Illustrative Resources 



2. Develop mathematical thinking and communications skills.
Read mathematics with understanding and communicate
mathematical ideas with clarity and coherence through writing and speaking.
Professor Robert Lee Moore's method of teaching at the Over the past five years there has been a growing effort
on the ProblemBased Learning (PBL) is both a curriculum and a
process. The curriculum consists of problems that have been selected and
designed to lead students to acquire critical knowledge, problemsolving
proficiency, selfdirected learning strategies, and team participation
skills. The problems are loosely structured in order to encourage students to
pursue various paths in the solution process. Websites with information about
PBL are at Samford
University, Pennsylvania State
University, Queensland University (Australia) and The Interdisciplinary Journal of
Problembased Learning. An openaccess, searchable database of the ways in which PBL
is being used by practitioners around the world is hosted by the University
of Brighton (UK). From 1999’2002, the Making Mathematics
project matched students and teachers in grades seven through twelve with
professional mathematicians who mentored their work on openended mathematics
research projects. Those involved with the project report that students
generated ideas, discovered patterns, posed questions, developed conjectures,
and built proofs of mathematical claims. They observed that when students
started to explore their own problems or to restate or repose old problems,
their impression that the world of mathematics is both finite and linear (the
classic algebrathroughcalculus sequence) was challenged. Problem posing was
a major focus in the project as students developed research projects. An article
from the project addresses how to create a new problem from an old one and
how to develop new questions for old problems in order to extend them. The MathPro Press
website provides online information about mathematical problems, problem
books, and problem journals, including an online searchable collection of
over 20,000 math problems and the collected problems of Stanley Rabinowitz
19632005. Research on Reasoning and Problem Solving
A large portion of Research
in Collegiate Mathematics Education III (A. Schoenfeld, J. Kaput
& E. Dubinsky, Eds.) focuses on aspects of a problemsolving course
taught for many years by Alan Schoenfeld. This course was designed in
response to findings (among them Schoenfeld’s, as described in his book Mathematical Problem Solving
and other of his articles) about common student attitudes and beliefs about
mathematics and proof as well as their own problemsolving abilities. A
fourpart paper by Abraham Arcavi, Cathy Kessel, Luciano Meira, and Jack
Smith addresses particular aspects of the classroom activity and Schoenfeld’s
teaching. A second paper by Manuel Santos discusses the course as a whole.
Finally, Alan Schoenfeld reflects on what these authors are saying about his
teaching. The three papers together ’provide a close look at a particular
example of ’good practice,’ a highly refined course and pedagogical approach
that over the years seems to succeed in teaching powerful problemsolving
skills.â? Case studies such as these illustrate good teaching methodologies
and provide a resource for instructors. Several papers by Alan Schoenfeld on mathematical thinking
and problem solving are available on his website.
In ’Learning
to Think Mathematicallyâ? he writes that his goals are ’(a) to outline and
substantiate a broad conceptualization of what it means to think
mathematically, (b) to summarize the literature relevant to understanding
mathematical thinking and problem solving, and (c) to point to new directions
in research, development and assessment consonant with an emerging
understanding of mathematical thinking and the goals for instruction outlined
here.â? Surveys of research on student learning in calculus can be
found in Changing
Calculus: A Report on Evaluation Efforts and National Impact from 1988’1998
by Susan Ganter, ’An overview of the calculus curriculum reform effort:
issues for learning, teaching, and curriculum development,â? by J.
FerriniMundy and K. Graham, American Mathematical Monthly 98 (1991),
and in the volumes of Research in Collegiate
Mathematics Education published
jointly by the AMS and the MAA. The Additional examples of research on reasoning and problem
solving are in Part
2, Section C.1. Activities to Help Students Learn to
Reason and Work Logically to Conclusions In his conclusion to ’Making the
transition to formal proofâ? (Educational Studies in Mathematics
27: 249266, 1994), Robert Moore, wrote: ’Until proof is integrated
throughout the school and university mathematics curricula in the United
States, I believe the abrupt transition to proof will continue to be a source
of frustration for undergraduate students and teachers.â? (For a brief summary
of this article, see Part 2, Section C.1.)
In a calculus or precalculus class, simply including
the phrase ’Justify your answerâ? or ’Explain your reasoningâ? on
quizzes, exams and homework problems can help students understand that
mathematical claims require justification. It should be made clear that
a significant amount of credit will be deducted if the justification is
missing or incorrect. In elementary courses, a single reason is often
sufficient to explain an answer (e.g., ’by the chain rule,â? ’by the ratio
test,â? ’by definition of xxâ?). Instructors can also encourage analytical
thinking through guided classroom activities. One strategy is to ask a
student to describe or explain a mathematical concept in a single sentence.
