the slow, tortured, inching toward natural math in the rest of the world :)

[Mike South]

This looks like a fun problem to think about and play with:

http://blog.mrmeyer.com/?p=4018

It's funny how they are trying to not "suck the air out of the room" by pushing "the math" on them immediately.  And I think the question the way it's posed the third time would be interesting to kids.  But the really funny part is this phrasing:  "We ask for all this risk-free student investment before we lower the mathematical framework down onto the problem."  I thought "BOOM!  Down comes the mathematical framework!  Hope it didn't smack you too hard!".

I'm not trying to be overly critical here--this guy is really working to have kids doing things that are actually intriguing to them, and that is to be applauded.  It just seems like he half-gets the idea that the kids should be invested in it, but then can't go all the way to letting them do the framework part themselves.  Much, or possibly all, of this is probably due to the fact that this is taking place in a classroom with the curriculum and expectations and schedule under huge external pressures, etc.

You can see what they're trying to do, and you wish you could just take away all the artificial barriers and let them succeed.

mike

[Maria Droujkova]

This glass rolling activity is a part of the series called "WCYDWT" or "What Can You Do With That" series.

From the page:
- Bad first question: What is the radius of the circle this cup makes when it is rolled across the floor?
- Dan's (the blog author) first question: Which of these (shows three very different glasses) rolls the largest circle?

I think the general idea of the series is to start qualitatively and then develop more formal, and more quantitative reasoning. For my part, I am mightily struggling with the transition from helping kids reason qualitatively in lively real situations to helping them create mathematical concepts, frameworks and theories ABOUT the situation. I find it challenging to pose intrinsic, appealing tasks that genuinely call for such reflection.

For example, as a part of the Early Algebra project, I am now collecting data on growth patterns, with Joan Moss from Institute of Child Study (http://www.oise.utoronto.ca/ICS/site_LaidlawCentre/ResearcherProfiles/profile_JoanMoss.shtml). The pattern approach met bitter criticism in the past for being a "dead end" leading to no formal mathematics. You can start strong, with students first making a variety of patterns, then moving to predicting next growth step (recursively). The activity of making patterns predictable is engaging enough. The activity of building a theory of different types of growth patterns, which is where formulas, types of functions and other relevant formal math resides, well... As a villager once told me when I asked for directions, "You can't get there from here." Well, I am researching possible paths toward the formal, all involving children authoring of course, but it's not obvious, intuitive, or easy. It goes sort of like this:

Making particular patterns:
Kids create a host of growth patterns, having incredible tactile and visual fun.

The guessing game:
"Can you make your pattern's growth predictable from step to step?" This is challenging, and leads to development of some nice math concepts, such as recursive formulas or constant growth.

Toward the theory of functional relationships:
"Can you make your pattern globally predictable? And by the way, make it different from those constant growth ones you've been making up to now..." - crickets

--
Cheers,
MariaD

Make math your own, to make your own math.

http://www.naturalmath.com social math site
http://groups.google.com/group/naturalmath future math culture email group
http://www.phenixsolutions.com empowering our innovations

[Sue VanHattum]

Hi Mike,

Interesting take on that. (I follow dy/dan's blog, and am enjoying these video lesson starters.)

>It just seems like he half-gets the idea that the kids should be invested in it, but then can't go all the way to letting them do the framework part themselves.

I'm curious how you would put together a lesson like this one. Or if lesson is the wrong word. Maybe I should ask how you would 'let them do the framework part themselves'?

I love thinking about this stuff. But I find it much harder to implement all this in the classroom.

Warmly,
Sue

---

Date: Thu, 11 Jun 2009 09:04:00 -0500
Subject: [NaturalMath] the slow, tortured, inching toward natural math in the rest of the world :)
From: mso...@gmail.com
To: natur...@googlegroups.com

---

Windows Live™: Keep your life in sync. Check it out.

[Mike South]

On Thu, Jun 11, 2009 at 10:28 AM, Sue VanHattum <suevan...@hotmail.com> wrote:

Hi Mike,

Interesting take on that. (I follow dy/dan's blog, and am enjoying these video lesson starters.)

