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7 August 2010Susanna S. Eppsepp@depaul.edu

Mathematical Association of AmericaMathFest: The Klein ProjectPittsburgh, PA

Issues in the Transition from Concrete to Formal Mathematics

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“Elementary Mathematics from an Advanced

Standpoint”by Felix Klein (1908, 1924)

When thinking about the relation between developments in advanced mathematics and K-12 education, Klein says that two questions should customarily be addressed:

1. “How much of all this should be taken up by the schools?”

2. “What should the teacher and what should the pupils know?”

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“Modern” Logic for Mathematics Instruction

Introduction of quantifiers and

Description of a variable as a placeholder, similar to a pronoun

Distinction between free and bound variables

Formulation of inference rules for quantified statements, concept of “natural deduction”

Careful thought about problems that arise when variables are used to express statements involving both and

History: Quantifiers were introduced formally in the late 19th century. Significant further development with relevance to mathematics education occurred into at least the 1950s. - Frege, C. S. Peirce, Schröder, Peano, Hilbert, Ackermann, Whitehead, Russell, Gödel, Gentzen, Jaśkowski, Tarski, Quine, Church, Copi, Montague, Suppes, et al.

The Transition to Operations with Letters

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Felix Klein: This represents such a long step in abstraction that one may well declare that real mathematics begins with operations with letters.

A. N. Whitehead: “The ideas of ‘any’ and ‘some’ are introduced to algebra by the use of letters.…it was not till within the last few years that it has been realized how fundamental any and some are to the very nature of mathematics…”

Alfred Tarski: “Without exaggeration it can be said that the invention of variables constitutes a turning point in the history of mathematics; with these symbols man acquired a tool that prepared the way for the tremendous development of the mathematical science and for the solidification of its logical foundations.”

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G. Frege (1893): The letter ‘x’ serves only to hold places open for a numeral that is to complete the expression… This holding-open is to be understood as follows: all places at which ‘ξ’ stands must be filled always by the same sign, never by different ones. I call these places argument-places…

W. V. Quine (1950?): The variables remain mere pronouns, for cross-reference; just as ‘x’ in its recurrences can usually be rendered ‘it’ in verbal translations, so the distinctive variables ‘x’,’y’, ‘z’, etc., correspond to the distinctive pronouns ‘former’ and ‘latter’, or ‘first’, ‘second’, and ‘third’, etc.

A. Church (1956): …a variable is a symbol whose meaning is like that of a proper name or constant except that the single denotation of the constant is replaced by the possibility of various values of the variable.

P. Halmos (1977): ‘He who hesitates is lost’. In pedantic mathematese this can be said as follows: ‘For all X, if X hesitates, then X is lost’.

S. Pinker (1994 – adapted to this example):The “He” in “He who hesitates is lost” does not refer to any particular person or group of people; it is simply a placeholder indicating that the “he” who is lost is the same as the “he” who hesitates.

Variables as Placeholders/Pronouns

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The Concept of Variable: a Unified Approach

We use a variable as a placeholder when we want to talk about a quantity but either --

Case 1: We know or hypothesize that it has certain values but we don’t know what those values are.

Ex: an unknown quantity, -- either to be found if possible (e.g., solving an equation) -- or to be reasoned with (e.g., when its existence is implied by a definition or deduced in a proof)

Case 2: We don't want to restrict it to a particular, concrete value because we want whatever we say about it to be equally true for all elements in a given set.

Ex: -- a symbol used to express an object in a universal statement (e.g., identity, function definition) -- a generic element in a proof

i.e, a “mathematical John Doe”

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Alfred Tarski (1941): “As opposed to the constants, the variables do not possess any meaning by themselves. …The ‘variable number’ x could not possibly have any specified property … the properties of such a number would change from case to case … entities of such a kind we do not find in our world at all; their existence would contradict the fundamental laws of thought.”

W. V. Quine ( 1950): “Care must be take, however, to divorce this traditional word of mathematics [variable] from its archaic connotations. The variable is not best thought of as somehow varying through time, and causing the sentence in which it occurs to vary with it. Neither is it to be thought of as an unknown quantity, discoverable by solving equations.

