May 8, 2008 Page 1

Mathematics Standards for Kindergarten: Summary of State-Level Attention and Focus

Paper Commissioned by the NRC Early Childhood Mathematics Committee

Barbara J. Reys, Kathryn B. Chval, John (Matt) Switzer

Center for the Study of Mathematics Curriculum University of Missouri

The No Child Left Behind (NCLB) legislation requires states to create challenging academic mathematics standards that specify what children are expected to know and be able to do. While some states have a history of producing and monitoring state standards for curriculum areas such as mathematics, for other states, the NCLB mandate introduced a new level of state curriculum regulation. As a result, most states have produced specific grade-level learning expectations (GLEs) articulated for elementary mathematics and these GLEs fulfill the requirement of NCLB. A study by Reys and Lappan (2006) concluded that there is wide variation in GLE articulation across grades K-8:

Mathematics learning goals vary along several dimensions including grain size (e.g., level of specificity), language used to convey learning goals (e.g., understand, explore, memorize, etc.), and the grade placement of specific learning expectations. In particular, while the set of learning expectations related to specific topics are similar across the K–8 state standards, the grade placement for any particular topic or learning expectation varies considerably. That is, state-level standards documents generally include similar goals for learning in the K–8 number and algebra strands. However, when the topics are introduced, their trajectory of development across grades and the grade at which students are expected to learn particular mathematical content differs dramatically across the states. (p. 112)

Even though there is considerable variation across K-8, it was unknown if such variation exists for GLEs written for the youngest students, namely those in kindergarten. What mathematics should kindergarten students learn? What is the nature of the kindergarten GLEs described by states? How many and what specific GLEs are realistic and appropriate for kindergartners? To determine the organization, content, and variability of learning expectations for kindergarten students across states, we analyzed the kindergarten GLEs from the 10 states with the largest student populations. The purpose of this paper is to report the findings from that analysis.

Mathematics Curriculum Standards for Kindergarten Since 2002, all but seven states (see Table 1) have developed new mathematics curriculum standards outlining the mathematics content that students at particular grades are expected to learn (and teachers are expected to teach). Many of the state standards documents are more specific than their earlier versions, outlining grade-specific learning expectations for all students.

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Table 1. Publication year of current state K-8 mathematics standards (or GLEs) (as of March 2008).

Publication Year

Number of states States

2008 3 NJ, WA*, WV 2007 11 CT*, FL, HI, IA*, ME, MN*, MS, MO*, RI*, SC, UT 2006 7 AK, DE, DC, GA, ID, KY, NV 2005 6 CA, CO, MI, NY, ND, TX 2004 10 AR, DoDEA, KS, LA, MD, MA, NE, NH, SD, VT 2003 4 AL, AZ, NC, WY 2002 4 MT, NM, OK, VA 2001 2 OH, TN

2000 or earlier 5 IL, IN, MT, PA, WI * Draft document under review. Some state standards are intended to be “models” for districts to utilize in shaping local curriculum specifications, particularly in states that value local control of educational decisions. In other states, the standards are mandatory, specifying the mathematics all students within the state are expected to learn at particular grades. While the accountability provisions of NCLB are primarily attached to grades 3-8, most states have outlined mathematics standards for primary grades. In fact, 41 of 52 state departments of education (including offices of education in the District of Columbia and the Department of Defense Education Agency) publish kindergarten-specific standards for mathematics as part of the state K-6; K-7 or K-8 mathematics standards document and four states articulate standards for kindergarten as part of a grade-band summary (see Table 2). Table 2. Summary of articulation of kindergarten mathematics standards by states

Kindergarten-specific standards

Kindergarten standards as part of grade-band

specification

No specification of Kindergarten standards

AL, AK, AZ, AR, CA, CT, DE, DoD, DC, FL, GA, HI, ID, IN, KS, LA, MD, MI, MN, MO, MS, NV, NH, NM, NC, ND, NY, OH, OK, OR, RI, SC, SD, TN, TX, UT, VT, VA, WA, WV, WY

NE (K-1), NJ (K-2), CO (K-4) , MA (PreK-K)

IA, IL, ME, MT, KY, PA, WI

41 4 7

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Organizing the Analysis of Kindergarten GLEs

This analysis is based on kindergarten GLEs from the ten states with the largest student populations that publish kindergarten-specific mathematics standards - California, Texas, New York, Florida, Ohio, Michigan, New Jersey, North Carolina, Georgia, and Virginia (see the reference list for documents analyzed). These states were selected for analysis because they represent approximately 50 percent of the U.S. school population and are therefore influencing the intended curriculum of a substantial school population. Given their size, these ten states are also most likely to influence textbook development. In recent years, there has been increased attention on the mathematics education of young children (Clements & Sarama, 2007). The Conference on Standards for Prekindergarten and Kindergarten Mathematics Education was held in 2000 in order to facilitate communication and coordinate efforts related to the development of mathematics standards, curricula, and teaching methods for young children (Clements, Sarama & DiBiase, 2004). The discussions and work during and following this conference resulted in a framework for thinking about mathematics in Pre-K to Grade 2 (Clements, 2004). To support our efforts to organize, code, and analyze the GLEs from 10 states, we utilized the “big mathematical ideas” (hereafter referred to as “Topics”) for each mathematical strand identified by Clements (2004) as shown in Table 3. It should be noted that Clements did not include probability; however it was added based on the fact that probability was included in two of the state documents. Utilizing this organization scheme presented the opportunity to determine which mathematical ideas were emphasized within and across states. Table 3. Mathematical ideas used to organize kindergarten GLEs*

