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Thompson, P. W., & Carlson, M. P. (in press). Variation,
covariation, and functions: Foundational ways of thinking in
mathematics. In J. Cai (Ed.), *Third handbook of research in
mathematics education*. Reston, VA: National Council of Teachers
of Mathematics. FOR COMMENT ONLY.

Yoon, H., Byerley, C., & Thompson, P. W. (in press). Teachers'
meanings for average rate of change in U.S.A. and Korea. In T.
Fukawa-Connelly (Ed.), *Proceedings of the 18th Meeting of the MAA
Special Interest Group on Research in Undergraduate Mathematics
Education*. Pittsburgh, PA: RUME.

Musgrave, S., Hatfield, N., & Thompson, P. W. (in press). Calculus
students' meaning for difference. In T. Fukawa-Connelly (Ed.), *Proceedings
of the 18th Meeting of the MAA Special Interest Group on Research in
Undergraduate Mathematics Education*. Pittsburgh, PA: RUME.

Moore, K. C., & Thompson, P. W. (in press). Shape
thinking and students' graphing activity. In T. Fukawa-Connelly
(Ed.), *Proceedings of the 18th Meeting of the MAA Special Interest
Group on Research in Undergraduate Mathematics Education*.
Pittsburgh, PA: RUME.

Joshua, S., Musgrave, S., Hatfield, N., & Thompson, P. W. (in
press). Frames of reference. In
T. Fukawa-Connelly (Ed.),

Musgrave, S., Hatfield, N. & Thompson, P. W. (in press). Teachers'
meanings for the substitution principle. In T. Fukawa-Connelly
(Ed.),

Thompson, P. W. (2015). Researching
mathematical meanings for teaching. In L. D. English & D.
Kirshner (Eds.), *Third handbook of international research in
mathematics education* (pp. 435-461). New York: Taylor &
Francis.

Byerley, C., & Thompson, P. W. (2014). Secondary
teachers' relative size schemes. In P. Liljedahl & C. C.
Nicol (Eds.), *Proceedings of the 38th Meeting of the
International Group for the Psychology of Mathematics Education*,
(Vol 2, pp. 217-224). Vancouver, BC: PME.

Musgrave, S., & Thompson, P. W. (2014). Function
notation as idiom. In P. Liljedahl & C. C. Nicol (Eds.), *Proceedings
of the 38th Meeting of the International Group for the Psychology of
Mathematics Education*, (Vol 4, pp. 281-288). Vancouver, BC:
PME. Retrieved from http://bit.ly/1p08TCG

Thompson, P. W. (2014). Schemes for thinking with magnitudes: A hypothesis about foundational reasoning abilities in algebra. In K. C. Moore, L. P. Steffe & L. L. Hatfield (Eds.), Epistemic algebra students: Emerging models of students' algebraic knowing. WISDOMe Monographs (Vol. 4, pp. 1-24). Laramie, WY: University of Wyoming.

Weber, E., & Thompson, P. W. (2014). Students'
images of two-variable functions and their graphs. *Educational
Studies in Mathematics*, 86.

Saldanha, L., & Thompson, P. W. (2014). Conceptual
issues in understanding the inner logic of statistical inference:
Insights from two teaching experiments. *Journal of
Mathematical Behavior*, 35, 1-30.

Thompson, P. W., Artigue, M., Torner, G., de Shalit, E. (2014). Collaboration
between mathematics and mathematics education. In M. Fried &
T. Dreyfus (Eds.), *Mathematics and mathematics education:
Searching for common ground* (pp. 313-333). New York: Springer.

Thompson, P. W. (2013, October). "Why
use *f*(*x*) when all we really mean is *y*?".
*OnCore, The Online Journal of the AAMT.*

Thompson, P. W. (2013). Constructivism
in mathematics education. In S. Lerman (Ed.), *Encyclopedia
of mathematics education [online]*. Berlin: Springer. doi:
10.1007/SpringerReference_313210 2013-05-10 00:00:07 UTC.

Thompson, P. W., Byerley, C., & Hatfield, N. (2013). A
conceptual approach to calculus made possible by technology. *Computers
in the Schools*, 30, 124-147.

Thompson, P. W. (2013). In
the absence of meaning. In K. Leatham (Ed.), *Vital
directions for research in mathematics education*, pp. 57-93. New
York: Springer.

