DEPARTMENT OF MATHEMATICS ASSESSMENT ... OF MATHEMATICS ASSESSMENT REPORT 2008{2009 DAVIS COPE, ANGELA HODGE, AND SEAN SATHER-WAGSTAFF (CHAIR) Contents Part I. Department of Mathematics

DEPARTMENT OF MATHEMATICS ASSESSMENT REPORT2008–2009

DAVIS COPE, ANGELA HODGE, AND SEAN SATHER-WAGSTAFF (CHAIR)

Contents

Part I. Department of Mathematics Assessment Strategy 3

I.1. Introduction 3

I.2. Assessment Scheme 3

I.3. Rationale 4

I.4. Changes to the Assessment Strategy 5

I.5. Structure of the report 5

I.6. Future assessment activities 6

Part II. Overview Statement 6

Part III. Data from Final Examinations, Analysis and Reflections 7

III.1. Math 270, Introduction to Abstract Mathematics, first section 7

III.2. Math 270, Introduction to Abstract Mathematics, second section 11

III.3. Math 270, Introduction to Abstract Mathematics, third section 15

III.4. Math 420, Abstract Algebra I, only section 19

III.5. Math 421, Abstract Algebra II, only section 21

III.6. Math 429, Linear Algebra, first section 23

III.7. Math 429, Linear Algebra, second section 25

III.8. Math 445/645, Differential Geometry, only section 29

III.9. Math 446/646, Introduction to Topology, only section 35

III.10. Math 450, Real Analysis I, only section 42

Date: September 17, 2010.1

2 DAVIS COPE, ANGELA HODGE, AND SEAN SATHER-WAGSTAFF (CHAIR)

III.11. Math 451, Real Analysis I, only section 44

III.12. Math 472, Number Theory, only section 46

III.13. Math 478, History of Mathematics, only section 49

III.14. Math 480, Applied Differential Equations, only section 51

III.15. Math 483, Partial Differential Equations, only section 54

III.16. Math 488, Numerical Analysis I, only section 57

Part IV. Departmental Reflection 59

Part V. Summary 60

DEPARTMENT OF MATHEMATICS ASSESSMENT REPORT 2008–2009 3

Part I. Department of Mathematics Assessment Strategy

I.1. Introduction

Departmental assessment consists of several interrelated goals. Departments are encour-aged to identify specific course and program objectives and to assess the degree to whichstudents are meeting these objectives. To achieve these goals, the Mathematics Depart-ment is continuing the assessment strategy initiated with the January 2009 report. Thisstrategy involves (1) the collection and analysis of summative assessment data from finalexaminations in certain mathematics courses (rotating annually through a cyclic protocol)and (2) faculty reports of formative assessment activities in these courses.

I.2. Assessment Scheme

We begin with the following model for course objectives in mathematics. We consider thelist of topics from the NDSU Course Bulletin for a given course. The desired outcomes ofthis course then take the following form: Students should be able to demonstrate throughwritten assignments, quizzes and exams that they have developed an understanding ofthe course topics. For instance, a student’s success in Math 165 will be measured by hisor her ability to solve problems related to the topics of limits, continuity, differentiation,Mean Value Theorem, integration, Fundamental Theorem of Calculus and applications.Instructors are encouraged to augment this list as necessary.

To assess students’ degrees of success in mathematics courses, we collect and analyzenumerical data from written final examinations given in all mathematics courses. Ourprocedures for collection and analysis of these data are described as follows.

For each course being assessed for this report, the instructor supplied the committee withthe following:

(a) A copy of the final examination;(b) A list of the course objectives being assessed by each final exam question;(c) For each student, the numerical scores for the individual questions on the exam;(d) The course syllabus for each course section being assessed; and(e) A description of formative assessment activities conducted for the course.

In addition, instructors for the course sections being assessed were asked to consider thefollowing rubric for assessing the degree of success in solving a given problem:

A. Completely correct;B. Essentially correct—student shows full understanding of solution and only makes a

minor mistake (e.g., wrong sign when calculating a derivative or arithmetic error);C. Flawed response, but quite close to a correct solution (appears they could do this type

of problem with a little review or help);


D. Took some appropriate action, but far short of a solution;E. Blank, or nothing relevant to the problem.

Using a scale of 0 to 100, these instructors were asked to describe the range of scoreswhich (according to their personal rubric) falls into each of the above categories.1 Forexample, one possible response is the following:

A. 100B. 80–99C. 50–79D. 30–49E. 0–29.

For each final exam question, the committee calculated the percentages of students inthe class whose numerical score satisfied the instructor’s degree-of-success criteria. Thecommittee then provided the instructor with these data and asked for the instructor’sreactions to the data. Specifically, the committee asked each instructor to answer thefollowing two questions:

1. What did you learn from these data?2. What will you do differently as a result of what you learned?

Finally, these data was presented to the department as a whole, and we collected individualfaculty members’ reactions to the data via the survey “Assessment of Student Learning:Self-Reporting of Levels of Implementation.”

I.3. Rationale

We choose to focus primarily on scores from final examinations for several reasons. First,because final exams in mathematics are often cumulative in nature, data from these examsshould contain information about each of the desired learning outcomes associated witha given course. Second, assessment requires a certain amount of data. However, we needto restrict ourselves to a reasonable amount that does not cause an undue burden for thedepartment or for the Assessment Committee.

With this report, we begin to include reports of formative assessment activities for tworeasons. First, in its review of our previous report, the University Assessment Commit-tee suggested the inclusion of such information. Second, and more importantly, theseactivities are important measures of the mathematics faculty’s commitment to studentlearning. It should be noted that most faculty have only described the formative assess-ment activities they conducted; they did not report the outcomes of this assessment. Wefeel that this fits with the intent of formative assessment.

1Our assessment of Math 480 uses a slight modification of this scale, at the instructor’s request.


I.4. Changes to the Assessment Strategy

In light of the University Assessment Committee’s review of the Mathematics Depart-ment’s most recent assessment report, we have made several changes in our assessmentstrategy.

(1) We now include faculty reports of formative assessment activities;(2) We have removed plans for assessment activities that are programmatic in nature, as

suggested in the review of last year’s report;(3) We have not included course rubrics from individual syllabi, as we feel that they do

not contribute to the report;2 and(4) We have included plans to assess our graduate courses, using the same strategies as

for our undergraduate courses.

It should be noted that we plan to continue including the statements of individual examquestions in the report. In the University Assessment Committee’s review of our mostrecent assessment report, it was suggested that the length of the report would be reducedby excluding these statements, “without loss of meaningful information.” We agree thatthis would reduce the length of the report. However, we feel that it is important, for themathematics faculty’s understanding of the data presented in the report, to include thesestatements. For instance, the specific wording of a given question or its difficulty maypresent a key to understanding student success (or lack thereof) with the question.

I.5. Structure of the report

Part I of this report is a description of the Mathematics Department’s assessment strategy.

Part II contains an overview of the report.

Part III contains the data (both quantitative and qualitative) collected by the committee.It is organized as follows. Each section of this part of the report is devoted to a specificsection of one of the courses assessed. Within each section, the first subsection containsthe final examination questions; after each question, we have listed the outcomes assessedby that question (as indicated by the instructor) and the degree-of-success data (as calcu-lated by the committee). The second subsection contains the instructor’s reaction to thedata. The final subsection contains a description of the instructor’s formative assessmentactivities for the course.3

Part IV of the report contains reactions of individual faculty members to the report.

Part V is a concluding summary.

2Course rubrics from syllabi are available by request.3The section for Math 478 follows a different format because (a) the list of outcomes only contains one

item, and (b) students made final presentations instead of taking final examinations.


I.6. Future assessment activities

The Assessment Committee plans to continue collecting and analyzing data in this fashionfor the next several years. This strategy will allow the department to amass a nontrivialamount of data; we expect that this will enable us to identify trends within the depart-ment. The committee plans to assess certain courses each year according to a cyclicprotocol. Specifically, the previous report contained an analysis of the data for Math 165to 266. The current report assesses undergraduate courses numbered 267 and higher. TheJanuary 2011 report will assess courses numbered 164 and lower, and the January 2012report will assess graduate courses. Then the cycle will repeat.

In future reports, we plan to consider further activities including the following:

(i) Establish explicit desired outcomes for students majoring or minoring in mathemat-ics, and identify methods for assessing these outcomes;

(ii) Establish explicit desired outcomes for students in mathematics courses who arenot majoring or minoring in mathematics, and identify methods for assessing theseoutcomes; and

(iii) Establish explicit desired outcomes for students completing graduate degrees inmathematics, and identify methods for assessing these outcomes.

