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.     

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.     

Calculator Use on NAEP: A Look at Fourth‐ and Eighth‐Grade Mathematics Achievement

This article summarizes research conducted on calculator block items from the 2007 fourth‐ and eighth‐grade National Assessment of Educational Progress Main Mathematics. Calculator items from the assessment were categorized into two categories: problem‐solving items and noncomputational mathematics concept items. A calculator has the potential to be used as a problem‐solving tool for items categorized in the first category. On the other hand, there are no practical uses for calculators for noncomputational mathematics concept items. Item‐level performance data were disaggregated by student‐reported calculator use to investigate the differences in achievement of those fourth‐ and eighth‐grade students who chose to use calculators versus those who did not, and whether or not the nation's fourth and eighth graders are able to identify items where calculator use serves as an aide for solving a given mathematical problem. Results from the analysis show that eighth graders, in particular, benefit most from the use of calculators on problem‐solving items. A small percentage of students at both grade levels attempted to use a calculator to solve problems in the noncomputational mathematics concept category (items in which the use of a calculator does not serve as a tool to solve the problem).     

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.     

Representations in applying functions

As part of a large research project—Heuristic Education of Mathematics: developing and investigating strategies to teach applied mathematical problem solving—inquiries were made into the question of the transfer of knowledge and skills from the area of functions to real-world problems. In particular, studies were made of the translation processes from one representation of a problem into another representation. Surprisingly, students often used informal methods not taught in their lessons. After a full year of teaching mathematics, including a lot of applied problem solving, a shift from informal methods to the analytical (expert) solution method was identified. There were also significant differences among the learning results of three instructional design conditions. This research was extended to consider implications of the use of the graphic calculator. Casual use of the graphic calculator diminished the application of analytical methods, but integrated use brought about an enrichment of solution methods.     

Affect and graphing calculator use

This article reports on a qualitative study of six high school calculus students designed to build an understanding about the affect associated with graphing calculator use in independent situations. DeBellis and Goldin's (2006) framework for affect as a representational system was used as a lens through which to understand the ways in which graphing calculator use impacted students’ affective pathways. It was found that using the graphing calculator helped students maintain productive affective pathways for problem solving as long as they were using graphing calculator capabilities for which they had gone through a process of instrumental genesis (Artigue, 2002) with respect to the mathematical task they were working on. Furthermore, graphing calculator use and the affect that is associated with its use may be influenced by the perceived values of others, including parents and teachers (past, present and future).     

An investigation into whether student use of graphics calculators matches their teacher's expectations

This research examines students’ use of graphics calculators and investigates the extent to which the students’ use meets their teachers aim when using graphics calculators in the classroom. The teacher's use of her graphics calculator was analysed over a week using Key Record software. The teacher was questioned about her aims and expectations for the students when using a graphics calculator. As a result an interview schedule for students was constructed in order to determine whether the teacher's aims had been met. It was found that in general all of the teachers’ aims were met to some extent by most of the students.     

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.     

Students encountering obstacles using a CAS

The paper describes a pilot study on the use of computer algebra at upper secondary level. A symbolic calculator was introduced in a pre-examination class studying for advanced pre-university mathematics. With the theoretical framework of Realistic Mathematics Education and Developmental Research as a background, the study focused on the identification of obstacles that students encountered while using computer algebra. Five obstacles were identified that have both a technical and a mathematical character. It is the author's belief that taking these barriers seriously is important in developing useful pedagogical strategies.This revised version was published online in September 2005 with corrections to the Cover Date.     

VISUALISATION AND THE INFLUENCE OF TECHNOLOGY IN ‘A’ LEVEL MATHEMATICS: A CLASSROOM INVESTIGATION

The study reported here is part of a wider study, which aims to investigate the potential of the graphical calculator for mediating the development of students’ abilities to visualise the graphs of functions at GCE Advanced level. This paper focuses on how the graphical calculator influenced six particular students’ work with functions. Initial results have illuminated ways in which the technology can have a positive impact on students’ visualisation capabilities. It is proposed that visual thinking forms a significant part of many students’ mathematical reasoning, enabling students to derive richer meaning from given problems. It is suggested further that use of the technology mediates the development of students’ visual capacities, by helping to highlight the links between complementary modes of representation.     

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.     

Middle School Mathematics Teachers Learning to Teach with Calculators and Computers

This paper reports the results of a project in which experienced middle grades mathematics teachers immersed themselves in calculator and computer use for both doing and teaching mathematics and prepared themselves as leaders for communicating their knowledge to colleagues. Project evaluation included interviews with participants at the beginning and end of the project and evaluation forms completed at the end of the project. Pre-interviews indicated that virtually all of the participants had no experience using technology to teach mathematics. Many felt that technology was not likely to be as effective in helping students learn mathematics as other teaching techniques. Post-interviews indicated that all teachers were confident of their abilities to use some technologies in teaching mathematics. They acknowledged that technology was useful in developing conceptual understanding and that their role was to guide this conceptual development. The differences in participants' perceptions about how the project affected them yielded suggestions for future inservice efforts about technology.     

