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, CAMay 14, 2008
Zalman UsiskinThe University of [email protected]
GradeTop 10–20% of
Students
5EM 6 or Pre-Transition
Mathematics
Next 50% of
Students
6Transition
MathematicsEM 6 or Pre-Transition
Mathematics
Next 20% of
Students
7Algebra Transition
MathematicsPre-TransitionMathematics
Remainder ofStudents
8Geometry Algebra Transition
MathematicsPre-TransitionMathematics
9Advanced Algebra Geometry Algebra Transition
Mathematics
10Functions, Statistics,
and TrigonometryAdvanced Algebra Geometry Algebra
11Precalculus and
Discrete MathematicsFunctions, Statistics,
and TrigonometryAdvanced Algebra Geometry
12Calculus (Not available
through UCSMP)Precalculus and
Discrete MathematicsFunctions, Statistics,
and TrigonometryAdvanced Algebra
The UCSMP Curriculum for Grades 6-12
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 Offspringy = mx y – b = mx Slope-intercept form
y = mx y – y0 = m(x – x0) Point-slope form
y = ax2 y – k = a(x – h)2 Vertex formx2 + y2 = r2 (x – h)2 + (y – k)2 = r2 General circley = Asin x y = Asin(x – h) Phase shift
Defining the sine and cosine
cos x −sinxsinx cosx
⎡
⎣⎢
⎤
⎦⎥
(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 .
Deducing formulas for cos(x+y) and sin(x+y)
cos(x + y) −sin(x+ y)sin(x+ y) cos(x+ y)
⎡
⎣⎢
⎤
⎦⎥
cos x −sinxsinx cosx
⎡
⎣⎢
⎤
⎦⎥
cos y −sinysiny cosy
⎡
⎣⎢
⎤
⎦⎥
cos x cos y – sin x sin y −sinxcosy−cosxsinysinxcosy+ cosxsiny cosxcosy–sinxsiny
⎡
⎣⎢
⎤
⎦⎥
=
=
Rx+y
Rx ° Ry
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.
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.(-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
Some isomorphic properties
For all real numbers a and reals m and n:
= ma
ma + na = (m + n)a0a = 0n(ma) = (nm)am< 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
a+a+...+am addends
1 24 34 a•a•...•a
m factors1 24 34
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 Take-away, comparison (incl. error, change)
Multiplication Area (array), rate factor, size change
Division Ratio, rate
Powering Permutation, growth
Conception of algebra
Use of variables
Actions
Generalized arithmetic Pattern generalizers
Translate, generalize
Means to solve problems
Unknowns, constants
Solve, simplify
Study of relationships Arguments, parameters
Relate, graph
Abstract structure Arbitrary marks on paper
Manipulate, justify
Use meanings of operations
Addition Putting-together, slide
Subtraction Take-away, comparison (incl. error, change)
Multiplication Area (array), rate factor, size change
Division Ratio, rate
Powering Permutation, growth
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 Linear combination; constant- increase/constant-decrease
Quadratic Area; 2-dimensional arrays; acceleration
Exponential Permutation, growth
Polynomial
Basic uses of functions
Linear Linear combination; constant- increase/constant-decrease
Quadratic Area; 2-dimensional arrays; acceleration
Exponential Permutation, growth
Polynomial Annuities
NMAP statement
“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)
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
Dimensions of mathematical understanding
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 x^2 + 6x + c factor(x^2 + 6x + c)1 x^2 + 6x + 1 x^2 + 6x + 12 x^2 + 6x + 2 x^2 + 6x + 23 x^2 + 6x + 3 x^2 + 6x + 34 x^2 + 6x + 4 x^2 + 6x + 45 x^2 + 6x + 5 (x + 1)(x + 5)6 x^2 + 6x + 6 x^2 + 6x + 67 x^2 + 6x + 7 x^2 + 6x + 78 x^2 + 6x + 8 (x + 2)(x + 4)9 x^2 + 6x + 9 (x + 3)(x + 3)10 x^2 + 6x + 10 x^2 + 6x + 10
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
Dimensions of mathematical understanding
