Math Forum Discussions

Gregory D. Foley

Posts: 8
Registered: 12/8/04

Variable reply
Posted: Oct 14, 1996 5:26 PM

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Joanna O. Masingila
Assistant Professor
Syracuse University

Joana and group,

Below is a function and graphs bibliography that I put together some time
ago, with a few recent additions.

Greg

Gregory D. Foley, PhD
Associate Professor of Mathematics
Director of the South Central Calculus Coalition

Department of Mathematical and Information Sciences
Sam Houston State University
Huntsville, Texas 77341-2206 USA

Email address: mth_gdf@shsu.edu

Barclay, W. H. (1985). Graphing misconceptions and possible remedies
using microcomputer-based labs. Technical Report 85-5. Cambridge, MA:
Technical Research Centers.
Barr, G. (1980). Graphs, gradients and intercepts. Mathematics in School,
9(1), 5-6.
Bell, A., & Janvier, C. (1981). The interpretation of graphs representing
situations. For the Learning of Mathematics, 2(1), 34-42.
Bergeron, J. C., & Herscovics, N. (1982). Levels in the understanding of
the function concept. In G. Van Barnveld & H. Krabbendam (Eds.),
Conference on functions (Report 1, pp. 39-46), Enschede, The Netherlands:
=46oundation for Curriculum Development.
Bestgen, J. B. (1980). Making and interpreting graphs and tables: Results
and implications from national assessment. Arithmetic Teacher, 28(4),
26-29.
Brasell, H., & Rowe, M. (1989). Graphing skills among high-school physics
students. Paper presented at the annual meeting of the American
Educational Research Association, San Francisco.
Browning, C. A. (1989). Characterizing levels of understanding of
functions and their graphs (Doctoral dissertation, The Ohio State
University, 1988). Dissertation Abstracts International, 49, 2957A.
Browning, C. A. (1990). The computer/calculator precalculus (C2PC)
project and levels of graphical understanding. In F. Demana, B. K. Waits,
& J. G. Harvey (Eds.), Proceedings of the Conference on Technology in
Collegiate Mathematics (pp. 114-117). Reading, MA: Addison-Wesley.
Buck, R. C. (1970). Functions. In E. G. Begle & H. G. Richey (Eds.),
Mathematics education, the 69th yearbook of the National Society for the
Study of Education (pp. 236-259). Chicago: University of Chicago.
Buck, R. C. (1988). Computers and calculus: The second stage. In L. A.
Steen (Ed.), Calculus for a new century: A pump, not a filter [MAA Notes
No. 8] (pp. 168-172). Washington, DC: Mathematical Association of America.
Clement, J. (1985). Misconceptions in graphing. In. L. Streefland (Ed.),
Proceedings of the ninth international conference of the International
Group for the Psychology of Mathematics Education (Vol. 1, pp. 369-375).
Utrecht, The Netherlands: IGPME.
Clement, J. (1989). The concept of variation and misconceptions in
Cartesian graphing. Focus on Learning Problems in Mathematics, II(1-2),
77-87.
Clement, J. (?). Adolescents' graphing skills: A descriptive analysis.
(ERIC Document Reproduction Service No. ED 264 127) [An in-depth
examination of middle school children's graphing skills and
misconceptions.]
Cleveland, W. S., & McGill, R. (1985). Graphical perception and graphical
methods for analyzing scientific data. Science, 229, 828-833.
Cleveland, W. S., McGill, M. E., & McGill, R. (1988). The shape parameter
of a two-variable graph. Journal of the American Statistical Association,
83, 289-300. [Shape parameter (ratio of the horizontal and vertical
distances spanned by data) and its effect on visual perception are studied.
A method is suggested for choosing shape that maximizes accuracy of
judgement.]
Curcio, F. R. (1987). Comprehension of mathematical relationships
expressed in graphs. Journal for Research in Mathematics Education, 18,
382-393. [Studies show children should be involved in graphing activities
to build the schemata needed for mathematical comprehension.]
Davidson, P. M. (in press). Early function concepts: Their development and
relation to certain mathematical and logical abilities, Child Development.
Davis, R. B. (1982). Teaching the concept of function: Method and reasons.
In G. Van Barnveld & H. Krabbendam (Eds.), Conference on functions (Report
1, pp. 47-55). Enschede. The Netherlands: Foundation for Curriculum
Development.
Demana, F., & Waits, B. K. (1987). Problem solving using microcomputers.
College Mathematics Journal, 18, 236-241.
Demana, F., & Waits, B. K. (1988). Pitfalls in graphical computation, or
why a single graph isn't enough. College Mathematics Journal, 19, 177-183.
Demana, F., & Waits, B. K. (1988). The Ohio State University Calculator
and Computer Precalculus Project: The mathematics of tomorrow today! The
AMATYC Review,10(1), 46-55. [The article describes the Calculator and
Computer Precalculus (C2PC) Project.]
