University of Maryland Physics Education Research Group


Literature Search of Student Understanding in Mathematics

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Author(s) Title Reference Abstract
Avioli, John A computer enhanced calculus I course Primus, vol. 3 no. 1, page 99, March 1993 Computer labs using Calculus T/L were introduced into Calculus I classes at Christopher Newport University. A description of the labs, the instructor's experiences in using the labs, and student reactions are detailed.

Contains instructions for Lab I involving inputting functions, finding difference quotients, and sketching a graph.
Barber, Fredrick Technology, cooperative learning, and assessment in the teaching of ordinary differential equations Primus, vol. 4 no. 4, pages 337-46, December 1994 CReports on the use of technology to enhance the teaching of ordinary differential equations, gives examples of laboratory activities using cooperative learning, and discusses assessment of student learning.

MacMath, TI-81 graphing calculators, and Maple were used in the course.
Bauman, Steve
Martin, Bill
Assessing the quantitative skills of college juniors College Mathematics Journal v26 n3 p214-20, May 95 Describes a quantitative assessment project for mathematics at the university level, its findings, and its impact.

Includes two roles for the mathematics department,departmental needs and student capabilities revealed by assessment, three levels of quantitative expectations, and patterns of student performance on quantitative tasks.
Baxter, Nancy H. Understanding how students acquire concepts underlying sets MAA Notes #33 Kaput and Dubinsky (Eds)  
Blanchard, Paul Teaching differential equations with a dynamical systems viewpoint College Mathematics Journal v25 n5 p385-93, Nov 94 Revisions involving the use of the computer in the differential equations course at Boston University are discussed. Some discussion is given about student performance after a traditional course.

Mainly the paper discusses the strengths of the new course and how the computer makes it possible for students to get more out of the course
Boyce, William New directions in elementary differential equations College Mathematics Journal v25 n5 p364-71, Nov 94 Describes a revised differential equations course at RPI where the computer is used as tool. Paper describes how the use of computers allows students to develop skills such as conceptualization, exploration, and higher level problem solving.

Although making some aspects of the course easier the course is ultimately more complex, more challenging, and more open ended. Students now have the opportunity to solve non-standard equations and also have a number of ways to present solutions.
Brown, Stephen Designing instructional materials: guesswork or facts? Distance Education v2 n1 p7-22, March 81 Describes an evaluation of the mathematical skills and learning capabilities of students immediately prior to taking Open University foundation courses in technology

Also discusses the design of questionnaires used in the surveys, and strategies for designing instructional materials for students with a weak preparation in mathematics and technology. (EAO)
Carlson, David The linear algebra curriculum study group recommendations for the first course in linear algebra College Mathematics Journal n1 p41-46, January 1993 Presents five recommendations of the Linear Algebra Curriculum Study Group: (1) The syllabus must respond to the client disciplines; (2) The first course should be matrix oriented; (3) Faculty should consider the needs and interests of students;
(4) Faculty should use technology; and (5) At least one follow-up course should be required. Provides a sample core syllabus. (MDH)
Carlson, David Teaching linear algebra: must the fog always roll in? College Mathematics Journal n1 p29-40, January 1993 Proposes methods to teach the more difficult concepts of linear algebra. Examines features of the Linear Algebra Curriculum Study Group Core Syllabus, and presents problems from the core syllabus.

These problems utilize the mathematical process skills of making conjectures, proving the results, and communicating the results to colleagues. Presents five strategies to help teach linear algebra.
Carpenter, Thomas Results from the First Mathematics Assessment of the National Assessment of Educational Progress NCTM 1978 A report from the National Council of Teachers of Mathematics Project for Interpretive Reports on National Assessment.

The report examines data from four age groups and evaluates performance in a number of content areas.
Cornez, Richard A computer based calculus curriculum College Teaching v41 n2 p47, Spring 93 A University of Redlands (California) calculus course was redesigned to include computer demonstrations and homework assignments in which the computer could play an important role.