Strand Big Mathematical Ideas/Topics

Number & Operation

Counting Comparing and Ordering Grouping and Place Value Adding to/Taking away Composing and Decomposing Equal Partitioning

Geometry

Shape Locations and Directions Visualizations and Spatial Reasoning Transformations and Symmetry

Measurement Attributes, Units and Processes Techniques and Tools

Algebra Patterns

Data Analysis/Probability

Collect Data Classify/Organize Data Represent Data Use Information to Address Questions Probability

* Based on Clements, 2004.

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Content of Kindergarten Mathematics Curriculum Standards K-8 state mathematics standards are commonly organized by content strands including: Number & Operation; Geometry; Measurement; Data Analysis & Probability; and Algebra. In fact, nearly every state includes these five strands although they may be grouped or labeled slightly differently. Some states also specify learning goals within process strands (e.g., Problem Solving, Reasoning) but these are generally not specific to a grade level, rather they are specified by a grade-band such as K-4 or K-8. Each mathematics strand at the kindergarten level is further sub-divided by topics. For example, in the number strand, major topics include counting, comparing and ordering, and adding to/taking away. Within each topic, state standards specify what students should know and/or be able to do. We refer to these statements as grade-level learning expectations (GLEs); however this term is not used universally across state documents. Table 4 provides a summary of the number of GLEs for kindergarten in each of the 10 states according to the content strands identified in the National Council of Teachers of Mathematics Principles and Standards for School Mathematics (2000)—Number & Operations; Geometry; Measurement; Algebra; and Data Analysis & Probability. The most dominant strands for kindergarten are Number, Measurement, and Geometry. The number of GLEs for kindergarten across all 10 states ranges from 11 to 74. The variation is due, in part, to the fact that some states include multiple concepts within a single GLE while other states include more finer-grained GLEs. As noted, our count of Virginia kindergarten standards includes both “essential understandings” and “essential knowledge and skills.” The mean number of kindergarten GLEs per state is 29.6 (standard deviation: 17.8). By strand, the mean number of GLEs is: Number (11.4); Geometry (5.5); Measurement (6.0); Algebra (2.3); Statistics/Probability (3.4). Table 4. Summary of the number of GLEs in state standards by strand.

* Count includes “essential understandings” and “essential knowledge and skills.” To determine how many different kindergarten GLEs were represented in the 10 state standards documents, we examined each GLE, sorting by topic and specific focus. Based on this analysis, we identified a total of 103 different kindergarten GLEs across the 10 state documents. This set of 103 GLEs represents the union of the ten state documents and will subsequently be referred to

State Number Geometry Measurement Algebra Data Analysis/

Probability Total AZ 17 3 4 3 7 34 CA 5 4 2 1 2 14 FL 3 5 2 1 0 11 GA 13 9 10 0 1 33 MI 10 3 5 0 0 18 NC 9 4 2 2 2 19 NY 13 5 3 2 5 28 OH 13 2 4 4 3 26 TX 8 8 8 3 2 29

VA* 23 12 20 7 12 74 MEAN 11.4 5.5 6.0 2.3 3.4 28.6

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as the “combined set” of kindergarten GLEs (see Appendix A). In developing the combined set of GLEs, we constructed general statements that captured the main ideas represented in the state documents. We used the combined set of GLEs to code each individual GLE in all of the ten documents. This coding allowed us to identify common and uncommon learning expectations across states as well as identify particular mathematical strands/concepts that were emphasized across states. Appendix A includes a summary of the specific states as well as the number of states that include each of the 103 kindergarten GLEs in the combined set.

Emphasis by Content Strand Within the combined set of 103 GLEs in Appendix A, there are few learning expectations that are common across most of the 10 states. In fact, only 21 of the 103 GLEs appear in six or more of the ten state documents. A summary of the common and uncommon GLEs by strand is provided in this section. Summary of Emphasis Within the Number Strand As might be expected, much of the attention in the kindergarten state standards focuses on the development of number concepts (40 of the 103 GLEs in the combined set). Areas of emphasis include counting objects, reading and writing numerals, identifying ordinal numbers (e.g., first, second, etc.), comparing the relative size of groups of objects, and modeling and solving problems using addition and subtraction. Table 5 includes the set of nine learning expectations within the number strand included in at least six of the 10 states investigated. Surprisingly, there wasn’t one GLE that appeared in all 10 state documents. Table 5. Common learning expectations within the number strand.

