High‐School Students' Approaches to Solving Algebra Problems that are Posed Symbolically: Results from an Interview Study

A cross‐curricular structured‐probe task‐based clinical interview study with 44 pairs of third year high‐school mathematics students, most of whom were high achieving, was conducted to investigate their approaches to a variety of algebra problems. This paper presents results from three problems that were posed in symbolic form. Two problems are TIMSS items (a linear inequality and an equation involving square roots). The other problem involves square roots. We found that the majority of student pairs used symbol manipulation when solving the problems, and while many students seemed to prefer symbolic over graphical and tabular representations in their first attempt at solving the problems, we found that it was common for student pairs to use more than one strategy throughout the course of their solving. Students' use of graphing calculators to solve the problems is discussed.     

Crutch or Catalyst: Teachers' Beliefs and Practices Regarding Calculator use in Mathematics Instruction

This study investigated K‐12 teachers' beliefs and reported teaching practices regarding calculator use in their mathematics instruction. A survey was administered to more than 800 elementary, middle and high school teachers in a large metropolitan area to address the following questions: (a) what are the beliefs and practices of mathematics teachers regarding calculator use? and (b) how do these beliefs and practices differ among teachers in three grade bands? Factor analysis of 20 Likert scale items revealed four factors that accounted for 54% of the variance in the ratings. These factors were named Catalyst Beliefs, Teacher Knowledge, Crutch Beliefs, and Teacher Practices. Compared to elementary teachers, high school teachers were significantly higher in their perception of calculator use as a catalyst in mathematics instruction. However, the higher the grade level of the teacher, the higher the mean score on the perception that calculator use may be a way of getting answers without understanding mathematical processes. The mean scores for teachers in all three grade bands indicated agreement that students can learn mathematics through calculator use and using calculators in instruction will lead to better student understanding and make mathematics more interesting. The survey results shed light on teachers' self reported beliefs, knowledge, and practices in regard to consistency with elements of the National Council of Teachers of Mathematics Principles and Standards for School Mathematics (2000) technology principle and the NCTM use of technology position paper (2003). This study extended previous research on teachers' beliefs regarding calculator use in classrooms by examining and comparing the results of teacher surveys across three grade bands.     

On high-school students' use of graphic calculators in mathematics

When students are working with hand held technology, such as graphic calculators, we usually only see the outcomes of their activities in the form of a contribution to a written solution of a mathematical problem. It is more difficult to capture their process of thinking or actions as they use the technology to solve the problem. In this paper we report on two case studies that follow the progress of students as they solve mathematical problems. We use software that works in the background of the graphic calculator capturing the students' keystrokes as they use the calculator. The aim of the research studies described in this paper was to provide insights into the working styles of these students. Through a detailed analysis of their graphic calculator keystrokes, interviews and associated written solutions we will discuss the effectiveness of their solution strategies and the efficiency of their use of the technology and identify some barriers to the use of graphic calculators in mathematical problem solving.     

Middle‐School Mathematics Teachers' Beliefs in NCTM's Vision

This study examined the extent to which seventh‐ and eighth‐grade mathematics teachers are aware of National Council of Teachers of Mathematics (NCTM) standards documents, Curriculum and Evaluation Standards for School Mathematics and Principles and Standards for School Mathematics and agree with NCTM's vision of school mathematics as expressed in these documents. Quantitative data were collected through the Mathematics Standards Belief Survey (MSBS), a survey specifically designed to measure teachers' overall belief in NCTM's vision as well as in certain philosophical tenets of NCTM. Of the 82 seventh‐ and eighth‐grade mathematics teachers in the identified school district of Nevada, 73 (89.0%) participated in this study. The data revealed that, among seventh‐ and eighth‐grade mathematics teachers, secondary‐certified teachers had significantly higher MSBS scores than elementary‐certified teachers. A number of other findings were made, including significant differences among mean belief scores in the philosophical tenets of NCTM.     

Planning District‐Wide Professional Development: Insights Gained From Teachers and Students Regarding Mathematics Teaching in a Large Urban District

This paper describes the mechanism used to gain insights into the state of the art of mathematics instruction in a large urban district in order to design meaningful professional development for the teachers in the district. Surveys of close to 2,000 elementary, middle school, and high school students were collected in order to assess the instructional practices used in mathematics classes across the district. Students were questioned about the frequency of use of various instructional practices that support the meaningful learning of mathematics. These included practices such as problem solving, use of calculators and computers, group work, homework, discussions, and projects, among others. Responses were analyzed and comparisons were drawn between elementary and middle school students' responses and between middle school and high school responses. Finally, fifth‐grade student responses were compared to those of their teachers. Student responses indicated that they had fewer inquiry‐based experiences, fewer student‐to‐student interactions, and fewer opportunities to defend their answers and justify their thinking as they moved from elementary to middle school to high school. In the elementary grades students reported an overemphasis on the use of memorization of facts and procedures and sparse use of calculators. Results were interpreted and specific directions for professional development, as reported in this paper, were drawn from these data. The paper illustrates how student surveys can inform the design of professional development experiences for the teachers in a district.     