Comparison of results of this instruction with that using traditional methods suggest the approach using computers has merit.
Cuoco, Albert A. Multiple representations for functions MAA Notes #33 Kaput and Dubinsky (Eds)  
Damarin, Suzanne Examining a model for courseware development. Journal of Computers in Mathematics and Science v6 n1 p38-43, Fall 86 Describes the approach used in a seminar on designing instructional software for problem solving in mathematics. Explains how a five stage software development model was followed for producing materials. Includes student evaluations of the course.
Davis, Paul Asking good questions about differential equations College Mathematics Journal v25 n5 p394-400, November 1994 Describes principles for constructing mathematics exercises that challenge students. Presents examples of good problems in differential equations.
Davis, R.B. Learning mathematics: the cognitive science approach to mathematics education (ED245928) 1984 (BOOK) There has long been dispute in mathematics between the drill and practice orientation that focuses primarily on memorizing mathematics as meaningless rote algorithms and the approach based on understanding and making creative use of mathematics.

The book is essentially concerned with providing a deeper understanding of the thought processes that are involved in mathematical thinking; what goes on inside children's heads as they learn mathematics and do mathematical problem-solving.
Douglas, R.G. Toward a lean and lively calculus MAA Notes #6  
Dreyfus, Tommy
Vinner, Shlomo
Images and definitions for the concept of a function Journal for Research in Mathematics Education v20 n4 p356-66, July 1989 Examines aspects of the images and definitions that college students and junior high school teachers have for the concept of function. Provides a seven-item questionnaire. Describes and categorizes the various images and definitions of the concept.
Dreyfus, Tommy
Halevi, Tirza
Quadfun--a case study of pupil computer interaction Journal of Computers in Mathematics and Science v10 n2 p43-48, Winter 1990-91 Discussed is the classroom suitability of an open learning computer microworld centered around families of quadratic equations.

This microworld provides a framework for exploring questions about parabolas to establish the link between the graphical and the algebraic representations of quadratic equations.
Dreyfus, Tommy
Eisenberg, Theodore
Intuitions on functions Journal of Experimental Education v52 n2 p77-85, Winter 1984 Discusses a concept-oriented calculus course as an alternative to a traditional skill-oriented calculus course. Two main ideas of conceptual calculus are visual conceptualization and receptive argumentation. Includes typical examination questions.
England, James W.
Eisenberg, Theodore
Calculus and linear algebra in APL American Mathematical Monthly 85 5 371-6 May 78 This paper is a report on experimental courses in calculus and linear algebra which have been given for the past three years at Swarthmore College.

Computing is used to represent the ideas and concepts of calculus and linear algebra and to facilitate their analysis and exploration. APL is used as the notation.
Fahidy, Thomas Z. A second-year undergraduate course in applied differential equations Chemical Engineering Education v25 n2 p88-91, Spring 1991 Presents the framework for a chemical engineering course using ordinary differential equations to solve problems with the underlying strategy of concisely discussing the theory behind each solution technique without extensions to formal proofs.

Includes typical class illustrations, student responses to this strategy, and reaction of the instructor.
Ferren,Ann Reforming college mathematics College Teaching v40 n3 p87-90, Summer 92 In the process of strengthening undergraduate mathematics, educators at the American University (District of Columbia) surveyed similar institutions concerning mathematics placement, course requirements, and perceptions of student attitudes.

They found common concerns but no obvious solutions. It is concluded that curricular change must be accompanied by comprehensive policies and instructional change.
Ferrini-Mundy, Joan An overview of the calculus curriculum reform effort: issues for learning, teaching, and curriculum development American Mathematical Monthly v98 n7 p627, August 91 Following a short history of calculus reform efforts, this article discusses curriculum development and research on student learning with respect to the concepts of function, limits and continuity, the derivative, and the integral.

The paper also discusses the availability and influence of new technologies on curriculum development; the consideration of the role of the teacher; and research directions.
Ferrini-Mundy, Joan
Graham, Karen
Research in calculus learning: understanding of limits, derivatives, and integrals MAA Notes #33 Kaput and Dubinsky (Eds)  
Ferrini-Mundy, Joan Secondary school calculus: preparation or pitfall in the study of college calc Journal for Research in Mathematics Education v23 n1 p56, January 92 This study investigated the effects of various levels of secondary school calculus experience on performance in first-year college calculus, with focus on student performance on conceptual and procedural exam items.