Thompson, P. W. (2012). Advances
in research on quantitative reasoning. In R. Mayes, R. Bonillia,
L. L. Hatfield & S. Belbase (Eds.), *Quantitative reasoning:
Current state of understanding* WISDOMe Monographs (Vol. 2, pp.
143-148). Laramie, WY: University of Wyoming Press.

Byerley, C., Hatfield, N., & Thompson, P. W. (2012). Calculus
Student Understandings of Division and Rate. In S. Brown, S.
Larsen, K. Marrongelle, & M. Oehrtman (Eds.), *Proceedings of
the 15th Annual Conference on Research in Undergraduate Mathematics
Education* (pp. 358-363). Portland, Oregon: SIGMAA/RUME.

Weber, E, Tallman, M., Byerley, C., & Thompson, P. W. (2012). Introducing
the derivative via calculus triangles. *The Mathematics
Teacher, 104*(4), 274-278.

Thompson, P. W. (2011). Quantitative
reasoning and mathematical modeling. In L. L. Hatfield, S.
Chamberlain & S. Belbase (Eds.), *New perspectives and
directions for collaborative research in mathematics education*
WISDOMe Monographs (Vol. 1, pp. 33-57). Laramie, WY: University of
Wyoming Press.

Liu, Y., & Thompson, P. W. (2009). Teachers' understandings of proto-hypothesis testing. Pedagogies, 4(1).

Thompson, P. W. (2008). On professional judgment and the National Mathematics Panel Report. Educational Researcher, 38(9), 582-587.

Silverman, J., & Thompson, P. W. (2008). Toward a framework for the development of mathematical knowledge for teaching. Journal of Mathematics Teacher Education.

Thompson, P. W. (2008, July). *Conceptual
analysis of mathematical ideas: Some spadework at the foundation
of mathematics education*. Plenary paper delivered at the
32^{nd} Annual Meeting of the International Group for the
Psychology of Mathematics Education. In O.
Figueras, J. L. Cortina, S. Alatorre, T. Rojano & A. SÈpulveda
(Eds.), Proceedings of the Annual Meeting
of the International Group for the Psychology of Mathematics
Education, (Vol 1, pp. 45-64). MorÈlia,
Mexico: PME.

Thompson, P. W., & Silverman, J. (2008). The
concept
of accumulation in calculus. In M. Carlson & C. Rasmussen
(Eds.), *Making the connection: Research and teaching in
undergraduate mathematics* (pp. 117-131)*.* Washington, DC:
Mathematical Association of America.

Thompson, P. W. (2008, June). One approach to a coherent K-12 mathematics. Or, it takes 12 years to learn calculus. Paper presented at the Pathways to Algebra Conference, June 22-25, Mayenne, France.

Oehrtman, M. C., Carlson, M. P., & Thompson, P. W. (2008). Foundational
reasoning abilities that promote coherence in students'
understandings of function. In M. P. Carlson & C.
Rasmussen (Eds.), *Making the connection: Research and practice in
undergraduate mathematics *(pp. 150-171). Washington, DC:
Mathematical Association of America.

Liu, Y. & Thompson, P. W. (2007). Teachers'
understandings of probability. *Cognition and Instruction, 25*(2),
113-160*.*

Thompson, P. W., Carlson, M. P., & Silverman, J. (2007). The
design of tasks in support of teachers' development of coherent
mathematical meanings. *Journal of Mathematics Teacher
Education, 10*, 415-432*.*

Smith, J., & Thompson, P. W. (2007). Quantitative
reasoning and the development of algebraic reasoning. In J.
Kaput, D. Carraher, & M. Blanton (Eds.), *Algebra in the early
grades* (pp. 95-132)*.* New York: Erlbaum.

Saldanha, L., & Thompson, P. (2007). Exploring
connections between sampling distributions and statistical
inference: An analysis of students' engagement and thinking in the
context of instruction involving repeated sampling. *International
Electronic Journal of Mathematics Education. *(2), pp. 270-297.

Thompson, P. W., Liu, Y., & Saldanha, L. A. (2007). Intricacies
of statistical inference and teachers' understandings of them.
In M. Lovett & P. Shaw (Eds.), *Thinking with data* (pp.
207-231). Mahwah, NJ: Erlbaum.