LEARNING EXPECTATION NUMBER OF STATES

Represent “how many” in a set (using pictures, fingers, numerals, numbers words). 9  

Count “how many” in a set (up to X). 8  Produce a set of objects of specified size (up to X). 7  Model addition/subtraction situations (using drawings or manipulatives). 7  

Count up (to X). 6  Compare the relative size of sets of objects, more than/less than/equal to (X). 6  

Identify ordinal numbers. 6  Solve problems using addition. 6  Solve problems using subtraction. 6  

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The greatest emphasis within the number strand across the 10 states is placed on counting. Some states specify counting forward or backward to a particular number (e.g., to 10, 20, 30 or 100) and others include reference to counting on a number line. For example,

Count aloud, forward to 20 or backward from 10, in consecutive order (0 through 20). (AZ) Count forward from 1 to 30 and count backward from 10 to 1. (VA) Count orally to 100 by ones. Count to 30 by 2’s, 5’s and10’s using grouped objects as needed. (MI) Count by ones to 100. (TX) Count on (forward) and count back (backward) on a number line between 0 and 10. (OH)

Two states (MI and VA) include attention to skip counting by 2s, 5s and 10s in their kindergarten standards. In virtually every state, kindergarten students are expected to learn to count and represent how many objects are in a set using pictures, fingers, numerals or number words. Six of the 10 states include attention to identifying or naming ordinal numbers from 1st to 10th. More than half the states include expectations for modeling and solving problems involving “adding to” and “taking away.” Only two states (NC and NY) do not include learning expectations specifically denoting use of these two operations. However, NC does include the following learning expectation:

Solve problems and share solutions to problems in small groups. (NC) Only one state (MI) includes the expectation that kindergarten students will represent problem situations with a number sentence. Additionally, one state (AZ) indicates that students should be able to identify symbols such as +, − and =. Three states (GA, OH, NC) include expectations that kindergarten students partition, share or divide equally sets of objects.

Use informal strategies to share objects equally (divide) between two to three people or sets. (GA) Share equally (divide) between two people; explain. (NC) Partition or share a small set of objects into groups of equal size; e.g., sharing 6 stickers equally among 3 children. (OH)

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While there are some commonalities among number-related learning expectations across the 10 states, there are also many differences. Table 6 provides a list of the number learning expectations found in only one or two of the state documents. Table 6. Learning expectations from the number strand found in 1 or 2 state documents.

LEARNING EXPECTATION NUMBER OF STATES  

Understand the meaning of addition/subtraction. 2  Identify numerals (to 10, 20, 30, …) 2  Know that a number is larger than another number if more objects are required to represent it. 2  

Group items in a set by tens (place value). 2  Estimate the number of items in a collection. 2  Skip count (by 2s, 5s, 10s, …) 2  Know number of objects without counting (subitising). 1  Represent equivalence of two collections. 1  Count up and back on a number line. 1  Know that skip counting can be used to count a set of objects. 1  Recognize patterns in skip counting (by 2s, 5s, 10s, …) 1  Know rules of counting. 1  Determine how many more/less. 1  Create story problems. 1  Select appropriate operation to solve a problem. 1  Solve problems using estimation. 1  Recognize when an estimate is reasonable. 1  Represent situations with a number sentence. 1  Use mental computation. 1  Identify/use symbols: +, -, = 1  Recognize 1/2 of a whole. 1  

Summary of Emphasis on Geometry and Measurement Strands A substantial amount of attention in the state kindergarten standards is focused on geometry and measurement (43 of the 103 GLEs). Much of this attention addresses measurable attributes, units & processes involving length, area, weight, capacity, temperature, and time. Other key topics include identifying, describing, and naming 2-and 3-dimensional shapes and knowing the relative position of objects. A few states also emphasize familiarity with money (recognizing the name and value of coins) and/or measuring with non-standard and standard units. As was the case in the number strand, few learning expectations are common across the 10 states. Table 7 includes the set of 6 geometry and measurement learning expectations included in at least 6 of the 10 states investigated.

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Table 7. Common learning expectations within the geometry and measurement strands.

LEARNING EXPECTATION NUMBER

OF STATES

Compare weight of objects. 9  Sort, compare and/or order objects. 9  Know the relative position of objects. 8  Compare length of objects. 8  Identify and name 2-D shapes. 6  Know days of the week. 6  

In addition to the “common” learning expectations noted above, nine of the ten states (AZ is the exception) emphasize some aspects of time including knowing the names of months (5 states), parts of the day (5 states), seasons (4 states), ordering events by time (3 states), comparing time (2 states), understanding the concept of time (2 states) and identifying time of everyday events to the nearest hour (2 states). However, none of the states addresses all of these areas. Aside from the learning expectations related to time noted above, the majority of the learning expectations dealing with measurable attributes, units and processes emphasize sorting, comparing and ordering objects based on measurable attributes such as weight, length, area, and temperature. However, few states include learning expectations suggesting that students will learn to measure these attributes with standard or non-standard units. Instead, students sort, compare and order objects based on informal comparisons. For example,

