DEPARTMENT OF MATHEMATICS AND STATISTICS

 

Undergraduate Assessment and Graduate Assessment in Statistics

 I. Assessment Procedures

III. Changes to Procedures or Curriculum Based on Assessment

 II. Achievement of Departmental Objectives for Students   

 A. Undergraduate-- 
     B. Graduate

IV. Changes in Department’s Assessment Goals

 

 

I. Assessment Procedures

A.      Outcomes assessed

1.      The “Core”: The following classes are the ones most often taken by students in their core. As a result, emphases on assessment have been placed on these courses. Note: CCS = Course Content Standard. For all classes retention and overall success rates are of immense importance to us as we strive to reduce withdrawal rates in all of these classes to less than 20% and increase successful completion rates of the retained students to at least 75%.

 

o           MATH 1101

The following CCSs were assessed both Fall and Spring semesters.

1.   CCS 3: Linear Functions. Students will demonstrate the ability to:

a.     Determine when two real-world variables are related by a linear or piecewise linear function.

b.     Model the behavior of two real-world variables that are directly proportional or are related by a linear or piecewise linear function using tables, graphs, equations, or combinations thereof.

g.     Define the linear function and the general equation of the linear function.

 

2.   CCS 4: Exponential Functions. Students will demonstrate the ability to:

a.   Determine when two real-world variables are related by an exponential function.

b.   Model the behavior of two real-world variables that are related by an exponential function using tables, graphs, equations, or combinations thereof including such applications as population growth and decay, radioactive decay, simple and compound interest, inflation, the Malthusian dilemma, musical pitch, and the Rule of 70.

e.   Evaluate exponential functions.

f.    Determine the exponential equation model from the table or graphical model.

 

3.   CCS5: Logarithmic Functions. Students will demonstrate:

a.   The ability to determine when two real-world variables are related by a logarithmic function.

b.   The ability to model the behavior of two real-world variables that are related by a logarithmic function using tables, graphs, equations, or combinations thereof including such applications as pH and the decibel system.

c.   Their understanding of the natural logarithm.

 

4.   CCS 6: Polynomial and Quadratic Functions. Students will demonstrate the ability to:

d.   Determine when two real-world variables are related by a quadratic  function by calculating the average rate of change of the average rates of change.

e.   Model the behavior of two real-world variables that are related by a quadratic  function using tables, graphs, equations, or combinations thereof including such applications as maximum area for fixed perimeter, minimum perimeter for fixed area, free fall, maximum profit, and break-even analysis.

f.    Determine the vertex, axis of symmetry, and horizontal and vertical intercepts of quadratic functions in either the a-b-c or a-h-k forms.

 

o        MATH 1111

              Fall

1.   CCS 2: Understand linear functions and be able to identify, graph, and find equations of linear functions (including parallel and perpendicular lines).

2.   CCS 4: Understand, identify, graph, interpret and apply the following in applied settings. Specifically,

·        quadratic functions of the form y =  - determine the vertex and intercepts.

·        Polynomial functions where the polynomial is factorable.

·        Exponential functions of the form y =  and their transformations.

3.   CCS 5: Determine, both algebraically and graphically, solutions to the following types of equations and apply these solutions to concepts related to functions and other applications: Quadratic

Spring

1.   CCS 1: Understand the general definition of a function and be able to:

·         Illustrate a function verbally, graphically, with charts/tables, and with set notation

·         Determine the domain and range of a function

·         Identify where a function is increasing, decreasing or constant.

2.      CCS 4: Understand, identify, graph, interpret and apply the following in applied settings

·         Quadratic functions of the form y =  

§   Determine the vertex and intercepts.

·         Polynomial functions where the polynomial is factorable.

§   Students will be able to describe the end behavior of polynomials and the relationship between end behavior and the degree of the polynomial.

§   Students will be able to determine intercepts of factorable polynomials exactly.

§   Students will be able to use appropriate technology to approximate x-intercepts and local extrema of polynomials.

·         Inverse functions

§   Get a rule for an inverse function

§   Graph a function and its inverse

·         Exponential functions of the form y = and their transformations.

·         Logarithmic functions

§   Define a logarithm

§   Convert between logarithmic and exponential forms

§   Understand the inverse relationship between logarithmic and exponential functions

3.   CCS 5:            Determine, both algebraically and graphically, solutions to the following types of equations and apply these solutions to concepts related to functions and other applications:

·         Factorable polynomial

·         Simple exponential equations

 

 

o        MATH 1113

1.      CCS2: Algebraic Functions - Students will use functions and related concepts including: recognition of a function in either graphical, table, implicit, or explicit form; be able to find domains and ranges and determine if a function is one-to-one; perform operations of functions including composition, finding inverses, and finding difference quotients.

