Annual Review of Mathematics Program

Core Competency: Written Communication Assessment

May 2015

Names: Courtney Davis, Don Hancock, Kevin Iga, Kendra Killpatrick, Tim Lucas, David Strong, Don Thompson (author)

Program: Mathematics, B.S.

Our focus this year is on the core competency of written communication as fulfilled by our proof­writing Mathematics Major PLO’s #1 & #3. Accordingly, we looked at direct evidence of our students’ proof­writing in our upper division courses. We teamed up in collaborative pairs of mathematics faculty to look at one homework or midterm problem and one final exam question for competency in proof­writing. Each pair evaluated student work as a function of each course’s grading criteria, matched against the instructor’s solution set. This allowed us to measure achievement and improvement over the course of the semester for all sophomore, junior and senior students within our major. Each pair developed a brief summary of their findings, focusing on the nature of each homework/midterm exam/final exam question, the distribution of results, and an analysis of their data. Section VIII contains all of these analyses. Because the work spans seven different courses, including final exams that just concluded two weeks ago, we plan to conduct a full “closing the loop” discussion of our findings in the coming fall semester.

Here are the pairings:

Fall 2014 Course Team

Real Analysis I Hancock, Strong

Linear Algebra Davis, Lucas

Combinatorics Killpatrick, Thompson

Spring 2015 Course Team

Linear Algebra Killpatrick, Davis

Real Analysis II Hancock, Thompson

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Transition to Abstract Mathematics Iga, Killpatrick

Complex Analysis Lucas, Iga

I. Program Learning Outcomes

A student who completes a B.S. in Mathematics will satisfy the Mathematics Program Learning Outcomes. Mathematics Program Learning Outcomes:

A student who completes the mathematics degree should be able to:

PLO #1 Formulate mathematical proofs that are clear, correct, complete, and logical.

PLO #2 Demonstrate an understanding of the knowledge and skills central to the discipline of mathematics.

PLO #3 Demonstrate the ability to apply appropriate mathematical ideas to both abstract and real­world contexts.

PLO #4 Demonstrate a willingness to serve by having participated in co­curricular activities that are central to the broader mathematical community.

II. Alignment of PLOs with Institutional Learning Outcomes

Our program prepares students for lives of purpose, service and leadership by training them in the mathematical thinking, skills, discipline, and service opportunities that are needed in various professions.

Institutional Learning Outcomes PLO #1

PLO #2

PLO #3

PLO #4

ILO #1

Demonstrate expertise in an academic or professional discipline, display proficiency in the discipline, and engage in the process of academic discovery.

X X X

ILO #2

Appreciate the complex relationship between faith, learning, and practice.

ILO #3

2

Develop and enact a compelling personal and professional vision that values diversity.

ILO #4

Apply knowledge to real­world challenges. X

ILO #5

Respond to the call to serve others. X

ILO #6

Demonstrate commitment to service and civic engagement. X

ILO #7

Think critically and creatively, communicate clearly, and act with integrity.

X X X

ILO #8

Practice responsible conduct and allow decisions and directions to be informed by a value­centered life.

ILO #9

Use global and local leadership opportunities in pursuit of justice.

III. Student Learning Outcomes

Based on the syllabi, list each course number, title, and the name of the faculty member who taught the course.

Fall 2014

MATH Course Number

Course Title Faculty Member

Correctly Stated SLOs

SLOs Related to PLOs

150 Calculus I Davis, Lucas

151 Calculus II Iga

250 Calculus III Strong

3

260 Linear Algebra Davis

270 Foundations of Elementary Mathematics I

Killpatrick

316 Biostatistics Strong

370 Real Analysis I Hancock

440 Partial Differential Equations

Lucas

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IV. Curriculum Map for PLOs

For each course, indicate whether students will be Introduced to the PLO (I), Develop their skills related to the PLO (D), or demonstrate Mastery of the PLO (M) by entering I, D, or M under the appropriate PLO.