Thompson, P. W., Castillo-Chavez, C., Culbertson, R. J., Flores, A., Greely, R., Haag, S., et al. (2007). Failing the future: Problems of persistence and retention in science, techhnology, engineering, and mathematics majors at Arizona State University. Tempe, AZ: Office of the Provost.

Liu, Y. & Thompson, P. W. (2006, April). *Teachers'
understandings of probability and their implications for teacher
professional development.* Paper presented at the Annual
Meeting of the American Educational Research Association, San
Francisco, CA.

Saldanha, L., & Thompson, P. (2007). Exploring
connections between sampling distributions and statistical
inference: An analysis of students' engagement and thinking in the
context of instruction involving repeated sampling. *International
Electronic Journal of Mathematics Education.* Bahia, Brazil.

Saldanha, L., & Thompson, P. (2006). Investigating
statistical unusualness in the context of resampling. *Proceedings
of the International Congress on Teaching Statistics.* Bahia,
Brazil.

Thompson, P. W., & Liu, Y. (2005). Understandings
of margin of error. In S. Wilson (Ed.), *Proceedings of the
Twenty-seventh Annual Meeting of the International Group for the
Psychology of Mathematics Education*, Roanoke, VA. Vicksburg, VA:
Virginia Tech.

Liu, Y., & Thompson, P. W. (2005). Teachers'
understanding of hypothesis testing. In S. Wilson (Ed.), *Proceedings
of the Twenty-seventh Annual Meeting of the International Group for
the Psychology of Mathematics Education*, Roanoke, VA. Vicksburg,
VA: Virginia Tech.

Silverman, J., & Thompson, P. W. (2005). Investigating
the relationship between mathematical understanding and teaching
mathematics. In S. Wilson (Ed.), *Proceedings of the
Twenty-seventh Annual Meeting of the International Group for the
Psychology of Mathematics Education*, Roanoke, VA. Vicksburg, VA:
Virginia Tech.

Liu, Y., & Thompson, P. W. (2004). Teachers'
personal and pedagogical understanding of probability and
statistical inference. In D. McDougal (Ed.), *Proceedings of
the Twenty-sixth Annual Meeting of the International Group for the
Psychology of Mathematics Education.* Toronto: PME-NA.

Thompson, P. W., & Saldanha, L. (2003). Fractions
and multiplicative reasoning. In J. Kilpatrick, G. Martin &
D. Schifter (Eds.), *Research companion to the Principles and
Standards for School Mathematics* (pp. 95-114). Reston, VA:
National Council of Teachers of Mathematics.

Saldanha, L. & Thompson, P. W. (2002). Conceptions
of sample and their relationships to statistical inference. *Educational
Studies in Mathematics,* *51,* 257?270.

Thompson, P. W. (2002). Didactic
objects and didactic models in radical constructivism*.* In
K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), *Symbolizing
and Modeling In Mathematics Education*. Dordrecth, The
Netherlands: Kluwer.

Liu, Y., & Thompson, P. W. (2002). Randomness:
Rethinking the foundations of probability. In D. Mewborn (Ed.),
*Proceedings of the Twenty-fourth Annual Meeting of the
International Group for the Psychology of Mathematics Education*.
Athens, GA.

Saldanha, L. & Thompson, P. W. (2002). Students'
scheme-based understanding of sampling distributions and its
relationship to statistical inference. In D. Mewborn (Ed.), *Proceedings
of the Twenty-fourth Annual Meeting of the International Group for
the Psychology of Mathematics Education*. Athens, GA.

Thompson, P. W. (2001). Holistic
perspectives on instructional design -- A review of __Symbolizing
and communicating in mathematics classrooms: Perspectives on
discourse, tools, and instructional design__. *Journal for
Research in Mathematics Education*, 32(3), 321-327.

Thompson, P. W. & Saldanha, L. (2000). Conceptual
issues in understanding sampling distributions and margins of error.
*Proceedings of the Twenty-second Annual Meeting of the
International Group for the Psychology of Mathematics Education*.
Tuscon, Arizona.

Thompson, P. W. & Saldanha, L. (2000). Epistemological
analyses of mathematical ideas: A research methodology. *Proceedings
of the Twenty-second Annual Meeting of the International Group for
the Psychology of Mathematics Education*. Tuscon, Arizona.