2.      CCS3. Defining the Trigonometric Functions - Students will use circular and trigonometric functions and related concepts including: find exact values of the functions by using the unit circle, wrapping function, and special triangles; know the relationship between radian measure and degree measure and be able to convert from one unit to the other; know the definition of the six (6) trigonometric functions as related to the right triangle; distinguish between right angled and oblique triangles and recognize the appropriate method needed to solve the triangle (Law of Sines, Law of Cosines, Pythagorean Theorem)

3.      CCS4. Use of Trigonometric Functions - Students will demonstrate knowledge of and be able to use trigonometry. Specifically: (1) given one of the trig values of an angle in a certain quadrant, be able to find the other five trigonometric functions through identities not limited to Pythagorean, identity, reciprocal identities, even/odd identities and quotient identities, (2) solve oblique triangles using the Law of Sines, and Law of Cosines, and work related applied problems, (3) graph the basic six trigonometric functions, including sine and cosine functions with applied graph transformations; identify the domain, range, period, amplitude and phase shifts of the functions, (4) find the exact values of the inverse trig functions, (5) solve linear and quadratic trigonometric equations and equations with compound angles.

 

o        MATH 2211

Both Fall and Spring:

1.   CCS 6B. Differentiation. Students will demonstrate an understanding of the derivative at a point, derivative functions, and related concepts.

2.   CCS 8A. Applications. While applying analytic, algebraic, geometric, and algorithmic techniques to solving applied problems students will communicate how the problem is modeled by a mathematical formulation, and how to interpret the result of the mathematical analysis.

Spring only

3.   CCS 7B. Integration. Students will demonstrate an understanding of integration and related concepts including applying properties of integration related to elementary functions, operations on functions, and elementary substitutions;

 

                  2.   The Undergraduate Major

·         General Learning Outcomes

 

                  3.   Graduate Programs

 

B.     Elements of assessment

1.      The “Core”

·         Common problems were used on the final exams of each of the classes listed in (A). Copies of these questions are attached to this document in Appendix A. Note: 1101 instructors were given the option of including anywhere from 2 to all 4 problems on their final exam. Of the data available, only one instructor included all four while the rest selected the linear and exponential model problems.

·         “DWF” rates for these students will be used to examine retention and success rates.

                 

2.   The Undergraduate Major

·         General Learning Outcomes

o        Faculty were surveyed about the students who graduated during the fiscal year (see attached form in Appendix B)

·         Major Learning Outcomes

o        Faculty were surveyed about the students who graduated during the fiscal year (see attached form in Appendix B)

o        ??

                  3.   The Graduate Major

o        Faculty were surveyed about the students who graduated during the fiscal year (see attached form in Appendix B)

 

C.     Data collected

1.      The “Core” – Data is in Appendix C.

·         Copies of each student’s work from the final exam (prior to its being graded by the instructor) was provided to a central person (course coordinator) for each of the indicated courses. This work was graded by the coordinator using a rubric they developed and summary as well as individual class statistics were generated. Copies of the summary statistics for Fall 2004 and Spring 2005 are in Appendix C. Note: Only summary data from Fall is currently available while only data from 5 of 10 instructors of 1101 in the spring are available.

·         “DWF” rates (from STATWARE) for these students will be used to examine retention and success rates.

                  2.   The Undergraduate Major

·         General Learning Outcomes – faculty survey results (Summary Data in Appendix D)

·         Major Learning Outcomes

o        Survey results (Data in Appendix D)

o        Correlation statistics

o        ??

                  3.   The Graduate Major

·         General Learning Outcomes – survey results (Summary Data in Appendix D)

·         Course specific outcomes assessed (Sample Summary Data in Appendix D)

 

D.     Data Analysis

1.      The “Core”

·         MATH 1101

o        Approximately 64% of the students received a grade of 7 or higher on the linear model problem, while approximately 59% received a grade of 7 or higher on the exponential model problem.

o        Note: Two sections of MATH 1101 in the spring were taught with a different text that had supplemental work from MyMathLab (the software package used in the redesign of MATH 1111 and MATH 1113). The use of Excel was also a part of these sections and was used by the students to solve the common problems on the final. Student performance on these problems is given below:

7+

Traditional Students

Redesign Students

Linear

56.80%

88.14%

Exponential

59.22%

59.32%

Thus there was a tremendous improvement in the students’ ability to model linear data, though there was no real difference in their ability to model exponential data.

 

·         MATH 1111

o        Though student performance on the common assessment elements showed the traditional students (as a whole) performed better than the redesign students, this difference was small in most cases. The true impact, however of student performance is retention of information and the student’s performance in successive classes. Students will be tracked to determine the longitudinal impact of the redesign on their performance in future classes.

o        A c2 test for difference of distribution was performed on the grade distribution of the students in the redesigned sections with the historical distributions in both fall and spring. Also, a c2 test for difference of distribution was performed on the grade distribution of the students in the redesigned sections vs. the traditional sections in each of the particular semesters, with p-values summarized in the following table:

 

 

 

 

 

 

 

 

p-values for

Historical Distribution

Traditional Sections

Fall 2004

7.104E-24

0.000161

Spring 2005

0.002565

0.000832

 

 

 

 

 

 

 

 

Thus, it is easy to say there was a significant change in the grade distributions for the students participating in the redesigned sections of MATH 1111 as compared to the students who took the class under a traditional format, both historically and during the specific term.

o        A second object is to have 75% of students who complete the course earn a C or higher. Historically we have made great gains towards this goal and met the goal once prerequisite checking was fully implemented in Fall 2003.

