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• Author or Editor: Edward A. Silver
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## Cognitive Complexity of Mathematics Instructional Tasks in a Taiwanese Classroom: An Examination of Task Sources

We examined geometric calculation with number tasks used within a unit of geometry instruction in a Taiwanese classroom, identifying the source of each task used in classroom instruction and analyzing the cognitive complexity of each task with respect to 2 distinct features: diagram complexity and problem-solving complexity. We found that instructional tasks were drawn from multiple sources, including textbooks, tests, supplemental materials, and the teacher. Our analysis of cognitive complexity indicated that the instructional tasks frequently involved both diagram complexity and problem-solving complexity. Moreover, the geometric calculation with number tasks from nontextbook sources tended to be more cognitively demanding than those found in the textbooks. Implications of task analysis on geometry domain and textbook analysis studies are discussed.

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## Probing the Foundations of Algebra: Grade-4 Pattern Items in NAEP

Suppose that you asked fourth-grade students to predict whether the number 375 would ever appear in a sequence of numbers generated by multiplying 2s (4, 8, 16, 32, …). Would students recognize the impossibility of 375 appearing in the sequence? Would they give a clear, complete explanation as to why 375 would not appear?

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## Research into Practice: Teaching Mathematics and Thinking

One afternoon. eleven-year-old Michael stopped by the home of an adult friend and found her nearly buried in decorating books, charts. and samples. Michael's sudden appearance at the door was a welcome sight, and he was asked to assist his friend in the process of decorating her home office. Although quite sure he knew little about interior decorating, Michael agreed to lend a hand. The friends began to measure the room and calculate the areas of the ceiling, walls, and floor. They looked at wallpaper swatch books, paint-color charts, and rug samples from various manufacturers and discussed the differences in price, the expected coverage per gallon of paint and roll of wallpaper, and the relative quality and ea e of upkeep for different products. One tyle of wallpaper they liked had a horizontal stripe and required matching. After approximating how many rolls of striped wallpaper they would need to buy, they performed a similar approximation for a design that did not require matching. They looked at carpet amples, discussed the relative merits of light and dark colors and variou styles, and considered the issue of cost versus quality with respect to durability and stain resistance. Finally, they selected a set of materials to comple te the decorating project based on cost, product quality. maintenance required, and personal preference. After helping his friend with this task for several hours, Michael departed for his home and his dinner.

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## Gaining Insight into Students' Thinking through Assessment Tasks

The task in Figure 1A, which was given to middle school students, asked them to find the number of dots in the twentieth step of the pattern. As shown in figure 1b, the answers that students gave ranged from 20 dots to well over 60 million dots. If your students gave these answers without providing work or explanations, would you be able to tell how they obtained them? Probably not. Without looking closely at the students' work or explanations or talking to them about their solution strategies, it is difficult to understand how they were thinking about the pattern task and how their thinking could have produced such a wide range of answers.

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## Implementing the Professional Standards for Teaching Mathematics: Focusing on Worthwhile Mathematical Tasks in Professional Development: Using a Task from the National Assessment of Educational Progress

Worthwhile mathematical tasks engage the problem solver in sound and significant mathematics, elicit a variety of solution methods, and require mathematical reasoning. Such problems also prompt responses that are rich enough to reveal mathematical understandings. Just as good classroom practice engages students in worthwhile mathematical tasks, sound professional development does the same with teachers. Providing teachers with opportunities to engage in worthwhile mathematical tasks and to analyze the mathematical ideas underlying those tasks promotes the vision of the Professional Standards for Teaching Mathematics (NCTM 1991).

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## Results of the Third NAEP Mathematics Assessment: Secondary School

The recent publication of the Report of the National Commission on Excellence in Education (1983) has focused national attention on the state of education in the United States and the academic achievement of students. The results of the third mathematics assessment of the National Assessment of Educational Progress (NAEP) provide a basis for examining students’ performance in mathematics and how it has changed over the last decade.

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## Sense Making and the Solution of Division Problems Involving Remainders: An Examination of Middle School Students' Solution Processes and Their Interpretations of Solutions

In this study, about 200 middle school students solved an augmented-quotient division-with-remainders problem, and their solution processes and interpretations were examined. Based on earlier research, semantic-processing models were proposed to explain students' success or failure in solving division-with-remainder story problems on the basis of the presence or absence of an adequate interpretation provided by the solver after obtaining a numerical solution. In this study, students' solutions and their attempts and failures to “make sense” of their answers were analyzed for evidence that supported or refuted the hypothesized semantic-processing models. The results confirmed that the models provide a solid explanation of students' failure to solve division-with-remainder problems in school settings. More generally, the results indicated that student performance was adversely affected by their dissociation of sense making from the solution of school mathematics problems and their difficulty in providing written accounts of their mathematical thinking and reasoning.

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## The Third National Assessment: Minorities and Mathematics

The Third National Assessment of Educational Progress in Mathematics was conducted in 1982. Data are available on exercises given to national samples of white, black, and Hispanic 9-, 13-, and 17-year-olds. Although black and Hispanic students continued to score below the national level of performance, they made greater gains than their white counterparts since the second assessment in 1978, especially on exercises assessing knowledge and skill. Students in schools with heavy minority enrollments made greater-than-average gains. The more mathematics courses taken by 17-year-olds, black or white, the higher the level of achievement. The assessment results support increased efforts to improve the learning of mathematics by minority students.

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## Problem Solving: Tips For Teachers

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## Problem Solving: Tips For Teachers

It is fairly common for students to believe that every problem has one and only one answer, that there is a correct algorithm or procedure to follow in solving any mathematics problem, and that all mathematics problems should be solved quickly, if they can be solved at all. It is important to offer experiences that broaden students' conceptions of mathematics and problem solving.