Engaging the Honors Student in Lower-division Mathematics

Minerva CorderoTh JTheresa JorgensenBarbara Shipman

http://www.uta.edu/math/preprint/

Technical Report 2008-15

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Engaging the Honors Student in Lower-division Mathematics

Minerva Cordero1, Theresa Jorgensen2, Barbara Shipman3

Department of Mathematics The University of Texas at Arlington

Arlington, TX 76019-0408, USA 1. Introduction 1.1. Mathematics in the Honors Curriculum In 2005, the National Collegiate Honors Council developed and published a set of twelve recommendations for fully-developed honors college [15]. Those that refer to curriculum are that the program offer significant course opportunities across all four years of study, that the honors curriculum constitute at least 20% of a student's degree program, and that an honors thesis or project be required. To satisfy these recommendations, institutions offer the following honors opportunities: honors courses for general education requirements, honors courses in the major or minor discipline, special topic upper-division honors seminars, special topic lower-division honors seminars, special topic interdisciplinary honors seminars, honors senior thesis/creative projects, honors independent study courses, undergraduate research courses, honors study abroad opportunities, honors internships, and service learning courses. Peterson's Smart Choices: Honors Programs and Colleges (2005) [12] details admission standards, degree requirements, and general program descriptions for the 84 honors colleges in American institutions of higher learning. The average number of honors credits required is 25.1 credits, with a range from 10 to 68. The most common lower-division honors courses offered in mathematics are Honors Calculus I, Honors Calculus II, and honors mathematics for non-science majors, for example, Liberal Arts Honors Mathematics at the University of Texas at Arlington (UT-Arlington). Because of our experience in teaching these honors courses, our colleagues approach us with questions such as the following. How do I make a calculus course "honors"? What should I be prepared to expect from the students? What mathematics should be taught in a course for honors liberal arts majors, and how can it be taught to provide an "honors" experience? This article will address these and other questions regarding lower-division mathematics in the honors curriculum.

1 Associate Dean of the Honors College and Associate Professor of Mathematics, Member of the Academy of Distinguished Teachers, The University of Texas at Arlington 2 Assistant Professor of Mathematics, Honors faculty and Recipient of the 2006 Outstanding Honors Faculty Award, The University of Texas at Arlington 3 Associate Professor of Mathematics, Honors faculty and Member of the Academy of Distinguished Teachers, The University of Texas at Arlington

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The second paper in this sequence, “Designing contracts and honors thesis projects in mathematics,” [9] gives suggestions for creating honors contracts in junior and senior-level mathematics courses and advice on mentoring honors students in designing, writing, and presenting an honors thesis on mathematics. 1.2. The Honors Student In Honors Calculus, the students are typically entering freshmen, while in an honors course for liberal arts majors, they may be taking their final mathematics course before graduation. In both cases, however, the students have designated themselves as honors students, thus invoking high expectations of them by their instructors:

• Honors students should have the desire and motivation during their undergraduate years to educate themselves beyond the requirements of the degree they seek.

• Honors students should want to understand what they study in greater depth and

within a broader context, with a vision toward development of a career and lifelong learning.

• Honors students should be actively engaged in their learning, taking ownership of

their education. • An honors student should be in a class because he/she wants to be there.

It has been our experience in meeting a new group of honors students in Honors

Calculus or in an honors mathematics course for liberal arts majors that these qualities might not yet be well-developed; that is, the students might not be ready to actively learn from the first day of class. We have discovered that the qualities listed above must be taught and nurtured in the students as part of an honors education. Honors students, like most students, have heavy course loads and commitments outside of classes that may tempt them to not put enough time into homework, skip class occasionally, and not take the initiative on their own to excel to the best of their capabilities. Instructors of honors courses need to be aware of these pressures on the students and be armed with instructional strategies that will help develop the qualities that we expect of them.

Below, we outline our expectations of an honors course in mathematics and list some teaching techniques that we have found helpful in bringing honors students in lower-division mathematics courses to reach the expectations outlined above. Sections 2 and 3 illustrate explicitly how these techniques can be implemented in an honors course in mathematics for liberal arts majors, and in Honors Calculus, respectively. 1.3. Expectations of an Honors Course in Mathematics

The following are our goals for any honors course in mathematics. It is expected that the institution offering the honors program implement a class size limit of about 25 students so that these goals can be accomplished.

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• Ownership: Honors students should become the owners of the mathematics that they study. They must take the initiative in deciding whether mathematical statements are true or false, whether a question is worthy of investigation or not, and how new mathematical concepts should be formally defined.

• Communication: In an honors course, students should communicate and defend

their arguments, both formally and informally, both orally and in writing, to the instructor and to their classmates.

• Greater Maturity: An honors course should expand the students' view of what

mathematics is and how to think about it so that after they complete the course students can look back and be amazed at how their mathematical maturity has developed.

• Broader Context: An honors course in mathematics should give the student a

perspective on how the subject has developed and how the subject is still evolving. This can include how other disciplines have influenced the development of mathematics or how mathematics has driven advances in other sciences or in other fields of mathematics.

1.4. Teaching Strategies for a Lower-division Honors Course in Mathematics

Here we list some teaching techniques that we have found helpful in reaching the expectations of the students outlined in Section 1.2 and the expectations of the course outlined in Section 1.3. Sections 2 and 3 explain how these strategies can be implemented in an honors course in mathematics for liberal arts majors and in Honors Calculus.