 

 

A

B

C

D

F

A-F

AVG

ABC %

DF%

Fall 00

305

351

322

153

203

1334

2.30

73.3%

26.7%

Fall 01

244

250

244

106

120

964

2.41

76.6%

23.4%

Fall 02

205

301

207

93

139

945

2.36

75.4%

24.6%

Fall 03

159

201

146

57

66

629

2.52

80.4%

19.6%

 

However, it should be noted that the redesign of the College Algebra course has increased the success rate of our students in the fall as well.

 

Fall 2004

 

 

 

 

 

 

 

 

 

 

A

B

C

D

F

A-F

AVG

ABC %

DF%

Redesign

95

77

38

9

10

229

3.04

91.7%

8.3%

Traditional

147

169

104

36

47

503

2.66

83.5%

16.5%

All

242

246

142

45

57

732

2.78

86.1%

13.9%

 

Meeting the objective of having 75% of students who complete the course earn a C or higher has been more difficult in the spring than in the fall. Historically we have made great gains towards this goal and met the goal once prerequisite checking was fully implemented.

 

 

A

B

C

D

F

A-F

AVG

ABC %

DF%

200001

180

176

201

78

132

767

2.25

72.6%

27.4%

200101

224

196

152

93

162

827

2.27

69.2%

30.8%

200201

84

152

110

60

76

482

2.22

71.8%

28.2%

200301

101

124

117

78

112

532

2.05

64.3%

35.7%

200401

63

64

50

22

30

229

2.47

77.3%

22.7%

 

 

 

 

 

 

 

 

 

 

200501

42

76

55

30

38

241

2.22

71.8%

28.2%

 

However, again the redesigned sections of College Algebra have had a positive impact on the level of success of retained students, very nearly attaining the desired 75% success rate:

 

 

A

B

C

D

F

A-F

AVG

ABC %

DF%

Traditional

17

27

22

13

19

98

2.10

67.3%

32.7%

Redesign

25

49

33

17

19

143

2.31

74.8%

25.2%

 

·         MATH 1113

o        Student performance on the common assessment elements in the fall was disappointing with only about 50% of the students getting the algebra problem correct and significantly fewer students correctly completing the other two problems. This, however, provided us with baseline data against which we could compare the spring redesign students’ performance on the same type of questions.

 

In the spring, the redesign students significantly outperformed the traditional students on the common assessment problems, though overall success levels were still disappointing.

 

o        Historically we have experienced great difficulty in having students successfully complete Precalculus.

 

 

AVE

TOTAL

DF %

W/WF%

DWF %

200208

2.2

809

23.4%

22.9%

46.2%

200308

2.2

614

22.0%

18.9%

40.9%

200408

2.2

547

23.9%

21.2%

45.2%

 

Such DWF rates are, unfortunately, not uncommon. However, we are striving to bring these down to the rates of 20% withdrawal and 75% of retained students successfully completing the course. As can be seen in the table below, we have approximately 69% of our retained students successfully completing MATH 1113.

 

 

A

B

C

D

F

A-F

AVG

ABC %

DF%

200108

123

140

103

67

100

533

2.22

68.7%

31.3%

200208

149

141

115

62

127

594

2.21

68.2%

31.8%

200308

108

116

100

52

83

459

2.25

70.6%

29.4%

200408

90

110

97

78

53

428

2.25

69.4%

30.6%

 

In the spring the redesign efforts were extended to MATH 1113. Retention of students in the spring has been a huge issue with withdrawal rates regularly between 25 and 30%. However, in Spring 2005 this overall level dropped to 20%, thereby meeting this objective. The reason for this is the extremely low withdrawal rate of students involved in the redesign project.

 

 

All Sections

AVE

ABC %

DF %

W %

DWF %

 

200201

2.3

49.9%

19.6%

30.5%

50.1%

 

200301

2.4

56.5%

17.7%

25.8%

43.5%

 

200401

2.3

53.9%

20.7%

25.4%

46.1%

 

200501

2.1

55.6%

24.4%

20.0%

44.4%

 

Breakdown of Spring 2005

 

AVE

ABC %

DF %

W %

DWF %

Students who had Redesigned 1111 in Fall 04:

2.2

67.1%

25.6%

7.3%

32.9%

Students who had Redesigned 1111 in Fall 04 and Redesigned 1113 in Spring 05

2.4

74.6%

22.0%

3.4%

25.4%

 

A c2 test for difference of distribution was performed on the grade distribution of the students in the redesigned sections with the historical distribution. Also, a c2 test for difference of distribution was performed on the grade distribution of the students in the redesigned sections vs. the traditional sections in the spring semester.

 

p-values for

Historical Distribution

Traditional Sections

Spring 2005

0.002755

0.208895

 

·         MATH 2211

Different learning outcomes were assessed in the fall and spring. In the fall over 50% of the students gave a “C” answer or better on a fairly standard maximization problem.

 

In the spring, three questions were given and on only one (a u-substitution problem) did 50% of the students give a “C” answer or better. Student performances on the derivative by definition and the related rates problem were 38% and 25% respectively.

