Formative validation methods were used with over 90 mathematics teachers in Utah to design an assessment of the mathematical and mathematics teaching cognitive competencies of underprepared middle and secondary school mathematics teachers. The resulting written-response test produces a score for each of 12 subtests: mathematics teaching methods, number theory, algebra, geometry, analysis, trigonometry, statistics, sets, knowledge of algorithms, comprehension of communications, conceptualization, and application. A summative validation study using 47 subjects indicated that the test is valid and reliable. The test has been adopted by the Utah State Office of Education for use in making recommendations for granting mathematics teaching endorsements and for planning in-service programs.

### A. Edward Uprichard and E. Ray Phillips

In previous studies, researchers have attempted to generate learning hierarchies using task analysis based primarily on epistemological considerations (Gagné & Paradise, 1961; Gagné, 1962; Cox & Graham, 1966; Uprichard, 1970; Okey Gagne, & 1970; Harke, 1971; Riban, 1971; Phillips & Kane, 1973; Miller & Phillips, 1975). Studies of this type conducted in the early si xties provide substantial evidence to support the hierarchical structureofknowledge(Gagné & Paradise, 1961;Gagné & Brown, 1961:Gagné, 1962, 1963; Gagné, Mayor, Garstens, & Paradise, 1962; Gagné & Staff, 1965). An examination of results from recent studies (Niedermeyer, Brown, & SulLen, 1969; Brown, 1970; Phillips & Kane, 1973; Callahan & Robinson, 1973) suggests that optimal learning sequences can be developed by sequencing instructional materials according to validated learning hierarchies. However, both Gagné (1968) and Pyalte (1969) have pointed out that the determination of an optimal or hierarchical sequence of subtasks from simplest to most complex is not easi ly achieved.

### Annie Selden and John Selden

This article reports on an exploratory study of the way that eight mathematics and secondary education mathematics majors read and reflected on four student-generated arguments purported to be proofs of a single theorem. The results suggest that such undergraduates tend to focus on surface features of arguments and that their ability to determine whether arguments are proofs is very limited—perhaps more so than either they or their instructors recognize. The article begins by discussing arguments (purported proofs) regarded as texts and validations of those arguments, that is, reflections of individuals checking whether such arguments really are proofs of theorems. It relates the mathematics research community's views of proofs and their validations to ideas from reading comprehension and literary theory. Then, a detailed analysis of the four student-generated arguments is given and the eight students' validations of them are analyzed.

### E. Ray Phillips and Robert B. Kane

An initial hierarchy for rational-number addition was constructed. A test to assess mastery at each of the 11 levels was administered to 163 elementary school children, and the pass-fail relationships were analyzed using various procedures for validating a learning hierarchy from test data. To examine the adequacy of each validation procedure, programed instructional materials were sequenced according to the hierarchy generated by each procedure. 142 students were randomly assigned to 7 treatments, and groups were compared on achievement, transfer, retention, and time to complete the instructional sequence. Although no procedure was found to be consistently superior, the results indicate that the overall efficiency of the learning process can be affected by sequence manipulation and that effective learning sequences can be derived using learning hierarchies validated from test data.

### Rose Mary Zbiek

This study explored the strategies used by 13 prospective secondary school mathematics teachers to develop and validate functions as mathematical models of real-world situations. The students, enrolled in an elective mathematics course, had continuous access to curve fitters, graphing utilities, and other computing tools. The modeling approaches fell under 4 general categories of technology use, distinguished by the extent and nature of curve-fitter use and the relative dominance of mathematics versus reality affecting the development and evaluation of models. Data suggested that strategy choice was influenced by task characteristics and interactions with other student modelers. A grounded hypothesis on strategy selection and use was formulated.

### J. Murray Lee and Dorris May Lee

Is there a need for an instrument of guidance in the field of geometry? The answer from practically all school people who have had to meet the problem is, “Yes.” Geometry was placed in our course of study when the almost sole function of the school was to prepare for college work. It was expected that all pupils should take it as it was considered necessary to their future studies. Furthermore, the students at that time were a highly selected group, only those of high intellectual ability attempting this advanced work. Thus it was not unreasonable to expect all to study geometry with at least a moderate degree of success.

### Harry A. Greene and Ruth O. Lane

feachcrs in all fields of subject matter arc turning to objective tests as the only reliable means of determining the efficiency of their instruction. This interest on the part of the cla sroom teacher has had a most wholesome effect on the recent developments in educational measurements. As teachers themselves become better informed they grow more critical of existing devices, and in many cases have themselves made significant contributions. In fact, a very considerable portion of the material described herein is the product of a classroom teacher working under normal instructional conditions.

### Michele B. Carney, Jonathan L. Brendefur, Gwyneth R. Hughes and Keith Thiede

As mathematics teacher educators, it is imperative that we have high-quality tools that conceptualize and operationalize mathematics instruction for large-scale examination. We first describe existing instructional practice survey scales, including their conceptualization of practice and related validity evidence. We then present the framework and initial validity evidence for our mathematics instructional practice survey. Survey participants were inservice teachers in a statewide mandated mathematics professional development course. Statistical analyses indicate the items measure two constructs: social-constructivist and transmission-based instructional practice. Of particular interest is the result that these two constructs were negligibly correlated. This is in contrast to the generally accepted notion that social-constructivist and transmission-based instructional practices are the two polar ends of a single construct for describing instructional practice.

### Keith Weber

The purpose of this article is to investigate the mathematical practice of proof validation—that is, the act of determining whether an argument constitutes a valid proof. The results of a study with 8 mathematicians are reported. The mathematicians were observed as they read purported mathematical proofs and made judgments about their validity; they were then asked reflective interview questions about their validation processes and their views on proving. The results suggest that mathematicians use several different modes of reasoning in proof validation, including formal reasoning and the construction of rigorous proofs, informal deductive reasoning, and examplebased reasoning. Conceptual knowledge plays an important role in the validation of proofs. The practice of validating a proof depends upon whether a student or mathematician wrote the proof and in what mathematical domain the proof was situated. Pedagogical and epistemological consequences of these results are discussed.

### Nancy Ann Belsky

These activities are designed to appeal directly to students. No solutions are suggested for the activities so that students will look to themselves as the mathematical authority, thereby developing the confidence to validate their work.