A Blueprint for the Foundations of Algebracould be the first step in SPS math repair

Dear Seattle School Board Members, 3-22-2008

The recently released National Math Panel Report should be used in giving you appropriate guidance in the coming high school math adoption. The panel’s recommendations reveal that the primary materials from the last two math adoptions are in serious misalignment with the panel’s views. I urge you to consider not just a high school adoption at this time but how the current math direction of the SPS can be corrected. I hope you will find the following information useful.

The recommendations included in “The Final Report of the National Mathematics Advisory Panel” and the “Report of the Task Group on Conceptual Knowledge and Skills” released on March 13, 2008 should be carefully considered in any Seattle Schools Math Adoptions, especially the items listed here as they are the most pertinent to standards:

A.. Focused, Coherent Progression of Learning
....i...A focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics, should become the norm in elementary and middle school mathematics curricula. (Get someone to work on D44.00 and D45.00, this neglect is malpractice)
....ii...Avoid approaches that revisit topics superficially, year after year without closure. (This means Everyday Math should be avoided.)

B.. Authentic Algebra
....iii...As called out in NMAP Table 1: “The Major Topics of School Algebra”
....iv...While the Panel finds no basis in research for a preference of integrated versus single-subject approach, an analysis of high school mathematics standards suggests that high school students enrolled in mathematics courses using an integrated approach to mathematics may find it more difficult to take advanced mathematics course work (e.g., calculus or pre-calculus) in their senior year than high school students who are able to enroll in an Algebra II course in their sophomore or junior year.
....v...Universal availability of a basic math education for more students than at present preparing them to enroll in Authentic Algebra by Grade 8.
...vi...The Major Topics of School Algebra should be the focus for Algebra I and Algebra II standards in state curriculum frameworks.
...vii...Algebra II by Grade 11.

C.. K-8 Critical Foundations
viii. As called out in Table 2: K-8 Critical Foundations
ix. Yearly progress to match or beat by the grade level indicated in the critical foundations.
x. The Critical Foundations are not meant to comprise a complete mathematics curriculum leading to algebra; however, they deserve primary attention and ample time in any mathematics curriculum

D.. Use of Calculators
....xi. The Panel cautions that calculators impede the development of automaticity and adversely affect fluency in computation. (What can be the possible benefit to calculators prior to grade 6 vs. risks?)
xii. Calculators should not be used on test items designed to assess computational facility.

E.. Fluency with Whole Numbers
....xiii. Computational proficiency with whole number operations is dependent on sufficient and appropriate practice to develop automatic recall of addition and related subtraction facts, and of multiplication and related division facts.
xiv. Fluency with the standard algorithms for addition, subtraction, multiplication, and division.

F.. Fluency with Fractions
....xv. A major goal for K–8 mathematics education should be proficiency with fractions (including decimals, percents, and negative fractions), for such proficiency is foundational for algebra and, at the present time, seems to be severely underdeveloped.
....xvi. Ensure acquisition of conceptual and procedural knowledge of fractions (including decimals and percents) and of proportional reasoning. This should include:
1. Representing and ordering fractions on a number line
2. Judging equivalence and relative magnitudes of fractions with unlike numerators and denominators
3. Solving problems involving ratios and proportion.
4. Knowing that sums, differences, products, and quotients of fractions are fractions and carry out these operations confidently and efficiently.
5. Understand why and how (finite) decimal numbers are fractions and know the meaning of percentages

G.. Particular aspects of geometry and measurement, especially for middle grades
....xvii. Knowledge of similar triangles
....xviii. The slope of a straight line and of linear functions
....xix. The properties of two- and three-dimensional shapes and the use of formulas to determine perimeter, area, volume, and surface area.
....xx. Finding unknown lengths, angles, and areas (geometry and not trigonometry).

