31 January 2012

Algebra without Numbers

This post began with this Tweet from a high school math teacher (and "Ed Leadership" student)
Bored in Math Class
"Had 8 out of 9 stdts not complete an Alg I test 2day. Said we never did it b4. Unit started Jan 2. Had flash cards and cheat sheet. What now"
I responded
"Algebra is a method for finding unknowns from knowns in a logical way. You could use numbers, or real things...  mysteries are solved through algebra. Kids don't get it because we disconnect it from reality"
The answer?
"or because they don't do homework, take notes, participate, or pay attention"
Me again,
"I always say, kids make rational micro-economic decisions. If they see no value in the course, they will not invest in it"
And this response,
"then maybe they will see value in it when they take it again next year"
followed by
"I shouldn't have to reteach because they were to lazy to try or participate this time. It sucks."

Obviously I could write about many things here, from public disrespect for students to a bit of unfortunate egocentrism ("I shouldn't have to reteach"), but I'm going toward the math here, first, repeating an old joke...
A man walks into a pub with a dog and orders a pint. The barman says, "you can't bring that dog in here." But the man says, "This is different, this is a very special dog, I taught him to sing Grand Opera." The barman is impressed, "Well," he says, "lets hear him sing." The man pats the dog on the head, says, "Sing!" and the dog begins to howl. "I thought you told me you taught him to sing Grand Opera," the barman says. "I did," the man responds, "he just didn't learn."
"Teaching" means nothing without the learning. And in order for learning to take place, whatever is to be learned must be accessible, and attractive (in some way), and must occur in an educational space where students are physically and emotionally comfortable enough to allow for the cognitive discomfort which opens the pathways in the brain.

How does your classroom meets the needs of the teenage brain?
Teens make rational decisions, microeconomic decisions, about
what is worth investing in... Ever watch teachers at a boring
Professional Development Day?
I do need to note that I did try teaching in the exchange above, I offered many links, including these:
Real World Algebra: Budgeting and Lifestyle Design.
Algebra in the Real World Movies
Karl Fisch Student Algebra Blogs
Learning Algebra on the Right Side of the Brain
The Changing Way That Math Is Taught To Children
but clearly, I failed. I was not, of course, meeting this teacher where he or she was. First, this teacher really wasn't looking for pedagogical learning with the first "question," just as many of the students probably aren't looking for equations to learn when they enter that classroom. The teacher wanted to rant and complain about "the world being insufficiently cooperative," which is often an idea filling the teenage brain as well. Second, the teacher was not ready to make any big leap in terms of teaching. What was wanted was a method of forcing students to comply. By offering new kinds of lesson ideas, or alternative ideas to run a classroom, I was creating the same kind of disconnect between interests and curriculum which is clearly occurring in that classroom.

What I was not doing was attracting this teacher to new ideas. Which is what isn't happening in that classroom. And we can not get students interested in investing in our curriculum unless we can attract them to it, any more than I can get most teachers to give up their evenings to reading the works of Edward Said, Antonio Gramsci, and Michel Foucault, no matter how essential these works are to teaching all students effectively. So, if I want either to happen - the kids to come to the math or the teachers to come to an understanding of power - I have to "teach and reteach" trying this and trying that, looking for the hook that works. Going out fishing with just one kind of bait or lure can result in disappointment.

I cannot sit through a TED talk - rehearsed stage lecture plus PowerPoint will never hold my interest, but obviously it works for some folks. But give me a decent football (soccer) game on television and you not only have me for two hours, you can get me into everything from mathematics to culture. Put a Khan video lecture in front of me and I'll be a behavior problem, but ask me to find three different ways to explain the same thing online, and you may have your most engaged student. It all goes back to the "amazing" statement I almost always hear - stated as if it is a problem - in IEP meetings about boys, "He pays attention when he's interested in something." Duh, yes, I think the vast majority of us do.

Relevance... Why would anyone learn this? Start with an attractive purpose
and the work behind it becomes worth investing in.
Algebra without Numbers, Algebra without Computation

The idea here is that instead of presenting Algebra as a system of mathematics which is essential to learning some other things kids probably aren't interested in, or cannot imagine why they would be interested in, we present it as what it truly is: a system of formalized problem solving used to discover an unknown from knowns. It is, essentially, detective work, and we must let kids understand that. The numbers in Algebra are incidental, the concepts are important.

And I am suggesting that it is often essential to start without the numbers, largely because of the damage done to the interest in mathematics by mathematics education before kids get to algebra. Schools work so hard at making mathematics boring, disconnected, almost absurdly repetitious nonsense, that by the time they walk into an algebra class, they do so with a combination of dread and disinterest. If you don't break through those walls first, you might as well call in sick for the year.

Philo Vance on film, with William Powell as Vance, in The Kennel Murder Case (1933)
I've been falling asleep to Old Time Radio shows lately, and one really hokey old drama, Philo Vance, a detective series from 1948-1950 has been pretty entertaining. As I thought about Algebra I thought about an episode I heard last night, The Vanilla Murder Case (clicking will let you download the mp3), and thought about using this in an Algebra class.