Course Number PLO #1 PLO #2 PLO #3 PLO #4

Math 130 – Colloquium in Mathematics I

Math 151 ­ Calculus II I I

Math 250 ­ Calculus III D D

Math 260 ­ Linear Algebra I D D

Math 320 ­ Transition to Abstract Mathematics I I

Math 325 – Mathematics for Secondary Education

D

Math 335 ­ Combinatorics D

Math 340 ­ Differential Equations D D

Math 345 ­ Numerical Methods D

Math 350 – Mathematical Probability D D

Math 355 ­ Complex Analysis D

Math 365 ­ Automata Theory D

Math 370 ­ Real Analysis I D D

Math 380 ­ Algebraic Structures I D D

Math 480 ­ Algebraic Structures II M M

Math 450 – Mathematical Statistics M

Math 440 ­ Partial Differential Equations M

Math 470 ­ Real Analysis II M M

Co­curriculum M

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Design of the Mathematics major

A typical math major begins with the three semester calculus sequence. Very frequently, entering first years already have credit for one or two of these through AP or similar credit. Calculus is one of the foundations of the modern scientific and technological age. It relates ideas from classical geometry like area and tangency to dynamical ideas like rate of change and sensitivity to change, and is the basic language of physics and many other areas of science, engineering, and economics. The second and third semester of calculus also build the student’s ability to visualize in three dimensions.

During the Spring semester of the first year, math majors and prospective math majors should take Colloquium in Mathematics, a 1 unit seminar where students are exposed to an overview of what the math major is like beyond calculus. Math faculty present short introductions to different topics that students will encounter later in the major. Along the way, students get to meet many of the math faculty. Students also learn about career options for math majors and learn about other opportunities such as undergraduate research and other programs for math majors.

During the second year or third year, students take the Linear Algebra and Differential Equations courses. These courses mix the largely computational aspects of calculus with the conceptual and abstract thinking of upper division courses. Linear Algebra further develops the mathematics of three (or higher) dimensions, describing linear (straight) objects using tables of numbers called matrices. It lies at the intersection of the geometric and the symbolic approaches to mathematics, as well as the intersection of theoretical and applied mathematics. Differential equations is a course in projecting how a system will behave over time, when various influences on the system are known. In this course, students learn to model real­life problems mathematically, and see how these tools can yield profound insights about the natural world.

During the second semester of the second year, math students typically take Transition to Abstract Mathematics. This is a course that introduces students to the concept of proof. Students learn to understand mathematical proofs and write their own, with an eye not only to logical accuracy but also clarity. The Transition course is a prerequisite for most upper division math major courses.

During the third and fourth year, students will take a range of upper division courses. Two in particular are required of all math majors: Real Analysis I and Algebraic Structures I. These introduce students to abstract and proof­based approaches to continuous and discrete areas of mathematics, respectively. They are the foundation for many areas of advanced mathematics, including many areas of mathematics research. These courses are offered every other year, so students must take Real Analysis I one year and Algebraic Structures I the other year.

Students should have an in­depth year­long experience at the upper division level. Thus they choose at least one of a list of four year­long sequences:

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• Real Analysis I and II

• Algebraic Structures I and II

• Differential Equations and Partial Differential Equations, or

• Probability and Statistics.

The idea is for students to be introduced to mathematical ideas in one semester and explore them in depth in another semester. This also allows for learning of remarkable ideas that take some time to develop. The course numbering scheme puts the second semester of each of these courses in the 400 level, and students are required to take at least one 400 level course.

Beyond this, math majors must take three other upper division courses from a range of topics, depending on possible career choices, graduate school preparation, or general interest. For instance, students interested in teaching at the middle school to high school level would benefit from Mathematics for Secondary Education, which fulfills many of the goals set by the National Council of Teachers of Mathematics that are not fulfilled elsewhere in the curriculum. Students pursuing graduate school in pure mathematics would benefit from Real Analysis II, Algebraic Structures II, Complex Variables, and in general should take a wide range of mathematics courses beyond those required to graduate. Combinatorics is an introduction to discrete mathematics and can lead to many research opportunities here and in graduate school. For those interested in graduate school in applied mathematics, Real Analysis II, Partial Differential Equations, Complex Variables, Numerical Methods, Mathematical Probability, and Statistics would be worthwhile. Automata Theory has connections to Computer Science, Logic and Philosophy, and Linguistics, for students interested in those topics.

Each student’s mathematical experience is complemented by courses outside of mathematics. Physics I allows students to experience mathematics applied in a physical science. Introduction to Computer Programming teaches students how to program computers, which is fundamental to much of applied mathematics, and is often the basis for many career options for math majors.

List all of the courses in your Program which fulfill GE Requirements.

Math 150 – Calculus IMath 316 – Biostatistics

Indicate which of these are being assessed as part of the major and which are not.

We are only assessing upper division mathematics courses that have a largeproof­writing element so as to measure writing competency.

Indicate which of these are regularly taught in an International Program.

