Introducing the derivative via polynomials ?

Joost Hulshof & Ronald Meester (2010) suggest to introduce the derivative in highschool by means of polynomials (pdf p16-17). My problem is that they first hide the limit and then let it ambush the student. Thus:

  • When they say that “you can present the derivative for polynomials without limits” then they mean this only for didactics and not for mathematics.
  • But they are not trained in didactics, so they are arguing this as a hobby, as mathematicians with a peculiar view on didactics. They provide a course for mathematics teachers, but this concerns mathematics and not didactics.
  • They only hide the limit, but they do not deny that fundamentally you must refer to limits.
  • Eventually they still present the limit to maintain exactness, but then it has no other role than to link up to a later course (perhaps only for mathematicians).
  • Thus, they make the gap between “didactics” and proper “mathematics” larger on purpose.
  • This is quite different from the algebraic approach (see here), that really avoids limits, and also argues that limits are fundamentally irrelevant (for the functions used in highschool).

I have invited Hulshof since at least 2013 (presentation at the NVvW study day) to look at the algebraic approach to the derivative. He refuses to look into it and write a report on it, though he was so kind to look at this recent skirmish.

Hulshof refers to his approach perhaps as sufficient. It is quite unclear what he thinks about all this, since he doesn’t discuss the proposal of the algebraic approach to the derivative.

Let me explain what is wrong with their approach with the polynomials.

Please let mathematicians stop infringing upon didactics of mathematics. It is okay to check the quality of mathematics in texts that didacticians produce, but stop this “hobby” of second-guessing.

PM. A recent text is Hulshof & Meester (2015), “Wiskunde in je vingers“, VU University Press (EUR 29.95). Potentially they have improved upon the exposition in the pdf, but I am looking at the pdf only. Meester lists this book as “books mathematics” (p14). Hulshof calls it “concepts from mathematics” with “uncommon viewpoints” for “teacher, student” and for “education and curriculum”. When you address students then it is didactics. It is unclear why VU University Press thinks that he and Meester are qualified for this.

The incline

A standard notation for a line is y = c + s x, for constant c and slope s.

The line gives us the possibility of a definition of the incline (Dutch: richtlijn). An incline is defined for a function and a point. An incline of a function f at a point {a, f[a]} is a line that gives the slope of that function at that point.

It is wrong to say that the incline “has the same slope”. You are not comparing two lines. You are looking at the slope. You only know the slope of the function because of the incline (the line with that slope).

Incline versus tangent

The incline is often called the tangent. Students tend to think that tangents cannot cross the function, while tangents actually can. Thus incline can be a better term.

Hulshof & Meester refer in horror to the Oxford Advance Learner’s Dictionary, that has:

ERROR “Tangent: (geometry) a straight line that touches the outside of a curve but does not cross it. The cart track branches off at a tangent.”

I don’t think that “incline” will quickly replace “tangent”. But it is useful to discuss the issue with students and offer them an alternative word if “tangent” continues to confuse them. It is useful to start a discussion with students by mentioning the (quite universal) intuition of not-crossing. An orange touches a table, and doesn’t cross it. But mathematically it would be quite complex to test whether there is any crossing or not. Thus it is simpler to focus on the idea of incline, straight course, alignment.

When you swing a ball and then let go, then the ball will continue in the incline of the last moment. The incline captures that idea, by giving the line with that very slope.

I thank Peter Harremoës for a discussion on this (quite universal) confusion by students (and the OALD) and potential alternative terms. (Incline is still a suggestion.) (The word “directive” was rejected as too confusing with “derivative”. But Dutch “richtlijn” is better than raaklijn.)

Polynomials and their division

A polynomial of degree n has powers of x of size n:

p[x] = c + s x + c2 x² + … + cn xn.

In this, we take c = c0 and s = c1. For n = 1 we get the line again. We allow that the line has s = 0, so that we can have a horizontal line, which would strictly be a polynomial of n = 0. There is also the vertical line, that cannot be represented by a polynomial.

If p[a] = 0 then x = a is called a zero of the polynomial. Then (x a) is called a factor, and the polynomial can be written as

p[x] = (x aq[x]

where q[x] is a polynomial of a lower degree.

If p[a] ≠ 0 then we can still try to factor with (x a) but then there will be a remainder, as p[x] = (x aq[x] + r[x]. When we consider p[x] – r[x] then x = a is a zero of this. Thus:

 p[x] – r[x] = (x aq[x]

With polynomials we can do long division as with numbers. The following example is the division of x³ – 7x – 6 by x – 4 that generates a remainder.

purplemath-divisionIncline or tangent at a polynomial

Regard the polynomial p[x] at x = a, so that bp[a]. We consider point {a, b}. What incline does the curve have ?

(A) For the incline we have the line in {a, b}:

y b = s (x a)

(B) We have p[a] – b = 0 and thus x = a is a zero of the polynomial p[x] – b. Thus:

p[x] – b = (x aq[x]

(C) Thus (A) and (B) allow to assume y ≈  p[x] and to look at the common term x – a, “so that” (quotes because this is problematic):

s = q[a]

The example by Hulshof & Meester is p[x] = – 2 at the point {a, b} = {1, -1}.

p[x] – b = (x² – 2) – (-1) = – 1

Factor:  ( – 1) =  (x – 1) q[x]

Or divide: q[x] = ( – 1) / (x – 1)  = x + 1

Substituting the value = a = 1 in x + 1 gives q[a] = q[1] = 2.

H&M apparently avoid division by using the process of factoring.

Later they mention the limiting process for the division: Limit[x → 1, q[x]] = Limit[x → 1, ( – 1) / (x – 1)] = 2.


