Development of a Geometric Spatial Visualization Tool

This paper documents the development of the Geometric Spatial Assessment. We detail the development of this instrument which was designed to identify middle school students' strategies and advancement in understanding of four geometric concept domains (geometric spatial visualization, spatial projection, cardinal directions, and periodic patterns) after experiencing a carefully designed integrated lunar unit. Previous research with students using this lunar unit showed students making significant gains on lunar‐related concepts (both scientific and mathematical) on a Lunar Phases Concept Inventory (LPCI) ( Lindell & Olsen, 2002 ). Following the administration of single domain assessments, clinical interviews were conducted to ascertain students' problem solving strategies. Results allowed us to select four suitable multiple‐choice items per domain.     

The role of the graphic calculator in mediating graphing activity

The PIGMI (Portable Information Technologies for supporting Graphical Mathematics Investigations) Project ¹ investigated the role of portable technologies in facilitating development of students' graphing skills and concepts. This paper examines the impact of a recent shift towards calculating and computing tools as increasingly accessible, everyday technologies on the nature of learning in a traditionally difficult curriculum area. The paper focuses on the use of graphic calculators by undergraduates taking an innovative new mathematics course at the Open University. A questionnaire survey of both students and tutors was employed to investigate perceptions of the graphic calculator and the features which facilitate graphing and linking between representations. Key features included visualization of functions, immediate feedback and rapid graph plotting. A follow-up observational case study of a pair of students illustrated how the calculator can shape mathematical activity, serving a catalytic, facilitating and checking role. The features of technology-based activities which can structure and support collaborative problem solving were also examined. In sum, the graphic calculator technology acted as a critical mediator in both the students' collaboration and in their problem solving. The pedagogic implications of using portables are considered, including the tension between using and over-using portables to support mathematical activity.     

A Partnership Approach to Improving Student Attitudes About Sharks and Scientists

This article describes the methods and impact of a student–teacher–scientist research partnership on student attitudes. The partnership objective was to teach students about the diverse roles of sharks in the marine environment while personally connecting students with scientific study. Students (N = 229) participated in lessons about shark biology and helped the partnering scientist design experimental protocols and analyze data. A self‐selected subset of students also volunteered (n = 82) for a field component working with live hammerhead sharks (Sphyrna lewinii). Student surveys before and after the partnership suggested that negative attitudes about sharks are due largely to lack of exposure, and direct attention to students' stereotypes about sharks resulted in significant attitude improvement. Change in students' attitudes toward scientists, however, was minimal. Students' negative views of scientists did decline significantly, but their overall views of scientists were relatively positive to begin with. Also of interest was the students' unremitting association of scientists with specialized equipment and the students' lack of personal connection to scientific ways of examining the world, suggesting that partnerships may be more effective at personally connecting students with scientific process if they explicitly incorporate activities designed to improve students' view of themselves as scientists.     

Searching and exploring properties of geometric configurations using dynamic software

The decline in enrolments and interest in advanced mathematics studies is of growing concern internationally. Previous research suggests that a range of factors can influence students' academic decisions. The focus of the paper is on one of these potential sources of influence— students' perceptions of the tertiary mathematics learning environment. Data from two large-scale surveys (N = 1883) and from a smaller number of interviews (N = 71) with students enrolled in tertiary mathematics courses at five Australian universities are presented and discussed. Collectively, the survey results and the interview data reveal considerable variations in the quality of the teaching and student support available in different mathematics departments. Students' comments were constructive and offered valuable ideas for improving the existing situation, retaining current students and attracting others to mathematics.     