Students who had a year of secondary school calculus differed significantly in performance from those who had either no experience or only a brief introduction.
Fey, J. T. Technology and mathematics education: a survey of recent developments and important problems Educational Studies in Mathematics 20 (3) p237-267  
Hart, Eric W. A conceptual analysis of the proof-writing performance of expert and novice students in elementary group theory MAA Notes #33 Kaput and Dubinsky (Eds)  
Harel, Guershon Learning and teaching linear algebra: difficulties and an alternative approach to visualizing concepts and processes Focus on Learning Problems in Mathematics v11 n1-2, p139-141, Winter-Spring 89 Describes learning difficulties students may have with the basic notions of linear algebra and three phases of abstraction. Discusses results of a program based on the abstraction process.
Harel, Guershon Variations in linear algebra content presentations For the Learning of Mathematics -- an international journal of mathematics education v7 n3 p29-32, November 1987  
Heid, Kathleen New directions for mathematics instruction. 1989 yearbook (ED309989) 1989 (MICRO)  
Heid, Kathleen Resequencing skills and concepts in applied calculus using the computer as a tool Journal for Research in Mathematics Education v19 n1 p3, January 88 During the first 12 weeks of an applied calculus course, two classes of college students studied calculus concepts using graphical and symbol-manipulation computer programs to perform routine manipulations.

Three weeks were spent on skill development. Students showed better understanding of concepts and performed almost as well on routine skills.
Henderson, Ronald and Landesman A preliminary evaluation of student preparation for the study of calculus (ED282925) 1986 (BOOK)  
Kaput, J Multaplicative word problems and intensive quantities: an integrated software response (ED295787) 1985 (MICRO) Reports on a project that is examining some of the difficulties encountered in teaching word problems involving multiplication, division, and intensive quantities. Some of the various uses of these operations and their structures are considered.

Described are discoveries and assumptions regarding students' cognitive models of these operations, especially as they pertain to intensive quantities. The report also describes the project's computer software.
Kaput, J Technology in mathematics education research: the first 25 years in the JRME Journal for Research in Mathematics Education v25n6 p677, December 94 Discusses the interactions between technology and research in mathematics education, including computer-assisted instruction, role of technology in learning, use of computer environments to study students' mathematical concept formation.

research methodologies, interactive technologies, and lack of technology-related research in the "Journal for Research in Mathematics Education."
Karplus, Robert Continuous functions: Students' Viewpoints European Journal of Science Education v1 n4 p397-415, October-December 1979 Investigates how the concept of a mathematical function is applied in science. The procedures by which 400 students ranging in age from 11 to 18 years deal spontaneously with data pairs that describe a continuous functional relationship are identified.
Lackner, Lois M. A pilot study on teaching the derivative concept in beginning calculus by inductive and deductive approaches School Science and Mathematics 71 6 p563-567  
Lefton, Lew E.
Steinbart, Enid
Calculus & mathematica: an end-user's point of view Primus v5 n1 p80-96, March 95  
Mathews, John H. Solving differential equations with computer algebra systems software Journal of Computers in Mathematics and Science v10 n2 p79-86, Winter 90-91  
Monk, G. S. Students' understanding of functions in calculus courses Humanistic Mathematics Network newsletter, No. 2, 1988  
NA Arizona essential skills for mathematics. reformatted edition (ED367531) 1992 (BOOK)  
NA Proceedings: summer conference for college teachers on applied mathematics, university of missouri-rolla, 1971 (ED180766) 1971 (BOOK)  
Nicolai, Micheal B. A discovery in linear algebra Mathematics Teacher 67(5)  
Norman, Alexander F.
Mary K. Prichard
Cognitive obstacles to the learning of calculus: a kruketskiian perspective MAA Notes #33 Kaput and Dubinsky (Eds)  
Orton, A. Students' understanding of integration Educational Studies in Mathematics 14 (1983b) pp. 1-18  A clinical interviewing method was used to investigate students' understanding of elementary calculus. The analysis of responses to tasks concerned with integration and limits led to detailed data concerning the degree of understanding attained.