Demana, F., & Waits, B. K. (1990). Implementing the Standards: The role
of technology in teaching mathematics. Mathematics Teacher, 83, 27-31.
[Explanations are given to support the use of technology. Examples are
provided to emphasize points.]
Demana, F., Waits, B. K. (Primary Authors), Osborne, A., & Foley, G. D.
(1990). Precalculus mathematics, a graphing approach. Reading, MA:
Addison-Wesley.
Dickey, E., & Kherlopian, R. (1987). A survey of teachers of mathematics,
science, and computers on the use of computers in grades 5-9 classrooms.
Educational Technology, 6, 10-14. [The article reports a survey of 167
teachers in South Carolina regarding the extent and type of usage of
computers in math, science, and computer class.]
Dreyfus, T., & Eisenberg, T. (1982). Intuitive functional concepts: A
baseline study on intuitions. Journal for Research in Mathematics
Education, 13, 360-380. [A study was conducted and conclusions drawn
regarding intuitions of functional relationships.]
Dreyfus, T., & Eisenberg, T. (1983). Intuitions on functions. Journal of
Experimental Education, 52, 77-85. [The article describes a study used to
assess the pre-instructional intuitions students have about functions.]
Dreyfus, T., & Eisenberg, T. (1983). The function concept in college
students: Linearity, smoothness and periodicity. Focus on Learning
Problems in Mathematics, 5, 119-132.
Dreyfus, T., & Eisenberg, T. (1987). On the deep structure of functions.
Proceedings of the International Conference on the Psychology of
Mathematics Education, XI, 190-196. [A study of the software, Green Globs,
shows that there is no significant difference in achievement between those
who had a structured learning environment and those who did not. Some
success in the understanding of transformations was observed.]
Dugdale, S. (1982). Green globs: A microcomputer application for graphing
of equations. Mathematics Teacher, 75, 208-214.
Dunham, P. H., & Osborne, A. (1991). Learning how to see: Students
graphing difficulties. Focus on Learning Problems in Mathematics, 13(4),
35-49.
Even, R. D. (1989). Prospective secondary mathematics teachers' knowledge
and understanding about mathematical functions (Doctoral dissertation,
Michigan State University). Dissertation Abstracts International, 50,
642A. [Pre-service mathematics teachers in last stages of education lacked
appropriate understanding of many aspects of functions.]
Even, R., & Ball, D. (1989, March). How do prospective secondary teachers
think about concepts and procedures related to mathematical functions?
Paper presented at the annual meeting of the American Educational Research
Association, San Francisco.
=46oley, G. D. (1990). Accessing advanced concepts with technology. The
AMATYC Review, 11(2), 58-64.
=46oley, G. D. (1990). Using hand-held graphing computers in college
mathematics. In F. Demana, B. K. Waits, & J. G. Harvey (Eds.), Proceedings
of the Conference on Technology in Collegiate Mathematics (pp. 28-39).
Reading, MA: Addison-Wesley.
=46reudenthal, H. (1982). Variables and functions. In G. Van Barnveld & H.
Krabbendam (Eds.), Conference on functions (Report 1, pp. 7-20). Enschede,
The Netherlands: Foundation for Curriculum Development.
Goldberg, F. M., & Anderson, J. H. (1989). Student difficulties with
graphical representations of negative values of velocity. The Physics
Teacher, 4, 254-260.
Goldenberg, E. P. (1987). Believing is seeing: how preconceptions
influence the perceptions of graphs. Proceedings of the International
Conference on the Psychology of Mathematics Education, XI. [Perceptual
illusions and shifts of attention from one feature to another obscure some
of what educational use of graphs is supposed to elucidate.]
Goldenberg, E. P., Harvey, W., Lewis, P. G., Umiker, R. J., West, J., &
Zodhiates, P. (1988). Mathematical, technical, and pedagogical challenges
in the graphical representation of functions (Tech. Rep. No. 88-4).
Cambridge, MA: Harvard Graduate School of Education, Educational Technology
Center. [Chapter II was reprinted as Goldenberg, E. P. (1988).
Mathematics, metaphors, and human factors. The Journal of Mathematical
Behavior, 7, 135-173.]
Goldenberg, E.P. (1987). Believing is seeing: How preconceptions
influence the perception of graphs. Proceedings of the eleventh
international conference of the International Group for the Psychology of
Mathematics Education (Vol. 1, pp. 197-203). Montreal: IGPME.
Goldenberg, P. E. & Kliman, M. Metaphors for understanding graphs: What
you see is what you see. (TR88-22). Cambridge: Educational Technology
Center, Harvard Graduate School of Education. [Metaphors are used to
describe how students apply knowledge and experience to the graphing of
functions.]