Gender Differences in Learning Constructs,Shifts in Learning Constructs,and Their Relationship to Course Achievement in a Structured Inquiry,Yearlong College Physics Course for Life Science Majors

This study investigated differences and shifts in learning and motivation constructs among male and female students in a nonmajors, yearlong structured inquiry college physics course and examined how these variables were related to physics understanding and course achievement. Tests and questionnaires measured students' learning approaches, motivational goals, self‐efficacy, epistemological beliefs, scientific reasoning abilities, and understanding of central physics concepts at the beginning and end of the course. Course achievement scores were also obtained. The findings showed that male students had significantly higher self‐efficacy, performance goals, and physics understanding compared to females, which persisted throughout the course. Differential shifts were found in students' meaningful learning approaches, with females tending to use less meaningful learning from beginning to end of the course; and males using more meaningful learning over this time period. For both males and females, self‐efficacy significantly predicted physics understanding and course achievement. For females, higher reasoning ability was also a significant predictor of understanding and achievement; whereas for males, learning goals and rote learning were significant predictors, but in a negative direction. The findings reveal that different variables of learning and motivation may be important for females' success in inquiry physics compared to males. Instructors should be cognizant of those needs in order to best help all students learn and achieve in college physics.     

Efficacy of Different Concrete Models for Teaching the Part-Whole Construct for Fractions

The effectiveness of different concrete and pictorial models on students' understanding of the part-whole construct for fractions was investigated. Using interview data from fourth and fifth grade students from three different districts that adopted the Mathematics Trailblazers series, authors identified strengths and limitations of models used. Pattern blocks had limited value in aiding students' construction of mental images for the part-whole model as well as limited value in building meaning for adding and subtracting fractions. A paper fraction chart based on a paper folding model supported students' ability to order fractions with same numerators but was less useful in helping students on estimation tasks. The dot paper model and chips did not support fifth grade students' initial understanding of the algorithm.     

Real-World Contexts,Multiple Representations,Student-Invented Terminology,and Y-Intercept

One classroom using two units from a Standards-based curriculum was the focus of a study designed to examine the effects of real-world contexts, delays in the introduction of formal mathematics terminology, and multiple function representations on student understanding. Students developed their own terminology for y-intercept, which was tightly connected to the meaningfulness and implicit/explicit temporality of the contexts that students investigated as part of their classroom activities. This terminology held great promise for promoting the concept of y-intercept within a multiple representation environment. However, the teacher's interpretation of different activities and his assumptions about the transparency of different representations, as well as students' past experiences left the student-generated terminology and the concept of y-intercept disconnected from one another. This resulted in student-generated terminology that had limited applicability, a fragile understanding of y-intercept within different representations, and for some students, interference between their invented terminology and the concept of y-intercept itself.     

The Effects on Students' Cognitive Achievement When Using the Cooperative Learning Method in Earth Science Classrooms

This study investigated the effects of cooperative learning instruction versus traditional teaching methods on students' earth science achievement in secondary schools. A total of 770 ninth-grade students enrolled in 20 sections of a required earth science course participated in this nonequivalent control group quasi-experiment. The control groups (n= 10) received a traditional approach, while the experimental groups (n= 10) used cooperative strategies. Study results include (a) no significant differences were found between the experimental groups and the control groups when overall achievement (F= 0.13, p > .05), knowledge-level (F= 0.12, p > .05), and comprehension-level (F= 0.34, p > .05) test items were considered; and (b) students who worked cooperatively performed significantly better than students who worked alone on the application-level test items (F= 4.63, p < .05). These findings suggest that cooperative-learning strategies favor students' earth science performance at higher but not lower levels of cognitive domains in the secondary schools.     

Reducing abstraction: The case of constructing an operation table for a group

This paper describes students' mental processes while constructing an operation table for a group. More specifically, undergraduate students' approaches are analyzed as the students fill in an operation table for four elements—a, b, c, and d—in such a way that it represents a group of order four. The data are analyzed from the perspective of reducing abstraction, which aims to explain students' conceptions of abstract algebra concepts. From this perspective, most students' responses and conceptions can be attributed to their tendency to work on a lower level of abstraction than the level on which concepts are introduced in class.     