 

    1. The Undergraduate Major

·         The survey data was compiled and sorted by term of graduation, student, and instructor and type of course (core, required major class, elective major class). Averages by term and type of class were computed and a correlation analysis was performed between the outcomes and the students’ GPAs.

 

    1. The Graduate Major

·         The survey data was compiled and all averages were above 3 (out of 4).

 

General Comment: Critical Thinking Skills (in particular the ability to formulate new research questions) seems to be a challenge for our students, both undergraduate and graduate.

 

II. Achievement of Departmental Objectives for Students

 

A. Undergraduate

1.      The “Core”

·         MATH 1101

·         Redesign efforts in MATH 1111 and 1113 have had a tremendous, positive impact on retention with W and WF rates dropping dramatically and meeting the goal of less than 20% withdrawal.

·         Student performances on individual class assessments often failed to meet a hoped for 50% “pass” rate.

·         Achievement of 75% of the remaining students receiving an A, B, or C is near at hand.

 

 

2.      The Undergraduate Major

A comparison of the average values of student success in each of the learning outcomes categories was performed against the expected value of student success as determined by the alignment matrix created during the assessment planning period (located in Appendix D). In many instances students outperformed our expected values, especially in the areas of collaborative skills and technology. However, there were many instances that students did not perform up to expectations.

 

B. Graduate

 

III. Changes to Procedures or Curriculum Based on Assessment

 

A. Undergraduate

1.      The “Core”

·         Assessment efforts will expand to include the courses most commonly used by students in Area D, specifically, MATH 1070 and 2212.

·         Preliminary successes in as a result of the redesign efforts in MATH 1111 and MATH 1113 support the expansion of these efforts to all sections in Fall 2005. Utilization of the lab will be analyzed and a determination will be made as to the feasibility of expanding the redesign effort to other classes.

·         Oversight of the Core courses has been erratic at best and the Department’s attempt at implementing “course coordinators” has been equally erratic. To do a good job of ensuring students have the equitable opportunities to be successful and to have those efforts assessed in equitable ways, the Department’s Improvement of Instruction Committee has developed proposed policies (located in Appendix E) that will be considered by the faculty at its first departmental meeting in August.

·         As redesign efforts are leading to the use of common exams in MATH 1111 and 1113, the Department is considering the use of common exams in the calculus sequence, MATH 2211, 2212, and 2215.

 

2.      The Undergraduate Major

·         Efforts in the assessment of learning outcomes in the major have been slow to implementation. Consistent oversight will be necessary to ensure that the plan submitted is actually executed and data collected. Faculty members typically rely on their final exams to provide them a picture of their students’ abilities (this is anecdotal based on faculty responses to the surveys). More specific and explicit instruments need to be developed for both formative and summative assessment at both the course and program levels.

·         The surveys given to the faculty for input on our recent graduates attainment of learning outcomes seem to have provided consistent data (there were common elements on both surveys) lending some sense of credence to their administration. Discussions are underway to devise ways to gather this data more often (perhaps at the end of each semester or at the end of each class) and more efficiently (most faculty submitted paper copies of their responses leading to data entry time delays). Online submission of these responses seems reasonable, but will likely not be developed until the Department’s WebCT site is moved to Vista.

 

 

B. Graduate

·         Attempts at moving away from course oriented assessments to more global assessments will be undertaken.

 

IV. Changes in Department’s Assessment Goals

 

A.   Undergraduate

·The “Core”

o        Currently the goals of reducing withdrawal rates to 20% and increasing success rates of retained students to at least 75% will be kept in place and emphasis will be place on student success of particular learning outcomes.

o        The use of common assessment procedures (not just problems) will be implemented beginning with the core courses.

·The Undergraduate Major

o        Evaluation of student progress toward the Department’s major outcomes will be implemented more evenly and continuously with an eye on student performance matching more closely the Department’s expected values of student level of success.

 

B.   Graduate
Graduate Assessment in Mathematics

 

I. Assessment Procedures

 

            In 2004, the Department of Mathematics and Statistics adopted the following Mission Statement:

 

            Mathematics is one of the great unifying themes in our modern culture. It is a language, a science, an art form, and a tool of tremendous power. The Department of Mathematics and Statistics, in its courses for both majors and nonmajors, seeks to introduce students to this vast area of knowledge and to show them how mathematics can be used to solve problems.  Graduate education should deepen and intensify that knowledge, preparing its graduates to enter society as creative, scientifically literate citizens.

 

            The overarching goals of any program in mathematics are that mathematics instruction should:  (from MAA’s Source Book for College Mathematics Teaching , Schoenfeld, 1990)

 

Provide students with a sense of the discipline of mathematics.

Develop student’s understanding of important concepts in core areas of mathematics.

Develop student’s ability to explore problem situations in a range of settings, at several levels of difficulty, and with a variety of methods. 

Help students to develop a mathematical point of view – perceive and represent structure and structural relationships. 

Help student’s to develop the ability to read and use mathematical literature and reference material.

 

            This lead to the Department’s adoption of the following General Learning Outcomes for the M.S. Program.   The Assessment Plan for each outcome is also included in Appendix B.

