Download MAT 140 Discrete Mathematics I

Association for Women in Mathematics
Hay Minisymposium
New Orleans, LA
The Role of Logic in the K-12
Mathematics Curriculum
8 January 2011
Susanna S. Epp
[email protected]
1
Adding It Up: Helping Children Learn Mathematics
(2001)
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
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
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 x1
and x2 in X, if F (x1) = F (x2) then x1 = x2, or, equivalently, if x1  x2
then F (x1)  F (x2).
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?
4
The Language of Quantification, If-then, & And
Even in early grades, one can introduce students to quantification using
small finite sets. For example:
True or false?
If an object is white,
1. All the white objects are squares.
then it is a square.
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.
“…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)
5
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.
u
1

u
1


u

u
1
v
1
2

v
2


v

v
2
w
2
3

w
3


w

w
3
functions
not functions
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?
6
2. One-to-one Function Definition
A function F from X to Y is one-to-one if, and only if for all x1 and
x2 in X, if F (x1) = F (x2) then x1 = x2, or, equivalently, if x1  x2 then F
(x1)  F (x2).
Y
X
Y
X
X
Y
Y
X
1
u
u
u
1
1
u
1
2
v
v
v
2
2
v
2
3
w
w
w
3
3
one-to-one
not 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?
7
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.)
Y
1
2
3
F
X
u
v
w
Y F -1
u
v
w
X
1
2
3
graph of F
y
x
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.
8
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
a fortiori
Do all these numbers satisfy the original inequality? Yes!
9
Some Other Opportunities Not to Miss
1. 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 a b  a  b 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  2n = 6n.
10
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.”
11
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. 75-84.
2. 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. 188-197.
3. National Research Council, Adding It Up: Helping Children Learn
Mathematics, NRC Press, 2001.
4. Whitehead, A. N., An Introduction to Mathematics, Henry Holt
& Co., 1911. (http://books.google.com)
Thank you!
E-mail: [email protected]
12
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 100 = 1 and 10–n =1/10n.
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:
http://jeff560.tripod.com/mathsym.html.
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 1998. Vol. 91 (#4), p. 282
Question: Why is the answer always 5? Response:
2(n  7)  4
n  5
2
14
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
15
Klein: persistence of ____, tendency to accept generalizatins (e.g., laws of exponents, comm, assoc, dist laws)