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Kentucky’s Approach to Standards-Based Mathematics Education

Kentucky’s Approach to Standards-Based Mathematics Education

Presented December 21, 2002 at the “International Con

ference on Children’s Mathematical Development and Standards-Based Assessment” Ministry of Education Taipei Municipal Teachers College Taipei, Taiwan R.O.C.

Willis N. Johnson, Professor Mathematics Education University of Kentucky Curriculum & Instruction Department Lexington, Kentucky 40506-0017 Voice: (859)-257-3158 Fax: (815)-333-3822 Email: wjohnsn@uky.edu

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Kentucky’s Approach to Standards-Based Mathematics Education

Overview: This paper has three basic parts: Part I: Kentucky’s History with Standards; Part II: How Children Learn Mathematics; and III: The Methods Course for Teaching Elementary School Mathematics. It is not meant to be a comprehensive report on mathematics education in the United States. It is an example of what happens during the implementation of standards-based education in mathematics and the assessment of mathematics. This paper connects how children learn mathematics to the preparation of teachers of mathematics. Hopefully, this paper will provoke many questions that are necessary in any attempt to effect change. Part I: Kentucky’s History with Standards In the late 1980’s, a Kentucky court ruled that Kentucky’s approach to funding schools was unconstitutional. As a result, the governor and the state legislature were set on a course to provide a constitutional system of education across the state. The result was the Kentucky Education Reform Act of 1990 (KERA). Within this law were six statements that defined the focus of instruction for Kentucky schools. These statements defined the goals of education. The Governor of Kentucky had the responsibility to implement the law using advisory groups composed of representative stakeholder groups throughout the state. Eleven people were appointed as the task force to define the essential skills and knowledge that best meet the six statements called the six “learner goals”. The author of this paper was one of the eleven (Transformations, 1995). Over the period of eighteen months, the group of citizens coordinated by the author refined how mathematics contributed to individual abilities within each of the six “learner goals”. After much debate within and between the various content areas, the “academic expectations” were defined. One definition of “standards” is that which represents “the skills and knowledge that we agree are essential” (Gandel and Vranek, 2001). It might be safe to conclude, then, that Kentucky’s Academic Expectations are indeed Kentucky’s education standards. After debating an organizational scheme for mathematics (suggested by Lynn Arthur Steen, 1990), the mathematics group specified seven Academic Expectations: It is no accident that these seven statements align very closely with the standards of the National Council of Teachers of Mathematics (NCTM, 1989). Since that time, however, both the Kentucky Academic Expectations and the NCTM Standards have undergone revisions. It is safe to say now, more so than before, that there is greater alignment between the NCTM Standards and Kentucky’s Academic Expectations. The next phase of this standards-based approach focused on selecting and designing appropriate assessments. The result was the implementation of open-ended assessment techniques. A goal was set to get every school to have a performance profile at the proficient level within twenty years of implementing reform. An examination of Figure 1 (below) will show why many existing standardized testing programs were insufficient to document and reinforce this profile.

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Kentucky’s Approach to Standards-Based Mathematics Education

Figure 1 Kentucky Department of Education Descriptors of the Four Performance Levels for Mathematics Teacher Developed – December 1, 1999

Level 1: Novice (N1) Student demonstrates limited understanding of problems. (N2) Student attempts to apply skills, concepts, and relationships to solve problems. (N3) Student demonstrates a limited understanding of core content. (N4) Student rarely or ineffectively uses mathematical terminology and/or representations (symbols, graphs, tables, diagrams, models). (N5) Student uses no or inappropriate reasoning. Level 2: Apprentice (A1) Student demonstrates basic or partial understanding of problems and attempts to use strategies (may include but not limited to use of tables, diagrams, simpler problems), but solutions may be incomplete or incorrect. (A2) Student applies skills, concepts and relationships to solve problems some of the time. (A3) Student demonstrates a partial understanding of core content. (A4) Student uses some mathematical terminology and/or representations (symbols graphs, tables, diagrams, models), but may be unclear. (A5) Student uses appropriate reasoning some of the time. Level 3: Proficient (P1) Student demonstrates a general understanding of problems and accurately uses appropriate strategies (may include but not limited to use of tables, diagrams, simpler problems), giving correct solutions most of the time. (P2) Student applies skills, concepts, and relationships to solve problems most of the time. (P3) Student demonstrates understanding of core content. (P4) Student uses appropriate and accurate mathematical terminology and/or representations (symbols graphs, tables, diagrams, models) effectively. (P5) Student uses appropriate reasoning most of the time. Level 4: Distinguished (D1) Student demonstrates a clear understanding of the problems by accurately and efficiently using appropriate strategies (may include but not limited to use of tables, diagrams, simpler problems), giving correct solutions. (D2) Student consistently applies skills, concepts, and relationships to solve problems. (D3) Student demonstrates comprehensive understanding of core content. (D4) Student uses appropriate and accurate mathematical terminology and representations (symbols, graphs, tables diagrams, models) in a clear and concise manner. (D5) Student uses appropriate reasoning consistently.

