7
Designing Curriculum, Instruction, Assessment, and Professional Development

The previous chapter describes seven principles that support learning with understanding. This chapter explores the implications of those principles for the intentional and systemic design of four key elements of the educational system—curriculum, instruction, assessment, and professional development—to promote learning with understanding within the context of advanced study. It is critical to recognize that programs for advanced study share many of the objectives of other programs in the same discipline; these design principles, therefore, also apply to the design and development of mathematics and science courses at all levels.

While each of the four key elements is addressed separately here, in practice they work together synergistically and need to be aligned in mutually supportive ways. Without such alignment and interdependence, deep conceptual understanding is more difficult to achieve. For example, if teachers focus on teaching “big ideas” but the related assessments measure students’ knowledge of discrete facts, it is impossible to know the extent to which students genuinely understand core concepts. The systemic and dynamic relationship among the four elements also means that changes in one element affect and require changes in the others.

In addition, it is essential to recognize the critical role of the learning environment in fostering learning with understanding. The learning environment of the school and the classroom in which these components of educational programs interact affects the degree to which teachers can integrate curriculum, instruction, and assessment to promote learning with understanding (National Research Council [NRC], 2000b).



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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools 7 Designing Curriculum, Instruction, Assessment, and Professional Development The previous chapter describes seven principles that support learning with understanding. This chapter explores the implications of those principles for the intentional and systemic design of four key elements of the educational system—curriculum, instruction, assessment, and professional development—to promote learning with understanding within the context of advanced study. It is critical to recognize that programs for advanced study share many of the objectives of other programs in the same discipline; these design principles, therefore, also apply to the design and development of mathematics and science courses at all levels. While each of the four key elements is addressed separately here, in practice they work together synergistically and need to be aligned in mutually supportive ways. Without such alignment and interdependence, deep conceptual understanding is more difficult to achieve. For example, if teachers focus on teaching “big ideas” but the related assessments measure students’ knowledge of discrete facts, it is impossible to know the extent to which students genuinely understand core concepts. The systemic and dynamic relationship among the four elements also means that changes in one element affect and require changes in the others. In addition, it is essential to recognize the critical role of the learning environment in fostering learning with understanding. The learning environment of the school and the classroom in which these components of educational programs interact affects the degree to which teachers can integrate curriculum, instruction, and assessment to promote learning with understanding (National Research Council [NRC], 2000b).

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools CURRICULUM A curriculum for understanding is intentionally designed around the organizing principles and essential concepts of the domain and provides opportunities for in-depth exploration in a variety of contexts (design principles for curriculum are summarized in Box 7-1). Such a curriculum emphasizes depth of understanding over breadth of coverage. It is designed to provide genuine opportunities for high-quality instruction and multiple points of entry into mathematics and science (Au and Jordan, 1981; Brown, 1994; Heath, 1983; Tharp and Gallimore, 1988). Research reveals that experts’ knowledge is organized around core concepts or organizing principles that guide their thinking in their area of exper- BOX 7-1 Principles of Curriculum for Understanding A mathematics or science curriculum for advanced study that promotes learning with understanding: Structures the concepts, factual content, and procedures that constitute the knowledge base of the discipline around the organizing principles (big ideas) of the domain. Links new knowledge to what is already known by presenting concepts in a conceptually and logically sequenced order that builds upon previous learning within and across grade levels. Focuses on depth of understanding rather than breadth of content coverage by providing students with multiple opportunities to practice and demonstrate what they learn in a variety of contexts. Includes structured learning activities that, in a real or simulated fashion, allow students to experience problem solving and inquiry in situations that are drawn from their personal experiences and real-world applications. Develops students’ abilities to make meaningful applications and generalization to new problems and contexts. Incorporates language, procedures, and models of inquiry and truth verification that are consistent with the accepted practice of experts in the domain. Emphasizes interdisciplinary connections and integration and helps students connect learning in school with the issues, problems, and experiences that figure prominently in their lives outside of the classroom.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools tise; expert knowledge is not simply a list of facts and formulas (Chi, Feltovich, and Glaser, 1981; Kozma and Russell, 1997; NRC, 2000b; see also Chapter 6, this volume). Therefore, in designing a curriculum for understanding, the key concepts and processes of the discipline should be clearly identified, explicated, and organized in a coherent fashion around the big ideas (Mintzes, Wandersee, and Novak, 1998; National Council of Teachers of Mathematics [NCTM], 1995; NRC, 1996). In addition, the interrelationships among topics should be clearly articulated to provide a framework teachers can use in developing and setting goals for their students’ learning (American Association for the Advancement of Science [AAAS], 2001). The organization of curriculum plays a critical role in helping students reconstruct misconceptions and see connections between what they are currently learning and what they have learned before. Curriculum for understanding represents more than a collection of activities or bits of information: it provides for the holistic performance of meaningful, complex tasks in increasingly challenging environments (Resnick and Klopfer, 1989). A curriculum for understanding takes the shape of topical strands that are highly interconnected in ways that are consistent with the knowledge structure used by experts in tackling complex tasks in their discipline (Marin, Mintzes, and Clavin, 2000).1 The deep disciplinary understanding of experts encompasses a vast amount of knowledge, but generally only a subset of that knowledge is used in the solution of any given problem. Experts not only have acquired extensive and deep knowledge and conceptual understanding, but also are skilled at discerning, identifying, and retrieving knowledge that is relevant to the solution of a particular problem. Their knowledge is organized into meaningful patterns and structures and is conditionalized (situated), meaning that what they know is accompanied by a specification of the contexts in which it is useful (Glaser, 1992; Simon, 1980). Many curricula and instructional materials, however, are not designed to help students conditionalize their knowledge. For example, textbooks are more likely to tell students how to do something than to help them understand the conditions under which doing it will be useful (Simon, 1980, p. 92). Having students work in laboratory settings is a familiar strategy for helping them develop conditionalized knowledge that supports problem solving. Well-designed laboratory experiences also encourage students to apply their knowledge and skills to concrete, real-world problems or novel situations (Resnick, 1994). 1   See also the proceedings of From Misconceptions to Constructed Understanding, the Fourth International Misconceptions Seminar, at http://www.mlrg.org/proc4abstracts.html (November 27, 2001).