A. Church (1956): Mathematical writers do speak of “variable real numbers,” or oftener “variable quantities,” but it seems best not to interpret these phrases literally. Objections … have been clearly stated by Frege and need not be repeated here at length. The fact is that a satisfactory theory has never been developed on this basis, and it is not easy to see how it might be done.

Are Variables “Variable Quantities”?

Comment: Sometimes a variable is defined to be “a quantity that can change.” Indeed, the very word “variable” suggests changeability. But it is not the x or the y that changes; it is the values that that may be put in their places.

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Variables Used in Functional Relationships

Ex: “y = 2x + 1”

For each possible change in the value of x the equation defines a corresponding change in the value of y.

Ex: “the function f (x) = 2x + 1”

The relation defined by corresponding to any given real number the real number obtained by multiplying that number by 2 and adding 1. I.e., no matter what real number is placed in box , f () = 2 + 1.

Problem: Saying “the function f (x) = 2x + 1” conflates the function (as a relation) with its value at x.

Ex: The slope of x 2 at x = 3 is

How should a student interpret this? Cf. x + y = y + x

What does this mean?

In essence, we ask students to learn that what we mean is different from what we say.

But: when we speak of “the valueof x ” or “the value of y ” we mean the values that are put in their places.

[ ]( )

.

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633x

d xxdx x

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Variables Used to Express Unknown Quantities

When we say, “Solve ,” what we mean is “Find all numbers (if any) that can be substituted in place of x in the equationin a way that makes its left-hand side equal to its right-hand side.”The role of x as a placeholder in a situation like this is sometimes highlighted by replacing x by an empty box: .

4 3x x

4 3x x

4 3

For comparison with U.S., see: TIMSS video Hong Kong 4

http://www.rbs.org/catalog/pubs/pd57.php/

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The Logic of Equation Solving

“When we solve an equation, we operate with the unknown(s) “as if it were a known quantity. . .A modern mathematician is so used to this kind of reasoning that his boldness is now barely perceptible to him.” --Jean Dieudonné (1972)

Given an equation, we ask:

Is it true

for some value(s) of the variable(s)? direct proof

for no value(s) of the variable(s)? proof by contradiction

for all value(s) of the variable(s)? generalizing from the generic particular

Start the same way in all three cases: Suppose there is a value of the variable that satisfies the equation, and deduce properties it must satisfy.

Davis, S. and Thompson, D. R. To encourage "algebra for all," start an algebra network. The Mathematics Teacher. Apr 1998. Vol. 91 (#4), p. 282 11

Q: Why is the answer always 5? A:

Comment: These students didn’t know how to simplify the expression on the left.

2( 7) 45

2n

n

Variables Used to Express Universal Statements

Davis, S. and Thompson, D. R. To encourage "algebra for all," start an algebra network. The Mathematics Teacher. Apr 1998. Vol. 91 (#4), p. 282 12

Variables Used to Express Universal Statements

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Variables Used to Express Universal Statements

Ex: The distributive property for real numbers states that for all real numbers a, b, and c, ac + bc = (a + b)c. This means that no matter what real numbers are substituted in place of a, b, and c, the two sides of the equation ac + bc = (a + b)c will be equal.

Comment: To learn to apply the distributive property in a broad range of situations, it is important to understand that the a, b, and c are just placeholders (aka dummy variables). They could be replaced by any three letters. Or we could represent the property by writing ( + ) = + where any three real numbers (or expressions that can represent real numbers) can be placed in the boxes.

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Bound Variables and Scope

Example: An integer is even if, and only if, there is an integer k so that the integer equals 2k.

Problem: Bound variables that jump beyond their bounds.

Proposed proof that the sum of any two even integers is even: Suppose m and n are any even integers. For an integer to be even means that there is an integer k so that the integer equals 2k . Thus

m + n = 2k + 2k = 4k … etc.

Auxiliary Definitions: For an integer to be even means that •there exists an integer a so that the integer equals 2a. •there exists an integer m so that the integer equals 2m. •it equals twice some integer. •it equals 2 , for some integer that can be placed into the box.•it equals 2 (some integer).

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Existential Instantiation

Two main uses:

1. In applying a statement of the form x (y such that P (x,y ))

[Ex: Every even integer equals twice some integer.]

