Mathematics: Undergraduate Programs
The major
The mathematics major at the University of Arizona serves a wide variety of students. Roughly a quarter to a third of the students who earn undergraduate degrees in mathematics go on to graduate study, and half of these pursue graduate studies in the mathematical and statistical sciences. Each graduating class has one or two students who pursue professional degrees in law, medicine and dentistry. Of those students who pursue nonmathematical graduate programs, economics, engineering, computer science, and education are represented. A number of graduates enter the precollege teaching profession. Students not continuing their study in school enter the workforce in a variety of fields, some not traditionally associated with mathematics. In recent years, training in mathematics has become fundamental to new sectors of our economy including computing, information technology, banking, insurance, and communications and the life sciences.
In response to the growing need for mathematics in a variety of professions, the department has created several different paths to the mathematics major. All mathematics majors take the same core courses, but the last fivesix courses are chosen with a view towards a student’s career plans. The Bachelor of Science and the Bachelor of Arts in Mathematics have seven options:

Comprehensive Option, for students who enjoy the abstraction of mathematics or who plan to pursue graduate study in mathematics or applied mathematics

Computer Science Option, for students interested in connections between mathematics and computer science, or planning to go to graduate school in the computational sciences

Economics or Business Option, for students interested in applications of mathematics to economics and finance, or planning to go to graduate school in those areas.

General/Applied Mathematics Option, for students who intend to enter the job market upon graduation or who plan to pursue graduate studies in engineering and other academic areas

Probability and Statistics Option, for students interested in going into actuarial work or applied statistics on graduation, or continuing on to graduate school in probability or statistics

Life Sciences Option, for students considering a career in medicine or interested in applications of mathematics to the life sciences, as well as for students wanting to attend graduate school in the life sciences

Mathematics Education option, for students preparing to teach mathematics at the secondary level. The option has two components: a set of courses in mathematics, and a set of courses in teaching and learning mathematics in secondary schools. As part of the program, students earn certification to teach in the State of Arizona
Foundations
In addition to the program for Mathematics majors, the department also has a very substantial service mission providing instruction in foundational mathematics courses for students in science and engineering as well as to the wider university community. Depending on the requirements of their majors and their level of interest, students may take courses in calculus, differential equations, probability and statistics or discrete methaematics as well as more basic courses in algebra and trigonometry. Throughout these courses, there is an emphasis on conceptual understanding and learning to apply mathematics to real problems as well as on computational techiniques.
General Outcomes
While we attempt to define these categories uniquely, it should be understood that they are far from mutually exclusive.

Theoretical Understanding. This is the traditional learning outcome that we expect from students in upper division “pure” mathematics courses. Theorems and proofs, which are the mainstay of mathematical research, are the vehicle for elucidating the content of these courses. The reasoning skills that are required to work with theorems and proofs are very high level abstract critical thinking skills.

Conceptual Understanding. This is the learning outcome that is expected in courses that use mathematical principles to understand real world concepts. Rate of change would be an example of such a concept that is of fundamental importance in calculus yet is an indispensible tool of mathematical modeling. Formulation and evaluation of appropriate models is another example of high level critical thinking that is an essential part of the train of mathematics students.

Consolidation of Concepts. This learning outcome of all of our upper division courses brings together ideas from various lower level courses. A more abstract and broader view of these ideas is part of the pedagogy, thereby consolidating that knowledge.

Real World Applications. The learning outcome is an exposure to mathematics as it applies, for example, to economics, life sciences, physical sciences, business, or engineering.

Problem Solving. This outcome is expected in all mathematics courses. Understanding the problem, using appropriate mathematical tools, exactly defining the parameters and developing a plan of attack are important skills for all students.

Computational Skills. This outcome is paramount in lower division elementary courses but is also a component of those upper division courses that require calculations, whether it be by computer or by pencil and paper.

Computer Usage. This learning outcome is found in many classes because technology isa part of everyday mathematics. Some courses, however, rely heavily on computers.

Group Work. As mentioned above, collaborative learning is a learning outcome that is common to many courses. In some courses it is an explicit requirement.

Proof Writing. The writing component is also a feature of more and more courses. It was always part of the theoretical courses that require proofs but it is now a part of courses that have projects.

Expository Writing. This, too, is writing mathematics, but it is more informal. Expository writing can occur in mathematics laboratory settings or in weekly problem solving assignments.
Course Specific Outcomes

All of the mathematics courses that are part of the major also serve as service courses for the rest of the university. Enrollment in mathematics courses bears this out. The pedagogy and the instructional goals of these courses take into account the dual nature of mathematics courses, to serve the mathematical needs of the university community and to prepare undergraduate mathematics majors to achieve their goals.

The course outcomes naturally divide into two separate phases, the lower division courses and the upper division courses.