Part II. Overview Statement

The Department of Mathematics Assessment Committee collected numerical scores forindividual questions from final examinations from Fall 2008 and Spring 2009 for Mathe-matics courses numbered 270 (3 of 3 sections), 420 (1 of 1 section), 421 (1 of 1 section),429/629 (2 of 2 sections), 445/645 (1 of 1 section), 446/646 (1 of 1 section), 450 (1 of 1section), 451 (1 of 1 section), 472 (1 of 1 section), 478 (1 of 1 section), 480 (1 of 1 section),483 (1 of 1 section), and 488 (1 of 1 section).

These data were analyzed according to a rubric determined in collaboration with theindividual instructors. The instructors also described formative assessment activities forthe courses being assessed. The instructors and other faculty members were asked toreflect on the data.


Part III. Data from Final Examinations, Analysis and Reflections

III.1. Math 270, Introduction to Abstract Mathematics, first section

III.1.1. Final exam.

1. (10) Using correct set builder notation rewrite the following set:

{· · · ,−7,−4,−1, 2, 5, 8, · · · },

Outcomes assessed: sets

Degree of success: A 31% B 7.5% C 23% D 31% E 7.5%

2. (25) State complete definitions:

(a) countably infinite,

Outcomes assessed: infinite sets

Degree of success: A 46% B 0% C 15% D 8% E 31%

(b) transitive relation,

Outcomes assessed: relations


(c) tautology,

Outcomes assessed: symbolic logic


(d) A = B for sets A and B,



(e) inverse image of D under f : A→ B, f−1(D),

Outcomes assessed: functions

Degree of success: A 31% B 0% C 7.5% D 7.5% E 54%

3. (25) State any one of the Principles of Mathematical Induction.

Outcomes assessed: math induction and its equivalents



4. (10) Negate the statement (∀x ∈ N )(∃y ∈ Z)(x+ y = 0)

Outcomes assessed: symbolic logic


5. Consider the proposition that for integers a, b with a 6= 0, b 6= 0, if a|b and b|c, thena|c.

(a) (20) Construct a thorough “know-show” table for a proof

Outcomes assessed: methods of proof


(b) (10) What is the corresponding “method of proof”?



6. (40) Prove that for all positive integers n,n∑i=1

i =n(n+ 1)

2.

Outcomes assessed: methods of proof; math induction and its equivalents


7. (30) Prove that for all positive real numbers x, if x is irrational, then 3√x is irrational.

Outcomes assessed: methods of proof; number systems


8. (30) Prove that if r is a real number such that r2 = 3, then r is an irrational number.

Outcomes assessed: methods of proof; number systems



III.1.2. Reflection.

1. What did you learn from these data?

I learned that the students didn’t understand set builder notation to the extent that I feltthey should.

I learned that the students came away from the course with a reasonably good knowledgeof the definitions.

I learned that on Problems 4–7, the students were quite well versed. In contrast they didnot do well at all on Problem 8. On Problem 3 requiring the statement of the Principleof Mathematical Induction, a very important topic to Math 270, the results were mixed,while the results of Problem 6 indicated a reasonably good knowledge of how to use thisprinciple.

2. What will you do differently as a result of what you learned?

I will consider emphasizing more the set builder notation. Problem 8 required an argue-ment by contradiction. I will make sure that the students see more of that. Since thestudents did reasonably well on the Priciple of Mathematical Induction problem, I willbe less concerned about the students inability to state the principle than if they had notdone the problem as well. Nevertheless, I will give additional emphasis to its statement.

III.1.3. Formative Assessment. Instructor collected information from students via in-troductory questionnaire at beginning of semester. Instructor also collected feedbackon the course from the students about half-way through the semester. (Questions areincluded on next page.)


MATHEMATICS 270–INTRODUCTION TO ABSTRACT MATHEMATICS

QUESTIONAIRE, DUE 29 AUGUST

NAME: ID #:

1. What is a statement or proposition?

2. What is a conditional statement?

3. What does it mean for a set of numbers to be closed under an operation?

4. What are dates of the two hour exams and the date and time of the final exam?

5. Will you be expected to contribute bluebooks? If so, how many will you owe if youcontribute on 15 October 2008?

6. What is the prerequisite for this course? How have you satisfied this prereq?

7. What do you want your instructor to know about you? (at least one item requested)(Continue on the back if necessary.)

MATHEMATICS 270

FORMATIVE EVALUATION – DUE 27 OCTOBER 2008

1. Are you satisfied that you are learning proof? If not what changes will improve thecourse for you?

2. How do you feel about the pace of the course?

3. How formally do you wish sections to be covered? Is it helpful when the section isoutlined by a fellow student (or you, yourself)? Any suggestions for improving studentsummaries?

4. How formally do you wish that problems be covered?

5. Do you wish to have regular quizzes in the future?

6. What do you like best about the way the course Math 270, Introduction to AbstractMathematics, is conducted?

7. What do you like least about the way the course Math 270, Introduction to AbstractMathematics, is conducted?

8. Was the College Bowl activity (of a few weeks ago) helpful to you?

9. Was the Math Down activity (of 20 October) helpful to you?


III.2. Math 270, Introduction to Abstract Mathematics, second section

III.2.1. Final exam. Each question is worth 20 points.

1. Complete each definition:

(a) The converse of the conditional statement P → Q is. . .

Outcomes assessed: propositions; symbolic logic

(b) The contrapositive of the conditional statement P → Q is. . .

Outcomes assessed: propositions; symbolic logic

(c) Two functions f : A→ B and g : C → D are equal provided that. . .


(d) A function f : A→ B is an injection provided that. . .


(e) A function f : A→ B is a surjection provided that. . .



2. Answer true or false for each of the following. You may justify your answers, but youare not required to do so.

(a) If n is a natural number, then the relation “congruence modulo n” is an equivalencerelation on Z.

Outcomes assessed: equivalence relations

(b) Every function has an inverse function.


(c) If f : A→ B and g : B → C are bijections, then (g ◦ f)−1 = g−1 ◦ f−1.Outcomes assessed: functions

(d) If f : A→ B and g : B → C are functions such that g ◦ f is a surjection, then f isa surjection.


(e) If f : A→ B is a bijection, then f−1 : B → A is a bijection.




3. You may justify your responses to any of the following, but you are not required to doso.

(a) Write the statement of the Division Algorithm.

Outcomes assessed: number systems

(b) Let R+ = {r ∈ R | r > 0}. Write the negation of the following statement insymbolic form in which the negation symbol is not used:(∀ε ∈ R+)(∃δ ∈ R+)(∀x ∈ R)(|x− 1| < δ → |x2 − 1| < ε).

Outcomes assessed: symbolic logic; quantifiers

(c) Give an example of a function that is not an injection.


(d) Give an example of a function that is not a surjection.


(e) List the distinct congruence classes modulo 7.



4. Define the relation ∼ on Z as follows: For a, b ∈ Z, a ∼ b if and only if 2 | (a + b).Prove that ∼ is an equivalence relation.



5. Let f : R→ R be defined by f(x) = x2 + 3x+ 2. Define the relation ∼ on R as follows:For a, b ∈ R, a ∼ b if and only if f(a) = f(b). (We have already shown that ∼ is anequivalence relation.) Use the roster method to describe [0], the equivalence class of 0.Justify your answer.



6. Let f : A→ B be an injection, and let S, T ⊆ A. Prove that f(A∩B) = f(A)∩ f(B).

Outcomes assessed: functions; sets



7. (a) Prove that, for each integer a, if a 6≡ 0 (mod 11), then a2 6≡ 0 (mod 11).


(b) Prove that the real number√

11 is an irrational number.



8. Let f : Z× Z→ Z be defined by f(a, b) = ab.

(a) Is f an injection? Justify your answer.(b) Is f a surjection? Justify your answer.(c) Is f a bijection? Justify your answer.(d) Does f have an inverse function? Justify your answer.



9. (a) Show that

(1

3+

1

2+

7

6

)is a natural number.

(b) Show that

(8

3+

4

2+

14

6

)is a natural number.

(c) Prove that, for each natural number n, the quantity

(n3

3+n2

2+

7n

6

)is a natural

number.



10. (a) Prove that, for all a, b ∈ Z, we have (a+ b)2 ≡ (a2 + b2) (mod 2).(b) Prove that, for all a, b ∈ Z, we have (a+ b)4 ≡ (a4 + b4) (mod 2).(c) Prove that, for all a, b ∈ Z and for all n ∈ N, we have(a+ b)(2

n) ≡ (a(2n) + b(2

n)) (mod 2).




11. [20 points (bonus)] Prove that there are infinitely many prime numbers. [Hint: Sup-pose that there are only finitely many prime numbers p1, p2, . . . , pn. Derive a contradictionby showing that the number (p1p2 · · · pn) + 1 has no prime factor.]





Students still struggle with various aspects of functions: injectivity, surjectivity, bijec-tivity, images of sets, and functions of several variables. This does not surprise me, asstudents usually struggle with these things.