Compare the length, weight, and capacity of objects by making direct comparisons with reference objects (e.g., note which object is shorter, longer, taller, lighter, heavier, or holds more). (CA) Compare length and weight of objects by comparing to reference objects, and use terms such as shorter, longer, taller, lighter, heavier. (MI) Order events based on time. For example: activities that take a long or short time. (OH)

Few of the ten states include learning expectations addressing measuring techniques and tools. Four states (CA, MI, VA and AZ) expect students to identify appropriate tools to measure. Three states (NY, OH and VA) expect students to measure with non-standard units:

Compare the length of two objects by representing each length with string or a paper strip. (NY)

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Measure length and volume (capacity) using uniform objects in the environment. For example, find: how many paper clips long is a pencil; how many small containers it takes to fill one big container using sand, rice, beans. (OH) Compare and describe weights of two objects (as heavier or lighter), using direct comparison or nonstandard units of measure (e.g., book, cubes, new pencil, paper clip, block). (VA)

Ohio’s learning expectation regarding measuring with non-standard units, noted above, is the only state expectation addressing measuring length and volume. Virginia is the only state with a learning expectation related to measuring with standard units:

Develop an understanding of measuring with nonstandard and standard units of measure. (VA)

All ten states include attention to shapes. However, the nature and extent of the focus on shapes differs. At one extreme, Ohio includes the general learning goal describe common shapes while Texas includes eight learning expectations related to shapes including identifying common 2- and 3-dimensional shapes by name, describe attributes of shapes and classify shapes by attributes. Six states (CA, TX, FL, GA, NC and VA) expect students to identify 2-D shapes. Five of these six states, VA is the exception, also expect students to identify 3-D shapes. There is considerable variation in the types of 2-D and 3-D shapes students are expected to identify. For example,

Identify and describe common geometric objects (e.g., circle, triangle, square, rectangle, cube, sphere, cone). (CA) Identify, name, describe and sort basic two-dimensional shapes such as squares, triangles, circles, rectangles, hexagons, and trapezoids. (FL) Identify, name, describe, and sort three-dimensional shapes such as spheres, cubes and cylinders. (FL) Recognize and name the following basic two-dimensional figures: triangles, rectangles, squares, and circles. (GA) Recognize and name the following three-dimensional figures: spheres (balls), and cubes. (GA) Identify, build, draw, and name triangles, rectangles, and circles; identify, build, and name spheres and cubes. NC) Describe, identify, and compare circles, triangles, rectangles, and squares (a special type of rectangle). (TX) Identify a circle, triangle, square, and rectangle. (VA)

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Five of the 10 states (OH, MI, GA, VA and AZ) give some attention in kindergarten to learning about money, including identifying and knowing the value of coins (penny, nickel, dime). Georgia includes the largest set of learning expectations (3) related to money:

Count out pennies to buy items that together cost less than 30 cents. (GA) Identify coins by name and value (penny, nickel, dime, and quarter). (GA) Make fair trades involving combinations of pennies and nickels or pennies and dimes. (GA)

In summary, there are some commonalities among geometry and measurement learning expectations across the 10 states; however, there are also many differences. For example, 17 of the 43 geometry and measurement GLEs are found only in one or two states (see Table 8). Table 8. Geometry and measurement learning expectations found in only one or two state documents.

LEARNING EXPECTATION NUMBER OF STATES

Draw 2-D shapes. 2  Compare temperature. 2  Compare time. 2  Understand concept of time. 2  Identify time of everyday events to the nearest hour   2  Determine value of collection of coins. 2  Compose/decompose shapes. 1  Complete spatial visualization tasks and puzzles 1  Measure with standard unit. 1  Measure length. 1  Measure volume. 1  Know how to use a calendar. 1  Know how to use a clock. 1  Explore symmetry in 2-D and 3-D shapes 1  Describe physical properties of coins. 1  Know value of coins (penny, nickel, dime, ...). 1  Show equivalence of combinations of coins. 1  

Summary of Emphasis on Algebra Strand As noted in Appendix A, 5 of the 103 kindergarten GLEs identified in the set of 10 state standards are included within the algebra strand. Four of these learning expectations focus on identifying, creating, extending or describing patterns related to numbers and/or geometry and at least six states include specific learning goals related to these areas (see Table 9). For example,

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one state (GA) includes a learning expectation that students make generalizations from patterns. It states:

Extend a given pattern, and recognize similarities (such as color, shape, texture, or number) in different patterns.

Table 9. Learning expectations within the kindergarten algebra strand.

LEARNING EXPECTATION NUMBER OF STATES

Extend a pattern (number, geometric, …) 10  Identify a pattern. 6  Create a pattern (number, geometric, …) 6  Describe a pattern (number, geometric, …) 6  Make generalizations from patterns 1  

Summary of Emphasis on Data Analysis and Probability Strand Each of the 10 states includes at least one GLE related to data analysis or probability within their kindergarten standards. The most common learning expectations (included in at least half the states) are related to formulating questions, collecting, and displaying data (see Table 10). Less common are GLEs related to reading and understanding graphical representations or probability concepts. In fact, only two states (VA and AZ) include GLEs related to probability. They are:

Make arrangements that represent the number of combinations that can be formed by pairing items taken from 2 sets, using manipulatives. (AZ) Describe verbally, pictorially, and/or with tally marks the outcome of dropping a two-colored counter or using a multicolored spinner (e.g., the number of times the red side of the counter landed up compared to the number of times the counter was dropped). (VA) Conduct investigations of probability through hands-on activities such as dropping a two-colored counter or using a multicolored spinner. (VA) Develop an understanding of chance (likelihood of an event) through informal investigations with spinners and two-colored counters. (VA)

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Table 10. Learning expectations within the kindergarten data analysis/probability strand.