 

Graduates should demonstrate knowledge of the discipline.

            This includes the ability to understand research problems in one or more areas of

                        mathematics and statistics. 

            Students should have an appreciation for the history of the subject, and the

                        sequence of results that has led to the current state of development of one

                        or more areas of mathematics and statistics.

            Assessment plan:

                        Ask questions during the thesis defense or on the general examination to

                                    assess the student’s proficiency.

                        Students should be tracked after graduation, to determine if their

                                    employment is within the field.  This is one indication that the

                                    student’s proficiency was sufficient to warrant employment.

 

Graduates should demonstrate advanced quantitative reasoning and problem solving ability.

            This includes numerical, combinatorial and statistical competency.

            The following existing courses specifically address this goal:  Stat 8561 (Linear

                        Statistical Analysis I), Math 8440 (Combinatorics) and  CSC/Math 8620                              (Numerical Linear Algebra).           At least one of those courses is required for

                        each M.S. candidate:

            Assessment plan:

                        Students must satisfactorily complete degree requirements.

 

Graduates should demonstrate advanced critical thinking skills.

            This includes the ability to see connections across fields within mathematics and

                        statistics as well as the ability to see applications of mathematics and

                        statistics to other disciplines.

            Students should develop a mathematical intuition about “how things work” in one

                        or more field within the discipline.

            This also includes the ability to draw conclusions from data, and to develop an

                        appropriate approach to solving problems.

            Students should be able to extend solution methods to problems not exactly like in

                        the book, both in a theoretical and applied setting.

            Assessment plan:

                        Ask questions during the thesis defense or on the general examination to

                                    assess the student’s proficiency.  This is particularly important on

                                    non thesis general examinations: it may be the only opportunity to

                                    measure such goals.

 

Graduates should demonstrate communication skills, both oral and written.

            This includes the ability to explain ideas to nonspecialists.

            Assessment plan:

                        Thesis students orally defend the thesis, as well as provide the thesis

                                    document.

                        Non-thesis students must produce a paper as part of their degree

                                    requirements.  Consideration should be given to having this paper

                                    orally defended.

 

II. Achievement of Departmental Objectives for Students

 

A. Undergraduate

B. Graduate

 

            Starting in 2005, the Department is beginning the use of the following assessment instrument at each thesis defense.  In Fall, 2005, meetings are scheduled to produce specific learning outcomes for graduate courses.

 

III. Changes to Procedures or Curriculum Based on Assessment

 

A. Undergraduate

B. Graduate

 

            The Department to date has made no changes to its procedures or curriculum based on assessment.  After sufficient data has been collected, changes could be considered.

 

IV. Changes in Department’s Assessment Goals

 

            The Department to date has proposed no changes in its goals for assessment, nor its goals for the M.S. programs.

 

 

 

 

 

Appendix A:

 

Common Questions Used on Final Exams

 

MATH 1101

 

Common Final Exam Assessment Questions (Aligned with Content Standards). The same questions were used both terms. Instructors were asked to select at least two questions to include on their final.

 

CCS 3.            Linear Functions. Students will demonstrate the ability to:

a.         Determine when two real-world variables are related by a linear or piecewise linear function.

b.         Model the behavior of two real-world variables that are directly proportional or are related by a linear or piecewise linear function using tables, graphs, equations, or combinations thereof.

g.         Define the linear function and the general equation of the linear function.

 

Q1: The cost of a rental van for a day depends on the number of miles it is driven. Find an equation that models the relationship between the Cost, C, and the number of miles driven, x. Use the data in the table below.

Number of Miles, x

Cost, C

0

$15.00

50

$17.50

100

$20.00

150

$22.50

200

$25.00

 

3 points for determining that the relationship is linear.

1 point for writing y = b + mx

2 points for determining that slope = 0.05

1 point for determining that b = 15

3 points for determining the equation C = 15 + 0.05x

 

CCS 4.            Exponential Functions. Students will demonstrate the ability to:

            a.         Determine when two real-world variables are related by an exponential function.

b.         Model the behavior of two real-world variables that are related by an exponential function using tables, graphs, equations, or combinations thereof including such applications as population growth and decay, radioactive decay, simple and compound interest, inflation, the Malthusian dilemma, musical pitch, and the Rule of 70.

e.         Evaluate exponential functions.

f.          Determine the exponential equation model from the table or graphical model.

 

Q2: $4,000 is invested in a savings account that is compounded annually. One year later, the account balance is $4,180.

 

1.  2 points      Find the interest rate in percentage form.

2.  5 points      Write a function that represents the account balance as a function of time in years.

3.  3 points      Find the account balance 16 years after the initial investment.

 

CCS 5.            Logarithmic Functions. Students will demonstrate:

a.         The ability to determine when two real-world variables are related by a logarithmic function.

b.         The ability to model the behavior of two real-world variables that are related by a logarithmic function using tables, graphs, equations, or combinations thereof including such applications as pH and the decibel system.

c.         Their understanding of the natural logarithm.

 

Q3: A table represents the simple logarithmic function y = log b x (for b greater than one) if successive y-values all differ by one and the ratio (larger to smaller) of the corresponding x-values is constant (always the same). If the table passes this test, the constant ratio of the x-values is b, the base of logarithm for that function.

