Tuesday, January 03, 2006

And a bit more about math...

I will admit to feeling a bit defensive* upon reading this comment:

This is why engineering schools are dominated by Asian students who know the theory AND the calculation process.

I mean, point taken, there are places where they manage to teach both the concept and the math, and probably in middle school, and clearly I do want my kids to compete with anyone out there. At the same time, they come to me with extremely weak math skills, which my school's math teachers are doing their darnedest to bring up to and beyond grade level. And there are many, many ways in which the science I teach supports that effort. We measure things all the time. I taught them to measure in basic SI units before they saw it in math. We compute averages, we describe relationships between variables, we graph early & often, we learn formulas and sometimes figure out how to manipulate the formula to tell us something different...


There are a limited number of days in a week, and I have found myself rather often bogged down in teaching the kids to handle math concepts and getting stuck on one topic for much longer than I intended or thought appropriate. Usually, it happens when I don't realize exactly how far back I will have to go to bring them up to speed on the particular math concept. In the end, I usually beat it into the them one way or another, but I sometimes feel like they lose sight of the science concept in all the backing up and practice math problems.

So I'm learning to pick my battles. There are times when the math is so fundamental to understanding the science that I must teach it. There are times when the math is just not that hard (F=ma) and worth throwing into the mix. There are times when the math concept itself is really important (precise measurement; graphing) or piggybacks nicely with what the math teachers are doing at that time, thus it is worth it. But there are times when I want to focus on teaching a science idea well, not taking twice as long teaching math algorithms so-so and science ideas so-so.

Personally, in middle school, I was taught all the technical stuff about simple machines, but understood none of it intuitively or deeply. I don't feel that I was taught how to think it through, just how to plug in numbers here and there. Now I am trying to sort out the best approach for presenting this material - mechanical advantage, efficiency, simple machines - so that the kids get it in their bones. Math will play some role, since we will do experiments to calculate the amount of force needed to lift something straight up versus up an inclined plane, and the distance over which it moves, and so on. I m still figuring out exactly what role.

Also, it's worth noting that mathematical thinking doesn't always need numbers. For example, if I introduce a formula and the kids can explain how it shows the ratio between two variables (what happens to force if distance increases?) that is a math idea, regardless of whether or not we put real numbers in.

Personally, I wish I'd been taught calculus with as few numbers and practice problems as possible, and far more graphs, descriptions, and essay questions. I have this sense that it is a beautiful way of describing phenomena, making predictions, solving problems on a theoretical plane. Later, the numbers and symbols can go in.

*Lately, I feel really thin-skinned in terms of what people are saying here. I think it's just me, being too sensitive for a myriad of personal reasons. Criticism is good. But please phrase it gently or at least directly, with a minimum of sarcasm. And remember that the comments are not intended to be a platform for stump speeches on your favorite topic. If every comment you make always has to do with the same thing, you can get your own blog for free at Blogger.


Blogger Polski3 said...

Good post. I recall trying to teach estimation while working with animal populations in Life Science classes years ago.....

Somebody correct me if I am wrong here, but don't US schools try to teach so much more "stuff" than many schools in other parts of the world? Could this be one reason many students from other nations are forging ahead while ours slink towards enlightened ignorance? When will it be time to get the politically correct crap out of our schools ?

9:17 PM  
Anonymous Muriel said...

Ms. Frizzle said :
Personally, I wish I'd been taught calculus with as few numbers and practice problems as possible, and far more graphs, descriptions, and essay questions. I have this sense that it is a beautiful way of describing phenomena, making predictions, solving problems on a theoretical plane.

I found your comment very interesting. Once again, it shows how diverse our experiences with learning maths can be and how differently all of our minds work.

I was taught calculus and all advanced math in a very traditional, learn the formulas and proofs and do exercises and problems, way. And I loved it. It all made sense to me and as of today I have a weird tendency to put any problem I encounter in equations to solve it.

It doesn't mean I was taught in the best possible way. It means I was taught in a way that appealed to my mind. And I'm sure many other students didn't enjoy the classes half as much because it didn't appeal to theirs.

All this to say that as a teacher, I have to be very careful to remember that even though I would have liked a concept taught in a particular way, this doesn't mean it's the most appropriate way for all my students. One of the hardest things we have to do is to find a way to cater to all the types of minds in our classrooms.

I haven't figured out how yet...

2:00 AM  
Anonymous Anonymous said...

polski3 -- I thought the US curriculum was lighter than other countries. I might be wrong. I feel there are so many distractions, pull outs, to a large extent mainstreaming kids who shouldn't be mainstreamed, etc. that public school teachers are constantly swimming up stream.