Thompson, P. W. (2000). What
is required to understand fractal dimension? *The Mathematics
Educator, *10(2), 33-35.

Thompson, P. W. (2000). Radical
constructivism: Reflections and directions. In L. P. Steffe
& P. W. Thompson (Eds.), *Radical constructivism in action:
Building on the pioneering work of Ernst von Glasersfeld* (pp.
412-448). London: Falmer Press.

Steffe, L. P., & Thompson, P. W. (2000). Teaching
experiment methodology: Underlying principles and essential elements.
In R. Lesh & A. E. Kelly (Eds.), *Research design in
mathematics and science education* . Dordrecht, The Netherlands:
Kluwer.

Steffe, L. P., & Thompson, P. W. (2000). Interaction
or intersubjectivity? A reply to Lerman. *Journal for
Research in Mathematics Education*.

Thompson, P. W. (1999). Remarks on representations, conventions, and common meanings. Panel for the PME-NA XXI Working Group on Representations, Cuernavaca, Mexico.

Thompson, P. W. (1999). Representations
and evolution: A discussion of Duval's
and Kaput's
papers. In F. Hitt (Ed.). *Proceedings of the Twenty-first Annual
Meeting of the Psychology of Mathematics Education, North America*.
Cuernavaca, Mexico: Centro de Investigación y de Estudios Avanzados.

Cortina, J., Saldanha, L., & Thompson, P. W. (1999). Multiplicative
conceptions of arithmetic mean. In F. Hitt (Ed.). *Proceedings
of the Twenty-first Annual Meeting of the International Group for
the Psychology of Mathematics Education*. Cuernavaca, Mexico:
Centro de Investigación y de Estudios Avanzados.

Saldanha, L., & Thompson, P. W. (1998). Re-thinking
co-variation from a quantitative perspective: Simultaneous
continuous variation. In S. B. Berensah & W. N. Coulombe
(Eds.), *Proceedings of the Annual Meeting of the Psychology of
Mathematics Education - North America*. Raleigh, NC: North
Carolina State University.

Thompson, P. W., & Cobb, P. (1998). On
relationships between psychological and sociocultural perspectives.
In S. Berenson (Ed.). *Proceedings of the Proceedings of the
International Group for the Psychology of Mathematics Education,
Plenaries* (pp. 3-32). Raleigh, NC: North Carolina State
Universty Press.

Thompson, P. W. (1998, March 30). *Multiplicative
relationships among fraction, measurement, multiplication, and
division*. Patrick W. Thompson. [1999, March 17].

Thompson, P. W. (1996). Imagery
and the development of mathematical reasoning. In L. P. Steffe,
P. Nesher, P. Cobb, G. Goldin, & B. Greer (Eds.), *Theories of
mathematical learning* (pp. 267-283). Hillsdale, NJ: Erlbaum.

Thompson, A. G., & Thompson, P. W. (1996). Talking
about rates conceptually, Part II: Mathematical knowledge for
teaching. *Journal for Research in Mathematics Education, 27*(1),
2-24.

Thompson, P. W. (1995). Constructivism,
cybernetics, and information processing: Implications for research
on mathematical learning. In L. P. Steffe & J. Gale (Eds.),
*Constructivism in education* (pp. 123-134). Hillsdale, NJ:
Erlbaum.

Dugdale, S., Thompson, P. W., Harvey, W., Demana, F., Waits, B.,
Kieran, C., McConnell, J. W., & Christmas, P. (1995). Technology
and algebra curriculum reform: Current issues, potential directions,
and research questions. *Journal of Technology in
Mathematics, 14*(3), 325-358.

Thompson, P. W. (1995). Notation,
convention, and quantity in elementary mathematics. In J. Sowder
& B. Schapelle (Eds.), *Providing a foundation for teaching
middle school mathematics* (pp. 199-221). Albany, NY: SUNY Press.

Thompson, P. W., & Ball, D. L. (1995). Research and practice. *Journal
for Research in Mathematics Education, 26*(4), 300-303.

Kaput, J. J., & Thompson, P. W. (1994). Technology
in mathematics education research: The first 25 years in *JRME*.
*Journal for Research in Mathematics Education, 25*(6), 676-684.

