What do we teach and who should we teach it to? (II)

In previous posts I discussed technology in schools and also the question of what should we be teaching the kids in high school math – specifically Algebra II.

A number of recent articles such as “Building the Digital District”, and “Out With Textbooks, in With Laptops for an Indiana School District”, discuss the effects and advantages of digital conversion for school districts (Mooresville N.C. and Munster, Indiana). Both districts provide individual laptops to all students, grades 3 -12 and 5 – 12 respectively.

In these articles, the administrators and district teachers list the following advantages they see following the introduction of technology: (a) a rise in number of students who scored “proficient”, (b) savings through not having to buy print textbooks, calculators, encyclopedias, and maps, (c) allowing teachers to review the students’ performance electronically, almost in real time, and thus provide targeted assistance. The articles also mention that students tend to focus more when they use their computers, without quantifying this effect.

Another push for technology in math education is in Conrad Wolfram’s 2010 TED talk at Oxford. Wolfram argues that computers’ role in schools (in math education actually) is to take the drudgery out of computation. He says “the math that most people are actually doing in school today is … applying procedures to problems they don’t really understand, for reasons they don’t get…the problem we have in math education is not that computers might dumb it down, but that we have dumbed down problems right now.”

It seems to me that these are two significantly different points of view. On one hand we have educators putting technology in the service of an existing curriculum, on the other we have people like Wolfram who argue that the existing curriculum is “dumb” and that technology can help us “smarten” it.

Where does this difference come from? Could it be that ordinary teachers and administrators are under the pressure of the “standards”, and for them, literally their livelihoods depend on how the kids do on the “standards”, whereas people like Wolfram come in with experience of how the real world uses mathematics and are not under the pressure of any handed down math “standards”? Could it be that we, the teachers, are teaching “dumb” problems because of inertia? After all, this is how and what we were taught.

Pretty much everybody now agrees that NCLB has damaged education by forcing too many teachers to teach to the test. However, even if the testing part were to go away, the question does remain: what should we teach the kids (in Algebra II)? This also begs the question: what are we preparing these kids for (mathematically that is)?

The Algebra II curriculum has not stayed frozen. Over the years the emphasis in Algebra II has changed. Here is an illustration. I picked three books from my library. “Second Course in Algebra” by Weeks and Adkins (1962), “Modern Algebra Two” by Dressler and Rich and “Algebra 2” by Dolciani et al (1983) – these were all considered top textbooks in their time. I supplemented these by our current textbook – a bloated text, written by a committee and published by Glencoe (2008). I then looked at the percentage of pages in each book devoted to (a) Fractions and Rational Expressions and (b) Matrices and Determinants. The result is as shown in this graph.

What is happening here? From the late sixties on, computers were being introduced more and more in industry, engineering and science and as a result linear algebra became a very significant and practical area of mathematics. Finding that (huge) inverse matrix quickly and economically became more and more important. Ergo – we should teach kids about matrices and determinants.

Never mind that nobody solves even a 3X3 by hand in the real world, never mind that if you are ever going to do anything with matrices and linear algebra, you will do it on the computer/calculator – we are going to “expose” the kids to this increasingly important area of mathematics. (In our district we don’t do transformations with matrices or even solve a system of two equations with the inverse – we just show the kids how to find the inverse. How long will they remember this? What will they think of math as a subject when we throw things like matrices at them and then don’t use it?) Oh yes and by the way, a fundamental area such as fractions and rational expressions – those we will spend less time on.

This is not the only example. How often, in real world applications, do we use absolute value equations? What about systems of inequalities? Linear programming and optimization you say? Does anybody do linear programming by hand in the real (non linear) world?

What does one do, when one has more experience in mathematics (Ph.D. in Engineering and a career in engineering) than most teachers, but is handed down a curriculum drawn by a committee that was probably trying to cover the whole field of mathematics?  What does one do when one is familiar with the Singapore curriculum, the East European curriculums, the French curriculum – who are all producing superior college bound seniors? Does one become a “guerrilla teacher” sneaking in thinking problems, tailoring the curriculum so that it makes sense, or does one soldier on doing to math teaching what we do to operations with fractions – the LCD?


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