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools Students presented with vast amounts of content knowledge that is not organized into meaningful patterns are likely to forget what they have learned and to be unable to apply the knowledge to new problems or unfamiliar contexts (Haidar, 1997). Curriculum for understanding provides ample opportunity for students to apply their knowledge in a variety of contexts and conditions. This helps them transfer their learning to new situations and better prepares them for future learning (Bransford and Schwartz, 2000). Providing students with frequent opportunities to apply what they learn in multiple contexts requires a reallocation of instructional time. Allowing time for in-depth learning means decisions must be made about what knowledge is of most worth. For this reason, the curriculum needs to specify clearly the appropriate balance between breadth and depth of coverage in terms of student learning outcomes. It is well accepted that students draw on their families, communities, and cultural experiences to create meaning and understanding. When curriculum is designed to build on students’ experiences, teachers are able to engage students’ prior knowledge, expose and restructure their knowledge and remediate misconceptions, and enhance motivation to learn. If students are able to draw on their cultural, social, and historical experiences in problem-solving situations, they are more likely to deepen their understanding. This can be accomplished by design through structured activities that, in real or simulated fashion, allow students to experience problem solving and inquiry in situations drawn from their personal experiences. An effective curriculum allows for incorporating socialization into the discourse and practices of academic disciplines and provides frequent opportunities for students to apply the modes of inquiry and truth verification strategies and processes characteristic of each domain. An appreciation of the distinctive features of disciplines, however, should not lead to their isolation from each other or from the everyday world. Rather, strong curriculum design emphasizes interdisciplinary connections, integration, and authenticity in the relationship between learning in and out of school. These features not only make learning more challenging, exciting, and motivating, but also help students develop their abilities to make meaningful connections by applying and transferring knowledge from one problem context to another. The emphases of a curriculum for supporting learning with understanding are presented in Table 7-1. INSTRUCTION Instruction in advanced courses in mathematics and science should engage students in a variety of learning activities that are purposefully designed to connect with what they already know and motivate them to work