 

In the x-y tables below, x represents the vertical leap in centimeters of professional basketball players and y represents the corresponding hamstring tendon strength factor.

 

a.         Apply the test outlined above to the tables below and identify the one table that represents the logarithmic function of the form y = logbx. Round ratios to the nearest thousandth.

                          Table 1                                      Table 2                                         Table 3

 

x

y

 

x

y

 

x

y

 

1.225

2.99

 

11

5

 

4.998

1.609

 

2.505

3.99

 

13

6

 

13.585

2.609

 

5.215

4.99

 

16

7

 

36.929

3.609

 

11.665

5.99

 

20

8

 

100.384

4.609

 

4 points for selecting Table 3

 

b.         Determine the equation of the form y = logbx from the one table above that corresponds to a logarithmic function.

2 points for either of these responses: y = log 2.718 x or y = ln x

c.         Determine the hamstring tendon strength factor when the vertical leap is 40.25 centimeters. Show work.

            2 points for responding 3.695 or 3.70 or 3.7

1 point for a response within plus or minus 0.2 based on approximation from the table or graph

d.         Determine the vertical leap in inches corresponding to a hamstring strength factor of 3.00. Show work.

2 points for responding 20.079 or 20.08 or 20.1 (using y = log 2.718 x) or 20.086 or 20.09 or 20.1 (using y = ln x)

1 point for a response within plus or minus 0.2 based on approximation from the table or graph

 

CCS 6.           Polynomial and Quadratic Functions. Students will demonstrate the ability to:

d.         Determine when two real-world variables are related by a quadratic  function by calculating the average rate of change of the average rates of change.

e.         Model the behavior of two real-world variables that are related by a quadratic  function using tables, graphs, equations, or combinations thereof including such applications as maximum area for fixed perimeter, minimum perimeter for fixed area, free fall, maximum profit, and break-even analysis.

f.          Determine the vertex, axis of symmetry, and horizontal and vertical intercepts of quadratic functions in either the a-b-c or a-h-k forms.

 

Q4: The table below gives the average number of customers, A(t), who will sign up for a new cell phone service plan t weeks after the plan is made available.

t (weeks)

A(t) (customers)

0

0

5

650

10

1100

15

1350

20

1400

25

1250

 

(a)         4 points           This data is best modeled by a ______________________ function.  EXPLAIN!

                              Use linear, quadratic, exponential or logarithmic to fill in the blank.

 

(b)   2 points             Determine the analytical model (equation) that gives A(t) in terms of t. SHOW WORK!

 

(c)   2 points             Using the model from (b), determine the time in weeks after the release of the plan, when no customers will sign up for the cell phone service. SHOW WORK!

 

(d)   2 points              Sketch the graph represented by the table above.

 

MATH 1111 – Fall 2004

Common Final Exam Assessment Questions (Aligned with Content Standards)

CCS 2: Understand linear functions and be able to:

·         Identify, graph, and find equations of linear functions (including parallel and perpendicular lines).

Q1: Find an equation for the line with y-intercept 3 that is perpendicular to the line

CCS 4: Understand, identify, graph, interpret and apply the following in applied settings

·         Quadratic functions of the form y =  

§         Determine the vertex and intercepts.

 

Q2: Determine the vertex of

(a) A (-4, 11)                  (b) B (-4, 18)               (c) C (4, 6)                  (d) D (4, 8)

·         Polynomial functions where the polynomial is factorable.

Q3: Find a polynomial function with zeros: 0, 1, -2

(a)                    (b)

(c)                      (d)

·         Exponential functions of the form y = and their transformations.

 

Q4: If a small island has a population of 25,000 people and if the population grows continuously at a rate of 2.8%, what will be the population in 10 years? Compute answer to 3 significant digits.

(a) 33,100                        (b) 411,000      (c) 31,800        (d) 25,700

CCS 5: Determine, both algebraically and graphically, solutions to the following types of equations and apply these solutions to concepts related to functions and other applications:

·         Quadratic

 

Q5: An object is shot upward with an initial velocity of 240 feet per second so that its height s (in feet) above the ground after t seconds is given by  For what values of t will the object be at least  416 feet above the ground?

(a) [2, 5]               (b) [2, 13]                     (c) [3, 5]                       (d) [3, 13]

MATH 1111 – Spring 2005

The first 15 questions on every student’s final exam were common:

 

CCS 1:   Understand the general definition of a function and be able to:

·         Illustrate a function verbally, graphically, with charts/tables, and with set notation

·         Determine the domain and range of a function

·         Identify where a function is increasing, decreasing or constant.

 

1)         Find g(a+1) when g(x) = 2x + 4.

A) 2a + 4         B) ½ a + 4       C) 2a – 6         D) 2a – 1

 

2)         An equation that defines y as a function of x is given. Solve for y in terms of x, and replace y with the function notation f(x).

9x2 + 7y = 6

            A) f(x) = 6 – 9x2          B) f(x) = -9x2 +        C) f(x) =          D) f(x) =

 

Perform the requested operation or operations.

