07 July 2012


Following up on yesterday's post on science phobia ~ science and math are intimately intertwined (math being the language of science).  So it is understandable (and regrettable) that a poor understanding of one field implies the probability of a poor understanding of the other.  This has been confirmed emphatically in recent years, as American students' science and math performance has fallen, while the performance of students from other developed and developing nations has risen to overtake us.  I fully support young people and adults of all nations becoming literate in science and math.  I do not support American young people and adults falling by the wayside, as others pass us by.  We have too much to contribute.

One reason for this downhill slide may be the very technology which our prosperity provides.  When I was young, our tools were a pencil, a piece of paper, and our minds.  We practiced until (a) we had the multiplication tables memorized, (b) we began to grasp the abstract concepts behind each operation, and (c) many of us could perform the math in our heads.  The closest we came to having technological tools were a compass, ruler, and protractor in geometry, and a slide rule in trigonometry (state of the art in those days, used by engineers in the space program).

Those days are gone.  Today schools allow students to rely on sophisticated hand-held calculators.  Punch in the numbers, and up pops the result.  What's wrong with that?  Just this ~ in a learning setting, students need to understand HOW math works in order to grasp fundamental concepts, and in order to have a foundation for asking intelligent questions.  In the classroom, a calculator is a crutch which discourages real learning.

When I was a teacher (math, algebra, biology, environmental studies), no calculators were allowed.  Further, I required my students to show their work on paper, not just write down an answer.  I had to know that they understood the process by which they arrived at that answer.  Otherwise it could be just a lucky guess, or cheating.  That's how I learned math, that's how I taught math.

Some of the best math teachers in the nation are returning to this very approach to learning, according to Konstantin Kakaes in a Slate article called Why Johnny Can't Add Without a Calculator.  One such teacher, Vern Williams, "doesn't just prefer his old chalkboard to the high-tech version.  His kids learn from textbooks that are decades old ~ not because they can't afford new ones, but because Williams and an handful of his like-minded colleagues know the old ones are better .... His preferred algebra book, he says, is 'in-your-face algebra.  They give amazing outstanding examples.  They teach the lessons.'  The modern textbooks, he says, contain hundreds of extraneous, confusing and often outright wrong examples, instead of presenting mathematical ideas in a coherent way.  The examples bloat the books to thousands of pages and disrupt the logical flow of ideas."

What about tech tools like calculators?  "According to an October 2011 report, 89 percent of high school math teachers think their students are ready for college-level mathematics.  But only 26 percent of post-secondary [university] teachers think the students are ready once they get there.  This shortfall in mathematical preparation for college-bound students has existed for a long time, but it is being exacerbated by the increasing use of technology.  College-level math classes almost never use graphing calculators (see image above), while high school classes invariably do.  College professors want their students to understand abstract concepts.  Technology advocates claim their products help teach students such abstractions, but in practice they simply don't."

And here's a point I stressed in yesterday's post ~ "Math and science can be hard to learn, and that's OK.  The proper job of a teacher is not to make it easy, but to guide students through the difficulty by getting them to practice and persevere.  'Some of the best basketball players on Earth will stand at that foul line and shoot foul shots for hours and be bored out of their minds,' says Williams.  Math students, too, need to practice foul shots ~ adding fractions, factoring polynomials.  And whether or not the students are bright, ' once they buy into the idea that hard work leads to cool results,' Williams says, 'you can work with them.'

"Educational researchers often present a false dichotomy between fluency and conceptual reasoning.  But as in basketball, where shooting foul shots helps you learn how to take a fancier shot, computational fluency is the path to conceptual understanding.  There is no way around it."

Note ~ the best reading and mathematics learning software packages have no measurable effect on test scores.  Sloppy studies commissioned by the purveyors of such software, folks who have a financial stake in the outcome, are unfortunately taken at face value by many teachers and administrators around the country.  No rigorous study has found technology to be effective in enhancing students' learning of math.  Just the opposite is true ~ math software is limited, narrow, and does not allow for questions or for alternative, equally valid paths to a solution.

"Computer technology, while great for many things, is just not much good for teaching, yet.  Paradoxically, using technology can inhibit understanding how it works.  If you learn how to multiply 37 by 41 using a calculator, you only understand the black box.  [You don't understand the process of multiplication.]  Maybe one day software will be smart enough to be useful, but that day won't be any time soon, for two reasons.  The first is that education, especially of children, is as much an emotional process as an imparting of knowledge ~ there is no technological substitute for a teacher who cares.  The second is that education is poorly structured.  Technology is bad at dealing with poorly structured concepts.  One question leads to another leads to another, and the rigid structure of computer software has no way of dealing with this.  Software is especially bad for smart kids, who are held back by its flexibility.

"The real shortfall in math and science education can be solved not by software or gadgets but by better teachers .... The new technology makes it easier than ever for teachers to avoid learning the subject .... A teacher who plans his own lecture is forced toward mastery of the material, but one who downloads a PowerPoint presentation doesn't have to know anything beyond how to download the presentation. It is a miracle of efficiency ~ empty calories."

When I taught remedial summer math classes, the first task was the students' taking a placement test.  If their results showed a gap in understanding first-year algebra, or long division, or fractions, that's where that individual's summer work began.  As predictable as the sunrise was the inevitable complaint, "But I already TOOK that."  I had to explain that the student may have been exposed to it at some point, but they didn't learn it.  And without the foundation, they would not have the tools for more advanced work.  So, we practiced.  And practiced.  And with luck and perseverance, at some point comes the reward which makes every true teacher's heart leap ~ that moment when you can actually see a light go on over the student's head, when they really, really get it, inside their own brains.  No calculator needed.

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