PA Coalition for World Class Math

                       Design Principles for K-12 Mathematics Standards and Assessments

United States Coalition for World Class Math

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In this document, we offer a set of principles to guide the development of strong mathematics standards and assessments from kindergarten through high school. Standards should be clear, accurately stated, rigorous, and grade-level specific, increasing in complexity logically and coherently from grade to grade, while concisely identifying the most important topics each year. They should develop the three pillars of mathematics learning: computational fluency, problem-solving skills, and conceptual understanding. They should reflect the best of what is taught internationally. Strong standards are the framework for a sound mathematics curriculum. A sound K-12 curriculum ensures that all students can take credit-bearing mathematics courses in their freshman year if, after high school, they choose to continue their education at a two or four-year college.

1.  All students should be expected to master foundational concepts and skills – especially in arithmetic – that are prerequisite to an authentic Algebra I course in a logical progression from grade to grade in the elementary and middle school years. The Final Report of the National Mathematics Advisory Panel (NMAP) should be the guiding document for appropriate mathematical content.

2.  The K-7 standards should be designed to prepare as many students as possible for an authentic Algebra I course in Grade 8. K-7 standards should be based on the "Critical Foundations of Algebra" described on pages 17-19 of the NMAP’s Final Report. Standards for authentic Algebra I and Algebra II courses should be based on "The Major Topics of School Algebra" described on pages 15-16 of the NMAP’s Final Report.

3.  Standards-based alternatives could be written for less prepared students and for alternate paths after algebra and geometry for high school students, depending on student achievement, interests, and career goals. For example:

a.  The standards document could outline the possibility of a two-year course spanning Grade 7 and Grade 8 based on Grade 7 standards for students who, at the end of Grade 6, are judged to need more time to master foundational concepts and skills for Algebra I.

b.  The standards document could outline a two-year course spanning Grade 8 and Grade 9 based on authentic Algebra I standards for students completing Grade 7 who are judged to need two full years to master Algebra I standards.

4.  As emphasized by the National Mathematics Advisory Panel, "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. Any approach that continually revisits topics year after year without closure is to be avoided." Placement of the standards should reflect the grade level at which mastery is expected, and standards should not be repeated from year to year.

a.  The sequence of the standards should be logical and hierarchical, following the structure of mathematics itself and should be modeled after the strong standards in California, Indiana, and Massachusetts.

b.  "Benchmarks for the Critical Foundations" (pages 19-20 in the National Mathematics Advisory Panel’s Final Report) and recommendations from the National Council of Teachers of Mathematics’ Curriculum Focal Points should guide grade-level placement.

c.  Concepts and skills, once mastered, should be used in subsequent years with a minimum of review.

5.  In order to focus on building solid foundations for the more advanced mathematics – including algebra – that occurs in Grades 8-12, extraneous topics including aspects of geometry such as tessellations, nets, statistical approaches to geometric properties, much of data analysis, probability and statistics, and non-algebraic concepts such as pattern recognition should not be present in the K-7 standards.

6. In Grades K-7, the distribution of content by strand should be stated explicitly as percentages at each grade level and that distribution should change as students move up through the grades.

a.  Early grades should concentrate on the arithmetic of whole numbers and measurement, with a limited amount of geometry and graphing. Certain aspects of algebra, as well as preparation for algebra, should be present from the earliest grades, as is the case with the California and Massachusetts standards.

b.  Students should be expected to acquire automatic recall of basic number facts at least to 10 x 10 and 10 + 10.

c.  Students should be expected to understand and use the standard algorithms of whole-number arithmetic in the early elementary grades (i.e., addition, subtraction, multiplication, and long division).

d.  Students should be expected to understand and use the standard definitions for operations with fractions in conjunction with the standard algorithms of whole number arithmetic to compute sums, differences, products and quotients of fractions, including fractions expressed as decimals and percents.

e.  The algebra strand should gain emphasis in the middle grades, focusing on the content specified by the National Mathematics Advisory Panel.

