Introducing two more relevant references, both having similar cultural following to the works mentioned in the article:

“Red Mars” trilogy with the idea of science-driven, purposeful socioeconomic change

and

“Here comes everybody” with the idea of cognitive surplus, measured in Wikipedias (100 million hours of volunteer work) http://www.herecomeseverybody.org/2008/04/looking-for-the-mouse.html ” When a TV reporter interviewed the author about social media, he writes, “she shook her head and said, “Where do people find the time?” That was her question. And I just kind of snapped. And I said, “No one who works in TV gets to ask that question. You know where the time comes from. It comes from the cognitive surplus you’ve been masking for 50 years.”

Studies show watching TV (or YouTube videos) does take a lot of brain energy. The theory where “learning is the derivative of beauty” comes from may explain why. “Refuseniks” (funny word) who don’t spend their cognitive surplus in this manner aren’t necessarily to be helped out of their state. Maybe we can guide them toward some open road they vaguely long to take.

Here is an interesting piece of data. Despite lack of fast internet, software and good computers at home, poor kids are more likely than rich kids to author and share content online (page 10 of this report http://www.pewinternet.org/~/media/Files/Reports/2005/PIP_Teens_Content_Creation.pdf.pdf). What can we make of it?

Regarding spectators vs Maria’s REFUSEnicks.

Seems like my engineers spectate from ignorance, whereas Maria’s learners are much more caught-up in an aversion to math or maybe an aversion to the whole enterprise.

Possibly Maria’s refuseniks are headed elsewhere, not “fast-math” but “slow-food” & “slow-math” and craft; which might be a thing of beauty too…[see the New Yorker reference]

My some-time taxi driver in San Jose talks excitedly of his son’s enthusiasm for math and something like tool design and getting his hands dirty in practice work sites, and what’s going to happen when he graduates into his craft.

Father & son both love the open road: Pura Vida is what they say here over and over again, which translates poorly into English as… Pure Life.
—
See
Shop Class as Soulcraft: An Inquiry Into the Value of Work which is reviewed in the current New Yorker: http://www.newyorker.com/arts/critics/atlarge/2009/06/22/090622crat_atlarge_sanneh?currentPage=all

“Interest,” in this theory, is rather poetically described as the first derivative of beauty, equivalent to learning.

I have no idea what that means or if it has any practical implications, even, but “the first derivative of beauty” is a helluva thing to ponder. Thanks.

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.

Here is my take so far…

I read from the point of view of a guy who in a earlier career taught engineers in factories to use math [applied descriptive statistics] to find solutions to problems embedded in real concrete “stuff” with cause and effect analysis, frequency distributions and average and range charts.

Every time I started a class, to a person, each new engineer hated going. They didn’t want to go to another math [factory statistics] class.

They would say: “Steve, we had statistics in engineering class, I got a great set of class notes.”

It stumped me for a long time – how to engage them.

One day I retorted: “let me ask you this, if you were at a Dentist getting a root canal and just before the laughing gas you asked the Doc how many root canals you’ve done and he said – well none but I got a great set of class notes. Would you get a root canal done from that Dentist?

No? Well, what we teach in class is factory root canals.

Let me see your last set of notes, how you addressed the problem, distinguished cause & effect how you made a testable hypothesis, how you collected data, let’s see your calculations and charts. They’d say – well we didn’t have to do THAT, we just had to use the formulas and crank the software program.

The point of that story is it is about handling reluctant learners and inviting them into the game.

Although they were skilled practitioners in engineering school – the engineers are novices in a factory that applies math to raise quality and reduce cost for the company.

So…I noticed that in new learning environments novices are mostly “Tell Oriented”; but in a familiar learning environment practitioners are almost always “Question Oriented.”

So?

First, it seems to me “remedial” is a little harsh – maybe reluctant would be more helpful. These learners seem to me – as you describe them – to be spectators watching the math game; they are not IN your math game, and some have never ever been in a math game, period. How do you know they are “remedial”?

Second, most of this conversation has been a rather lively Shared Inquiry among practitioners who are “in the math game”.

A portion of this conversation has been an Advocacy Dispute involving displays of credentials and positions, challenges and back and forth verbal debate + sometimes a little snarky sharpshooting.

To my view none of these pairs:
– Inquiry or Advocacy
-Ask or Tell orientation
-Novice or Practitioner
are good or bad strategies.

Where the heat comes from and what lowers the quality of the conversation is missing the signals in the game. People are listening for inquiry instead they get advocacy or visa versa. They miss the signal calls. They run terrible plays.

This is a little like suiting up for football and showing up for a baseball game.

It is the basic root question about learners: how do you get people to suit up for the right game and start playing by the rules?

In the factory my goal was to teach engineers to make terrific “root canals”…

…but one time when my younger daughter Margot was in a 3rd grade math class the class made “stone” [chicken] soup, did taste testing, did frequency distributions of the ingredients and colored the frequency charts posted next to the taste tests…but that is another story

Please don’t stop the conversation now, seems like you were just getting started…

When students learn a new concept or idea about once every day or two, there is this sense of “surprise” on a daily basis. Students may feel that they don’t have enough experience on the first idea before we move onto the next one, because that is what the district pacing guide says we are supposed to do. Couple this with holes in their foundational knowledge, it is no wonder students feel so frustrated and often give up.

I will have to keep this idea in my head when planning for next school year.

How and when you use that information (which you try to update periodically) is up to you. You many never find anything useful for a given student; for some you may find useful things that somehow the right situation for never seems to arise. But the point is to try to get these resources as you can and think about them periodically: if opportunity presents itself, great.

Further, I never suggested or would suggest that it’s only about applications. I don’t believe that and doubt that anyone would get far with me trying to make that point. I’m speaking to a particular dilemma you face and one alternative to what you are doing (not instead of, but in addition to).

I have no definitive answer to your last question. If you are forced to do something prematurely, you have to scaffold that topic as best you can. And probably give it as minimal a treatment as the state assessment tools indicate you can. Unless, of course, you can find a way to connect that topic to what you’ve already done. I tried to make some suggestions along those lines in a previous post regarding issues of asymptotes, graphs, and ideas about rational numbers extended to algebraic expressions. But as you ask WHITHER RAES, I can’t help you there, at least in that my opinion is as good or bad as yours in that regard: if you HAVE to teach them, then you have to.

By meeting the needs of the students and giving them more time on task, could such sacrifices in curriculum pay out greater dividends later? What are such expected dividends?

What is the loss of forsaking rational expressions and equations if such students never enter higher level mathematics greater than Geometry? Would we be teaching students that enough is enough and just getting by is sufficient and that it is ok to avoid difficult learning?

Arguments can be made in both directions. Both have a sound basis and are valid. But as professionals, how do we decide? What is best?

Though much of my current work involves context-rich mathematics, not all mathematics pins itself to a neat context. Jason’s example notwithstanding (and, if I’m not mistaken, “work problems” also involve the harmonic mean) RAEs resist context like few concepts I have encountered in Algebra.

MPG entered this thread endorsing individual contact and real-world context – the wheelchair-bound grandparent, the real cost of leaving a lightbulb on. I won’t contradict him on individual contact, but I’m not sure that real-world context should be a primary motivator of our math curriculum. I see value, I guess, in teaching the abstractions of math, just as I see value in teaching the abstractions of philosophy, though they both may lack immediate real-world context.

MPG seems to be saying, “Take every student separately. Some need more context. Some need more abstraction.” I won’t contradict that either. But class sizes larger than three necessitate a formalized, adopted math curriculum. Which is kind of a drag but, yeah, I’m speaking pragmatically here. Whither rational expressions?