What Are Some Organizing Principles
Around Which One Can Create a
Coherent Pre-college Algebra Program?
Critical Issues in Education:
Teaching and Learning Algebra
MSRI, Berkeley, CA
May 14, 2008
Zalman Usiskin
The University of Chicago
z-usiskin@uchicago.edu
The UCSMP Curriculum for Grades 6-12
Grade
Top 10–20% o f
Students
5
EM 6 or Pre-Transition
Mathematics
Next 50% of
6
Transition
Mathematics
EM 6 or Pre-Transition
Mathematics
Next 20% of
Algebra
Transition
Mathematics
Pre-Transition
Mathematics
Remainder of
Students
Geometry
Algebra
Transition
Mathematics
Pre-Transition
Mathematics
Advanced Algebra
Geometry
Algebra
Transition
Mathematics
10
Functions, Statistics,
and Trigonometry
Advanced Algebra
Geometry
Algebra
11
Precalculus and
Discrete Mathematics
Functions, Statistics,
and Trigonometry
Advanced Algebra
Geometry
12
Calculus (Not available
through UCSMP)
Precalculus and
Discrete Mathematics
Functions, Statistics,
and Trigonometry
Advanced Algebra
7
8
9
Students
Students
The algorithmic approach
• 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
Major Organizing Principles for Algebra
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.
An example of the deductive approach
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.
Major Organizing Principles for Algebra
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.
Theorems about Graphs
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.
Some Corollaries of the
Graph Translation Theorem
Parent
y = mx
y = mx
y = ax2
x2 + y2 = r2
y = Asin x
Offspring
y – b = mx
y – y0 = m(x – x0)
y – k = a(x – h)2
(x – h)2 + (y – k)2 = r2
y = Asin(x – h)
Slope-intercept form
Point-slope form
Vertex form
General circle
Phase shift
Defining the sine and cosine
(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
 cos x  sin x 
Rx is 
.

 sin x cos x 
Deducing formulas for
cos(x+y) and sin(x+y)
 cos(x  y)  sin(x  y) 
 sin(x  y) cos(x  y) 


 cos x  sin x   cos y  sin y 
=  sin x cos x   sin y cos y 



Rx+
y
Rx ° Ry
 cos x cos y – sin x sin y  sin x cos y  cos x sin y 
= sin x cos y  cos x sin y cos x cos y – sin x sin y 
Major Organizing Principles for Algebra
1.
2.
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.
3. Use geometry. Transformations provide a
powerful set of ideas for dealing with graphs
of functions and trigonometry.
Field properties (typical arrangement)
For all real numbers a, b, and c:
a + b is a real number.
a+b=b+a
a + (b + c) = a + (b + c)
0 such that a + 0 = a.
ab is a real number.
ab = ba
a(bc) = ab(c)
1 such that a•1 = a.
(-a) such that a + (-a) = 0.
(1/a) such that a•(1/a) = 1.
a(b + c) = ab + ac
Some isomorphic properties
For all real numbers a
and reals m and n:
a+a+...+a = ma
14 2 43
m addends
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:
a•a•...•a
14 2 43 = xm
m factors
xm • xn = xm+n
x0 = 1
(xm)n = xmn
m<0 and x<1  xm > 1
More isomorphic ideas
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
Major Organizing Principles for Algebra
3. 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.
Major Organizing Principles for Algebra
5. Consider the students. A course for all
students cannot assume they all have the
background, motivation, and time that we
would prefer.
6. 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.)
Uses of Numbers
counts
measures
ratio comparisons
scale values
locations
codes and identification
Use meanings of operations
Addition
Putting-together, slide
Subtraction
Division
Take-away, comparison (incl. error,
change)
Area (array), rate factor, size
change
Ratio, rate
Powering
Permutation, growth
Multiplication
Conception of
Use of
algebra
variables
Generalized arithmetic Pattern
generalizers
Means to solve
Unknowns,
problems
constants
Study of relationships Arguments,
parameters
Abstract structure
Arbitrary
marks on paper
Actions
Translate,
generalize
Solve,
simplify
Relate,
graph
Manipulate,
justify
Use meanings of operations
Addition
Putting-together, slide
Subtraction
Division
Take-away, comparison (incl. error,
change)
Area (array), rate factor, size
change
Ratio, rate
Powering
Permutation, growth
Multiplication
Using the growth model
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.
Basic uses of functions
Linear
Quadratic
Exponential
Polynomial
Linear combination; constantincrease/constant-decrease
Area; 2-dimensional arrays;
acceleration
Permutation, growth
Basic uses of functions
Linear
Exponential
Linear combination; constantincrease/constant-decrease
Area; 2-dimensional arrays;
acceleration
Permutation, growth
Polynomial
Annuities
Quadratic
NMAP statement
“The use of ‘real-world’ contexts to introduce
mathematical ideas has been advocated… A
synthesis of findings from a small number of highquality studies indicates that if mathematical ideas
are taught using ‘real-world’ contexts, then students
performance on assessments involving similar ‘realworld’ 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
General theorems for solving sentences
in one variable
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
c
1
2
3
4
5
6
7
8
9
10
x^2 + 6x + c
x^2 + 6x + 1
x^2 + 6x + 2
x^2 + 6x + 3
x^2 + 6x + 4
x^2 + 6x + 5
x^2 + 6x + 6
x^2 + 6x + 7
x^2 + 6x + 8
x^2 + 6x + 9
x^2 + 6x + 10
factor(x^2 + 6x + c)
x^2 + 6x + 1
x^2 + 6x + 2
x^2 + 6x + 3
x^2 + 6x + 4
(x + 1)(x + 5)
x^2 + 6x + 6
x^2 + 6x + 7
(x + 2)(x + 4)
(x + 3)(x + 3)
x^2 + 6x + 10
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!
z-usiskin@uchicago.edu