3)         f(x) = 4x – 9, g(x) = 8x – 6. Find (f – g)(x).

 

            A) – 4x – 3       B) 12x – 15      C) 4x + 3         D) – 4x – 15

 

4)         f(x) = , g(x) = 8x – 13. Find (x).

 

            A) 8      B) 2    C) 2    D) 8

 

5)         The graphs of the functions f and g are shown. Use these graphs to find g(f(4)).

 

 

 

 

 

 

 

 

 

 

 

           

A) 4                  B) 8                 C) 6                 D) -2

 

7)         Find the domain and range of the function f(x) = 1 – x2.

            A) Domain: ; Range:               B) Domain: ; Range:

            C) Domain: ; Range:                         D) Domain: ; Range:           

 

 

 

CCS 4: Understand, identify, graph, interpret and apply the following in applied settings

·         Inverse functions

§         Get a rule for an inverse function

§         Graph a function and its inverse

9)         If f is one-to-one, find an equation for its inverse: .

            A)       B)    C)      D) Not a one-to-one function

 

10)       The graph of a function f is given. Use the graph to find value of f -1(2).

 

 

 

 

 

 

 

 

 

            A) 2                  B)                C) 6                 D)

 

CCS 4:      Understand, identify, graph, interpret and apply the following in applied settings

·         Quadratic functions of the form y =  

§         Determine the vertex and intercepts.

6)         Identify the vertex of the parabola .

            A) (1,0)            B) (6,1)            C) (1,6)            D) (0,6)

 

·         Polynomial functions where the polynomial is factorable.

§         Students will be able to describe the end behavior of polynomials and the relationship between end behavior and the degree of the polynomial.

§         Students will be able to determine intercepts of factorable polynomials exactly.

§        Students will be able to use appropriate technology to approximate x-intercepts and local extrema of polynomials.

8)         Factor f(x) into linear factors given that k is a zero of f(x) = ; k = -5 (multiplicity 2)

            A)                               B)

            C)                              D)

 

 

CCS 5:   Determine, both algebraically and graphically, solutions to the following types of equations and apply these solutions to concepts related to functions and other applications:

·         Factorable polynomial

·         Simple exponential equations

 

11)       Solve .

            A) {6}               B) {- 4} C) {4}               D) {125}

 

12)       For the polynomial, , one zero is 5 – find all others.

            A) 1 + 2i, 1 – 2i            B) 1 + i, 1 – i                C) 1 + , 1 –      D) -1 + 2i, -1 – 2i

 

CCS 4: Understand, identify, graph, interpret and apply the following in applied settings

·         Exponential functions of the form y = and their transformations.

·         Logarithmic functions

§         Define a logarithm

§         Convert between logarithmic and exponential forms

§         Understand the inverse relationship between logarithmic and exponential functions

13) Graph the exponential function  using transformations where appropriate.

A)                                             B)                   

 

 

 

 

 

 

 

C)                                            D)

 

 

14) Evaluate .

 

            A) -2                B) 9                 C) – 9              D) 2

 

15) Write  in logarithmic form.

 

            A) = 3            B)           C)           D)         

 

 

 

 

 

 

MATH 1113 – Fall 2004 and Spring 2005

Common Final Exam Assessment Questions (Aligned with Content Standards)

The vast majority of students taking MATH 1113 go on to take MATH 2211, if not both MATH 2211 and 2212. Hence, the assessment elements this year were very “skill” oriented; i.e., the students were asked to perform common tasks found in either MATH 2211 or MATH 2212.

 

CCS2. Algebraic Functions - Students will use functions and related concepts including: recognition of a function in either graphical, table, implicit, or explicit form; be able to find domains and ranges and determine if a function is one-to-one; perform operations of functions including composition, finding inverses, and finding difference quotients.

Q1: If , find and simplify .  (Fall 2004)

 

      For the function , find  in simplest form. (Spring 2005)

 

CCS3. Defining the Trigonometric Functions - Students will use circular and trigonometric functions and related concepts including: find exact values of the functions by using the unit circle, wrapping function, and special triangles; know the relationship between radian measure and degree measure and be able to convert from one unit to the other; know the definition of the six (6) trigonometric functions as related to the right triangle; distinguish between right angled and oblique triangles and recognize the appropriate method needed to solve the triangle (Law of Sines, Law of Cosines, Pythagorean Theorem)

 

Q2: Write cos(sin-1 x) as an algebraic expression in x free of trigonometric or inverse trigonometric functions.

       

        For u > 0, write the expression  as an algebraic (non-trigonometric) expression in u. (Spring 2005)

 

CCS4 Use of Trigonometric Functions - Students will demonstrate knowledge of and be able to use trigonometry. Specifically: (1) given one of the trig values of an angle in a certain quadrant, be able to find the other five trigonometric functions through identities not limited to Pythagorean, identity, reciprocal identities, even/odd identities and quotient identities, (2) solve oblique triangles using the Law of Sines, and Law of Cosines, and work related applied problems, (3) graph the basic six trigonometric functions, including sine and cosine functions with applied graph transformations; identify the domain, range, period, amplitude and phase shifts of the functions. (4) find the exact values of the inverse trig functions, (5) solve linear and quadratic trigonometric equations and equations with compound angles.