The instructor then writes the sentence verbatim on the board and asks the
class to elaborate or clarify the explanation until it describes the concept
to their satisfaction. This process gives students practice in expressing
mathematics carefully, and the resulting sentence provides them with a model
for their own work. In another activity, pairs of students make one or
two ’formalâ? problem presentations per term, which they first rehearse with
the instructor, a process that usually requires about ten minutes per pair. In the talk he gave after winning an MAA Haimo award for
distinguished teaching, Herb Wilf of the In recent years a number of Java applets and Flash
applications have been developed to help teach students about logic and
proof. A sampling is given below:
Strategies for Problem Solving The following excerpt provides problemsolving guidance
for a wide range of students in collegelevel courses. It is from ’The
Mathematical Education of Prospective Teachers of Secondary School
Mathematics,â? by J. FerriniMundy and B. Findell, in CUPM Discussion Papers about
Mathematics and the Mathematical Sciences in 2010: What Should Students Know? . The ideas in the excerpt are
attributed to unpublished working papers of Stanley and Callahan, The
InDepth Secondary Mathematics Institute, Texas Education Agency and the
Texas Statewide Systemic Initiative of the Charles A. Dana Center at the University of Texas at Austin. In the Spring 2000
MER Newsletter, Marjorie Enneking of Portland State University offers the
following problemsolving advice: ’Mathematics involves penetrating
techniques of thought that all people can use to solve problems, analyze
situations, and sharpen the way they look at their world.â? She gives students
the ’Top 10 Lessons for Life: (1) Just do it. (2) Make mistakes and fail, but
never give up. (3) Keep an open mind. (4) Explore the consequences of ideas.
(5) Seek the essential. (6) Understand the issue. (7) Understand simple
things deeply. (8) Break a difficult problem into easier ones. (9) Examine
issues from several points of view. (10) Look for patterns and similaritiesâ? Robert Talbert from Franklin College reported on the
experience of using the Internet and email when he assigns ’Reading
Questionsâ? to his students. Graded on a scale of 1’10 (which includes
grammar, mathematical correctness, and clarity of exposition), each
assignment requires students to ’write a concise yet complete outline of this
section,â? and then define terms, perhaps do a simple calculation, and answer
some conceptual questions. ’Questions are handed out (and posted on the
course web site) the class meeting before the reading topic is covered in
class. Students read the section, answer the questions, and then come into
class ready to engage on that topic.â? Talbert reported that ’This has
dramatically improved the kind of instruction I can give in class. I had
always been frustrated by spending (wasting?) time on things such as writing
definitions on the board or going over some subtle but easily understandable
point the book makes ’ having the students read has allowed me to spend more
time on nittygritty kinds of examples and more lengthy group problems, which
the students appreciate.â? Although he usually requires Calculus I and
Calculus III students to submit work every day, he said he found it less
difficult than it sounds (and even fun) and very revealing to read what
students write. He said he is often amazed what the students can figure out
on their own if they just try, adding that he believes these assignments
train students to be good readers of mathematics and teach them the right way
to do mathematics: jump in and try it on their own first, improve with the
help of the teacher, and then consolidate their accomplishments with
practice. He admitted that this method is not a perfect assessment tool, and
is timeconsuming to write and grade, but he observed that the students have
come to like it after some initial grumbling. He says that he now cannot
imagine doing another course without Reading Questions. In ’Requiring
Student Questions on the Textâ?
Bonnie Gold reported that to get students to read their textbooks, some
calculus classes at Monmouth University require students to come to
class with three questions about the section to be discussed that day .