I was surprised by the degree of success for question 9. The students seemed to understandbasic induction problems like this during the semester. On the other hand, question 10was also an induction problem, and they did fine on that one. I am not sure what makesquestion 9 harder than question 10.

When I graded this exam, I was surprised that most students did not even attempt toanswer question 11, especially given the fact that the hint contained most of the solution.Perhaps the students ran out of time? Perhaps they assumed that they did not need theextra credit?


I need to brainstorm with my colleagues about effective ways to help the students learnabout functions.

III.2.3. Formative Assessment. I performed analyses of the scores from the midtermssimilar to the analysis of final exam scores for the assessment report. Scores on the firstmidterm were very high, showing that students were comfortable with the material fromthe first third of the course. Scores on the second midterm were lower, especially onthe two questions dealing with rationality/irrationality. As a result, I spent more timeworking on these ideas. I included a similar question (number 7) on the final exam, andscores were much higher.


III.3. Math 270, Introduction to Abstract Mathematics, third section


1. (4 points) For the following give your answers as “true” or “false” and justify youranswers.

a) The domain of a reflexive relation R ⊂ A× A cannot be a proper subset of A.b) The inverse of any relation is a relation.c) Every equivalence relation is a reflexive relation.d) If R is an equivalence relation on a nonempty set A, then for every a ∈ A it is true

that [a] 6= ∅.

Outcomes assessed: functions and relations; equivalence relations


2. (6 points) State the definitions of any two of the following: Relation, function,transitive relation, equivalence relation.



3. (16 points) For the following, provide examples with the prescribed properties.

a) A statement that can be proved by E-PMI but not PMI.b) A relation which is not a function.c) A relation which is reflexive and symmetric but not transitive.d) An equivalence relation with at least two distinct equivalence classes.

Outcomes assessed: mathematical induction; functions and relations; equivalence rela-tions



4. (12 points) Consider the following statements

i) x ∈ R =⇒ (x3 > 1 ∨ x2 > 1),ii) (x ∈ Z ∧ x > 0) =⇒ (∃y ∈ R)(y =

√x),

iii) (∀n ∈ Z)(√n− 2 > 0).

Negate the statement (iii); write the converse of the statement (ii); write the con-trapositive of the statement (i).

Outcomes assessed: symbolic logic; propositions; quantifiers


5. (14 points) Prove any two of the following statements using different methods ofproof.

a) For each x ∈ R, if 0 < x < π2, then sinx+ cosx > 1.

b) For all x, y ∈ R, if x 6= 0 and y 6= 0, then√x2 + y2 6= x+ y.

c) For all x, y ∈ R, if x > 0 and y > 0, then 2x

+ 2y6= 4

x+y.



6. (8 points) Prove one of the following statements:

a) A− (B ∪ C) = (A−B) ∩ (A− C)b) If X ⊂ Y , then A×X ⊂ A× Y .



7. (20 points) Use PMI or E-PMI to prove any two of the following statements:

a) For any n ∈ N, 3 divides n3 + 23n.b) For any n ∈ N, n > 2, 3n > 1 + 2n.c) For any n ∈ N, n > 6, 2n > (n+ 1)2.d) For any n ∈ N, n > 3, (1 + 1

n)n < n.

Outcomes assessed: methods of proof; mathematical induction



8. (20 points) Let R and S be relations on R defined by

xRy if and only if |x| < y; and xSy if and only if y = x2 + 1.

a) Determine if these relations are reflexive and symmetric.b) Is any one of them a function? Explain.c) Find R−1 and S−1 and their domains.



9. (20 points) Pick your favorite realtion R or S on A below

aRb⇐⇒ 2a+ 3b ≡ 0(mod5), where A = Z,

(a, b)S(c, d)⇐⇒ ad = bc, where A = {(x, y) ∈ Z2 : y 6= 0},and

a) show that it is an equivalence relation,b) exhibit at least two distinct equivalence classes,c) find its domain, range and inverse.

Outcomes assessed: equivalence relations; number systems


Bonus. (10 points) State and profe the PMI. (You can assume the Well Ordering Prin-ciple, which states that: every nonempty subset of N has a least element.)

Outcomes assessed: mathematical induction




I learned from the data that: a) students have a good understanding of the conceptsrelated to symbolic logic, propositions, quantifiers, methods of proof, sets and PMI (theirsuccess rate in relevant test questions is 80% in average); b) their understanding of con-cepts related to relations and equivalence relations needs improvements (their success ratein relevant test questions is 54% in average).



Having learned this fact, next time I’ll make sure to emphasize the importance of conceptsrelated to relations and equivalence relations, and spend a little more time on thesesubjects. In the flow of the subjects of this course, these concepst are covered towardsthe end of the semester; this also contributes to the student understanding, sometimes,negatively.

I should plan the delivery of the subjects much carefully so that adequate time is spentto cover and investigate concepts related to relations and equivalence relations.

III.3.3. Formative Assessment. In addition to usual test, quiz and assignments, I alsocoducted a pre-post test.


III.4. Math 420, Abstract Algebra I, only section


1. (20p) Let R be a commutative ring with 1 6= 0.

(a) For I, J ideals of R, denote I + J = {x+ y | x ∈ I, y ∈ J}.Prove that I + J is the smallest ideal containing I ∪ J .

(b) Let m1, m2 be two maximal ideals of R with m1 6= m2. Prove that m1 +m2 = R.

Outcomes assessed: rings; ideals


2. (20p) Let R, S be commutative rings and let φ : R→ S be a ring homomorphism.

(a) Prove that if J is an ideal of S, then φ−1(J) is an ideal of R.(b) If I is an ideal of R, is it always true that φ(i) is an ideal of S? (Justify your

answer.)

Outcomes assessed: homomorphisms; rings; ideals


3. (10p) Let R = {a + bi | a, b ∈ Z} ⊆ C. Show that R is a commutative ring. Find allthe units in R.

Outcomes assessed: rings; integers


4. (10p) Let G be a group and define

Z(G) := {g ∈ G | gx = xg for all x ∈ G}.Prove that Z(G) is a subgroup of G. Is it always true that Z(G) is a normal subgroup ofG?

Outcomes assessed: Groups



5. (10p) Prove that the ring C[X, Y ]/(X2 + Y 2) is not an integral domain.

Outcomes assessed: integral domains


6. (10p) Let G be a group with no proper, nontrivial subgroups, and assume that G hasmore than one element. Prove that G is isomorphic to Zp for some prime p.

Outcomes assessed: Groups


7. (10p) prove that the group Z2008 × Z2 is not cyclic.

Outcomes assessed: Groups; cyclic groups


8. (10p) Let H and K be finite subgroups of a group G whose orders are relatively prime.Prove that H ∩K = {e}.Outcomes assessed: Groups; subgroups




Absolutely nothing new. I have graded all of those exams, so I already knew what thestudents were able to do and what they were not able to do.


I am not sure what I will do differently (I will not teach this course very soon). ButI would certainly consider emphasizing deeper results in group theory during the firstsemester of the sequence (Math 420).

III.4.3. Formative Assessment. n/a


III.5. Math 421, Abstract Algebra II, only section


1. (10p) Find the degree of Q(213 , 5

14 ) over Q.

Outcomes assessed: field extensions


2. (a) (10p) For any positive intebers a and b show that Q(√a,√b) = Q(

√a+√b).

(b) (5p) Find the minimal polynomial fo α =√

2 +√

3 over Q.

(c) (10p) Find a polynomial with rational coefficients g(X) such that α−1 = g(α).

(d) (5p) Find a polynomial f(X) ∈ Q[X] such that f(α) =√

2.

Outcomes assessed: fields; field extensions

Degree of success: A 12.5% B 0% C 37.5% D 50% E 0%

3. (10p) Find the degree [L : Q] where L is the splitting field of x4 − 2 over Q.

Outcomes assessed: field extensions

Degree of success: A 62.5% B 0% C 0% D 37.5% E 0%

4. (10p) Show that there is no simple group of order 56.

Outcomes assessed: simple groups; Sylow theorems

Degree of success: A 25% B 25% C 12.5% D 25% E 12.5%

5. (10p) Let Φ8(X) be the 8th cyclotomic polynomial. Prove that Φ8(X) = X4 + 1. If nis a positive integer, what is Φ2n(X)?

Outcomes assessed: cyclotomic polynomials

Degree of success: A 25% B 25% C 12.5% D 25% E 12.5%

6. (10p) Find the Galois group of the splitting field of X4 − 2 over Q.

Outcomes assessed: Galois Theory

Degree of success: A 37.5% B 0% C 37.5% D 12.5% E 12.5%


7. (10p) How many nonisomorphic abelian groups with 2008 elements are there?

