# Assessment Plan 2015

# CALIFORNIA STATE UNIVERSITY, CHICO ANNUAL PROGRAM ASSESSMENT REPORT

BS, Department of Mathematics and Statistics

(4 Options: Applied Mathematics, Pure Mathematics, Mathematics Education, Statistics)

Date: 09/23/15

# I. Assessment of Student Learning Outcomes

**Name and Contact Information of Program Assessment Coordinator: Vladimir Rosenhaus**

**Department of Mathematics and Statistics Holt 219, phone 898-4108. ** vrosenhaus@csuchico.edu

# 2. Goal Statements and Student Learning Outcomes

**[General Content] Graduates are proficient in performing basic operations on fundamental mathematical objects and have a working knowledge of the mathematical ideas and theories behind these operations.**

** **

GC1 Demonstrate basic skills and conceptual understanding of differential, integral, and multivariable calculus.

GC2 Demonstrate basic skills and conceptual understanding as relating to fundamental mathematical objects introduced in our degree core, such as, sets, functions, equations, vectors, and matrices.

GC3 Demonstrate more technical skills and more in-depth and broader conceptual understanding in core mathematical areas (such as, analysis, geometry/topology, algebra, applied math, statistics), relevant to their option in the major.

# [Critical Thinking/Problem Solving] Graduates use critical thinking and problem solving skills to analyze and solve mathematical & Statistical problems.

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PS Interpret and translate problems into appropriate mathematical language; then solve problems by applying appropriate strategies and interpreting the results.

# [Communication] Graduates communicate mathematics effectively in a manner appropriate to career goals and the mathematical maturity of the audience.

Com1 Demonstrate the ability to effectively and accurately write on mathematical topics relevant to their mathematics option and appropriate to their audience.

Com2 Demonstrate the ability to effectively and accurately speak on mathematical topics relevant to their mathematics option and appropriate to their audience.

# [Proof Proficiency] Graduates have a basic proficiency in the comprehension and application of proofs.

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PP Students can read mathematical proofs, extract the key ideas used in the proof, and convey the logic behind the proof; they can also write their own rigorous and logically correct proofs.

# [Technology] Graduates know how to use technology tools (e.g., graphing calculators, computer algebra systems) appropriate to the context of the problem.

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Tech Students use technology to manipulate mathematical objects (e.g., functions equations, data sets, etc.), to conduct mathematical explorations, to model problem scenarios, and to analyze mathematical objects.

# [Life-long Learner] Graduates are aware of the important role of mathematics and have the interest and ability to be independent learners and practitioners.

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LL1 Students demonstrate the ability to apply mathematics and statistics to new contexts (e.g., in other classes, the workplace, graduate school, or classes they teach).

LL2 Students recognize and appreciate the role that mathematics can play in their futures and in society in general.

# Course Alignment Matrix:

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See attached Excel Spreadsheet

# 4. Learning Outcome(s) Assessed in AY 2014-15 (Year 8 of Assessment Plan)

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PS-M, GC2-IPM, PS-M, Tech-PM

# 5. Assessment Methodology Used

We assessed the SLO’s at the level of mastery (I, P, M, for Introductory, Progressing, Mastery) indicated above by embedding assessment items in tests and final exams for Math 350 (F14), Math 260 (F14), Math 230(F14), Math 305 (S15), and Math 351

(S15). As far as possible, we used assessment items similar to those in Year 3 of our assessment plan. Math 260 courses were taught by two instructors who contributed to developing the instruments and agreed to make them common to their respective exams. All instructors also contributed to constructing rubrics that we applied to determine performance at the Exemplary, Proficient, Acceptable, and Unacceptable levels.

In appendix A we collect assessment items, rubrics, raw scores and sorted data for all class assessments. Student work is available upon request. When possible, we used the rubrics of Year 3 as a guide, updating as needed. Rubrics for new items follow the same general style with descriptions of performance at the Exemplary, Proficient, Acceptable, and Unacceptable levels.