Goldenberg, P. E. (1988). Mathematics, metaphors, and human factors:
Mathematical, technical, and pedagogical challenges in the educational use
of graphical representation of functions. Journal of Mathematical
Behavior, 7, 135-173. [Using the Function Analyzer, differences between
student perceptions of graphed functions and the perceptions of
knowledgeable adults are discussed. Conclusions suggest that careful, well
thought-out use of graphing software must be employed.]
Harvey, J. G. (1990). Changes in pedagogy and testing when using
technologies in college-level mathematics courses. In F. Demana, B. K.
Waits, & J. G. Harvey (Eds.), Proceedings of the Conference on Technology
in Collegiate Mathematics (pp. 40-54). Reading, MA: Addison-Wesley.
Heid, M. K., Sheets, C., Matras, M. A., & Menasian, J. (1988, April).
Classroom and computer lab interaction in a computer-intensive algebra
curriculum. Paper presented at the annual meeting of the American
Educational Research Association, New Orleans.
Hight, D. W. 91968). Functions: Dependent variables to fickle pickers.
Mathematics Teacher, 61, 575-579.
Janvier, C. (1978). The interpretation of complex cartesian graph
representing situations -- Studies and teaching experiences. Doctoral
dissertation. University of Nottingham.
Janvier, C. (1980). Reading cartesian graphs for understanding. Short
communication presented at ICME IV. International Congress on Mathematical
Education, Berkeley, CA.
Janvier, C. (1981a). Difficulties related to the concept of variables
presented graphically. In C. Comiti (Ed.) Proceedings of the fifth
international conference of the International Group of Psychology of
Mathematics Education (pp. 189-193). Grenoble, France: IGPME.
Janvier, C. (1982). Approaches to the notion of function in relation to
set theory. In G. Van Barnveld & H. Krabbendam (Eds.). Conference on
functions. =B7Report 1, pp. 114-124). Enschede. The Netherlands:
=46oundation for Curriculum Development.
Janvier, C. (1983). Teaching the concept of function. Mathematical
Education for Teaching. 4(2), 48-60.
Janvier, C. (1987). Representation and understanding: The notion of
function as an example. In C. Janvier (Ed.), Problems of representation in
the teaching and learning of mathematics (pp. 67-71). Hillsdale, NJ:
Erlbaum.
Kaput, J. J. (1986). Information technology and mathematics: Opening new
representational windows (Tech. Rep. No. 86-3). Cambridge, MA: Harvard
Graduate School of Education, Educational Technology Center.
Kaput, J. J. (1986). Information technology and mathematics: Opening new
representational windows. Cambridge: Educational Technology Center,
Harvard Graduate School of Education. [Descriptions of the Geometric
Supposer, an algebra/graphing software, and a ratio-proportion software are
given. Research and pedagogical issues involved in implementing such
software into the curriculum are also discussed.]
Kaput, J. J. (1988). Looking back from the future: A history of computers
in mathematics education, 1978-1998. Cambridge: Educational Technology
Center, Harvard Graduate School of Education. [The article summarizes what
has happened in the past decade regarding education and technology, then
provides an utopian vision of technology for the next decade.]
Karplus, R., (1979). Continuous functions: Students' viewpoints.
European Journal of Science Education.. 1(4), 397-413
Kent, G. S. (1987). The effect of computer graphics on achievement in the
teaching of functions in college algebra. (Doctoral dissertation,
University of South Florida). Dissertation Abstracts International, 48,
587A. [A positive effect on achievement is suggested in the teaching of
functions and graphs when the microcomputer is used as a student-teacher
utility and lessons are given to enhance understanding.]
Kerslake, D. (1977). The understanding of graphs. Mathematics in
Schools, 6(2) 56-63. [A test of graphing abilities reveals students'
difficulties in producing and interpreting graphs.]
Kerslake, D. (1981). Graphs. In K. M. Hart (Ed.), Children's
understanding of mathematics concepts: 11-16 (pp. 120-136). London: John
Murray.
Krabbendam, H. (1982). The non-qualitative way of describing relations and
the role of graphs: Some experiments. In G. Van Barnveld & H. Krabbendam
(Eds.), Conference on functions (Report 1, pp. 125-146). Enschede. The
Netherlands: Foundation for Curriculum Development.
Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs,
and graphing: Tasks, learning, and teaching. Review of Educational
Research, 60, 1-64.
Linn, M. C., Layman, J. W., & Nachmias, R. (1987). Cognitive consequences
of microcomputer-based laboratories: Graphing skills development.
Contemporary Educational Psychology, 12, 244-253. [Instruction based on a
chain of cognitive accomplishments with use of microcomputer-based
