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Math Content-Specific Criteria

Math Content-Specific Criteria

This section lays out the core or minimal content material that students in fourth, eighth, and twelfth grade should know, as well as fifteen-year-old students (above and beyond eighth-grade expectations). It also includes an explanation of the proper treatment of data analysis, statistics, and probability in K-12 education.

Fourth Grade

Addition and subtraction

  1. Students should know the single-digit addition facts and the corresponding subtraction facts.
  2. They should be able to add and subtract whole numbers fluently and understand the algorithms they use.
  3. They should understand the inverse nature of addition and subtraction.

Multiplication and division

  1. Students should know the single-digit multiplication facts and be fluent with multiplying whole numbers.
  2. They should understand the algorithms they use.
  3. Students should be introduced to division and understand the inverse nature of multiplication and division.

Measurement

  1. Students should understand some forms of measurement. In particular, they should know and understand the formulas for the area and perimeter of a rectangle and know how to measure distances to the nearest centimeter and half-inch using rulers.

Problem solving

  1. Students should be able to use the above essential material to solve complex multi-step exercises and word problems.
  2. In addition to the above core that should be mastered, they should be given an introduction to other material. Students should be introduced to fractions and decimals and be able to compare the magnitude of simple fractions and decimals.
  3. Students should be able to use commutativity and associativity.
  4. They should be able to do elementary estimations.
  5. They need a basic vocabulary for geometry, including parallel and perpendicular lines.
  6. They should be exposed to very elementary statistics and probability including reading and making basic graphs and tables.

Eighth Grade

Arithmetic

  1. Students should be fluent with the four arithmetic operations with rational numbers and decimals, without a calculator, and they should understand the operations.

Ratios

  1. Students should understand and be able to use rates, ratios, proportions, and percentages.

Geometry and measurement

  1. Students should have a good geometry vocabulary, know about angles associated with triangles and parallel lines, understand similar and congruent triangles, and be able to compute areas, perimeters and volumes of various geometrical shapes, including circles.
  2. They should know the Pythagorean Theorem and the coordinate system.

Problem solving

  1. Students should be able to use the above essential material to solve complex multi-step exercises and word problems.
  2. In addition to the essentials above, students should have an elementary knowledge of data analysis, statistics, and probability. (See below, “How much DASP do students need?”)
  3. They should understand absolute value and be able to solve elementary linear equations.

Fifteen year-olds

Fifteen year-olds should have mastered eighth-grade material and they should have the equivalent of a year of algebra as well. They should understand and be able to use roots, reciprocals, and powers. They should be able to solve equations and inequalities that are linear or involve the absolute value. Students should know about slope and forms of linear functions and how to graph them. Students should be fluent with the arithmetic of polynomials and rational expressions, including elementary factoring. Students should be able to use the above essential material to solve complex multi-step exercises and word problems. In addition to the essentials above, students should be introduced to quadratic equations and their solutions.

Twelfth Grade

Minimal twelfth grade expectations, in addition to the expectations for fifteen-year-olds, include a working knowledge of geometry containing constructions, definitions, and proofs of the major results in Euclidean geometry. Students should understand logarithmic and exponential functions as well as basic trigonometry and trigonometric functions. Students should also be able to completely analyze quadratic equations, inequalities, and functions. Students should know basic statistics and probability, particularly the counting arguments involving combinations and permutations. Students should be able to use this essential material to solve complex multi-step exercises and word problems.

HOW MUCH DASP DO STUDENTS NEED?

Data Analysis, Statistics, and Probability (DASP), play a prominent role in many sets of standards. Certainly students should be able to read and construct various kinds of graphs and displays for data, compute elementary probabilities, and know the basic descriptors for data such as percentiles, mean, and median. In addition, they should be able to use combinations and permutations in their counting arguments for probability and have some concept of random variability. Many standards, however, go far beyond these basic needs. Justification for this is usually slim, frequently based on a vague notion that a good citizen should know this material. Presently, many standards include deep mathematical content that is far above the level of mathematics accessible to K-12 students. This results in superficial coverage that often lacks mathematical coherence. In addition, the “grain size” of DASP standards tends to be smaller than for other areas of math standards, i.e., more detail is devoted to them. This increases the number and percentage of standards dedicated to DASP and gives them more prominence than they deserve.

A common example, frequently observed in middle school standards, is a variant on “eye-balling” a line of best fit for a scatter-plot. More specifically, the eighth grade NAEP framework has this standard: “Visually choose the line that best fits given a scatterplot.” “Visually choosing” (i.e., eye-balling) is not a mathematical activity. Further, the concept of “least squares” is mentioned in the NAEP twelfth grade standards, though it is inappropriate to include it. That’s because most students do not have the requisite math background to see this for what it is: Finding the closest point on a plane in n-dimension Euclidean space to a point not on that plane using the Pythagorean Theorem to compute the distances. This is material from a college-level linear algebra or multivariable calculus course, courses that generally follow calculus. Another example is when students are exposed to the normal distribution, but its origin and connection to the binomial distribution—college-level material to be sure—is not covered, leading to superficial coverage of the former. There are many other examples where students lack mathematical understanding with regard to what is actually occurring in a DASP problem.

Further, recent research has shown that DASP does not prepare students for placement exams in college or for college level material,3 nor is it material requested by college mathematics teachers.4 Certainly students headed for scientific and technological careers will be required to take college-level courses in statistics and probability, as will anyone else who needs this material in college. In the end, though, knowledge of arithmetic, ratios and algebra will be much more important to all students than DASP content void of a conceptual base. After all, students should possess the mathematical foundation that allows them to understand what they are doing.

Read moreABOUT THIS REPORT

Our review ofthe NAEP Framework

Our review ofState Standards