Some conclusions were drawn concerning the teaching of integration and limits.
Orton, A. Students' understanding of differentiation Educational Studies in Mathematics 14 (1983a) pp. 235-250 A clinical interviewing method was used to investigate students' understanding of elementary calculus. The analysis of responses to tasks concerned with differentiation and rate of change led to data concerning the degree of understanding attained.

Some conclusions were drawn concerning the teaching of differentiation and rate of change.
Ratay, Gabriella M. Student performance with calculus reform at the united states merchant marine academy Primus v3 n1 p107-111, March 93  
Raymond, Anne M. Assessment in mathematics education: what are some of the alternatives in alternative assessment Contemporary Education v66 n1 p13, Fall 94 Four means of assessing student learning that are used more and more frequently in mathematics classrooms include group and individual problem-solving exams, group projects, written reflections, and self-assessment.

The article describes the use of the four elements in one mathematics content course for elementary education majors.
Rosamond, Frances A. The role of emotion: expert and novice mathematical problem-solving MAA Notes #33 Kaput and Dubinsky (Eds)  
Selden, John et al. Even good calculus student's can't solve nonroutine problems MAA Notes #33 Kaput and Dubinsky (Eds)  
Schurle, Arlo W. Does writing help students learn about differential equations? Primus v1 n2 p129-36 Jun1991 Reports on the differences in two sections of a college course on differential equations. Results on a common examination showed that writing assignments substituted for traditional homework assignments did not improve test performance.

However, survey results indicated that students felt the writing assignments, in lieu of additional homework, improved their comprehension.
Silver, Edward A.
Randall I. Charles (eds)
The Teaching and Assessing of Mathematical Problem Solving Research Agenda for Mathematics Education Series. Volume 3 (1988) This document is the product of one of four NCTM Research Agenda Project conferences held during 1987
Steen, L.A. Forces for change in the mathematics curriculum Research Agenda for Mathematics Education Series. Volume 3 (1988) The school mathematics curriculum: Raising national expectations. UCLA, 1986
Struik, Ruth Rebekka
Roberta Flexer
Self-paced calculus: a preliminary evaluation American Mathematical Monthly 84 2 p129-134  
Thompson, A.G
P.W. Thompson
Talking about rates conceptually, part II: mathematical knowledge for teaching Journal for Research in Mathematics Education v27 n1 p2-24  
Thompson, W.P. Experience, problem solving, and learning mathematics: considerations in developing mathematics curricula teaching and learning mathematical problem solving (189-236), Hillsdale, NJ lawrence erlbaum associates, 1985  
Tucker, Alan The growing importance of linear algebra in undergraduate mathematics College Mathematics Journal v24 n1 p3-9 Jan 1993 Discusses the theoretical and practical importance of linear algebra. Presents a brief history of linear algebra and matrix theory and describes the place of linear algebra in the undergraduate curriculum.
Wang, Tse-Wei A course on applied linear algebra Chemical Engineering Education v23 n4 p236-41 Fall89 Abstract: Provides an overview of a course, "Applied Linear Algebra," for teaching the concepts and the physical and geometric interpretations of some linear algebra topics.

Describes the philosophy of the course, the computer project assignments, and student feedback. Major topics of the course are listed.
White, Paul
M. Mitchelmore
Conceptual knowledge in introductory calculus Journal for Research in Mathematics Education v27 n1 p79-95 Responses to word problems involving rates of change were collected. The number of students who could symbolize rates of change in complex and non complex situations is discussed.

Analysis shows three main categories of error all in which variables are treated as symbols to be manipulated rather than as quantities to be related. The abstract-apart and abstract-general concept of a variable is also discussed.
Zwick, Rebecca Assessment of differential item functioning for performance tasks Journal of Educational Measurement v30 n3 p233-51 Fall93  

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