I welcomed the luxury of spending time away from the classroom, to read, think, talk, and write about teaching math.Kate Nonesuch, More Complicated Than It Seems
I was awake early. And so, like many of you would, I took the time to cue up some April Wine and think about math. In particular, I was thinking about how 6 equal-sided triangles make up a circle, and wondering if that has anything to do with 6-sided figures being the easiest to pack closely together.
You know, thinking about math.
Forever ago, when I was learning about blogging and mathematics at the same time, I decided to create a math blog. Not a "teaching math" blog, you understand. That would be something different. I just wanted a place to spend time, well, reading, thinking, talking, and writing about math.
New to these worlds - to blogging, but also to math - I went looking for other math blogs. What I found was disheartening.
On the one hand, there were many teaching-math sites full of good ideas and bad ideas. On the other, there were "advanced" math sites that used words and symbols I didn't and don't understand. No where could I find conversations about ordinary maths.
(By the way, this seems also to be the case for books on mathematics. Either they are about complex, high-end maths, or they are about teaching math. Oh. And then there is a third category of books that are about patterns and stats - part of math yes, but, um... only part. More about that later.)
Hey! Have you guys read Kate's stuff on teaching math? Stuff like Family Math Fun! or Changing the Way We Teach Math? I just ask because your time might be better spent with those resources. 'Cause, these days, I'm feeling especially grumpy about talking about math.
The proximate cause was a 2010 National Institute for Literacy paper called Algebraic Thinking in Adult Education written by Myrna Manly and Lynda Ginsburg. Manly and Ginsburg are usually sensible sorts, so I was excited to find the paper online. Then I read it, and bore the crushing disappointment badly.
How to begin.
Should I start with this:"According to the National Survey of America’s College Students (American Institutes for Research 2006)... 30 percent of students earning two-year degrees and 20 percent of those earning four-year degrees... were unable to complete such tasks as calculating... whether their car has enough gas left to make it to the next gas station"? I guess college campuses must be stuffed with out of gas cars. Or maybe the football team just shoves them to the side of the road.
But here's the thing: if this paper is going to be about the math skills of college students and/or how secondary schools do or don't prepare them, it isn't a paper that addresses my role as a facilitator of adult basic education. This isn't the reason I visit The National Institute for Literacy's website.
Then there is the authors' focus on long-term job preparation. They point out a (pre-crash) 2005 U.S. Department of Labor paper that promises jobs in biotechnology, geospatial industries, health care, financial services and "the skilled trades" to people with the right "mathematical knowledge and skills... who are comfortable with ideas and abstractions, can adapt flexibly to changes, and can generalize and synthesize."
Manly and Ginsburg ask, "What kind of mathematics instruction for adults would enable them to meet the demands of this predicted future?" I sometimes wonder what kind of mathematics instruction will enable us to meet the twin collapse of the U.S. and Chinese economies after a joint failure to innovate in the face of runaway global warming. But, more to the point, I'm not trying to prepare adults for university in order to address predicted labour demands. I'm trying to help them reach their immediate learning goals and, sometimes, find a job.
To their credit, the authors were brave enough to ask the age-old question, "Why do we have to learn this?" in a section titled Why is Algebra Important in Adult Education? Sadly, their answer boils down to "to satisfy formal academic requirements for advancement," which is pretty much what I say. Except I say, "So you can pass the GED test."
Knowing that this is cold comfort, they hasten to add, "as well as to meet the genuine skill demands of home and work." But I don't believe this, and I doubt many others will.
Their argument, at length, is that algebraic thinking or "the more sophisticated understanding of mathematical relationships imparted by algebra" is "useful in navigating life’s decision-making challenges." The "the skills and procedures of algebra are important," they say, but less so than "a genuine grasp of the mathematical concepts that studying algebra provides."
Quantitative demands can be both complex and pervasive, reaching into an individual’s daily life as citizen, worker, parent, and consumer. Often it is the big ideas of algebra, not the procedural details, which people draw upon when making wise decisions.
In the workplace, workers often are not aware when they use an overarching mathematical concept. For example, researchers have shown that the concept of proportionality was applied in nearly every workplace they studied (Hoyles et al. 2002; Marr and Hagston 2007; Selden and Selden 2001). Nurses use proportionality when determining the correct dose of medication, and cosmetologists use it when mixing solutions, but few recognize that they are using “school math” because the mathematical ideas are so deeply embedded into the context of the job. Adults often say they have never used the algebra they learned in school. That may be true for the rote aspects of manipulating symbols, but they likely are using the mathematical reasoning and problem-solving aspects of algebra unconsciously.Let's set aside the "unconsciously" (which is frightfully close to "instinctively"). There's surely a fallacy here. The other day, someone asked me how to determine the mechanical advantage of a particular first-class lever. I didn't know, even though I've been using these levers effectively since about the age of three. Why didn't I know? Maybe I wasn't told. Maybe I was told but didn't care. Probably because knowing the "overarching... concept" wasn't a necessary part of getting the job done.
I'm aware there is a fertilization process involved with corn that relates to both the physical structure of the plant and the ground and air temperature the plant encounters at a certain stage in its development. I don't know much more than that, despite having grown lots of backyard corn as a teenager. And I didn't know it when I was growing the corn (I learned it later while reading about the impact of climate change on agriculture). What was important, in those days, was effective tactics for keeping the raccoons out of the corn patch.
What I'm saying is science or math theories and theorems may underlie our daily practices, but that doesn't mean we know them, or want to know them, or need to know them, or will do a better job for knowing them.
Mostly, that kind of academic knowledge is useful at test time.
So, anyway, I read this, filled the margins with complaints, realized I had only made it to page three, and thought, "Rats!"
Then I glanced at the clock, and saw we had 20 minutes left to the morning. Looking about, in a fit of pique, I caught some helpless learner's eye and demanded, "Do you want to learn how to do algebra right now?"
"Do you want to learn algebra!?!"
"In the next twenty minutes."
So I took twenty minutes and taught him. He was pleased enough with the whole thing to ask for homework, which I (feeling smug) supplied.
The next day, after another couple of hours of algebra, he said, "I gotta tell you."
I used to really, really hate math. Like, before this... like now I don't mind because of the way you showed me. But I used to really hate math. But now I don't.
Which was all a bit embarrassing given my somewhat shady motives in the whole affair. But, what are you going to do?
Anyway, I've only reached page three of this document, and I suppose I should carry on, but maybe I shouldn't either.
I could go back and re-read Kate's stuff.
Or maybe I'll just draw some 6-sided shapes and triangles. You know, I was thinking about them this morning - hexa-whatchamacallits. I was thinking about them because I'd read in an essay on evolutionary patterns that these shapes packed more tightly than others. And I remembered learning about how you could divide a circle up into 6 triangles. I was listening to Electric Jewels and wondering if the 6's were a coincidence.
Sometimes I think about stuff like that.
Read and think and talk and write about math.