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Last edited 3/10/2008 10:55:00 AM

COGNITIVE DISSONANCE incomplete

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.

Contents

Metaphorical contamination. 1

Semantic contamination. 1

Bad names. 2

Only if 3

Vacuous implication. 3

Number theory. 3

Formal analogy. 3

Metaphorical contamination

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.

Semantic contamination

Many math objects have names that are ordinary English words.  (See names.)   So the person learning about them is faced with two inputs:

¨  The definition of the word as a math object.

¨  The meaning and connotations of the word in English.

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 this is a bad idea:

Bad names

¨  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. 

¨  The name may have been badly chosen.   Some mathematicians have been totally sloppy about the way they chose names.  For example, nothing about the English words “group” and “field” suggest anything about having binary operations.   For other examples see quotient and subset.  See also the discussion here.

Series

Let’s look at the word “series” in more detail:

In ordinary English, a series is a bunch of things, one after the other. 

¨  The World Series is a series of up to seven games, coming one after another in time. 

¨  A series of books is not just a bunch of books, but a bunch of books in order. 

·   In the case of the Harry Potter series the books are meant to be read in order. 

·   A publisher might publish a series of books on science, named Physics, Chemistry, Astronomy, Biology, and so on, that are not meant to be read in order, but the publisher will still list them in order.  (What else could they do?)

In mathematics an infinite series is an object expressed like this:

where the  are numbers.  It has partial sums

For example, if  is defined to be  for positive integers k, then

This infinite series converges to , which is , about 1.65.  (This is not obvious.  See Zeta function (MW, Wi)).

¨  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,  (k = 1,2,…), in other words

is an infinite sequence, not an infinite series.

Only if 

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 rains, I will carry an umbrella” and “It will rain only if I carry an umbrella”.   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.             

Vacuous implication

Inequality

Number theory

Formal analogy

        Analogy may suggest new theorems or ways of doing things.  But it is fallible.  What happens particularly often in abstract math is applying a rule to a situation where it is not appropriate.  This is an easy trap to fall into when the notation in two different cases has the same form  that is an example of formal analogy.

Matrix multiplication

If r and s are real numbers then the products rs and sr are always the same number; multiplication of real numbers is commutative.  However, subtraction is not commutative.  In general, .  This causes no trouble because the operations are written differently.

¨  The product of two matrices M and N is written MN, just as for numbers.  But matrix multiplication is not commutative.  For example,

¨ 

¨  Because rs = sr for numbers, the formal similarity of the notation suggests MN = NM, which is wrong.

¨  If the product of two numbers is 0, then one or both of the numbers is zero.  But that is not true for matrix multiplication:

¨ 

¨   

Canceling sine

Beginning calculus students have already learned algebra. 

¨  They have learned that an expression such as xy means x times y. 

¨  They have learned to cancel like terms in a quotient, so that for example

¨  They have learned to write the value of a function f at the input x by f(x). 

¨  They have seen people write sin x instead of sin(x) but never really thought about it.

¨  So they write

(This happens fairly often in freshman calculus classes.  But you wouldn’t do that, would you?