These can be questions asking for clarification of a point discussed in
the textbook, questions about an issue raised in the textbook, or
questions posed in ’Jeopardy style,â? i.e., questions answered by a
particular paragraph or example in the textbook. The questions are used
to start the day’s lesson. Specific Techniques to Improve Students’
Ability to Read Mathematical Writing In ’Helping
Undergrauates Learn to Read Mathematicsâ? Ashley Reiter, Maine School of
Science and Mathematics, wrote about handouts and followup assignments she
created to provide specific advice on how to read definitions and theorems
for mathematics majors at the University of Chicago. She reports that when
her department used these, students gradually developed the ability to read
mathematics on their own, which facilitated their transition to becoming
independent learners. Laura Taalman wrote Problem
Zero: Getting Students to Read Mathematics, which describes her
experience requiring students to write a brief outline of the sections of the
text corresponding to each day’s lessons. Carl Cowen (American Mathematical
Monthly, 1991) describes
having students work in class to analyze pieces of mathematical writing and
then testing their ability to read mathematics with understanding. For the
tests, he presented students with a new small ’theoremâ? accompanied by a
brief proof. The proof was followed by a question that would be impossible to
answer without understanding the proof. Tevian Dray of Oregon State University (http://www.math.oregonstate.edu/~tevian/)
asks students to diagram mathematical writing by labeling the various symbols
and expressions. He reports that this exercise helps them learn to read
mathematics. Students can use the process to translate from words to symbols
and from symbols to words. An example he gives, excerpted from Student Use
of Visualization in UpperDivision Problem Solving (Browne, unpublished
dissertation at OSU, 2001, p. 97), is shown below. In ’Does
Calculus Reform Work,â? Joel Silverberg wrote, ’Student interviews,
discussions, and dialog quickly revealed that what the student sees when
looking at a graph is not what the teacher sees. What students hear is not
what the instructor thinks they hear. Almost nothing can be taken for
granted. Students must be taught to read and interpret the text, a graph, an
expression, a function definition, a function application.â? In terms of
student articulation, he stated, ’Interviews revealed that the frequent use
of pronouns often masks an ignorance of, or even an indifference to, the nouns
to which they refer. The weaker student has learned from his past experience
that an instructor will figure out what ’it’ refers to and assume he means
the same thing.â? Some faculty members respond to this phenomenon by
forbidding students to use the word ’itâ? in their writing or speaking. Kathleen Snook (1997) emphasized the importance of
listening carefully to students. She asked, ’How many times during a
classroom discussion does a teacher think ’well, the student said xxx, but
she really meant yyy’ and assume the student simply wasn’t very articulate?
How many other students in the class also thought xxx? How many now have a
misconception because the point was not clarified?â? In her work interviewing
students and conducting faculty development workshops, she has found that the
interaction described above happens quite frequently. Although students may
not be very articulate, they usually say exactly what they are thinking. She
argues that development of articulation goes hand in hand with development of
understanding of the mathematical topics under study. Further information is
available from her at kathleen.snook@verizon.net.
Sandra Frid (1994) investigated three different approaches
to calculus instruction, focusing on their impact on students’ language use
and sources of conviction. With all three approaches, students used
symbols less frequently than they used technical mathematics or everyday
language. Although they were able to perform standard symbolic operation,
they generally chose not to use symbols to describe or explain a mathematical
concept. The study found that students’ use of everyday language is a
significant factor in their mathematical learning. Discussions of differences in the use and meaning of
everyday and mathematical language can be found in Speaking
Mathematically: Communication in Mathematics Classrooms by David Pimm
(Routledge and K. Paul, 1987), A Handbook of
Mathematical Discourse by Charles Wells, and ’The Logic of
Teaching Proofâ? by Susanna S. Epp. Mathematical Writing Assignments
Every year, we buy ten cases of paper at $35 each; and every year we sell
them for about $1 million each. Writing well is very
important to us.  Bill Browning, President of Applied Mathematics,
Inc. Bruce Crauder (speech at David
R. Stone of Georgia Southern University reported that he has given the
following examples of writing assignments in Calculus I: (1) Write a complete
description of how to solve a maxmin problem. (2) Write a complete
description of how to solve a related rates problem. (3) Write a complete
description of how to graph a function. He expects complete sentences,
general principles rather than specific examples, and several paragraphs on
each. He suggests that assignments be started early in the semester and
repeated several times before requiring a significant piece of writing for a
grade. He has been surprised at how his class explanations, which seem
crystalclear, become garbled when students put the ideas into writing and
practice! For instance, he has found that students rarely begin an
optimization problem by deciding what quantity is to be maximized or
minimized, although he modeled and discussed that approach in class. He
comments that reading student responses really drives home the points that
(1) just "telling" is not the same as having students learn, and
(2) working many examples and homework problems does not necessarily
guarantee that students will be able to formulate a plan of attack for such
problems. Stone suggests that similar questions can be asked in later
courses: ’Show me your plan for deciding whether or not a series converges.