The realities of cost-benefit analyses provide a history of what has been attempted to “drive” instruction. This history is rich with a battery of machine and non- machine scored testing strategies. All of the approaches have been classified as “high stakes” testing. The variety of approaches used has won many supporters in some areas and many non-supporters in other areas. Performance-events testing is one of these approaches. The use of performance events in mathematics has evolved from long-term portfolios to more short-term, on-demand tasks. Initially, both portfolios and on-demand tasks were used. Items to be machine scored provided only a small portion of the assessments. The alignment between instruction and assessment is constantly evolving. There are still many problems to be addressed. There are still problems related to the amount of time to get results back to schools. There are still problems related to getting students to be accountable for their performance. And, as expected, there are problems that are aggravated by the rapidly changing teacher workforce and more mandated testing by the federal government. Hopefully, the needs and drives of humans will not be overlooked in this quest for higher performance levels. In 1989, the American Association of the Advancement of Science raised a number of questions to guide the common core of learning in science, mathematics, and technology that should be accomplished by the end of high school. These questions were related to utility,

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social responsibility, intrinsic value of the knowledge, philosophical value, and childhood enrichment. -Utility. Will the knowledge or skill significantly enhance long-term employment or educational prospects and personal decision making? -Social responsibility. Will the content help citizens participate intelligently in making social and political decisions? -Intrinsic value of the knowledge. Does the content have pervasive cultural or historical significance? -Philosophical value. Does the content help individuals ponder the enduring questions of what it means to be human? -Childhood enrichment. Will the content enhance the unique experiences and values of childhood? The standards movement in the United States began to blossom during this time. Have things changed as a result? Are we implementing a standards-based curriculum that is consistent with the above questions? The answer is both “yes” and “no”. In Kentucky and throughout much of the United States, “many educators continue to cover the content in the books, and their students continue to memorize the related vocabulary and algorithms— an inefficient and ineffective mention-and-move-on instructional strategy. U.S. schools and colleges devote huge amounts of classroom time to reviewing and re-teaching the same material every year because students don't learn it the first, second, or third time.” . . .” Recent evaluations of science and mathematics textbooks reveal similar weaknesses; most texts ignored or obscured the most important ideas by focusing instead on technical terms and trivial details” (American Association for the Advancement of Science, 2000). Further, “many models of curriculum design seem to produce knowledge and skills that are disconnected rather than organized into coherent wholes" (p. 138). In differentiating between novice learning and expert learning, the report notes that "it is the network, the connections among objectives, that is important" (pp. 138–139). Research also finds that superficial coverage of many topics won't help students develop the competencies that will prepare them for future learning and work (National Research Council, 2000). For more enlightenment on related questions about the content worth knowing in science, mathematics, and technology consult the writing of George Nelson, director of Project 2061 (Nelson, 2001). An organization known as Achieve studies and documents the history of the implementation of standards in the United States. Achieve found that for standards to have an impact they must be 1) rigorous, reasonable, teachable and 2) they must be measured by tests that meet the following criteria: a. If it's not in the standards, it shouldn't be on the test. b. When the standards are rich and rigorous, the tests must be as well. c. Tests should become more challenging in each succeeding grade. The standards in Kentucky have these characteristics. And most importantly, how students learn mathematics drives almost all discussions about mathematics except one: the role of the student in high-stakes assessment. Part II: How Children Learn Mathematics To implement KERA (1990), Kentucky mandated the implementation of the non-graded primary school. This has had mixed results. The theory was to provide an environment that