Q3: If , find the exact value of . (Fall 2004)

        Find the EXACT value of . (Spring 2005)

MATH 2211 - Common Final Exam Assessment Question

CCS 6B. Differentiation. Students will demonstrate an understanding of the derivative at a point, derivative functions, and related concepts.

CCS 8A. Applications. While applying analytic, algebraic, geometric, and algorithmic techniques to solving applied problems students will:

·         Use appropriate technology;

·         Communicate how the problem is modeled by a mathematical formulation, and how to interpret the result of the mathematical analysis.

Fall 2004 (Faculty were asked to select one of the following problems to include on their final exam.)

 

1. One side of a rectangular field is bounded by a straight river. The other three sides are bounded by straight fences. The total length of the fence is 800 feet. Determine the dimensions of the field given that the area is a maximum.

 

2. A rectangular garden 200 square feet in area is to be fenced off against rabbits. Find the dimensions that will require least amount of fencing if one side of the garden is already protected by a barn.

 

3. A rectangular playground is to be fenced off and divided into two parts by a fence parallel to one side of the playground. Six hundred feet of fencing is used. Find the dimensions of the playground that will enclose the greatest total area.

 

Spring 2005

CCS 6B. Differentiation. Students will demonstrate an understanding of the derivative at a point, derivative functions, and related concepts including:

Q1: Consider the function .

a.      Find  by definition.

b.      Find the equation of the line tangent to the graph of f(x) at the point (1, f(1)).

 

Q2: The top of a 25 foot ladder, leaning against a vertical wall, is slipping down the wall at the rate of 1 foot/minute. How fast is the bottom of the ladder slipping along the ground when the bottom of the ladder is 7 feet away from the base of the wall?

CCS 7B. Integration. Students will demonstrate an understanding of integration and related concepts including:

Q3: Evaluate .

 

 

 

 

 

 

 

Appendix B:

 

Surveys Utilized in Data Collection

 

General Learning Outcomes Assessment Form for

BS Degree Students

 

Name: _____________________________________________                              ID#: _________________________

Indicate the extent to which the student fulfilled each learning outcome: 

What classes did you have this student for?

 

_________________________________________________________________

Not Applicable

Poor

Good

Very Good

Excellent

Goal I.  Communication

  1. Students communicate effectively using appropriate writing conventions and formats.

0

1

2

3

4

2.      Students communicate effectively using appropriate oral or signed conventions and formats.

0

1

2

3

4

 

 

 

 

 

 

Goal II. Collaboration

 

 

 

 

 
  1. Students participate effectively in collaborative activities.

0

1

2

3

4

 

 

 

 

 

 

Goal III. Critical Thinking

 

 

 

 

 
  1. Students formulate appropriate questions for research. 

0

1

2

3

4
  1. Students effectively collect appropriate evidence.   

0

1

2

3

4
  1. Students appropriately evaluate claims, arguments, evidence and hypotheses.

0

1

2

3

4
  1. Students use the results of analysis to appropriately construct new arguments and formulate new questions.

0

1

2

3

4

 

 

 

 

 

 

Goal IV.  Contemporary Issues

 

 

 

 

 
  1. Students effectively analyze contemporary issues within the context of diverse disciplinary perspectives.

0

1

2

3

4
  1. Students effectively analyze contemporary multicultural, global, and international questions.

0

1

2

3

4

 

 

 

 

 

 

Goal V.  Quantitative Skills

 

 

 

 

 
  1. Students effectively perform arithmetic operations, as well as reason and draw appropriate conclusions from numerical information.

0

1

2

3

4
  1. Students effectively translate problem situations into symbolic representations and use those representations to solve problems.

0

1

2

3

4

 

 

 

 

 

 

Goal VI.  Technology 

 

 

 

 

 
  1. Students effectively use computers and other technology appropriate to the discipline.

0

1

2

3

4

 

Instructor:________________________________________ Date: ________________

 

Director of Undergraduate Studies: ____________________ Date: ________________

 

Department Chair: _________________________________ Date: ________________

Learning Outcomes in the Major Assessment Form for

BS Degree Students

 

Name: _____________________________________________                              ID#: _________________________

Indicate the extent to which the student fulfilled each learning outcome: 

 

What classes did you have this student for?  _____________________________________________________

A graduate of a baccalaureate program in mathematics should:

Not Applicable

Poor

Good

Very Good

Excellent

 

1.  Excel in the use of basic quantitative skills including

Symbolic representation.

0

1

2

3

4

Symbolic manipulation.

0

1

2

3

4

Modeling.

0

1

2

3

4

Pattern recognition.

0

1

2

3

4

Problem solving.

0

1

2

3

4

Quantitative reasoning.

0

1

2

3

4

Estimation.

0

1

2

3

4

2.  Demonstrate content knowledge of core areas of mathematics including

 

 

 

 

 

Algebraic, order, and completeness properties of the real number system.

0

1

2

3

4

Analysis of functions from Rn to Rm.

0

1

2

3

4

Algebra of linear functions from Rn to Rm.