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  1. Analysis of State-Level Mathematics Curriculum Standards Barbara J. Reys University of Missouri Center for the Study of Mathematics Curriculum

  2. What mathematics should students learn and when should they learn it?

  3. No Child Left Behind Each state is required to: • adopt challenging academic content standards that will be used by the State, its local educational agencies, and its schools. • measure the achievement of students in mathematics against the standards in each of grades 3 through 8.

  4. Publication of Newest State-Level Mathematics Standards(as of 4-10-06) 2006 4 states (ID, MS, NH, NV) 2005 9 states 2004 13 states 2003 8 states (UT) 2002 4 states 2001 4 states 2000 2 states pre-2000 7 states

  5. http://mathcurriculumcenter.org

  6. Mathematics Core Curriculum Utah State Office of Education (2003) “The Core Curriculum represents those standards of learning that are essential for all students. They are the ideas, concepts, and skills that provide a foundation on which subsequent learning may be built . . . Although the Core Curriculum standards are intended to occupy a major part of the school program, they are not the total curriculum of a level or course.”

  7. Grade-Level Learning Expectations (GLE) • GLE documents describe mathematics learning expectations for specific grades • 43 states have GLE documents • Most common grades: K-8 (37 states) • Others: K-7, 3-8 or 3-10 (5 states)

  8. Increased Specificity, Authority, and Influence For many states, the most recent standards documents represent increased specificity of learning goals compared to previous state standards documents. The learning goals carry additional “weight” since they are tied to annual assessments in grades 3-8. Teachers and state department leaders acknowledge the increased influence of state standards in determining curriculum focus at the classroom level.

  9. Differences in GLE Documents • Organization and Language of GLEs. • Level of specificity or grain size of learning expectations. • Grade placement of key topics.

  10. Utah Core Curriculum • Intended Learning Outcomes • Standards • Objectives • Indicators

  11. Example GLEs (Basic Number Combinations) • Know the addition facts (sums to 20) and the corresponding subtraction facts and commit them to memory. (CA, gr. 1) • States and uses with efficiency and accuracy basic addition facts with sums from 0 to 20 and corresponding subtraction facts. (KS, gr. 2) • Recall basic addition and subtraction facts through 18. (ID, gr. 3)

  12. Number of GLEs(grain size) Grade *Does not include the Intended Learning Outcomes (ILOs) in grades 3-6. Mean number of LEs per grade level for the 42 state GLE documents: 47 LEs

  13. Our Focus To what extent are the elementary and middle school (K-8) grade-level learning expectations described in state-level mathematics curriculum standards similar in terms of content and grade placement? Number Algebra Reasoning

  14. Our Analysis • Focused on state GLE documents from 43 states. • K-8, Number and Operation Strand • Not comprehensive, not evaluative • Descriptive • Chose particular topics within specific strands for analysis • Utilized an “organic” or bottom-up approach with “learning expectations” as the unit of analysis

  15. Number & Operation Strand • Fluency with basic number (single-digit) combinations • Multi-digit whole number computation • Fraction computation • Fraction concepts • Computational estimation • Allowance for calculators as computation tools

  16. Addition of Basic Number Combinations

  17. Multiplication of Basic Number Combinations

  18. Common Grade Placement of Basic Number Combinations

  19. Multi-digit Whole Number Computation

  20. Example GLEs (Multi-digit Whole Number Addition) • Using pictures, diagrams, numbers or words, demonstrate addition and subtraction of whole numbers with 2-digit numbers (CO, gr. 3) • Add and subtract two three-digit whole numbers(AZ, gr. 3) • Find the sum of any two addends with three or fewer digits, including monetary amounts, and describe the process used. (UT, gr. 3) • The student will solve problems involving the sum or difference of two whole numbers, each 9,999 or less, with or without regrouping, using various computational methods, including calculators, paper and pencil, mental computation, and estimation (VA, gr. 3)

  21. Coding Categories(whole number computation) • Operation • Addition, subtraction, multiplication, division • Specification of size of number • Number of digits • Representations • Explain, model, represent • Tools/Methods • Mental, calculator, manipulative, paper/pencil • Specification of algorithm • Standard/traditional, constructed, multiple strategies • Initial, intermediate, culminating GLE

  22. Addition of Whole Numbers

  23. Multiplication of Whole Numbers

  24. Grade Placement of Culminating GLE for Whole Number Computation

  25. Fluency with Fraction Computation

  26. Example GLEs • Add, subtract, and multiply fractions. (UT, gr. 5) • Add and subtract common fractions and mixed numbers with unlike denominators. (GA, gr. 5) • Demonstrate computational fluency with addition, subtraction, multiplication, and division of decimals and fractions. (AL, gr. 6) • Add, subtract, multiply and divide common fractions and mixed numbers as well as fractions where the common denominator equals one of the denominators. (MN, gr. 6)

  27. Progression of GLEs (AZ) • Demonstrate addition and subtraction of fractions with like denominators (halves) using models. (gr. 1) Initial GLE • Demonstrate addition and subtraction of fractions with like denominators (halves and fourths) using models. (gr. 2) • Add or subtract fractions with like denominators (halves, thirds, fourths, eighths, and tenths) appropriate to grade level. (gr. 3) • Add or subtract fractions with like denominators, no regrouping. (gr. 4) • Add or subtract proper fractions and mixed numbers with like denominators with regrouping. (gr. 5) • Add or subtract proper fractions and mixed numbers with unlike denominators with regrouping. (gr. 6)Culminating GLE

  28. Addition and Subtraction of Fractions

  29. Multiplication of Fractions

  30. Division of Fractions

  31. When do states expect students to proficiently add, subtract, multiply and divide fractions?* *For this summary, we used the culminating learning expectation that indicated students were working with common and uncommon denominators when adding and subtracting fractions.

