Do textbook series in arithmetic develop the mathematical concepts and skills in accordance with current learning theory? This is an important question for the junior high school mathematics teacher. The answer determines the extent to which the teacher may rely on the textbook for guidance in the learning sequence. Since, for many students, mathematics instruction ceases at ninth grade, mathematical strands considered by teachers and textbook writers to be important must be brought to a culminating point by that time.

### G. W. Brown

### Edited by Lucien B. Kinney and Dan T. Dawson

### Anderson Norton and Jesse L. M. Wilkins

Piagetian theory describes mathematical development as the construction and organization of mental operations within psychological structures. Research on student learning has identified the vital roles of two particular operations–splitting and units coordination–play in students' development of advanced fractions knowledge. Whereas Steffe and colleagues (e.g., Steffe, 2001; Steffe & Olive, 2010) describe these knowledge structures in terms of fractions schemes, Piaget introduced the possibility of modeling students' psychological structures with formal mathematical structures, such as algebraic groups. This paper demonstrates the utility of modeling students' development with a structure that is isomorphic to the positive rational numbers under multiplication–the splitting group. We use a quantitative analysis of written assessments from 58 eighth grade students to test hypotheses related to this development. Results affirm and refine an existing hypothetical learning trajectory for students' constructions of advanced fractions schemes by demonstrating that splitting is a necessary precursor to students' constructions of 3 levels of units coordination.

### Martin A. Simon, Nicora Placa and Arnon Avitzur

Tzur and Simon (2004) postulated 2 stages of development in learning a mathematical concept: participatory and anticipatory. In this article, we discuss the affordances for research of this stage distinction related to data analysis, task design, and assessment as demonstrated in a 2-year teaching experiment. We describe our modifications to and further explicate and exemplify the theoretical underpinnings of these stage constructs. We introduce a representation scheme and use it to trace the development of a concept from initial activity, through the participatory stage, and to the anticipatory stage.

### James H. Beaird

In *Educational Comment* 1965, edited by Edward B. Wickes and Thomas C. Gibney, College of Education, The University of Toledo (Ohio), appear six articles which give some provocative insight into some fundamental issues of learning theory and mathematics instruction.

### Eugene D. Nichols and Robert W. King

Programmed instruction, now well into its second decade of existence, has made a place for itself in the educational community. There is no longer any doubt that students can and do learn using programmed material. Many programs have been written and are being used at all academic levels from preschool to graduate school. Programmed instruction has also been widely used in educational research to investigate various aspects of learning theories and teaching practices.

### Jeffrey A. Frykholm and Mary E. Pittman

Throughout the past several years, middle-grades mathematics curricula have undergone a significant shift. Recently developed curriculum programs based on both recommendations of the NCTM and contemporary learning theories now emphasize problem solving, critical thinking, mathematical connections, and mathematical communication in ways that they did not before. As these powerful curriculum programs continue to find a stronghold in our middle schools, new implications and roles for both teachers and students are becoming clear.

### Frances R. Curcio, Alice F. Artzt and Merna Porter

One of the greatest challenges for secondary mathematics teacher-educators is preparing future teachers to support reform efforts that lead to high–quality teaching. In particular, careful lesson planning, anticipation of student misconceptions, and constructive reflection on a lesson after instruction are critical concerns not only for novice teachers but also for experienced teachers. One way to help preservice teachers begin to appreciate the importance of planning and reflecting entails college faculty collaborating with exemplary school teachers in integrating and connecting learning theories with teaching practice.

### Sally K. Roberts

One lesson we learned early on in child psychology courses is that learning progresses from the concrete to the semiconcrete, or representational, and finally to the abstract or symbolic level. At first glance, this seems to be not only logical but also a given fact of life. This vision of learning is a linear progression. When applied to mathematics, this learning theory leads to the assumption that the use of manipulatives and hands-on learning experiences should precede procedural symbol manipulation. It also leads to the belief that using manipulatives can ensure understanding of more abstract representations.

### George Gadanidis

Mathematics education suffers from a condition that resembles schizophrenia. One of its personalities is exhibited in the day-to-day realities of classroom learning; another is evident in journal articles, in-service presentations, and other such forums where educators present alternative realities of learning. For the purposes of this article, these personalities will be labeled, respectively, as the practice and theory of mathematics education. This article focuses on the latest form of the theoretical personality of mathematics education, constructivism, by asking what is constructivist learning theory and what does it imply for the practice of learning mathematics?

### David Kirshner

*The End of Ignorance* (TEOI), laced with anecdote, is published for a lay audience, by a nonacademic press, with no real foundation in learning theory. Yet every mathematics education researcher—perhaps every educational researcher—needs to have John Mighton's book clearly within his or her sights. Mighton's curriculum approach, JUMP Math, threatens to leapfrog over the head of the mathematics education establishment, delivering mathematical learning to every child, from the profoundly learning disabled on up, at up to three or four grade levels beyond their current grade placement.