• Discovery: Students should be prompted during class and on homework

assignments to discover answers to mathematical questions by experimenting with examples, making conjectures, and defending their arguments to their peers.

• Cooperative learning: A significant portion of the work in class and outside of

class should be assigned as group work to be done by groups of three or four students. We have found this to produce homework papers of better quality, and, since group work results in fewer papers to grade, this allows more time for instructive grading of the homework.

• Accountability: In order for discovery and cooperative learning techniques to be

successful, students need to faithfully come to class, prepared and ready to learn. Some incentives we have used to encourage this in a constructive way are the following:

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o Participation: Give points for participation in classroom group activities, and update students often on the status of their participation score.

• Presentation: Assign one to three short review problems at the end of each

class. Call on students “at random” to explain their solutions at the beginning of the next class and deduct points if the student is not ready to present when called. Update students often on their presentation score.

• Homework Completion: Give points for completion of homework

activities that will be used in subsequent class meetings. For example, points can be given for making models of platonic solids and bringing the models to the next class meeting.

• Assessment: Exams should be closely tied to the concepts discussed in depth

during class and on the homework so that students will see that their participation in class and diligence on the homework brings rewards on the exams. The purpose of a well-written exam is not to stump the students with new questions but to verify that the concepts have been understood. Dedicated participation in the course should produce excellent grades on the exams, which will be encouraging to those who participate actively.

2. Mathematics for the Honors Liberal Arts Student Many students, even honors students, enter mathematics courses with a fear of the subject. At UT-Arlington, we teach an honors mathematics course designed for honors liberal arts students. In this course, fear of mathematics is the invisible gorilla in the room at the beginning of the semester. The most represented majors in this class tend to be English and journalism, and as a whole, the students do not have much confidence in their ability to do mathematics. However, since the course is designed so that the students can discover and explore topics in mathematics that they (and even undergraduate mathematics majors) may have never heard of, they seem to leave behind many of their mathematical hang-ups and open their minds to the possibility of enjoying mathematics. The mathematical situations that we study are often simple to state but incredibly rich in their depth. The students encounter and interact with mathematical areas that have open, yet understandable questions. They are expected to do mathematics that they initially believe to be far beyond their abilities, and it is amazing how they rise to the occasion. Students in the classes that we have taught recently have made the following comments on their course evaluations: "This course made math fun, in spite of my being a liberal arts major!" "Math was exciting and something new. I learned how to think outside of the box. It was very different from anything I have ever taken." "It was a great class - especially the less traditional areas of mathematic ‘intrigue’." Honors mathematics for liberal arts majors offers the opportunity to study all sorts of mathematics that are accessible to students at the college freshman level but have been omitted from the high school mathematics curriculum because they do not fit in any

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obvious way with the already crowded traditional sequence of algebra, geometry, and trigonometry. Textbooks that we have used as a source for topics, discovery problems, and projects include The Heart of Mathematics, An Invitation to Effective Thinking [1], To Infinity and Beyond [11], and Knots and Surfaces: A Guide to Discovering Mathematics [6]. We also supplement with personal favorites, articles from journals such as the Mathematics Magazine, the American Mathematical Monthly, and the Notices of the AMS (see [7], [10] and [14], for example,) and occasionally an interesting movie on mathematics. Some of the topics we have included in the past are graph theory, knot theory, the mathematics of voting, fair division, cryptography and coding theory, Fibonacci numbers, the Golden Rectangle, and notions of infinity. The course also includes a few topics that the students have either studied or heard of, but treats them in a new way. The following project is a good starting point to give the students a full flavor of the course. Activity 2.1: The Pythagorean Theorem

The statement: Students invariably recall having studied the Pythagorean Theorem, and many of them are able to state it and use it correctly. To begin this exercise, the class should be asked to state the theorem and give a few examples where the lengths of the sides are integers (Pythagorean triples). The instructor can prompt the class to suggest finding such triples by listing the squares of the first fifteen or so positive integers and checking to see which two squares have a sum that is equal to another. The class should then be asked whether the theorem holds for right triangles where the lengths of one or more sides are not integers and to give some examples.

The question: The challenge now comes when the class is asked, "How do you know that the Pythagorean Theorem is true? Is there a right triangle for which it does not work?" For many students, this may be the first time they have considered the question of "why" in mathematics. Now, not only are they confronting the question, but they will be asked to discover a solution themselves and defend their answers. The class as a whole should be given a few minutes to think about this question and offer any thoughts they may have. The purpose of this phase is to bring the students to realize that throughout their study of mathematics, they have been using formulas without understanding why they are true. They should now be curious about finding an explanation for the Pythagorean Theorem.

Group discovery: The students gather in groups of three to four around tables and work with cut-outs to devise a geometric proof of the Pythagorean Theorem. The book The Heart of Mathematics, An Invitation to Effective Thinking [1] comes with a kit that contains cut-outs of four identical right triangles and one square. These five shapes can be arranged in multiple ways. Two possibilities are (1) as a large square whose edges are the hypotenuses of the four right triangles, as in Figure 1, and (2) as two concatenated squares, as in Figure 2. The groups are asked to place their cut-outs, one set per group, on the table. It is then agreed, for purposes of consistency, to denote the length of the long leg of each right triangle as a, the length of the short leg as b, and the length of the hypotenuse as c.

The group assignment consists in three parts: (1) arrange the shapes as one large square and calculate the area of the square; (2) arrange the shapes in a different

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configuration and use it to calculate the area again, and (3) think about how the Pythagorean Theorem can be explained by these results.

