Produced by Charles Wells Revised 2017-03-11 Introduction to this website website TOC website index blog Back to top of Understanding Math chapter
In some situations you may have conflicting information from different sources about a subject. The resulting confusion in your thinking is called cognitive dissonance.
It may happen that a person suffering cognitive dissonance suppresses one of the ways of understanding in order to resolve the conflict. For example, at a certain stage in learning English, you (small child or non-native-English speaker) may learn a rule that the past tense is made from the present form by adding "-ed". So you say “bringed” instead of “brought” even though you may have heard people use “brought” many times. You have suppressed the evidence in favor of the rule.
Some of the ways cognitive dissonance can affect learning math are discussed here
We think about math objects using metaphors, as we do with most concepts that are not totally concrete. The metaphors are imperfect, suggesting facts about the objects that may not follow from the definition. This is discussed at length in the section on images and metaphors here.
Mathematicians think of the real numbers as constituting a line infinitely long in both directions, with each number as a point on the line. But this does not mean that you can think of the line as a row of points. Between any two points there are uncountably many other points. See the article on real numbers for more about this.
One of the most intransigent examples of metaphorical contamination occurs when students think about countably infinite sets. Their metaphor is that a sequence such as the set of natural numbers $\{0,1,2,3,4,\ldots\}$ "goes on forever but never ends". The metaphor mathematicians have in mind is quite different: The natural numbers constitute the set that contains every natural number right now.
An excruciating example of this is the true statement "$.999\ldots=1.0$." The notion that it can't be true comes from thinking of "$0.999\ldots$" as consisting of the list of numbers \[0.9,0.99,0.999,0.9999,0.99999,\ldots\] which the student may say "gets closer and closer to $1.0$ but never gets there".
Now consider the way a mathematician thinks: The numbers are all already there, and they make a set.
The proof that $.999\ldots=1.0$ has several steps. In the list below, I have inserted some remarks in red that indicate areas of abstract math that beginning students have trouble with.
The problem that occurs with the word "definition" in this case is that a definition appears to be a dictatorial act. The student needs to know why you made this definition. This is not a stupid request. The act can be justified by the way the definition gets along with the algebraic and topological characteristic of the real numbers.
Each one of these steps should be made explicit. Even the Wikipedia article, which is regarded as a well written document, doesn't make all of the points explicit.
Many math objects have names that are ordinary English words. (See names.) So the person learning about them is faced with two inputs:
It is easy and natural to suppress the information given by the definition (or part of it) and rely only on the English meaning. But math does not work that way:
|
If another source of understanding |
The connotations of a name may fit the concept in some ways and not others. Infinite cardinal numbers are a notorious example of this: there are some ways in which they are like numbers and other in which they are not.
For a finite set, the cardinality of the set is the number of elements in the set. Long ago, mathematicians started talking about the cardinality of an infinite set. They worked out a lot of facts about that, for example:
The teacher may even say that there are just as many points on the real line as in the real points. And know-it-all math majors will say that to their friends.
Many students will find that totally bizarre. Essentially, what has happened is that the math dictators have taken the phrase "cardinality" to mean what it usually means for finite sets and extend it to infinite sets by using a perfectly consistent (and useful) definition of "cardinality" which has very different properties from the finite case.
That causes a perfect storm of cognitive dissonance.
Math majors must learn to get used to situations like this; they occur in all branches of math. But it is bad behavior to use the phrase "the same number of elements" to non-mathematicians. Indeed, I don't think you should use the word cardinality in that setting either: you should refer to a "one-to-one correspondence" instead and admit up front that the existence of such a correspondence is quite amazing.
Let’s look at the word “series”in more detail. In ordinary English, a series is a bunch of things, one after the other.
In mathematics an infinite series is an object expressed like this:
\[\sum\limits_{k=1}^{\infty }{{{a}_{k}}}\]where the ${{a}_{k}}$ are numbers. It has partial sums
\[\sum\limits_{k=1}^{n}{{{a}_{k}}}\]For example, if ${{a}_{k}}$ is defined to be $1/{{k}^{2}}$ for positive integers $k$, then
\[\sum\limits_{k=1}^{6}{{{a}_{k}}}=1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\frac{1}{36}=\frac{\text{5369}}{\text{3600}}=\text{ about }1.49\]This infinite series converges to $\zeta (2)$, which is about $1.65$. (This is not obvious. See the Zeta-function article in Wikipedia.) So this "infinite series" is really an infinite sum. It does not fit the image given by the English word "series". The English meaning contaminates the mathematical meaning. But the definition wins.
The mathematical word that corresponds to the usual meaning of "series" is "sequence". For example, $a_k:=1/{{k}^{2}}$ is the infinite sequence $1,\frac{1}{4},\frac{1}{9},\frac{1}{16}\ldots$ It is not an infinite series.
"Only if" is also discussed from a more technical point of view in the article on conditional assertions.
In math English, sentences of the form $P$ only if $Q$” mean exactly the same thing as “If $P$ then $Q$”. The phrase “only if” is rarely used this way in ordinary English discourse.
Sentences of the form "$P$ only if $Q$” about ordinary everyday things generally do not mean the same thing as “If $P$ then $Q$”. That is because in such situations there are considerations of time and causation that do not come up with mathematical objects. Consider “If it is raining, I will carry an umbrella” (seeing the rain will cause me to carry the umbrella) and “It is raining only if I carry an umbrella" (which sounds like my carrying an umbrella will cause it to rain). When "$P$ only if $Q$" is about math objects, there is no question of time and causation because math objects are inert and unchanging.
Students sometimes flatly refuse to believe me when I tell them about the mathematical meaning of “only if”. This is a classic example of semantic contamination. Two sources of information appear to contradict each other, in this case (1) the professor and (2) a lifetime of intimate experience with the English language. The information from one of these sources must be rejected or suppressed. It is hardly surprising that many students prefer to suppress the professor's apparently unnatural and usually unmotivated claims.
The following cause more minor cognitive dissonance.
Besides the examples given above, you can find many others in these two works: