Throughout the history of American education, learning to write proofs has been an important objective of the geometry curriculum for college-bound students. At the same time proof writing has also been perceived as one of the most difficult topics for students to learn. Until recently, the extent of students’ difficulties with writing proofs has been largely a matter of conjecture, for little research has been conducted in this area.

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- Author or Editor: Sharon L. Senk x

### Sharon L. Senk

This study investigated relations between van Hiele levels, achievement in writing geometry proofs, and achievement in standard geometry content. Two hundred forty-one secondary school students who were enrolled in full-year geometry classes were tested in the fall for van Hiele level of thinking and entering knowledge of geometry, and in the spring for van Hiele level, standardized geometry achievement, and proof-writing achievement. Proof-writing achievement correlated significantly, .5 with fall van Hiele level, .6 with spring van Hiele level and with entering knowledge of geometry, and .7 with standardized geometry achievement in the spring. Proof-writing achievement also varied significantly with van Hiele level when either entering knowledge of geometry or geometry achievement in the spring was used as a covariate. The predictive validity of the van Hiele model was supported. However, the hypothesis that only students at Levels 4 or 5 can write proofs was not supported.

## Implementing the Assessment Standards for School Mathematics

### Using Rubrics in High School Mathematics Courses

### Denisse R. Thompson and Sharon L. Senk

Recommendations in the *Curriculum and Evaluation Standards* for *School Mathematics* (NCTM 1989) and in the *Assessment Standards for School Mathematics* (NCTM 1995) encourage teachers to incorporate into their curriculum and assessment practices more tasks that require students to construct their own responses, as opposed to primarily using tasks for which a response is provided, such as true-orfalse or multiple-choice tasks. Constructed responses enable students to demonstrate their depth of understanding of mathematics and give teachers greater insight into their students' knowledge of concepts. But when students are required to write about mathematics or explain their solution strategies, teachers want to know how to score such responses. Teachers have therefore become more interested in issues related to rubrics. A rubric is a set of guidelines for evaluating students' responses to one or more tasks. *A general rubric* is a broad outline that indicates vatious levels of performance and the factors that teachers should consider when specifying performance levels; a *task-specific rubric* interprets the general rubric for a specific task and specifies the particular mathematical aspects of the task that determine each level of performance (NCTM 1995; California Mathematics Council 1993).

### Sharon L. Senk and Daniel B. Hirschhorn

Geometry as a subject uniquely furnishes a language for describing our physical world. It also gives a way visually to represent concepts and relations in other branches of mathematics. Although debate might always ensue on whether geometry should be a full-year secondary school course, the importance of geometry throughout a student's mathematics education seems to have broad acceptance. Consequently, it is not surprising that in the *Curriculum and Evaluation Standards for School Mathematics* (NCTM 1989) we find an explicit standard on geometry for all levels K–4, 5–8, and 9–12. In fact, for grades 9–12, two standards on geometry are included—one focusing on a synthetic approach, the other on an algebraic approach.

### Denisse R. Thompson and Sharon L. Senk

We examine the performance of 8 pairs of 2nd-year algebra classes that had been matched on pretest scores. One class in each pair used the UCSMP *Advanced Algebra* curriculum, and the other used the 2nd-year-algebra text in place at the school. Achievement was measured by a multiple-choice posttest and a free-response posttest. Opportunity-to-learn (OTL) measures were used to ensure that items were fair to both groups of students. UCSMP students generally outperformed comparison students on multistep problems and problems involving applications or graphical representations. Both groups performed comparably on items testing algebraic skills. Hence, concerns that students studying from a *Standards*-oriented curriculum will achieve less than students studying from a traditional curriculum are not substantiated in this instance.

### Sharon L. Senk and Denisse R. Thompson

This Brief Report describes a secondary analysis of the solutions written by 306 second-year algebra students to four constructed-response items representative of content at this level. The type of solution (symbolic, graphical, or numerical) used most frequently varied by item. Curriculum effects were observed. Students studying from the second edition of the University of Chicago School Mathematics Project's (UCSMP) *Advanced Algebra* curriculum used a higher percentage of graphical and numerical strategies than comparison students. Achievement and choice of strategy were also related. Both UCSMP and non-UCSMP students who used symbolic or graphing strategies were generally successful on the quadratic comparison item; UCSMP students who used graphing strategies were also successful on items dealing with logarithm properties and a quadratic application.

### Gwendolyn J. Johnson, Denisse R. Thompson and Sharon L. Senk

The authors provide a framework for investigating proof-related reasoning in high school algebra and precalculus textbooks and suggest ways to increase students' opportunity to learn proof-related reasoning.

### Denisse R. Thompson, Charlene E. Beckmann and Sharon L. Senk

Cunently much discussion is occurring within the mathematics-education community regarding assessment. In attempting to develop the mathematical power of students, the *Assessment Standards for School Mathematics* (NCTM 1995, 29) encourages teachers to make several changes in their assessment practices. Among these are the following:

### Daniel B. Hirschhorn, Denisse R. Thompson, Zalman Usiskin and Sharon L. Senk

The University of Chicago School Mathematics Project (UCSMP) was begun in 1983 as an attempt to implement the recommendations of many reports to improve school mathematics. The national reports available at the time (e.g., NACOME [1975); NCTM [1980]; CBMS [19821; College Board [19831; NCEE [1983)) called for a curriculum of broader scope that would include statistics, probability, and discrete mathematics and that would give strong attention to applications, use the latest in technology, and emphasize problem solving. To accomplish the curricular revolution recommended by these reports, it was essential that new, appropriate materials be written. History had shown that neither materials written for the best students, such as those from the new-math era, nor materials written for the slower students, such as those popular in the backto-basics movement, were appropriate for the vast majority of students without major revisions (Usiskin 1985). Thus UCSMP started with the goal of developing mathematics for all grades K–12 that *would* be appropriate for the majority of students in the middle.

### Sharon L. Senk, Charlene E. Beckmann and Denisse R. Thompson

The assessment and grading practices in 19 mathematics classes in 5 high schools in 3 states were studied. In each class the most frequently used assessment tools were tests and quizzes, with these determining about 77% of students' grades. In 12 classes other forms of assessment, such as written projects or interviews with students, were also used, with performance on such instruments counting for about 7% of students' grades averaged across all 19 classes. Test items generally were low level, were stated without reference to a realistic context, involved very little reasoning, and were almost never open-ended. Most test items were either neutral or inactive with respect to technology. Written projects usually involved more complex analyses or applications than tests did. The teachers' knowledge and beliefs, as well as the content and textbook of the course, influenced the characteristics of test items and other assessment instruments. Only in geometry classes did standardized tests appear to influence assessment.