Outcomes assessed: structure of finite abelian groups

Degree of success: A 87.5% B 12.5% C 0% D 0% E 0%

8. (10p) Late p be a prime number. What is the splitting field of Xp −X over Zp?Outcomes assessed: fields; field extensions

Degree of success: A 50% B 0% C 0% D 12.5% E 37.5%



Absolutely nothing new. I have graded all of those exams, so I already knew what thestudents were able to do and what they were not able to do.


I would cover more serious results in factorization and have that extensively reflected inthe final exam. (Having 2 extra weeks of no classes during the Spring of 2009 certainlydid not help.)



III.6. Math 429, Linear Algebra, first section


Problem 1. Find the determinant and the rank of the matrix

A =

1 −1 −1 3 00 4 0 1 00 2 1 5 1−2 1 1 3 00 1 0 0 1

Outcomes assessed: Vector spaces; linear transformations


Problem 2. The real polynomials of degree at most n form a linear space over R withrespect to usual addition and multiplication by a scalar. Let Vn denotes this linear space.

a) What is the dimension of Vn? Why?

b) for n = 3, are the vectors 3t3− t2 + 3t+ 1, t3 + t, t+ 1, t2 + t+ 1 linearly independentin V3?

c) Let F : Vn → Vn be a linear map given by the formula F (p(t)) = ddtp(t). Find

dimensions of the range and of the kernel(nullspace) of F .

Outcomes assessed: Vector spaces


Problem 3. Let A, B be 2×2 matrices. Prove that tr(AB) = tr(BA). Does the formulastill hold for n× n matrices for any n > 1?

Outcomes assessed: linear transformations


Problem 4. Find the eigenvectors of the matrix

B =

3 0 00 3 30 0 3

Outcomes assessed: eigenvalues and eigenvectors



Problem 5. For which real values of α is the matrix

C =

(1 4α 1

)diagonalizable?

Outcomes assessed: eigenvalues and eigenvectors; canonical forms


Problem 6. Let A be a 3× 3 matrix with real entries. Prove that if all eigenvalues of Aare equal to zero then A3 = 0.

Outcomes assessed: canonical forms




It was generally interesting to see the data. One striking fact I noticed is that for everysingle problem, the result in line B is 0%, that is no student has solved any of the problems“essentially correctly”.


This made me to think that I need to arrange a better balance in writing final examproblems; and I took this into account already in the Final Exam in Fall 2009 (for thesame course Math 429). By studying final exam works, I can say that the result is muchbetter and more balanced.



III.7. Math 429, Linear Algebra, second section


1(a). (7) State complete definitions: Ker(L)



1(b). (7) State complete definitions: Linear map



1(c). (7) State complete definitions: The fourier coefficient of g with respect to f , wheref(x) = sin kx.

Outcomes assessed: applications


1(d). (7) State complete definitions: Eigenvector



1(e). (7) State complete definitions: Characteristic polynomial PA(t)



2(a). (13) Complete the statements of the following results: Theorem I.3.2. Let V be avector space. Suppose that one basis has n elements and another basis has m elements.Then . . .




2(b). (13) Complete the statements of the following results: Theorem IV.3.4. Let V,W,Ube vector spaces. Let B,B′,B′′ be bases for V,W,U respectively. Let F : V → W andG : W → U be linear maps. Then . . .



2(c). (13) Complete the statements of the following results: Theorem V.1.1 SchwarzInequality. For all u, v ∈ V we have . . .

Outcomes assessed: inner product spaces


3(a). (13) Let A and I be n × n matrices, where I is the identity (unit) matrix. Provethat if A4 = 0, then I − A is invertible.



3(b). (20) Prove that if C is a convex subset of vector space Rn, then L(C), the imageof C under L, is also convex, where L : Rn → Rn is a linear map.



3(c). (20) Let V be the linear space of all polynomials of degree three or less. Prove ordisprove that S = {p(x) ∈ V : p(1) = 0} is a subspace of V .



3(d). (13) Let V be a vector space with basis {v1, v2, v3} and W be vector space withbasis {w1, w2}. What is a basis for V ×W , the direct product of V and W .




3(e). (20) Prove that the number of odd permutations of Jn = {1, . . . , n} for n > 2 isequal to the number of even permutations of Jn.

Outcomes assessed: cardinality


3(f). (13) Prove Theorem V.1.2. Triangle Inequality. If u, v ∈ V , then ‖v + w‖ 6‖v‖+ ‖w‖. Hint: Schwarz Inequality.

Outcomes assessed: inner product spaces


3(g). (27) Prove Theorem VIII.2.1. Let V be a finite dimensional vector space, λ be anumber, and A : V → V be a linear map. Then λ is an eigenvalue of A if and only ifλI − A is not invertible.

Outcomes assessed: Vector spaces; eigenvalues and eigenvectors




I learned that the students came away from the course with a great deal of knowledge ofthe definitions. The students’ knowledge of the theorems was not as thorough as it was inthe case of the definitions, and in turn the students knowledge of and/or ability to comeup with proofs was not as strong as their knowledge of theorems.


I will encourage better knowledge of theorems. One way to do this is to emphasizetheorems more on quizzes. To encourage better performance on proofs, I will attempt toemphasize them more, again through quizzes, and perhaps by covering fewer of them.

III.7.3. Formative Assessment. Instructor collected information from students via in-troductory questionnaire at beginning of semester. (Questions are included on the nextpage.)


MATHEMATICS 429 – LINEAR ALGEBRA

INVENTORY, DUE 16 JANUARY

NAME: ID #:

1. Complex numbers. Let i =√

?1

(a) (a+ bi)(c+ di) =? (Simplify.)

(b) ‖a+ bi‖ =? (length, modulus, sometimes |a+ bi|)2. As real vectors, (a, b, c) + (d, e, f) =?

3. If a is a real and (x, y, z) is a real vector, then a · (x, y, z) =?

4. As vectors are (0, 1, 2) and (0, 2, 4) linearly independent? What about (1, 0, 0) and(0, 1, 1)?

5. Let U and W be subspaces of vector space V . What is U + W , the sum of twosubspaces?

6. Every linear transformation may be represented by what mathematical object?

7. Let tA mean the transpose of A. In terms of that when will a matrix be symmetric?

8. If A is a squre matrix with eigenvalue λ and eigenvector x 6= 0, what equation is true?

9. What is a convex set?


III.8. Math 445/645, Differential Geometry, only section

III.8.1. Final exam. (Each question was worth 5 points.)

1. These are short answer questions–do any 15 of the 20 parts.

(a) Give the definition of a curve in a geometric surface M . Give the definition of a curvesegment in a geometric surface M Describe how they are different.

Outcomes assessed: Basic properties of curves

Degree of success:

445: A 0/1 B 1/1 C 0/1 D 0/1 E 0/1

645: A 1/3 B 2/3 C 0/3 D 0/3 E 0/3

combined: A 1/4 B 3/4 C 0/4 D 0/4 E 0/4

(b) Give the definition of a geometric surface (i.e. a 2-dimensional riemannian manifold).

Outcomes assessed: Basic properties of surfaces

Degree of success:

445: A 0/2 B 0/2 C 1/2 D 0/2 E 1/2

645: A 0/2 B 1/2 C 0/2 D 1/2 E 0/2

combined: A 0/4 B 1/4 C 1/4 D 1/4 E 1/4

(c) Give the definition of a tangent vector field on a geometric surface M . Give thedefinition of a frame field for M .


Degree of success:

445: A 0/2 B 2/2 C 0/2 D 0/2 E 0/2

645: A 1/3 B 1/3 C 1/3 D 0/3 E 0/3

combined: A 1/5 B 3/5 C 1/5 D 0/5 E 0/5


(d) Give the Frenet formulas for a unit-speed curve in a surface M ⊆ R3.

Outcomes assessed: Basic properties of curves; Frenet equations

Degree of success:

445: A 0/2 B 1/2 C 1/2 D 0/2 E 0/2

645: A 1/1 B 0/1 C 0/1 D 0/1 E 0/1

combined: A 1/3 B 1/3 C 1/3 D 0/3 E 0/3

(e) Given a frame field {E1, E2} on a geometric surface M , define the dual forms θ1 andθ2. How does one use the first structural equations to determine the connection form ω12

from this information.

Outcomes assessed: Basic properties of surfaces; intrinsic geometry of surfaces

Degree of success:

445: A 0/2 B 2/2 C 0/2 D 0/2 E 0/2

645: A 1/1 B 0/1 C 0/1 D 0/1 E 0/1

combined: A 1/3 B 2/3 C 0/1 D 0/3 E 0/3

(f) Define a patch x : D → M and a 2-segment y : R → M . Describe how they aredifferent.


Degree of success:

445: A 2/2 B 0/2 C 0/2 D 0/2 E 0/2

645: A 1/3 B 2/3 C 0/3 D 0/3 E 0/3

combined: A 3/5 B 2/5 C 0/5 D 0/5 E 0/5

(g) Both 1-forms and 2-forms on a surface M can be described as functions on the tangentvectors of M . Give the details for this particular description of the 1-forms and 2-forms.


