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Critical Issues in Education: Teaching and Learning Algebra MSRI, Berkeley, CA May 14, 2008

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What Are Some Organizing Principles Around Which One Can Create a Coherent Pre-college Algebra Program?

Critical Issues in Education: Teaching and Learning AlgebraMSRI, Berkeley, CAMay 14, 2008

Zalman UsiskinThe University of [email protected]

The UCSMP Curriculum for Grades 6-12

•The sum of two like terms is their common factor multiplied by the algebraic sum of the coefficients of that factor. (p.13)

- When removing parentheses preceded by a minus sign, change the signs of the terms within the parentheses. (p. 15)
- To divide a polynomial by a monomial: (1) Divide each term of the polynomial by the monomial. (2) Connect the results by their signs. (p. 21)
- The product of two binomials of the form ax + b equals the product of their first terms, plus the algebraic sum of their cross products, plus the product of their second terms. (p. 30)
Source: A Second Course in Algebra, Walter W. Hart, 1951

1. The algorithmic approach: The content is sequenced by skills following prescribed rules (algorithms), and in such a way that when you come upon a new skill, you are either putting together previously-learned skills or given a new rule.

Assume the ordered field properties of the real numbers. Then, mainly from the distributive property of multiplication over addition ( real numbers a, b, c, a(b + c) = ab + ac), we can deduce the following:

•ax + bx = (a + b)x

- -(a + b) = -a + -b
- a/x ± b/x ± c/x = (a ± b ± c)/x
- (ax + b)(cx + d) = acx2 + (bc + ad)x + bd.

1. The algorithmic approach: The content is sequenced by skills following prescribed rules (algorithms), and in such a way that when you come upon a new skill, you are either putting together previously-learned skills or given a new rule.

2. The deductive approach: Deduce the rules as theorems from the ordered field properties of the real (and later, complex) numbers, and in so doing change the view of mathematics from a bunch of arbitrary rules to a logical and organized system.

Graph Translation Theorem: In a relation described by a sentence in x and y, the following two processes yield the same graph:

(1) replacing x by x – h and y by y – k;

(2) applying the translation T: (x, y) (x + h, y + k) to the graph of the original relation.

Graph Scale-Change Theorem: In a relation described by a sentence in x and y, the following two processes yield the same graph:

(1) replacing x by x/a and y by y/b;

(2) applying the scale change S: (x, y) (ax, by) to the graph of the original relation.

ParentOffspring

y = mxy – b = mx Slope-intercept form

y = mxy – y0 = m(x – x0)Point-slope form

y = ax2y – k = a(x – h)2Vertex form

x2 + y2 = r2(x – h)2 + (y – k)2 = r2General circle

y = Asin xy = Asin(x – h)Phase shift

(cos x, sin x) = Rx(1, 0), where Rx is the rotation of magnitude x about (0, 0).

Rπ/2(1, 0) = (0, 1), from which a matrix for

Rx is .

Rx+y

=

Rx° Ry

=

1. The algorithmic approach: The content is sequenced by skills following prescribed rules (algorithms), and in such a way that when you come upon a new skill, you are either putting together previously-learned skills or given a new rule.

- The deductive approach: Deduce the rules as theorems from the ordered field properties of the real (and later, complex) numbers, and in so doing change the view of mathematics from a bunch of arbitrary rules to a logical and organized system.
- Use geometry. Transformations provide a powerful set of ideas for dealing with graphs of functions and trigonometry.

a + b is a real number.

a + b = b + a

a + (b + c) = a + (b + c)

0 such that a + 0 = a.

(-a) such that a + (-a) = 0.

ab is a real number.

ab = ba

a(bc) = ab(c)

1 such that a•1 = a.

(1/a) such that a•(1/a) = 1.

a(b + c) = ab + ac

For all real numbers a and reals m and n:

= ma

ma + na = (m + n)a

0a = 0

n(ma) = (nm)a

m< 0 and a< 0 ma > 0

For all positive reals x and reals m and n:

= xm

xm • xn = xm+n

x0 = 1

(xm)n = xmn

m<0 and x<1 xm > 1

Additive idea:

negative numbers

Linear functions

Arithmetic sequences

2-dimensional translations

Multiplicative idea:

numbers between 0 and 1

Exponential functions

Geometric sequences

2-dimensional scale changes

- Use geometry. Transformations provide a powerful set of ideas for dealing with graphs of functions and trigonometry.
4.Use isomorphism covertly. Use properties in one structure to suggest and work with properties in a second structure (e.g., <+, •> and <R, +>, or matrices and transformations.

- Consider the students. A course for all students cannot assume they all have the background, motivation, and time that we would prefer.
- Sequence by uses. Employ uses of numbers and operations to develop arithmetic, and employ uses of variables to move from arithmetic to algebra.(Go to http://socialsciences.uchicago.edu/ucsmp/ , click on Available Materials, scroll down to and download Applying Arithmetic: A Handbook of Applications of Arithmetic.)

counts

measures

ratio comparisons

scale values

locations

codes and identification

If a quantity is multiplied by a growth factor b in every interval of unit length, then it is multiplied by bn is every interval of length n. (nice applications to compound interest, population growth, inflation rates)

b0 = 1 for all b since in an interval of length 0 the quantity stays the same regardless of the growth factor.

bm • bn = bm+n because an interval of length m+n comes from putting together intervals of lengths m and n.

“The use of ‘real-world’ contexts to introduce mathematical ideas has been advocated… A synthesis of findings from a small number of high-quality studies indicates that if mathematical ideas are taught using ‘real-world’ contexts, then students performance on assessments involving similar ‘real-world’ problems is improved. However, performance on assessments more focused on other aspects of mathematics learning, such as computation, simple word problems, and equation solving, is not improved .” (p. xxiii and p. 49)

Dimensions of mathematical understanding

Skill-algorithm understanding (Algorithms)

from the rote application of an algorithm through the selection and comparison of algorithms to the invention of new algorithms

Properties - mathematical underpinnings understanding (Deduction, Isomorphism)

from the rote justification of a property through the derivation of properties to the proofs of new properties

Uses-applications understanding (Uses)

from the rote application of mathematics in the real world through the use of mathematical models to the invention of new models

Representations-metaphors understanding (Transformations)

from the rote representations of mathematical ideas through the analysis of such representations to the invention of new representations

For any continuous real functions f and g on a domain D:

(1)If h is a 1-1 function on the intersection of f(D) and g(D), then f(x) = g(x) ˛ h(f(x)) = h(g(x)).

(2)If h is an increasing function on the intersection of f(D) and g(D), then f(x) < g(x) ˛ h(f(x)) < h(g(x)).

If h is a decreasing function on the intersection of f(D) and g(D), then f(x) < g(x) ˛ h(f(x)) > h(g(x)).

Exploring the factoring of x2 + 6x + c

Dimensions of mathematical understanding

Skill-algorithm understanding (Algorithms, CAS)

from the rote application of an algorithm through the selection and comparison of algorithms to the invention of new algorithms

Properties - mathematical underpinnings understanding (Deduction, Isomorphism)

from the rote justification of a property through the derivation of properties to the proofs of new properties

Uses-applications understanding (Uses)

from the rote application of mathematics in the real world through the use of mathematical models to the invention of new models

Representations-metaphors understanding (Transformations)

from the rote representations of mathematical ideas through the analysis of such representations to the invention of new representations

Thank you!