# 6. Assessment Results

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Student Learning Outcome |
Where sample is from (sample size) |
Measure |
% Exemplary + Proficient by item |

GC2-I |
MATH 350 (5) |
Embedded Test 1 Problem 1 |
88 |

GC2-I |
MATH 350 (5) |
Embedded Test 1 Problem 2 |
59 |

GC2-I |
MATH 350 (37) |
Embedded Test 2 Problem 1 |
55 |

GC2-I |
MATH 350 (39) |
Embedded Test 3 Problem 1 |
74 |

GC2-P |
MATH 350 (5) |
Embedded Test 1 Problem 3 |
64 |

GC2-P |
MATH 350 (37) |
Embedded Test 2 Problem 2 |
18 |

GC2-P |
MATH 350 (39) |
Embedded Test 3 Problem 2 |
34 |

GC2-M |
MATH 350 (5) |
Embedded Test 1 Problem 4 |
12 |

GC2-M |
MATH 350 (37) |
Embedded Test 2 Problem 3 |
55 |

GC2-M |
MATH 350 (39) |
Embedded Test 3 Problem 3 |
18 |

PS-M |
MATH 260 (8) |
Embedded Test 1 Problem 1 |
88 |

PS-M |
MATH 260 (8) |
Embedded Test 2 Problem 1 |
88 |

PS-M |
MATH 260 (8) |
Embedded Test 2 Problem 2 |
88 |

Tech-P |
MATH 230 (13) |
Embedded Test 1 Problem 2 |
42 |

Tech-P |
MATH 230 (12) |
Embedded Final Exam Problem 2 |
80 |

Tech-P |
MATH 230 (13) |
Embedded Test 1 Problem 3 |
36 |

Tech-M |
MATH 230 (13) |
Embedded Test 1 Problem 4 |
16 |

Tech-M |
MATH 230 (12) |
Embedded Final Exam Problem 3 |
18 |

PS-M |
MATH 351 (13) |
Embedded Test 1 Problem 3 |
36 |

** **

PS-M |
MATH 351 (14) |
Embedded Test 2 Problem 4 |
0 |

PS-M |
MATH 351 (14 ) |
Embedded Test 3 Problem 4 |
50 |

GC2-I |
MATH 305 (23) |
Embedded Test 1 Problem 13 |
50 |

GC2-P |
MATH 305 (21) |
Embedded Final Exam Problem 1 |
43 |

GC2-M |
MATH 305 (22) |
Embedded Test 1 Problem 4 |
43 |

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**Analysis / Interpretation of Results**

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**Fall, 2014 Assessment**

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**Math 350**: Math 350 is the first course in a three semester sequence required for statistics majors. It is required for statistics majors and many math majors take it as an elective. Engineer students also often take this class to fulfil a math requirement for their major.

**Test 1 Problem 1**: This problem involves solving a probability problem using Bayes’ Theorem. The following steps were evaluated:

- whether they used the correct formula
- whether they calculated the correct conditional distribution
- whether they had the correct final answer

Most students were able to solve this problem. 88% of the students got full points for this problem while only 12% missed this question.

**Test 1 Problem 2**: This question involved finding conditional probability for disjoint events. For this problem students generally got full credit or no credit, therefore, they were scored simply on whether or not they got it correct.

59% of the students got this correct and 41% did not get this correct.

**Test 1 Problem 3**: This question involved finding the union of 2 events when the events are independent. The students were scored on the following criteria:

- whether or not they recognized that the events were independent
- whether or not they used the correct formula
- whether or not they calculated the intersection correctly

64% of the students got this answer completely correct while the remainder had at least one mistake. About 14% lost all points on this question.

**Test 1 Problem 4**: This problem was a counting probability problem. They were graded on the following criteria:

- whether or not they used permutations or combinations
- whether they got the denominator correct or incorrect
- whether they got the numerator correct
- whether they used the correct formula

These sorts of problems are generally harder for students. Only 12% got this problem completely correct while 29% got either the denominator or numerator incorrect. 7% of students did not receive get anything correct.