Degree of success:

445: A 0/1 B 0/1 C 1/1 D 0/1 E 0/1

645: A 0/3 B 1/3 C 1/3 D 1/3 E 0/3

combined: A 0/4 B 1/4 C 2/4 D 1/4 E 0/4


(h) State Stokes Theorem.

Outcomes assessed: applications

Degree of success:

445: A 0/2 B 2/2 C 0/2 D 0/2 E 0/2

645: A 1/2 B 0/2 C 0/2 D 1/2 E 0/2

combined: A 1/4 B 2/4 C 0/4 D 1/4 E 0/4

(i) Give two equivalent definitions for an orientable surface.


Degree of success:

445: A 0/1 B 1/1 C 0/1 D 0/1 E 0/1

645: A 1/2 B 1/1 C 0/2 D 0/2 E 0/2

combined: A 1/3 B 2/3 C 0/3 D 0/3 E 0/3

(j) Give a formula for the shape operator of a surface M ⊆ R3.


Degree of success:

445: A 0/1 B 1/1 C 0/1 D 0/1 E 0/1

645: A 1/1 B 0/1 C 0/1 D 0/1 E 0/

combined: A 1/2 B 1/1 C 0/2 D 0/2 E 0/2

(k) Give the definition of principal curvature, Gaussian curvature, and mean curvaturefor a surface M ⊆ R3. How are they related.

Outcomes assessed: Basic properties of curves; Basic properties of surfaces; intrinsicgeometry of surfaces

Degree of success:

445: A 0/1 B 1/1 C 0/1 D 0/1 E 0/1

645: A 0/2 B 2/2 C 0/2 D 0/2 E 0/2

combined: A 0/3 B 3/3 C 0/3 D 0/3 E 0/3


(l) Give the definition of an isometry of surface s : M → N and a local isometry t : M → N .Describe how they are different.

Outcomes assessed: intrinsic geometry of surfaces

Degree of success:

445: A 0/2 B 1/2 C 1/2 D 0/2 E 0/2

645: A 1/3 B 2/3 C 0/3 D 0/3 E 0/3

combined: A 1/5 B 3/5 C 1/5 D 0/5 E 0/5

(m) Give the definition of Gaussian curvature for a geometric surface.


Degree of success:

445: A 0/2 B 1/2 C 0/2 D 0/2 E 1/2

645: A 2/3 B 1/3 C 0/3 D 0/3 E 0/3

combined: A 2/5 B 2/5 C 0/5 D 0/5 E 1/5

(n) Give the definition of total curvature of a compact surface M oriented by an areaform dM .


Degree of success:

445: A 1/2 B 1/2 C 0/2 D 0/2 E 0/2

645: A 3/3 B 0/3 C 0/3 D 0/3 E 0/3

combined: A 4/5 B 1/5 C 0/5 D 0/5 E 0/5

(o) Give the definition of a complete geometric surface.


Degree of success:

445: A 0/1 B 1/1 C 0/1 D 0/1 E 0/1

645: A 2/3 B 1/3 C 0/3 D 0/3 E 0/3

combined: A 2/4 B 2/4 C 0/4 D 0/4 E 0/4


(p) State the Gauss-Bonnet theorem for a compact orientable surface M .

Outcomes assessed: Gauss-Bonnet Theorem

Degree of success:

445: A 1/1 B 0/1 C 0/1 D 0/1 E 0/1

645: A 1/1 B 0/1 C 0/1 D 0/1 E 0/1

combined: A 2/2 B 0/2 C 0/2 D 0/2 E 0/2

(q) Describe all of the compact connected surfaces up to diffeomorphism.

Outcomes assessed: Gauss-Bonnet Theorem; applications

Degree of success:

445: A 0/1 B 0/1 C 1/1 D 0/1 E 0/1

645: A 1/3 B 0/3 C 2/3 D 0/3 E 0/3

combined: A 1/4 B 0/4 C 3/4 D 0/4 E 0/4

(r) Give the definition of a geodesic curve. Given a geometric surface M and two points pand q in M , give a condition on M which ensures there is a geodesic connecting p and q.

Outcomes assessed: Basic properties of curves; Basic properties of surfaces; geodesics

Degree of success:

445: A 1/2 B 0/2 C 1/2 D 0/2 E 0/2

645: A 2/3 B 1/3 C 0/3 D 0/3 E 0/3

combined: A 3/5 B 1/5 C 1/5 D 0/5 E 0/5

(s) Give the definition of a covering map. Give two examples of such a map.


Degree of success:

445: A 1/2 B 0/2 C 1/2 D 0/2 E 0/2

645: A 2/2 B 0/2 C 0/2 D 0/2 E 0/2

combined: A 3/4 B 0/4 C 1/4 D 0/4 E 0/4


(t) Describe all of the complete connected geometric surfaces of constant positive curva-ture. Describe all of the complete connected geometric surfaces of constant zero curvature.Why is there no such nice description in the case of negative constant curvature.

Outcomes assessed: Gauss-Bonnet Theorem; applications

Degree of success:

445: A 0/2 B 0/2 C 2/2 D 0/2 E 0/2

645: A 2/3 B 0/3 C 1/3 D 0/3 E 0/3

combined: A 2/5 B 0/5 C 3/5 D 0/5 E 0/5



This confirmed that the students were familiar with the fundamental definitions/conceptsin the course, at approximately the A/B level. I believe this is consistent with what Iexpect from such a course. It was interesting that given the option of doing only a certainnumber of problems there was a wide mix of the questions answered.


I believe that next time I teach the course I would like to do some more emphasis (andassessment of) the computational aspects of differential geometry with perhaps computermodules associated to the various concepts.



III.9. Math 446/646, Introduction to Topology, only section

III.9.1. Final exam. (Each part was worth 3 points.)

1. Complete 20 of the following 25 questions:

(a) Give the definition of a topology on a space X.

Outcomes assessed: topological spaces

Degree of success:

446: A 9/10 B 0/10 C 1/10 D 0/10 E 0/10

646: A 1/1 B 0/1 C 0/1 D 0/1 E 0/1

combined: A 10/11 B 0/11 C 1/11 D 0/11 E 0/11

(b) Explain what the discrete topology and the trivial topology on a space X are.


Degree of success:

446: A 8/10 B 0/10 C 1/10 D 0/10 E 1/10

646: A 1/1 B 0/1 C 0/1 D 0/1 E 0/1

combined: A 1/11 B 0/11 C 1/11 D 0/11 E 1/11

(c) Explain what it means for a topological space to be connected.

Outcomes assessed: topological spaces; connectivity

Degree of success:

446: A 9/10 B 0/10 C 0/10 D 0/10 E 1/10

646: A 1/1 B 0/1 C 0/1 D 0/1 E 1/1

combined: A 10/11 B 0/11 C 0/11 D 0/11 E 1/11

(d) Give the definition of a continuous function.

Outcomes assessed: continuity

Degree of success:

446: A 9/10 B 0/10 C 1/10 D 0/10 E 0/10

646: A 1/1 B 0/1 C 0/1 D 0/1 E 0/1

combined: A 10/11 B 0/11 C 1/11 D 0/11 E 1/11


(e) Define what it means for a sequence in a topological space to converge.


Degree of success:

446: A 2/10 B 0/10 C 6/10 D 0/10 E 2/10

646: A 0/1 B 0/1 C 0/1 D 0/1 E 1/1

combined: A 2/11 B 0/11 C 6/11 D 0/11 E 3/11

(f) Define an open cover, and what it means for a topological space to be compact.

Outcomes assessed: compactness

Degree of success:

446: A 9/10 B 0/10 C 1/10 D 0/10 E 0/10

646: A 1/1 B 0/1 C 0/1 D 0/1 E 0/1

combined: A 10/11 B 0/11 C 1/11 D 0/11 E 0/11

(g) Give two equivalent definitions for a connected topological space.

Outcomes assessed: connectivity

Degree of success:

446: A 3/6 B 0/6 C 2/6 D 0/6 E 1/6

646: A 0/0 B 0/0 C 0/0 D 0/0 E 0/0

combined: A 3/6 B 0/6 C 2/6 D 0/6 E 1/6

(h) Define what it means for a topological space to be Hausdorff.

Outcomes assessed: Hausdorff property

Degree of success:

446: A 9/10 B 0/10 C 0/10 D 0/10 E 1/10

646: A 1/1 B 0/1 C 0/1 D 0/1 E 0/1

combined: A 10/11 B 0/11 C 0/11 D 0/11 E 1/11


(i) Give an example of a compact space which is not finite.


Degree of success:

446: A 8/8 B 0/8 C 0/8 D 0/8 E 0/8

646: A 1/1 B 0/1 C 0/1 D 0/1 E 0/1

combined: A 9/9 B 0/9 C 0/9 D 0/9 E 0/9

(j) Give an example of a locally compact Hausdorff space which is not compact.