H.. Less focus should be placed on the patterns, data and probability, especially in the early grades.
....xxi. In the Major Topics of School Algebra set forth in the report, patterns are not a topic of major importance; therefore, patterns should be de-emphasized.
....xxii. In asking for redesign of the NAEP, the Panel noted the importance of fractions for the conceptual understanding of probability. The Panel questioned the appropriateness of items related to probability within NAEP at Grade 4.
....xxiii. The Panel recommends that the strand on Data Analysis and Probability in the NAEP at Grade 4 emphasize well-organized representations of data pictorially and numerically and be re-titled as “Data Display”.
Sincerely, Danaher M. Dempsey, Jr.


Table 1: The Major Topics of School Algebra
.... (EM and CMP2 are not preparing enough students adequately for these topics.)

Symbols and Expressions
• Polynomial expressions
• Rational expressions
• Arithmetic and finite geometric series

Linear Equations
• Real numbers as points on the number line
• Linear equations and their graphs
• Solving problems with linear equations
• Linear inequalities and their graphs
• Graphing and solving systems of simultaneous linear equations

Quadratic Equations
• Factors and factoring of quadratic polynomials with integer coefficients
• Completing the square in quadratic expressions
• Quadratic formula and factoring of general quadratic polynomials
• Using the quadratic formula to solve equations

Functions
• Linear functions
• Quadratic functions—word problems involving quadratic functions
• Graphs of quadratic functions and completing the square
• Polynomial functions (including graphs of basic functions)
• Simple nonlinear functions (e.g., square and cube root functions; absolute value; rational functions; step functions)
• Rational exponents, radical expressions, and exponential functions
• Logarithmic functions
• Trigonometric functions
• Fitting simple mathematical models to data

Algebra of Polynomials
• Roots and factorization of polynomials
• Complex numbers and operations
• Fundamental theorem of algebra
• Binomial coefficients (and Pascal’s Triangle)
• Mathematical induction and the binomial theorem

Combinatorics and Finite Probability
• Combinations and permutations, as applications of the binomial theorem and Pascal’s Triangle

I urge you to begin Math Repair mode with the adoption of the Algebra Readiness Textbook and Courseware from the Mind Research Institute. At least 40% of entering 9th graders in Seattle High Schools would benefit from these materials, as these children are too unskilled to undertake the Authentic Algebra recommended by the National Math Panel. These materials could also be effectively used for many children in SPS middle schools. These materials in addition to preparing students for Authentic Algebra will close the Math Achievement Gap that has continually widened in the SPS over the last decade.


I urge you to look at the textbook “A Blueprint for the Foundations of Algebra” and the accompanying software. This can be the foundation of building a strong math foundation for SPS students. It can be the anchor point for real Authentic Algebra and a successful High School math program.

Recommendation: The Benchmarks for the Critical Foundations in Table 2 should be used to guide classroom curricula, mathematics instruction, and state assessments. They should be interpreted flexibly, to allow for the needs of students and teachers.

Table 2: Benchmarks for the Critical Foundations

Fluency With Whole Numbers
1) By the end of Grade 3, students should be proficient with the addition and subtraction of whole numbers.
2) By the end of Grade 5, students should be proficient with multiplication and division of whole numbers.

Fluency With Fractions
1) By the end of Grade 4, students should be able to identify and represent fractions and decimals, and compare them on a number line or with other common representations of fractions and decimals.
2) By the end of Grade 5, students should be proficient with comparing fractions and decimals and common percents, and with the addition and subtraction of fractions and decimals.
3) By the end of Grade 6, students should be proficient with multiplication and division of fractions and decimals.
4) By the end of Grade 6, students should be proficient with all operations involving positive and negative integers.
5) By the end of Grade 7, students should be proficient with all operations involving positive and negative fractions.
6) By the end of Grade 7, students should be able to solve problems involving percent, ratio, and rate and extend this work to proportionality.

