When I think about student engagement, I always try to compare where I was most engaged (anything with problems and critical thinking) to where I was least engaged - Math. That's interesting, because I had some engaging teachers who taught me math, and a number of my friends are Math teachers. There must be something there when so many people I like and respect use so much of their time and energy to work on it...
For years, I pegged my low potency in Math to my dyslexia. How can you feel able when 6 and 9 or 3 and 8 look the same and you are always getting basic arithmetic wrong? When you are always making mistakes, I told myself, it is hard to be engaged. Yet as the same time, I actually sought out other things because they weren't easy. I loved English and History, although my writing was full of spelling mistakes that never failed to illicit comments from my teachers about how I did it wrong. I also really liked all the sciences, and liked the applied Math I found in them.
I have been reading Dan Meyer's blog dy/dan since meandthedoor sent it to me. She sent out another link to Dan's TEDx talk and I watched it. I think Dan has it right on the money. I have been gradually realizing that I didn't hate Math because I made computational errors, I hated it because I didn't see it as problems or creative thinking. This is a little ironic, as my father (yup, mathematician) and one of my best friends (physicist), are always "going on" about how math is the language of all the interesting questions in the world.
I use my mathematics education all the time - basic arithmetic, but also geometry when I build things and statistics because they interest me. Permutations and combinations were my favorite unit in all of my K-12 math education because they were about prediction and I could see how I was using them to problem solve choices I was making playing cards. I guess every time my math is problem solving I like it, but in my education it rarely was. Even word problems were just identifying the formula and subbing in the variables.
I know the many good math teachers I know are not surprised that students don't feel they can do it and just want a formula. I was always way better at memorizing the formula and doing the evens or odds in the text than I was at understanding the why. But understanding the why is what would have made Math relevant for me, and I would have persevered through lack of potency if I was really thinking critically. I just wasn't doing much thinking most of the time.
Dan talks early on in the video about how textbooks are a big part of the problem. They were for me. Since they gave me only the variables I needed, they were like a badly written mystery. You know, the kind where the only minor characters you meet will be significant later? You don't do any thinking there, you just sub in Mr. Scott in as the murder. If I had been figuring out what I needed to know and excluding irrelevant information, the why would have been automatic, as would my sense of engagement. I have always liked a challenge, but math just seemed to be low level thinking.
As my husband is teaching math to our daughters, I can see that texts are starting to change, but the problems Dan identifies are still firmly in place. I'd sure welcome some insights from my math loving friends on this one. Is Dan right in your experience? How do you think we should teach math? I have always liked a good problem - and maybe it is worth starting to do the math.
When I was in school the pattern that was the most effective was:
ReplyDelete1) teach me how to use the tools i needed (getting the hang of new principles, not sure how it could be done without some boring wrote practice)
2) show me how I can use those tools to solve real problems, and that's what assignments needed to evaluate.
If you skip step #2, it goes in one ear and out the other, for me anyway.
Sigh. I know our current math curriculum makes no sense. It is relevant to no one outside the field of "school mathematics." I used to have this great song and dance I did about how math was much like going to the gym for your brain - not particularily fun or relevant, but really good for your body/brain. I don't even believe myself anymore (for the record, I also don't go to the gym anymore either).
ReplyDeleteOne of my many problems, is that I grew up loving school math. I still love school math. I know it's useless, I know it's boring to a large portion of the population, but there is something about the nature of the work that I enjoy. When I was young my parents bought me workbooks to do at home and I would sit for hours and do math questions. When I took an extra calculus class for my accreditation a few years ago, I did way more than the required homework to ensure I was understanding all the nuances of potential questions. It was really frustrating at the time, but also very rewarding for me.
I am a terrible (and terrified) math problem solver. Going to math PD will often give me anxiety because if you give me an open ended problem I rarely have an idea of where to start. All these brilliant minds around me will be throwing out crazy guesses and arguing with one another and I sit quietly wishing I grasped half the math savy of these coworkers. I'm fairly sure if you polled current math teachers, there would be a lot just like me.
I'm not sure if I had more exposure to "real" problem solving if I would be more comfortable. I do know that nowhere in my mathematical education was it ever acceptable to have any other answer than the "right" one. I was never asked to solve a problem my own way or explain my reasoning until my first master's course last summer.
This is troublesome for a few reasons. I'm not super sure how to model or guide a process that I only started really understanding a year ago. Maybe this is actually a great thing, because when I force myself to solve problems I make all sorts of mistakes and it's likely really good for my studetns to see this (but it's hard). I'm also not 100% sure how to construct these good problems. I am starting to find resources for them, but my creativity here is lacking since I'm not a creative problem solver I'm also not naturally a creative problem inventor.
I do recognize when I've gotten it right though - on days when all my students walk out of my room still trying to figure out "the answer." The teacher high from those moments is way greater than the one you recieve sharing individual successes with students, and that tells me all the hard stuff is worth it.
Interesting - I always figured we didn't do real problems because we were doing textbook (school math). It never occured to me that the skill sets are not the same. As you've been working on real problems, has it informed what you do in school math? When the students are walkign out the door still talking is that also happening with well taught school math?
ReplyDeleteI think as the new Sask Curr. role out, many math teachers will struggle with the exact issue you suggest.
Physics has exactly the same issues with how it is taught and it drives me crazy.
ReplyDeleteThe plug-and-chug approach where students are trained to match the word problem to a formula and then whack buttons on their calculator is absolutely horrible. It's everything that science is not...