None of the upper division mathematics courses are taught regularly in any of theInternational Programs.

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V. Alignment of PLOs with Core Competencies

Place a check mark to indicate which PLOs develop each of the Core Competencies.

PLO#1 PLO#2 PLO#3 PLO#4

Critical Thinking X X X

Information Literacy X X

Oral Communication X X

Written Communication X X

Quantitative Skills X X

VI. Assessment of Writing Competency

We are measuring student improvement in writing competency among our math majors in their upper division courses. The writing that students produce in these courses belongs to the mathematical proofs that they are asked to write as a part of their homework, midterm exams, and their final exam. Thus, all of the data on writing competency in our major is available as embedded assessment data and direct evidence.

We met in the fall of 2014 to determine what other evidence to collect in order to measure mathematical proof­writing. After lengthy discussion, our unanimous sentiment in terms of using our findings to improve our program is that direct evidence is the best measure. Moreover, we have the luxury of being able to measure this evidence longitudinally because of our research design, thus allowing us to see direct evidence of improvement in writing competency.

In addition, the rubric that we apply to analyzing all proof­writing is based on the telos of each proof ­ i.e. the solution set created by each course instructor and disseminated to students after completion of their proof­writing. The solution set represents the ideal/correct/canonical form of the proof that can be used as a point of comparison. This is how students become involved in discussing proof­writing ­ they read the instructor solution set and discuss its solutions in class and during office hours. In effect, having a single ideal solution to each assigned proof represents proof that the proof is correct. No other evidence is required.

Mathematical proofs constitute a writing genre that is unique in the liberal arts. These proofs must follow several strict guidelines of deductive or inductive reasoning, all of which are

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assessed by our faculty teams: a) correct use of mathematical notation, b) correct use of all assumptions given in the proposition/theorem being proven, c) correct logic within each proof, relying on previously established theorems, d) demonstration that the given proposition/theorem is in fact true, either by direct demonstration or reductio ad absurdum, therefore completing the proof itself.

VII. Assessment Plan

For each PLO, list the year it will be assessed and the direct and indirect forms of evidence that will be used in the assessment. Direct evidence arises from performance­based evaluations such as observation and student work samples. Indirect evidence arises from measures of perceived value such as surveys or questionnaires. Authentic evidence arises from measure of a student's ability to apply his or her learning and knowledge in real world applications. See http://jfmueller.faculty.noctrl.edu/toolbox/whatisit.htm#definitions

Our tentative plan for each PLO, by year, is:

PLO Number

Assessment Schedule

Direct Evidence

Indirect Evidence

Authentic Evidence

PLO1 2014­2015 Majors’ Coursework N/A N/A PLO2 2013­2014 ETS – MFT Math Alumni Survey ETS test, UG

Research PLO3 2016­2017 340 Projects Alumni Survey UG Research PLO4 2015­2016 List of Talks,

Service Participation Graduate Survey TBD

VIII. Status Report of Assessment Activities

Here are the results of our writing competency data collection and analysis for six distinct courses based on analysis of data from all of the students in each course, as authored by each faculty pair.

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Fall 2014 – Math 260, Linear Algebra

Courtney Davis and Tim Lucas

Course Description Systems of linear equations and linear transformations; matrix determinant, inverse, rank, eigenvalues, eigenvectors, factorizations, diagonalization, singular value, decomposition; linear independence, vector spaces and subspaces, bases, dimensions; inner products and norms, orthogonal projection, Gram­Schmidt process, least squares; applications; numerical methods, as time follows. Prerequisite: C­ or better in MATH 250 or concurrent enrollment.

Discussion Math 260 Linear Algebra is a required course for Mathematics majors, Computer Science/Mathematics majors, and the Applied Mathematics minor. It is generally taken by sophomores and juniors. The teaching of rigorous proof­writing technique is not the primary focus of this course; however, introductory proof­writing skills are developed through course materials, assignments, and exams.

We kept in mind the following when determining our expectations for proof­writing by Linear Algebra students: (1) aside from this class, some students have no background whatsoever in proof­writing, while some have taken multiple proof­focused courses such as Math 320 (Transitions to Abstract Mathematics); (2) many students are majoring in fields other than Mathematics, and thus do not have the same general mathematical background as the Mathematics majors (the Mathematics majors tend to be more adept in reading and writing proofs); (3) while proof­writing is a component of this course, it is not our main focus. It is also important to note that we have indicated that PLO #1 is introduced in Math 260 and students are not expected to have developed competence or mastery of this difficult skill at this time. Thus, we decided that a reasonable bar for success would be that a majority of students score at least 70% on proof questions (that is, a C–, the grade required to move on to other courses for which Math 260 is a prerequisite).