After about 15 minutes, some groups will have found both configurations, perhaps aided by focused questions from the instructor. The instructor should make sure that the groups have also calculated the area in both cases. The benefit of these two configurations is that the areas to be computed are all for rectangular shapes.

Class discussion: After each of the groups has discovered at least one of the

configurations and each configuration has been found by at least one group, the students convene again as a class to report their results. The instructor can begin by asking the students in a group that has correctly solved part (1) to draw their pattern on the board

c

Figure 1

a

b

Figure 2

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and find its area. Here the students should be asked to explain how they know that the figure is actually a square. Next, a group that has found the second pattern is asked to draw its result beside the first. In Figure 2, the students should be asked to justify how they know that the two “squares” are indeed squares. This can be deduced from the first arrangement, and the instructor should make sure that the class understands how this is seen. The class can then be asked explain how the two figures provide a pictorial proof of the Pythagorean Theorem. In this way, the students become the owners of the result and can believe it because they have discovered and proved it with each others’ help.

Writing mathematics: To develop students’ ability to write mathematics accurately and clearly, the students may be asked to explain their proof of the Pythagorean Theorem, in writing, to a friend who may not clearly remember the theorem and certainly would not be able to explain why it is true. The assignment can be given (to be done individually or in groups of two or three) as follows.

(a) Give a clear statement of what the Pythagorean Theorem says, and give a few examples illustrating it.

(b) Tell the reader that you are about to give a pictorial proof of why this theorem is true, and describe the set of cut-outs that you will use to do this.

(c) Sketch the two arrangements of the cut-outs and explain how you can determine the area of each.

(d) Explain how this provides a proof of the Pythagorean Theorem. Broader context: After seeing that the Pythagorean triples, such as (3, 4, 5) and

(5, 12, 13), provide integer solutions of the equation x² + y² = z², the class will have an appreciation for the question of whether the corresponding equation, with the squares replaced by cubes, has integer solutions. The problem can be given as a challenge exercise for homework: find a triple of integers (x, y, z) that satisfies x³ + y³ = z³ and bring it to the next class meeting. This can be assigned to one or two groups, and other groups can be asked to find a triple satisfying xⁿ + yⁿ = zⁿ for other values of n >2.

Of course, the next class meeting may be met with disappointment at the number of solutions found. This provides a lead-in to a discussion of the history behind Fermat’s Last Theorem and its eventual solution in 1992. A great way to conclude this study is with a viewing of the 1997 NOVA production, “The Proof,” which documents the story of Fermat’s Last Theorem in a way that has inspired awe and excitement of mathematics in students who have viewed this production in several iterations of our honors course.

Other variations: There are other geometric configurations of squares and triangles from which one can prove the Pythagorean theorem. This can be assigned as a bonus problem for interested students or as an end-of-class project. ■

For honors students, communication is not a problem, but communicating

mathematically is a new twist. One way in which students' mathematical confidence grows is in the realization that a mathematical argument does not need to consist in a two-column proof, but in a convincingly rigorous argument. These students love to discuss ideas, and so it becomes natural for them to build their understanding of the mathematics by verbalizing it. It is wonderful to hear students who considered themselves math-phobic at the beginning of the semester, heatedly and reasonably arguing about mathematics. The following example outlines how one can structure a

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discussion about whether it is possible to take something away from a set and still have a set of the same cardinality.

Activity 2.2: Two countably infinite sets, Part I

The context: This exercise can be presented before any discussion of finite or infinite cardinalities. Traditionally, before infinite cardinalities are introduced, the students are shown that in comparing two finite sets, one can determine that they have the same size by constructing a one-to-one pairing between the sets (rather than counting each set and comparing the two numbers). Presenting the following exercise before any discussion of one-to-one correspondences gives the honors students the opportunity to explore their own thoughts about counting and come up with their own arguments and ideas before being exposed to the ways that mathematicians have, after decades of work, agreed to understand counting.

The question: Does the set of natural numbers, N = {1, 2, 3, ...}, contain more elements than the set of even natural numbers, 2N = {2, 4, 6, ...}? The students are first asked as a class to give their initial opinions on this so that various points of view may be proposed for discussion.

Initial responses: Three common responses are the following. (1) No, because both sets are infinite. (2) Yes, because N contains twice as many numbers as 2N. (3) It doesn’t make sense to compare the sizes of infinite sets.

The instructor will recognize a misconception about cardinality in each of these responses. The first correctly claims that the N is not larger than 2N but incorrectly attributes this to the fact that both sets are infinite. The second response incorrectly assumes that a proper subset has a smaller cardinality than the original set. It also fails to recognize that how the elements of a set are denoted (that is, what their names are, or what the elements are called) has no influence on how many elements there are. The third response tries to avoid both dilemmas by claiming that it is not possible to reasonably compare the sizes of two infinite sets. These three responses should be written on the board for the class to consider in the next step.

Group discussion: The class is now given some time to discuss these options in groups of four. The assignment is for each person in the group to make a clear argument for or against each of the three responses and to present these arguments to the group. The arguments should be based on clear mathematical reasons, not on personal opinions or emotional inclinations. The goal is for each group to decide on one of the responses and together, to formulate an argument for their case to present to the class. The group should also formulate a clear argument against each of the other two responses. If a group remains sharply divided, then two dissenting opinions may be crafted. All arguments, once agreed upon by the group (or subgroup), should be expressed clearly in writing. This part of the exercise may take up to 20 minutes.

