Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.



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


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 773412206 USA
Email address: mth_gdf@shsu.edu
Barclay, W. H. (1985). Graphing misconceptions and possible remedies using microcomputerbased labs. Technical Report 855. Cambridge, MA: Technical Research Centers. Barr, G. (1980). Graphs, gradients and intercepts. Mathematics in School, 9(1), 56. Bell, A., & Janvier, C. (1981). The interpretation of graphs representing situations. For the Learning of Mathematics, 2(1), 3442. 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. 3946), 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), 2629. Brasell, H., & Rowe, M. (1989). Graphing skills among highschool 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. 114117). Reading, MA: AddisonWesley. 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. 236259). 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. 168172). 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. 369375). Utrecht, The Netherlands: IGPME. Clement, J. (1989). The concept of variation and misconceptions in Cartesian graphing. Focus on Learning Problems in Mathematics, II(12), 7787. Clement, J. (?). Adolescents' graphing skills: A descriptive analysis. (ERIC Document Reproduction Service No. ED 264 127) [An indepth 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, 828833. Cleveland, W. S., McGill, M. E., & McGill, R. (1988). The shape parameter of a twovariable graph. Journal of the American Statistical Association, 83, 289300. [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, 382393. [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. 4755). Enschede. The Netherlands: Foundation for Curriculum Development. Demana, F., & Waits, B. K. (1987). Problem solving using microcomputers. College Mathematics Journal, 18, 236241. Demana, F., & Waits, B. K. (1988). Pitfalls in graphical computation, or why a single graph isn't enough. College Mathematics Journal, 19, 177183. 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), 4655. [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, 2731. [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: AddisonWesley. Dickey, E., & Kherlopian, R. (1987). A survey of teachers of mathematics, science, and computers on the use of computers in grades 59 classrooms. Educational Technology, 6, 1014. [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, 360380. [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, 7785. [The article describes a study used to assess the preinstructional 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, 119132. Dreyfus, T., & Eisenberg, T. (1987). On the deep structure of functions. Proceedings of the International Conference on the Psychology of Mathematics Education, XI, 190196. [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, 208214. Dunham, P. H., & Osborne, A. (1991). Learning how to see: Students graphing difficulties. Focus on Learning Problems in Mathematics, 13(4), 3549. Even, R. D. (1989). Prospective secondary mathematics teachers' knowledge and understanding about mathematical functions (Doctoral dissertation, Michigan State University). Dissertation Abstracts International, 50, 642A. [Preservice 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), 5864. =46oley, G. D. (1990). Using handheld 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. 2839). Reading, MA: AddisonWesley. =46reudenthal, H. (1982). Variables and functions. In G. Van Barnveld & H. Krabbendam (Eds.), Conference on functions (Report 1, pp. 720). 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, 254260. 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. 884). 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, 135173.] 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. 197203). Montreal: IGPME. Goldenberg, P. E. & Kliman, M. Metaphors for understanding graphs: What you see is what you see. (TR8822). 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, 135173. [Using the Function Analyzer, differences between student perceptions of graphed functions and the perceptions of knowledgeable adults are discussed. Conclusions suggest that careful, well thoughtout use of graphing software must be employed.] Harvey, J. G. (1990). Changes in pedagogy and testing when using technologies in collegelevel mathematics courses. In F. Demana, B. K. Waits, & J. G. Harvey (Eds.), Proceedings of the Conference on Technology in Collegiate Mathematics (pp. 4054). Reading, MA: AddisonWesley. Heid, M. K., Sheets, C., Matras, M. A., & Menasian, J. (1988, April). Classroom and computer lab interaction in a computerintensive 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, 575579. 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. 189193). 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. 114124). Enschede. The Netherlands: =46oundation for Curriculum Development. Janvier, C. (1983). Teaching the concept of function. Mathematical Education for Teaching. 4(2), 4860. 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. 6771). Hillsdale, NJ: Erlbaum. Kaput, J. J. (1986). Information technology and mathematics: Opening new representational windows (Tech. Rep. No. 863). 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 ratioproportion 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, 19781998. 