‘Lies to children’ in mathematics and statistics education

Question

In teaching, we sometimes necessarily oversimplify concepts. Terry Pratchett famously referred to this as Lies to children:

A lie-to-children is a statement that is false, but which nevertheless leads the child’s mind towards a more accurate explanation, one that the child will only be able to appreciate if it has been primed with the lie.

In online discussions about lies to children, most examples I have come across are in the natural sciences, most frequently physics and chemistry. There are some examples on the Wikipedia page as well. Does anyone know of any good examples in mathematics and statistics, particularly at the undergraduate level or higher?

I don’t mean examples where we as teachers simplify and give hand-waving arguments because the level required for a rigorous proof is far too high. What I have in mind are explanations that are completely and utterly incorrect, but somehow still helpful.

Answer 1

Young children 5-8 years old, are taught to subtract the smaller number from the bigger number. They are told that you can't subtract a bigger number from a smaller number. This lie has its advantages and helps cement the order of the numbers in a subtraction sentence. Then when students learn about negative numbers they can easily unlearn the previous "lie" that you can't subtract a smaller number from a bigger number.

Of course, there is often a bright 7 year old who has been taught about negative numbers at home who objects to not being able subtract bigger numbers from smaller numbers.

Answer 2

We usually teach:

$$\int\frac1xdx=\ln{|x|}+c$$

Whereas it should be:

$$\int\frac1xdx = \begin{cases} \ln{x}+c_1 & x>0 \\ \ln{(-x)}+c_2 & x<0 \end{cases}$$

Why don't we teach the correct version? Using the "wrong" version usually leads to the right final answer, whereas using the correct version would probably overburden struggling calculus students.

Answer 3

The idea that “a number” means “this decimal expansion”, rather than the expansion being a way of representing a number that has some more set-theoretic definition. It's the de facto truth for everyone in the world who isn't going into rigorous mathematics, but thinking of the decimal as “the number” and other properties as things that are true about it is the root of the classic confusion around $0.999... = 1$.

When I was introduced to the concepts at the undergraduate level, we couldn’t even get through the basic definitions without acknowledging that $0.999... = 1.000...$ as an assumption of how the representation is constructed to allow us to prove that numbers map (almost) uniquely in both directions.

This relates closely to Patrick Stevens' answer – a straightforward understanding of rational and real numbers works intuitively, but the formalism has to go the other way from the intuition.

Answer 4

The average

where the lie-to-children is the word "the".

Ask anyone what "the average" of a set of values is, and immediately you'll be told the arithmetic mean. That's how it's taught initially, and that's what everyone falls back to by default.

A little further on in primary school, you're taught that actually there are three kinds of averages - mean, median and mode. But only three.

Study actual statistics though, and you get onto least-squares, standard deviation and other ways to know how confident you are about your "average", fitting to polynomials or other functions, Bayesian statistics, and so on.

Answer 5

Whether or not the derivative $\frac{dy}{dx}$ is a fraction. Similarly, what, exactly, are $dy$ and $dx$?

This actually goes through several iterations of lies:

1. We first hammer it into Calc I students that $\frac{dy}{dx}$ is not a fraction, but instead the limit of fractions and that we write it as a fraction for intuition. We teach that $dx$ should be thought of as "an infinitesimal change in $x$" and sometimes even mention that this isn't strictly accurate, but does mirror the intuition of Newton, Leibniz, and others during the historical development of infinitesimal calculus.

2. Somewhere in a differential equations or physics course, we teach students to manipulate $\frac{dy}{dx}$ as a fraction. But only in special circumstances and that it's not really a fraction; it just plays one on TV. All of these fraction manipulations are simply algebraic shorthand for things like change of variables in integration when solving a separable differential equation.

3. Somewhere in a differential geometry/calculus on manifolds course we introduce the idea that $\frac{dy}{dx}$ is, in fact, a fraction relating vectors in tangent spaces. And simultaneously introduce the idea that a derivative is a linear mapping between tangent spaces and not a fraction. And that $dx^i$ is simply a map that returns the $i$-th coordinate of a vector in a tangent space.

Answer 6

"Random variable."