The proof question on the final exam that was assessed for all students is:

Let be . Prove that Nul is a subspace of and therefore is a vectorA x nn A Rn space. Be precise, and clearly state any key conclusions along the way.

This problem was chosen both to test each student’s ability to write a proof and to test his or her knowledge of subspaces in relation to vector spaces. The problem was a 10­point problem, for which they could receive 4 points for understanding what must be proven and the remaining points by showing the proof itself.

Twelve students took the final exam. The scores ranged from 4 to a full 10 points. Thus, every student received at least 4 points, the minimum required to show he or she understood the requirements for a subspace. The average and median scores were 7.8/10 (78%) and 9/10 (90%), respectively. The mode was a perfect 10, which was earned by 4 students (one­third of the

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class). Seven of 12 students (58%) earned a score of 7 (70%) or better on this problem. Thus, a majority of students did, in fact, earn 70% or better on this proof problem.

A total of three proof questions, plus a bonus proof question, were given on this final exam. The averages overall for the assigned proof problems were 78%, 84.5% and 69%. These tested subspaces, inner products, and symmetric matrix eigenvalues, respectively.

The bonus final exam question was the following.

Let T: be the transformation . (Notice and→ RR3 3 ([x x x ]) x x 0 ]T 1 2 3 = [ 2 1 x1 swap places.) Is a linear transformation? Prove yes or no.x2 T

This was given as a follow­up to a low­scoring question from the first midterm exam of the semester. The 8 students who attempted the final exam bonus problem earned an average score of 2.9/4 (71.8%). 5/8 (62.5%) of the students who answered the final exam bonus problem received scores of at least 3 (75%). We have not included students who did not answer the question because it was optional, which makes for an imperfect, but still useful, comparison.

This can be contrasted with the earlier midterm exam. On the first midterm, before students had much experience working with proofs, the following question was given.

State the definition of a linear transformation. Then determine if is a linear transformation. Show all work.([x x ]) x x ]T 1 2 = [ 2 1

If we eliminate the students who dropped the class and thus did not later take the final exam, the average on this 9­point problem was 3.625/9 (40.3%), with only 3 of 12 students earning at least 6.3/9 (70%). Furthermore, if we compare only the students who answered the bonus question, the average on the midterm exam question is 39.6% with only 2 of 8 (25%) of students earning at least 70%. Notice that the average of this last group is lower than the whole­group average; this is because both strong and weak students skipped the final bonus problem. While an imperfect comparison, the data show that students scored better on the final exam bonus question than on the midterm question. This suggests that the students developed a better understanding of proof­writing between the first and final exams.

Thus, overall, the students are performing satisfactorily, with the majority of students earning at least 70% on the proof questions by the final exam. Furthermore, development of proof­writing skills is apparent over the course of the semester. While some work can be done to introduce proof­writing more strongly from the start, overall we are pleased with the performance of the students in this class. We believe that the course has satisfied the goal of introducing students to proof­writing techniques and that students develop a satisfactory level of proof­writing in this course.

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***

Spring 2015 ­ Math 320 Transition to Abstract Mathematics

Kevin Iga & Kendra Killpatrick

Course Description Bridges the gap between the usual topics in elementary algebra, geometry, and calculus and the more advanced topics in upper­division mathematics courses. Basic topics covered include logic, divisibility, the Division Algorithm, sets, an introduction to mathematical proof, mathematical induction and properties of functions. In addition, elementary topics from real analysis will be covered including least upper bounds, the Archimedean property, open and closed sets, the interior, exterior and boundary of sets, and the closure of sets. Prerequisite: C­ or better in MATH 151. (PS, RM, WI)

Discussion Demographics: 18 students 10 sophomores 5 juniors 3 seniors

Two questions were evaluated on Midterm 3 (taken on March 23) and two questions were evaluated on the final exam (taken April 30). They were rated in the following manner: Correctness: out of 10 Readability: out of 2

Midterm Question 1 Suppose A ⊆ B and A⊆ C. Prove A ⊆ B∩ C.