  32. What messages regarding calculators and technology are conveyed within the state standards documents?

  33. Fordham Foundation (The State of Math Standards, 2005) • Calculators. “One of the most debilitating trends in current state math standards is their excessive emphasis on calculators. Most standards documents call upon students to use them starting in the elementary grades, often beginning with Kindergarten.”

  34. Messages regarding calculators and technology • Searched 42 state GLE documents (K-8). • 20 state documents include a statement regarding calculators/technology within the introductory material. • 32 state documents mention “calculator” or “technology” within the set of GLEs.

  35. Summary of attention to calculators/technology in state-level mathematics GLE documents.

  36. Introductory Comments • Technology will be a fundamental part of mathematics teaching and learning. (KS) • Extensive reliance on calculators runs counter to the goal of having students practice [computational and procedural skills]. More to the point, it is imperative that students in the early grades be given every opportunity to develop a facility with basic arithmetic skills without reliance on calculators . . . It should not be assumed that caution on the use of calculators is incompatible with the explicit endorsement of their use when there is a clear reason for such an endorsement. Once students are ready to use calculators to their advantage, calculators can provide a very useful tool not only for solving problems in various contexts but also for broadening students’ mathematical horizons. (CA)

  37. Introductory Comments • While learning mathematics, students will be actively engaged in using concrete materials and appropriate technologies such as calculators and computers. (UT, gr. K-2) • Students must use technology and other mathematical tools, such as manipulative materials, to develop conceptual understanding and solve problems as they do mathematics. (UT, gr. 3-6) • Technology: Must be integrated into the curriculum and used appropriately as part of mathematical instruction; Provides visual images when teaching mathematical ideas and concepts; Facilitates the organization and analysis of data; Facilitates efficiency and accuracy in computation. Enhances the investigation and modeling of a wide variety of mathematical concepts. (UT, gr. 7-12)

  38. Common Messages: Introductory Comments • Appropriate use of calculators/technology is encouraged. • Calculators/technology can support increased understanding. • The existence of calculators/technology does not diminish the need for computational fluency. • Calculators/technology can support effective teaching. • Calculators/technology are commonly used in the workplace, therefore students should learn to use these tools to solve problems. • Teachers are responsible for appropriate and effective use of calculators/technology.

  39. Review of learning expectations referring to calculators/technology • Compiled a set of 451LEs from state documents that include “calculator” or “technology” or both (about 3% of all LEs) • 21 LEs from 7 states indicate that students should NOT use calculators • 34 LEs focused on computer technology (software) rather than calculators. • 396LEs were used for this analysis • The mean number of LEs referencing calculators was 12.8 per state and 1.4 per grade.

  40. Example GLEs (reference to calculators and/or technology) • Use appropriate technology to gather and display data sets and identify the relationships that exist among variables within the data set. (ID, gr. 7) • Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers. (NJ; gr. 3,4,5,6) • Use technology, including calculators, to understand quantitative relationships, e.g., for skip counting and pattern exploration. (NY; gr. K,1,2,3,4) • Solve problems using the four operations with whole numbers, decimals, and fractions. Determine when it is appropriate to use estimation, mental math strategies, paper and pencil, or a calculator. (UT; gr. 5,6)

  41. Allowance for calculators and/or technology

  42. Role of calculator/technology

  43. What mathematics are fourth graders in the U.S. expected to learn?

  44. Analysis of 4th Grade GLEs • Goal was to document the level of consensus regarding mathematics GLEs at one grade level (we chose 4th grade). • Focused on GLE documents from the ten most populous states that publish such documents (CA, TX, NY, FL, OH, MI, NJ, NC, GA, VA)

  45. Method • Collected the 10 state documents, combined and sorted all GLEs (492 total) by content strand. • Searched for common themes across GLEs and developed list of “substrands”. • Sorted all GLEs into substrands and eliminated duplicates. • Developed list of “distinct” GLEs (108 total) and coded all 4th grade GLEs to determine commonality across the 10 states. • Summarized findings by content strand.

  46. Distinct Set of 4th Grade GLEs(with duplicates removed) Total: 108

  47. Common GLEs across all 10 documents(4 of 108) • Read, write, compare, and order whole numbers. • Read, write, compare and order decimals. • Add and subtract decimals. • Solve applied problems involving whole number multiplication and division.

  48. Examples of Unique GLEs(28 of 108) • Use concrete materials and symbolic notation to represent numbers in bases other than base ten, such as base five. • Compare decimal number system to the Roman numeral system (using the Roman numerals I, V, X, L, C, D, and M.) • Use models to identify perfect squares to 100.

  49. GLEs common to at least 6 of the 10 states