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Chapter Four Cognitive Abilities While NAEP was designed to monitor, assess, and report student achievement nationally, an inevitable effect of this monitoring and reporting is clearly improvement in mathematics learning. If real change in the mathematics curriculum is to take place, the manner in which assessment is conducted will also have to change. Assessment activities often are the primary sources from which students discern what teachers really value and what teachers really want them to know. As a result, over time, the portions of the curriculum that are tested become the portions of the curriculum that receive greater emphasis both in teachers' and in students' allocations of effort and time. Central to the NCTM Standards description of the features of students' performance that should be assessed is "mathematical power." Mathematical power is characterized as a student's overall ability to gather and use mathematical knowledge through exploring, conjecturing, and reasoning logically; through solving nonroutine problems; through communicating about and through mathematics; and through connecting mathematical ideas in one context with mathematical ideas in another context or with ideas from another discipline in the same or related contexts. Assessing a student's mathematical power requires many different indicators over time. As power develops beyond the general mathematical abilities of conceptual understanding, procedural knowledge, and problem solving, it is important to ensure that measures are taken of a student's ability to reason in mathematical situations, to communicate perceptions and conclusions drawn from a mathematical context, and to connect the mathematical nature of a situation with related mathematical knowledge and information gained from other disciplines or through observation. It is the total interaction of all of these abilities that defines a student's overall mathematical power at a given time. The mental skills of reasoning, communicating, and connecting lie at the foundation of each of the content strands and each of the mathematical abilities featured in prior NAEP assessments. These relationships, illustrated in Chapter Two, indicate the multidimensional nature of mathematical power. Mathematical power can be viewed from a variety of perspectives. Students may encounter a new problem in an old context or an old problem in a new context. When first attempts to solve a problem fail, the student may reexamine the information, rework it, and then reapply it to the situation in a more productive fashion. The process of revising an approach to a problem based on reasoning, gathering new information, and making connections with other ideas is a dynamically growing and changing ability. This feature of mathematical power can be viewed through student performance within a particular content strand at the conceptual, procedural, and problem-solving levels of ability. Equally, a particular concept, procedure, or problem context might be viewed across the different strands. In the latter case, families of items are particularly helpful in assessment. The use of hand calculators allows students to quickly pursue alternative paths and check to see if they either provide fruitful new information or reconfirm judgments made through other approaches. Students display their mathematical power through the formulation of lines of attack on problems and the way in which they reason through situations involving a multitude of possibilities. It is here that the recommendation that students experience a number of extended open-ended items requiring construction of responses is important. Through a student's report of his or her thinking, the questions of the relevance of approach, nature of reasoning, and ability to solve problems becomes less a high inference guess and more a conclusion that can be drawn from evidence. This is especially true when the collected evidence includes the communication of a student's approach and when partial credit for student efforts is awarded in the scoring of an item. Finally, mathematical power is a function of students' prior knowledge and experience and the ability to connect that knowledge in productive ways to new contexts. This aspect of power can be measured with the multiple-choice items and through analysis of the ways in which students develop their responses to the constructed-response items on the assessment. Information related to these features of students' development is as difficult to isolate and statistically extract from the data as the mathematical abilities featured in the past NAEP assessments in mathematics. However, they are important aspects of the mathematical development of students. As such, the three features of mathematical power (reasoning, communication, and connections) will be used as underlying threads for item construction and overall test design. For the 1996 and 2000 assessment, these threads may not be specifically reported, although they will be represented in the overall way the assessment is conceived and developed. The dimensions of general mental abilities associated with mathematics and used in past NAEP assessments are conceptual understanding, procedural knowledge, and problem solving. These three areas are specifically identified as primary foci for assessment, and they received focal attention in the design of the 1990 and 1992 assessments. Conceptual understanding can be viewed simply as a measure of a student's knowing "that" or "about," while procedural knowledge can be viewed as a student's knowing "how." These two abilities combined provide a base for the capability to recognize and understand a situation, to formulate a plan to confront the