Geometry and Measurement

1) By the end of Grade 5, students should be able to solve problems involving perimeter and area of triangles and all quadrilaterals having at least one pair of parallel sides (i.e., trapezoids).
2) By the end of Grade 6, students should be able to analyze the properties of two-dimensional shapes and solve problems involving perimeter and area, and analyze the properties of three-dimensional shapes and solve problems involving surface area and volume.
3) By the end of Grade 7, students should be familiar with the relationship between similar triangles and the concept of the slope of a line.
Source: National Mathematics Advisory Panel, 2008.
(please let the above be reflected in D44.00 and D45.00 by next Fall)

A Need for Coherence
There seem to be two major differences between the curricula in top-performing countries and those in the U.S.—in the number of mathematical concepts or topics presented at each grade level and in the expectations for learning. U.S. curricula typically include many topics at each grade level, with each receiving relatively limited development, while top-performing countries present fewer topics at each grade level but in greater depth. In addition, U.S. curricula generally review and extend at successive grade levels many (if not most) topics already presented at earlier grade levels, while the top-performing countries are more likely to expect closure after exposure, development, and refinement of a particular topic. These critical differences distinguish a spiral curriculum (common in many subjects in U.S. curricula) from one built on developing proficiency—a curriculum that expects proficiency in the topics that are presented before more complex or difficult topics are introduced. (Everyday Math is a spiral curriculum not built on developing proficiency.)

The Singapore standards (Singapore Ministry of Education, 2006) provide an established example of curriculum standards designed to develop proficiency in a relatively small number of important mathematics topics, as validated by a recent analysis (Ginsburg et al., 2005). The desirability of emphasizing fewer important mathematics topics in greater depth has also been recognized by some U.S. educators. (but not those making math curriculum decisions in Seattle )

In 2005, the Fordham Foundation report on state mathematics standards (Klein et al., 2005) ranked state mathematics curriculum standards based on mathematics content, clarity, and reasoning, as well as negative qualities, assigning different weights to each criterion for the overall assessment. The standards of California, Indiana, Massachusetts, Alabama, New Mexico, and Georgia achieved the highest ranking. The curricular profiles of the standards of these six states do, on the whole, provide an emphasis on fewer important topics per year than most states; but compared with the “A+ countries” (Singapore, Japan, Korea, Hong Kong, Flemish Belgium, and the Czech Republic), they all spend a great deal of time in the primary grades on topics other than arithmetic.

A more recent development in the national discussion is the publication of Focal Points (National Council of Teachers of Mathematics, 2006), which offers curricular direction to teachers and administrators by suggesting areas of emphasis for the concepts, skills, and procedures that connect important mathematics topics from grade to grade, and form the foundation for more advanced mathematics, beginning with Algebra. The message of Focal Points is also one of curriculum coherence with an emphasis on fewer important topics per year. Focal Points does not represent a set of standards but calls for a curriculum which reduces the number of important topics per year. In effect, Focal Points asks for greater emphasis on key topics, particularly with whole numbers and fractions and particular aspects of geometry and measurement. Yet Focal Points still implies more time on non-number topics, especially in the primary grades, than is the case in the A+ countries but less than the intended mathematics curriculum as represented in the frameworks of the six states.
The Panel also notes that a state’s (or a country’s) mathematics standards, however highly their quality may be judged, cannot ensure high student achievement. For example, the six leading states in the Fordham study exhibit a wide range of student achievement on the 2007 NAEP mathematics tests for Grades 4 and 8. The quality of a state’s assessments and the extent to which its standards drive sound school curricula, as well as appropriate programs for teacher preparation and professional development, are intervening variables that strongly influence achievement. They may well override the quality of the standards. (It appears that the Everyday Math alignment with failed state standards was the primary motivation for its adoption -- Now what??? -- Mind Institute software could be a start to repairing the Reform Math damage done to students in grades K-8)
-------------------------------------------------------------------------------------------------
MIND Research Institute’s Algebra Readiness is a new, one year curriculum that rebuilds a solid math foundation for struggling middle and high school students who would otherwise enroll in, and fail, Algebra I. The innovative learning sequence and distinctive visual diagrams of its textbook fully explain and interconnect all the essential math concepts and skills from grades 2 through 7 which form the foundation for success in Algebra I. The text is fully integrated with MIND’s unique Spatial Temporal courseware, which engages students who struggle in conventional math programs.

http://www.mindinst.org/cont/edu/stmath.php

Please click the above link to find out what will bring math success for ELL and most students who are unprepared for Algebra in SPS schools.