ridiculous to the general public. Consider a set operation called Inclusion Map. Here is one silly-looking definition!
Definition: Inclusion Map
Given a subset B of a set A,
the injection f: B –> A
defined by f(b)=b
for every b that is a member of B.
This definition seems to describe a function that does not do anything whatsoever, in fancy terms. What is it for?!
March is the Disability Awareness Month, which I did not know until the good people in CVS announced their Caremark program and blogging contest. The way a society treats every member, in all the diversity of styles and abilities, is the reflection of its humanistic advancement.
Math is often used inhumanely. Many people see math as an abstract torture tool, because it repeatedly causes them mental anguish. No less frequently, math is used as a gatekeeper, a way to prevent large groups of people from access to careers or programs that often have nothing to do with the entrance test’s mathematics. This is not right, and we need to solve these problems.
I thought more people hate math than love math. However, Google search says otherwise! The phrase “love math” returns 449,000 hits and the phrase “hate math” returns 145,000 hits. So, about three times as many math love as math hate! More love than hate is good, but how does math compare to other human endeavors?
Love math = 449,000 Hate math = 145,000 Love/hate ratio = 3.09
Love physics = 42,400 Hate physics = 15,500 Love/hate ratio = 2.74
Fewer people express their emotions about physics, and also the emotions are slightly more even than math emotions. How about chemistry?
Love chemistry = 83,300 Hate chemistry = 14,800 Love/hate ratio = 5.63
Among those who expressed their feelings, chemistry has a stronger popularity ratio than math or physics.
However, sciences lag incredibly far behind some of the arts. Consider:
Love music = 14,000,000 Hate music = 268,000 Love/hate ratio = 52.24
About thirty times more people confessed love to music than love to math, while there are about the same number of math and music haters.
Today, Katya (10) and I looked at this SAT problem:
The square of the result of adding 7x and y is equal to the result of subtracting the square roof of 4x from y.
Which of the following is an equation for the statement above?
(A)
(B)
(C)
(D)
(E)
Katya knew all the ideas involved, but it took us some half an hour to work through their names and symbols, connecting our private math language to the standard language of SAT mathematics. We drew some squares, messed with powers, joked about Ancient Greeks saying everything in words, colored the text to help translate it and in general had a jolly good time. At the end, the very elated Katya said: “At first, this problem looked very scary and did not make any sense, but now I can quickly understand every part.” I replied with a reference to a “Babylon V” episode we watched a couple of days ago.
This is a very early brainstorm to help me make sense of a feeling. The internet communications are feeling richer than before, and I am trying to figure out why. Researchers say that if you measure the information exchange in face to face conversations, word content accounts for 30-40% of it. The rest of information is in gestures, intonations, body positioning and so on. I will use the word “gestures” to denote all these powerful nonverbal ways to convey tons of information quickly. The places I frequent on the web still mostly use text, with static avatars and occasional photo or video illustrations. But people feel much more real than they did in the early days of blogging.
I think “semantic web” features such as history and trace aggregation, and the ability to “meet” the same person in many web contexts, may serve as an adequate analogy for gestures. Emoticons were designed for something similar, but there is no way they will carry more information than the text they accompany, so they fail as “gesture substitutes.” Say, when I first meet a person on Twitter, I check out the web site: this is like quickly glancing at the person’s clothes and the general appearance and style while meeting face to face. The analogy breaks for people without web sites: I don’t see such a person as naked, but rather as less of an entity. Such people aren’t quite as real!
Quickly checking out the person’s last few blog entries is like getting acquainted with their voice’s inflictions and tone. It takes a bit of time, but it tells you a lot about the person. LinkedIn, Facebook, wikis, nings behaviors - made visible because there is history - allow to accumulate more and more ideas of the person’s pace, style, position.
Moreover, running into the same person on a few of your networks invokes the rather pleasant feeling your are neighbors. Here is that colleague in your ning, in your mailing list, and you microblog reader - it feels like seeing the same familiar face as you jog in the park, shop at the local grocery store, and visit the post office. The general rule of number psychology: “Three is many.” Interacting with the same person on three or more different networks, you will probably feel neighborly.
My conclusion is that it’s important to leave multiple traces as you engage in your web activities, to help people perceive you more fully. It’s probably a good idea to use many different networks together while building online communities, to promote that feeling of strong neighborhood.
In the book - I did not know it got a Nobel - games consist of “playing a bead” symbolizing an entity of culture - a chemical formula, a theorem, a music or art element. The leader of the game is called Magister Ludi, translated as “game master” or “elementary school teacher.” After Magister Ludi played each bead, the group meditated on its meaning and its role in the game up to it.
Here is a seminar I’d like to run. Each bead is a cultural entity, captured as a rich media web object. After the Magister Ludi plays a bead, there is a pause for participants to do their web searches about the bead and share interesting links. Shared object format: the new object you found; your interpretation/feeling/thought on it (meditation made visible); your link from the last bead to this next one. Magister Ludi plays after all the participants, in the same format.
I think it can run on existing software (suggestions are appreciated), but maybe we’ll have to write a module for it. I’d like it to be shiny, like glass beads - those are among my top five favorite objects in the universe.
Of course, we should webcast games live, to open participation to the world. The result of the webcast can be captured, as well. Thinking Elluminate.
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!
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.”
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.
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.
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.
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!
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.