Presentation of arguments: Once all the groups have produced their written statements, each group is asked in turn to write its argument for its chosen position on the board under the statement it supports and give a brief explanation. If one group’s explanation seems to be “the same” as an argument that has already been recorded, it should still be written down, since slight changes in wording may have dramatic effects on meaning. After all the groups have recorded their statements, the counter-arguments

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can be presented and recorded (perhaps in a different color) under the supporting arguments for each statement.

Reflection: After hearing all arguments for and against each of the three initial responses, the students are given a few minutes to quietly reflect on what they have heard and to possibly modify their position.

The instructor will also need some time to reflect on what the students have said in order to make a logical transition to the next stage of the exercise. Usually there are one or two strongly stated arguments for and against each of the first two responses. It should be apparent to the instructor that some key opinions that the students will have presented take into account what the numbers in each set are rather than focusing exclusively on how many elements there are. This observation is the motivation for the next sequence of questions. Further questions: The following questions can now be directed to the class as a whole: (a) Do the names of the students in a class affect how many there are? (b) Do the magnitudes of the numbers in a set affect how many there are? (c) Do the sets {2, 4, 6, ..., 200} and {5, 10, 15, ..., 500} have the same size? How can you show this?

Students will generally agree that the answer to questions (a) and (b) is "No -- what the elements are does not affect how many there are." The students should then recognize without much help that constructing a one-to-one pairing between two sets in part (c) is an intuitive and natural way to show that one set contains exactly as many elements as the other. Armed with this observation and a clear acceptance that what the elements are does not affect their quantity, the students are ready to accept as reasonable and correct the following definition.

Definition: Two sets have the same cardinality if there exists a one-to-one correspondence between them. The class should be asked to give examples of sets that have the same cardinality according to this definition. At this point, many of the students will see the obvious pairing between N and 2N, concluding correctly that the sets have the same cardinality. It is instructive here to offer the counter-argument that 2N can be mapped into N by sending n to n. Does this contradict the definition? This is a good opportunity to emphasize the importance of reading mathematical statements very carefully.

Homework: At this point, the students may have complained that their heads hurt and will need to take a reprieve from infinity before tackling homework problems. Appendix A contains a worksheet that will help the students test and solidify their understanding of cardinality. ■

One of our favorite questions is the following. True or False? After all the work

that mathematicians have put into defining infinite cardinalities, it is still true that there are more natural numbers than even natural numbers. Surprisingly, after working through Activity 2.1, several students admitted that they still secretly believed that there are more natural numbers than even natural numbers, despite openly acknowledging that the two sets have the same size. To help the students out of this uncomfortable position, the following sequel to Activity 2.2 may be presented.

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Activity 2.3: Two countably infinite sets, Part II The two statements: The conflicting statements to be considered are first written on the board and named, for example, as follows:

Definition 1: Two sets have the same cardinality if there exists a one-to-one correspondence between them. Claim 1: The set of natural numbers N contains more elements than the set of even natural numbers, 2N.

Thought experiment, Part A: The following imaginary experiment is proposed to

the class. Suppose the set C consists of infinitely many circular disks, each with a red side and a blue side. On the red side, the disks are numbered in a one-to-one pairing with the counting numbers. On the blue side of each disk, the number 2k is written, where k is the number that appears on the red side. The set S consists of infinitely many square pieces, each with a red side and a blue side; the red and blue sides are numbered exactly as the circular disks in the set C. Now imagine that the circles and squares are lined up on an infinitely long piece of glass as shown below. The number that appears on each circle or square is indicated. Here is the view from the front of the window: FRONT VIEW:

The class is asked to quietly think about the following two questions for a minute

and decide on the answers. Question A1: Using Definition 1 as the basis of judgment and looking at the view above, which are more numerous: the circles, the squares, or neither? Question A2: Using Claim 1 as the basis of judgment and looking at the view above, which are more numerous: the circles, the squares, or neither?

The class will observe that Claim 1 finds the circles to be more numerous, while Definition 1 concludes that the two sets have the same size.

2 4 6 8 . . .

3 421 . . .

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Thought experiment, Part B: The students are then invited to view the same line-up from the back side of the window: BACK VIEW:

Again, the class is asked to quietly think about the following two questions for a

minute and decide on the answers. Question B1: Using Definition 1 as the basis of judgment, and looking at the back view, which are more numerous: the circles, the squares, or neither?

Question B2: Using Claim 1 as the basis of judgment, and looking at the back view, which are more numerous: the circles, the squares, or neither?

Claim 1 now finds the squares to be more numerous, contradicting its previous position. Defintion 1, however, is consistent, concluding again that the two sets have the same size.

Conclusion: The students can now see that Claim 1 contradicts itself, whereas Definition 1 gives a consistent answer. They now have a conflict with their “secret belief” as it stands, independently of its inconsistency with Definition 1. This exercise challenges the secret belief on its own grounds, thus leading the students to abandon it in their own minds. ■

As illustrated in the activities above, we let the students decide whether comparing the sizes of infinite sets is an interesting question or whether it even makes sense to talk about it. Further questions that may be considered in a similar manner are the following. Are some infinite sets larger than others? If we add more elements to an infinite set, must the set become larger? What do "larger" and "smaller" mean when we are talking about infinite sets?