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools TABLE 7-1 Characteristics of Curriculum for Student Understanding Less Emphasis on More Emphasis on Lists of topics to be covered Underlying disciplinary principles and processes sequenced to optimize learning Isolated concepts presented in relation to a single context or topic with no the connection to other topics Continuity and interdisciplinary integration through emphasis on relationships of unifying concepts and processes to many topics Individual teachers working in isolation Collaborative teams including content and pedagogy experts Coverage of as many topics as possible in a limited and fixed amount of time Understanding of concepts in expanded and more flexible time periods Example: A curriculum guide specifies that 8 percent of the advanced biology course should include topics related to cell energy, including cellular respiration and photosynthesis. The teacher schedules 1 week for a unit on cellular respiration. During this time, students carry out a variety of activities and laboratory exercises in class and complete out-of-class assignments. At the end of the week they take a unit test. The class goes on to the next unit, photosynthesis. Example: A curriculum specifies four statements of essential knowledge about the production and utilization of energy in cells that are critical to building an understanding of the processes of photosynthesis and respiration. Prior relevant knowledge possessed by students is assessed using quizzes or student concept maps. The teacher schedules 3 days for activities that focus on this knowledge. A rubric is written so that both teacher and students know how the knowledge will be demonstrated. Students carry out a variety of activities in class and complete out-of-class assignments. At the same time, they must demonstrate their knowledge of each of the four specified outcomes before they can receive credit for the unit. Students continue to work on essential knowledge and present evidence when they are ready, even as activities for the next unit may be beginning. Curriculum developed for one course at a time without articulation among the levels of schooling Curriculum that is well articulated between the elementary grades and between high school and college and makes recognized connections with other disciplines A rigid, prescribed, static curriculum Curriculum that can be adapted to meet the diverse needs of students and situations No relationship between NRC’s National Science Education Standards/NCTM’s Principles and Standards for School Mathematics and programs for advanced study or college courses Standards-driven changes in courses, leading to advanced high school and introductory college courses through modification of content and pedagogy at the advanced course and college levels   SOURCES: Adapted from ACS (1997); NRC, (1996); and NSTA, (1996). toward developing deeper understanding. Instruction should focus students on the central concepts and fundamental principles of the discipline. It also should assist them in constructing a framework for organizing new information as they explore concepts in depth and in a variety of contexts and develop problem-solving strategies common to the discipline (Novak, 1991). The design principles for instruction are summarized in Box 7-2.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools BOX 7-2 Principles of Instruction for Understanding Teaching for conceptual understanding in advanced mathematics and science courses: Maintains students’ focus on the central organizing themes and underlying concepts of the discipline. Is based on careful consideration of what students already know, their ideas and ways of understanding the world, and the patterns of practice they bring with them into the classroom. Focuses on detecting, making visible, and addressing students’ often fragile, underdeveloped understandings and misconceptions. Reflects an understanding of differences in students’ interests, motivations, preferences, knowledge, and abilities. Is designed to provide the appropriate degree of explicitness for the situation and the abilities of the learners. Recognizes students’ preferences for and varying abilities to process different symbol systems, such as language (written and spoken), images, and numerical representations, by employing multiple representations during instruction. Engages students in worthwhile tasks that provide access to powerful mathematical and scientific ideas and practices; moves students to see past the surface features of problems to the deeper, more fundamental principles; and develops their conceptual understanding and skills. Structures learning environments in which students can work collaboratively to gain experience in using the ways of thinking and speaking used by experts in the discipline. Orchestrates classroom discourse so that students can make conjectures, present solutions, and argue about the validity of claims, thus helping them explore old understandings in new ways, reveal misconceptions, and generalize and transfer their learning to new problems or more robust understandings. Provides explicit instruction in metacognition as part of teaching in the discipline. Uses various kinds of formal and informal formative assessments to monitor students’ understanding and target instruction effectively. Creates expectations and social norms for the classroom that allow students to experience success and develop confidence in their abilities to learn.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools The selection of instructional strategies and activities should be guided by knowledge of learners and should recognize and build on individual differences in students’ interests, understandings, abilities, and experiences. Instruction should take into account common naive concepts held by students, as well as the effects of their cultural and experiential backgrounds on their learning. It also should reflect the teacher’s own strengths and interests and consideration of available local resources. Activities and strategies should be continually adapted and refined to address topics arising from student inquiries and experiences. It is important for instruction in advanced courses in mathematics and science to engage students in inquiry using a variety of activities and strategies, including experimentation, critical analysis of various sources of information, and the application of technology in problem solving. In this way, students combine knowledge in the domain with reasoning and thinking skills as they are engaged collaboratively in asking questions, constructing, testing and analyzing explanations, communicating the explanations, and considering alternatives (Townes and Grant, 1997). One of a teacher’s primary responsibilities is to select and develop significant and meaningful problems, learning experiences, projects, and investigations for students. Learning experiences are worthwhile when they represent concepts and procedures; foster skill development, reasoning, and problem solving; and help students make connections among mathematical and scientific ideas and to real-world applications (NCTM, 1991). Such experiences prompt the learner to see past the surface features of a problem to deeper, more fundamental principles. They “lend themselves to multiple solution methods, frequently involve multiple representations, and usually require students to justify, conjecture, and interpret” (Silver and Smith, 1996, p. 24). The design of such tasks is complex, requiring teachers to take account of students’ knowledge and interests, of the ways students learn particular mathematical or scientific ideas, and of common points of confusion and misconceptions about those ideas (Borko et al., 2000). Accomplishing this complex endeavor requires a qualified teacher. The National Board for Professional Teaching Standards (NBPTS) describes a qualified teacher as one who effectively enhances student learning and demonstrates the high level knowledge, skills, abilities, and commitments reflected in the following five core propositions:2 Teachers are committed to students and their learning. Teachers know the subjects they teach and how to teach those subjects to students. 2   See http://www.nbpts.org (November 22, 2001).