(A big decision tree might be most appropriate here.) How do you decide
whether some vectors form a basis of a space? Describe your process for
solving a system of linear congruences.â? He admits, ’The downside, of course,
is that it takes a lot of your time to read and write comments on such
assignments. I believe it is time well spent.â? Gavin
LaRose, University of Michigan wrote that he has required students to
write up a certain number of problem solutions with great care. ’For a couple
of semesters I experimented with assigned written problems and portfolios for
students in my calculus courses. These went through several incarnations, of
which the most successful were a weekly assignment in which students picked a
problem they had done for homework in the previous week and found difficult,
and reworked it as a ’written’ problem, which means that they had to explain each
step of their work ’ as we would were we writing a paper or textbook. In
addition, they submitted a short explanation of how the mathematics that they
were doing in that problem was related to the mathematics that we had covered
in the preceding week.â? LaRose assigned these ’writtenâ? problems in all his
classes one semester and reports that he ’just about died with the grading
load.â? The following semester he tried essentially the same thing, but with
the written problems being moved into the homework. ’Each assignment I would
pick a problem which required a written explanation,â? and weighted that
problem double. He continued allowing students to pick problems but only did
so every two weeks. LaRose found the grading load much more manageable. Although the website Tools for Understanding,
funded through the US Department of Education, is intended to be a resource
for secondarylevel mathematics teaching, it contains a section on
writing in mathematics courses that can be useful at the college level as
well. Because the sophomorelevel discrete mathematics course
taught by Rochelle Leibowitz, Wheaton College, serves as a bridge between
’computational mathematics and computer scienceâ? on the one hand and
’theoretical mathematics and computer scienceâ? on the other, ’the emphasis is
on writing algorithms and mathematical proofs.â? As a result, Leibowitz
obtained a ’writing intensiveâ? designation for the course. Leibowitz provides
’individual responses to students’ writing by making comments, corrections,
and suggestions on their writing style as well as on the mathematical content
of their answers.â? Each class begins with students putting solutions to
problems on the board. This is often followed by lively discussion,
especially if ’solutionsâ? are incomplete or incorrect. Leibowitz reports that
’students learn that writing and doing mathematics are one and the same. They
come to appreciate that writing mathematics is an essential survival skill for
any mathematician.â? (Quotes from ’Writing Discrete(ly)â? in Discrete Mathematics in
the Schools, J. G. Rosenstein, D. S. Franzblau, and F. S. Roberts,
eds., AMS/NCTM Publication, 1997.) Guidelines for Mathematical Writing J.J. Price wrote an influential article ’Learning
Mathematics Through Writing: Some Guidelinesâ? (Coll. Math. J., 20(5),
393401, 1989), which has influenced many others. An adaptation
of his article, by Eliza Berry and Jeff Lawson from the University of Alberta, is freely available on the Internet. Annalisa Crannell, Steve Maurer, Swarthmore College, wrote A
Short Guide to Writing Mathematics, which is available online in its
entirety by request. He also has an article, ’Advice for
Undergraduates on Special Aspects of Writing Mathematics,â? first
published in PRIMUS, with sections entitled Introduction, What Kind of
Mathematics Paper?, Know Your Reader, Titles, Introduction, Divisions into
Sections, Theorems, Definitions, Examples, Figures, Big Little Words (let,
thus, so), When to Give Credit, Complicated Mathematical Expressions,
Displays, Two Common Mistakes, Miscellaneous, and References. Giving students explicit guidelines for their written work can reduce the amount of time needed to evaluate their writing. Annalisa Crannell (Franklin and Marshall College) has students staple a checklist to their papers. J.J. Price (Purdue University) includes dos and don’ts in his article ’Learning Mathematics Through Writing: Some Guidelinesâ? (Price, Coll. Math. J., 20(5), 393401, 1989). Bruce Crauder (Oklahoma State University) and others provide students with a few exemplary problem solutions, whose style they are encouraged to emulate. Melvin Henriksen (Harvey Mudd College) and Jennifer Szydlik (University of Wisconsin at Oshkosh) report that grading students’ first efforts severely results in dramatic improvement. They say that students generally forget their initial dismay and appreciate the progress they have made by the time course evaluations are administered. And the better students perform, the easier it is to grade their work. Although initial efforts to require writing in mathematics classes may have been at the grassroots level within the mathematics community, more ’writing across the curriculumâ? programs have emerged at various institutions. Print resources include the MAA publications Writing in the Teaching and Learning of Mathematics (Meier and Rishel, 1998), Using Writing to Teach Mathematics (Sterrett, 1990), and Learning to Teach and Teaching to Learn Mathematics (Delong and Winter, 2001).