Through discussions as outlined in the previous activities, students are led to formulate, with the instructor's guidance, reasonable and self-consistent definitions of mathematical concepts. They will recognize that they have no alternative but to accept the often counterintuitive outcomes. There are exactly as many even integers as integers.

1 2 3 4 . . .

6 842 . . .

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And, there are exactly as many points between 0 and 1 as there are on the whole infinite real line! Someone encountering such statements for the first time may think that mathematics is simply outlandish, but our honors students, from any major, can now explain that these are indeed properties of infinite sets and that statements to the contrary would be self-contradictory.

The essays "Cantor's new look at the infinite" and "To infinity and beyond" from the collection of essays in To Infinity and Beyond [11] offer a historical perspective on the development of the mathematical concept of cardinality. Students in our classes have been relieved to learn that the same questions that gave them headaches in class have given the most brilliant mathematicians at least as hard a time and caused at least as many arguments among them. Even Kronecker, Georg Cantor's mentor, refused to accept Cantor's rigorous and ingenuous formulation of cardinality.

In our course, we also train the students to write mathematics with precision and clarity so that someone who does not already know the mathematics can read their work and understand the question, the method of solution, and the conclusion. Writing assignments may be given to groups of two or three so that the students can check each others’ writing for completeness and clarity. A writing assignment is usually based on a problem that has been discussed at length in class, where the students have already explained their solutions verbally. The paper should include the following components.

• Statement of purpose. The introduction should state the purpose of the paper, which could be to present a mathematical problem, explain its solution, verify that the solution works, and give examples to illustrate the results.

• Statement of the problem. This should include, as appropriate, clearly labeled diagrams, definitions of concepts used in stating the problem, and a few illustrative examples.

• Definitions. Before jumping into solving equations or analyzing a table or diagram, it is necessary to tell the reader what the equations mean and what the entries and notation in the table or diagram are. This section should define all variables (with units if appropriate) and state the meaning of any equations, tables, or diagrams that will be considered.

• Explanation of the solution. Take the reader carefully through each step of the solution, keeping in mind that the reader is assumed not to already know how to solve the problem.

• Statement of the result. After guiding the reader through the solution and arriving at the final result, it should be clearly stated what has just been shown. This will confirm that the arguments presented have indeed answered the problem posed at the beginning.

• Verification of the solution. Unless the problem is to prove a statement, it is usually possible to check the solution in some way. This may be as simple as “plugging in” the answer to see that it works, or it may involve verifying a strategy for winning a game by testing it on examples that cover various possible cases.

In writing projects such as these, there is opportunity to help the students (many

of whom are English majors) to write precisely. What the students write should express what they mean, but often it does not. To help the students with this, we also work on

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shorter writing exercises during class, in which the students are asked to get in their groups, take out a piece of paper, and formulate in writing, without the help of the textbook or notes, a definition or mathematical result that has either been discussed previously in class or that they have seen in their previous mathematics education. Some examples are the following. (1) Write down what it means for a number to be prime. (2) Explain clearly, in writing, how to put a fraction into lowest terms. (3) Give a good definition of a function. (4) State in writing, without equations, what a circle is. (5) What is the least common multiple of two positive integers? (6) State how to find the greatest common divisor of two positive integers.

After each group has produced a written statement, the students are asked to write them on the board. Each statement should be read carefully and taken at face value for exactly what it says, without interpreting it according to what the writer “meant to say” or according to what the reader “knows that the writer means.” When a statement does not correctly express what is being defined, the instructor should present some examples that satisfy what is written but do not agree with what is intended. Consider, for example, Statement A: “To find the greatest common divisor of two positive integers, multiply the factors that go into both numbers.” This means, then, that to find the greatest common divisor of 12 and 18, we multiply 2, 3 and 6 to obtain 36. The students will readily agree that this is not correct, but that this is what the statement says to do. They might then modify this to Statement B: “To find the greatest common divisor of two positive integers, multiply the prime factors that go into both integers.” Now, to find the greatest common divisor of 24 and 36, Statement B tells us to multiply 2 and 3 to obtain 6. The students will agree that this is not correct either, and the process continues until a correct statement is written.

After many weeks of working together as a class in this way, the students come to know each other well and have found peers with common interests and goals. The class often culminates in formal small group presentations on topics chosen by the groups. Some of the recent topics on which students have become "the house experts" include chaos, interconnections between mathematics and music, deciphering bar-codes, and the role of game theory in jury selection and medical decision-making. 3. Honors Calculus

A sequence of Honors Calculus courses is a wonderful opportunity to build a mathematical learning community among students. UT-Arlington offers a year-long sequence, Calculus I and Calculus II, as honors courses. Roughly the same group of students takes both semesters of the course, so they receive an academic year of exposure to the same instructor and the same peers. As well, many of the students, who are primarily mathematics and science majors, share common schedules in their other courses. So the learning community that gets fostered in their calculus course spills over and supports interactions in their other courses.

In a traditional calculus class, much of the time is spent understanding and practicing techniques for the computation of limits, derivatives, and integrals. As all mathematics instructors know, it's quite easy for the students to lose the forest for the trees. In an honors course, one has the luxury of requiring and expecting the students to

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pull back and understand how the topics fit into their cumulative mathematical knowledge. All four primary expectations of an honors mathematics course can be developed in the context of an Honors Calculus sequence – ownership, communication, mathematical maturity, and broader context.