LEARNING EXPECTATION NUMBER OF STATES

Formulate questions, gather data to address question. 6 Display data in an organized way. 6 Collect data. 5  Organize data. 3  Use/interpret a pictograph. 3 Interpret charts/graphs. 2 Describe data. 1  Understand meaning/purpose of graphs/tables. 1  Read different graphical representations. 1  Know “more likely/less likely” situations (probability). 1 Make all possible combinations from two sets of objects. 1 Collect data from probability experiments 1

Summary of Common Content Learning Expectations Based on our analysis, there are some common ideas emphasized at the kindergarten level across the 10 states. These include: counting, identifying patterns, comparing the size of objects, and collecting and displaying data (see Table 11 for a list of the most common GLEs within the 10 state standards analyzed). While there is some commonality, there is also considerable variation across all of the mathematical strands. In fact, 47 of the 103 GLEs are included in only 1 or 2 state documents. Examples include:

Count up and back on a number line. (1 state)

Skip count (by 2s, 5s, 10s, …) (2 states)

Solve problems using estimation. (1 state)

Understand the meaning of addition/subtraction. (2 states)

Show equivalence of combinations of coins. (1 state)

Know how to use a calendar. (1 state)

Know “more likely/less likely” situations (1 state)

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Table 11. Common learning expectations across all 10 state documents (included in at least 6 states).

STRAND LEARNING EXPECTATION

NUMBER OF

STATES

Number

Represent “how many” in a set (using pictures, fingers, numerals, numbers words). 9  

Count “how many” in a set (up to X). 8  Produce a set of objects of specified size (up to X). 7  Model addition/subtraction situations. 7  Count up (to X). 6  Compare the relative size of sets of objects. 6  Identify ordinal numbers. 6  Solve problems using addition. 6  Solve problems using subtraction. 6  

Geometry/ Measurement

Compare weight of objects. 9  Sort, compare and/or order objects. 9  Know the relative position of objects. 8  Compare length of objects. 8  Identify and name 2-D shapes. 6  Know days of the week. 6  

Algebra

Extend a pattern (number, geometric, …) 10  Identify a pattern. 6  Create a pattern (number, geometric, …) 6  Describe a pattern (number, geometric, …) 6  

Data/Probability Formulate questions, gather data to address question. 6 Display data in an organized way. 6

Emphasis on Process Standards Of the ten state documents reviewed for this analysis, three (Florida, North Carolina, and Virginia) make no mention of process standards at the kindergarten level. Two states (Arizona and Massachusetts) provide a general description of process standards in the introductory material of their K-6 or K-8 document. These descriptions emphasize the importance of communication, problem solving, reasoning and proof, connections and representations – the process strands outlined in the NCTM Principles and Standards for School Mathematics (2000). For example, the following general comment appears in the Arizona Academic Content Standards (2003):

Communication, problem solving, reasoning and proof, connections and representation are the process standards as described in the Principles and Standards for School Mathematics from the National Council of Teachers of Mathematics (NCTM). These process standards are interwoven within all the content strands of the Arizona Mathematics Standard. The process standards

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emphasize ways to acquire and use the content knowledge. (p. viii) Arizona also includes a strand called “Structure & Logic.” At the kindergarten level, there were two GLEs within this strand and one of them was related to process standards. It states, “Provide rationale for classifying objects according to observable attributes (color, size, shape, weight, etc.).” Three states (Georgia, New York and Texas) include identification of specific standards by process strand. Although they are numbered/coded for kindergarten, they are very similar, if not identical, at each grade, K-8. See Table 12 for a list of K-8 process standards from the Georgia document. Table 12. Process standards for grades K-8 included in the Georgia Mathematics Performance Standards (2006). MKP1. Students will solve problems (using appropriate technology). a. Build new mathematical knowledge thorough problem solving. b. Solve problems that arise in mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving. MKP2. Students will reason and evaluate mathematical arguments. a. Recognize reasoning and proof as fundamental aspects of mathematics. b. Make and investigate mathematical conjectures. c. Develop and evaluate mathematical arguments and proofs. d. Select and use various types of reasoning and methods of proof. MKP3. Students will communicate mathematically. a. Organize and consolidate their mathematical thinking through communication. b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and

others. c. Analyze and evaluate the mathematical thinking and strategies of others. d. Use the language of mathematics to express mathematical ideas precisely. MKP4. Students will make connections among mathematical ideas and to other disciplines. a. Recognize and use connections among mathematical ideas. b. Understand how mathematical ideas interconnect and build on one another to produce a

coherent whole. c. Recognize and apply mathematics in contexts outside of mathematics. MKP5. Students will represent mathematics in multiple ways. a. Create and use representations to organize, record, and communicate mathematical ideas. b. Select, apply, and translate among mathematical representations to solve problems. c. Use representations to model and interpret physical, social, and mathematical phenomena.

