kitchen table math, the sequel: Natl Math Panel: increase algebra in 8th grade

Friday, March 14, 2008

Natl Math Panel: increase algebra in 8th grade

All school districts should ensure that all prepared students have access to an authentic algebra course—and should prepare more students than at present to enroll in such a course by Grade 8. The word authentic is used here as a descriptor of a course that addresses algebra consistently with the Major Topics of School Algebra (Table 1, page 16). Students must be prepared with the mathematical prerequisites for this course according to the Critical Foundations of Algebra (page 17) and the Benchmarks for the Critical Foundations (Table 2, page 20).

FINAL REPORT
p. xviii

Table 1: Major Topics of School Algebra
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

Recommendation: Proficiency with whole numbers, fractions, and particular aspects of geometry and measurement should be understood as the Critical Foundations of Algebra. Emphasis on these essential concepts and skills must be provided at the elementary and middle grade levels.
Recommendation: The coherence and sequential nature of mathematics dictate the foundational skills that are necessary for the learning of algebra. The most important foundational skill not presently developed appears to be proficiency with fractions (including decimals, percents, and negative fractions). The teaching of fractions must be acknowledged as critically important and improved before an increase in student achievement in algebra can be expected.

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 threedimensional 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:
FINAL REPORT, 2008.

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