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools Teachers are responsible for managing and monitoring student learning. Teachers think systematically about their practice and learn from experience. Teachers are members of learning communities. To promote understanding, explicit instruction in metacognition should be integrated into the curriculum. Thus, instruction should create tasks and conditions under which student thinking can be revealed so that students, with their teachers, can review, assess, and reflect upon what they have learned and how. Additionally, teachers should make their reasoning and problem-solving strategies visible to students whenever possible (Collins and Smith, 1982; Lester et al., 1994; Schoenfeld, 1983, 1985). Effective instruction in advanced courses should involve building and nurturing a community of learners. A community of learners encourages students to take academic risks by providing opportunities for them to make mistakes, obtain feedback, and revise their thinking while learning from others with whom they are engaged in inquiry and cooperative problem-solving activities. To nurture the capacity of students to generalize and transfer their learning to new problems, teachers must help students explore old understandings in new ways. To this end, teachers must draw out misconceptions in order to challenge and displace them (Blumenfeld, Marx, Patrick, Krajcik, and Soloway, 1997; Caravita and Hallden, 1994; Jones, Rua, and Carter, 1998; NRC, 2000b; Pearsall, Skipper, and Mintzes, 1997;). Since intrinsic motivation is self-sustaining, instruction should be planned so as to maximize the opportunity for developing a strong intrinsic motivation to learn. Students benefit when they can experience success and develop the confidence of a successful learner—one who has the tools to ask relevant questions, formulate problems and reframe issues, and assess his or her own knowledge and understanding (Alaiyemola, Jegede, and Okebukola, 1990; Stipek, 1998). Table 7-2 illustrates the emphases of instructional practices to support learning with understanding. ASSESSMENT Educational assessments can be designed for any number of purposes, from conducting large-scale evaluations of multiple components of educational programs to measuring individual students’ mastery of a specified skill. Understanding assessment results requires that the user draw inferences from available data and observations that are supported by the assessment. Three key concepts related to assessments—reliability, validity, and fairness—underlie a user’s ability to draw appropriate inferences from the

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools TABLE 7-2 Characteristics of Instruction for Student Understanding Less Emphasis on More Emphasis on Presenting scientific knowledge through lecture, text, and demonstration, with activities centered on the teacher Guiding students through active and extended inquiry and facilitating student-centered learning The same learning experiences for all students Appropriate matching of strategies and learners based on awareness of individual student’s prior knowledge, abilities, and interests Strategies that have students working alone Strategies that incorporate collaboration among students and foster the development of classroom learning communities “One-size-fits-all” instructional strategies Multiple strategies designed to enhance understanding Establishing blocks of time and designing lessons such that all students are required to learn the same thing in the same way at the same rate Flexible scheduling and learning experiences that provide students with enough time, space, resources, guidance, and feedback for learning Students doing numerous, often simplistic and unconnected laboratory activities and being exposed to many different procedures Students conducting extended investigations and inquiry and having opportunities to progress through cycles of assessment and revision Focusing on elaborate, equipment-intensive laboratory exercises Focusing on interactions between students and materials, as well as teacher–student and student– student interactions Laboratory exercises in which students are provided with all relevant background and procedures and are asked to follow the steps, fill in the data, and answer a few questions, after which the class moves on Laboratory and inquiry experiences in which students are challenged to formulate questions that can be answered experimentally, propose and support hypotheses, plan procedures, design data tables and data analyses, evaluate and discuss results, and repeat experiments with modifications Teachers monitoring laboratory work to ensure that the steps of the procedure are being followed correctly Teachers acting as facilitators for laboratory experimentation; advising students on what essential measurements must be taken; discussing sample sizes; suggesting equipment that is available for use during the experiment; coaching students in techniques and protocols; and, within the limits of safety, allowing students to make mistakes and try again Example: The teacher’s lecture centers on presenting and explaining rate expressions and factors that affect forward and reverse reactions. Keq calculations are shown. The teacher demonstrates color changes in a reversible reaction. LeChatelier’s principle is reviewed, and students predict equilibrium shifts on the basis of hypothetical changes in reactants, products, and conditions. Student misconceptions about the nature of equilibrium remain uncovered and unchallenged. Example: The teacher poses a question: “If the rate of a forward reaction is faster than the rate of a reverse reaction, will the system ever come to equilibrium?” Students are asked to construct a model for such a system, in which particles can be moved between two containers at different rates, and to use this model to collect data needed to answer the question. The common student misconception that equilibrium means equal amounts in each container is challenged as students develop an understanding of the principle of equilibrium.   SOURCES: Adapted from ACS (1997); NRC (1996); NSTA (1996).