Thompson, P. W., & Thompson, A. G. (1994). Talking
about rates conceptually, Part I: A teacher's struggle. *Journal
for Research in Mathematics Education, 25*(3), 279-303.

Thompson, A. G., Philipp, R. A., Thompson, P. W., & Boyd, B. A.
(1994). Calculational
and conceptual orientations in teaching mathematics. In A.
Coxford (Ed.), *1994 Yearbook of the NCTM* (pp. 79-92). Reston,
VA: NCTM.

Thompson, P. W., & Sfard, A. (1994). Problems
of reification: Representations and mathematical objects. In D.
Kirshner (Ed.). *Proceedings of the Annual Meeting of the
International Group for the Psychology of Mathematics Education -
North America, Plenary Sessions* Vol. 1 (pp. 1-32). Baton Rouge,
LA: Lousiana State University.

Thompson, P. W. (1994, April). *Bridges
between mathematics and science education*. Paper presented
at the Research blueprint for science education conference, New
Orleans, LA.

Thompson, P. W. (1994). Concrete
materials and teaching for mathematical understanding. *Arithmetic
Teacher, 41*(9), 556-558.

Thompson, P. W. (1994). The
development of the concept of speed and its relationship to concepts
of rate. In G. Harel & J. Confrey (Eds.), *The
development of multiplicative reasoning in the learning of
mathematics* (pp. 179-234). Albany, NY: SUNY Press.

Thompson, P. W. (1994). Images
of rate and operational understanding of the Fundamental Theorem of
Calculus. *Educational Studies in Mathematics, 26*(2-3),
229-274.

Thompson, P. W. (1994). Students,
functions, and the undergraduate mathematics curriculum. In E.
Dubinsky, A. H. Schoenfeld, & J. J. Kaput (Eds.), *Research in
Collegiate Mathematics Education, 1* (Vol. 4, pp. 21-44).
Providence, RI: American Mathematical Society.

Thompson, P. W. (1993). Quantitative
reasoning, complexity, and additive structures. *Educational
Studies in Mathematics, 25*(3), 165-208.

Thompson, P. W. (1993). Yes,
Virginia, some children do grow up to be mathematicians. [Review
of *Advanced Mathematical Thinking*, D. Tall (Ed.)]. *Journal
for Research in Mathematics Education, 24*(3), 279-284.

Fraivillig, J. L., Fuson, K. C., & Thompson, P. W. (1993).
Microworld support of children's understanding of multidigit addition.
In S. Ohlson, P. Brna, & H. Pain (Eds.), *Proceedings of the
World Conference on AI in Education* Vol. 1. Edinburgh, Scotland:
University of Edinburgh.

Thompson, P. W., & Thompson, A. G. (1992, April). *Images
of rate*. Paper presented at the Annual Meeting of the
American Educational Research Association, San Francisco, CA.

Thompson, P. W. (1992). *Blocks
Microworld* *5.1*. Computer Program for Macintosh.
Santa Barbara, CA: Intellimation. (Click here
to see user documentation, which also describes and justifies design
decisions.)

Thompson, P. W. (1992). Notations,
conventions, and constraints: Contributions to effective uses of
concrete materials in elementary mathematics. *Journal for
Research in Mathematics Education, 23*(2), 123-147. (Click here
for full program documentation.)

Thompson, P. W. (1991). Getting
ahead, with theories: I have a theory about this. In R.
Underhill & C. Brown (Eds.), *Proceedings of the Annual Meeting
of the Psychology of Mathematics Education, North America, Plenary
Lectures* Vol. 1 (pp. 240-245). Blacksburgh, VA: Virginia Tech.

Thompson, P. W. (1991). To
experience is to conceptualize: Discussions of epistemology and
experience. In L. P. Steffe (Ed.), *Epistemological
foundations of mathematical experience* (pp. 260-281). New York:
Springer-Verlag.

Thompson, P. W. (1987, 1991). *Word Problem Analyst* *2.1*.
Computer Program for Macintosh. San Diego, CA: San Diego State
University.

Thompson, P. W., & Thompson, A. G. (1990). Salient
aspects of experience with concrete manipulatives. In *Proceedings
of the 14th Annual Meeting of the International Group for the
Psychology of Mathematics* Vol. 3 (pp. 337-343). Mexico City.