Observing student working styles when using graphic calculators to solve mathematics problems

Some research studies, many of which used quantitative methods, have suggested that graphics calculators can be used to effectively enhance the learning of mathematics. More recently research studies have started to explore students’ styles of working as they solve problems with technology. This paper describes the use of a software application that records the keystrokes made by students as they use calculators, in order to enable researchers to gain better insights into students’ working styles. The recordings obtained from this software can be replayed to observe how students have actually used their calculator in tackling a problem. The paper describes three pilot studies from quite different contexts, in which the software reveals how the calculators have been used by the students. In all of these studies the software provides insights into the working that would have been very difficult to obtain without the record of the keystrokes provided by the software.     

Effects of State Tests on Classroom Test Items in Mathematics

Classroom tests from nine eighth‐grade mathematics teachers were collected from the 2003–04 and 2005–06 school years. These years represent one school year prior to the eighth‐grade Ohio Achievement Test (OAT) in mathematics being implemented and the year after the eighth‐grade OAT in mathematics was implemented, respectively. In addition, teachers were interviewed to determine factors that influence classroom assessment practices. Classroom assessment data were compared between the two years, and interview data were examined, to investigate the impact that the new state test was having on classroom assessment practices. An average of 87% of teachers' classroom assessment items were at the lowest depth of knowledge level during both years. Teachers relied heavily on curriculum materials for their test items, and these items tended to only assess students ability to recall basic facts or perform straightforward procedures. The presence of a state test did not entice teachers to assess students at higher depth of knowledge levels.     

Computational Estimation Performance on Whole‐Number Multiplication by Third‐ and Fifth‐Grade Chinese Students

Four hundred and three 3rd‐ and 5th‐grade Chinese students took the Multiplication Estimation Test or participated in the interview on it, designed to assess their computational estimation performance on whole‐number multiplication. Students perform better when tasks are presented visually than orally. Third graders tend to use rounding based while fifth graders tend to use written algorithm based strategies, but boys' and girls ‘performances do not differ. It is concluded that students often will not estimate simply at the request to estimate if an exact answer is within their mental computation capability, and a two‐step process is suggested for helping students decide what route to take when given arithmetic problems.     

Technology Use in the Teaching of Mathematics and Science in Elementary Schools

Principals from 80 elementary schools, in predominantly rural Western Pennsylvania, completed and returned a 22-item questionnaire designed to assess the nature and extent of technology use for the teaching of elementary mathematics and science within their buildings. Technologies included calculators, microcomputers, overhead projection panels, videotape, and interactive video. Microcomputers were being used in at least some elementary grade levels for mathematics or science instruction in 84% of the schools. Teachers used microcomputers more frequently in mathematics (82.5%) than in science instruction (55%). Principals reported a lower frequency of calculator use with 63.3% of the schools having some grade levels where calculators were used in mathematics lessons, and 21.5% of schools in science lessons. Further analyses of data suggest that microcomputer and calculator use is more common in the intermediate grades (3–6) than in the primary grades (K-2).     

A Comparison of Calculator Use in Eighth‐Grade Mathematics Classrooms in the United States,Japan, and Portugal: Results From the Third International Mathematics and Science Study

The study used data from the Third International Mathematics and Science Study to determine trends in calculator use among Population 2 (13‐year‐olds) students in Japan, the United States, and Portugal. While relatively high levels of calculator use were observed for the US and Portugal, virtually no calculator use was found for the Japanese sample. Hierarchical Linear Model analysis determined a statistically significant negative relationship between students' frequency of calculator use and student performance in Japan; no statistically significant relationship was detected for the US and Portuguese samples. US student achievement was positively associated with each of the five reported ways in which calculators are used; however, a statistically significant negative relationship was found between student performance and Japanese students' use of calculators on tests. Plausible explanations are explored.     

The Effects of Non‐CAS Graphing Calculators on Student Achievement and Attitude Levels in Mathematics: A Meta‐Analysis

Forty‐two studies comparing students with access to graphing calculators during instruction to students who did not have access to graphing calculators during instruction are the subject of this meta‐analysis. The results on the achievement and attitude levels of students are presented. The studies evaluated cover middle and high school mathematics courses, as well as college courses through first semester calculus. When calculators were part of instruction but not testing, students' benefited from using calculators while developing the skills necessary to understand mathematics concepts. When calculators were included in testing and instruction, the procedural, conceptual, and overall achievement skills of students improved.     