laboratories increase learning and enable researchers to predict graphing
performance.]
Lovell, K. (1971). Some aspects of growth of the concept of a function.
In M. F. Rosskopf, L. P. Steffe, & S. Taback (Eds.). Piagetian cognitive
development research and mathematical education (pp. 12-33). Washington,
DC: National Conference of Teachers of Mathematics.
Malik, M. A. (1980). Historical and pedagogical aspects of the definition
of function. International Journal of Mathematics Education in Science and
Technology: 11(4), 489-492.
Mansfield, H. (1985). Points, lines, and their representations. For the
Learning of Mathematics, 5(3), 2-6.
Markovits, A., Eylon, B., & Bruckheimer, M. (1983). Functions: Linearity
unconstrained. In R. Hershkowitz (Ed.), Proceedings of the seventh
international conference of the International Group for the Psychology of
Mathematics Education (pp. 271-277). Rehovot, Israel: Weizmann Institute
of Science.
Markovits, Z., Eylong, B., & Bruckheimer, M. (1986). Functions today and
yesterday. For the Learning of Mathematics, 6(2), 18-28.
Marnyanskii, I. A. (1975). Psychological characteristics of pupils'
assimilation of the concept of a function. In J. Kilpatrick, I. Wirszup,
E. Begle, & J. Wilson (Eds.), Soviet studies in the psychology of learning
and teaching mathematics (Vol. 13, pp. 163-172). Chicago: University of
Chicago Press.
McDermott, L., Rosenquist, M., & vanZee, E. (1987). Student difficulties
in connecting graphs and physics: Example from kinematics. American
Journal of Physics, 55(6), 503-513.
McKenzie, D. L., & Padilla, M. J. (1984, April). Effects of laboratory
activities and written simulations on the acquisition of graphing skills by
eighth grade students. Paper presented at the meeting of the National
Association for Research in Science Teaching. New Orleans, LA.
McKenzie, D. L., & Padilla, M. J. (1986). The construction and validation
of the Test of Graphing in Science (TOGS). Journal of Research in Science
Teaching, 23(7), 571-579.
Mokros, J. R., & Tinker, R. F. (1986). The impact of microcomputer-based
labs on children's ability to interpret graphs. Unpublished manuscript,
Technical Education Research Centers and Harvard University, Cambridge, MA.
Mokros, J. R., & Tinker, R. F. (1987). The impact of microcomputer-based
labs on childrens ability to interpret graphs. Journal for Research in
Science Teaching, 24(4), 369-383.
Monk, G. S. (1987). Students' understanding of functions in calculus
courses. Unpublished manuscript, University of Washington, Seattle.
Monk, G. S. 91989, March). A framework for describing student
understanding of function. Paper presented at the annual meeting of the
American Educational Research Association, San Francisco.
Osborne, A., Demana, F., Waits, B. K., & Foley, G. D. (1989, August).
Annual report-Mathematics through technology: Establishing concepts and
skills of graphing and functions in grades 9 through 12. Presented to the
National Science Foundation concerning Project No. TPE-8751353.
Padilla, M. J., McKenzie, D. L., & Shaw, E. L. (1986). An examination of
the line graphing ability of students in grades seven through twelve.
School Science and Mathematics, 86(1), 20-25. [Skills needed to draw and
interpret graphs are identified. A test of graphing ability was
administered to 625 students. Students scored well on "lower order"
questions, but poorly on "higher order" graphing tasks.]
Ponte, J. P. M. da. (1984). Functional reasoning and the interpretation
of Cartesian graphs. Dissertation Abstracts International, 45, 1675A.
Preece, J. (1983a). Graphs are not straightforward. In T. R. G. Green &
S. J. Payne (Eds.), The psychology of computer use: A European perspective
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Preece, J. (1983b). A survey of graph interpretation and sketching errors.
(CAL Research Group Tech. Rep. No. 34). Milton Keynes, England: Institute
of Educational Technology, The Open University, Walton Hall.
Reed, S. K., & Evans, A. C. 91987). Learning functional relations: A
theoretical and instructional analysis. Journal of Experimental
Psychology: General, 116(2), 1-13.
Schoenfeld, A. H. (1988). Grapher: A case study in educational technology,
research, and development. Unpublished manuscript, University of
California, Berkeley, CA.
Schultz, K., Clement, J., & Mokros, J. (1986). Adolescent graphing skills:
A descriptive analysis. Paper presented at the annual meeting of the
American Educational Research Association, San Francisco.
Schwartz, J. L. (1988). Intensive quantity and referent transforming
arithmetic operations. In J. Hiebert & M. Behr (Eds.), Number concepts and
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of Teachers of Mathematics, and Hillsdale, NJ: Erlbaum .