Ms Frizzle-- My mother, a teacher over 50 years ago, was talking about her education training yesterday. She wanted to be a high school history teacher but ended up teaching grades five and six - all the subjects. She knows she was (and is) weak in math. She said she knows she failed her students who needed help in math. Seems like nothing has changed. For students to have a good solid grasp of math before the get to 7th grade, seems to me the K-6 teachers need a solid understanding of the math (and science) they will be teaching.

Don't know if this helps or not --


6:17 AM  
Anonymous Anonymous said...

Math, the language of science, is no different than other languages in many ways. Fluency in math, like language, does not arise from knowing the names of its parts. Math, like speaking and writing, is best learned by doing.

However, students' receptiveness is another matter, and one that is too often far beyond the reach of classroom teachers.

8:20 AM  
Anonymous Anonymous said...

Ms. Frizzle,

I think your blog is wonderful, and although I have criticized it from time to time, the fact is that you're an amazing writer, thinker, and I'm sure equally good in the classroom.

I quit teaching a year ago, and I still read your blog because it was an inspiration to me when I was struggling in the classroom.


6:04 PM  
Blogger jonathan said...

Math is such an amazing problem in elmeentary education in this country, and in this society as a whole.

Where there are convenient opportunities, or where it is necessary, you teach the math. Good, and thank you, and all that.

The failing is real. But it would be foolish to hold middle school science teachers responsible.

Where you can, the biggest problems the kids have are fractions and ratios (one commentator seems to think ratios are simple; this is not true from the kiddies' perspectives.)

The bigger picture is not pretty. I've heard adults ask a numbers question and say "I never could do any math" and chuckle, and get a sympathetic "neither could I" in return. Can you imagine someone joking "I never could read" and getting "me neither" in response??

We keep shifting curricula between back to basics and concept-driven stuff. This has been going on for almost half a century, and lately the shifts have been coming more quickly (more disruptively).

In elementary schools we have many common branch teachers whose understanding is insufficient to teach the math curriculum du jour. We have many whose understanding is insufficient to teach any math at all.

And when our kids fall behind in basic computational skills, they are less able to cope with fractions, which are a major step more abstract (and when they come to us with mixed or poor fraction skills, this is usually a reasnable indicator of some limitations on their ability to do algebra).

So, science teacher, if you help out, great. And you really worry about this? Thank you. But we should accept that your subject comes first.


6:43 PM  
Anonymous Anonymous said...

As the previous poster noted, the biggest obstacle to math is the attitude of the students, or ex-students.

I've heard people make the same "I can't do math" comment for decades. When the statement is made in public, it's almost a call to create an instant support group for people who are marginally proud of their self-fulfilling assessment.

It is a tragedy that one subject with clearly defined rules -- at least through high school -- daunts so many students.

7:27 PM  
Anonymous Anonymous said...

What a fascinating comment:

---Personally, I wish I'd been taught calculus with as few numbers and practice problems as possible, and far more graphs, descriptions, and essay questions.

I have this sense that it is a beautiful way of describing phenomena, making predictions, solving problems on a theoretical plane. Later, the numbers and symbols can go in.

you have this SENSE??? As in, you don't know, or you do know calculus?

I think you mean that that sense is your intution. That sense is the same sense you lacked in middle school wrt simple machines. Why didn't you have that sense in middle school? Because you can't start from the intuition, and even after being taught how to play with the machines, the intuition wasn't there yet.

All of the "explaining" how beautiful calculus is never TEACHES calculus--because to understand calculus you must DO CALCULUS. Sometimes, it's even the case that teaching all that "beautiful" part just makes students mistake their high-level meta knowledge for REAL knowledge, and that's really frustrating to a lot of students.

Intution comes over time, from practice. You can't understand the graphs or descriptions or essays without first grokking what's underneath. And you don't get intution in a week or a month.

It's like you're suggesting that the way to have been taught to ride a bicycle would have been with graphs, descriptions, and essay questions. Really? Do you think you could have been given good enough graphs, descriptions, and essay questions to help you understand how to ride a bike without being ON a bike?

The more we try to give the meta-level sense, the epistemology rather than the facts, the more we erode the student's chances of building that intution on their own. No, it's not black-or-white, contructivism vs. reductionism. But there's a reason why children don't learn to derive 6 times 4 is 24 from being first taught the beauty of prime factorization.

Calculus really makes sense the third time you learn it. If you're lucky, you learn it once in a math course, a second time in physics course, and again in a math course, and THEN, FINALLY you get it. But you have to be using the numbers and symbols the whole time for it to click in your head.

3:00 AM  

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