...because, as we all know, a random variable is neither random nor a variable. It is a real-valued function. But if we tried to introduce the concept, the feeling, of a stochastic, uncertain, incompletely known environment using such a deterministic terminology as "real-valued function", we would certainly fail. Hence, random variable it is.

Answer 7

I was introduced to the real numbers as "all the points on a [two-sided infinitely long] line".

• At best this is a circular definition. It's certainly very sloppy, and those words could be used to describe many different objects.
• The real-world intuition is incorrect. By the time you're looking at a small enough scale for it to matter whether you've got $\mathbb{R}$ or $\mathbb{Q}$, space has stopped behaving like $\mathbb{R}^3$.

But it's intuitively obvious what it means. Eventually you'll be introduced to "Dedekind- or Cauchy-completion of the rationals" to formalise what it means for something to be a "point on a line" in the intended sense.

Answer 8

$ $

$\LARGE\mathbb R$

Students are introduced to real numbers long before they are ready for the formal definition. At second level they are primed for dealing with fractions and not fractions and told there are numbers like $\pi$ where the decimal expansion goes on forever.

The formal definition is in terms of completion of a metric space or Dedekind cuts. I was in my early twenties when I first encountered it.

Interesting fact: The metric space definition is already circular if you insist on defining a metric as a function $d: X \times X \to \mathbb R$. This assumes $\mathbb R$ is defined and cannot be used to create the definition. Your homework is to repair this problem.

Answer 9

I was told in high school that Euclidean geometry can be derived from the five postulates written by Euclid, but this is not the case. Several of Euclid's proofs have holes in them and one can create models of the five axioms that do not satisfy those results (e.g., in $\mathbb{Q}^2$ a line can be closer to the center of a circumference than the length of its radius and still not inersect the circle).

A rigorous foundation of Euclidean geometry was done by Hilbert in 1899 and it requires 20 axioms, which exclude the above models and make all classical theorems valid.

Answer 10

A lot of things around the limitation and construction of number spaces come to mind such as:

• You cannot divide 5 by 2.
  You cannot take the square root of a negative number.
  You must not divide by zero.

• Real numbers hold more reality than imaginary numbers.

• Imaginary numbers hold some reality because they have applications in electrical engineering, quantum physics, …

• Mathematicians invented complex numbers because they wanted to take the square root of negative numbers.

  (There may be some truth to this one. It’s not a good motivation though.)

• We use complex numbers to be able to solve more equations.

• Infinity is not a valid result and you cannot perform computations with it.

Whereas the truth is more like:

• All number spaces (and infinity) are artificial constructs.
• “Advanced” number spaces (and infinity) allow us to treat certain problems more efficiently or avoid irrelevant pathologies.
• Taking the square root of a real number (and similar) is most often not something we do for its own sake, but something that comes up in application and is either a way to an useful answer or not.
• The restrictions of number spaces reflect restrictions of real-life or inner-mathematical applications and it’s our duty to apply them as appropriate.

Answer 11

My biggest pet peeve:

"A vector is a quantity with both magnitude and direction"

One is required to say this to pick up marks on on A-level physics exam, say, despite it being very, very wrong and arguably placing emphasis on the wrong concepts. I am not yet at university, but I imagine there is a large culture shock when students are first exposed to linear algebra. That said, introducing any other notion seems to be very counter productive to the high school education, and with regards to one actually needs to be able to do, the "magnitude-and-direction" lie is successful.

Many examples of this can be found in high school mathematics curricula. Unrigorous notions or incorrect definitions of the derivative, even of "increasing/decreasing function", integral, etc. are all lies that tend to cause students some difficulty unlearning in the first years of university but are essential for teaching the applications of high school maths to the typical high schooler.

Answer 12

I think sometimes when introducing $\pi$ to children they are told that it's exactly $3.1415$ or some other arbitrary number of decimals, or even that it equals $22/7$

Answer 13

We can define irrational numbers to be those numbers which are not rational, and then define the real numbers to be the union of the rationals and irrationals.

The problem with this definition is that it first requires use to have defined the set of real numbers.

Here is a related discussion:

Why does the widespread erroneous definition of "irrational number" persist without being taught?

