University of Nebraska Kearney

Mathematics and Statistics Assessment Report 2006

Submitted Fall, 2006

University of Nebraska at Kearney Department of Mathematics and Statistics
Student Assessment Academic Year 2005 – 2006

The student assessment plan of the Department of Mathematics and Statistics at the University of Nebraska at Kearney is based on the mission’s statement of the Department. The missions of the Department now include providing quality instruction in mathematics, mathematics education, and general studies courses to students majoring in mathematics or mathematics education, and to the entire student population, together with research, other forms of scholarship and service in all of these areas.

Students who successfully complete undergraduate degrees offered by the Department of mathematics and Statistics at the University of Nebraska at Kearney should be able to demonstrate an understanding and ability to apply the concepts and processes of mathematics and statistics in a variety of careers in government and business, in secondary school teaching, in graduate study, and with proper planning, in a variety of professional programs. In addition, graduates will develop an understanding of the importance and scope of professional ethics and play active and ethical roles as participants, and leaders in industry, education, graduate schools, and society.

Based on this mission, the Department has established a number of student learning objectives, and has developed a variety of direct and indirect measures to assess the Department’s success in achieving them.

In the 2005/06 academic year, the Assessment Program of the Department of Mathematics and Statistics included the collection of data in the 2005 Fall semester to assess the success of the departmental programs and the general studies program.

A. Departmental Programs Assessment

The following measuring instruments were used in the academic year 2005 – 06 to collect data in order to assess the progress of students enrolling in any of the programs of study offered by the Department: B.A. in Math, B.S. in Math, B.S. in Math Comprehensive, B.S.Ed. in 7-12 Math Subject Endorsement, or B.S.Ed. in Math Field Endorsement towards the goals and objectives of these programs of study, and to determine how well graduates from these programs were mathematically prepared to perform their duties in their current positions.

I. Capstone Experience

Assessment Report for Advanced Calculus Math 460 (WI)

During the first two weeks of the fall term, the Department of Mathematics and Statistics administers an assessment test to all students registered for our capstone course Advanced Calculus (MATH 460). This course is required for all mathematics and mathematics education majors; most students take it during their Junior or Senior years. Questions on the assessment test come from Calculus I (MATH 115), Calculus II, (Math 202), Calculus III (Math 260), and Foundations of Mathematics (MATH 250).

The primary purpose of the assessment test is to determine how well students are learning and retaining the fundamental concepts from MATH 250 and the calculus sequence (MATH 115-Math 260). In part, we use the results of the assessment test to determine what topics need to be given special attention in Advanced Calculus (Math 460 WI).

In the spring 2004 term, the departmental Assessment committee approved an assessment test that was to be administered in the 2005 fall term. The test covers symbolic logic (MATH 250), set theory, function concepts, and ideas and concepts of elementary calculus, especially the Newton quotient.

Results of Math 460 Assessment Test, Sept. 2005

Assessment Report
Advanced Calculus (MATH 460 WI)
Fall, 2005
Barton Willis 

Introduction: 

Each fall term the department of Mathematics gives an assessment test to the members of the MATH 460 (Advanced Calculus) class. The test is given during the first two weeks of the term. The primary intent of the test is to determine how well students are prepared for MATH 460; additionally, we are interested in knowing how prepared students are for additional study of mathematics, and how well students (mainly mathematics education majors) are prepared to teach calculus.

The questions on the test are grouped into three categories: questions la, 1b, 2, and 3 cover logic; questions 4, 5a, and 5b cover sets and functions; and questions 6a, 6b, 7, 8a, 8b, 9a, 9b, and 10 cover basic calculus facts. Each question was graded either correct or incorrect. I graded the questions fairly strictly, but a minor mistake did not necessarily earn a score of zero. A example of a minor mistake is something like miss copying a symbol from one line to the next.

Results: 

In fall 2005, fourteen students took the assessment test. The score in each table shows the number of students that gave a correct answer to the question.

Discussion: 

Except for the question on one—to--one functions (question 5b), students did well on the logic questions and the questions on sets and functions. It’s clear that students are thoroughly learning this material in MATH 250.

On the remainder of the test, the students did poorly. Only two of fourteen students could identify

(a difference quotient) as the slope of a secant line (question 9a); however, four of fourteen students recognized the limit of a difference quotient as the slope of a tangent line (question 9b). Some students did realize that the limit of the difference quotient defines the derivative, but the question 9b asked for the geometric meaning of the derivative. Two students responded that the limit of the limit of the difference quotient had something to do with area.