From a functional perspective, math is like Lego, clay, or a sketchpad. It's a modeling device -- a way to think about a problem, lay out your ideas about how something works, and construct a virtual working model that you can chase into the unknown.
The language of mathematics is the single most stunning work of beauty that our species has developed. It is literally the collaborative work of untold generations of people working together over thousands of years to expand, polish, and explore what is, at its foundation, a common expressive language with shocking power. No other creative work of mankind is even in the same league. Yeah, I know how I sound, but that /is/ what lies at the soul of mathematics.
Troll the math department for an invited speaker. I guarantee there are a one or two grad students and/or profs that:
a) are passionate about their subject, and
b) can effectively communicate that passion and give kids a glimpse of what mathematics _really_ is.
For example, have someone sketch out the construction of the real number line from the basic axioms. Done properly, it's akin to breathing on a window in deep winter and watching the mist crystallize into a zillion snowflakes on the glass.
I suspect a lot could be learned from successful (verbal) language teaching techniques too. Mathematics is referred to as a language for a very good reason. It is interesting, and perhaps quite telling, that I would describe my experience learning French in ways that mirror how others describe math and physics class. I scored well in my classes, but it was by memorizing verb conjugation tables, lists of nouns, and rules for plugging disjoint fragments of french in the right order to get the right answer.
Only in my final high-school year did I have a french teacher that took an immersive conversational french approach. She also took time to describe and teach some of the French culture. This included showing slides of her trips, having french-themed picnics, etc. (and always speaking entirely in french). She zoomed out and showed how the language connected with something that existed outside of french class. That's six years of french if I count primary and secondary school, and I got one decent semester. I sometimes wonder if that teacher was a total anomaly, or if she represented the leading edge of a new teaching approach... Perhaps I digress.
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One of the difficulties with teaching mathematics is that all of the interesting high-level stuff is constructed, literally without exception, from the tedious(?) low-level stuff. There really aren't as many low-level details as many think, but getting them right is extremely important. It is a little like building a mortar-free wall from clay. The bulk of the time is spent working on the bricks -- a pretty boring task. But, if you're sloppy with the bricks you'll never get higher than a few rows before the whole construct starts to fail. Math (and physics) share that feature. If you don't understand the basics you will never progress beyond a certain point.
ReplyDeleteThe conventional approach (in my experience) is to have the student drill on bricks for 5 years, and follow up by constructing a bunch of flat, featureless, boxes and walls. The boxes won't hold anything, and the walls don't support anything. They're just exercises to prove you can stack your bricks.
Move back and forth from a high level overview of how the bits you're teaching connect with the whole. Confront the students with a physical problem that students can grasp: launching a water-rocket, building a model bridge (out of pasta or balsa), running a casino, etc. Then zoom in to the details and run through the geometry/trig, math/calculus, combinatorics/probability needed to make it fly farthest, hold the most weight, make a guaranteed profit. Collaborate with the physics class. Talk to mathematicians who really know their stuff -- there is a lot of beauty in pure math. I have no doubt that engaging teaching sections could be constructed from pure geometry and basic algebra. And talk to the language teachers -- that could be really
interesting.
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ReplyDelete@Brad, So are you going to come to the next conference? Sounds like you're just the type of person to give the talk you were describing ;)
ReplyDeleteThoughtful posts, Brad. The bricks metaphor is a good one for thinking about it. I think the zooming out helps, especially when I am doing something with my hands and can visualize it.
ReplyDeleteSo here are a couple of questions I have been thinking about that trouble me:
1. If in 5 years I get to do more than make bricks, why are my 11 and 10 year olds still doing only that? How could I as the average teacher with okay understanding of Math help my students to love and understand it rather than engage in rote activity?
2. In a world with Google, I question the value to testing the date of the Battle of Hastings. I am also starting to question the value of automaticity outside of some critical thinking. Do you think 5 years of automatic brick making is esential child labour where math is concerned? Can we do somthing else?
One of the ways I know if I am teaching somthing well is if those who do not have natural apptitude start to like the subject or start doing well in it. Sometimes I think that the way we teach math precludes the understanding that it is a powerful language unless the student came into the class instinctively understanding that.
I'd really like more of our students to feel the way you do about Math and less of them to feel the way I did.
I must have been unclear about the brickmaking. While I understand how the method became established, and recognize the value in (some) drilling, I do not encourage 5 years of labor before students can do something interesting. As you say, that results in a lot of students coming out hating 'math' because all they've seen is the drudgery. I would hope that having experts come in to reveal some of the beauty and integrating applied math to highlight the utility would help with that issue.
ReplyDeleteI can't see drilling disappearing completely though. Math may be a language, but it is a demanding one. If I am sloppy with the syntax, then I am guaranteed to fail. 'Pidgin' math is a broken tool. That said, there are many ways to interleave the drilling practice with applied practice. I'm also curious about how natural language teaching technique has evolved. Historically that involved a lot of drilling too -- perhaps there are more successful approaches now.
I would really like to see more teaching units where the math is used to explore and support a 'real-life' project similar to the ones I mentioned. I do understand the difficulty behind developing such projects so that students and teachers don't get left behind. There must be pre-written textbooks or equivalent around that provide, or could be adapted to clearly lay out, such a project step by step. Hitting up the university physics (or math) department might be useful. This kind of development work would also seem like an excellent training project for upper-year college Education students. Once a unit is identified or developed, I would suggest dedicating PD session(s) to training math/physics teachers on that specific unit.
[I realize that this is all much easier said then done...]