Every mathematics major in the College of Science takes the same core classes in mathematics. The lower division core consists of calculus, differential equations, linear algebra and introduction to proofs.
Lower Division Phase Outcomes

Calculus and differential equations represent the traditional tools of mathematics. These courses form the foundation for many of the applications of mathematics and thus prepare students to apply their mathematical knowledge. The courses also introduce students to the major theoretical questions that form the foundation of modern mathematics.

Students should understand the important ideas of differentiation and integration and how these ideas are tied to physical constructs.

The notion of limit is introduced at this level and will be addressed in the upper division courses. At this level, students should be able to give a geometric description of the limit and be able to do calculations. They are also expected to be able to be able to identify and correct common conceptual or mathematical errors.

Linear algebra is the course that ties in high school ideas with the more abstract ideas of modern mathematics. At this point students should begin the process of abstraction, making unexpected connections, and becoming comfortable with working in higher dimensions. They are also required to be able to make arguments to "show" (not quite prove) why certain mathematical properties hold.

Formal Mathematical Reasoning and Writing begins the serious study of the foundations of mathematical proof. A great deal of time is spent in this course in getting students to produce correct and precise arguments. Students should become comfortable with the rigorous notion of proof, the need for correct definitions, and the ability to construct proofs and to critically evaluate whether a proof is correct.

All of these courses should serve to motivate all students, not just mathematics majors, to the continued study of mathematics.
Upper Division Phase Outcomes

A set of courses for the Comprehensive option is designed to prepare students for graduate study in mathematics. The expected outcome is that students will understand the framework of modern mathematics and that they will be prepared for graduate study in the mathematical and statistical sciences.

Another set of courses is designed to prepare students to use mathematical tools to investigate other phenomena. An outcome that we envision is that students will have a sufficient knowledge of mathematics that they will know what mathematical tools apply in the situation that they are in. Students are expected to learn to create and critically evaluate models and methods for solving problems coming from a variety of application fields.

All mathematics courses are designed to increase the problem solving skills of the students. An increased attention to detail is an outcome that is expected.

In some course content, technology is indispensible. In these courses, students are expected to become proficient in the use of technology to model complex situations. It is also expected that students will understand the limitations of the software and hardware being employed in their scientific investigations and will be able to critically evaluate appropriate software and mathematical tools for those complex modeling situation.

In some course content, abstraction is indispensible. In these courses students are expected to see the interconnections between the traditionally distinct areas of mathematics, algebra, analysis and geometry.

In all mathematics courses students are expected to communicate their results, in both written and oral form.

Consolidation of information is an expected outcome. The amount of mathematics that students learn is vast, yet distinct ideas are consolidated into a body of knowledge that is manageable.
The math department has identified a shorter list of program learning objectives for the math major based on the larger outcomes discussed above. The table below shows the outcomes as columns and identifies how those outcomes are assessed by math 323 instructors and by math 485 instructors as well as based on self assessments in exit surveys.
Students will be able to… 

Major Courses 
Define mathematical terms precisely. 
Recognize when arguments are valid, and identify logical gaps and flaws. 
Create valid proofs. 
Critically evaluate and extend selected mathematical models in the current scientific literature. 
Apply computational methods and mathematical concepts from to analyze scientific problems 
Effectively communicate results to nonspecialized audiences in written and verbal form 
Math 323 
Rubric item Exams 
Rubric item Homework and Exams 
Rubric item Homework and Exams 

Math 485 
Rubric item Project 
Rubric item Report on Project 
Rubric item Poster Presentations and Talks 