No student gave a correct response to the question about the Riemann sum; most students did attempt to answer this question.

Conclusion: 

Students are learning and retaining the material from MATH 250. It’s not necessary to spend much class time in MATH 460 on sets and the basic properties of functions.

Students did poorly on the questions involving concepts from MATH 115. I don’t believe that the cause is that we are doing a poor job; instead, I think that some concepts from calculus simply take quite some time to internalize. Often, understanding only comes after learning about a concept from a new and more abstract perspective. This is what MATH 460 is all about. The final examination for MATH 460 had three questions that involved the definition of the derivative. The thirteen students students that took the final all earned full credit on these questions. Unfortunately, we were only able to devote about five lectures on the topic of Riemann sums. This was not enough time; only one third of the class gave correct answers to the two questions on the final that involved the Riemann sum.

Recommendations: 

1. Twelve of fourteen students answered question 6a (find the value of the limit
   
   correctly. Although there is a great deal of lovely mathematics behind this question, getting the correct answer doesn’t require any understanding of continuity or the limit. Even students with faulty beliefs (if a limit can’t be found by direct direct substitution, it doesn’t exist) will get the correct answer. I suggest replacing this question with something like:
   Let F : R R. If what property must have?
    
2. There is too much overlap between questions 9a and 9b; either one should be dropped, or they should be blended into one question. It’s likely that some students knew that the derivative is the slope of a tangent line, but they didn’t know what was meant by the “geometric meaning.” Maybe the test next year could ask the same question using somewhat different language.

A copy of the test follows.

Advanced Calculus, Fall 2005 Name:_____________________
Assessment Test, 26 August

1. This exam has questions 1 through 10 with a total of 75 points. the Box to the left of each question gives the point value for the question.
    
2. You needn’t fill in your name.
    
3. Do as well as you can; your score will not hurt or help your course grade.
    
4. If your are asked to explain your answer, do so using complete English sentences; if no explanation is requested, respond only with the work you used to answer the question.
    
5. After you hand in this examination, it becomes property of the University. Your identity will not appear in the assessment report.

1. For each statement, give its negation. Here A and B are both subsets of the real numbers.
   
   
   1. 5 For every , it is true that
       
   2. 5 For at least one , it is true that
       
    
2. 5 Use a truth table to show that  is a tautology.¹ 
3. 5 Write the contrapositive of the statement: If a function F is differentiable at 5, then F is continuous at 5.
    
4. 5 Let A and B be sets. Prove that , where is the complement of A and  is the complement of .
    
5. Let F: 
   
   
   1. 5 Show that the function F is one-to-one.
       
   2. 5 Show that the function F is onto.
       
    
6. Find the value of each limit.
   
   
   1. 5 
       
   2. 5  .
       
    
7. 5 Find a  R that makes the function  continuous on R.
   
    
8. Find a formula for the derivative of F.
   
   
   1. 5 F(x) = cos(sin(x)),
       
   2. 5 F(x) = 
       
    
9. 5 Let F : R—R be a function.
   
   
   1. For any x  0, explain the geometric meaning of the quotient .
      
       
   2. 5 Assuming  exists and is a real number, what is a real number, what is the geometric meaning of the limit?
       
    
10. 5 Express the definite integral  as a limit of a Riemann sum. Form the sum using either inscribed or circumscribed rectangles.² 

_________________________________________________________________
¹The symbol ~ means “negation;” the symbol V means “or;” and the symbol A means “and.”
²Equivalently, either use the left—point sums or right—point sums.
 

II. Alumni Assessment Survey

Every 4 – 6 years, the Department sends out an assessment instrument to individuals who graduated from any of our programs either four, five, or six years previously. The purpose of this assessment is to elicit meaningful observations from the past graduates of our programs of study relative to how their respective programs met the demands of their chosen pursuits in the three to six years time period following their graduation. The purpose of this instrument is to help the Department assess the growth of graduates from our programs toward students learning objectives. The following are the results of the Alumni Assessment Survey for graduates from 1994 through 2002. The Department plans to send out the survey in the fall of 2008

UNK Math/Stat Program Assessment Survey Results
For
Graduates from 1996 through 2002

In January, 2005, 129 copies of the Mathematics and Statistics Programs Assessment Survey were mailed to graduates from the Department’s programs for the period extending from 1996 through 2002. The graduates were asked to respond to the items included in the survey and return their completed surveys to the Department of Mathematics and Statistics. Only 52 surveys of the original 129 were returned. Among these 52 surveys, 17 respondents did not complete items 4, 5a, 5b, and 5c. Since the assessment of the programs depends on these items, all 17 surveys are not included in the analysis of the results of the survey.