Thompson, P. W. (1990). *Over & Back* *1.0*. Computer
Program for Macintosh. San Diego, CA: San Diego State University.

Thompson, P. W. (1990). A theoretical model of quantity-based reasoning in arithmetic and algebraic. Center for Research in Mathematics & Science Education: San Diego State University.

Thompson, P. W. (1989). Artificial
intelligence, advanced technology, and learning and teaching algebra.
In C. Kieran & S. Wagner (Eds.), *Research issues in the
learning and teaching of algebra* (pp. 135-161). Hillsdale, NJ:
Erlbaum.

Thompson, P. W. (1989, July). *Mathematics
software*. Paper presented at the Apple Computer Higher
Education Conference on "Designing for Learning", Cuptertino, CA.

Thompson, P. W., & Dreyfus, T. (1988). Integers
as transformations. *Journal for Research in Mathematics
Education, 19*, 115-133.

Thompson, P. W. (1988). Quantitative
concepts as a foundation for algebra. In M. Behr (Ed.). *Proceedings
of the Annual Meeting of the North American Chapter of the
International Group for the Psychology of Mathematics Education*
Vol. 1 (pp. 163-170). Dekalb, IL.

Thompson, P. W., & Thompson, A. G. (1987). Computer
presentations of structure in algebra. In N. Herscovics & C.
Kieran (Eds.), *Proceedings of the 11th Annual Meeting of
International Group for Psychology of Mathematics Education* Vol.
1 (pp. 248-254). Montréal: University of Quebec, Montréal.

Thompson, P. W. (1987). Mathematical
microworlds and intelligent computer-assisted instruction. In G.
Kearsley (Ed.), *Artificial Intelligence and Education* (pp.
83-109). New York: Addison-Wesley.

Thompson, P. W. (1986). Logo
as a medium for thinking about thinking. In R. Noss & C.
Hoyles (Eds.), *Proceedings of the Second International Conference
on Logo and Mathematics Education* Vol. 1 (pp. 209-215). London:
London Institute of Education.

Thompson, P. W. (1985). Computers in research on mathematical problem
solving. In E. Silver (Ed.), *Teaching and learning mathematical
problem solving: Multiple research perspectives* (pp. 417-436).
Hillsdale, NJ: Erlbaum.

Dreyfus, T., & Thompson, P. W. (1985). Microwolds
and Van Hiele levels. In *Proceedings of the Ninth Annual
Meeting of the International Group for the Psychology of Mathematics
Education*. .

Thompson, P. W. (1985). Experience,
problem solving, and learning mathematics: Considerations in
developing mathematics curricula. In E. Silver (Ed.), *Teaching
and learning mathematical problem solving: Multiple research
perspectives* (pp. 189-243). Hillsdale, NJ: Erlbaum.

Thompson, P. W. (1985). A Piagetian approach to transformation
geometry via microworlds. *Mathematics Teacher, 78*(6), 465-472.

Thompson, P. W. (1985). Understanding
recursion: Process approximates Object. In S. Damarin (Ed.). *Proceedings
of the 7th Annual Meeting of the North American Group for the
Psychology of Mathematics Education* (pp. 357-362). Columbus, OH:
Ohio State University.

Thompson, P. W. (1984). Content versus method. *College
Mathematics Journal, 15*(5), 394-395.

Thompson, P. W. (1984). Microworld environments for teaching
mathematics to future elementary school teachers. In *Proceedings
of the De Anza Conference on Computers in Higher Education* Vol.
1 (pp. 5-11). .

Thompson, P. W. (1982). *A theoretical framework for understanding
young children's concepts of whole-number numeration.*
Unpublished Doctoral dissertation, University of Georgia, Department
of Mathematics Education.

Steffe, L., Thompson, P., & Richards, J. (1982). Children's
counting in arithmetical problem solving. In T. Carpenter, T. Romberg,
& J. Moser (Eds.), *Addition and subtraction: A cognitive
perspective* (pp. 83-97). Hillsdale, NJ: Erlbaum.

Thompson, P. W. (1982). Were
lions to speak, we wouldn't understand. *Journal of
Mathematical Behavior, 3*(2), 147-165.

Thompson, P. W. (1979, April). *The
teaching experiment in mathematics education research*.
Paper presented at the NCTM Research Presession, Boston, MA.