Graphing Calculator Use in Algebra Teaching

This study examines graphing calculator technology availability, characteristics of teachers who use it, teacher attitudes, and how use reflects changes to algebra curriculum and instructional practices. Algebra I and Algebra II teachers in 75 high school and junior high/middle schools in a diverse region of a northwestern state were surveyed. Forty of the 75 schools (53%) returned a total of 109 individual surveys. Results indicated that: (1) While 78% of teachers have some access to the technology, only 28% use it regularly. (2) Statistically significant relationships exist between use and age, years of experience, teaching assignment, and teaching level. (3) Respondents view graphical solution methods as secondary to symbolic methods. (4) Teachers are more receptive to using technology to supplement rather than expand the curriculum.     

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.     

Graphing in Groups: Learning About Lines in a Collaborative Classroom Network Environment

This article presents a design experiment in which we explore new structures for classroom collaboration supported by a classroom network of handheld graphing calculators. We describe a design for small group investigations of linear functions and present findings from its implementation in three high school algebra classrooms. Our coding of the problem-solving efforts of six student focus pairs in this environment over the course of several class sessions indicates that these students tended to move from exploratory and visual to more analytic means of establishing lines of a specified slope. As they adopted these analytic approaches, they were also more likely to enact their strategies jointly. In closer examination of emerging analytic strategies in episodes selected from the work of one of the pairs, we argue that the processes by which these students discovered the need for coordinated action on their respective points, and came to establish mathematical meaning for the relations between their coordinate locations as slope, were overlapping and intertwined.     

Middle Grades Students' Algebraic Understanding in a Reform Curriculum

The National Council of Teachers of Mathematics' Curriculum and Evaluation Standards in 1989 was pivotal in mathematics reform. The National Science Foundation funded several curriculum projects to address the vision described in the Standards. This study investigates students' learning in one of these Standards‐based curricula, the Connected Mathematics Project (CMP). The authors of CMP believe that the teaching and learning of algebra is an ongoing activity woven through the entire curriculum, rather than being parceled into a single grade level. The content of the study investigates students' ability to symbolically generalize functions. The data regards the solutions of four performance tasks dealing with three different types of relationships—linear, quadratic, and exponential situations—completed by five pairs of eighth‐grade students. The major finding claims that middle to high achieving students who had 3 years in the CMP curriculum demonstrated achievement in five strands of mathematical proficiency of a significant piece of algebra.     

Learning to Teach: Prospective Teachers' Evaluation of Students' Written Responses on a 1992 NAEP Graphing Task

This article describes how prospective elementary teachers examined, analyzed, and evaluated four students' written responses on a graphing task for an end‐of‐course performance assessment in a mathematics methods course. Also, they described teaching strategies that built on what students know and do not know, as shown in the fourth‐grade students' work. This course assessment provided evidence of the prospective teachers' pedagogical content knowledge. Two themes emerged in the context of this final course project: the importance of process and correct answers and the usefulness of creating rubrics.     

The Effect of Calculator‐Based Ranger Activities on Students' Graphing Ability

Middle school students can learn to communicate with graphs in the context of appropriate Calculator‐Based Ranger (CBR) activities. Three issues about CBR activities on graphing abilities were addressed in this study: (a) the effect of CBR activities on graphing abilities; (b) the extent to which prior knowledge about graphing skills affects graphing ability; (c) the influence of instructional styles on students' graphing abilities. Following the use of CBR activities, students' graphing abilities were significantly more developed in three components _ interpreting, modeling, and transforming. Prior knowledge of graphing skills on the Cartesian coordinate plane had little effect on students' understanding of graphs. Significant differences, however, were found in students' achievement, depending on instructional styles related to differentiation, which is closely connected to transforming distance‐time graphs to velocity‐time graphs. The result of this study indicates that the CBR activities are pedagogically promising for enhancing graphing ability of physical phenomena.     

Understanding Equivalence of Symbolic Expressions in a Spreadsheet-Based Environment

Use of spreadsheets in a beginning algebra course was investigated mainly with regard to their potential to promote generalization of patterns. Less is known about their use in promoting understanding and learning of transformational activities. The overall purpose of this paper is to consider the conceptual aspects of learning a transformational skill (use of the distributive law to produce equivalent algebraic expressions) in a learning sequence composed of both spreadsheets and paper-and-pencil activities. We conducted a sequence of classroom activities in several classes, and analyzed the students’ work on a spreadsheet activity and on an assessment activity by both qualitative and quantitative methods. The findings indicate both encouraging benefits and some potential sources of difficulties caused by the use of spreadsheets at initial stages of learning symbolic transformations.

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.     

Assessing Elementary Understanding of Multiplication Concepts

This article summarizes the basic concepts of multiplication and provides some evidence that the traditional third‐grade curriculum and instruction emphasizing memorization of multiplication facts produces much less understanding of the basic concepts of multiplication than a standards‐based curriculum and instruction emphasizing construction of number sense and meaning for operations. This study also describes a collection of assessment tasks that provided meaningful evidence of children's understandings of basic multiplication concepts, including understandings of the relationships between multiplication and addition.     

David's Understanding of Functions and Periodicity

This is a study of David, a senior enrolled in a high school precalculus course. David's understandings of functions and periodicity was explored, through clinical interviews and contextualized through classroom observations. Although David's precalculus class was traditional, his understanding of periodic functions was unconventional. David engaged in sense making behaviors even though these behaviors were not encouraged or explicitly taught. A careful analysis of his work revealed that David's understandings of functions, function notation, and periodicity were compartmentalized. However, David was able to skirt compartmentalization through flexibility in problem solving, translation between representations, and transfer of mathematical information from one representation to another.     

Exploring Students' Conceptual Understanding of the Averaging Algorithm

Conceptual understanding of arithmetic average includes both an understanding of the computational algorithm and the statistical aspects of the concept. This study focused on the examination of 250 sixth-grade students' understanding of the arithmetic average by assessing their understanding of the computational algorithm. The results of the study showed that the majority of the students knew the “add-them-all-up-and-divide” averaging algorithm, but only about half of the students were able to correctly apply the algorithm to solve a contextualized average problem. Students were able to use various solution strategies and representations to solve the average problem. Those who used algebraic and arithmetic representations were better problem solvers than those who used pictorial and verbal representations. This study not only suggests that the average concept is more complex than the simplicity suggested by the computational algorithm, but also indicates the need for teaching the concept of average, both as a statistical idea for describing and making sense of data sets and as a computational algorithm for solving problems.     

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.