Correctness: 10/10: 14 students 9/10: 4 students

Readability: 2/2: 15 students 1/2: 3 students

Midterm Question 2: Prove the following for all integers n ≧ 1, by induction. ∑k=1n (2k­1) = n2

Correctness: 10/10: 10 students

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9/10: 6 students 8/10: 2 students Readability: 2/2: 12 students 1/2: 6 students Final Question 1: Prove the following formula for n∈N using induction: ∑k=1n k = n(n+1)/2 Correctness: 10/10: 7 students 9/10: 4 students 8/10: 1 student 7/10: 2 students 6/10: 2 students 5/10: 1 student 4/10: 1 student Readabililty: 2/2: 11 students 1/2: 7 students Final Question 2: Suppose f : A → B is a function and A1 and A2 are subsets of A. Prove f(A1∩ A2) ⊆ f(A1) ∩ f(A2). Correctness: 10/10: 8 students 9/10: 5 students 7/10: 3 students 6/10: 2 students Readability: 2/2: 11 students 1/2: 7 students In summary, all but 4 students scored at least 7 out of 10 on correctness on all problems (we kept track of scores for each person), with a majority scoring 9/10 or higher. Our goal for the writing standard is to have at least 75% of the class achieving at least 70% or better on the writing score and we appear to have achieved this. For readability, no one scored 0, but scores of 1 varied from 17% to 39%. Still, the majority received 2 points. Readability is important since it is more a matter of the writing style of the proof than the mathematical correctness.

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***

Fall 2014, Math 335 ­ Combinatorics

Kendra Killpatrick & Don Thompson Course Description Topics include basic counting methods and theorems for combinations, selections, arrangements, and permutations, including the Pigeonhole Principle, standard and exponential generating functions, partitions, writing and solving linear, homogeneous and inhomogeneous recurrence relations and the principle of inclusion­exclusion. In addition, the course will cover basic graph theory, including basic definitions, Eulerian and Hamiltonian circuits and graph coloring theorems. Throughout the course, learning to write clear and concise combinatorial proofs will be stressed. Prerequisites: C­ or better in MATH 151 and MATH 320 or concurrent enrollment in MATH 320 or consent of the instructor. Discussion Math 335 – Combinatorics is an upper division elective normally taken by Math or Math Education majors. The course involves both solving combinatorial problems and writing combinatorial proofs, a new skill that students are introduced to in this course. On the final exam, one of the questions specifically addressed the writing of a combinatorial proof and this is the question we examined to see if students had mastered the skill of writing this type of a proof. Drs. Killpatrick and Thompson analyzed proof­writing by students enrolled in Math 335 ­ Combinatorics during the fall 2014 term. This course is an advanced elective for mathematics majors, all of whom have taken the calculus sequence and the department’s Transitions course which focuses on proof­writing and the process of increasing a student’s mathematical maturity. Math 335 lists proof­writing as one of its learning outcomes, stating that this course seeks: To develop the ability to write combinatorial proofs. Combinatorial proofs vary greatly from algebraic proofs in that equations and formulas are seldom used in an elegant combinatorial proof. Instead, combinatorialists appeal to basic counting arguments to prove complicated formulas. To facilitate the development of proof­writing abilities, students will often be asked to work in small groups in class to construct the proof of a theorem or to solve a set of problems and then present their solutions to the class. It is very important to note that combinatorial proofs, on which this course is primarily focused, represent a very specific way of thinking and writing, not found in other advanced mathematical courses nor found in writing in the humanities, social sciences, religion, communication or other

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disciplines represented at Seaver College. Thus, our report highlights the degree to which students demonstrate and improve in their ability to think and communicate combinatorially.

For this analysis, we gathered up a homework set of proofs that the students submitted on September 18, 2014 as well as a proof from their December final exam.

Homework Analysis

Students completed five homework problems from section 5.5 of Alan Tucker’s Applied Combinatorics book. All questions call for proofs of binomial identities that represent increasing complexity and sophistication. Questions ranged from committee­selection identities to combination/permutation counting to set partitions to binomial expansions and their derivatives to three­dimensional block walking. Each of these proofs requires that the student a counting technique in two different and equivalent ways without appealing to specific examples, but demanding the use of mathematical abstraction and Bloom’s cognitive taxonomy at three levels: analysis, evaluation, and synthesis. Algebraic proofs, a potential form of argument for these identities, are nearly impossible to generate, especially as problem complexity increases. Thus, these five homework exercises test, at a very early point in the semester, a student’s ability to understand combinatorial cognition and to express their understanding in clear and precise written form.

Students could score 3 points per problem for a maximum of 15 points on this assignment. The class scores were 11.5, 14, 14, 15, 13, and 5.5. The same students then completed proof­writing problems in their final exam, described below, allowing for an analysis of improvement in proof­writing.

