Structures of Constant Rate Word Problems: A functional approach analysis

Many recent projects have lately explored function-based approaches to algebra. For the last five years we developed and tested such an approach to algebra, the 'Visual Mathematics'¹ curriculum, in grades seven to nine. While teaching this curriculum we learned from our students' work that while functions form an important basis for solving various sorts of algebra word problems, they introduce some tension in dealing with other sorts of word problems. The data accumulated during our experiments led us to conjecture that viewing algebra rate problems through a function lens offers new ways to look at similarities and differences among word problems. This article presents a theoretical framework for a systematic investigation of the structures of algebra word problems and their relationship to function-based solutions. We think that for those who take a function approach to algebra, this paper suggests new insights into the categorization of word problems. This revised version was published online in July 2006 with corrections to the Cover Date.     

Constructing and Consolidating of Algebraic Knowledge within Dyadic Processes: A Case Study

Studies of knowledge constructing often focus on the analysis of a single episode, without considering enough the history of the learners, or the future learners' trajectories with regard to the concepts learned. This paper presents an example of knowledge constructing within the context of peer learning. We show how the design of the task and the tools available to the students afford the constructing of conceptual knowledge (the phenomenon of exponential growth and variation, as it is expressed in its numerical and graphical representations). We trace the constructing of knowledge through a series of dyadic sessions for a few months in a classroom environment. We show that knowledge is constructed cumulatively, each activity allowing for the consolidating of previous constructs. This pattern indicates the nature of the processes involved: knowledge constructing and consolidating are dialectical processes, developing over time, when new constructs stem from old ones already consolidated, which gain consolidation through the new construction, creating a new abstract entity. We also discuss the potential of the tool the students used (a spreadsheet program) to such processes of learning mathematics.     

The development of dynamic inquiry performances within an open inquiry setting: A comparison to guided inquiry setting

Dynamic inquiry learning emphasizes aspects of change, intellectual flexibility, and critical thinking. Dynamic inquiry learning is characterized by the following criteria: learning as a process, changes during the inquiry, procedural understanding, and affective points of view. This study compared the influence of open versus guided inquiry learning approaches on dynamic inquiry performances among high‐school biology students. We hypothesized that open inquiry students who engage in the inquiry process from its initial stage, participating in the decision making process of asking inquiry questions and planning all aspects of the inquiry, will outperform students who experienced guided inquiry, in terms of developing dynamic inquiry performances. Students were divided into two groups: guided and open inquiry learning approaches. Both groups were followed throughout their 2‐year inquiry learning process. The data sources included interviews, students' inquiry summary papers, logbooks, and reflections. A quantitative content analysis of the two groups, using a dynamic inquiry performances index, revealed that open inquiry students used significantly higher levels of performances in the criteria “changes during inquiry” and “procedural understanding.” However, the study's results indicated no significant differences in the criteria “learning as a process” and “affective points of view.” The implementation of dynamic inquiry performances during inquiry learning may shed light on the procedural and epistemological scientific understanding of students conducting inquiries. © 2009 Wiley Periodicals, Inc. J Res Sci Teach 46: 1137–1160, 2009     

Examining the psychometric properties of the Infant–Toddler Environment Rating Scale-Revised Edition in a high-stakes context

The psychometric properties of the Infant–Toddler Environment Rating Scale-Revised Edition (ITERS-R) were examined using 153 classrooms from child-care centers where resources were tied to center performance. An exploratory factor analysis revealed that the scale measures one global aspect of quality. To decrease redundancy, subsets of items were selected randomly and by experts who rated items according to ease of administration and importance to quality. The shorter subsets demonstrated good discriminant validity, adequate to good psychometric properties, and high associations to the full ITERS-R score. They also demonstrated similar associations to staff education and staff-to-child ratio, as the full instrument. The best assessment of quality was demonstrated by the shortened subset that included items that assess both structural and process features of quality. Multilevel-analyses indicated that classrooms from the same providers score more similarly on ITERS-R than classrooms from other providers. The implications for using the ITERS-R in high-stakes contexts are discussed.     

