Computational media: the universal acid of mathematics teaching (7)

Posted by: Gary Ernest Davis on: August 4, 2010

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Gizem Karaali (Pomona College) and Bruce Yoshiwara (Los Angeles Pierce College) have an informative and thoughtful article on the use of Wolfram|Alpha in teaching and learning mathematics.

They ask a number of specific questions, stimulated by student and instructor use of Wolfram|Alpha:

• Do students need to master the skill of integration by parts or partial fraction decomposition in a calculus class just because we want them to find anti derivatives?
• Do algebra students need to master factoring polynomials simply for solving equations and rewriting expressions?
• Do any students need synthetic division?

To my mind these questions are examples of how computational media acts as a universal acid of mathematics teaching, eating away at our models and world views, compelling us to re-think our assumptions about the practice of mathematics, and what is valued in that practice, by whom, and under what circumstances.

In this post I want to look at just one of these questions: “Do any students need synthetic division?”

I suspect that for many people, including many teachers of mathematics, this is aÂ  question that only a real math geek would answer affirmatively. For example:

• “They sure do. As a backdrop for little BÃ©zout’s theorem.”
• “I wouldn’t be able to cubic bezier y-at-x if I wasn’t a synth. div. machine – I like to show students these examples.”
• “If basic prowess is aimed at, then not being able to factor/divide is a disaster.”
• “It’s a valuable tidbit for understanding polynomials; the rings of polynomials are crucial for algebraic geometry.”

Why do these responses seem geeky? It is largely, I feel, because of the use of terms such as “little BÃ©zout’s theorem”, “cubic Bezier”, “rings of polynomials are crucial for algebraic geometry”, and “not being able to factor/ divide is a disaster.”

Giving a reason why something is important that uses highly technical terms is a turn-off for most people. They rightly suspect that the supposed importance is being passed off to a higher realm of knowledge, to which they presently have little or no knowledge. It’s a version ofÂ  the “you might need it some day” argument. Also claiming that not knowing something is a “disaster” rightly leads many people to suspect the writer is exaggerating: nothing disastrous seems to have happened to them yet as a result of their relative ignorance, so how is it a disaster?

If you, dear reader, want to know about little BÃ©zout’s theorem (aka the polynomial remainder theorem), cubic Beziers, why rings of polynomials are crucial for algebraic geometry, andÂ  why not being able to factor/ divide is a disaster, then, I agree: you need to know about synthetic division of polynomials. But here’s a crucial distinction to make: you might need to know about these things, but do you need to be proficient in using them?

Not all mathematicians are good at all things at all times, under all circumstances, not even those of eternal world renown. So the question at issue is this: Is it so important that students are proficient at synthetic division of polynomials that to pass it off to Wolfram|Alpha would be robbing students of a necessary basic skill?

Let us see what sort of problems might face a person skilled at polynomial division.

The polynomial $p(x)=5 + 4 x + x^2 - x^3 + 9 x^4 + x^5$Â  – a monic polynomial with integer coefficients – is irreducible in the ring $Z[x]$ of all polynomials with integer coefficients. Here, “irreducible” means that $p(x)$ cannot be expressed as a product of two polynomials of degree 1 or greater, both of which have integer coefficients, and “monic” means that the coefficient of the highest power of $x$ is 1.

The quotient ring $Z[x]/$ can be thought of inÂ  a concreteÂ  way as the polynomial remainders after division by $p(x)$. Â These remainders form a field: as well as addition and multiplication, division – except by 0 – is always possible. This results from the irreducibility of $p(x)$.

So, here is a polynomial of degree 9: $q(x)=-8 + 10 x + x^2 + x^3 - 8 x^4 - 7 x^5 + 5 x^6 - 9 x^7 - 2 x^8 + x^9$.Â  What is the remainder when $q(x)$ is divided by $p(x)$?

Do we really want students to be proficient at such divisions, or would we be happy if they got a machine, or Wolfram|Alpha to do it?

What does Wolfram|Alpha give as an answer?

$\frac{-8 + 10 x + x^2 + x^3 - 8 x^4 - 7 x^5 + 5 x^6 - 9 x^7 - 2 x^8 + x^9}{5 + 4 x + x^2 - x^3 + 9 x^4 + x^5}$.