Final Exam Analysis

The question on the final exam was:

Give a combinatorial proof that

There were 7 students in the class during the Fall 2014 semester. The question was worth 10 points. 3 of the 7 students scored a perfect 10 points and 1 student scored 9/10 points. The student who missed 1 point made a simple calculation error in concluding her proof, but she had the correct proof technique and the correct proof (up to this small notational error). The two students who scored 6 and 7 points both had the correct approach to the problem (a combinatorial proof versus an algebraic proof). Both students were able to correctly interpret the

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right hand side of the equation in a combinatorial fashion, but were unable to give a correct combinatorial interpretation of the left hand side. These students were able to clearly write more than half of the proof and had the correct proof technique. One student scored a 2 on this problem. This student did use the correct proof technique (i.e. she tried a combinatorial proof as opposed to an algebraic proof), but did not use a correct combinatorial interpretation for either side of the equation. Overall, we believe that 4 of the 7 students had mastered the skill of writing combinatorial proofs in a clear and correct way by the end of the semester. Two students demonstrated that the skill was still developing, although they could write more than half of the proof correctly, and one student demonstrated an inability to write a combinatorial proof, although this student did demonstrate an understanding of what type of proof a combinatorial proof is.

*** Spring 2015 ­ Math 355, Complex Variables

Kevin Iga & Tim Lucas

Course Description An introduction to the theory and applications of complex numbers and complex­valued functions. Topics include the complex number system, Cauchy­Riemann conditions, analytic functions and their properties, complex integration, Cauchy’s theorem, Laurent series, conformal mapping and the calculus of residues. Prerequisites: C­ or better in MATH 250 and MATH 320 or concurrent enrollment in MATH 320 or consent of the instructor. Discussion Complex Variables is an upper division elective course that is taken by sophomore, junior and senior mathematics majors and minors. Professor Lucas taught this course during the spring term of 2015. An important theme in the course is to extend results from calculus and analysis on the real line to the complex plane. For example, students are asked to use their understanding of limits of real functions to define the limit of a complex function. Similarly, they are asked to use their understanding of the definition of sequence convergence for real numbers to define sequence convergence for complex numbers. It is important to note 9 of the 11 students have taken an introduction to proof­writing through Math 320.

On the first exam we chose to assess the following question:

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For the final exam we chose to analyze the following similar question:

Before grading the problem, Professor Lucas and Professor Iga developed a rubric both of the proofs. On a scale from 1­4 the scores, we assigned the following scores: 4 represented a completely correct proof, 3 represented a proof that followed correct logic with minor details missing, 2 represented some understanding of the definitions without an understanding of how to put them together and 1 represented little to no understanding. Our expectation is that 75% of the students would show a sufficient understanding of proof­writing with a score of 3 or 4. The scores for each of the proofs are given in the table below.

Score Proof 1 Proof 2 4 5 7 3 4 1 2 2 2 1 0 1

For the first proof 9 out of 11 students (82%) received a score of 3 or 4 and for the second proof 8 out of 11 students (73%) received a score of 3 or 4. These numbers are close to our proficiency expectations. Although 2 students saw their scores decline from the first test to the final exam, 3 students improved their scores and 6 students maintained their scores.

Overall we are pleased with the results of this assessment. The majority of the students demonstrated an understanding of basic proof techniques that were introduced in Math 320: Transition to Abstract Mathematics and their proof­writing did improve throughout the course.

***

Fall 2014 ­ Math 370, Real Analysis I

Don Hancock & David Strong

Course Description Rigorous treatment of the foundations of real analysis; metric space topology, including compactness, completeness and connectedness; sequences, limits, and continuity in metric spaces; differentiation, including the main theorems of differential calculus; the Riemann integral and the fundamental theorem of calculus; sequences of functions and uniform convergence. Prerequisites: C­ or better in MATH 250 and MATH 320 or consent of instructor. Discussion