>It just seems like he half-gets the idea that the kids should be invested in it, but then can't go all the way to letting them do the framework part themselves.

I'm curious how you would put together a lesson like this one. Or if lesson is the wrong word. Maybe I should ask how you would 'let them do the framework part themselves'?

I love thinking about this stuff. But I find it much harder to implement all this in the classroom.

I agree completely.  I perhaps should have lead off with this instead of burying it at the end: 

"Much, or possibly all, of this is probably due to the fact that this is taking place in a classroom with the curriculum and expectations and schedule under huge external pressures, etc." 

I don't know that there is much you can do.  My disrespect is of the system that won't let kids develop naturally, not the people that are trying to work within it to change it.

This guy is trying to get the kids to invest in it, etc--but because he has external mandates that every kid in his classroom must learn X by time Y, there's really no way to avoid "lowering the framework" on them if you want to keep your job.

So when I said that he "half-gets" it, that was really not my place to say, he may very well want to let everyone go at their own pace, but the external requirements forbid him from implementing that way.

When you let reading happen naturally, for example, some kids don't start until in their teens.  Flexible timetables are not really part of the school curriculum.  I agree that what he is doing is an improvement, a vast improvement, over a list of problems/worksheets, etc, and I applaud anyone who is working to make what happens in a classroom closer to what would happen in a natural setting with uncoerced learners.

In other words, the answer to "how would I approach this" is that I would let anyone uninterested wander off :).

Ok, let me try a little harder.  Maybe have a game where you roll the glass from a fixed starting point and try to get it to knock something over.  Have a bunch of different sized glasses, and then after people have played a while ask them what they can say about strategies to pick a glass to hit a particular target.  And, seriously this time, I would expect interest to wane as stuff got more complicated, and I would let anyone not interested in going further drop off.  But if I did find someone that was interested, I wouldn't have any qualms about either the approach of showing them equations that would work or seeing if they can figure that out on their own.  The idea of wrapping paper around the glass and seeing the point of the resulting cone, I really like.  Kids could come up with a construction-type approach to make an accurate preditction--trace the outline of the glass, extend the lines until they meet, bingo, the radius.

Having that under their belts, then, later, maybe, when something else comes up with similar triangles, I might say, hey, remember that glass-rolling game?  Here's a way you can get the answer on a glass without bothering with the construction.

But I wouldn't expect a large fraction of a randomly-chosen population to be interested in this particular math, any more than I would expect them all to like a particular kind of music.  I would be happy to point people to music that they might like based on what I've observed, or even show them things they might not have considered but that with a stretch they may actually start enjoying, and open up a whole new vista to them.  It doesn't bother me in the least that some people don't like They Might Be Giants, even though they are practically the best band ever :).

So, yes, I'm happy that people are working to align the people in a coercive situation with the path their brain might take if it were not coerced.  I think in the end that will have better outcomes for the coercive classes, and, in an important way, make the coercive situations less coercive (if someone "makes" you go to a class you find intriguing and interesting, after all, the coercion reduces to near mootness).  As long as the coercion is standing by outside, though, you are going to find yourself bumping into it all the time, and that's just reality.  It was the starkness of that contrast that led me to post--how quickly he had to go from intrigue and interest to imposing the method of solution.

Even when you do have to push everyone in the same direction, I think it's worthwhile to make it as interesting as possible--it can, at least, give them a chance to see that there might be interesting things about math, which is a thought that has literally never entered into many minds.  That would be a wonderful step in the right direction.

mike

[Sue VanHattum]

Thanks for the great detail in your reply, Mike!

I teach at a community college, because I love teaching and I love math. But of course the students don't generally like math much, and do feel coerced. They're not coerced in the same way they were in high school, but if they want to transfer to a 4-year school, they need the equivalent of algebra II (called intermediate algebra) plus one "college level course". And to even get an AA degree, they now need the intermediate algebra.

So I face the dilemma you describe so well all the time. I love this, but why should they?

Warmly,
Sue

[Maria Droujkova]

On Thu, Jun 11, 2009 at 12:50 PM, Mike South <mso...@gmail.com> wrote:

In other words, the answer to "how would I approach this" is that I would let anyone uninterested wander off :).