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Finally, the California and Ohio documents include process standards organized within one strand (“Mathematical Reasoning” in the California document and “Mathematical Process Standard” in the Ohio document) for each grade. See Table 13 for a list of process standards from the California document. They are common across grades K and 1. Likewise, the Ohio document includes a list of process standards that are common to grades K-2. Table 13. Process standards for grades K-1 included in the Mathematics Framework for California Public Schools (2005). Mathematical Reasoning (same for K, 1) 1.0 Students make decisions about how to set up a problem: 1.1 Determine the approach, materials, and strategies to be used. 1.2 Use tools and strategies, such as manipulatives or sketches, to model problems. 2.0 Students solve problems in reasonable ways and justify their reasoning: 2.1 Explain the reasoning used with concrete objects and/or pictorial representations. 2.2 Make precise calculations and check the validity of the results in the context of the problem.

Resources Influencing the State Mathematics Standards A survey of state Department of Education personnel conducted in 2005 asked respondents to identify resources utilized in the development of their most recent state mathematics standards. The most frequently cited resource was the NCTM Principles and Standards for School Mathematics (PSSM) (2000), with 28 of 35 states reporting that it had a great or strong influence. These respondents also indicated that the organizational structure of PSSM (common content strands and goals across multiple grades) was replicated in the state document with the five content strands of PSSM serving as the headings for many state GLE documents. Another influence was a focus on representation, a process standard emphasized in PSSM. Some of the language of PSSM (e.g., computational fluency) was also adopted within new state standards. Respondents also indicated that the grade-band placement of topics in PSSM was helpful in identifying developmentally appropriate content for particular grade levels. While most states regularly review and update their state mathematics standards, they complete the writing and review stages using different time schedules/cycles. As noted in Table 1, 41 states have published new mathematics standards since the release of PSSM; however only 11 states have published new state mathematics standards since the publication of NCTM’s Curriculum Focal Points in the fall of 2006. Therefore, many states have not had an opportunity to utilize the CFP recommendations to update or revise their existing GLE documents. During the summer of 2007 we contacted representatives from the 11 states (Arizona, Florida, Iowa, Minnesota, Missouri, Nevada, New Mexico, Tennessee, Utah, Washington, and West Virginia) that have published, updated, revised or reviewed their state standards since the fall of 2006, inquiring about the influence of CFP. Each of the 11 state Department of Education staff reported that CFP was used as a resource in the revision. In some cases, CFP was used to

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identify the grade at which particular learning goals would receive attention. In other cases, CFP was used to organize state GLEs around “big ideas” of mathematics. Eight states (Washington, Utah, Tennessee, New Mexico, Nevada, Minnesota, Florida, and Iowa) reported a “significant” impact of CFP on their new state mathematics GLEs. In Missouri and Arizona, the three “focal points” for each grade served as a starting point in creating their K-8 mathematics standards.

Summary The role and influence of state standards has increased with the advent of NCLB. State Department of Education staff confirm that state standards influence not only state or district assessments, but also the selection and use of textbooks, classroom practice, and professional development activities (Reys, et al, 2005). The staff also indicate that the new standards carry greater weight and authority than previous standards, primarily because of their tie to new or expanded state-mandated annual assessments in mathematics. Indeed, teachers and school administrators are paying close attention to the curriculum outlined in the state standards. Given the level of attention paid to state standards and their influence on teaching and assessment, it is important to understand how they differ across states and the impact of such differences.

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Mathematics Supervisors: What influence has NCTM’s Curriculum Focal Points had on state-level specification of learning goals for mathematics? Retreived on March 30, 2008 from: http://www.mathcurriculumcenter.org/reports_research.php.

Clements, D.H. (2004). Major themes and recommendations. In D.H. Clements, J. Sarama, &

A.-M. DiBiase (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education (pp. 7-72). Mahwah, NJ: Erlbaum.

Clements, D. H., Sarama, J., & DiBiase, A.-M. (Eds.). (2004). Engaging young children in

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March 15, 2008 from http://www.fldoestem.org/page221.aspx.

Georgia Department of Education. (2006). Georgia performance standards. Retrieved March 15, 2006 from http://www.georgiastandards.org/math.aspx.

Michigan Department of Education. (2005). Michigan grade level content expectations (GLCE). Retrieved March 15, 2008 from

http://www.michigan.gov/mde/0,1607,7-140-28753_33232---,00.html. National Council of Teachers of Mathematics. (2000). Principles and Standards for School

Mathematics. Reston, VA.: The National Council of Teachers of Mathematics, Inc. National Council of Teachers of Mathematics. Curriculum Focal Points for Prekindergarten

through Grade 8 Mathematics. Reston, Va.: Author, 2006. New Jersey Department of Education. (2008). New Jersey core curriculum content

standards for mathematics. Retrieved March 15, 2008 from http://education.state.nj.us/cccs/?_standard_matrix;c=4.