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As part of the ongoing assessment of our math majors’ ability to write mathematical proofs, professors Hancock and Strong analyzed a proof question from a mid­semester homework assignment and another from the final exam for Math 370 (Real Analysis I). Math 370 is a required upper division course, which was taught to 14 students in Fall 2014 by Professor Hancock. For each problem, proof­writing competency was evaluated by assigning to each student a score of 3, 2, or 1, corresponding to high, medium, and low achievement of written communication competency. Before grading began, professors Hancock and Strong discussed a general rubric for assigning points and hoped that at least 80% of the students would score a 2 or above. A score of 3 was viewed as corresponding to a letter grade of A, a score of 2 as B or C level work, and 1 as D or F work. To earn a 3, a student had to write a basically flawless proof, with suitable level of clarity and detail. A proof that had only a minor flaw in logic or mathematics earned a 2, while a student earning a 1 exhibited one or more major flaws, such as making significant unsupported statements, using definitions or theorems incorrectly, or drawing erroneous conclusions. The following question was assessed from the mid­semester homework assignment, during a time when the topic of compactness was being studied: “In an arbitrary metric space, prove that if every sequence has a convergent subsequence then every infinite subset has a cluster point.” This problem was completed by 13 students, while the remaining student left the problem blank. For the 13 students who attempted the problem the distribution was seven 3s, four 2s, and two 1s. Thus, 11 of the 14 students demonstrated at least a medium­level of achievement, only slightly below the 80% threshold we had hoped for. The students who earned a 1 recognized what the hypothesis and conclusion were, but were unable to articulate the critical observation that an infinite set must contain a sequence of distinct points. Although the four who earned a 2 displayed reasonable writing skills, they exhibited some minor misunderstanding as to what a cluster point means. For the final exam, the problem that was assessed concerned a continuity property for integral functions, which is generally considered as part of the Fundamental Theorem of Calculus: “Suppose that f is integrable (and thus bounded) on an interval I = [c,d], and a is some fixed

point in I. If , for all x in I, then use an epsilon­delta argument to prove that F is continuous on I. ” This problem was chosen because it tied together several key ideas from the latter half of the course (including epsilon­delta arguments, continuity, and properties of integrals), yet the solution was discussed in class and is of moderate difficulty. On the final exam this problem was worth 10 points out of 150, and when professor Hancock originally graded it the distribution was six 10s, three 9s, three 8s, one 5, and one 2. When professors Hancock and Strong assessed it using the 1­3 scale, there were nine students assigned a 3, three a 2, and two a 1, so roughly 85% exhibited at least medium­level written work, which exceeded expectations. Of the nine students at level 3, six had flawless proofs, while the remaining students’ only gap was a failure to use an upper bound for the absolute value of f instead of just for f. The three students who were rated as a 2 all included some unnecessary statements in their proofs, or had minor mathematical errors, but their logic was sound. Finally, for the two who were given a 1, one of them incorrectly assumed that f was continuous (and hence F

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differentiable), while the other demonstrated little understanding of what an epsilon­delta argument entails. It is worth noting that the analysis course is only offered every other year, and is populated by students with significantly varying levels of experience writing proofs. Coupled with the fact that real analysis is viewed as one of the most challenging undergraduate math courses, this helps explain why student performance in Math 370 is traditionally more diverse than in some other math classes. Overall, we are pleased with the results of our assessment and do not recommend changes at this time.

***

Spring 2015 ­ Math 470, Real Analysis II

Don Hancock & Don Thompson Course Description Convergence and other properties of series of real­valued functions, including power and Fourier series; differential and integral calculus of several variables, including the implicit and inverse function theorems, Fubini’s theorem, and Stokes’ theorem; Lebesgue measure and integration; special topics (such as Hilbert spaces). Prerequisite: C­ or better in MATH 370. Discussion Math 470 is a continuation of Real Analysis I. Thus, students who continue into part two of the yearlong sequence are mature mathematical thinkers and writers. By the time students have reached this advanced course, their proof­writing should be at a mastery level. We found that to be the case with this particular instantiation of Real Analysis I. Accordingly, we measure proof­writing as a function of two proofs, encountered by the students during a mid semester homework problem and again during their final exam. The first proof was taken from a homework assignment in the second month of the semester: Suppose f: is measurable, and g: is continuous. Prove that g f is measurable→ RA (A)→R f ∘ in A. This homework problem was awarded a score of 0, .5, or 1. Each of the five students earned a score of 1, demonstrating their proof­writing competency on a relatively simple problem. In effect, students have already achieved the desired competency. What changes for them over the course of the semester is not the writing exercise itself, but the complexity of the material. By the end of the course, students are mastering concepts that are easily found in a graduate course in Real Analysis. The final exam contained the following problem for our evaluation:

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Suppose that A1 A2 A3 ... is an expanding sequence of measurable subsets of A. ⊆ ⊆ ⊆

Prove that m( ) = (A )⋃∞n=1 n (m(A ) )limn → ∞ n

This problem was worth 10 points, and the students earned the following scores: 10, 10, 9, 8, & 8. The errors that students committed were largely cosmetic and not functionally problematic. Thus, overall, the students stayed at approximately the same level of high achievement between their homework problem and final exam problem, demonstrating that they have achieved mastery of mathematical writing competency.