The results of the 35 respondents who completed all the items of the survey were as follows:

Note that some graduates have changed jobs and probably more than once since their graduation. The number of graduates who changed jobs and how many times jobs were changed are not clear from their responses. In addition, some respondents have more than one position, and again, the number of those occupying more than one position is not clear from their responses.

B. General Program Mathematics Assessment

The specific General Studies objectives relating to the Department of Mathematics and Statistics include: A student will be able to:

1. apply functions and other mathematical relationships to the analysis of a wide variety of applied problems,
2. use one or more of a variety of mathematical methodologies, including but not limited to algebraically based deduction, statistical processes, the limit, the derivative, or the integral, to define, analyze, and solve a diverse selection of applied problems,
3. use the “language” of mathematics and/or statistics to effectively communicate problems analysis and solutions and think critically and use reasoning in solving applied problems,
4. demonstrate the ability to express formal, mathematical relationships using logical analyses and differing forms of mathematical reasoning,
5. demonstrate the ability to utilize mathematical techniques in order to define problems and to search for strategies for testing solutions, and
6. demonstrate the ability to manage and interpret numerical data using the appropriate mathematics tools.

Students in three general studies math courses, Math 115 (Calculus I With Analytic Geometry), Math 123 (applied Calculus I), and Stat 241, Elementary Statistics were given additional items on the final exam, which were linked to specific general studies objectives. In each of these three courses, the additional items were constructed by a committee consisting of three faculty members of the Department. Each members of each committee was teaching at least one section of the course.

In tables (1), (2) , and (3) shown below, the percentage of students who passed each question in each of these three courses ( i.e. received 60% of the points ) is listed.

It must be noted that the sample of students for these questions includes Those who might end up failing the course anyway. So the percentage of students who passed each item and passed the course would be higher. The questions themselves were constructed by a committee of three faculty members in each case, and thus were not pre-tested for difficulty level nor standardized on a large sample. Nonetheless, the proportion of students passing each item ranges from 47% to 83% for Math 115, from 79% to 97% for Math 123, and from 28% to 95% for Stat 241.

The questions and the analyses for each of the three courses are given below.

I. FALL SEMESTER 2005

ASSESSMENT REPORT FOR MATH 115, CALCULUS I
WITH ANALYTICAL GEOMETRY

RICHARD L. BARLOW, MATH 115 COORDINATOR

Assessment Procedure: 

In the fall semester of 2005, the Department of Mathematics & Statistics offered three sections of Math 115. The three sections were offered at 08:00-08:50 (Niemann), 09:05-09:55 MWF & 9:30 – 10:20 TT (Barlow), and 12:20-01:10 MW & 12:30 – 01:45 TT(Heckman) using the same textbook Calculus, Seventh Edition, by Larson, Hostetler, and Edwards. Each section covered the same five chapters (Chapters 1 – 5 more or less).

To assess the success of Math 115, the committee (Barlow, Heckman, & Niemann) believed it would be best for all three sections to be included in the assessment procedure. As such, each final examination would include six identical problems in addition to the other problems each professor used to test his class. The question placement on the three individual final examinations was dependent upon the professor instructing the class, but each final did include these six common questions. The six common questions were written to assess the important calculus concepts deemed necessary for the Math 115 student to have the knowledge of calculus needed for future work and the successful mastery of the content of Math 115. The six questions were written by the three Math 115 professors jointly.

General Studies Objectives: 

Across the range of disciplines and courses offered, the General Studies Program is designed to develop and demonstrate the following abilities:

1. the ability to locate and gather information,
2. the capability for critical thinking, reasoning, and analyzing,
3. effective communication skills including the ability to read, speak, and write effectively, using the materials, ideas, and discourse modes of specific academic areas,
4. an understanding of the experiences and values of groups and cultures which have been historically under-represented.

General Studies Student Objectives: 

Upon completion of the General Studies Program, students will be able to demonstrate:

1. the ability to locate and gather information,
2. the capability for critical thinking, reasoning, and analyzing,
3. effective communication skills including the ability to read, speak, and write effectively, using the materials, ideas, and discourse modes of specific academic areas,
4. an understanding of the experiences and values of groups and cultures which have been historically under-represented.