Exit Survey/Interview Selfevaluation 
x 
x 
x 
x 
x 
x 
University and Departmental Data for the major
Each year, the Math Center and the Undergraduate Committee review the data from surveys that have been administered by the University and by data collected by the department on courses. These surveys included the incoming freshmen survey and data that the Math Center collects about the characteristics of the incoming freshmen class. Enrollment characteristics of our calculus classes are important.
An important question to assess is whether the undergraduate program is producing mathematics majors that are competitive nationally. One way of assessing this question is by collecting data on the number of mathematics majors who are accepted into summer internship positions during the summer. Another important question is the number of students accepted into postgraduate study as well as joining the workforce. The number of mathematics majors receiving national awards is also indicative of quality. The Math Center attempts to collect this data.
Instructor Outcomes Rubric after Formal Mathematical Reasoning and Writing
Administered: Each Semester Starting Spring 2016
Target Population: Students completing math 323
Participation Rate: 100%
The program learning objectives identifies above are assessed at the end of the Formal Mathematical Reasoning and Writing class, Math 323. Instructors evaluate outcomes for students based on the criteria below:
Objective 
4 
3 
2 
1 
Define mathematical terms precisely. 
Almost always fluently states precise definitions of a wide variety of mathematical terms 
Usually captures key ideas but sometimes does not use precise language 
Sometimes captures key ideas but rarely uses precise language 
Typically fails to correctly state key features 
Construct proofs that follow directly from a definition. 
Routinely produces correct and precise arguments 
Usually constructs simple implications from definitions, but sometimes unable to identify which components of a definition lead to the desired conclusion 
Sometimes constructs simple implications from definitions, but frequently struggles to identify which components of a definition lead to the desired conclusion, or sometimes demonstrates lack of understanding of definitions 
Struggles to construct straightforward implications from components of definitions, or frequently demonstrates lack of understanding of definitions 
Recognize when arguments are valid, and identify logical gaps and flaws (false statements). 
Almost always able to identify logical gaps and flaws 
Usually able to identify logical flaws, but sometimes fails to recognize assumptions that have not been established by a valid argument 
Sometimes fails to recognize logical flaws and often fails to recognize assumptions that have not been established by a valid argument 
Frequently fails to identify logical gaps and flaws 
Produce valid proofs using the techniques of mathematical induction, contradiction, contrapositive, and construction. 
Able to identify an appropriate proof strategy and almost always able to create a correct proof using multiple strategies as appropriate 
Able to identify an appropriate proof strategy, but often makes errors in the execution of the proof 
Can construct a correct proof if strategy is specified, or can identify a strategy but often fails to correctly construct the proof 
Unable to identify an appropriate strategy, and also often cannot construct a proof when a strategy is specified 
Outcomes after Modeling Capstone (Math 485)
Administered: Each Spring starting 2016
Target Population: Students completing math 485
Participation Rate: About 100% of those enrolled in 485, about 60% of all majors
Objective 
4 
3 
2 
1 
Critically evaluate and extend selected mathematical models in the current scientific literature 
Able to balance simplicity versus accuracy in selecting an appropriate model, and able to employ iterative model refinement 
Can identify simplifying assumptions, and modify an existing model 
Able to identify simplifying assumptions, and evaluate their appropriateness. Also able to relate a scientific problem to a mathematical formulation. 
Unable to make the connection between the scientific problem and a mathematical formulation. 
Apply computational methods and concepts from prerequisite mathematical content (e.g. linear algebra and differential equations) to analyze scientific problems 
In the context of a scientific problem, is able to identify and apply mathematics from a variety of previous courses 
In the context of a scientific problem, is consistently able to apply mathematical tools once they are identified. 
In the context of a scientific problem, is sometimes able to apply mathematical tools once they are identified. 
Unable to apply identified mathematical tools from prerequisite content 
Work together in small groups to read, understand, and reproduce work from articles in the current scientific literature 
All members of the group contribute substantially and can explain their results. 
All members of the group can explain their results, but some of the group members did not make substantial contributions. 
Some members of the group cannot explain their results. 
Group does not work as a team 
Effectively communicate results to nonspecialized audiences in written and verbal form (at poster session) 
Able to present results in a clear, succinct, and comprehensible way, including the motivation for the study 
Able to present results correctly and completely 
Some results not explained correctly 
Paraphrases results from published works, without the ability to explain 
Senior Exit Survey
Administered: Each Semester
Target Population: Graduating Seniors
Participation Rate: About 25%
Findings
Fall 2016 and Spring 2017
Math 323 Formal Mathematical Reasoning and Writing:
Math 323 Fall 2016 and Spring 2017  4  3  2  1 
Define…  34%  39%  22%  5% 
Construct direct…  21%  39%  33%  8% 
Recognize valid…  16%  39%  35%  10% 
Produce valid …  19%  39%  31%  11% 
Math 485 Mathematical Modeling:
Math 485 Spring 2017  4  3  2  1 
Critically evaluate…  36%  50%  14%  0% 
Apply…  52%  32%  16%  0% 
Work together…  25%  61%  14%  0% 
Effectively communicate…  45%  32%  23%  0% 
Exit Survey:
Exit Survey Dec 2016 and May 2017  Strongly Agree  Agree  Neither  Disagree  Strongly disagree 
Define…  48%  48%  4%  0%  0% 
Recognize valid …  44%  56%  0%  0%  0% 
Produce valid…  36%  56%  4%  4%  0% 
Critically evaluate...  40%  44%  16%  0%  0% 
Apply…  40%  52%  8%  0%  0% 
