Communicating Mathematics A Workshop by Dr. Kris Green for Webster School District Middle and High School Teachers on March 22, 2000

Who Am I? 4 Assistant Professor in the MST Department at St. John Fisher College Project NExT fellow 4 Ph.D. in Applied Mathematics from University of Arizona 4 B.S. in Engineering Physics from University of Tennessee, Knoxville

Overview of the Workshop 4 Reasons for communicating mathematics 4 Theoretical perspectives on communicating mathematics 4 Activities I use to encourage students to communicate mathematics 4 My observations on effectiveness of these 4 Resources

Reasons for Communicating Mathematics 4 NYS MST Learning Standards (explore, describe, analyze, communicate) 4 The Work Force needs it (engineering teams, publication and dissemination) 4 To make informed decisions (online trading, phone services, etc.) 4 Easily incorporates new technologies (electronic portfolios, desktop publishing, spreadsheets, graphics applications) 4 Demonstrates mastery/points out weaknesses

Theoretical Perspectives 4 Algebra vs. Geometry –Algebra is actually more Verbal/Linguistic –Geometry is more Logical/Mathematical 4 Multiple Intelligences (Gardner) –Writing provides more ways to bring the verbal/linguistic intelligence into class with support from the logical/mathematical and visual intelligences (as well as the others to a lesser degree)

Some of My Activities 4 Journal Writing –Based on readings from the text or other sources (MAA Resources for Calculus, v. 5) –Includes information as well as “feelings” and interpretations –Involves a dialogue between student and myself –Students are given questions to answer (these are “big, important” questions)

More of My Activities 4 Class Reflections - standard questions to be answered in journals after each class –What was the most important thing you learned today? –What are you still confused about? –Were you on time for class? –Rate you participation in class (1-10). Why? –What will you do to prepare for the next class?

More Activities 4 Writing Assignments –What is Mathematics? –Families of Functions –Calculus Murder Mystery –Synthesis of journal entries showing progress 4 Projects –Essentially these are longer problem sets that involve an application of the mathematics

More Activities 4 Exams (2-3 per semester, plus final) –Q1: four “non-standard” multiple choice questions –Q2: four short answer/short calculation –Q3: essay question - based on journals –Q4 and Q5: free response, longer calculation problems 4 Concept Mapping

More Activities 4 In class group work –To derive properties of some objects (gradient) –“Sketch the graph of a function which has as many of the following properties as possible, and then justify that your function has the properties you claim it does.” –Working as a class to build a graph (erf(x)) –Finding the shortest distance (geodesics) –Explaining and evaluating the “Mystery Number System”

More Activities 4 Two-column calculations –Left 3/4 of page for computations (line by line) –Right 1/4 of page for explanations of each major step (reasoning, definitions, theorems, etc) 4 Visual proofs in geometry –(with Dr. Allen Emerson) to combine the visual and logical/axiomatic approach to geometry

Even More Activities 4 Independent Study Project –John is physically handicapped –John has studied algebra, computer science –Project Read some of the mathematics education literature on the understanding of functions Write a paper that compares/contrasts these views Write a reflection on where his understanding falls

Observations 4 Journals have shown me where the book/class/etc have caused confusion in the use of an idea (integration as “giving back the original function” or only applying to “getting distance from velocity”) 4 Gets more students involved in the class -- the Murder Mystery received many positive comments

More Observations 4 These types of assignments are easily more time-consuming to grade 4 “To check grammar or not to check grammar, that is the question” 4 Often yields (pleasantly) surprising results

Resources for Math Reading 4 MAA Resources for Calculus, volume 5, Readings for Calculus 4 The Mathematical Tourist, Ivars Peterson 4 Codes, Puzzles and Conspiracy, Dennis Shasha 4 Flatland, Edwin Abbott 4 The Adventures of Penrose, the Mathematical Cat, Theoni Pappas

Resources for How to Implement Writing in Mathematics 4 Writing in the Teaching and Learning of Mathematics, John Meier and Thomas Rishel -- this book is okay for its purpose 4 Using Writing to Teach Mathematics, edited by Andrew Sterrett -- not as good a resource 4 Other teachers!

Ideas for Good Writing Assignments and Projects 4 Play on Words (the “Families of Functions” project) 4 PRIMUS (Journal from the US military Academy) 4 Mathematics Teaching in the Middle School 4 The Mathematics Teacher 4 Colleagues in Other Disciplines (History, Science, Economics, Art, Music)

More Ideas 4 Any of the resources in the “Readings” section 4 Scientific American (particularly the “Mathematical Recreations” of Martin Gardner) 4 Old Isaac Asimov’s Science Fiction Magazine articles by Martin Gardner

More Ideas 4 The NYS MST Learning Standards themselves (also the online resources)  Sample Web-based projects are available from the “Virtual Teacher Resources” under the MST homepage located at  Online computing resources are located at

Online Ideas  Chevron Corporation: under Educational Programs at  Discovery Channel School located at  Exemplars materials can be viewed and ordered from