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools results. A brief discussion of each is provided in Box 7-3 (for further detail see, American Educational Research Association [AERA]/American Psychological Association [APA]/National Council on Measurement in Education [NCME], 1999; Feldt and Brennan, 1993; Messick, 1993; NRC, 1999b). Assessment is a critical aspect of effective teaching and improved education (NBPTS, 1994; NCTM, 1995; NRC, 2001a; Shepard, 2000). It is important to note, however, that assessment does not exist in isolation, but is closely linked to curriculum and instruction (Graue, 1993). Thus as emphasized earlier, curriculum, assessment, and instruction should be aligned and integrated with each other, and directed toward the same goal (Kulm, 1990; NCTM, 1995; Shepard, 2000). In advanced mathematics and science, that goal is learning with understanding. This section reviews design principles for two types of assessments: those that measure student achievement at the end of a program of study, such as AP Physics, and those that are used by teachers to provide feedback to students, guide instruction, and monitor its effects throughout the course of study (see Box 7-4 for a summary of the design principles for assessment). To guide instruction, teachers need assessments that provide specific BOX 7-3 Reliability, Validity, and Fairness Reliability generally refers to the stability of results. For example, the term denotes the likelihood that a particular student or group of students would earn the same score if they took the same test again or took a different form of the same test. Reliability also encompasses the consistency with which students perform on different questions or sections of a test that measure the same underlying concept, for example, energy transfer. Validity addresses what a test is measuring and what meaning can be drawn from the test scores and the actions that follow (Cronbach, 1971). It should be clear that what is being validated is not the test itself, but each inference drawn from the test score for each specific use to which the test results are put. Thus, for each purpose for which the scores are used, there must be evidence to support the appropriateness of inferences that are drawn. Fairness implies that a test supports the same inferences from person to person and group to group. Thus the test results neither overestimate nor underestimate the knowledge and skills of members of a particular group, for example, females. Fairness also implies that the test measures the same construct across groups.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools BOX 7-4 Principles of Assessment for Understanding The following principles of assessment can be applied both to assessments designed to assist learning (curriculum-embedded or formative assessments) and those designed to evaluate student achievement at the end of a unit of study (summative assessments). Effective assessments for measuring students’ learning with understanding in advanced mathematics and science are: Based on a model of cognition and learning that is derived from the best available understanding of how students represent knowledge and develop competence in a domain. Designed in accordance with accepted practices that include a detailed consideration of the reliability, validity, and fairness of the inferences that will be drawn from the test results (see Box 7-3). This is especially important when the assessment carries high stakes for students, teachers, or schools. Aligned with curriculum and instruction that provide the factual content, concepts, processes, and skills the assessment is intended to measure so the three do not work at cross-purposes. Designed to include important content and process dimensions of performance in a discipline and to elicit the full range of desired complex cognition, including metacognitive strategies. Multifaceted and continuous when used to assist learning by providing multiple opportunities for students to practice their skills and receive feedback about their performance. Designed to assess understanding that is both qualitative and quantitative in nature and to provide multiple modalities with which a student can demonstrate learning. information about what their students are learning and what they do and do not understand. Of primary importance if a test is to support learning is that students be given timely and frequent feedback about the correctness of their understandings; in fact, providing such feedback is one of the most important roles for assessment. There is a large body of literature on how classroom assessment can be designed and used to improve learning and instruction (see for example, Falk 2000; Shepard 2000; Wiggins, 1998; Niyogi, 1995). Concept maps, such as those discussed in Box 6-2 in Chapter 6, are one example of an assessment strategy that can be used to provide timely

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools and informative feedback to students (Edmondson, 2000; Rice, Ryan, and Samson, 1998). End-of-course tests are too broad and too infrequently administered to provide information that can be used by teachers or students to inform decisions about teaching or learning on a day-to-day basis. The power of such tests lies in their ability to depict students’ attainment of larger learning goals and to provide comparative data about how the achievement of one student or one class of students compares with that of others.3 Ultimately, end-of-course tests are often used to shape teachers’ instructional strategies in subsequent years. Thus, the content of the tests should be matched to challenging learning goals and subject matter standards and serve to illustrate what it means to know and learn in each of the disciplines. Because advanced study programs in the United States are strongly influenced by high-stakes assessment, the committee is especially concerned with how this form of assessment can be structured to facilitate learning with understanding. It is well known that such assessments, even coming after the end of instruction, inevitably have strong anticipatory effects on instruction and learning. Thus if high-stakes assessments fail to elicit complex cognition and other important learning outcomes, such as conceptual understanding and problem solving, they may have negative effects on the teaching and learning that precede them. In designing such assessments, then, both psychometric qualities and learning outcomes should be considered. If end-of-course tests are to measure important aspects of domain proficiency, test makers need to have a sophisticated understanding of the target domain. They must understand the content and the process dimensions that are valued in the discipline and then design the test to sample among a broad range of these dimensions (Millman and Greene, 1993). Doing so is complicated, however, by the fact that an assessment can only sample from a large universe of desirable learning outcomes and thus can tap but a partial range of desirable cognitions. Consequently, concerns will always arise that a particular assessment does not measure everything it should, and therefore the inferences drawn from it are not valid. Similarly, the selection of tasks for an assessment may be criticized for measuring more than is intended; an example is word problems on mathematics tests that require high levels of reading skill in addition to the mathematics ability that is the target of the assessment. To ensure the validity of inferences drawn from tests, a strong program of validity research must be conducted on all externally designed and administered tests. The higher the stakes of the test, the more critical is this research and the more frequently it must be reviewed (AERA/APA/NCME, 1999). 3   In the case of such tests as the AP and IB examinations, the results are used additionally to guide decisions about college placement and credit.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools Test design and construction includes consideration of which forms of complex thinking fairly reflect important aspects of domain proficiency. Assessments that invoke complex thinking should target both general forms of cognition, such as problem solving and inductive reasoning, and forms that are more domain-specific, such as deduction and proof in mathematics or the systematic manipulation of variables in science. Because metacognition is such an important component of experts’ performance, both classroom-based and end-of-course assessments should be designed to evaluate students’ use of metacognitive strategies (NRC, 2001a). Given that the goals of curriculum and assessment for advanced study are to promote deep understanding of the underlying concepts and unifying themes of a discipline, effective assessment should reveal whether students truly understand those principles and can apply their knowledge in new situations. The ability to apply a domain principle to an unfamiliar problem, to combine ideas that originally were learned separately, and to use knowledge to construct new products is evidence that robust understanding has been achieved (Hoz, Bowman, and Chacham, 1997; Perkins, 1992). Meaningful assessment also includes evidence of understanding that is qualitative and quantitative in nature, and provides multiple modalities and contexts for demonstrating learning. Using multiple measures rather than relying on a single test score provides a richer picture of what students know and are able to do. The characteristics of assessments that support learning with understanding are presented in Table 7-3. TEACHER PROFESSIONAL DEVELOPMENT One of the most important factors influencing student achievement, if not the most important, is teacher expertise (see Shepard, 2000; National Commission on Mathematics and Science Teaching for the 21st Century, 2000; National Center for Education Statistics [NCES], 2000a; Darling-Hammond, 2000). Thus, the key to implementing the committee’s vision of learning with deep conceptual understanding is having highly skilled teachers who can effectively put into practice the strategies suggested earlier in the discussion of instruction for understanding. This observation is particularly true when one is implementing well-structured external programs that build on the regular curriculum already in place at a school. Most teachers, even those regarded as excellent, would have to change their beliefs and practices significantly to teach in a manner consistent with the committee’s conceptual framework (Haidar, 1997; Jones et al., 1998; Ryder, Leach, and Driver, 1999; Schoon and Boone, 1998; Southerland and Gess-Newsom, 1999). Such change cannot occur unless teachers are given ample opportunity and support for continual learning through sustained professional development, as