Blocking landing techniques in volleyball and the possible association with anterior cruciate ligament injury

The number and type of landings performed after blocking during volleyball matches has been related to the potential risk of ACL injury. The aim of the present study was to determine whether gender affects the frequency of specific blocking landing techniques with potential risk of ACL injury from the perspective of foot contact and subsequent movement after the block used by volleyball players during competitive matches. Three matches involving four female volleyball teams (fourteen sets) and three matches involving four male volleyball teams (thirteen sets) in the Czech Republic were analyzed for this study. A Pearson chi-square test of independence was used to detect the relationship between gender and different blocking techniques. The results of the present study showed that gender affected single-leg landings with subsequent movement in lateral direction and double-leg landings. Although the total number of landings was lower for male athletes than for female athletes, a larger portion of male athletes demonstrated single leg landings with a subsequent movement than female athletes. Single leg landings with a subsequent movement have a higher potential risk of ACL injury.     

The Effect of Integrating Movement into the Learning Environment of Kindergarten Children on their Academic Achievements

The aim of this study was to test the notion that integrating movement into the learning environment contributes to the academic achievements of kindergarten students. One hundred and sixty 4–6 year-old kindergarten students participated in the study for 145 days, which included pre- and post-intervention tests in language, mathematics, and non-verbal intelligence. The three interventions consisted of (a) a mindful movement—integrating movement in academic learning, (b) a movement for its own sake—allowing children free movement without providing academic instruction, and (c) a control condition—children engaging in their regular academic environment activities. The findings revealed that the mindful movement intervention resuted in the highest improvement in the academic achievement tests. Children engaging in movement for its own sake did not differ from the control condition. The findings support the notion that mindful movement enhances kindergarten children’s academic achievements. Possible explanations for this outcome are discussed.     

Reductive thinking in computer science

This paper discusses the role of reduction in computer science and describes a study on undergraduate students' perception of the concept of reduction. Specifically, based on an analysis of students' answers to questions addressing different computer science topics, we present several findings regarding the ways in which undergraduate students conceive of and apply reduction. These findings can be interpreted within the framework of the tension that exists between the following two factors: The need to think in terms of high levels of abstraction, on one hand, and the fact that reduction introduces a new approach to be used in problem-solving situations, on the other. In addition the paper suggests several teaching applications.     

High school computer science education paves the way for higher education: the Israeli case

The gap between enrollments in higher education computing programs and the high-tech industry’s demands is widely reported, and is especially prominent for women. Increasing the availability of computer science education in high school is one of the strategies suggested in order to address this gap. We look at the connection between exposure to computer science in high school and pursuing computing in higher education. We also examine the gender gap, in the context of high school computer science education. We show that in Israel, students who took the high-level computer science matriculation exam were more likely to pursue computing in higher education. Regarding the issue of gender, we will show that, in general, in Israel the difference between males and females who take computer science in high school is relatively small, and a larger, though still not very large difference exists only for the highest exam level. In addition, exposing females to high-level computer science in high school has more relative impact on pursuing higher education in computing.     

Exploring young children’s self-efficacy beliefs related to mathematical and nonmathematical tasks performed in kindergarten: abused and neglected children and their peers

This article reports on young children’s self-efficacy beliefs and their corresponding performance of mathematical and nonmathematical tasks typically encountered in kindergarten. Participants included 132 kindergarten children aged 5–6 years old. Among the participants, 69 children were identified by the social welfare department as being abused and/or neglected. Individual interviews were conducted where children were asked to assess their self-efficacy regarding sorting tasks, mathematics tasks, and reciting the alphabet. Children were then requested to perform each of the tasks. Results revealed that no significant differences were found between the abused and neglected children and their peers regarding their self-efficacy beliefs and performances for any of the tasks. For some of the tasks, children were able to correctly assess their performance, while for other tasks, children overestimated their performance. Possible reasons for these outcomes are discussed.     

Problem solving strategies and mathematical resources: A longitudinal view on problem solving in a function based approach to algebra

This study is an attempt to analyze students' construction of function based problem solving methods in introductory algebra. It claims that for functions to be a main concept for learning school algebra, a complex process that has to be developed during a long period of learning must take place. The article describes a longitudinal observation of a pair of students that studied algebra for 3 years using a function approach, including intensive use of graphing technology. Such a long observation is difficult to carry out and even more difficult to report. We watched for three years classrooms using the ‘Visual-Math’ sequence, and sampled students that exhibited various levels of mathematics achievement. The analysis method presented here is a non-standard case study of a pair of lower achievers students and their work is often juxtaposed to the work of other pairs participating in the study. The students' attempts to solve a linear break-even problem is analyzed along three interviews which present the development of the use of mathematical resources and the patterns of problem solving at different learning phases. Beyond describing solving attempts, the article offers terms for describing and explaining what and how do learners appreciate and make out of solving introductory school algebra problems over a three years course. This revised version was published online in July 2006 with corrections to the Cover Date.