Department of Mathematics & Statistics Additional Objectives: 

In addition to the four general objectives noted above, there are specific objectives relevant to each of the major categories within the General Studies Program. For the Department of Mathematics & Statistics the student will

1. demonstrate the ability to manage and interpret numerical data using the appropriate mathematical tools,
2. demonstrate the ability to express formal, mathematical relationships using logical analyses and differing forms of mathematical reasoning,
3. demonstrate the ability to utilize mathematical techniques in order to define problems and to search for strategies for testing solutions,
4. apply functions and other mathematical relationships to the analysis of a wide variety of applied problems,
5. use one or more of a variety of mathematical methodologies, including but not limited to algebraically based deduction, statistical processes, the limit, the derivative, or the integral, to define, analyze, and solve a diverse selection of applied problems,
6. use the language of mathematics and/or statistics to effectively communicate problem analysis and solutions, think critically and use reasoning in solving applied problems.

Assessment Questions: 

As stated previously, each of the three Math 115 sections had six common problems used to measure the Department of Mathematics & Statistics objectives. They were by question number:

1. Given some information about a function, the student was asked to sketch the graph of that function. (Objectives a,b,d,e,f)
2. Given a function, the student was asked to find a formula of its derivative where the student needed to apply the product rule to correctly answer the question. (Objectives e & f)
3. Given a function, the student was asked to find a formula of its derivative where the student needed to apply the quotient rule to correctly answer the question. (Objectives e & f)
4. Given a function, the student was asked to find a formula of its derivative where the student needed to apply the chain rule to correctly answer the question. (Objectives e & f)
5. To answer this question, the student needed to understand the concept of the antiderivative. (Objectives c & e)
6. To answer this question, the student needed to use a u-substitution to solve an antiderivative. (Objectives c & e)

Assessment Results: 

Each of the three Math 115 professors placed the six identical questions at an appropriate position of his final examination. Some were at the beginning of that final exam and some were dispersed throughout the final examination. The point total for each problem varied depending upon the professor’s point assignment. For the purpose of standardizing the results, each of the six common problems were standardized to a five point perfect score with proportional partial credits as assigned by each professor. In all, 66 student scores for the three sections of Math 115 are included in this report. The results were:

Summary: 

Most of the students were able to efficiently find the derivatives required in questions 2 through 4 with at least 50 of the 66 students receiving perfect scores. Questions 5 and 6 were integration methods which were taught later in the semester which might account for the lower scores. Question 1 entailed using knowledge in a different form than many of the students were accustomed; hence, some found that problem more difficult to get a perfect score. All in all, the 66 students appeared to achieve our goals and knew a lot of calculus at the conclusion of the three sections. Since these questions appeared on the final examinations during the university final exam week in December 2005, the assessment survey included only those students completing the course. The assessment question results pretty well fell in line with the course grade results.

II. FALL SEMESTER 2005

ASSESSMENT REPORT FOR MATH 123, APPLIED CALCULUS

DONALD NIEMANN, MATH 123 COORDINATOR

Math 123 Assessment Questions as related to Objectives 

The learning outcomes specific to Mathematics/Statistics Courses are as follows:

Students will demonstrate the ability to:

1. apply functions and other mathematical relationships to the analysis of a wide variety of applied problems.
2. use one or more of a variety of mathematical methodologies, including but not limited to algebraically based deduction, statistical processes, the limit, the derivative, or the integral, to define, analyze, and solve a diverse selection of applied problems.
3. use the “language” of mathematics and /or statistics to effectively communicate problem analysis and solutions.
4. think critically and use reasoning in solving applied problems.

Assessment Questions and Student Performance 

Question #1. (6 points)

Find the x-value for all relative extreme values and state the extreme values of the function

F(x) = x⁴ – 8 x + 3

 

Points Awarded:	6	5	4	3	2	1	0
# of Students:	20	2	9	6	0	0	1

Question # 2. (12 points)

Find the derivative of each of the following functions:

a. b. G(x) = (8 x – 3 x²)³ c. H(x) = x² e^3x 

 

Points Awarded:	12	11	10	9	8	7	6	5	4	3	2	1
# of Students:	17	8	3	1	7	0	1	0	0	0	1	1

Question #3. (6 points)

Explain and sketch the graph of the following function without the aid of a calculator.

 

 

Points Awarded:	6	5	4	3	2	1	0
# of Students:	20	0	11	2	6	0	0

Question #4. (4 points)

How much should be deposited in an account paying 7% interest compounded monthly in order to have a balance of $15,503.77 in a period of three years?

 

Points Awarded:	4	3	2	1	0
# of Students:	37	1	1	0	0

Summary: 

Percent of Students attaining 60% or better of the possible number of points on each question (No. of Students: 39) 

Question #1: 79% of the students attained a score of 60% or better of the possible points.