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools TABLE 7-3 Characteristics of Assessment for Understanding Less Emphasis on More Emphasis on Summative examinations—unit tests and the final examination Formative assessments—ongoing assessment of teaching and learning High-stakes tests that impact college entrance and placement and drive unintended program change Learning assessments that drive program changes in the direction of the goal for advanced studies Assessments that measure students’ ability to recall facts Assessments that evaluate understanding and reasoning Assessment as something that is done to students and that provides information to teachers about students Students participating in developing and analyzing the results of assessments Private communication of students’ ideas and understandings to the teacher/examiner Classroom discourse including argument and explanation of students’ ideas and understandings Example: During a unit on cell structure, biology students participate in lecture/discussions, complete reading assignments, use electron micrographs to examine cell ultrastructure, and conduct related practical investigations involving microscopic examination of a variety of cell types and scale drawings. At the end of the unit laboratory write-ups are collected, and a unit test is administered. Example: Students participate in varied assessment activities throughout the unit. A brief oral examination tests understanding of a reading assignment. A teacher uses a check sheet during microscope work to assess skills. Laboratory data records are evaluated individually as the students are working on the investigations. A short multiple-choice test on identifying and naming cell parts is given, marked, and discussed. Students prepare a concept map illustrating the relationship between cell structure and function and are asked to explain their thinking to a small group. The group gives feedback, and each student performs a self-assessment of the quality of his or her concept map. The class develops a rubric that will be applied to a unit examination essay question comparing prokaryotic and eukaryotic cell structures. A unit test, of which the essay is one component, is given. Assessing discrete, easily measured information Assessing what is most highly valued—deep, well-structured knowledge Example: A multiple-choice question asking students to select the weak base given a set of chemical formulas. Example: A free-response question involving this scenario: The student is given four 0.10 M solutions labeled A, B, C, D; a conductivity probe; and a pH probe. One of the solutions is a weak acid, one a strong acid, one a weak base, and another a strong base. The student is asked to characterize each solution, describing both the method developed to solve the problem and the results. Student dependence on using algorithms Students communicating thought processes One right or wrong answer Partial credit for various subtasks and many possible paths to a successful outcome Unintended uses of results Clear relationships between the decisions and the data   SOURCES: Adapted from ACS (1997); NRC (1996); NSTA (1996).

