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*Every course should incorporate activities that will help all students progress in developing analytical, critical reasoning, problem-solving, and communication skills and acquiring mathematical habits of mind. More specifically, these activities should be designed to advance and measure students’ progress in learning to:*

*State problems carefully, modify problems when necessary to make them tractable, articulate assumptions, appreciate the value of precise definition, reason logically to conclusions, and interpret results intelligently;**Approach problem solving with a willingness to try multiple approaches, persist in the face of difficulties, assess the correctness of solutions, explore examples, pose questions, and devise and test conjectures;*

*Read mathematics with understanding and communicate mathematical ideas with clarity and coherence through writing and speaking.*

**Discovery Learning/Inquiry-Based Learning/Problem-Based Learning**

Professor Robert Lee Moore's method of teaching at the

Over the past five years there has been a growing effort on the * Educating Undergraduates in the Research University, Reinventing Undergraduate Education: A Blueprint for America's Research University* advocated the appropriateness and use of IGL in undergraduate education. IGL capitalizes on one of the key strengths of research universities, the expertise of its faculty in research, and it responds to a point made by John Dewey almost a century ago, that learning is based on discovery guided by mentoring, rather than the simple transmission of information. The NC State website Faculty Center for Teaching and Learning contains information on the IGL program, including an extensive set of resources. A description of how NC State is using the IGL method in the Foundations of Advanced Mathematics, Abstract Algebra, and Introduction to Analysis courses is included in Part 2, Section C.1.

Problem-Based 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, problem-solving proficiency, self-directed 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 Problem-based Learning. An open-access, 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 open-ended 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 algebra-through-calculus 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 1963-2005.

**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 problem-solving 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 problem-solving abilities. A four-part 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 problem-solving 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. Ferrini-Mundy 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 *The Concept of Function: Aspects of Epistemology and Pedagogy* (Harel & Dubinsky, 1992); ’Students, Functions, and the Undergraduate Curriculumâ? (Thompson, 1994); ’On Understanding How Students Learn to Visualize Function Transformationsâ? (Eisenberg & Dreyfus, 1994); and ’An Investigation of the Function Conceptâ? (Carlson, 1998). The Special Interest Group of the MAA on Research in Undergraduate Mathematics Education (SIGMAA on RUME) is a good source for research on student understanding, logical reasoning, and problem solving. Further discussion of the concept of function is found in Part 1, Section 3.

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: 249-266, 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:

* Truth table constructor (Brian S. Borowski, Seton Hall):

* Experimentation with AND, OR, and NOT gates (Geoffrey De Smet)

* Logic Cafe, Online courseware for symbolic logic (John F. Halpin)

* Logic Daemon, a web-based proof checker, and Quizmaster, a set of interactive logic quizzes, to accompany *Logic Primer* (MIT Press, 2000, by Colin Allen and Michael Hand)

* Logic Circuit Lab (from *The Most Complex Machine* by David Eck,

* Proof Checker (Robert StÃ¤rk)

_ Tarski's World Applet (Robert StÃ¤rk)

* Tilomino Tutorial (Neil Deakin)

* Flash applications from *Introduction to Discrete Mathematics: Mathematical Reasoning with Puzzles, Patterns and Games* by D. Ensley & W. Crawley, John Wiley and Sons (Construct a counterexample, Test understanding of a proof, Unscrambling a proof)

* For additional activities designed to help students reason and work logically to conclusions, see the items in this section under ’Mathematical Writing Assignmentsâ? and Part 2, Section C.2.

**Strategies for Problem Solving**

The following excerpt provides problem-solving guidance for a wide range of students in college-level courses. It is from ’The Mathematical Education of Prospective Teachers of Secondary School Mathematics,â? by J. Ferrini-Mundy 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 In-Depth 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.

’The mathematical content involved in extended analyses of problems can be expressed as a set of mathematical ’principles.’ These include:

* Selecting parameters to represent key quantities in a problem situation. Typically the parameters replace numerical values of some of the quantities that are used in the initial statement of the problem. There are many sub-themes here, such as: (a) considering which quantities to parameterize; (b) being alert for ways to generalize the results being found, and at the same time looking for important special cases; (c) replacing a variable *x* that has a particular range 0 x L with a ... variable *p* with a range 0 p * Coaxing expressions into their most useful forms. Again, there are many sub-themes (a) collapsing separate occurrences of the independent variable; (b) making use of ratios in particular and dimensionless factors in general.

* Representing relationships in a situation in several different kinds of ways to get different insights. Examples are diagrams, graphs, tables, and formulas.

* Looking for connections to different kinds of mathematics. For example: (a) looking for geometrical interpretations of analytic results, and conversely (b) looking to connect discrete mathematics with continuous mathematics.

* Anticipating, asking, and answering many of the sorts of questions that may occur to a reader who is trying to understand the ideas. Standard treatments often bypass such questions since they are not part of the most efficient and elegant presentation.â?

In the Spring 2000 MER Newsletter, Marjorie Enneking of Portland State University offers the following problem-solving 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?

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.

*Mathematical Reasoning: Writing and Proof *by Ted Sundstrom are expected to read a couple of pages on their own at the beginning of each section and do ’preview activitiesâ? to prepare for the more extensive work to be done in the section.

**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 follow-up 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 Upper-Division 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**

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 max-min 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 crystal-clear, 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 secondary-level mathematics teaching, it contains a section on writing in mathematics courses that can be useful at the college level as well.

Because the sophomore-level 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), 393-401, 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, *PRIMUS*. The table of contents is as follows:

1. Why Should You Have To Write Papers In A Math Class?

2. How is Mathematical Writing Different?

3. Following the Checklist

1) Clearly restate the problem to be solved.

2) State the answer in a complete sentence that stands on its own.

3) Clearly state the physical assumptions that underlie the formulas.

4) Provide a paragraph that explains how you will approach the problem.

5) Clearly label diagrams, tables, graphs, or other visual representations of the math (if these are indeed used).

6) Define all variables used.

7) Explain how each formula is derived, or where it can be found.

8) Give acknowledgment where it is due.

9) In this paper, are the spelling, grammar, and punctuation correct?

10) In this paper, is the mathematics correct?

11) In this paper, did the writer solve the question that was originally asked?

4. Good Phrases to Use in Math Papers

5. Helpful Hints for the Computer

6. Other Sources of Help

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.

**Assessing Students’ Skills in Writing Mathematics**

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), 393-401, 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).

The following are some of the websites that offer information, bibliographic references, and resources about writing in mathematics: The Math Forum: Writing/Comm in Math, Writing to Learn Mathematics: An Annotated Bibliography, Selected Bibliography on Writing Across the Curriculum: Mathematics, Articles on Writing Across the Curriculum’Math, Writing in Mathematics with Dr. Annalisa Crannell, Franklin & Marshall College, and the Grinnell College Writing Lab. Other resources include MAA journals and *PRIMUS *(many articles of which can be found on Find Articles).

More information and resources on developing mathematical thinking and communication skills are located in Part 2, Sections B.2 and C.1.