Analogies and Reconstruction of Probability Knowledge

The effectiveness of utilizing analogies to effect conceptual change in students' alternative probability concepts was investigated. Forty-one senior high school mathematics students were engaged in a knowledge reconstruction process regarding their beliefs about common everyday probability situations, such as sports events or lotteries. The students were given situations similar to those shown in previous research to reveal alternative mathematical conceptions. They were also given analogous researcher-generated anchoring situations that had been pretested and found to elicit mathematically acceptable responses. The cognitive dissonance produced by the conflicting responses motivated students to reconstruct their knowledge. The results of the investigation showed that analogies can be effective in producing a desired conceptual change in high school students' probability concepts.     

Students' Understanding of the Particulate Nature of Matter

The particulate nature of matter is identified in science education standards as one of the fundamental concepts that students should understand at the middle school level. However, science education research in indicates that secondary school students have difficulties understanding the structure of matter. The purpose of the study is to describe how engaging in an extended project‐based unit developed urban middle school students' understanding of the particulate nature of matter. Multiple sources of data were collected, including pre‐ and posttests, interviews, students' drawings, and video recordings of classroom activities. One teacher and her five classes were chosen for an indepth study. Analyses of data show that after experiencing a series of learning activities the majority of students acquired substantial content knowledge. Additionally, the finding indicates that students' understanding of the particulate nature of matter improved over time and that they retained and even reinforced their understanding after applying the concept. Discussions of the design features of curriculum and the teacher's use of multiple representations might provide insights into the effectiveness of learning activities in the unit.     

Students' Concepts‐ and Theorems‐in‐Action on a Novel Task About Similarity

Similarity is a fundamental concept in the middle grades. In this study, we applied Vergnaud's theory of conceptual fields to answer the following questions: What concepts‐in‐action and theorems‐in‐action about similarity surfaced when students worked in a novel task that required them to enlarge a puzzle piece? How did students use geometric and multiplicative reasoning at the same time in order to construct similar figures? We found that students used concepts of scaling and proportional reasoning, as well as the concept of circle and theorems about similar triangles, in their work on the problem. Students relied not only on visual perception, but also on numeric reasoning. Moreover, students' use of multiplicative and proportional concepts supported their geometric constructions. Knowledge of the concepts and ideas that students have available when working on a task about similarity can inform instruction by helping to ground formal introduction of new concepts in students' informal prior experiences and knowledge.     

Limits and continuity: some conceptions of first-year students

A research method consisting of written tests and individual interviews was introduced to examine first-year university students' understanding of fundamental calculus concepts. Six hundred and thirty students from three South African universities were subjected to the tests pertaining to this study. Several misconceptions underlying students' understanding of calculus concepts were identified. This paper deals mainly with some of the common errors and misconceptions relating to students' understanding of ‘limit of a function’ and ‘continuity of a function at a point’.     

Understanding a Teacher's Reflections: A Case Study of a Middle School Mathematics Teacher

The purpose of this research was to understand how one teacher reflected on different classroom situations and to understand whether the teacher's approach to these reflections changed over time. For the purposes of this study, we considered reflection as the teacher's act of interpreting her own practices and students' thinking to make sense of student understanding and how teaching might relate to that understanding. We investigated a middle school mathematics teacher's reflection on her students while watching videotapes of her classroom and categorized the reflection as Assess, Interpret, Describe, Justify, and Extend. The results show a higher percentage of Extend instances in later interviews than in earlier ones indicating the teacher's increasing attention to her own teaching in how her students developed their understanding. In addition, her reflection became clearer and better integrated as defined by the Cohen and Ball's triangle of interactions.     