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools well as the opportunity to try out and reflect upon new approaches in the context of their own classrooms (Putnam and Borko, 1997). Because the standards and frameworks driving reform efforts do not provide specific guidelines for teaching, the implementation of reform visions, including that described in this report, poses a considerable challenge for teachers and school administrators responsible for curriculum and instruction. Moreover, the changes demanded of teachers are not a simple matter of learning new teaching strategies: “Learning to practice in substantially different ways from what one has oneself experienced can occur neither through theoretical imaginings alone nor unguided experience alone” (Darling-Hammond, 1999a, p. 227). Thus the success of current reform efforts—in secondary mathematics and science, as well as other curricular areas—in fostering learning with understanding depends on creating opportunities for teachers’ continual learning and providing sufficient professional development resources to exploit these opportunities (Darling-Hammond, 1996, 1999b; Sykes, 1996). Unfortunately, current professional development for U.S. teachers can be described only as inadequate: Sadly and short-sightedly, however, professional development is too often treated not as a necessity but as a luxury item on the school budget. Many people erroneously believe that teachers are not working unless they are standing in front of a classroom. In fact, preparation time, individual study time, as well as time for peer contact and joint lesson planning, are vital sources of both competence and nourishment for all teachers. But teachers are granted precious little time for any of these activities…. High quality professional development ought to be the lifeblood of American teaching; instead, it is used only to provide the occasional transfusion. (National Commission on Mathematics and Science Teaching for the 21st Century, 2000, p. 27) Nothing has promised so much and been so frustratingly wasteful as the thousands of workshops and conferences that led to no significant change in practice when teachers returned to their classrooms. (Fullan, 1991, p. 315) To teach advanced mathematics or science well, teachers need to know their subjects deeply and extensively, know their students and understand how they learn, and know the pedagogical techniques specific to their subjects. These three domains of professional knowledge form the core content of professional development for teachers (NCTM, 2000; Schulman, 1986). Teachers use this knowledge to listen carefully to students and examine their work in order to identify understandings and misunderstandings and frame appropriate learning activities for each student (NCTM, 1991; NRC, 1996). Box 7-5 summarizes the design principles for professional development of teachers.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools BOX 7-5 Principles of Effective Professional Development Effective professional development for teachers of advanced study in science and mathematics: Focuses on the development of teachers’ subject matter knowledge, knowledge of students, and subject-specific pedagogical knowledge. Emphasizes deep conceptual understanding of content and discipline-based methods of inquiry. Provides multiple perspectives on students as learners. Develops teachers’ subject-specific pedagogical knowledge. Treats teachers as active learners who construct their own understandings by building on their existing knowledge and beliefs. Is grounded in situations of practice. Takes place in professional communities where teachers have the opportunity to discuss ideas and practices with colleagues. Uses with teachers’ instructional strategies and assessment practices that teachers are expected to use with students. Is most effective when teachers take an active role. Is an ongoing, long-term effort spanning teachers’ professional lives. Professional development should emphasize more than the fundamental facts, concepts, and procedures of a discipline. It also should help teachers understand the particular methods of inquiry in their discipline, know discipline-specific ways to reason and communicate, and understand the relationships of the discipline to other school subjects and to societal issues (NCTM, 1991; NRC, 1996). Professional development should also help teachers understand students as learners by providing opportunities to examine students’ thinking about mathematics and science. Excellent professional development addresses such issues as how students think and behave, what they already know and believe, and what they find interesting. Issues of race, gender, age, and socioeconomic background also are germane, as is an understanding of students’ common preconceptions and misconceptions. In addition, teachers should expect to receive a strong foundation of pedagogical content knowledge from effective professional development. Such knowledge might include instructional and classroom organization strategies, materials and resources

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools (including technology), alternative ways to represent concepts and strategies or assess student understanding, and methods of fostering classroom discourse and communication. High-quality professional development treats teachers as teachers should treat their students, including acknowledging that learning is an active process wherein learners construct new understandings based on what they already know and believe. There is considerable evidence that existing knowledge and beliefs play an important role in how teachers learn to teach, how they teach, and how they think about teaching in new ways (Cohen and Ball, 1990; Prawat, 1992; Putnam and Borko, 1997). If professional development is to support meaningful change in teaching and teachers, it must address teachers’ existing knowledge and beliefs, just as teachers are expected to address prior knowledge in their students in order to promote learning with understanding. What teachers know and believe will influence their interpretation and enactment of new ideas for teaching. Moreover, professional development instructors are role models who contribute to teachers’ evolving visions of what and how to teach. Although they ought to model the kinds of practices they expect teachers to use in their own classrooms, all too often teacher educators, like K–12 educators, revert to what they know best—the ways they were taught themselves. The result is professional development activities that promote traditional views of schooling and uninspired, didactic teaching methods. Because knowledge is integrally connected with the contexts in which it is acquired and used, teacher learning ought to be situated in practice. There are numerous ways to accomplish this, with advantages for different components of teacher learning (Putnam and Borko, 1997). One approach is to conduct professional development at the schools—and largely in the classrooms—where participants teach. Doing so enables the teacher and staff developer to work together to interpret classroom events in light of various ideas about teaching, learning, and subject matter. A second approach is to have teachers bring experiences from their classrooms, such as samples of students’ work or videotapes of classroom activities, to workshops. Away from the classroom, teachers have opportunities to develop a reflective perspective on teaching by considering different points of view on their own teaching practice and that of others. Other activities, such as detailed study of the discipline, may best occur away from the classroom. Settings such as summer workshops free teachers from the daily demands of having to think about the immediate needs of their students and classrooms. A combination of experiences situated in different settings may be the most powerful way to foster meaningful change in teachers’ thinking and practice (Putnam and Borko, 1997). The recommendations of the National Commission on Mathematics and Science Teaching for the 21st Century (2000)