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Kentucky’s Approach to Standards-Based Mathematics Education

would increase the probability of success when the child entered fourth grade. The instructional environment was to be interdisciplinary, use flexible groups, learning centers, social interaction, independent work, authentic assessment, active hands-on work, use developmentally appropriate practices, use qualitative reporting methods, provide opportunities for students to use their strengths, have professional teamwork, have multiage/multi-ability classrooms, provide a safe and supportive environment, and opportunities for “making sense” of things. Justifications for the above characteristics were attributed to the works of Benjamin Bloom, Erik Erikson, Howard Gardner, Leslie Hart, Jean Piaget and Lev Vygotsky. For a more detailed explanation on the role these theories played, please consult the Kentucky Elementary Learning Profile (1996). The implications for teaching mathematics in Kentucky’s primary schools are consistent with the Principles and Standards for School Mathematics (NCTM, 2000). The goals of which are for all students to: -learn to value mathematics; -become confident in their ability to do mathematics; -become mathematics problem solvers; -learn to communicate mathematically; and -learn to reason mathematically. Instruction and assessments in mathematics are to be characterized by tasks that require the student to solve authentic and challenging problems and be able to communicate and model their reasoning. There should be many opportunities for students to explore and investigate as they solve problems. The specific content is developed within the following strands (standards): number, computation, geometry, measurement, probability, statistics, and algebra (NCTM Principle and Standards, 2000 and the Kentucky Core Content for Assessment). Successful completion of mathematics in the Kentucky primary school means the student has achieved competency in the above strands (standards) by the end of third grade. The Kentucky Elementary Learning Profile (KELP) (Appendix A) illustrates the descriptors and record keeping (Appendix B) recommended (but not mandated) to be used in assisting teachers in their selection of instructional and assessment resources. In most schools, the school’s council is responsible for such decisions. Each school council is composed of the principal, teachers, and a parent. As presented earlier, the state of Kentucky measures the progress of whole-school populations toward the goal of proficiency. Fifth grade is the year that the “high-stakes” accountability assessments in mathematics are given. Sample items that were used in previous years are in Appendix C. [There was a portfolio part of the mathematics assessments but it was abandoned. However, portfolios are still a major component of the assessments of language arts development.] The instructional environment for mathematics that best builds on how children learn mathematics uses many attributes from the child’s world. The richest environment is indicated by the data entries into the child’s profile (Kentucky Elementary Learning Profile) (See Appendix B). Among these are:

? ? conversations with parents/guardians and students (to understand and record the learning that takes place at home), observations of children while they are involved in school-based learning experiences

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? ? ? ? ? ? ? ? ? ? ? ? ?

recorded by using concise notes showing specific information and developmental milestones, varied work samples which show growth throughout the year(s), a student’s reflections of his/her own learning, a student’s best performances: posing a question and researching to get an answer; communicating through oral and written language; communicating through an aesthetic project, performance, or reflection of appreciation; reading for literary experience, to gain information, to perform a task, and to discriminate among messages; solving a real-life problem using computation and problem solving skills; completing and presenting a long-term project which integrates subject matter; participating in performing and reporting a group project; developing a Lifeline representing and reflecting the personal growth and learning; developing a personal well-being plan or project, a student’s growth in specific subject skills as evidenced in his/her performance(s).

A variety of curricula approaches can be used to help children “make sense” of the many symbols, rules, and procedures of mathematics. Through a careful and deliberate study of number, space, patterns, order and the use of explorations, investigations, and problemsolving, the child is eventually empowered as a mathematician. PART III: The Methods Course for Teaching Elementary School Mathematics Ideally, one-on-one instruction should be used to train and mentor teachers. One-on-one instruction maximizes opportunity for learning in the university and elementary school classroom. This has a high probability of success if the National Council of Teachers of Mathematics (NCTM) Principles characterize the instructional context. These principles are 1. Equity, 2.Curriculum, 3.Teaching, 4.Learning, 5.Assessment, and 6.Technology. To effect such a reality with one instructor and thirty students is the challenge. The goal for the instructor is to approximate, as close as is possible, the ideal. This can be accomplished by using a variety of inter-related strategies. Goals for the Course By the end of the semester, the hope is for significant progress in attaining the following: -Pre-service teachers will have a high degree of preference for choosing to teach mathematics. -Pre-service teachers will have a higher propensity to engage in discussions about ideas used to teach mathematics. -Pre-service teachers will verbalize context and connections for teaching mathematical ideas. The Methods and Practicum Semester The methods courses at the University of Kentucky have three parts: six weeks of on-campus preparation; five weeks to practice teaching in a school; and four additional weeks of oncampus preparation. The pre-service teachers take the science, language arts, social studies, and mathematics methods courses during this semester. Among the many ideas that are