Outcomes assessed: Hausdorff property; compactness

Degree of success:

446: A 8/8 B 0/8 C 0/8 D 0/8 E 0/8

646: A 1/1 B 0/1 C 0/1 D 0/1 E 0/1

combined: A 9/9 B 0/9 C 0/9 D 0/9 E 0/9

(k) Define what it means for a topological space to be normal.

Outcomes assessed: Hausdorff property

Degree of success:

446: A 8/9 B 0/9 C 1/9 D 0/9 E 0/9

646: A 1/1 B 0/1 C 0/1 D 0/1 E 0/1

combined: A 9/10 B 0/10 C 1/10 D 0/10 E 0/10

(l) Define the product topology on a finite number of spaces.


Degree of success:

446: A 10/10 B 0/10 C 0/10 D 0/10 E 0/10

646: A 1/1 B 0/1 C 0/1 D 0/1 E 0/1

combined: A 11/11 B 0/11 C 0/11 D 0/11 E 0/11


(m) Give an example of a quotient topology on a quotient space.


Degree of success:

446: A 9/9 B 0/9 C 0/9 D 0/9 E 0/9

646: A 1/1 B 0/1 C 0/1 D 0/1 E 0/1

combined: A 10/10 B 0/10 C 0/10 D 0/10 E 0/10

(n) Give an example of a space which is not locally compact.


Degree of success:

446: A 3/4 B 0/4 C 0/4 D 0/4 E 1/4

646: A 0/0 B 0/0 C 0/0 D 0/0 E 0/0

combined: A 3/4 B 0/4 C 0/4 D 0/4 E 1/4

(o) Explain what it means for two loops to be homotopic.

Outcomes assessed: homotopy

Degree of success:

446: A 7/9 B 0/9 C 2/9 D 0/9 E 0/9

646: A 1/1 B 0/1 C 0/1 D 0/1 E 0/1

combined: A 8/10 B 0/10 C 2/10 D 0/10 E 0/10

(p) Explain the construction of the fundamental group for a path connected topologicalspace X.

Outcomes assessed: fundamental group

Degree of success:

446: A 2/3 B 0/3 C 1/3 D 0/3 E 0/3

646: A 0/0 B 0/0 C 0/0 D 0/0 E 0/0

combined: A 2/3 B 0/3 C 1/3 D 0/3 E 0/3


(q) Give the definition of a knot.

Outcomes assessed: knots

Degree of success:

446: A 4/7 B 0/7 C 1/7 D 0/7 E 2/7

646: A 0/1 B 0/1 C 1/1 D 0/1 E 0/1

combined: A 4/8 B 0/8 C 2/8 D 0/8 E 2/8

(r) Explain the coloring number of a knot and compute a nontrivial example.

Outcomes assessed: knots

Degree of success:

446: A 10/10 B 0/10 C 0/10 D 0/10 E 0/10

646: A 0/1 B 0/1 C 1/1 D 0/1 E 0/1

combined: A 10/11 B 0/11 C 1/11 D 0/11 E 0/11

(s) Define what it means for a space to be path connected.


Degree of success:

446: A 8/10 B 0/10 C 2/10 D 0/10 E 0/10

646: A 1/1 B 0/1 C 0/1 D 0/1 E 0/1

combined: A 9/11 B 0/11 C 1/11 D 0/11 E 0/11

(t) Give an example of a connected space which is not path connected.


Degree of success:

446: A 5/6 B 0/6 C 1/6 D 0/6 E 0/6

646: A 1/1 B 0/1 C 0/1 D 0/1 E 0/1

combined: A 6/7 B 0/7 C 1/7 D 0/7 E 0/7


(u) Explain what the fundamental group of the unit circle is and why.

Outcomes assessed: fundamental group

Degree of success:

446: A 1/3 B 0/3 C 1/3 D 0/3 E 1/3

646: A 0/0 B 0/0 C 0/0 D 0/0 E 0/0

combined: A 1/3 B 0/3 C 1/3 D 0/3 E 1/3

(v) Define what it means for X to be locally compact.


Degree of success:

446: A 3/8 B 0/8 C 3/8 D 0/8 E 2/8

646: A 0/1 B 0/1 C 1/1 D 0/1 E 0/1

combined: A 3/9 B 0/9 C 4/9 D 0/9 E 2/9

(w) Explain why the continuous image of a connected set is connected.

Outcomes assessed: continuity; connectivity

Degree of success:

446: A 9/9 B 0/9 C 0/9 D 0/9 E 0/9

646: A 0/1 B 0/1 C 0/1 D 0/1 E 1/1

combined: A 9/10 B 0/10 C 0/10 D 0/10 E 1/10

(x) Explain why the continuous image of a compact set is compact.

Outcomes assessed: continuity; compactness

Degree of success:

446: A 6/7 B 0/7 C 1/7 D 0/7 E 0/7

646: A 1/1 B 0/1 C 1/1 D 0/1 E 1/1

combined: A 7/8 B 0/8 C 1/8 D 0/8 E 0/8


(y) Give an example of a Hausdorff space X a non-Hausdorff space Y and a continuousmap f : X → Y .

Outcomes assessed: Hausdorff property; continuity

Degree of success:

446: A 4/4 B 0/4 C 0/4 D 0/4 E 0/4

646: A 1/1 B 0/1 C 0/1 D 0/1 E 0/1

combined: A 5/5 B 0/5 C 0/5 D 0/5 E 0/5



I would have liked to see higher achievement on question (g), especially considering theresults on questions (s), (t), and (w) which are related in content.


I believe that next time I teach this course I will need to focus on some more illustrativeexamples. I also think that in a course like this a more focussed syllabus may be moreappropriate. In addition based on student feedback from this course I believe that I willhave a required book rather than expecting the students to take notes.

III.9.3. Formative Assessment. I gave an anonymous survey midway through the se-mester specifically adressing the issue of pacing.


III.10. Math 450, Real Analysis I, only section


1. (20 points) Find the supremum of the set

A =

{6n− 8

10n− 5

∣∣∣∣n ∈ Z+

}.

Outcomes assessed: Sequences in R; convergence in R


2. (10 points) Let S be the collection of all sequences whose terms are the integers 0 and1. Prove that S is uncountable.

Outcomes assessed: Sequences in R


3. (20 points) Determine all the accumulation points of the set of irrational numbers, anddecide whether this set is open or closed.

Outcomes assessed: N/A


4. (20 points) Let A be a subset of a metric space S.

(i) Prove that if A is complete, then A is closed;

(ii) Assuming that S is complete, prove that if A is closed then A is complete.



5. (10 points) Let f : [0, 1]→ R be defined by

f(x) =

{1 if x ∈ [0, 1] ∩Q,

0 if x ∈ [0, 1]\Q.

Prove that f is NOT continuous anywhere in the interval [0, 1].

Outcomes assessed: continuity



6. (20 points) Let f : [a, b]→ R be three times differentiable on [a, b]. Assuming that

f(a) = f ′(a) = f(b) = f ′(b) = 0

prove that there exists a point c ∈ (a, b) such that f ′′′(a) = 0.

Outcomes assessed: differentiability; applications




The students have perfect solutions for the HW problems but they do not know how tosolve similar problems given at the exam.


In the future I will ask my colleagues to not solve my students’ HW problems.



III.11. Math 451, Real Analysis I, only section


1. (10 points) Let f : [a, b]→ R be a step function. Show that f is a function of boundedvariation on [a, b].



2. (10 points) Let f : [a, b]→ R be a continuous function on [a, b], and g be a function ofbounded variation on [a, b]. Show that F : [a, b]→ R defined by

F (x) =

∫ x

a

f(t)dg(t)

is a bounded variation on [a, b].

Outcomes assessed: Riemann-Stieltjes integration


3. (20 points) Let f : [a, b]→ R be a continuous function on [a, b]. Show that there existξ, η ∈ (a, b) such that ξ < η, and

f(η) =η − ab− η

f(ξ).

Outcomes assessed: Riemann integration


4. (20 points) Let α, β ∈ R, α < β and f : [α, β]→ R defined by

f(x) =√

(x− α)(x− β).

Compute ∫ β

α

f(x) dx.

Outcomes assessed: Riemann integration



5. (20 points) Let f : [0, 3] → R be a function with the property that there exists asequence of polynomials {Pn}n of degree 6 3, such that

Pn(x)→ f(x),∀x ∈ [0, 3].

Show that f is a polynomial of degree 6 3.

Outcomes assessed: convergence theorems


6. (20 points) Construct a convergent sequence of integrable functions on [a, b] such thatthe limit function is not integrable on [a, b].

Outcomes assessed: Riemann integration; convergence theorems




The students have perfect solutions for the HW problems but they do not know how tosolve similar problems given at the exam.


