Stephen J. Pape

The Role of Representation(s) in Developing Mathematical Understanding

Stephen J. Pape is assistant professor of education at The Ohio State University; Mourat A. Tchoshanov is associate professor of education at the University of Texas at El Paso. E LLEN, A 4-YEAR-OLD CHILD, IS PLAYING with several toys. As she sets the toys out to play, she names the first as one, the second as two, and so on. These words are symbols for the position of the toys in the series the child is enumerating. In the beginning, she may not understand that the last number stated is the number of toys all together. These number words may simply be words the child has learned to utter as she touches each object in a series of objects. Later, at around 5 or 6 years old, the child begins to understand that the last number named in this game is the number of toys that are in the set and finally that there is a numeral that represents the number of elements in the set. Once learned, the numeral and its name (e.g., 5 and five) become the external representations that are the conventions for the internal abstraction, the number of elements in the set. Thus, the number name, five, and the numeral, 5, are the external representations that act to stimulate an image, the internal representation, of a set of five objects.

Spouse support, coping, and mood among individuals with cancer

A model of the relations between spouse support, coping, and positive and negative mood was examined with 221 individuals with cancer using LISREL analyses. A moderating effect for patient life expectancy was predicted for disease prognosis. Results indicated that spouse criticism was associated with negative mood indirectly through avoidant coping strategies, and spouse support was associated with positive mood indirectly through positively-focused coping. Results did not support a moderating influence for life expectancy upon the association between spouse behaviors and patient coping. The results of this study are discussed in terms of their implications for psychosocial interventions to reduce psychological distress among individuals with cancer.

Functional impairment, marital quality, and patient psychological distress as predictors of psychological distress among cancer patients' spouses.

This study investigated contextual determinants of psychological distress among 197 spouses of cancer patients. It was hypothesized that higher levels of patient functional impairment would lead to greater patient distress. Patient distress, in turn, would lead to lower spouse marital satisfaction and ultimately to higher spouse distress. Spouses completed measures of distress and marital quality at three time points. Cancer patients rated their functional impairment and psychological distress at the same time points. Results indicated that at all time points, greater patient impairment was associated with higher levels of patient distress, which, in turn, was related to lower marital satisfaction. However, marital quality was related to spouse distress at only 1 time point, but spouse distress was directly associated with patient distress at each time point. Implications for cancer patients and spouses are discussed.

Compare word problems: Consistency hypothesis revisited☆

Abstract Compare problems involve relationships between quantities. In inconsistent language (IL) formats, relational terms conflict with solution operations. Lewis and Mayer’s (1987) consistency hypothesis, which explains likely translations of IL problems to match problem solvers’ schemas, is examined Using think-aloud and recall data. Supporting hypotheses, students committed more errors overall and reversal errors specifically (i.e., used opposite operation) on IL than their consistent language (CL) counterparts and recalled IL problems as CL problems especially following reversal errors. Counter to expectations, language consistency did not result in differences in behaviors, reading or response times, or quality of recall. Additional analyses indicate a strong impact of fraction-of-a-number relational terms and provide evidence of difficulties forming representations and solving problems involving fraction-of-a-number construction.

Prime Online: Exploring Teacher Professional Development for Creating Inclusive Elementary Mathematics Classrooms:

Enhancing all students’ academic performance continues to be a national priority, and although achievement gains have been made overtime, shortfalls in mathematics learning for students with disabilities (SWD) remain. Research reveals that a substantial portion of the variability in students’ mathematics achievement gains is due to the teacher. To address the need for teacher professional development (PD) in mathematics for SWD and other struggling mathematics learners, we designed and studied Prime Online—a yearlong, online, PD program with support from an Institute of Education Sciences (IES) Goal 2 Development and Innovation research grant. In this article, the development process and an exploratory study are discussed. Study findings suggest that Prime Online positively influenced general and special education teachers’ reported beliefs and practices, and their learning of mathematics content for teaching, and generated high teacher satisfaction ratings. No difference in the performance of SWD on a state accountability measure of mathematics was found. Implications for further research are discussed.

Encouraging Sixth-Grade Students' Problem-Solving Performance by Teaching through Problem Solving.

Abstract This teaching experiment provided students with continuous engagement in a problem-solving based instructional approach during one mathematics unit. Three sections of sixth-grade mathematics were sampled from a school in Florida, U.S.A. and one section was randomly assigned to experience teaching through problem solving. Students’ problem-solving performance and performance on a unit test were analyzed. The intervention had a positive effect on students’ problem-solving performance whereas the comparison group experienced no changes. ANCOVA analyses suggest that intervention students solved more problems on the posttest than their peers, but the comparison group outperformed the intervention group on the unit test.

Incorporating practitioner inquiry into an online professional development program: the Prime Online experience

Engagement in practitioner inquiry by classroom teachers is a promising mechanism for teacher professional learning. While much has been learned about the positive role inquiry can play in traditional professional development efforts, we know less about the impact of inquiry in a rapidly advancing technological age that includes the proliferation of online professional development institutes and programs. The purpose of this article is to explore the role of inquiry in online professional development by providing a detailed description of a year-long program called Prime Online. Supported by US federal funding, this program was designed to help teachers develop the mathematics content and pedagogical content knowledge necessary to implement research-based practices in inclusive elementary classrooms within a framework of practitioner inquiry. A detailed description of this program is presented followed by an examination of the inquiry projects completed by the participants. Implications of this effort for teacher learning and online professional development endeavors are offered.