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), 397413 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 studentteacher utility and lessons are given to enhance understanding.] Kerslake, D. (1977). The understanding of graphs. Mathematics in Schools, 6(2) 5663. [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: 1116 (pp. 120136). London: John Murray. Krabbendam, H. (1982). The nonqualitative way of describing relations and the role of graphs: Some experiments. In G. Van Barnveld & H. Krabbendam (Eds.), Conference on functions (Report 1, pp. 125146). 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, 164. Linn, M. C., Layman, J. W., & Nachmias, R. (1987). Cognitive consequences of microcomputerbased laboratories: Graphing skills development. Contemporary Educational Psychology, 12, 244253. [Instruction based on a chain of cognitive accomplishments with use of microcomputerbased 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. 1233). 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), 489492. Mansfield, H. (1985). Points, lines, and their representations. For the Learning of Mathematics, 5(3), 26. 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. 271277). Rehovot, Israel: Weizmann Institute of Science. Markovits, Z., Eylong, B., & Bruckheimer, M. (1986). Functions today and yesterday. For the Learning of Mathematics, 6(2), 1828. 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. 163172). 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), 503513. 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), 571579. Mokros, J. R., & Tinker, R. F. (1986). The impact of microcomputerbased 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 microcomputerbased labs on childrens ability to interpret graphs. Journal for Research in Science Teaching, 24(4), 369383. 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 reportMathematics through technology: Establishing concepts and skills of graphing and functions in grades 9 through 12. Presented to the National Science Foundation concerning Project No. TPE8751353. 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), 2025. [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 (pp. 4156). London: Academic Press. 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), 113. 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 operations in the middle grades (pp. 4152). Reston, VA: National Council of Teachers of Mathematics, and Hillsdale, NJ: Erlbaum . Schwartz, J. L. The representation of function in the algebraic proposer. Proceedings of the International Conference on the Psychology of Mathematics Education, XI, 235241. [This article illustrates a variety of functional representations employed by The Algebraic Proposer.] Shaw, J. M. 91984). Making graphs. Arithmetic Teacher, 31(5), 711. Silberstein, E. P. (1986). Graphically speaking . . . . The Science Teacher, 53(5), 4145. [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. American Educational Research Journal. 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 Barnveld & H. Krabbendam (Eds.), Conference on functions (Report 1, pp. 151164). Enschede, The Netherlands: Foundation for Curriculum Development. Tall, D. (1985). Understanding the calculus. Mathematics Teaching, 110, 4953. [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, 4851. [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, 5054. [Tall discusses how software may be incorporated into calculus curriculum.] Thomas, H. L. (1975). The concept of function. In M. Rosskopf (Ed.), Children's mathematical concepts (pp. 145172). New York: Columbia University, Teachers College. Vergnaud, G., & Errecalde, P. (1980). Some steps in the understanding and the use of scales and axis by 1013 yearold students. In R. Karglus (Ed.), Proceedings of the fourth international conference for the Psychology of Mathematics Education (pp. 285291). Berkeley: University of California. Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20, 356366. Vinner, S., (1983). Concept definition, concept image and the notion of function. International Journal of Mathematics Education in Science and Technology, 14(3), 293304. Vonder Embse, C. B. (1987). An eye fixation study of time factors comparing experts and novices when reading and interpreting mathematical graphs (Doctoral dissertation, The Ohio State University, 1987). Dissertation Abstracts International, 48, 1141A1142A. (ERIC Document Reproduction Service No. ED 283 672) Vonder Embse, C. B. (1990). Research on reading and interpreting computer generated graphs using eyetracking technology. In F. Demana, B. K. Waits, & J. G. Harvey (Eds.), Proceedings of the Conference on Technology in Collegiate Mathematics (pp. 278281). Reading, MA: AddisonWesley. Wagner, S. (1981). Conservation of equation and function under transformations of variable. Journal for Research in Mathematics Education, 12, 107118. Waits, B. K., & Demana, F. (1988). Microcomputer graphing: A microscope for the mathematics student. School Science and Mathematics, 88(3), 218224. [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. Weaver, D. (?). Function plotters for secondary math teachers. A MicroSOFT Quarterly Report. (ERIC Document Reproduction Service No. ED 290 631). [The article summarizes available graphing utilities. (Somewhat outdated)] Woodward, E., & Byrd, F. (1984). Make up a story to explain the graph. Mathematics Teacher, 77, 3234. [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 international, Psychology of Mathematics Education (Vol. 3, pp. 252260. 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, 255261. [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.] Zimmermann, W., & Cunningham, S. (Ed.). (1991). Visualization in teaching and learning mathematics [MAA Notes No. 19]. Washington, DC: Mathematical Association of America.