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Kentucky’s Approach to Standards-Based Mathematics Education

covered in the mathematics methods class are the five content standards of the National Council of Teachers of Mathematics (NCTM), the five NCTM process standards, the six NCTM Principles; and the six NCTM Professional Standards for Teaching Mathematics. These ideas are consistent with and supportive of the Learner Goals and Academic Expectations of the Kentucky Education Reform Act (KERA) of 1990. Written specifically into the law are the following six learner goals: 1. Use basic communication and mathematics skills for purposes and situations they will encounter throughout their lives. 2. Apply core concepts and principles from mathematics, the sciences, the arts, the humanities, social studies, practical living studies, and vocational studies to what they will encounter throughout their lives. 3. Become self-sufficient individuals. 4. Become responsible members of a family, work group, or community, including demonstrating effectiveness in community service. 5. Think and solve problems in school situations and in a variety of situations they will encounter in life. 6. Connect and integrate experiences and new knowledge from all subject matter fields with what they have previously learned and build on past learning experiences to acquire new information through various media sources. Addressing the NCTM Content Standards: The NCTM organizes the K-12 curriculum into five areas or Standards: Number & Operations, Algebra, Geometry, Measurement, and Data Analysis. The State of Kentucky uses different labeling and grouping for the same things: Number/Computation, Geometry/Measurement, Probability/Statistics, and Algebraic Thinking. Pre-service teachers practice these ideas throughout their on-campus training. The technique used to facilitate this “practice” is a modified Jigsaw approach to cooperative learning (Aronson, 1978). Pre-service teachers are assigned to groups of 5-6 members. Each group member is assigned “minilessons” to teach to their group members. The instructor observes looking for misconceptions of the mathematics and for variations used by group members. The instructor then conducts a whole-class discussion. A general strategy used to begin classes is to give the class a problem to solve. A Think-Pair-Share strategy asks these pre-service teachers to work on the problem alone, then with a partner or small group, and finally some individuals are asked to share their solution and reasoning. The above approach to focusing pre-service teachers’ thinking on the NCTM Standards and the Kentucky Core Content Strands should result in certain habits of mind. First, there are a variety of ways to help people make sense of mathematics. Second, we learn by doing. Third, interacting with others can help clarify and broaden one’s perspectives. Fourth, a questioning and problem-solving approach can help motivate learning. Fifth, the human brain is susceptible to practices of the learning theories known as “constructivism” (Vygotsky, Piaget and others) and “Classical Conditioning” (Pavlov, 1927). Why such an eclectic approach? Basically, pre-service teachers from the University of Kentucky usually find teaching positions throughout Kentucky and throughout the world. There are many philosophies that dictate what pre-school and elementary school children will experience in classrooms. And, the Third International Mathematics and Science Study (TIMSS) supports the argument that instructional strategies used in US elementary schools