Controlling Choice: Teachers,Students, and Manipulatives in Mathematics Classrooms

This research study examines the instructional practices of 10 middle grades teachers related to their use of manipulatives in teaching mathematics and their control of mathematics tools during instruction. Through 40 observations of teaching, 30 interviews, and an examination of 67 written documents (including teachers' plans and records), profiles were developed that describe how teachers used and controlled manipulatives during instruction. Results showed that teachers used a variety of manipulatives and other mathematics tools over the course of the year‐long study. Teachers reported using a mathematics tool (manipulative, calculator, or measuring device) in 70% of their lessons, and this self‐report was verified by observations in which teachers used mathematics tools in 68% of their lessons. During a 3‐ to 4‐month period of “free access,” in which students had some measure of control in their selection and use of the mathematics tools, the students used manipulatives spontaneously and selectively. During free access, teachers exhibited various behaviors, including posting lists of items on containers, assigning group leaders to manage tools, and negotiating the control of the mathematics tools during instruction.     

The difficulty of “length × width”: Is a square the unit of measurement?

In individual interviews, 220 students in grades 4, 6, 8, and 9 were given one task, and 72 eighth graders were given three tasks to answer two questions: (a) Is a square the unit of measurement for an area for students in grades 4-8? and (b) Does a square have a space-covering characteristic for students in grade 8? The answers to both questions were No (except for eighth (and ninth) graders in advanced sections of mathematics). The difficulty of “length × width” is explained in light of Piaget's theory, and educational implications are discussed.     

Singaporean students' mathematical thinking in problem solving and problem posing: an exploratory study

This study explored Singaporean fourth, fifth, and sixth grade students' mathematical thinking in problem solving and problem posing. The results of this study showed that the majority of Singaporean fourth, fifth, and sixth graders are able to select appropriate solution strategies to solve these problems, and choose appropriate solution representations to clearly communicate their solution processes. Most Singaporean students are able to pose problems beyond the initial figures in the pattern. The results of this study also showed that across the four tasks, as the grade level advances, a higher percentage of students in that grade level show evidence of having correct answers. Surprisingly, the overall statistically significant differences across the three grade levels are mainly due to statistically significant differences between fourth and fifth grade students. Between fifth and sixth grade students, there are no statistically significant differences in most of the analyses. Compared to the findings concerning US and Chinese students' mathematical thinking, Singaporean students seem to be much more similar to Chinese students than to US students.     

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.     

A Multi‐Method Investigation of Mathematics Motivation for Elementary Age Students

This paper presents the results of a multi‐method study examining elementary students with high self‐reported levels of mathematics motivation. Second‐ through fifth‐grade students at a Title One school in the southeastern United States completed the Elementary Mathematics Motivation Instrument (EMMI), which examines levels of mathematics motivation across three subscales: (a) Math Anxiety, (b) Self‐Efficacy, and (c) Value of Math. Results from this quantitative phase were used to identify a sample for a qualitative phase examining how students who report high levels of motivation perceive mathematics. The resulting qualitative phase utilized a phenomenological design to explore mathematics motivation for a particular set of students in a fifth‐grade setting. Findings indicate that elementary students with high mathematics motivation value mathematics as a present and future oriented discipline and value teachers that deemphasize testing as a measure of success.     

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.     

The Graphing Calculator: A Survey of Classroom Usage

The purpose of this study was to investigate secondary mathematics teachers' use of the graphing calculator in their classrooms, The study examined whether algebra teachers are currently using this technology in their classrooms, their perceptions toward the technology, and any changes in the curriculum or instructional practices. A survey methodology was used in this study. The findings indicated that the use of the graphing calculator is still controversial to many algebra teachers. Teachers of algebra I were using graphing calculators to a significantly lesser degree than teachers of algebra H. However, modifications of the algebra curriculum are beginning to appear in classes using graphing calculators. Finally, a majority of algebra teachers responded that the graphing calculator was a motivational tool.     

Promoting Percent as a Proportion in Eighth‐Grade Mathematics

The literature provides many and varied suggestions for promoting conceptual understanding of percent and performing percent calculations. The diversity of ideas provides a wide selection but offers little clarity on the true nature of percent. From the premise that percent is fundamentally a proportion, this study incorporated a proportional approach for percent problem solving within an instructional program on percent. Classroom research with eighth‐grade students indicated that the method was readily adopted by students and helped them experience success in percent problem solving, with percent problem solving proficiency maintained over a delayed period. It is hypothesized that the method has the potential to promote students' conceptual knowledge of percent as a proportion and the multiplicative structure of percent, as well as to build proportional knowledge.