T- 520 Computers and the Learning & Teaching of Mathematics
General Course Information
The purpose of this course is to give students a different vision of what primary and secondary school mathematics might be but is not now. The necessary mathematical background for the course is that which a competent high school student with a course in algebra and geometry can be expected to have acquired.
The course is outlined below. The phrases in boldface are software titles that will serve to structure the class discussion. These pieces of software are available during the term on the computers in the computer laboratory. Students are expected to spend of the order of two to three hours per week outside of class working with these pieces of software as well as with selected applets on the web.
In addition to attending lectures, students registered in the course are expected to work on the problems that will be distributed throughout the semester. Students are encouraged to work on the problem in groups of n=2 or 3 (not n=1 or n >3). Problems are to be turned in and will be discussed in class. They will not be formally graded.
The grade for the course will be based on the basis of participation in class discussion and either a term paper on an appropriate topic or an implemented piece of software useful to a mathematics learner and/or teacher.
Proposals for term paper topics or software designs are due before the Thanksgiving break.
There is a substantial literature in this field. Perhaps the most useful pieces of pertinent literature, in terms of providing an intellectual framework for the course are
The International Journal of Computers for Mathematical Learning, Volume 4, Nos. 2-3, 1999
Chazan, Daniel, Beyond Formulas in Mathematics and Teaching: Dynamics of the high school algebra classroom, 2000, Teachers College, New York
An eclectic collection of pertinent materials is on reserve in the library. In addition, an extended bibliography of readings is available for interested students. There is a small collection of photocopied assigned readings assembled into a course packet that may be purchased at Gnomon Copy.
Students should feel free to contact me with any questions. I can be reached unreliably at my office in Longfellow Hall at the Graduate School of Education - the relevant phone number is 495-9373. However, I can be reached reliably by email – JUDAH_SCHWARTZ@HARVARD.EDU - I try to answer my email promptly. When necessary and desirable I will be happy to meet with you.
Course Outline
Part I - Introduction & Elementary School Mathematics
Session 0 - Introduction
who am I? who are you?
a quick tour of some of the pedagogic puzzles of elementary/secondary mathematics
Session 1 - What Do You Do with a Broken Calculator?
on the pedagogic strategy of using less than a full complement of intellectual tools
Session 2 - The View From The Top - Get Off The Plane (3D only)
on the issue of visual representation and the ubiquity of 2D-3D problem
Session 3 - A is to B as C is to ... - Power Drill - The Algebraic Proposer
quantity with meaning - dimensional analysis
nominal & adjectival quantity, discrete & continuous quantity, intensive & extensive quantity
Session 4 - The Algebraic Proposer - The Algebra Sketchbook
on the qualitative understanding of function & the mathematical modeling of one's surround
Readings for Part I of the course
Can Technology Help Us Make The Mathematics Curriculum Intellectually Stimulating and Socially Responsible?
- Schwartz (paper on reserve in library and in course packet).The Visual Display of Quantitative Information - Tufte - Cheshire Press, Cheshire, CT.
Semantic Aspects of Quantity - Schwartz (paper on reserve in library and in course packet).
On the Need for a Bridging Language For Mathematical Thinking - Schwartz & Yerushalmy - For the Learning of Mathematics, Vol. 15, No. 2, 1995. (paper on reserve in library and in course packet).
Part II - The Mathematics of Shape & Space
Session 5 - Get Off The Plane - Oxxo's Xox
going beyond the sensory - thinking spatially in many dimensions
Session 6 - The Geometric superSupposer
software as intellectual mirror - the role of conjecture in learning, teaching & making mathematics
Session 7 - The Geometer's Sketchpad
on the implicit pedagogical assumptions in software interface design
Readings for Part II of the course
Flatland : a romance of many dimensions
- Abbott, HarperPerennial, 1994.The Fourth Dimension : Toward a Geometry of Higher Reality - Rucker - Houghton Mifflin, 1984.
Intellectual Mirrors: A Step in the Direction of Making Schools Knowledge-Making Places - Schwartz - Harvard Educational Review, Feb. 1989.
The Geometric Supposer: What Is It A Case Of? - Schwartz, Yerushalmy & Wilson - Lawrence Erlbaum Associates, Hillsdale, NJ, 1993.
Part III - The Mathematics of Function - Algebra, Pre-Calculus and Calculus
Session 8 - Tools for Doing Mathematics I - Derive, MathCad, spreadsheets, etc.
general purpose mathematical tools - how well suited are they to the classroom?
Session 9 - Tools for Learning Mathematics I - The Function Analyzer - Calculus Unlimited
Functions of one variable
Session 10 - Tools for Learning Mathematics II - Explorations in Analytic Geometry -
Functions of two variables
Session 11 - Tools for Learning Mathematics III - Function Family Register
families of functions & invariants
Session 12 - Tools for Learning Mathematics IV - Unsolving...
Comparing functions using equations and inequalities -
What you may not do to both sides of an equation or inequality
What you may do to one side of an equation or inequality
Readings for Part III of the course
Shuttling between the Particular & the General: Reflections on the Role of Conjecture & Hypothesis in the Generation of Knowledge in Science & Mathematics
- Schwartz - in Software Goes To School: Teaching for Understanding in the Age of Technology, D. Perkins, J.L. Schwartz, M.S. Wiske, M.M. West, (eds.), Oxford University Press, New York, 1995.Getting Students to Function in Algebra - Schwartz - (paper on reserve in library and in course packet).