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results in high achievement on the TIMSS’ criteria for elementary mathematics. Thus the challenge is to be general rather than specific to an approach preferred by the instructor. However, there are some identified elements related to high performing schools. The NCTM have collected them under the notion of Principles: Equity, Curriculum, Teaching, Learning, Assessment, and Technology. Equity. Excellence in mathematics education requires equity— high expectations and strong support for all students. At the University of Kentucky, the number of pre-service students that have a history of success in mathematics and are comfortable learning mathematics is estimated to be around 25%. The number of pre-service teachers that have been successful but choose not to pursue further mathematics study is estimated to be around 25%. But the estimate of the number of pre-service teachers that have struggled with mathematics and will avoid further mathematics study is around 50%. Historical surveys of how much these pre-service teachers look forward to teaching mathematics has a similar distribution. One of the challenges, then, in the methods class is to influence dispositions toward teaching mathematics. The approach is to provide many opportunities to teach and to make sense of mathematics. Pre-service teachers practice with both peers and school children. As a result, post-course surveys also support this approach. When compared to pre-course surveys, the historical pattern shows that a significant number of pre-service teachers tend to raise mathematics as a subject that they prefer to teach. Hopefully, this positive outlook will have some transfer to their expectations for their future students. But equity in teaching mathematics is more than having a positive attitude or high expectations. More specifically, equity is best supported by focusing on the remaining five Principles. Curriculum. A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades. The methods class starts out by asking students to give their definition of mathematics. Almost invariably, pre-service teachers make statements to the effect that “mathematics is the study of numbers and rules to solve problems in a variety of situations.” Only a few preservice teachers develop a more comprehensive definition by the end of the course. At that time, most pre-service teachers will have amended their thinking to include “the use of numbers to measure, to do geometry, and solve problems in a logical manner.” Although there is much effort to challenge their thinking by forcing connections between the Kentucky content Standards/Strands and the NCTM process standards (Problemsolving, Reasoning & Proof, Communication, Connections, and Representation), preservice teachers insist that numbersense is the context for mathematical thinking. Perhaps it is sufficient at this point that new teachers see mathematics as making sense and that teachers are convinced of its importance in helping children now and throughout their lives. For the curriculum to be effective, the teacher should use the NCTM Professional Standards for Teaching Mathematics as a decision- making tool. These six standards focus on tasks, discourse, environment, and analysis. Specifically, they are: 1. Worthwhile Mathematical Tasks; 2.Teacher's Role in Discourse; 3. Students' Role in Discourse; 4. Tools for Enhancing Discourse; 5. Learning Environment, and 6. Analysis of Teaching and Learning The following three NCTM Principles provide focus for these professional standards for teaching mathematics.

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Kentucky’s Approach to Standards-Based Mathematics Education

Teaching. Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well. A primary focus in the methods course is to provide questions, problems, and situations that enhance students’ problem-solving skills. The pre-service teacher is presented with many opportunities to reflect on approaches to keep students mentally active and focused on mathematical thinking. Preservice teachers test approaches for balancing discourse by teachers and students. Judgments must be made regarding the tasks, tools, and feedback used to enhance learning. Learning. Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge. Among the many things that teachers need from their students is risk-taking. Students must be willing to share their reasoning. The teacher must really listen and be willing to modify their own thinking. In addition, students must be willing to ask questions, offer suggestions, and be willing to correct faulty thinking. This is best accomplished in a relatively “safe” environment. Assessment. Assessment should support the learning of important mathematics and furnish useful information to both teachers and students. As implied above, assessment is an ongoing activity. Many feedback mechanisms must be built in to maximize the use of time, to keep students mentally active, and aware of their progress. The instructional system must reward learning –not punish it. Technology. Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning. Hands-on, visual, and electronic aids can all be wastes of time and talent if used inappropriately. In mathematics, such tools must be used to foster meaning and to make it easier for the mind to organize complex systems. Instruction that begins with problems and/or questions often requires interpreting/translating or “acting out”. The four-step approach to solving problems is a reasonable approach: 1. What, exactly, is the problem? What details are important? What are the limitations on the solutions? Why is this situation a problem? 2. List all possible solutions. For each, give the possible good and bad points of the solution. 3. Choose a solution. 4. Check it out. Does it make sense? Does it work? If it doesn't work, go back to one of the previous steps and try again. Models or tools give us insight into what another is thinking. Models or tools give us common things to talk about. And it’s this common language that helps us to talk about the “general case”. That is what we can “see” only in the mind’s eye. Further, electronic technology may help us to better approximate one-on-one instruction. The student can “learn on their own”. The technology can help the student answer important questions, make sense of ideas that may not be understood, or strengthen memory in a variety of ways that are not threatening. Conclusion Statement: The pre-service teacher learns more than a daily routine that is mirrored in every classroom. He or she learns that doing meaningful mathematics is more than just memorizing and using rules to move meaningless symbols around on paper. The pre-service teacher sees that a deliberate orchestration of the above instructional principles can result in high achievement of Standards in just about any instructional environment. REFERENCES:

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Achieve, Inc. is an independent, bipartisan, nonprofit organization that helps states raise academic standards, measure performance against those standards, establish clear accountability for results and strengthen public confidence in our education system. www.achieve.org American Association for the Advancement of Science. (1989). Science for all Americans. New York: Oxford University Press. American Association for the Advancement of Science. (1993). Benchmarks for science literacy. New York: Oxford University Press. American Association for the Advancement of Science. (2000). Evaluation of science and mathematics textbooks online [Online]. Available: www.project2061.org Aronson, E., N. Blaney, C. Stephin, J. Sikes & M. Snapp (1978). The jigsaw classroom. Beverly Hills, CA: Sage Publishing Company. Commonwealth Achievement Test System (CATS) Released Mathematics Assessment Items are available online at http://www.kde.state.ky.us/oapd/ttp/ri/mathematics%20index.asp Gandal, Matthew and Vranek, Jennifer Standards (2001): Here Today, Here Tomorrow. Educational Leadership Volume 59 Number 1 September 2001 George D. Nelson Choosing Content That's Worth Knowing (2001). Educational Leadership Volume 59 Number 2 October 2001 Kentucky Elementary Learning Profile documents are available online at: http://www.kde.state.ky.us/osle/extend/primary/kelp.asp and http://www.kde.state.ky.us/osle/extend/primary/descript.asp National Council of Teachers of Mathematics (NCTM 2000). Principles and Standards for School Mathematics. Reston, VA: NCTM. National Council of Teachers of Mathematics (NCTM 1991). Professional Standards for Teaching Mathematics. Reston, VA: NCTM. National Research Council. (2000). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press. On the Shoulders of Giants: New Approaches to Numeracy. Lynn Arthur Steen, Editor; Mathematical Sciences Education Board, National Research Council, 1990 Pavlov, I. (1960). Conditioned reflexes. Trans. 1927. New York: Dover Transformations: Kentucky's Curriculum Framework. Frankfort, KY. Kentucky Department of Educatio n, September 1995 Transformations: Kentucky’s Curriculum Framework, Kentucky Department of Education, 1995. http://www.kde.state.ky.us/oapd/curric/Transformations/Transf_I.pdf Vygotsky, L (1978). Mind and Society. Cambridge, MA: Harvard University Press.

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Kentucky’s Approach to Standards-Based Mathematics Education

APPENDIX A

The Kentucky Elementary Learning Profile (KELP) Descriptors

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K ENTUCKY ELEMENTARY LEARNING PROFILE

A Part of Kentucky’s Model Primary Assessment System Student Name Last Teacher(s) Name School District/County First Middle Name used

As part of the Kentucky Education Reform Act of 1990, a Primary Program was designed for students from the time they enter school until they enter fourth grade. The attributes of this Primary Program are: developmentally appropriate practices, multi-age/multi-ability classrooms, continuous progress, authentic assessment, qualitative reporting methods, professional teamwork, and positive parent/guardian involvement. The Kentucky Elementary Learning Profile (KELP) is the model assessment instrument designed by the Kentucky Department of Education to correspond with the Primary Program. The KELP instrument is designed to document a student’s real learning, growth, and development during the primary years. The KELP instrument, along with the progress report and Learning Descriptions, is designed to be a comprehensive primary assessment system. The KELP allows for documentation of: ? conversations with parents/guardians and students (to understand and record the learning that takes place at home), ? observations of children while they are involved in school-based learning experiences recorded by using concise notes showing specific information and developmental milestones, ? varied work samples which show growth throughout the year(s), ? a student’s reflections of his/her own learning, ? a student’s best performances: ? posing a question and researching to get an answer; ? communicating through oral and written language; ? communicating through an aesthetic project, performance, or reflection of appreciation; ? reading for literary experience, to gain information, to perform a task, and to discriminate among messages; ? solving a real-life problem using computation and problem solving skills; ? completing and presenting a long-term project which integrates subject matter; ? participating in performing and reporting a group project; ? developing a Lifeline representing and reflecting the personal growth and learning; ? developing a personal well-being plan or project, ? a student’s growth in specific subject skills as evidenced in his/her performance(s). For more information concerning Kentucky’s primary program and/or the Kentucky Elementary Learning Profile, please contact the Early Learning Branch in the Kentucky Department of Education, 500 Mero Street, Frankfort, Kentucky 40601 or visit the Primary Web Page at: http://www.kde.state.ky.us/osle/extend/primary/default.asp A handbook is available to provide explanations and support in the use of this instrument.