Different instructors have chosen different texts from which to build the course. One instructor may choose a reform calculus text [8], while another may use a more standard text, such as [13]. However, it is the structure of the course rather than the textbook that provides the enrichment expected in an honors class.

Each week of Honors Calculus consists of three hours of lecture and two hours of lab. Instead of conducting recitation in the traditional way, by simply going over homework problems, each lab session is structured around a worksheet of problems, written by the instructor, which are much more challenging and far-reaching than those assigned from the textbook. (For some examples of lab worksheets, see the appendix.) The students work in small, self-selected groups to find satisfactory solutions to the problems. The instructor serves as a facilitator, posing questions that might help the students gain a deeper understanding of the problems, but rarely answering questions. This serves many purposes. First, the students develop working relationships with the other students in class, relationships that hopefully will carry through to other courses that they will take together. Second, the students develop a sense of mathematical confidence. That is, since they become accustomed to the instructor responding to all of their questions with more questions, they start to develop an ability to know when they are on the right track and begin to answer their own leading questions. This lab structure is modeled in part after the format utilized by the Emerging Scholars Program at the University of Texas at Austin and the MathExcel Program at the University of Nebraska-Lincoln. For a complete set of worksheets see [3] and [4] in the references. To emphasize accountability, each student must submit his solutions to the lab worksheets the following week. The students’ grades on these problems comprise one component of their course grade.

The problems that the students work on in lab sessions are varied. There are two types of problems that we have used over and over. The first type requires a "flip side" of understanding. Here the students are put into the role of the teacher, in that they need to create problems or find examples fitting given specifications to illustrate key ideas. By building their own examples, rather than simply applying theorems or results derived in class, the honors students develop a more complete understanding of the concepts and an ownership of those ideas. An example of a problem of this type is the following.

Example 3.1: The sum rule for limits?

The Problem: Do there exist two functions ( )f x and ( )g x and a constant c such that lim ( )

x cf x

→does not exist, lim ( )

x cg x

→does not exist, but lim( ( ) ( ))

x cf x g x

→+ does

exist? Either find an example, or prove that no such example exists. The Context: To solve this problem, it is not enough to just know how to compute

a limit. The students must put their knowledge of functions into the context of limits to try to create a counterexample to a common "mistaken theorem" that calculus students often try to apply. First they must think about how to build a function for which the limit does not exist at some value x c= . Then they need to consider how they can "fix" the "bad behavior" by adding another function. This problem can be adapted or expanded by

15

considering a different limit rule, e.g. the limit of the product of functions or the limit of the composition of functions, and also by changing the limit behavior of the constituent functions.

A Solution: There are infinitely many possible solutions to this problem, but a

simple solution would be to let 1( )f xx

= and 1( )g xx

= − and consider their behavior at

0.x = ■ Here is an example of another problem from this category.

Example 3.2: Constructing function examples of indeterminate forms The Problem: For each of the following situations, find a function that satisfies

the given requirements, if possible. The limits below can be taken to be one-sided if it is convenient for your example.

a) ( )f π has the form 0∗∞ and lim ( ) .x

f xπ→

= ∞

b) ( )g π has the form 0∗∞ and lim ( ) 0.x

g xπ→

=

c) ( )h π has the form 0∗∞ and lim ( ) 5.

xh x

π→=

The Context: This problem requires students to not only know how to apply

l'Hôpital's Rule, but to really understand what an indeterminant form means, and again, draw on their knowledge of functions to create an example. Moreover, they need to explicitly exhibit the indeterminant nature of the form 0∗∞ by constructing examples of functions with a prescribed limit. This is not an easy problem for students, even honors students, but by building these examples from the ground up, they gain a better understanding of the concept of indeterminant forms. The instructor can guide student groups to the solution by questioning the students as they attempt to build the functions, for example, she may ask the students what parameters in their functions they have control over and how changes in the parameters will change the function behavior. This problem can be adapted by changing the indeterminate form to one of the other classes, e.g. 0 00 / 0, / ,1 ,0 , ,∞∞ ∞ ∞−∞ ∞ .

A Solution: Again, there are many solutions to this problem. Here are some possibilities:

a) 1

( ) ( ) xf x x e ππ −= −

b) ( ) ( ) ln( )g x x xπ π= − − −

c) ( ) 5sin tan( )2

h x x x π= − ■

The second problem type that is emphasized on the lab worksheets asks students

to explain concepts in their own words, for example, “Give an explanation of the Mean

16

Value Theorem that a pre-calculus student could understand. Your explanation should be both verbal and pictorial.” These may also be interpretation problems, such as explaining what a derivative means. For instance,

Example 3.3: A derivative in practical terms

The Problem: A company's revenue from computer sales, R, measured in thousands of dollars, is a function of advertising expenditure, a, also measured in thousands of dollars. Suppose ( ).R f a= Explain what the statement '(301) 2f = means in practical terms.

The Context: It is important, especially on these interpretation problems, to require the students to really answer the question. Students will often solve the previous problem by saying that '(301) 2f = means the derivative of f at 301 is 2. It becomes the instructor's job to draw out the answer by asking a sequence of scaffolding questions designed to guide the groups to fully consider the problem. Here are some examples of scaffolding questions for this problem.

