MariaD’s blog

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::

An excellent collection!

With about 70% or lower high school graduation rate, and the technology changes, there is a huge push toward computer learning. The majority of existing learning systems are “ebooks” - note the letter “e” in there, indicating the early nineties style. The system provides explanatory text and then gives exercises: either multiple choice or “type an answer in the window” and tells you if the answers are right. But this style is going to change, hopefully.

Just think about the scale of the changes, though. The behemoth that is the textbook production and classroom management and teacher training industry is built on the assumption of a group of people doing math in lockstep, simultaneously, together. This is simply not true about computer learning systems. They are asynchronous, most of the time, or there is a strong monetary, design and logistics push for them to be. This one variable - learning being distributed in time - breaks the lockstep pedagogy in incomparably more powerful ways than any psychological, educational and parental movements that ever tried to break it. Developmental psychologists’ pleas for individualized instruction, lolconstructivism, or homeschooling - all affected the behemoth very little, though homeschooling of course made a huge difference for families practicing it. But now the behemoth goes supernova. Computer learning systems are leading to education being distributed in time, and that, in turn, does away with most of the lockstep pedagogy. And who even has any expertise outside of it, eh? Precious few. May we live in interesting times.

A supernova, scattering Brown Nebula against an ancient planet, by by Boris Bello.

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.