1 8 January 2011 Susanna S. Epp Association for Women in Mathematics Hay Minisymposium New Orleans, LA The Role of Logic in the K-12 Mathematics Curriculum

Adding It Up: Helping Children Learn Mathematics Mathematical proficiency, as we see it, has five components, or strands: conceptual understanding—comprehension of mathematical concepts, operations, and relations procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and appropriately strategic competence—ability to formulate, represent, and solve mathematical problems adaptive reasoning—capacity for logical thought, reflection, explanation, and justification productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. These strands are not independent; they represent different aspects of a complex whole. 2 (2001)

A Sampling of Missed Opportunities 1. Vertical line test (to check whether a graph is the graph of a function) 2. Horizontal line test (to check whether a function is one-to-one) 3. Finding a formula for the inverse of a function (Solve for x in terms of y and interchange x and y.) 4. Test point method for solving (x – a)(x – b) > 0 and (x – a)(x – b) < 0. Comment Each of these methods provides a way for students to obtain correct answers without real understanding of the concepts involved by circumventing their difficulties with logic (quantifiers, if-then, “and,” and “or”) and complex linguistic phrases. Thus they do not contribute to developing students’ understanding of mathematical discourse or to preparing a solid basis for more advanced mathematical activity. 3

The Underlying Formalities Are Pretty Formal 4 1. A function F from X to Y is a correspondence that relates every element of X to some element of Y, but no element of Y is related to more than one element of X. 2. A function F from X to Y is one-to-one if, and only if for all x 1 and x 2 in X, if F (x 1 ) = F (x 2 ) then x 1 = x 2, or, equivalently, if x 1  x 2 then F (x 1 )  F (x 2 ). 3. Given a one-to-one correspondence F from X to Y, the inverse function F -1 is defined as follows: for all y in Y, F -1 (y) is the (unique) element x in X that is related to y by F. 4. If a product of two real numbers is positive, then either both are positive or both are negative. If a product of two real numbers is negative, then one is positive and the other is negative. (So, to solve a quadratic inequality, use the logic of “and” and “or.”) What can help students develop an ability to understand these statements?

The Language of Quantification, If-then, & And Even in early grades, one can introduce students to quantification using small finite sets. For example: 5 True or false? 1. All the white objects are squares. 2. All the square objects are white. 3. No square objects are white. 4. There is a white object that is larger than every gray object. 5. Every gray object has a black object next to it. 6. There is a black object that has all the gray objects next to it. 7. All objects that are not small are not gray. These examples also introduce the concepts of counterexample and the non-equivalence between a statement and its converse. If an object is white, then it is a square. “…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 North Whitehead (1911)

1. Definition of Function: The Value of Finite Examples A function F from X to Y is a correspondence that relates every element of X to some element of Y, but no element of Y is related to more than one element of X. 6 123123 uvwuvw 1212 uvwuvw 123123 uvwuvw not functions functions 123123 uvwuvw Questions: Is there any element in X that is not related to some element of Y? Is there some element of X that is related to more than one element of Y?

2. One-to-one Function Definition A function F from X to Y is one-to-one if, and only if for all x 1 and x 2 in X, if F (x 1 ) = F (x 2 ) then x 1 = x 2, or, equivalently, if x 1  x 2 then F (x 1 )  F (x 2 ). 7 one-to-onenot one-to-one Questions: Are there any two elements in X that are related to the same element of Y? Are any two distinct elements of X related to two distinct elements of Y? Is every element of Y related to at most one element of X? 123123 uvwuvw X Y 1212 uvwuvw X Y 123123 uvwuvw X Y 123123 uvuv X Y

3. Definition of Inverse Function 3. If the function F from X to Y is a one-to-one correspondence, then F -1 is defined as follows: for all y in Y, F -1 (y) is the (unique) element x in X such that F (x) = y. (Or: F -1 (y) is the (unique) element of X that is related to y by F.) 8 uvwuvw 123123 F -1 XYX 123123 uvwuvw FY Linguistic precursors Definition of square root: If a is a nonnegative real number, the square root of a is the (unique) nonnegative real number that, when squared, equals a. Definition of logarithm: If b and x are positive real numbers, the logarithm with base b of x is the (unique) exponent to which b must be raised to obtain x. graph of F y x

