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

1. 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 Chicagoz-usiskin@uchicago.edu

2. The UCSMP Curriculum for Grades 6-12

3. 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

4. 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.

5. 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.

6. 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.

7. 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.

8. Some Corollaries of the Graph Translation Theorem Parent Offspring y = 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 form x2 + y2 = r2 (x – h)2 + (y – k)2 = r2 General circle y = Asin x y = Asin(x – h) Phase shift

9. 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 Rx is .

10. Deducing formulas for cos(x+y) and sin(x+y) Rx+y = Rx° Ry =

11. 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. • 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.

12. 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. Field properties (typical arrangement) For all real numbers a, b, and c: a(b + c) = ab + ac

13. 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 Some isomorphic properties

14. 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 More isomorphic ideas

15. Major Organizing Principles for Algebra • 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.

16. Major Organizing Principles for Algebra • 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.)

17. Uses of Numbers counts measures ratio comparisons scale values locations codes and identification

18. Use meanings of operations

20. Use meanings of operations

21. 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.

22. Basic uses of functions

23. Basic uses of functions

24. 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)

25. 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

26. 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)).

27. Exploring the factoring of x2 + 6x + c

28. 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

29. Thank you! z-usiskin@uchicago.edu