What are the units of the number 301? What are the units of the number 2? What does f measure? Do you have any information about the value of (301)f ? Do you need it? How would the computer company use information about f'? If the company is already spending $301,000 on advertising, would it be wise

for the company to increase its advertising expenditures? If '(301) 0.3f = , would your answer to the previous question change? The Solution: A complete answer to this problem should include the following:

'(301) 2f = means that if the company is already spending $301,000 on advertising and it spends a little bit more on advertising, it would expect its revenue to increase by approximately twice the amount of increase in advertising expenditure. For instance, if it spends $301,100 on advertising, it would expect its revenue to go up by about $200, so it would make back the extra $100 it spent on advertising, plus $100 more. ■

These problems seek to make connections between the computational and conceptual ideas of calculus and hone the honors students' communication and justification abilities. For some sources of good problems, see references [8] and [5] (particularly "Using geometric models to predict convergence" on pages 95-98).

Students are expected to periodically present their ideas and solutions to problems to the rest of the class, further reinforcing the need to communicate mathematics. To encourage the students to keep up with the regular homework assignments, one lab each week starts with ten minutes of presentations of homework problems by students. Three homework problems assigned the previous week are selected by the instructor. The students are not informed ahead of time which problems will be presented. Three students are randomly selected to present homework problems. Each student is given the option to present. If the student is not at lab that day, she gets zero points for her presentation. If the student is at lab, but does not feel ready or willing to present the

17

requested problem, she gets two points for attendance. If the student presents a solution, five points are given. The atmosphere during the presentations is not stressful or high pressure. It is meant to be an opportunity for the students to get practice at communicating mathematics and refining their solutions. Again, it is important to emphasize to the students that mathematics needs to be communicated and that it is often by offering one's work for criticism from one's peers that mistakes are found. This is true of the way professional mathematicians do mathematics, and it can be true of an undergraduate mathematics classroom as well.

A major component of the Honors Calculus I and Calculus II courses is an extensive collaborative project culminating in a professionally written report. The students work in small groups of four, and most of their work on the project is done outside of class. The groups are chosen by the instructor to ensure that each group contains a mix of student abilities and that the students’ schedules align for at least two available working hours each week. All groups work on the same project. The students are given a month and a half to complete the project, and the final product is expected to be of high mathematical quality and well-written in all respects. Crucial to the success of these group projects is a timeline for completion. A minimal timeline should include 1) a date for an initial meeting with the instructor to discuss the group’s preliminary ideas, 2) the date by which the first draft is due to be submitted and reviewed, and 3) the final submission date. Each group submits a single final paper, which should be mathematically typeset and include appropriate diagrams. Students should be encouraged to address their paper to a reader who is superior to them in position, a boss for instance, but equal to them in knowledge of calculus. As well, their paper should be rich in context, explanation, and prose. Often students expect that a mathematical paper should look like the solution to a homework problem, with nothing but numbers and symbols. It is helpful to give the students a model project write-up from a previous semester, or a sample of exemplary mathematical writing at the college level.

There are many sources where rich calculus project problems can be found, see for example [2]. A recent Calculus I semester group project was based on designing a suspension bridge to satisfy prescribed dimensions. The students needed to determine the length of a catenary supporting the bridge, using only their knowledge of Calculus I. They had not yet encountered the arc length formula, and through the project, the students developed the formula. Another Calculus I project involved measuring the volume of wine in a barrel with a bung rod after finding the optimal barrel dimensions (see [13]). A recent Calculus II project had the groups finding the generating function of the Fibonacci sequence via Taylor series, and another revolved around employing power series and clever use of trigonometric identities to find more efficient ways to compute many digits of π. Such group projects reinforce the cooperative atmosphere of the classroom, requiring the students to work together extensively on their own time. As well, the final papers the groups produce are creative and entertaining to read, which is always a delight for the instructor.

As anecdotal evidence of the successful creation of an atmosphere of enjoyable community learning, we note that when the instructor of the Honors Calculus class of Fall 2006 arrived at her classroom to administer the final exam, she found a room decorated with streamers and confetti, a buffet of home-baked cupcakes, and students blowing bubbles, lying in wait to celebrate their final exam. The students truly feel like

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the class is their own and their peers are their collaborators (though not on the final exam). 4. Conclusion

While incorporating mathematics into the honors curriculum comes in different guises, the common thread is high expectations regarding motivation, enthusiasm, creativity, participation, and modes of thinking, of both the students and the instructor. This brings into play more options for student interaction and assessment, ones that incorporate high standards of communication, encourage student ownership of the mathematics being learned, and give students a real feel for the profession of mathematics. Though the ultimate goal is to create an enriched learning situation for the students, the successful engagement of honors students in mathematics is hugely rewarding for the instructor, and is an experience we hope this monograph facilitates.

References [1] Burger, E. and Starbird, M., The Heart of Mathematics, An Invitation to Effective Thinking, 2nd ed., Key College Publishing, 2005 [2] Crannell, A., LaRose, G., Ratliff, T., Rykken, E., Writing Projects for Mathematics Courses: Crushed Clowns, Cars & Coffee to Go, MAA, 2004 [3] Epperson, J., “Calculus I Worksheets,” Treisman Workshop Resources Worksheet Archive, http://math.sfsu.edu/hsu/workshops/resources.html (last updated 2002). [4] Epperson, J., “Calculus II Worksheets,” Treisman Workshop Resources Worksheet Archive, http://math.sfsu.edu/hsu/workshops/resources.html (last updated 2002). [5] Epperson, J., Pace, D., Childs, K., eds. Supporting and Strengthening Standards-Based Mathematics Teacher Preparation. The Charles A. Dana Center, The University of Texas at Austin, 2004 [6] Farmer, D. and Stanford, T., Knots and Surfaces: A Guide to Discovering Mathematics, American Mathematical Society, 1996 [7] Gerstein, L., Pythagorean triples and inner products, Mathematics Magazine 78(3):205-213 (2005) [8] Hughes-Hallett, D., Gleason, A., McCallum, W., et al. Calculus, Single Variable, 4th ed., John Wiley and Sons, Inc., 2005