***

Spring 2015 ­ Math 260, Linear Algebra

Courtney Davis & Kendra Killpatrick

Course Description Systems of linear equations and linear transformations; matrix determinant, inverse, rank, eigenvalues, eigenvectors, factorizations, diagonalization, singular value, decomposition; linear independence, vector spaces and subspaces, bases, dimensions; inner products and norms, orthogonal projection, Gram­Schmidt process, least squares; applications; numerical methods, as time follows. Prerequisite: C­ or better in MATH 250 or concurrent enrollment. Discussion Linear Algebra, Math 260, is a course for advanced freshman (those coming in with significant AP Calculus credit) or sophomore math majors or minors. Students take the course after completing with first two or three semesters of calculus and generally prior to taking Math 320 ­ Transition to Abstract Mathematics which is the course that teaches students how to write proofs. Some students take the course concurrently with Math 320 and these students are generally better at writing proofs than the other students in the class. proof­writing is considered an emerging skill at the Math 260 level ­ we want to expose our students to good proof­writing techniques and ask them to learn how to write very specific and basic kinds of proofs, but proof­writing is not the focus of this course.

To assess the writing component of Math 260, Dr. Killpatrick (the professor for the Spring 2015 section of the course) assigned students the following proof­writing problem first as a homework assignment, then later the same question appeared on both the midterm and the final so that we could track student progress.

Problem: Given an m x n matrix A, prove that Nul (A) is a subspace.

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Students were allowed to use any materials on the homework assignment (books, help from other students, etc.), but had no additional materials allowed on the midterm or the final. A grading rubric was developed by Dr. Davis and Dr. Killpatrick. Students earned 2 points for correctly identifying the 3 parts necessary to complete the proof that Nul(A) is a subspace. Then students earned 2 points for correctly proving that 0 is in Nul(A) and 3 points each for proving that Nul(A) is closed under addition and scalar multiplication.

The homework assignments, midterm and final were then graded separately by Dr. Killpatrick and Dr. Davis (who often teaches the course and taught the course in Fall 2014) based on the rubric above. To establish inter­rater reliability, scores were compared on each assignment. On the homework question, Dr. Davis scored an average of 7.19 while Dr. Killpatrick scored an average of 6.79. Once assignments were separately graded, Dr. Davis and Dr. Killpatrick met to jointly determine scores and determined that 9 of 16 students completed the assignment with a score of 7 or above (3 students did not complete the assignment).

On the midterm and final exam, Dr. Davis and Dr. Killpatrick graded the exams not only on mathematical correctness (out of 10 points) but also on readability (out of 2 points) and then met in person to compare scores and determine a final score for each student. The readability score is a measure of the written coherence of the proof, not simply the mathematical correctness. It is possible to construct a mathematically correct proof that still does not read well and we would like students to develop both skills. On the midterm, Dr. Davis gave an average mathematical score of 4.3 and Dr. Killpatrick gave an average score of 4.38 while on the final Dr. Davis gave an average mathematical score of 8.53 while Dr. Killpatrick gave an average score of 8.32.

On the midterm, only 5 of 19 students scored 7 or more out of 10 on the mathematical score and only 5 out of 19 scored a 2 out of 2 on the readability score. This is a lower percentage than we would like to see since our goal is to have 70% of our students achieving a score of 70% or better on the proof­writing questions. However, on the final exam, 16 out of 19 students (84%) achieved a score of 7 or more on the mathematical portion and 10 out of 19 scored a 2 on the readability score. These scores on the final exam exceeded our expectations for students in Math 260 since we simply expect students proof­writing skills to be emerging at this point in their mathematical career and not fully developed. Since for many students this is their first experience with writing proofs, it is natural that the early scores on the midterm are not great but that they really learn throughout the semester how to write clear and logically correct proofs and can demonstrate this skill by the final exam.

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IX. Closing the Loop – We plan to close the loop during our weekly mathematics faculty meeting discussions in the fall of 2015, a time when we can look at the entire body of work accumulated in AY2014­15 and decide how best to proceed. To do so any sooner is logistically prohibitive.

X. Changes in Response to Previous Action Items

N/A

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