If we try to define irrational numbers as "numbers that are not rational" then we unwittingly capture numbers like $\sqrt{-1}$, quaternions, $\aleph_0$, surreal numbers, etc.

Answer 14

Anyone who was taught about Venn Diagrams was probably told that a set is a collection of objects.

This is, of course false. But, it is a handy way to think about sets, and even when you get around to getting to a more formal notion of sets- you'll still think of them as collections of objects, while reasoning about them.

Answer 15

As OP asks for examples "particularly at the undergraduate level or higher", here is a meta-mathematical "lie-to-children" that undergraduates tend to learn (often probably implicitly) when they major in mathematics:

Validating a mathematical result means checking how every single step in the argument follows logically from the previous steps.

The skill to check how every single step of a proof follows from the previous steps is obivously something that one needs to learn well. So I don't think one can avoid that the students get, at some point, the impression that this consecutive validation of steps is the essence of checking a mathematical result. However, when more experienced mathematicians read a result and its proof, many of them approach it quite differently then reading every single step in the proof in detail.

Consequences of this (maybe unavoidable) "lie-to-children" can be observed on many occasions (for instance, on several Stackexchange sites):

• Some (many?) people seem to be under the impression that a mathematical theory were essentially a logical house of cards that collapses once one removes a single piece. Experience shows that this is not the case, though: mistakes in research papers do occur on a regular base and at varying degrees of severity, yet mathematics (or subfields of it) is far from collapsing. Several reasons for this are discussed in the answers to this MathOverflow post.

• Many PhD students believe that the "canonical" way to referee a mathematical research paper is to check every single step in all the proofs. However, this is hardly what one does in practice. In this post I described how checking proofs during peer review tends to work in practice.

Answer 16

Independence in probability theory is not independence in usual sense because it does not take into account causation.

Addition1: The notion of independence is well-known en.wikipedia.org/wiki/Independence_(probability_theory). For example, events $A$ and $B$ are called independent if $\mathbb{P}(AB)=\mathbb{P}(A)\mathbb{P}(B)$. As we can see, this is not at all the same as causal independence or smth.like that, because there is not a word about causes in this definition. And if smb. speaks about independence, for example "Student John's grade in physical education does not depend on the size of Napoleon's army", it doesn't look like he means that $\mathbf{P}(AB) = \mathbf{P}(A)\mathbf{P}(B)$ for the corresponding events.

Addition2: This is not the first time someone put a dislike, failing to formulate a single objection against the obviously correct thesis)

Answer 17

1-2-3-4-5-6-7-8-9-10... Well, not if you are using binary...

The angles on a triangle add up to 180 degrees... Only if the triangle is on a flat surface.

Multiplication is repeated addition. Have you ever multiplied, say, (-3)*(-4) and wondered about how you were adding up a negative number of times? No?

Most of elementary mathematics when you get to abstract algebra, really. Adding two positive numbers always gives a positive, larger number? Nope, sorry, we are in Z7 today so 4+4=1. (Or we are actually adding time--if it is currently 10pm and your homework is due in 11 hours, 10+11=9am.)

Subtraction is adding a negative number. Unless you are subtracting a negative number, oops.

IMO math is absolutely crammed full of lies to children, to the point where any elementary math teacher who uses the words "always", "never", "every", etc., is probably automatically fibbing.

Answer 18

$$\frac{1}{0} = \infty$$

I was told this by my Math teacher in my 8 or 9 grade. I only knew at that time about infinity was that it is a very large number; so large that no can ever write it.

I don't expect it to be very common though.

Answer 19

Students are often taught the chain-rule as a trivial cancellation law. In reality, there are intricacies within the chain-rule.

Answer 20

In the US, UK, and former UK colonies, at certain primary-school grades, you can't leave your final answer as an improper fraction. You'll lose marks for answering

$$\frac{3}{4}+\frac{1}{2}=\frac{5}{4}.$$

Students are taught that there is something "improper" and wrong about leaving the above as your final answer.

For full marks, you must answer

$$\frac{3}{4}+\frac{1}{2}=\frac{5}{4}=1\frac{1}{4}.$$

Some grade levels later, students discover this "rule" was nonsense.

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This seems to be mainly a US/UK thing: See What is the rationale for distinguishing between proper and improper fractions?.