situation, to arrive at a solution to the problem the situation presents, and to reflect upon the solution. The later stages can be thought of as facets of problem solving. However, as recommended in Chapter One, the role these dimensions of students' mathematical power will play in the new assessment should change from one of a direct matrix feature to one of a design characteristic that assists in providing balance to the overall assessment. The NAEP design for the 1996 and 2000 assessments should certainly continue to focus on conceptual understanding, procedural knowledge, and problem solving in bringing some balance to the assessments for grades 4, 8, and 12. In particular, it is recommended that the overall mixture of assessment items for each grade level include at least one-third of the items measuring each of the abilities of conceptual understanding, procedural knowledge, and problem solving. As with the mathematical content strands, mathematical abilities are not separate and distinct factors of an individual's ways of thinking about a mathematical situation. These abilities are, rather, descriptions of the ways in which information is structured for instruction and the ways in which students manipulate, reason with, or communicate their mathematical ideas. As a consequence, there can be no singular or unanimous agreement among educators about what constitutes a conceptual, a procedural, or a problem-solving item. What can be classified are the actions a student is likely to undertake in processing information and providing a satisfactory response. Thus, within the content strands, assessment tasks will be classified according to the ability categories they most closely represent in terms of the type of processing they might be expected to require. Further, the mathematical power features of reasoning, communication, and connections will be woven through the specifications to provide an added level of richness to the assessment tasks. The following discussions of conceptual understanding, procedural knowledge, and problem solving are given to illustrate the primary features the NAEP assessment should employ in trying to capture features of cognitive activities that combine to empower a student in mathematical situations. Students demonstrate conceptual understanding in mathematics when they provide evidence that they can recognize, label, and generate examples and nonexamples of concepts; use and interrelate models, diagrams, manipulatives, and varied representations of concepts; identify and apply principles (that is, valid statements generalizing relationships among concepts in conditional form); know and apply facts and definitions; compare, contrast, and integrate related concepts and principles to extend the nature of concepts and principles; recognize, interpret, and apply the signs, symbols, and terms used to represent concepts; or interpret the assumptions and relations involving concepts in mathematical settings. Conceptual understanding reflects a student's ability to reason in settings involving the careful application of concept definitions, relations, or representations of either. Such an ability is reflected by student performance that indicates the production of examples, common or unique representations, or communication indicating the ability to manipulate central ideas about the understanding of a concept in a variety of ways. Students demonstrate procedural knowledge in mathematics when they select and apply appropriate procedures correctly; verify or justify the correctness of a procedure using concrete models or symbolic methods; or extend or modify procedures to deal with factors inherent in problem settings. Procedural knowledge includes the various numerical algorithms in mathematics that have been created as tools to meet specific needs efficiently. Procedural knowledge also encompasses the abilities to read and produce graphs and tables, execute geometric constructions, and perform noncomputational skills such as rounding and ordering. These latter activities can be differentiated from conceptual understanding by the task context or presumed student background -- that is, an assumption that the student has the conceptual understanding of a representation and can apply it as a tool to create a product or to achieve a numerical result. In these settings, the assessment question is how well the student executed a procedure or how well the student selected the appropriate procedure to effect a given task. Procedural knowledge is often reflected in a student's ability to connect an algorithmic process with a given problem situation, to employ that algorithm correctly, and to communicate the results of the algorithm in the context of the problem setting. Procedural understanding also encompasses a student's ability to reason through a situation, describing why a particular procedure will give the correct answer for a problem in the context described. In problem solving, students are required to use their accumulated knowledge of mathematics in new situations. Problem solving requires students to recognize and formulate problems; determine the sufficiency and consistency of data; use strategies, data, models, and relevant mathematics; generate, extend, and modify procedures; use reasoning (spatial, inductive, deductive, statistical, or proportional) in new settings; and judge the reasonableness and correctness of solutions. Problem-solving situations require students to connect all of their mathematical knowledge of concepts, procedures, reasoning, and communication/representational skills in confronting new situations. As such, these situations are, perhaps, the most accurate measures of students' proficiency in mathematics.
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