**Test 2 Problem 1: **This problem involved using the properties of a probability distribution in order to solve for an unknown. The criteria used to grade this problem were:

- whether or not they recognized if it was a discrete variable
- whether they set up the equation correctly
- whether they solved for the unknown correctly

55% of the students had no mistakes on this question while about 41% had at least one major mistake.

**Test 2 Problem 2: **This problem was a counting problem using binomial distribution. They were graded on the following criteria:

- whether they recognized that it is a binomial problem
- whether or not they used the correct combinatoric formula
- whether or not they multiplied by 2 since there were 2 scenarios

Only 18% of students received full credit for this problem. 21% had only a small error while 61% of students had at least one major error.

**Test 2 Problem 3: **This problem involved finding a probability given a joint distribution. The criteria that was used to grade this problem was as follows:

- setting up the integration correctly
- integrating correctly
- whether the bounds of the integration were correct

55% got this answer completely correct, 18% were acceptable and 26% had at least 2 major errors.

**Test 3 Problem 1**-This problem involved calculating a probability given a moment generating function. The criteria used to grade this problem are as follows:

- whether or not they recognized it as a Poisson
- whether they got the parameter correct for a Poisson
- whether they computed the probability correctly

74% got this answer correct and 21% made at least one major error.

**Test 3 Problem 2-**This question involved calculating a probability using the normal distribution. There were 3 parts to this question. Criteria for grading this question were as follows:

- Did they use the correct formula
- Did they know to use the results of the Central Limit Theorem
- Did they use the binomial formula correctly in the third part
- Did they use the normal distribution table correctly

Only 34% of students got this question completely correct. 47% did used the wrong distribution or formula and 13% had 3 or more major errors.

**Test 3 Problem 3: **This problem involved determining the moment generating function from a uniform distribution. The criteria for grading this problem were as follows:

- Whether or not they set up the integration correctly
- Whether the bounds of the integration were correct
- Whether they integrated correctly

Only 18% got full credit for this question and 56% had so little correct that they received no credit. 25% received partial credit for this question.

Raw data for our analysis is available in Appendix A (M350). Copies of students’ work are available upon request.

**MATH 260: **The Math 260 course is required for Applied Mathematics, General Mathematics, and Statistics, majors, and also taken by some Mathematics Education majors. It is also a required course for most Engineering majors. Two instructors who taught three sections of the course in Fall 2014 together prepared problems common to their two tests for assessment.

Problem 1. This is a problem dealing with calculations within the population model. Students were expected to demonstrate their ability to solve a differential equation, and interpret the meaning of its solution. The following steps were evaluated:

- Setting up correct differential equation for population growth

- Correct solution of this equation

- Correct use of data for finding a rate of growth

- Correct expression for determining a population at any time t

- Correct evaluation of time, and the world population in 2020

Most students were able to handle well this problem: 50% of the majors scored at the Exemplary and 37.5% at the Proficient level.

Raw data for our analysis is available in Appendix M260. Copies of students’ work are available in Appendix M260_tests.

Problem 2. This is a problem for finding the general solution of a linear second-order non-homogeneous differential equation. Students were expected to demonstrate their ability to find a general solution of a linear homogeneous differential equation with constant coefficients, and construct a special solution to a given non-homogeneous equation.

Most students were able to solve this differential equation correctly: 75% of the majors scored at the Exemplary and 12.5% at the Proficient level.

Problem 3. Students were offered to find the form of the general solution of a linear second-order non-homogeneous differential equation. Students were expected to obtain the correct form of the solution using the method of undetermined coefficients without having to evaluate these coefficients. One of the complications of the problem was the interference between the homogeneous solutions and the right hand side of the equation.

Most students were able to solve this differential problem correctly: 37.5 % of the majors scored at the Exemplary and 50.0 % at the Proficient level.