Schwartz, J. L. The representation of function in the algebraic proposer.
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Mathematics Education, XI, 235-241. [This article illustrates a variety of
functional representations employed by The Algebraic Proposer.]
Shaw, J. M. 91984). Making graphs. Arithmetic Teacher, 31(5), 7-11.
Silberstein, E. P. (1986). Graphically speaking . . . . The Science
Teacher, 53(5), 41-45. [Silberstein explains why students have
difficulties producing and interpreting graphs of scientific data.]
Stein, M. K., 7 Leinhardt, G. (1989). Interpreting graphs: An analysis of
early performance and reasoning. Unpublished manuscript, University of
Pittsburgh, Learning Research and Development Center, PA.
Stein, M. K., Baxter, J., & Leinhardt, G. (in press). Subject matter
knowledge and elementary instruction: A case from functions and graphing.
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Swan, M. (1980). The language of graphs. Nottingham, England: University
of Nottingham, Shell Centre for Mathematical Education.
Swan, M. (1982). The teaching of functions and graphs. In. G. Van
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Tall, D. (1985). Understanding the calculus. Mathematics Teaching, 110,
49-53. [Computer graphics can be employed to approach many calculus
concepts to promote better understanding.]
Tall, D. (1986). A graphical approach to integration. Mathematics
Teaching, 114, 48-51. [A graphical approach to calculus provides insights
to the beginning calculus students as well as lays a foundation for more
powerful insights into formal mathematical analysis.]
Tall, D. (1987). Whither calculus. Mathematics Teaching, 118, 50-54.
[Tall discusses how software may be incorporated into calculus curriculum.]
Thomas, H. L. (1975). The concept of function. In M. Rosskopf (Ed.),
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of California.
Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept
of function. Journal for Research in Mathematics Education, 20, 356-366.
Vinner, S., (1983). Concept definition, concept image and the notion of
function. International Journal of Mathematics Education in Science and
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Vonder Embse, C. B. (1987). An eye fixation study of time factors
comparing experts and novices when reading and interpreting mathematical
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Vonder Embse, C. B. (1990). Research on reading and interpreting computer
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& J. G. Harvey (Eds.), Proceedings of the Conference on Technology in
Collegiate Mathematics (pp. 278-281). Reading, MA: Addison-Wesley.
Wagner, S. (1981). Conservation of equation and function under
transformations of variable. Journal for Research in Mathematics
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Waits, B. K., & Demana, F. (1988). Microcomputer graphing: A microscope
for the mathematics student. School Science and Mathematics, 88(3),
218-224. [The article describes a Microsoft BASIC graphing program
designed for the Apple II computer and gives examples illustrating
potential uses of this program.]
Wavering, M. J. (1985, April). The logical reasoning necessary to make
line graphs. Paper presented t the annual meeting of the National
Association for Research in Science Teaching, French Lick Springs, Indiana.
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outdated)]
Woodward, E., & Byrd, F. (1984). Make up a story to explain the graph.
Mathematics Teacher, 77, 32-34. [Students exhibit enthusiasm and graphical
understanding when given an opportunity to make up a story to explain the
given graphs.]
Yerushalmy, M. (1989). The use of graphs as visual interactive feedback
while carrying out algebraic transformations. Actes de la 13e conference
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Zaslavsky, O. (1987). Conceptual obstacles in the learning of quadratic
functions. Unpublished doctoral dissertation, Technion, Haifa, Israel.
Zaslavsky, O. (1990, April). Conceptual obstacles in the learning of
quadratic functions. Paper presented at the annual meeting of the American
Educational Research Association, Boston.
Zehavi, N., Gonen, R., & Taizi, N. (1987). The effects of microcomputer
software on the intuitive understanding of graphs of quantitative
relationships. Proceedings of the International Conference on Psychology
of Mathematics Education, XI, 255-261. [Using a specially designed
software package (Dots and Rules), an experiment is conducted to determine
the effectiveness of preparation of students to study linear equations.
Results indicate such software is moderately effective.]
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