Improving senior secondary school students’ attitude towards mathematics through self and cooperative-instructional strategies

This study investigated the effects of self and cooperative-instructional strategies on senior secondary school students’ attitude towards Mathematics. The moderating effects of locus of control and gender were also investigated. The study adopted pre-test and post-test, control group quasi-experimental design using a 3 ×?2 ×?2 factorial matrix with two experimental groups and one control group. Three hundred and fifty SSS II students from six purposively selected secondary schools in Ijebu-North Local Government Area of Ogun State were the subjects. Three instruments were developed, validated and used for data collection. Analysis of Covariance (ANCOVA) and Scheffé post hoc analysis were the statistics used for data analysis. Findings showed that the treatments had significant main effect on students’ attitude towards Mathematics. The participants exposed to self-instructional strategy had the highest post-test mean attitude score. The study found no significant main effects of locus of control and gender on the participants’ attitude towards Mathematics. It was concluded that Mathematics teachers should be trained to use self and cooperative learning packages in the classroom, since the strategies are more effective in improving students’ attitude towards Mathematics than the conventional method.     

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.     

Filling in the gaps: modelling incomplete CBL data using a graphing calculator

This article discusses the real-world problem-solving lesson that emerged when a high school math teacher used a motion detector with a CBL and graphing calculator to obtain the 'bounce' data of a ping-pong ball. While practising the 'bounce' data collection--a series of diminishing parabolas--the teacher accidentally pulled the motion detector away and then, realizing his mistake, pulled it back. The resulting data showed a series of parabolas, but one was missing. The teacher used this opportunity to create a lesson in which his students collect 'bad data' and then fill in the missing parabola using critical components of parabolas, such as the latus rectum and the vertex, and using matrices. The article provides all the necessary directions, formulas, and names of resources needed to replicate the lesson. The creation of this lesson demonstrates that a serendipitous error can create a genuine and authentic problem-solving activity for math students.     

Toward an Algebra Acquisition Support System: A Study Based on Using Graphic Calculators in the Classroom

This article discusses the results of a study that focused on using graphic calculators. The algebraic code of the calculator was used to introduce 11- to 12-year-old students to algebraic language as a tool for modeling and solving problems, relating this to their previous arithmetical experience and their evolving use of symbolic language. This study provided empirical evidence for the potential of conceiving algebra as a language and teaching it as a language-in-use, supported by the graphic calculator. The teaching approach was based on Bruner's (1983) research on natural language acquisition. Bruner claimed that natural language is taught and that the adult shapes the environment such that children can learn the rudiments of their mother tongue through its use, without needing to know syntactical rules and definitions. The main aim of this study was to investigate the outcomes of an algebra learning scheme based on Bruner's theory of language acquisition.     

One-step and multi-step linear equations: a content analysis of five textbook series

Being able to solve one-step and multi-step linear equations is a hallmark of algebraic proficiency in the United States and abroad. The purpose of this paper is to report on a textbook comparison study regarding the treatment of linear equations in five textbook series that are popular in the United States: Center for Mathematics Education Project Algebra 1 and Algebra 2 (CME), Core-Plus Mathematics Program Courses 1–3 (CPMP), Glencoe Algebra 1 and Algebra 2, Interactive Mathematics Program Years 1–3 (IMP), and University of Chicago School Mathematics Project Algebra and Advanced Algebra (UCSMP). Data are reported for the following curriculum variables: content, cognitive behavior, real-world context, and tools (technology and manipulatives). Sequencing of the content, and links between symbolic and functions-based approaches are discussed. Based on the data presented, what students experience relative to setting up and solving one-step and multi-step linear equations is likely to be markedly different, depending on which textbook is used in their classrooms. Implications for practice and ideas for further research are discussed.     

Assessing the Use of Graphing Calculators in College Algebra: Reflecting on Dimensions of Teaching and Learning

This article considers various aspects of teaching and learning stemming from the integration of graphing calculator use in a semester-long college algebra course. The project examined four class sections, in which two instructors each taught one section using graphing calculators and one section using a traditional approach. Achievement and attitude data showed no significant differences for treatment or instructor; a significant difference in achievement was found for gender. Students in the calculator sections responded to an open-ended questionnaire about their use of the calculator and were generally supportive of the technology. Students were in agreement about specific topics for which the technology was most useful. Overall, findings indicated that the technology use had a positive impact on various dimensions of student learning.