No Child Left Behind Act of 2001, Pub. L. No. 107-110, 115 Stat. 1425 (2002). North Carolina Department of Public Instruction. (2003). Mathematics standard course

May 8, 2008 Page 18

of study and grade level competencies. March 15, 2008 from http://www.ncpublicschools.org/curriculum/mathematics/scos/.

Ohio Department of Education. (2001). Academic content standards. Retrieved March 15, 2008

from http://www.ode.state.oh.us/GD/Templates/Pages/ODE/ODEDetail.aspx?Page=3&TopicRelationID=1210&Content=32581.

Reys, B.J., Dingman, S., Sutter, A., & Teuscher, D. (2005). Development of state-level

mathematics curriculum documents: Report of a survey. Retrieved August 15, 2006 from http://mathcurriculumcenter.org/reports_publications.php

Texas Education Agency. (1998). Texas essential knowledge and skills for mathematics.

Retrieved January 3, 2006 from http://www.tea.state.tx.us/rules/tac/chapter111/index.html.

University of the State of New York: State Education Department. (2005). New York state

mathematics core curriculum. Retrieved March 15, 2008 from http://www.emsc.nysed.gov/ciai/mst/mathstandards/mathcorepage.htm.

Virginia Department of Education. (2002). Mathematics standards of learning

curriculum framework. Retrieved March 15, 2008 from http://www.doe.virginia.gov/VDOE/Instruction/Math/math_framework.html.

May 8, 2008 Page 19

Appendix A. Coding for state-level mathematics GLEs, by strand, topic and state.

STRAND: Number  

Topic GLE CA  

TX  

NY  

FL  

OH  

MI  

GA  

NC  

VA  

AZ  

Stat

es

with

GLE

 

N

UM

BER

S/

NU

MER

ALS

Identify numerals (to 10, 20, 30, …)                 *   *   2  Read numerals (to 20, 30, … 100).         *   *     *   *     4  Write numerals (to 20, 30, … 100).         *       *   *   *   4  Identify ordinal numbers.   *   *         *   *   *   *   6  Know that a number is larger than another number if more objects are required to represent it. *                 *     2  

CO

UN

TIN

G

Use one-to-one correspondence.   *         *       *     3  Count "how many" in a set (up to 10, 20, ...). *     *     *   *   *   *   *   *   8  Represent "how many" in a set (using pictures, fingers, numerals, numbers words). *   *   *   *     *   *   *   *   *   9  

Know that the last number counted represents the set (cardinality).   *   *     *         *     4  

Produce a set of objects of specified size (10, 20, ...).   *   *   *   *   *   *       *   7  Know number of objects without counting (subitising).         *             1  Estimate the number of items in a collection.             *   *       2  Represent equivalence of two collections.               *       1  Count up (to 20, 30, … 100)   *   *     *   *       *   *   6  Count back (from 10, 20, … 100)     *             *   *   3  Count up and back on a number line.         *             1  Skip count (by 2s, 5s, 10s, …)           *         *   2  Know that skip counting can be used to count a set of objects.                 *     1  Recognize patterns in skip counting (by 2s, 5s, 10s, …)                 *     1  Know rules of counting.         *             1  

CO

MPA

RIN

G &

O

RD

ERIN

G Compare and order numbers.         *   *     *       3  

Compare the relative size of sets of objects, more than/less than/equal to (X). *   *   *     *   *   *         6  

Determine how many more/less.                   *   1  

GR

OU

PIN

G

AN

D P

LAC

E V

ALU

E

Group items in a set by tens (place value).           *       *     2  

May 8, 2008 Page 20

STRAND: Number (cont.)  

Topic GLE CA  

TX  

NY  

FL  

OH  

MI  

GA  

NC  

VA  

AZ  

Stat

es

with

GLE

 

               

AD

DIN

G T

O/T

AK

ING

AW

AY

Understand the meaning of addition/subtraction.                 *   *   2  Create story problems.             *         1  Solve story problems.           *   *   *       3  Explain/describe solutions to problems.           *   *   *       3  Select appropriate operation to solve a problem.                   *   1  Model addition/subtraction situations (drawings, manipulatives). *   *       *   *   *     *   *   7  

Solve problems using addition. *       *   *     *     *   *   6  Solve problems using subtraction. *     *   *     *     *   *   6  Solve problems using estimation.                 *   1  Recognize when an estimate is reasonable. *                     1  Represent situations with a number sentence.           *           1  Use mental computation.                   *   1  Identify/use symbols: +, -, =                   *   1  

CO

MPO

SIN

G

/DEC

OM

POSI

NG

Create number combinations/partners (sums to 10).         *   *   *       *   4  

EQU

AL

PAR

TITI

ON

S

Use the idea of "fair shares".   *       *     *   *       4  

Recognize 1/2 of a whole.   *                   1  

May 8, 2008 Page 21

STRAND: Geometry/Measurement  

Topic GLE

 TX

 