Raw data for our analysis is available in Appendix A (M260). Copies of students’ work are available upon request.

**MATH 230. **This course is an introduction to computational Mathematics. It is a required course for Math Education majors and recommended for Applied Mathematics majors, as a prerequisite for the required course Math 461: Numerical Analysis. The course provides basic knowledge of the computer algebraic software Mathematica, and prepares students to use computer to solve mathematical problems.

# PROGRESSING LEVEL

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**Problem 1 (**Question 2: Midterm Exam Fall 2014)

Application of the theory of numbers. Create a list of 100 random two-digit positive integers that are not prime numbers. b. Among them give the list of all distinct integers that are squares.

**Results: **Most students were able to solve this problem correctly: 42% of the majors scored at the Exemplary and 33% at Acceptable level.

**Problem 2 **(Question 2 -Final Exam Fall 2014)

Application of Calculus II – Area between two curves. Students needed to compute the area between two given curves up two 4 digits after the decimal point.

**Results: **Most students were able to solve this problem correctly: 80% of the majors scored at the Exemplary and 20% at Acceptable level.

**Problem 3 **(Question 3 – MidTerm Exam)

Application of Calculus I and II: Zeros of a function and area under a curve. Approximate the area under a given curve up to 2 digits after the decimal point:

**Results: **Most students were able to solve this problem correctly: 36% of the majors scored at the Exemplary and 45% at Acceptable level.

# MASTERY LEVEL

**Problem 1 **(Question 4 – MidTerm Exam) Application of optimization and geometry.

Plot 10 random points inside a given square, and find the two points that are the furthest away from each other. Draw the line segment connecting those two points.

**Results: **More than half of students were able to solve this problem correctly: 16% of the majors scored at the Exemplary and 42% at Acceptable level.

**Problem 2 **(Question 3 - Final Exam)

Writing a program that calculates all the solutions (real and complex) of the quadratic equation. Consider all different cases, and give examples for illustration

**Results: **More than half of students were able to solve this problem correctly: 18% of the majors scored at the Exemplary and 36% at Acceptable level.

**Problem 3 **(Question 4. Final Exam)

Writing a program that reproduce given picture. This question required a knowledge on the geometry of circles and trigonometry in addition to programming. (Write a program that draws on a light background five concentric circles of different color, and add colored layers of small disks of equal radius next to each other).

**Results: **Only 27% of students were able to solve this problem correctly: 18% of the majors scored at the Exemplary and 9% at Acceptable level.

Raw data for our analysis is available in Appendix A (M230). Copies of students’ work are available upon request.

# Spring, 2015 Assessment

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**MATH 305: **Math 305 is requires for all students taking the mathematics major with the Mathematics Education option, and some Statistics option students also take the course. Two students enrolled in the course in Spring 2015 were senior Computer Science majors, and their department chair had allowed this course instead of Math 350 to meet graduation requirements without schedule conflict.

# Midterm 1 problem:

This is a problem dealing with the relationship between standard deviation and percentile. Students were evaluated on both their ability to determine a percentile knowing a score (part a) and to determine the score knowing the percentile (part b). Students were evaluated on correct use of the equation correct interpretation of z-scores. 50% of students solved this problem at an Exemplary level and 27% solved it at an Acceptable level.

# Midterm 1 Results:

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Mean: 76%, Median: 76%, Standard Deviation: 11.5%, Min: 57%, Max: 98%.

# Midterm 2 problem:

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**Overview: **This problem focuses on binomial distribution, specifically programming to find it, interpreting the outcome, and analyzing its use in context. 77% of students were able to solve this problem: 41% at the Exemplary level and 36% at the Acceptable level.