Question #2: 92% of the students attained a score of 60% or better of the possible points.

Question #3: 79% of the students attained a score of 60% or better of the possible points.

Question #4: 97% of the students attained a score of 60% or better of the possible points.

III. FALL SEMESTER 2005

ASSESSMENT REPORT FOR STAT 241,
ELEMENTARYSTATISTICS

POLLY AMSTUTZ, STAT 241 COORDINATOR

Assessment for Statistics 241-02
Instructor: Polly Amstutz

Question 1: Dependent t-test; t-distribution

1a. Set up null and alternative hypotheses (objectives 1 and 3) (2 points)

1b. Choose correct test statistic and calculate its value. (6 points)

1c. Choose correct distribution, set up critical region and make decision (objective 3). (4 points)

Question 2: Estimating the difference of two proportions (objectives 1,2,3, and 4)

2a. Construct a confidence interval estimate. (6 points)

2b. Find a point estimate. (2 points)

Question 3: Test for the difference of two proportions; Z-distribution

3a. Set up null and alternative hypotheses (objectives 1 and 3). (2 points)

3b. Choose the correct test statistic and calculate its value. (objectives 1, 2, and 4). (6 points)

3c. Choose correct distribution set up critical region and make decision (objective 3). (4 points)

Question 4: Goodness of Fit test; Chi-Square distribution

4a. Set up null and alternative hypotheses (objectives 1 and 3) (2 points)

4b. Choose correct test statistic and calculate its value (objectives 1, 2, and 4). (8 points)

4c. Choose correct distribution, set up critical region and make decision (objective 3). (4 points)

Question 5. ANOVA-test; F-distribution

5a. Set up null and alternative hypotheses (objectives 1 and 3). (2 points)

5b. Choose correct test statistic and calculate its value (objectives (1, 2, and 4). (8 points)

5c. Choose correct distribution, set up critical region and make decision (objective 3). (4 points)

Assessment questions for Statistics 241-02
Instructor: PoIIy Amstutz
Fall 2005

Question 1: A teacher was interested in finding out whether a special study program would increase the scores of students on a national exam. Twelve students were selected and paired according to lQ and scholastic performance. One student from each pair was randomly selected to participate in the special program, with the other student participating in the standard program. Then the students took the national exam The results were as follows. Use a 10% level of significance and test the claim that the special study program in more effective in raising the national exam score.

1a. Set up null and alternative hypotheses

1b. Choose the correct statistic and calculate its value

1c. Choose the correct distribution, set up the critical region and make decision

Question 2. Out of a random sample of 100 adult Americans who did not attend college, 35 of them said they believe in extraterrestrials. However, out of a random sample of 100 adult Americans who did attend college, 48 claim that they believe in extraterrestrials. Construct the 90% confidence interval to estimate the difference in the proportions of these groups that believe in extraterrestrials.

2a. Construct a confidence interval estimate

2b. Find the point estimate for the difference in proportions of the two groups.

Question 3. Is the drug Zyban more effective in helping smokers quit their habit than a nicotine patch? In a study of 223 Zyban users, 109 quit smoking, while 80 of 223 quit smoking by using a nicotine patch. Test the claim that Zyban is more effective than a nicotine patch in helping smokers quit smoking. Use a 0.10.

3a. Set up null and alternative hypotheses

3b. Choose the correct test statistic and calculate its value

3c. Choose the correct distribution, set up the critical region and make decision

Question 4. A researcher wanted to see whether the modes of transportation to work by workers had changed over the past five years. Five years ago, 70% of the workers had driven alone, 20% had been in a carpool, 8% used public transportation, and the rest used other modes. A random sample for the current year yielded the following results:

Mode 0
alone 320
carpool 130
public 35
other 15

Is there sufficient evidence to indicate that the modes of transportation used to get to work have changed? Use a 0.05 level of significance.

4a. Set up the null and alternative hypotheses

4b. Choose the correct test statistic and calculate its value

4c. Choose the correct distribution, set up the critical region and make decision

Question 5. A production plant manager claimed that there was no difference in mean times to complete an assembly line job between plants A, B, C. Samples from each of the plants yielded the following data, where a data value represents the time in minutes to complete the job. Test the claim. Use 0.05

Fill in an ANOVA table and test for equality of means.

5a. Set up the null and alternative hypotheses

5b. Choose the correct test statistic and calculate its value

5c. Choose the correct distribution, set up the critical region and make decision

Assessment for 241-01
Fall 2005
Polly Amstutz

Assessment
Statistics 241-01, Fall 2005
Instructor: Polly Amstutz