Permission is granted to copy any part of the Kentucky Elementary Learning Profile for educational use in Kentucky schools.

INTRODUCTION Mathematics is the study of number, space, pattern, and order. Children use mathematics to

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Kentucky’s Approach to Standards-Based Mathematics Education

meet their own needs, purposes, or intentions. Mathematics involves exploration, investigation, and problem solving and has its own set of symbols, rules, and procedures. A child learns mathematical ideas more by constructing the ideas than by acquiring them through listening and memorization. Focus should be placed on the meaning being communicated rather than the form of the communication. BECOMING A MATHEMATICIAN Young children are very concrete and egocentric, and their mathematics involves objects and experiences that are real and familiar. As children grow, develop, and learn about their worlds, they become less egocentric and more global in their thinking. Their mathematics also becomes more abstract and global as they grow and develop. A child develops knowledge of numbers gradually over time. Initially a child learns about single digit numbers how to use them to represent quantities in the environment, and how they are related to other numbers. The child solves his or her own problems involving small numbers, common shapes, and simple language by inventing procedures. As the child grows and develops, he/she learns about larger numbers using place value ideas and constructing written procedures for operating on these larger numbers, e.g., adding, multiplying. As children develop knowledge about whole numbers, they begin to construct ideas about fractions, decimal fractions, and negative numbers. As children’s environments widen, they solve more general problems involving larger numbers, more complex shapes, and more sophisticated language; children, however, continue to invent methods for finding solutions. CONNECTION TO NCTM STANDARDS The mathematics continuum outlined on the following pages reflects the developmental nature of children. It also reflects the major ideas set forth by the National Council of Teachers of Mathematics (NCTM) in their groundbreaking 1989 document, Curriculum and Evaluation Standards for School Mathematics. In particular, the cont inuum supports the five general goals set forth by NCTM for all students: ? they learn to value mathematics; ? they become confident in their ability to do mathematics; ? they become mathematical problem solvers; ? they learn to communicate mathematically; and ? they learn to reason mathematically. PORTFOLIO CONNECTION Careful attention has been given to the alignment of the Mathematics Learning Description with the standards and expectations established for fourth and fifth graders and their Mathematics Portfolios. THE MATHEMATICS LEARNING DESCRIPTION The Mathematics Learning Description is divided into the components Problem Solving, Communication/Connections , Number Concepts, Spatial Concepts, and Numerical Procedures. However, each component affects and interacts with the other components in a highly integrated fashion. A child constructs mathematical ideas involving all five components simultaneously. Problem Solving includes problem complexity (features of a problem that determine the degree of difficulty, such as the size and types of numbers, the operations involved, the presence or absence of necessary data, and the number of variables that affect the solution of

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the problem), reasoning/patterns (looking for relationships and drawing conclusions), and strategies (methods a student uses to solve a problem, such as guess, check, and revise; act it out; use models; draw pictures or diagrams; make a list, table or chart). Communication/Connections includes communication (using oral and/or written means to represent and understand ideas and relationships, or to convey results of doing mathematics) and connections (recognizing relationships among different mathematical concepts, using mathematics in other subjects, and using mathematics in every day life). Number Concepts includes number meanings (knowledge, understanding, and use of whole numbers, fractions, negative numbers, and decimal fractions), number relationships (how numbers are related to each other, including greater than and less than relationships and partwhole relationships), and the relative magnitude of numbers (mentally ordering the numbers from 0?1,000,000,000 and mentally ordering commonly used fractions, decimals, and negative whole numbers). Spatial Concepts includes geometry (properties and relationships involving 2- and 3dimensional figures) and measurement (the process of attaching a number to an object/figure with respect to one of the following attributes of the object/figure, length, area, weight, volume, time, temperature, and angle). A Numerical Procedure is a well-defined sequence of steps, involving numbers, to achieve a specified purpose, such as: adding multi-digit numbers, finding the median, using estimation strategies, e.g., rounding, front end. Executing a procedure can be entirely mental; it can involve recording results on paper, using calculators or manipulatives; or it can involve a combination of these processes. The relationship of these five mathematics-learning components is evident when we consider two children faced with the problem of sharing three cookies fairly. To solve the problem together, they must communicate their mathematical ideas. They use spatial concepts when they divide the cookies into equal parts and they develop a procedure for doing this. They confront the number concept of ?± half?°in this problem, which is connected to their lives. one As this cookie-sharing problem indicates, children develop and connect mathematical ideas in many areas simultaneously.

