The mathematics program in too many of today's schools is textbook dominated, con. cemed primarily with the manipulation of symbols, and, all too often, largely removed from the real world of the child. Mathematics enjoys (or suffers from, depending upon one's perspective) a reputation which is clearly less than inspiring. Why are there such large-scale negative attitudes towards mathematics? Endless repetition, meaningless memorization, a never-ending series of worksheets or practice exercises, and a gen. eral lack of interest and understanding are but a few of the reasons that might be listed. One problem with a textbook-dominated approach to teaching and learning is that young people do not exist in a twodimensional world. Children exist in a world that they are very eager to learn about until that eagerness and enthusiasm is dulled by the establishment of numerous artificial situations to which they are required to respond. What is needed is a mathematics program that is very much alive and vibrant, and relevant and meaningful; a program that parallels the way in which young learners are alive and vibrant and continually searching for relevance and meaning as they seek to understand the world around them.

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## A model for the construction and sequencing of laboratory activities

### Thomas R. Post

## The energy crisis: an opportunity for meaningful arithmetical excursions

### Thomas R. Post

The energy crisis, expected by at least some segments of our society for several years, has arrived. Its precise causes are unclear and perhaps always will remain so, but it is likely to have profound effects on many aspects of our economy as well as on our environment. Some of these effects may be positive in nature. For example, expanded use of smaller, more economical automobiles; renewed efforts to locate and develop additional sources of energy; and the evolvement of a more energy-conscious citizenry are likely to result.

## One Point of View: Fractions and Other Rational Numbers

### Thomas R. Post

The results of national and international assessments indicate that students have significant difficulties in learning about rational numbers, in 1979 only 24 percent of the nation's thirteen-year-olds could estimate the sum of 12/13 and 7/8 given the following possibilities: 1, 2, 19, 21, and I don't know. Fifty-five percent selected either 19 or 21 as the estimated sum!

## Fractions: Results and Implications from National Assessment

### Thomas R. Post

Despite the recent introduction of the calculator and emphasis on the metric system, fractions continue to occupy an important place in school mathematics programs. Persuasive arguments for continued emphasis can be made not only because of their social utility but also because of their place in the development and structuring of mathematical ideas to follow. Accordingly, the National Assessment of Educational Processes (NAEP) allocated a substantial portion of the second mathematics assessment to fractions.

## An Experimental Study of the Effectiveness of a Formal versus an Informal Presentation of a General Heuristic Process on Problem Solving in Tenth-Grade Geometry

### Thomas R. Post and Michael L. Brennan

It was the purpose of this study to investigate the effects of various modes of presentation of general heuristic processes on students' problem-solving ability. It was felt that if students are to become effective problem solvers they must be formally instructed in the established techniques of, and approaches to, problem solving. It was hypothesized that these techniques would both faci litate students' efficiency in solving problems and improve their ability to solve various kinds of problems. This study, then, was an attempt to formally instruct students in a particular heuristic problemsolving process so that awareness and ability could be developed, and subsequently to evaluate the effects of this instruction on their ability to solve problems.

## Critique: Teaching Problem-Solving Heuristics: A Response

### Michael L. Brennan and Thomas R. Post

The improvement of problem-solving ability continues to be an elusive, yet important, goal of mathematics education. A variety of individual attempts have been made to improve student performance in this area. The Journal for Research in Mathematics Education (JRME) has performed a valuable service to the profession by facilitating interaction between individuals who have made or are contemplating such attempts. Only through such professional interaction will old ideas be reshaped and new ones generated. In this manner perhaps we shall someday provide definitive answers to questions we know to be of mutual concern. The following comments constitute our response to Kulm's critique of our previous JRME article (Post % Brennan, 1976).

## Initial Fraction Learning by Fourth- and Fifth-Grade Students: A Comparison of the Effects of Using Commercial Curricula With the Effects of Using the Rational Number Project Curriculum

### Kathleen A. Cramer, Thomas R. Post, and Robert C. delMas

This study contrasted the achievement of students using either commercial curricula (CC) for initial fraction learning with the achievement of students using the Rational Number Project (RNP) fraction curriculum. The RNP curriculum placed particular emphasis on the use of multiple physical models and translations within and between modes of representation—pictorial, manipulative, verbal, real-world, and symbolic. The instructional program lasted 28–30 days and involved over 1600 fourth and fifth graders in 66 classrooms that were randomly assigned to treatment groups. Students using RNP project materials had statistically higher mean scores on the posttest and retention test and on four (of six) subscales: concepts, order, transfer, and estimation. Interview data showed differences in the quality of students' thinking as they solved order and estimation tasks involving fractions. RNP students approached such tasks conceptually by building on their constructed mental images of fractions, whereas CC students relied more often on standard, often rote, procedures when solving identical fraction tasks. These results are consistent with earlier RNP work with smaller numbers of students in several teaching experiment settings.

## Brief Reports: Using Microcomputers with Fourth-Grade Students to Reinforce Arithmetic Skills

### Carol Carrier, Thomas R. Post, and William Heck

Microcomputers in elementary school classrooms will soon be as common as handheld calculators and reading stations. Luehrmann (1981) states that within 3 years the average school will have 16 microcomputers. The acceptance and availability of reasonably priced computer technology for the classroom and the home suggests that elementary teachers will be encouraged to incorporate computer-based activities into many areas of the curriculum. Although some of the most intriguing applications of computers in instruction at the elementary level can be seen in tutorials, si mulations, or interactive systems such as Logo, many teachers will continue to use the computer primarily to give children skill-building experiences in ma thematics and other school subjects.

## Construct a Sum: A Measure of Children's Understanding of Fraction Size

### Merlyn J. Behr, Ipke Wachsmuth, and Thomas R. Post

This report from the Rational Number Project concerns the development of a quantitative concept of rational number in fourth and fifth graders. In a timed task, children were required to select digits to form two rational numbers whose sum was as close to 1 as possible. Two versions of the task yielded three measures of the skill. The cognitive mechanisms used by high performers in individual interviews were characterized by a flexible and spontaneous application of concepts of rational number order and fraction equivalence and by the use of a reference point. Low performers tended either not to use such cognitive mechanisms or to apply concepts in a constrained or inaccurate manner.

## Teachers', Principals', and University Faculties' Views of Mathematics Learning and Instruction as Measured by a Mathematics Inventory

### Thomas R. Post, William H. Ward Jr., and Victor L. Willson

Differences and similarities among the views of secondary mathematics teachers (n=199), high school principals (n=160), and college mathematics education professors (n=117), as reflected by responses on an inventory concerned with cognitive and affective aspects of mathematics learning and instruction, were examined. The response patterns of each group analyzed by factor analysis or analysis of variance methods were found to have some unique characteristics: however, teachers were found to be more similar to principals than to college professors in both factor structure and individual item responses. Implications of this finding for the transmission of innovation in instruction are discussed.