Not an informative answer! Mathematica will do better if we use the “PolynomialMod” command:

$\textrm{PolynomialMod}[-8 + 10 x + x^2 + x^3 - 8 x^4 - 7 x^5 + 5 x^6 - 9 x^7 - 2 x^8 + x^9, 5 + 4 x + x^2 - x^3 + 9 x^4 + x^5]$

$=-37633 - 25960 x - 4675 x^2 + 8043 x^3 - 68611 x^4$

Is this what we want our students to be proficient at, or are we happy to let the software do it?

Of course no one in schools, or likely not even in university or college, gives anyone but the most able students such complicated synthetic divisions.

So, we know that we really do NOT want our students to be proficient at synthetic division. What we MIGHT want is that they should be proficient enough that they understand what we are talking about when we discuss, but do not carry out, division one polynomial by another. In other words, we want students to practice small and simple examples on which we can test the students for “knowledge” to convince ourselves that we have taught them an important and useful skill: “important” and “useful” because we, or someone else will, or might, use them later.

Where is any evidence that students will do worse at understanding a concept such as division of polynomials if they use software to do the grunt work for them? Is it evidence similar to saying that a farmer does not really understanding plowing a field unless she does it with an ox and a hand plow: that a tractor just takes away all the understanding and competence in real plowing?

What sort of intelligent questions, requiring thought and an intelligent response, might we ask about polynomial division if we allow software to do the heavy lifting?

4 Responses to "Computational media: the universal acid of mathematics teaching (7)"

While I don’t at all disagree with your overall point, I don’t think the farmer and plow analogy is an illuminating one. Someone who farms understands what his problem and the desired outcome are–soil that needs to be tilled and planted. He’ll use whatever tool he deems best from those available to him.

In mathematics, a student has a much more limited sense of what the problem and the desired outcome are. In fact, how a student is presented mathematical tools (and which ones) will greatly influence her understanding of what mathematics is and what it means to be good at it. While it’s clear that we should prefer the farmer choose the more advanced technology to achieve his goals, it is not so obvious that using the latest mathematical technology is the best way to mold a young mind for mathematics.

That’s only to say the premise isn’t obvious. I would love to see new technologies used well to help people to better grasp mathematics, and I work toward this in my own classroom. But your analogy makes a path that seems possible (but perhaps difficult to traverse) a foregone conclusion.

If a student of agriculture learns modern methods and techniques of agronomy, and modern technologies to implement those methods, should not students of mathematics do the same with regard to professional mathematical practice?

If the question you’ve just posed is an attempt at refining the rhetorical point you were trying to make in your post, then I think it works better–that the parallel is more correct and more clear.

If instead your intention is to just give me back what you’ve said before, I don’t know if I’ve made my point clear. I certainly agree that someone at an advanced stage of studying either agriculture or mathematics should be using cutting-edge methods. I also think that changing technologies should have an impact on the education of all students, no matter how young. All I mean to say is that using combines and calculators is not a wholesale replacement for planting lima beans in cups and making and arranging piles of pebbles (or counting blocks).

Coming to an understanding of what’s important in any field requires engagement with pushing things and concepts around with one’s bare hands, as well as with the far-ranging glimpses provided by new methods and technologies. Students must stand on the shoulders of giants, but they must also stand on their own two feet. Providing the right balance of these opportunities is what we teachers have to make happen.

Gary,

Have you tried the “grid” method of polynomial division? My limited experience with this method is that students respond well to it. It seems to me that it activates other ways of thinking about polynomial division that touches on deeper and more useful aspects of the operation, and also makes the whole process seem more playful.

I have not had much success tracking down resources / texts for this division method – a while ago I tried to describe it here:

http://mathrecreation.blogspot.com/2009/03/dividing-polynomials-grid-method.html

We sometimes adopt a particular algorithm due tradition, perceived efficiency, or ease of type-setting, rather than choosing a method that best reflects the underlying concepts at play. The widespread use of synthetic division compared to the limited use of the grid method, may be an example of this.

Thanks,

— Dan