2. When existence is hypothesized

[Ex: Does there exist a number x such that the LHS of this equation equals its RHS?]

If we know that an object exists, then we may give it a name

as long as we are not already using the name for another object in our current discussion.

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The Dependence Rule**Arsac & Durand-Guerrier, 2005

In the sentence“For all x in set D, there exists a y in set E such that…” the value of y depends on the value of x.

“Proof: Suppose n is any odd integer. By definition of odd, n = 2k + 1 for any integer k…”

Theorem: For all functions f : X Y and g : Y Z, if f and g are onto, then so is g of.

“Proof : By definition of onto, given any y in Y, there is an x in X with f (x) = y. Also by definition of onto, given any z in Z, there is a y in Y with g (y) = z. So g of (x) = g

(f (x)) = g (y) = z, and so g of is onto.”

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Universal Generalization (Generalizing from the Generic Particular)

“Mathematics, as a science, commenced when first someone, probably a Greek, proved propositions about ‘any’ things or about ‘some’ things without specification of definite particular things.”

Alfred North Whitehead (1861-1947)

If we can prove that a property is true for a particular, but arbitrarily chosen, element of a set, then we can

conclude that the property is true for every element of the set.

i.e., a generic element of the set

Ex: Is a sum of odd integers always even?Answer: Yes. Suppose m and n are any [particular but arbitrarily chosen] odd integers. We will show that m + n is even.

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Universal Instantiation

= (k + (k + 2))2k+2

If a property is true for all elements of a set, then it is true for any particular element of the set.

Etc.

Two main uses:

1. Every time we do algebra

Ex: Simplify

k2k+2 + (k + 2)2k+2

2. Extensively in explanation/justification/proof

Ex: Is a sum of odd numbers always even?“... So m + n = 2(r + s + 1), where r + s + 1 is an integer, and this number is even because any integer that can be written as 2 (some integer) is even.”

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Existential Generalization

Main use: Counterexamples

Example: True or false? The quantity n 2 + n + 41 is always prime.

(Euler)

Answer: False, because 412 + 41 + 41 = (41)(43), which is not prime.

That is: There exists an integer n such that n 2 + n + 41 is not

prime.

If we know that a certain property is true for a particular object, then we may conclude that

“there exists an object for which the property is true.”

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What We Say vs. What We Mean

What we say What we meanthe value of x the quantity that is put in place of xas the value of x increases as larger and larger numbers are put in place of xas the value of x increases, the value of y increases If larger and larger numbers are put in place of x, the

corresponding numbers that are put in place of y become larger and larger

where x is any real number for all possible substitutions of real numbers in place of x

Let n be any even integer. Imagine substituting an integer in place of n but do not assume anything about its value except that it is an even integer.

By definition of even, n = 2k for some integer k. By definition of even, there is an integer we can substitute in place of k so that the equation n = 2k will be true. (Note that there is only one such integer; its value is n/2.)

the function x2 the function that relates each real number to the square of that number. In other words, for each possible substitution of a real number in place of x, the function corresponds the square of that number.

where x is some real number that satisfies the given property

There is a real number that will make the given property true if we substitute it in place of x.

A general linear function is a function of the form f(x) = ax + b where a and b are any real numbers.

A general linear function is a function defined as follows: for all substitutions of real numbers in place of a and b, the function relates each real number to a times that number plus b. Or: the function is the set of ordered pairs where any real number can be substituted in place of the first element of the pair and the second element of the pair is a times the first number plus b.

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Some References

1. Church, A. (1956), Introduction to Mathematical Logic, vol. 1.2. Frege, G. (1893), The Foundations of Arithmetic.3. Klein, F. (1908, 1924), Elementary Mathematics from an Advanced Standpoint.4. Pinker, S. (1994), The Language Instinct.5. Quine, W. V. (1950, 1982), Methods of Logic.6. Tarski, A. (1941, 1965), Introduction to Logic and to the Methodology of Deductive Sciences.7. Whitehead, A. N. (1911, 1958), An Introduction to Mathematics.

Also: Epp, S., What Is a Variable – Draft article (http://condor.depaul.edu/~sepp/WhatIsAVariable.pdf)

E-mail: sepp@depaul.edu

Thank you!