In the future I will ask my colleagues to not solve my students’ HW problems.



III.12. Math 472, Number Theory, only section

III.12.1. Final exam. Work on six of the following eight problems. Indicate clearlywhich problems you choose to have graded. (Each problem is worth five points.)

1. (a) Find all integer solutions of the system

3x ≡ 2 (mod 5)

x ≡ 2 (mod 3).

(b) Show that the system

3x ≡ 2 (mod 10)

8x ≡ 3 (mod 15).

has no solutions.

Outcomes assessed: Properties of integers

Degree of success (14 students):

A 64% B 7% C 29% D 0% E 0%

2. Assume that gcd(a, b) = 1. Show: If d|a+ b then gcd(a, d) = gcd(b, d) = 1.

Outcomes assessed: Properties of integers; prime numbers; Bezout or unique prime fac-torization


A 86% B 7% C 7% D 0% E 0%

3. Show that ϕ(7n) = 7ϕ(n) if and only if 7 divides n.

Outcomes assessed: number theoretic functions


A 69% B 0% C 23% D 0% E 8%

4. Show that 2 is a primitive root modulo 193. (All necessary congruences are on theback of this sheet.)

Outcomes assessed: primitive roots


A 50% B 7% C 43% D 0% E 0%


5. Assume that p ≡ 1 (mod 4). Find all p such that −5 is a quadratic residue modulo p.

Outcomes assessed: quadratic residues


A 29% B 7% C 43% D 21% E 0%

6. Let m,n ∈ N be relatively prime. Show that

mϕ(n) + nϕ(m) ≡ 1 (mod mn).

Outcomes assessed: number theoretic functions


A 0% B 50% C 0% D 0% E 50%

7. Show that ∑n6x

arctan(n) = x arctanx− 1

2ln(1 + x2) + h(x),

where |h(x)| is bounded by a constant.

Outcomes assessed: distribution of prime numbers: testing partial summation; techniqueneeded for investigation of asymptotic growth of prime counting function, but this problemis easier.


A 20% B 50% C 10% D 20% E 0%

8. Let µ(n) be the Mobius function, let ω(n) be the number of distinct prime factors ofn, and define λ(n) = (−1)ω(n). Show that

(µ ∗ λ)(n) =

{(−2)ω(n) if n is squarefree,

0 else.

(Hint: All functions in this identity are multiplicative.)

Outcomes assessed: prime numbers; multiplicative functions


A 33% B 0% C 0% D 0% E 67%




The student were generally comfortable with problems that required calculations withactual numbers, even when the questions involved applications of more abstract concepts,e.g., multiplicativity of arithmetic functions. A large percentage had difficulties with theconcept of quadractic residues and quadratic reciprocity. #6 required a combination ofthe Chinese Remainder Theorem and Euler’s theorem, and #8 asked for a verification ofan identity using convolution and multiplicativity. Both problems were perceived to bedifficult, and students elected not to work on them (they had to choose 6 out of 8).


These difficulties were to be expected. A future class would have more time to spend onthese topics; classes were cut short due to the flood in Spring 2009.



III.13. Math 478, History of Mathematics, only section

III.13.1. Summary. The students in this course were assigned an end of the semesterproject/presentation in which they had to connect significant events in their own lives (orin the lives of a make-believe person) to the important events, people, etc. in the historyof mathematics. In this presentation students had to demonstrate that they had met thecourse objectives, but they could be as creative as they wanted in doing so. There werealso additional components to the project, as listed in the project description.

III.13.2. Final project Instructions. You have spent the semester learning about thehistory of mathematics. Now its time to share your life history. The catch is that youmust relate the significant events in your life to the history of mathematics. Throughyour project and presentation it must be clear that you understand the material in thishistory of mathematics course. You can be as creative as you would like in designing yourartifacts that show your life in relation to the history of mathematics. For example, youmay bring a picture of yourself competing in a sporting event. Then you could relate thisto mathematicians competing to be the first to develop topic X.

a. Grading guidelines.i. Students life history is clear and complete. (20 pts)

ii. Students life history is related to important events in the history of mathe-matics. (20 pts)

iii. Student is creative in presenting these events.1. Creative in writing up the project. (10 pts)2. Creative in presenting the project. (10 pts)

iv. Students presentation is clear and easy to follow. (10 pts)v. Students presentation involves the class. (10 pts)

III.13.3. Outcomes assessed. Historical considerations emphasizing the source of math-ematical ideas, growth of mathematical knowledge, and contributions of some outstandingmathematicians.

III.13.4. Degree of Success Rubric.

A. Exceptional understanding of the course objectives. Students went above and beyondexpectations of the final presentation/paper.

B. Great understanding of the course objectives. Students met the objectives, but partswere missing on following directions for presentations/papers. For instance, the stu-dents may not have involved the class in their presentation or the organization of theirpaper was lacking.

C. Good understanding of the course objectives. These students met the requirementsfor the presentation/paper, but they did so by doing the bare minimum.


III.13.5. Degree of success. A 57% B 30% C 13%



From this assessment activity, I learned that the students did well in this course.


In the future, I will make sure to think about the course objectives and where they fitwith each portion of the final presentation/paper for the class.

III.13.7. Formative Assessment. Formative assessment was a regular part of this course.Every time the class met, the students presented the reading material to each other. Thenthey worked in groups on the problems. I went around the room continuously to assesstheir understanding of the material. I used this to help me plan for the next class period,so I could help them clear up any misunderstandings that were evident during either theirpresentations or while they worked on problems.


III.14. Math 480, Applied Differential Equations, only section

III.14.1. Refinement of Degree of Success.

A. Completely correct;

B. Essentially correct—student shows full understanding of solution and only makes aminor mistake (e.g., wrong sign when calculating a derivative or arithmetic error);

C. Flawed response, but quite close to a correct solution (appears they could do this typeof problem with a little review or help);

D.(a) Started in a correct or feasible direction, then lost their way;

D.(b) Something relevant;

E. Blank, or nothing relevant to the problem.

III.14.2. Final exam. (Problems 1–6 graded on scale 4–10; problem 7 made extra credit)

(1) For the Laguerre DE xy′′(x) + (1 − x)y′(x) + λy(x) = 0 on x > 0, find the powerseries solution centered at x = 0, give an explicit form for the coefficients in terms ofPochhammer’s symbol, and give the radius of convergence and an appropriate theoreticaljustification for your value. Show that the solution reduces to a polynomial of degree Nwhen λ = N .

Note: Define the function L(λ;x) := y(x) such that y(0) = 1. The functions L(n;x) forn = 0, 1, 2, . . . are the Laguerre polynomials.

Outcomes assessed: Power series expansions; special functions and their use

Degree of success:

A 67% B 0% C 22% D(a) 11% D(b) 0% E 0%

(2) Check your solution to (1) by showing that the Laguerre polynomials L(n;x) satisfythe recursion relation

(n+ 1)L(n+ 1;x) = (2n+ 1− x)L(n;x)− nL(n− 1;x) for n = 1, 2, 3, . . ..

Outcomes assessed: Power series expansions; special functions and their use

Degree of success:

A 22% B 11% C 0% D(a) 34% D(b) 22% E 11%


(3) Using the method of regular singular points, find a second solution centered at x = 0for the Laguerre DE. Give the indicial roots. Note: The solution has the form z(x)+logarithmic expression, where z(x) =

∑∞k=0 zkx

k. You should find the (nonhomogeneous)two-term recursion relation for the zk, but you do not need to solve the recursion explicitly.

Outcomes assessed: the method of Frobenius; special functions and their use

Degree of success:

A 22% B 0% C 22% D(a) 22% D(b) 34% E 0%

(4) Determine whether or not there is a relation analogous to that of Problem (2) betweenthe Laguerre solutions L(λ;x), that is, a relation of the form

f(λ)L(λ+ 1;x) = (g(λ)− x)L(λ;x) + h(λ)L(λ− 1;x) for some range of λ.

If so, give f(λ), g(λ), h(λ) explicitly. If not, explain clearly why not.

Outcomes assessed: the method of Frobenius; special functions and their use

Degree of success:

A 22% B 0% C 0% D(a) 33.5% D(b) 11% E 33.5%

(5) Show that the Laguerre polynomials L(n, x) satisfy∫ +∞0

L(m;x)L(n;x)e−x dx = 0for m 6= n. Hint: Use the self-adjoint form of the Laguerre DE.

Outcomes assessed: special functions and their use; orthogonal expansions; orthogonalpolynomials; Sturm-Liouville theory

Degree of success:

A 0% B 22% C 33.5% D(a) 11% D(b) 33.5% E 0%

(6) For the DE y′′(x)+x2p−2y(x) = 0 on x > 0 with p > 0, consider changes of independentvariable z = bxa and obtain a representation of solutions in the form y(x) = f(x)w(z(x)),where z2w′′(z) + zw′(z) + (z2 − v2)w(z) = 0 is the Bessel DE. Here p is given, and allother relevant quantities (a, b, v, etc.) should be stated in terms of p.