As said, the H&M approach is convoluted. They have no background in didactics and they hide the limit (rather than explaining its relevance since they still deem it relevant).

Mathematically, they might argue that they don’t divide but only factor polynomials.

  • But, when you are “factoring systematically” then you are actually dividing.
  • When you use “realistic mathematics education” then you can approximate division by trial and error of repeated subtraction, but I don’t think that they propose this. See the “partial quotient method” and my comments.
  • Addendum December 22: there is a way to look only at coefficients, Ruffini’s Rule, in wikipedia called Horner’s method. A generalisation is known as synthetic division, which expresses that it is no real division, but a method of factoring. (MathWorld has a different entry on “Horner’s method“.) See the next weblog entry.

When dividing systematically, you are using algebra, and you are assuming that a denominator like x – 1 isn’t zero but an abstract algebraic term. Well, this is precisely what the algebraic approach to the derivative has been proposing. Thus, their suggestion provides support for the algebraic approach, be it, that they do it somewhat crummy and non-systematically, whence it is little use to refer to this kind of support.

Didactically, their approach is undeveloped. They compare the slopes of the polynomial and the line, but there is no clear discussion why this would be a slope, or why you would make such a comparison. Basically, you can compare polynomials of order n with those of order m, and this would be a mathematical exercise, but devoid of interpretation. For didactics it does make sense to discuss: (a) the notion of “slope” of a function is given by the incline, (b) we want to find the incline of a polynomial for a particular reason (e.g. instantaneous velocity), (c) we can find it by a procedure called “derivative”. NB. My book Conquest of the Plane starts with surface and integral, and only later looks at slopes.

A main criticism however is also that H&M overlooked the fundamental problem with the notion of a slope of a line itself. They rely on some hidden issues here too. I discussed this recently, and repeat this below.

PM. See a discussion of approximating a function by polynomials. Observe that we are not “approximating” a function by its incline now. At {a, b} the values and slope are exactly the same, and there is nothing approximate about this. Only at other points we might say that there is an “error” by looking at the incline rather than the polynomial, but we are not looking at such errors now, and this would be a quite different topic of discussion.

Copy of December 8 2016: Ray through an origin

Let us first consider a ray through the origin, with horizontal axis x and vertical axis y. The ray makes an angle α with the horizontal axis. The ray can be represented by a function as y =  f [x] = s x, with the slope s = tan[α]. Observe that there is no constant term (c = 0).


The quotient y / x is defined everywhere, with the outcome s, except at the point x = 0, where we get an expression 0 / 0. This is quite curious. We tend to regard y / x as the slope (there is no constant term), and at x = 0 the line has that slope too, but we seem unable to say so.

There are at least three responses:

(i) Standard mathematics then takes off, with limits and continuity.

(ii) A quick fix might be to try to define a separate function to find the slope of a ray, but we can wonder whether this is all nice and proper, since we can only state the value s at 0 when we have solved the value elsewhere. If we substitute y when it isn’t a ray, or example x², then we get a curious construction, and thus the definition isn’t quite complete since there ought to be a test on being a ray.




(iii) The algebraic approach uses the following definition of the dynamic quotient:

y // x ≡ { y / x, unless x is a variable and then: assume x ≠ 0, simplify the expression y / x, declare the result valid also for the domain extension x = 0 }

Thus in this case we can use y // x = s x // x = s, and this slope also holds for the value x = 0, since this has now been included in the domain too.

Line with constant

When we have a line y = c + s x, then a hidden part of the definition is that the slope is s everywhere, even though we cannot compute (y c) / x when x = 0. (One might say: “This is what it means to be linear.”)

When we look at x = a and determine the slope by taking a difference Δx, then we get:

b = c + s a

b + Δy = c + s (a + Δx)

Δy = Δx

The slope at would be s but is also Δy / Δx, undefined for Δx = 0

Thus, the slope of a line is either given as s for all points (or, critically for x = 0 too) (perhaps with a rule: if you find a slope somewhere then it holds everywhere), or we must use limits.

The latter can be more confusing when s has not been given and must be calculated from other resources. In the case of differentials dy = s dx, the notation dy / dx causes conceptual problems when s itself is found by a limit on the difference quotient.

  1. The H&M claim that polynomials can be used without limits is basically a didactic claim since they evidently still rely on limits (perhaps to fend of fellow mathematicians). This didactic claim is a wild-goose chase since they are not involved in didactics research.
  2. If they really would hold that factoring can be done systematically without division, then they might have a point, but then they still must give an adequate explanation how you get from (A) & (B) to (C). Saying that differences are “small” is not enough (not even for polynomials). Addendum December 22: see the next weblog entry on Ruffini’s rule.
  3. They present this for a “reminder course in mathematics” for teachers of mathematics, but it isn’t really mathematics and it is neither useful for teaching mathematics.
  4. A serious development that avoids limits and relies on algebraic methods, that covers the same area of polynomials but also trigonometry and exponential functions, is the algebraic approach to the derivative, available since 2007 with a proof of concept in Conquest of the Plane in 2011.
  5. It is absurd that Hulshof & Meester neglect the algebraic approach. But they are mathematicians, and didactics is not their field of research. I think that the algebraic method provides a fundamental redefinition of calculus, but I prefer the realm of didactics above the realm of mathematics with its culture of contempt for empirical science.
  6. The H&M exposition and neglect is just an example of Holland as Absurdistan, and the need to boycott Holland till the censorship of science by the directorate of the Dutch Central Planning Bureau has been lifted.
I wouldn't want to be caught before a blackboard like that (Screenshot UChicago)

I wouldn’t want to be caught before a blackboard like that (Screenshot UChicago)

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