7.  The organization of the standards should change at Grade 8.

a.  In grades K-7, standards should include multiple strands of mathematics, with their relative weight appropriately adjusted through the grades.

b.  For algebra and beyond, standards should be given for a single-subject course sequence (Algebra I, Geometry, Algebra II, Pre-calculus, etc.) and their components re-ordered for alternative integrated mathematics courses. The standards for the Geometry course should require students to do proofs and to understand postulates, theorems, and corollaries.

8.  Mathematical problems should have mathematical answers.      

a.  In general, students should learn techniques for problem solving that can be applied to many contexts. Problems should be contextualized in the "real world" only when the context is sensible and relevant and contributes to an understanding of the mathematics in the problem.

b.  Standards documents should include example problems. The level of difficulty of these problems should reflect mathematical complexity rather than non-mathematical issues.

9.   K-12 math standards should meet the criteria specified by the American Federation of Teachers. They should be:

a.  Clear and specific enough to provide the basis for a common core curriculum.

b.  Rooted in the content of mathematics.

c.  Clear and explicit about the content and the complexity students are to learn.

d.  Measurable and objective.

e.  Comparable in rigor to the standards of the "A+" countries, with grade-level specificity.

10.  Standards documents should appropriately emphasize the attainment of procedural fluency. Students must be competent in performing all K-7 tasks without using a calculator.

11.  Standards documents should only address mathematical content; language pertaining to pedagogy should be excluded.

12.  Effective assessments are necessary to verify student learning and guide the implementation of a standards-based curriculum. To that end, grade-level assessments should be tailored to grade-level standards, and should assess both the depth and the breadth of student learning of the important mathematics that is embodied in the standards. Students should have a personal stake in these assessments; hence assessments should deliver individual and reliable results.

Mathematics assessments in K-12 should: 

a.  reliably measure the breadth and depth of the content knowledge and skills defined in the grade-level standards;

b.  provide individually reliable scores to students and parents;

c.  assess both struggling and advanced learners adequately; i.e., test items should assess high as well as low performance adequately.

d.  focus on and adequately represent the "Critical Foundations of Algebra" in elementary and middle school assessments. Toward this end, they should minimize the number of questions or, at some grade levels, eliminate them altogether, on such topics as: 

  1. patterns, e.g., "the next number in the sequence is ...," since such questions cannot be phrased rigorously at this level;
  2. probability and inferential statistics, which can and should be postponed to high school or college courses (meaning that graphs, and concepts like "median" and "mean," should continue to be taught); 
  3. transformations of geometric figures, which can be treated only superficially in K-12;

e.  weight strands in ways that prioritize the mathematics content needed for Algebra I and more advanced high school mathematics courses;

f.   make "real-life" test items linguistically simple, to maximize access for English language learners;

g.  avoid the types of errors, confusions, or other issues in "real-world" problems noted by the National Mathematics Advisory Panel’s Assessment task group;

h.  disallow use of calculators on testing at least through Grade 6;

j.  focus on efficient assessment techniques to minimize assessment burden; and

k.  assess both procedural skills (i.e., computation) and problem solving (i.e., word problems).

13.  Scoring of items on mathematics assessments must take care to avoid conflating proficiency in prose writing with proficiency in mathematics.

14.  As emphasized by the National Mathematics Advisory Panel, "mathematicians should be included in greater numbers, along with mathematics educators, mathematics education researchers, curriculum specialists, classroom teachers, and the general public, in the standard-setting process (setting cut scores) and in the review and design of mathematical test items for state, NAEP, and commercial tests."

15.  Mathematics professors who direct majors and programs for students who will use significant mathematics in science, engineering, and other technical fields should be included in all activities involving development of standards and assessments. Their professional judgment as to the K-12 mathematics content needed to prepare students for success in such majors and programs should play a significant part in determining the content of K-12 standards and assessments.

 Revised 6/30/09