Asking mathematical questions mathematically

It is proposed that the style and format of the questions used by lecturers and tutors profoundly influence students' conceptions of what mathematics is about and how it is conducted. By looking at reasons for asking questions, and becoming aware of different types of questions which mathematicians typically ask themselves, we can enrich students' experience of mathematics. Drawing on recent work by Watson and Mason stimulated by the ideas of Zygfryd Dyrszlag, the paper proposes that mathematical themes, powers, heuristics and activities generate a mathematical discourse which is not always represented in the questions students are asked, and that the real purpose of questions is to provoke students into construal, into constructing their own stories which constitute meaning and understanding, and which equip them for the future. The use of questions of whatever type depends on both scaffolding and fading their use with and in front of students, so that students internalize the questions into their own learning and doing of mathematics. The framework directed—prompted—spontaneous is proposed as an alternative to scaffolding—fading for informing interactions with students.     

A Modeling Perspective on the Teaching and Learning of Mathematical Problem Solving

This study analyzed the processes used by students when engaged in modeling activities and examined how students' abilities to solve modeling problems changed over time. Two student populations, one experimental and one control group, participated in the study. To examine students' modeling processes, the experimental group participated in an intervention program consisting of a sequence of six modeling activities. To examine students' modeling abilities, the experimental and control groups completed a modeling abilities test on three occasions. Results showed that students' models improved as they worked through the sequence of problem activities and also revealed a number of factors, such as students' grade, experiences with modeling activities, and modeling abilities that influenced their modeling processes. The study proposes a three-dimensional theoretical model for examining students' modeling behavior, with ubsequent implications for the teaching and learning of mathematical problem solving.     

Applying Mathematical Concepts with Hands‐On,Food‐Based Science Curriculum

This article addresses the current state of the mathematics education system in the United States and provides a possible solution to the contributing issues. As a result of lower performance in primary mathematics, American students are not acquiring the necessary quantitative literacy skills to become successful adults. This study analyzed the impact of the Food, Math, and Science Teaching Enhancement Resource (FoodMASTER) Intermediate curriculum on fourth‐grade students' mathematics knowledge. The curriculum is a part of the FoodMASTER Initiative, which is a compilation of programs utilizing food, a familiar and necessary part of everyday life, as a tool to teach mathematics and science. Students exposed to the curriculum completed a 20‐item researcher‐developed mathematics knowledge exam (intervention n = 288; control n = 194). Overall, the results showed a significant increase in mathematics knowledge from pretest to posttest. These findings suggest that the food‐based science activities provided the students with the context in which to apply mathematical concepts to an everyday experience. Therefore, the FoodMASTER approach was successful at improving students' mathematics knowledge while building a foundation for becoming quantitatively literate adults.     

Examining the Conceptual Organization of Students in an Integrated Algebra and Physical Science Class

The cross-disciplinary context of density and slope was used to compare the conceptual organization of students in an integrated algebra and physical science class (SAM9) with that of students in a discipline-specific physical science class (PSO). Analyses of students' concept maps indicated that the SAM9 students used a greater number of procedural linkages to connect mathematics and science concepts on the SAM9 students' maps than did the PSO students. The maps produced by SAM9 students also tended to show a more compartmentalized approach to thinking about the content of the two disciplines, a finding contrary to the researcher's original assertion. Traditional teaching territories and conceptual complexity were examined as possible explanations for the discrepancy between the predicted and actual outcomes.     

Fuzzy Logic,Neural Networks,Genetic Algorithms: Views of Three Artificial Intelligence Concepts Used in Modeling Scientific Systems

Students' conceptions of three major artificial intelligence concepts used in the modeling of systems in science, fuzzy logic, neural networks, and genetic algorithms were investigated before and after a higher education science course. Students initially explored their prior ideas related to the three concepts through active tasks. Then, laboratories, project work, use of computer modeling of scientific systems, and cooperative group work were used to help students construct key characteristics of each concept. Finally, they applied each concept in contexts different from that in which it had been previously studied. In postcourse interviews using a set of scenarios for each of the major course concepts, 49% of students' applications included key characteristics of the concepts studied versus an application of 5% in precourse interviews. Students' post interview applications were inconsistent even though they were more frequent, indicating a state of transition in their conceptual change. Applications were most consistent when used with scenarios deemed very familiar to the students, indicating the effects of context in conceptual change.