Republic of Mathematics blog

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?

Tanya Khovanova posted a short and perceptive review of, and recommendation for, Richard Rusczyk’s book “Introduction to Algebra“.

She was surprised to find the book apparently free of errors:

“I didn’t find any flaws in it — not in the first 15 minutes, and not even in the first hour. In fact, having used the book many times I have never found any mistakes. Not even a typo. This was disturbing. Is Richard Rusczyk human? It was such an unusual experience with an American math book, that I decided to deliberately look for a typo or a mistake.”

She wrote in considerable praise of the author’s style:

“I like this book for its amazing accuracy and clean explanations. There are also a lot of diverse problems in terms of difficulty and ideas. Richard Rusczyk has good taste. Many of the problems are from different competitions and require inventiveness. I like that there are a lot of challenge problems that go beyond the boring parts of algebra. Also, I like that important points of algebra are chosen wisely and are emphasized.”

Bu then she had an issue with the solution to problem 7.3.1 in the book. The problem is:

Five chickens can lay 10 eggs in 20 days. How long does it take 18 chickens to lay 100 eggs?

The solution given in the book, in a form shortened by Tanya, is:

The number of eggs is jointly proportional to the number of chickens and the amount of time. One chicken lays one egg in 10 days. Hence, 18 chickens will lay 100 eggs in 1000/18 days, which is slightly more than 55 and a half days.

She then remarks: “What is wrong with this solution? Richard Rusczyk is human after all.”

What information are we given?

5 chickens can lay 10 eggs in 20 days.

Assuming that chickens lay eggs at a uniform rate in appropriate units of time, and all the chickens lay at the same rate, we can use this information to figure out how many eggs 1 chicken can lay in 20 days :

1 chicken can lay 2 eggs in 20 days.

or, to put it another way:

1 chicken lays 1 egg in 10 days.

So,

18 chickens lay 18 eggs in 10 days.

To get 100 eggs from 18 chickens we have to know how many lots of 18 there are in 100. The answer is, of course, not a whole number: .

If we could had knowledge about how many eggs chickens laid in fractions of a day then we could say that:

18 chickens lay 100 eggs in days.

But we do not have such information: we do not even know how many eggs 1 chicken lays in 1 day  – of an egg? – the point being that while the days are divisible into fractional parts, the eggs are not, at least not as the chicken lays them (only afterward, when we make an omelet) . So we can only operate in units of 10 days, because that’s how long it takes a chicken to lay an egg. So, we can only say that:

18 chickens lay 90 eggs in 50 days, and 108 eggs in 60 days.

So, to get 100 eggs from 18 chickens we are going to have to wait 60 days -  at which point we will have 8 more eggs than the 100 we wanted.

The moral: fractional answers make sense only when the units of measurement can be divided arbitrarily. An egg, as it is being laid, is 1 egg. A chicken does not lay an egg or an egg.