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools provide one model for multifaceted professional development. Central to the commission’s recommendations are summer institutes designed to address pressing needs, such as enhancing teachers’ subject matter knowledge, introducing new teaching methods, and integrating technology into teaching. Also recommended are ongoing inquiry groups. Teachers need to form learning communities in which they can explore problems of practice that occur during the school year and engage in continuing discussion to enrich their knowledge of subject matter, students, and teaching. Professional development, as characterized here, typically occurs in professional communities, not in isolation. This model is consistent with a view of cognition as developing within a social context that provides the language, concepts, and modes of thought with which people make sense of the world. If teachers are to be successful in developing new practices, they need opportunities to participate in, as McLaughlin and Talbert (1993, p. 15) put it, “a professional community that discusses new teacher materials and strategies and that supports the risk taking and struggle entailed in transforming practice.” Through their participation in such professional communities, as well as through other means of communication, such as books, professional journals, and electronic networks, teachers come to understand and think in ways that are common to those communities while also helping the thinking of the community to develop and change. Because learning is a developmental process that takes time and hard work, teachers’ professional development needs to continue throughout their careers, and teachers need to accept responsibility for their own learning and growth. If teachers view themselves as agents of change, responsible for improving teaching and learning in their schools, they may be more likely to take advantage of high-quality professional development. Professional development, in turn, should support and encourage teachers in accepting such responsibilities. In sum, tomorrow’s students will have very different needs from those of today as a result of new knowledge in the various disciplines, new technologies, and new workplace demands. To meet those needs, teachers must constantly revise their practice and reflect on teaching and learning. To this end, teachers need professional development that provides opportunities for them to expand their knowledge, to experiment with new ideas about teaching and learning, to receive feedback about their teaching, and to work with others to effect positive changes in mathematics and science education (NCTM, 1991; NRC, 1996; Putnam and Borko, 1997). Lifelong learning experiences can provide teachers with the opportunity to continually consider and contribute to the evolving knowledge base of teaching and learning (NRC, 1996). Table 7-4 summarizes the emphases of professional development programs that promote understanding.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools TABLE 7-4 Characteristics of Effective Teacher Professional Development Less Emphasis on More Emphasis on Sporadic learning opportunities Continuous ongoing, connected learning experiences Transmission of knowledge and skills by lecture Inquiry into teaching and learning Isolation of teachers, classrooms, and disciplines Interdisciplinary inquiry that extends beyond the classroom Separation of science/mathematics content knowledge and teaching knowledge Discipline-specific pedagogy Separation of theory and practice Integration of theory and practice in school settings Individual learning, as teachers of an advanced subject are physically and temporally isolated from other teachers of the same advanced subject Collegial and collaborative learning in a professional community and time for teachers to interact during the school day and year Teacher as technician and consumer of knowledge Teacher as reflective practitioner and producer of knowledge Teacher as target of change Teacher as source and facilitator of change Reliance on external or internal expertise only Mix of internal and external expertise Fragmented, one-shot workshops Long-term, coherent plans incorporating 1- to 2-week summer institutes with follow-up during the school year   SOURCES: Adapted from ACS (1997); NRC (1996); NSTA (1996). FROM FRAMEWORK TO ADVANCED STUDY PROGRAMS The development of educational programs that foster learning with conceptual understanding, as outlined in this chapter, could be seen as idealization that cannot be achieved in practice; the committee sees it as an imperative. The changes in education proposed here represent a significant departure from current practice in most schools and from most of the preparation and professional development that teachers and school administrators have received. Implementing such changes will not require reforming, but rather transforming the entire social and institutional context for learning, including school culture, leadership practices, and pedagogical practice. The more aligned the institutional and social context for learning is with the committee’s conceptual framework for analyzing and designing advanced study, the more likely it is that the innovations we advocate will be sustained over time and enhance student achievement.

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Learning and Understanding: Improving Advanced Study of Mathematics and Science in U.S. High Schools The committee has come to a sobering conclusion: given what is now known about human learning, transformation of advanced study is imperative. Accordingly, the committee presents in this report a framework and a set of guidelines for assessing and designing advanced study. This framework was constructed on the basis of current knowledge about how people learn and about the nature of subject matter expertise. Advanced study programs that are aligned with this framework and the attendant principles will foster deep, robust conceptual understanding. They also will make such deep learning accessible to a broader range of students, because the principles of learning and of program design set forth here (1) recognize multiple ways of thinking and (2) support a wide range of perspectives and practices in the school curriculum. The complex, authentic, multidimensional learning opportunities that could be designed following the committee’s model could make it possible for all children to discover and use their unique strengths to engage in learning at a deep conceptual level.