# Midterm 2 Results:

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Mean: 80%, Median: 84%, Standard Deviation: 14.5%, Min: 0% (nothing submitted),

50%, Max: 98%

# Example problem from Final Exam:

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**Overview: **This problem deals with comparing data sets from two different groups using beginner and advanced methods while incorporating checks on single-variable analysis tools. Students are checked on: naming null and alternative hypothesis, comparing box plots, finding outliers, two-sample t-tests, and understanding and interpreting p-values and alphas and their significance. 42.5% of students could successfully complete this problem at the Exemplary level (43%) or Acceptable level (9.5%), with another 24% able to compute solutions but not interpret them.

# Final Exam Results

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Mean: 73.8%, Median: 74.5%, Standard Deviation: 18.0%, Min: 0% (nothing submitted),

35.5%, Max: 100%

Raw data for our analysis is available in Appendix A (M305). Copies of students’ work are available upon request.

**MATH 351**. Math 351 is the second semester of a 3 semester series that is required for statistics majors and minors. In addition, a few math majors take this class as an elective. It is a calculus based statistics course.

**Test 1 Problem 3**: This problem had students determine which estimator is most efficient by finding the variances of 2 estimators. The criteria used to grade this problem is as follows:

- Whether they correctly found the variances of each estimator
- Whether they were able to justify correctly which estimator was more efficient
- Whether they were able to identify the correct distribution of the pdf

36% of the students in this class received a near perfect score on this problem. Almost half of the students had a number of major mistakes.

**Test 2 Problem 4: **This problem involved finding a probability after identifying the correct distribution. The following criteria were used to grade this problem:

- Whether they correctly identified the chi-‐square distribution
- Whether they correctly manipulated the equation to use the chi-‐square distribution
- Whether they correctly solved the probability using the chi-‐square distribution

Although any students were on the right track for this problem, no student got it completely correct. 23.1% of the students correctly recognized the distribution but failed to find the probability correctly. About 54% of students used a different distribution to solve the problem.

**Test 3 Problem 4: **This problem dealt with using Bayesian methods to derive a posterior distribution when the data follow an exponential distribution and the parameter is a gamma distribution. The criteria that was used to grade this problem is as follows:

- Identify that Bayesian statistics was necessary to derive the posterior distribution
- Set up the conditional distribution correctly
- Integrate the denominator correctly
- Correctly identify the posterior distribution

About 50% of the students were able to complete this difficult problem without any major errors and 36% of students had at least one major error.

Raw data for our analysis is available in Appendix A (M351). Copies of students’ work are available upon request.

# 8. Planned Program Improvement Actions Resulting from Outcomes:

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No program improvements are planned for Math 230, 260, 305, 350, or 351 as a result of the SLO assessment.

# 9. Planned Revision of Measures or Metrics:

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For the most part, there are no planned revisions for the measures or metrics for any of our course assessments.

# 10. Planned Revisions to Program Objectives or Learning Outcomes (if applicable)

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No changes made or planned.

# 11. Changes to Assessment Schedule (if applicable)

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No changes made or planned.

# 12. Information for next year

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What learning outcome(s) are you examining next year and who will be the contact person?

2015-2016 will be Year 3 of a new Assessment schedule. In year three we collect data from Math 235, Math 330, and Math 420. The SLOs that will be assessed from these classes are:

235 — GC1 at the Progressing Level, GC3 at the Introductory and Progressing levels, PS at Mastery level, PP at the Introductory level, Com2 at the Introductory level.

330 — Com 1 at the Progressing Level. PP at the Introductory and Progressing levels.

420 – GC1 at the Mastery level, Com1 at the Mastery level, Com2 at the Mastery level, and PP at the Mastery level.

The contact people will be Colette Calmelet (Assessment Coordinator), Rick Ford (Department Chair), and Sergei Fomin (apprentice) Assessment Committee member.

# II. Appendices (please include any of the following that are applicable to your program)

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**Assessment Data Summaries (Test items, Rubrics, Raw Data)**

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**Assessment Schedule and Course Alignment Matrix**

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**Please submit completed reports electronically to your college assessment**** representative.**