Outcomes assessed: invariants

Degree of success:

A 0% B 0% C 33.3% D(a) 33.3% D(b) 33.3% E 0%


(7) The confluent hypergeometric DE has the form xw′′(x) + (b − x)w′(x) − aw(x) = 0on x > 0. Determine whether the DE is oscillatory or not on 0 < x < +∞, in particular,whether such behavior varies with a, b.

Outcomes assessed: special functions and their use; oscillation theorems

Degree of success:

A 0% B 0% C 0% D(a) 11% D(b) 11% E 78%



I have not learned anything beyond what simple inspection would give due to the smallsample size (9 students). Simple inspection did occur because, after all, I graded thepapers.


I don’t expect to proceed differently in the future because the essential step in teachingthe course is to find a balance or compromise between the designated course contentand the range of backgrounds in a given class. For this particular course, the class sizesare small and the backgrounds vary substantially, and the balance must be found on aclass-by-class basis.

III.14.4. Formative Assessment. I don’t do any formal “formative assessment” in thesense of handing out forms and collecting answers. What I do is listen to questions,especially in the lectures and when students come by the office, watch for what they findconfusing, and either clarify/correct in following lectures or - when it’s too late for thissemester - make a note to change it next semester.


III.15. Math 483, Partial Differential Equations, only section


1. (10 points) Solve the PDE

ux + uy + u = ex+2y

with the condition u(x, 0) = 0.

Outcomes assessed: Solution methods for potential equations; nonhomogeneous equations


2. (20 points) Prove the uniqueness of solutions to the diffusion problem with Neumannboundary conditions:

ut − kuxx = f(x, t) for 0 < x < l, t > 0

u(x, 0) = φ(x)

ux(0, t) = g(t)

ux(l, t) = h(t)

using the energy method.

Outcomes assessed: Solution methods for diffusion equations; nonhomogeneous equations;boundary conditions


3. (20 points) Solve the following problem for the homogeneous wave equation on thehalf-line:

utt = c2uxx for x ∈ (0,∞), t ∈ (0,∞)

u(x, 0) = ut(x, 0) = 0

ux(0, t) = h(t).

Express the solution in terms of c > 0 and h.

Note: Do NOT use separation of variables !

Outcomes assessed: Solution methods for wave equations; homogeneous equations; bound-ary conditions



4. (15 points) Let

D =

{(x, y) ∈ R2 :

x2

4+y2

9< 1

}′and let u be the solution to the Dirichlet problem{

−uxx − uyy = 1 in D

u = 0 on ∂D.

Use the maximum principle to show that

1 6 u(0, 0) 69

4.

Outcomes assessed: Solution methods for potential equations; boundary conditions


5. (20 points) (a) Use separation of variables to find the (unique) harmonic function inthe infinite strip S = {(x, y) ∈ R2 : 0 6 x 6 π, 0 6 y <∞} which satisfies the boundaryconditions

u(0, y) = u(π, y) = 0, u(x, 0) = h(x), limy→∞

u(x, y) = 0;

(b) Would the problem considered in part (a) still have a unique solution if we omittedthe condition at infinity limy→∞ u(x, y) = 0? Justify your answer.

Outcomes assessed: boundary conditions; separation of variables


6. (15 points) Let D be the wedge given in polar coordinates by

D = {r, θ) ∈ R2 : 0 < r < a, 0 < Θ < β}.

Use separation of variables to solve the problemuxx + uyy = 0 in D

u(r, 0) + u(r, β) = 0

u(a, θ) = h(θ).

Write the solution as a series but do NOT attempt to sum it.

Outcomes assessed: homogeneous equations; boundary conditions; separation of variables





The students did not do as well as I would have liked to see on some of the problems onthe final exam. But I knew that already. . . .


Remind the students (as I always do, in fact) that they should study harder for the finalexam, & throughout the semester. The final exam questions were, in large part, similarto the homework pbms as well as examples done in class.

III.15.3. Formative Assessment. I didn’t do any paper or online questionnaire for thatcourse. As with all of my courses, I did do an ongoing formative assessment, throughoutthe semester, which included asking questions in class and during office hours regard-ing the students’ understanding of the material covered in lectures, students’ backgroundregarding presumed knowledge which was actually needed for my course, students’ ex-pectations, level of difficulty of the homework and midterms, etc. I do use this type ofinformation to improve student learning in all my courses.


III.16. Math 488, Numerical Analysis I, only section

III.16.1. Final exam. (Each numbered question was worth 10 points.)

1. Let f(x) = ex + x.

(a) Find the polynomial of degree 2 that interpolates f(x) at x = −1, x = 0, x = 1.(b) Give an upper bound for the error of interpolation on the interval [−1, 1] for the

polynomial from part (a).

Outcomes assessed: interpolation


2. Let f(x) be a differentiable function, and let h > 0. Find an approximation to f ′′(x0)that uses h, f(x0), f(x0 + h), and f(x0 − h), and has an error order O(h2).

Outcomes assessed: numerical differentiation


3. Assume that g(p) = p for some a < p < b. Suppose that for all x1, x2 ∈ [a, b] we have|g(x1)− g(x2)| 6 1/2.

(a) Show that the fixed point algorithm pn = g(pn−1) converges to p, provided that p0is chosen between a and b. (Hint: consider |g(pn−1)− p|.)

(b) Under what conditions on g′(p) does (pn) converge linearly to p? Justify.

Outcomes assessed: Numerical solution of nonlinear equations


4. Consider the differential equation y′ = et + y, y(0) = 1.

(a) Construct the second order Taylor method which approximates the solution of thedifferential equation. (Start with the Taylor expression of y(t).)

(b) What is the local truncation error of this method?

Outcomes assessed: numerical solution of initial value problems for ordinary differentialequations



5. Consider the matrix

A =

1 1 −1 2−1 −1 1 52 2 3 72 3 4 5

.

Find a permutation matrix P , a lower triangular matrix L, and an upper triangular matrixU so that PA = LU .

Outcomes assessed: matrix factorization


6. Let f(x) be a continuous function defined on [−1, 1].

(a) What choice of points x0, . . . , x4 in [−1, 1] gives the least error for the interpolationof f(x) by polynomials of degree 6 4?

(b) What is the upper bound for the error of this approximation?

Outcomes assessed: interpolation; perfect interpolation




Computational problems were generally solved correctly. Students had great difficultieswith problems requiring proof techniques. This was particularly the case for the applica-tion of the mean value in problem 3 and the use of Taylor series to derive an approximationmethod for a first order differential equation in problem 4.


In a future class I would put additional emphasis on the mathematical principles behindthe numerical algorithms that the students encounter during class.



Part IV. Departmental Reflection

Response 1.

I have two reactions:

1. There were some good ideas for formative assessment during the semester and I planto take some of the ideas and use them in my own classes in the future.

2. It is a little disappointing that some faculty do not take assessment seriously, yetencouraging that some take it very seriously.

Response 2.

I found the report quite interesting. Usually each one of us has information about thecourse (or few courses) that we have taught, but not about the rest, and this reportprovided a panoramic view of the undergraduate courses. It is instructive to see wherethe students struggle. Most of the instructors’ reflections described ways to address theseissues in the future. In courses with several sections (like 270) it could be good for theinstructors to discuss together their ideas about what to do differently in the future, aseach one of them had several good ideas about it.

I realize that starting with 400-level courses, most instructors did not conduct a mid-termassessment. That could be something useful to do in the future.


Part V. Summary

The mathematics department at NDSU has designed and begun to implement a systematicassessment plan. The assessment plan will evolve as the department faculty members andthe assessment committee sees fit to continue to assess our courses.

We have begun by assessing if our students are meeting the course objectives currentlyoutlined in the Bulletin. Final examinations were analyzed question by question for eachsection of a course. Then the instructor for each course examined the data. We collectedinformation about what the instructors learned from these data and what they will dodifferently next time.

It was found that some professors learned a lot from the data and will teach and/or reviewmaterial for their exams differently in the future. Some professors, however, did not findthese data that helpful. Therefore, other assessment measures will be used in the future.The assessment committee will seek input from faculty as to what forms of data wouldbe most useful. It is predicted that formative assessment data collected throughout thesemester will be useful to help us determine what our students are learning throughout thesemester. All in all, we will continually seek to improve our assessment plan to evaluateboth student learning of course objectives and the overall success of our program.

Documents

DEPARTMENT OF MATHEMATICS ASSESSMENT ... OF MATHEMATICS ASSESSMENT REPORT 2008{2009 DAVIS COPE, ANGELA HODGE, AND SEAN SATHER-WAGSTAFF (CHAIR) Contents Part I. Department of Mathematics