9 4. Solving (x – a)(x – b) > 0: Using “And,” “Or” and Deductive Reasoning Task: Find all real numbers x for which x 2 - 3x + 2 > 0. Solution: Suppose x is a real number for which x 2 -3x + 2 > 0. Then (x – 2)(x – 1) > 0  [(x – 2) > 0 and (x – 1) > 0] or [(x – 2) < 0 and (x – 1) < 0]  [x > 2 and x > 1] or [x < 2 and x < 1]  x > 2 or x < 1 Do all these numbers satisfy the original inequality? Yes! a fortiori

Some Other Opportunities Not to Miss Incorporating work with “all,” “some,” and “no” in algebra courses a. (Russia, grade 3, age 9) For what values of the letters are the following equalities true? (a) 36  b = b (b) 10  c = 10 (c) 12  a = a  12 (d) a  a = a etc. b. Is true for all numbers a and b, for some numbers a and b, or for no numbers a and b ? 2. Introducing the concept of counterexample a. True or false? All middle school students are lazy. b. True or false? For all whole numbers n, 3  2 n = 6 n.

11 3. Keeping students focused on the real meaning of “solve the equation” a. Find all numbers that make the left-hand side equal to the right-hand side: (a) 2x + 6 = 20 (b) 2x + 6 = 2(x + 3) (c) 2x + 6 = 3(x + 4) – (x + 5) 4. Thinking of FOIL as a four-letter word. “To multiply two polynomials, compute the product of every term on the left with every term on the right and combine like terms.”

12 Some References 1. Epp, S., Logic and discrete mathematics in the schools, in Discrete Mathematics in the Schools, D. Franzblau and J. Rosenstein, eds., American Mathematical Society, Providence, 1997, pp Epp, S., The language of quantification in mathematics instruction, in Developing Mathematical Reasoning in Grades K-12, F. R. Curcio and L. V. Stiff, eds., National Council of Teachers of Mathematics, 1999, pp National Research Council, Adding It Up: Helping Children Learn Mathematics, NRC Press, Whitehead, A. N., An Introduction to Mathematics, Henry Holt & Co., ( Thank you!

Discussing the Reasons for Definitions and Notation Taught first class 43 years ago, still learning from my students MMT 400 Discussing place value, issue of why 10 0 = 1 and 10 –n =1/10 n. Students were very surprised (actually quite shocked) that I couldn’t give them a reason why these notations had to be defined in this way, that the definitions/notations are simply a matter of convenience. Discussed the historical development of the notation: Have a mathematical idea, want to decide on a definition or a notation for it. Issues of how to choose them. 13

Davis, S. and Thompson, D. R. To encourage "algebra for all," start an algebra network. The Mathematics Teacher. Apr Vol. 91 (#4), p Question: Why is the answer always 5? Response:

Abstract The Role of Logic in the K-12 Mathematics Curriculum How can we teach children important mathematical facts in ways that are both age-appropriate and intellectually honest? Some informal explanations are both helpful and suggestive of the mathematics that underlies the facts. Other explanations help students get right answers on tests but do not provide a sound basis for future understanding. This talk will examine examples of both kinds of explanations in the context of courses for prospective and in-service mathematics teachers. Negative numbers, Multiplication of fractions (repeated addition??) (See “Multiplication Overview” - If you have m groups, each of which contains n units of a quantity, then you have m x n units of the quantity. In a certain sense, times always means the same as “of.” m groups of n each gives a total of m x n units. Why does “A times B” mean “A of B”? Then use distributive law and add if appropriate. From class: “Each fraction is ‘of’ something. The ‘something’ is ‘the whole.’” Analyze why this is the same as multiplying the tops and the bottoms. Something about “some” and “all” Introducing the use of variables: “Let a be apples and p be pears.” Number magic Importance of definitions (examples from MMT400, class 1 etc.), def of fractional exponent, Solving an inequality of the form (x – a)(x – b) > 0 or (x – a)(x – b) > 0 Vertical line test Horizontal line test Typical directions for finding a formula for the inverse of a function: Start with f(x) = y; solve for x; interchange x and y. Inverse functions: Problem on an exam: Fill in the blank: If f : X  Y is a one-to-one correspondence and y is an element of Y, then f -1 (y) = ____. Developing the point-slope formula for the equation of a straight line Disappearance of mathematical induction from the h. s. curriculum Disappearance of proof from geometry Establishing an identity by starting with it and deducing a true statement Klein: persistence of ____, tendency to accept generalizatins (e.g., laws of exponents, comm, assoc, dist laws) 15