SNAPSHOT

This section provides a quick overview of the five mathematical components. It gives a general description or a starting point to help you identify the characteristics a student is exhibiting while learning mathematics. Few students will exhibit characteristics all at the same point on the continuum. The detailed narrative explanations on pages 6-15 will give you more specific information about the students’ mathematical development, often with examples.

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Kentucky’s Approach to Standards-Based Mathematics Education

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Appendix B

The Kentucky Elementary Learning Profile (KELP) Record Keeping Chart

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Kentucky’s Approach to Standards-Based Mathematics Education

PART A: CONVERSATIONS

A1 Record of conversation between student's parent(s)/guardian(s) and teacher(s)

Signatures: Parent(s)/Guardian(s)__________________________________ Teacher(s)__________________________________________

Date______/______/______ Date______/______/______

A2 Record of learning conversation between student and teacher(s)

Signatures: Student ___________________________________________ Date______/______/______ Teacher(s)___________________________________________ Date______/______/______

This section is adapted with permission from the original Primary Language Record, developed and copyrighted by the Centre for Language in Primary Education, Webber Row, London SE1 8QW, U.K.

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PART B: DIARY OF OBSERVATIONS

SOCIAL CONTEXT:

Date

Student’s Name ________________

Independent (I)

Student with Adult (S/A)

Anecdotal Notes

Pair (P)

Small Group (G)

Group Led by Adult (G/A)

Next S teps

Observations

This page may be reproduced as often as needed. Alternate pages, located in the Teacher Handbook, may be substituted for this part. You may write on this page or attach labels, index cards, sticky notes, etc. Anecdotal notes should include samples from various learning contexts including: literacy, mathematics, science, social studies, arts and humanities, design and construction, physical development, and dramatic and investigative play. The notes should also span the learning domains: social, emotional, physical, aesthetic, as well as cognitive.

This section is adapted with permission from the original Primary Language Record, developed and copyrighted by the Centre for Language in Primary Education, Webber Row, Lo ndon SE1 8QW, U.K.

18

Kentucky’s Approach to Standards-Based Mathematics Education

PART C: T YPES OF PERFORMANCES

Type of Performance* Entries Include d

(T here)

Student’s Name________________

Reflections Included

(T here)

Optional: Name of Project/Notes/Ideas

Pose a question and research to get an answer. Communicate through oral and written language (or an alternative form of language, i.e., Braille, sign language, etc. when applicable).

Communicate through an aesthetic project, performance, or reflection of appreciation.

Read for literary experience, to gain information, to perform a task, and to discriminate among messages.

Solve a real-life problem using computation and problem solving skills.

Complete and present a long-term project, which integrates subject matter.

Participate in performing and reporting a group project.

Develop a ?Lifeline? representing and reflecting the personal growth and learning. (may extend over several years)

Develop a personal well-being plan or project. (may extend over several years)

* Each performance entry must be accompanied with a reflection.

19

20

Kentucky’s Approach to Standards-Based Mathematics Education

21

Appendix C

Sample CATS Test Item and Analysis Grade 5

22

Kentucky’s Approach to Standards-Based Mathematics Education

23

24

Kentucky’s Approach to Standards-Based Mathematics Education

25

26

Kentucky’s Approach to Standards-Based Mathematics Education

27

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Kentucky’s Approach to Standards-Based Mathematics Education

29

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