But I wouldn't expect a large fraction of a randomly-chosen population to be interested in this particular math, any more than I would expect them all to like a particular kind of music.  I would be happy to point people to music that they might like based on what I've observed, or even show them things they might not have considered but that with a stretch they may actually start enjoying, and open up a whole new vista to them.  It doesn't bother me in the least that some people don't like They Might Be Giants, even though they are practically the best band ever :).

Even when you do have to push everyone in the same direction, I think it's worthwhile to make it as interesting as possible--it can, at least, give them a chance to see that there might be interesting things about math, which is a thought that has literally never entered into many minds.  That would be a wonderful step in the right direction.

There is more discussion at that blog post. I wish I could just plop that whole discussion here in a widget... well, in five years or so, right? Here is the link meanwhile http://blog.mrmeyer.com/?p=4018

I've been thinking about "lowering framework on students," Mike. I think the point is right there. It's not about selecting what particular type of learning content each student wants to consume. It's about realizing we are offering this producer-consumer roles in the first place.

In one of the studies I read today, there was data that lower socioeconomic classes, ages 12-18, are more likely to author digital content online than higher classes. Isn't this striking?!

Today, I wrote a long reply to another post on dy/dan blog, called "But How Do I Remediate THAT?" Dan writes, in part: "What I'm saying is that, when I play, for example, this fantastic loop of time lapse photography, my Algebra 1 students sit a few millimeters closer to the edges of their seats and lean a few degrees closer to the screen than do my Remedial Algebra students. They call out observations and deconstruct the movie in ways the remedial classes do not anticipate. In general, they seem eager to engage the unknown whereas my Remedial Algebra students seem to prefer that the unknown stay unknown, that life's unturned rocks stay unturned. This bears out even between sections of the same course. The length of a class' discussion of show and tell media correlates positively to the class' average grade."

Here's my long reply. I think it relates to the "lowering framework on students" conversation.
---
I want to make two closely related points.

1. Intellectual consumption may not be the universal good.

2. Beauty/interest/learning connection is a huge can of worms, too.

"How do I remediate that?" assumes "that" is a bad thing. It may be, but let's be clear that a value judgment and an assumption have been made right there. Ditto about high algebra grades.

Narrowly, the goal of a math teacher is to help students learn math, as measured by the people who hired the teacher. Broadly, as measured by the teachers' understanding of what math is about. It is easy to assume that reaching goals toward which we work is good in itself, especially if goals are so darn hard to reach. We got to examine the goals, though, to determine their own value...

This understanding of what math is about is cultural. For example, in a very bitter article "A Russian teacher in America," Toom says: "It is a most important duty of a teacher of humans to teach them to be humans, that is, to behave reasonably in unusual situations." http://www.de.ufpe.br/~toom/articles/engeduc/ARUSSIAN.PDF Then there is a girl Susan observed, who said, "I once asked one of my remedial students who worked at a local drugstore if she didn’t find her job boring since she had been working there for three years. I was surprised by her answer. She said, not at all. She said I like that I know exactly what I am supposed to do every day, and that there are no surprises."

According to information processing theory, human sense of beauty is based on seeing similarities (pattern, fractal, repetitions and so on) that reduce our information processing load. We are rather weak on memory and some other information processing capacities, so anything that reduces the load is highly valued. "Interest," in this theory, is rather poetically described as the first derivative of beauty, equivalent to learning. Something is interesting if it has a high potential to be compressed by the beauty algorithms (i.e. learning). http://en.wikipedia.org/wiki/Mathematical_beauty#Beauty_and_mathematical_information_theory

And that leads to Stevehar's point about "spectators" - except I think these learners REFUSE to be spectators. They don't want to consume/connoisseur mathematical beauty - similarity, fractality or pattern. They don't want interesting/surprising things, those onramps to learning, because the compression process (learning) is the derivative of the unwanted beauty consumption process.

Maybe instead of trying to offer different content for intellectual consumption, we can offer activities that aren't consumption. Then the two cultures can meet and make friends during co-production.

http://blog.mrmeyer.com/?p=4024