The murderer, "X" equals one set of facts (the whole of observations) minus another set of facts (the irrelevant).

So the class might begin by listening (which will be a bit more universally appealing than reading, but a little complicated for a generation not typically raised on audio-only storytelling, and thus, a little challenging) and collecting all the observations, as those observations begin to build up, some will begin to slide into the "irrelevant parentheses," others will be added together.

This is not a "side path" on the way to algebra, this is the path to algebra. The trick to algebra is understanding that formalized, logical thinking can help make sense of the world. That kind of thinking begins with attention to the issues in question. In the Vanilla Murder Case, in any mystery, there are many facts. Some facts cancel out other facts as we add them together. Some facts multiply the importance of other facts. If this does not sound familiar, you've been teaching arithmetic instead of mathematics.

We do this kind of "algebra without numbers" all the time. How do we decide who has not shown up yet? X = (all the people we expected to show up) - (all the people here) What is the best time to meet for this movie? X = (the people who can show up at this time) - (the people who can show up at that time) / (who are the people we really want to be there)

In other words, we are not just giving experience in this formalized structure of data collection and data assembly, we are proving that far from being some worthless foreign language, algebra is a basic part of our lives.

Once Algebra is something of value, the microeconomic decisions of the students change. Once you have brought the "cost" of engaging down for students, their microeconomic decisions change, and then, classroom behaviors change.

Next, you may want to bring coding into class, so that "right" and "wrong" and replaced with "works" or "doesn't work," which makes a whole lot more sense to most of us anyway. "It turns out that programming is just so much fun that students can't help but get engaged, which is a far cry from what usually happens in math class. Sure, we can make math fun with activities, and once in a while you hit upon a topic or a problem that kids are naturally drawn to. But much of the time I would loosely equate teaching math with pulling teeth, and programming couldn't be more different."

If you'd rather listen to a TED talk lecture than me, here's Conrad Wolfram...

"Conrad Wolfram says the part of maths we teach -- calculation by hand -- isn't just tedious,
it's mostly irrelevant to real mathematics and the real world."
"Right now in schools we're spending 80% of our time teaching students to do something by hand which computers do much better and faster. Calculating used to be the limiting step, but now it isn't. Computers have liberated math from calculation," Wolfram says.

In other words, concepts matter. And we need to change everything we do in mathematics classrooms.

- Ira Socol


Amy said...

I've been reading your blog for a while - first as a parent of a child with a disability, second as a graduate student in Cultural Studies. You are so right about engagement being critical to learning. A couple of years ago, I invited a teacher who had been commenting on my daughter's lack of focus to observe one of her horseback riding lessons. At the end of the 45-minute lesson, during which my daughter's attention never strayed, the teacher got my point. The school year went much better after that.

Thanks for the work you do - it's so important!

Cynthia Arrington said...

My name is Cynthia Arrington and I am a student in EDM310 at the University of South Alabama

As a Leadership Instructor for Freshman in an intercity high school, I clearly understand the need for innovative methods of teaching in a classroom where the young minds are in a cloud of cyberspace. What would you suggest if the school's budget limits the use of technology in the classroom?

Malyn said...

I love your approach and it reminded me of how I introduced Algebra to a class who groaned when I announced that it was going to be the next topic. See point 2 of this post.

Like you, I got them to listen. I used a youtube video but did not project on-screen. I wanted the students to detect the patterns in the music. I wanted them to see how these patterns helped me learn to play the music. I wanted them to see that Algebra is not just about numbers, it's about patterns and that Algebra itself is a representation (or manipulation) of such patterns. I consider it one of my best teaching highlights when more than half the class at year's end said that they loved Algebra....it was introduced to engage them.

I've also written a couple of posts re: Wolfram's computer-based maths but this comment has gone long enough so have just posted links.


Wm Chamberlain said...

Glad you are still enjoying old time radio mysteries :)

I can't believe we are still teaching content in a vacuum. I think the real problem is we have created artificial barriers between the content areas that cause students to fail to make connections. It is no wonder kids don't learn algebra when they are not given any real world applications or when it is made relevant with their background knowledge.

I am hoping to have a self contained classroom next year and my goal is to integrate all content into a single story. If all goes as planned I will be teaching fifth grade and using American history from pre-colonization until Reconstruction. The math possibilities are awesome. For example, I would use General Knox's expedition to bring back the canons from Fort Ticonderoga to teach simple machines (using a 10 foot long telephone pole). Naturally there would be a whole bunch of math involved including measurement as well as designing and building some kind of cart with wheels to move the "canon". This will be a real hands on application of math that is not only relevant to the narrative we are using, but also could be very valuable in the "real world" because they will be able to identify applications outside of school.

Em said...

My mother (you know, TravelerBlue) used to give us 'food math' to introduce concepts of multiplication and division and simple algebra. If there are fifteen tacos and five people, how many does each person get? It encouraged us to set up the problem in our mind and work through it logically. The best thing my calculus teacher did was make the proofs for each new concept really interesting. I don't remember what examples she used, but she explained how it worked and that was really helpful. To this day, I don't remember any of the trig functions I was supposed to have memorized, but I can go through the logic of how to derive them (if I have to).