NY  

FL  

OH  

MI  

GA  

NC  

VA  

AZ  

Stat

es

with

GLE

 

SHA

PE (2

- and

3-D

Spa

ce)

Describe common shapes. *   *     *   *         *     5  Compare shapes.   *           *   *   *     4  Compose/decompose shapes.             *         1  Draw 2-D shapes.               *   *     2  Identify and name 2-D shapes. *   *     *       *   *   *     6  Identify and name 3-D shapes. *   *     *       *   *       5  Identify and name shapes in real life.   *     *     *         *   4  Classify shapes by attributes.   *         *         *   3  Describe attributes of shapes   *   *             *     3  Describe spatial reasoning/relationships     *   *   *               3  Create models of concrete objects (2- or 3-D)         *       *   *   *     4  

LOC

ATI

ON

SD

IREC

TIO

NS

Know the relative position of objects.   *   *   *   *     *   *   *   *   8  

Know vertical/horizontal positions.     *             *   *   3  

VIS

UA

LIZA

TIO

N

SPA

TIA

L R

EASO

NIN

G

Complete spatial visualization tasks and puzzles               *       1  

MEA

S. A

TTR

IBU

TES,

UN

ITS

& P

RO

CES

SES

Compare area of shapes.   *       *           *   3  Compare capacity of solids. *   *       *   *   *         5  Compare length of objects. *   *   *   *   *     *   *   *     8  Compare temperature.   *               *     2  Compare time.   *       *             2  Compare weight of objects. *   *     *   *   *   *   *   *   *   9  Understand concept of time. *       *               2  Know days of the week. *   *       *     *   *   *     6  Know months of the year.   *       *     *   *   *     5  Order events by time.   *       *     *         3  Know parts of the day (morning, afternoon, …). *     *   *     *   *         5  Know seasons (fall, spring, …).         *     *   *   *     4  Sort, compare and/or order objects. *   *   *   *   *   *     *   *   *   9  Sort sets of objects in different ways.     *   *   *         *     4  Compare objects by measurable attributes.   *               *   *   *   4  Identify time of everyday events to the nearest hour   *           *           2  

May 8, 2008 Page 22

STRAND: Geometry/Measurement (cont.)  

Topic GLE CA  

TX  

NY  

FL  

OH  

MI  

GA  

NC  

VA  

AZ  

Stat

es

with

GLE

 

MEA

S.

TEC

HN

IQU

ES &

TO

OLS

Measure with non-standard unit.     *     *         *     3  Measure with standard unit.                 *     1  Measure length.         *             1  Measure volume.         *             1  Identify appropriate tools to measure. *           *       *   *   4  Know how to use a calendar.   *                   1  Know how to use a clock.                 *     1  

TRA

NSF

OR

M.

&

SYM

MET

RY

Explore symmetry in 2-D and 3-D shapes     *                 1  

MO

NEY

Describe physical properties of coins.                 *     1  Identify coins (penny, nickel, dime, …).         *     *     *   *   4  Know value of coins (penny, nickel, dime, …).         *             1  Determine value of collection of coins.             *     *     2  Show equivalence of combinations of coins.             *         1  

STRAND: Algebra

Topic GLE CA  

TX  

NY  

FL  

OH  

MI  

GA  

NC  

VA  

AZ  

Stat

es

with

GLE

 

PATT

ERN

S

Identify a pattern. *   *     *   *     *     *     6  Make generalizations from patterns.             *         1  Create a pattern (number, geometric, ...)   *   *     *   *     *     *   6  Extend a pattern (number, geometric, ...) *   *   *   *   *   *   *   *   *   *   10  Describe a pattern (number, geometric, ...) *     *     *   *       *   *   6  

May 8, 2008 Page 23

STRAND:  Data/Probability  

Topic GLE CA  

TX  

NY  

FL  

OH  

MI  

GA  

NC  

VA  

AZ  

Stat

es

with

GLE

 

CO

LLEC

T D

ATA

Collect data. *     *     *       *   *     5  

CLA

SSIF

Y/

OR

GA

NIZ

E

Organize data.         *       *   *     3  

Describe data.               *       1  

REP

RES

ENT Understand meaning/purpose of graphs/tables.                 *     1  

Read different graphical representations.                 *     1  Display data in an organized way. *   *   *     *       *   *     6  Use/interpret a pictograph.     *     *           *   3  Interpret charts/graphs   *                 *   2  

USE

INFO

TO

A

DD

RES

S Q

UES

TIO

N

Formulate questions, gather data to address question. *   *   *     *         *   *   6  

PRO

BA

BIL

ITY

Know "more likely/less likely" situations (probability).                 *     1  

Make all possible combinations from two sets of objects.                   *   1  

Collect data from probability experiments.                 *     1  

Topic GLE CA  

TX  

NY  

FL  

OH  

MI  

GA  

NC  

VA  

AZ  

Stat

es

with

GLE

 

OTH

ER Use appropriate terminology   *   *   *   *   *   *   *   *   *   9  

Least-colors problem                   *   1  Model problem situations.         *   *     *       3