Communicate…  40%  52%  4%  4%  0% 
Spring and Summer 2016
Math 323 Formal Mathematical Reasoning and Writing:
Math 323 Spring+Summer 2016  4  3  2  1 
Define…  38%  48%  20%  4% 
Construct direct…  32%  40%  34%  4% 
Recognize valid…  11%  33%  51%  16% 
Produce valid …  14%  38%  43%  14% 
Math 485 Mathematical Modeling:
Math 485 Spring 2016  4  3  2  1 
Critically evaluate…  62%  38%  0%  0% 
Apply…  54%  46%  0%  0% 
Work together…  62%  38%  0%  0% 
Effectively communicate…  69%  31%  0%  0% 
Exit Survey:
Exit Survey Spring+Summer 2016  Strongly Agree  Agree  Neither  Disagree  Strongly disagree 
Define…  57%  39%  4%  0%  0% 
Recognize valid …  54%  43%  4%  0%  0% 
Produce valid…  36%  54%  7%  0%  0% 
Apply…  46%  46%  7%  0%  0% 
Communicate…  50%  43%  7%  0%  0% 
University and Departmental Data
College of Science data of graduating seniors have shown that mathematics majors have had the highest cumulative GPA.
An important question is whether or not the core courses are adequately preparing the math majors for their upper division courses, thence graduation. Data on the important transition course in the core, math 323, shows that the majority of students who earn at least a C in this course complete graduation within three years.
The characteristics of the entering freshmen class has shown that a substantial percentage of these students arrive at the university with credit for first and second semester calculus. Based on this data, the Math Center has developed recruitment procedures to attract these students into the major and this has resulted in a dramatic increase in the quantity and quality of mathematics majors.
Mathematics majors are very successful in obtaining summer internship opportunities, as well as finding internship opportunities in research laboratories across the university. A number of mathematics majors have received national awards.
Exit Survey
Most respondents have favorable comments about the department, their teachers and their learning experiences. We continue to make efforts to maintain contact with alumni. Approximately 50 % of graduating seniors have had some kind of intensive research or teaching experience during their undergraduate time of study. Part of the exit survey is to determine postgraduation plans. The percentage of graduates pursuing graduate studies in the mathematical and statistical sciences is higher than the national average. National data of how many mathematics majors pursue postgraduate education in other fields is not available. There is a sizable percentage of our graduates who pursue this option. The percentage of graduates pursuing careers in precollege is at least 15%.
Changes Made on the Basis of Findings: Students have remarked in their Exit Surveys that research experiences give students an edge in the job market as well as graduate schools. The department has increased its efforts to help student find summer internships.
The elite graduate schools require applicants to have taken graduate courses as undergraduates and to have had research experiences. The department has advised students about this and more students are graduating having done some of these activities. Students who have gone on to work in industry or national labs have also reported to us the value that employers have placed on knowledge of statistics, and as a result probability and statistics has been included as a way of satisfying upper level sequence requirements for several of the major options.
University and Departmental Data for courses
Almost all of the math department's lower level courses which are taught using multiple sections are organized using common finals. These common finals are an important assessment tool not only for measuring how well individual students have mastered the skills and concepts that we teach, but also for the department to measure how well we are teaching. These common finals are graded jointly and simultaneously by all instructors who are teaching the those course that semester, and at the end of the grading session, the results are presented broken down question by question and section by section. These results are analyzed by the group and changes in the syllabus or the methods of teaching different parts of the material are recommended to the course coordinators for implemetation. The data from these exams has been collected and is very valuable in evaluating changes in the courses content and emphasis.
In addition, placement of new students into University of Arizona math classes has been based primarily on the outcome on a placement test (formerly the math readiness test, and more recently the online ALEKS placement test). Data on placement and outcomes (grades received as well as preliminary exam scores) is analyzed each year after the end of the Fall semester and where necessary, the placement levels are adjusted for the following Fall.
The department observed that students who performed poorly on the first exam in Calculus 1 (Math 124) had a very high D/F/W rate. In response, the department created Math 122 A/B. The first exam now corresponds to the end of Math 122A. Students who do poorly are tracked into Math 120R (Precalculus), and are givenmultiple opportunities to improve their final grade in Math 122A. Students who do well on that first exam continue on, well prepared, into Math 122B.
For upper level courses, there is a more informal assessment and feedback between the instructors of "downstream" courses and those teaching the prerequisite courses that is used to make sure that students have the necessary skills to succeed. This is encouraged and partially organized through the departments undergraduate committee.
Math 323 "Formal Mathematical Reasoning and Writing" is one of the most challenging courses for our majors. The department undertook a study of that course to determine how we could better meet the needs of students in 323. We combined those results with the assessments reported by instructors for all 323 students starting in Spring 2016. In response to student feedback and the assessments, we created a new one credit supplemental course which we call Math 396L "Wildcat Proofs Workshop." In that course students get more handson experience creating and critiquing mathematical arguments. The 396L course was originally only open to students coenrolled in 323, but after the successful roll out of that course we opened it to students in those upper division courses in which additional practice with mathematical argumentation could be useful.