Is synthetic division of polynomials something all students should be expected to be able to do proficiently?

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 a  question that only a real math geek would answer affirmatively. For example:

• “They sure do. As a backdrop for little Bézout’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 Bézout’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 of  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 Bézout’s theorem (aka the polynomial remainder theorem), cubic Beziers, why rings of polynomials are crucial for algebraic geometry, and  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   – a monic polynomial with integer coefficients – is irreducible in the ring of all polynomials with integer coefficients. Here, “irreducible” means that 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 is 1.

The quotient ring can be thought of in  a concrete  way as the polynomial remainders after division by .  These remainders form a field: as well as addition and multiplication, division – except by 0 – is always possible. This results from the irreducibility of .

So, here is a polynomial of degree 9: .  What is the remainder when is divided by ?

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?

.

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

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?