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[9] Jorgensen, T. and Shipman, B, Designing contracts and honors thesis projects in mathematics, preprint [10] Laugwitz, D., The historical development of infinitesimal mathematics, American Mathematical Monthly 104(7):654-663 (1997) [11] Maor, E., To Infinity and Beyond, Princeton University Press, 1987 [12] Peterson's Smart Choices: Honors Programs and Colleges, Thompson Peterson, Lawrenceville, New Jersey, 2005 [13] Strauss, M., Bradley, G., Smith, K., Calculus, 4th edition, Pearson Education, 2006 [14] Sumners, D., Lifting the curtain: Using topology to probe the hidden action of enzymes, Notices of the AMS 42(5):528-537 (1995) [15] http://www.nchchonors.org/basichonorscollegecharacteristics.aspx

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Appendix

Sample Honors Calculus I Worksheet

1. For each situation, sketch the graph of a function f that satisfies the given condition:

(a) limx→5 f(x) exists, however f is not continuous at x = 5.

(b) f is continuous on (−∞, 1) and on (1,∞), but f is not continuous on (−∞,∞).

(c) f has domain [0, 5] and is continuous on [0, 5), but is not continuous on [0, 5].

(d) f is continuous everywhere except at x = 5, at which point it is continuous from theright.

(e) f is discontinuous at x = 5, and f(5) = 1, however, this function f is such that iff(5) = 1 were changed to f(5) = 0, then f would be continuous at x = 5.

2. Draw the graph of an interesting, non-symmetric function f(x) which is differentiable on(−2, 4). Without doing any computations, that is, purely geometrically, order each of thefollowing from least to greatest for your function.

f ′(0),f(2) − f(0)

2, f ′(3), f ′(−1),

f(0) − f(−1)

1

3. Define a function f that has domain R, but is continuous NOWHERE.

4. For each of the following, you must completely justify your answer, with either a convincingargument or a counterexample.

(a) True or False. As x increases to 100, f(x) = 1/x gets closer and closer to 0, so the limitas x goes to 100 of f(x) is 0.

(b) True or False. limx→a f(x) = L means that if x1 is closer to a than x2 is, then f(x1)will be closer to L than f(x2) is.

(c) You are trying to guess limx→0 f(x). You plug in x = 0.1, 0.01, 0.001, ... and get f(x) = 0for all these values. In fact, you’re told that for all n ∈ N,f( 1

10n ) = 0. True or False.Since the sequence 0.1, 0.01, 0.001,... goes to 0, we know limx→0 f(x) = 0.

(d) If limx→a f(x) = 0 and limx→a g(x) = 0, then limx→af(x)g(x)

i. does not exist

ii. must exist

iii. not enough information

(e) The statement ”Whether or not limx→a f(x) exists depends on how f(a) is defined,” istrue

i. sometimes

ii. always

iii. never

(f) Suppose you have two linear functions f and g. The points (0, 6) and (a, 0) lie on the

graph of f , and the points (0, 3) and (a, 0) lie on the graph of g. Then limx→af(x)g(x)

is

i. 2

ii. does not exist

iii. not enough information

iv. 3

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Sample Honors Calculus II Worksheet

1. One of the most important functions in analysis is the gamma function,

Γ(x) =∫ ∞

0e−ttx−1dt, x > 0.

(a) Use integration by parts to prove that Γ(x + 1) = xΓ(x).

(b) Show that Γ(1) = 1. Conclude that Γ(n) = (n − 1)! for all natural numbers n.The gamma function provides a simple example of a continuous function which inter-polates the values of n! for natural numbers x.

2. (a) Give an example of a sequence which is bounded above but diverges.

(b) Give an example of a sequence bounded above by 1/2, below by -1/2, but which hasno limit.

(c) Give an example of a sequence {an} such that an+1 > an > 0 and limn→∞ an = 2.

(d) Give an example of a sequence {an} such that a2n > 1, a2n+1 < 1 and limn→∞ an = 1.

3. Find two different ways to prove that 1 = .999.... What does this imply about the uniquenessof numerical representation?

4. Is there a value of r for which the series

Σ∞k=0r

k =7

8?

Explain your answer.

5. Some copy machines will make reduced copies. Suppose you copy a page 8 inches wide andit comes out 3/4 as wide. Then you copy this, and so on, indefinitely. How far will theoriginal and all the copies reach if you lay them out side by side on a long table?

6. You are driving along in Texas Hill Country and are experiencing car trouble when you cometo the bottom of a steep hill. You begin to ascend the hill, but due to difficulties with yourcar, you begin rolling downhill. You manage to stop descending only after rolling downhillhalf the initial distance ascended. You start up again and ascend one third of the initialdistance upward before your car acts up again and forces you downhill one fourth of theinitial distance. This continues so that at the nth stage you are either rolling downhill onenth of the initial distance or moving uphill one nth of the initial distance. As this continuesindefinitely, prove or disprove that you will have moved a finite distance uphill/downhill.Prove or disprove that the total distance traveled is finite. Do you ever make it up the hill?