Fall M 333 Linear Algebra

Assessment Coordinator: Ryan Grady, David Ayala

According to the description of the Mathematics Program Learning Outcomes and Assessment, 20 students were assessed for the following outcomes in M 333. 

Evaluation

Students will produce rigorous proofs of results that arise in the context of linear algebra. Moreover, these proofs will use proper mathematical language, and meet standards of organization, coherence, logic, and style.

Results

Based on two multiple part final exam questions, 13 of the 20 students adequately demonstrated their understanding at an acceptable level or better. Many students in M333 did not have sufficient background in mathematical proof writing. Further, some of these students struggled with implementing mathematical definitions and confused the role of example versus proof.

Group Projects Exceptional Acceptable Marginal Unacceptable
LO 1 0-13 0-13 0-7 0-7
LO 3 0-13 0-13 0-7 0-7
LO 6 0-13 0-13 0-7 0-7

Recommendations

It is our recommendation that M242 become a pre-requisite for M333. This will allow students to more fully engage with the course content as opposed to being distracted by the basic mechanics of mathematical proofs.

Spring M 333 Linear Algebra

Assessment Coordinator: Kevin Wildrick, Lukas Geyer

According to the below description of Mathematics Program Learning Outcomes and Assessment, 12 students were assessed for Outcomes 1, 3, and 6 in M431.

Outcome 1

Effectively communicate mathematical ideas by precisely formulating them in proper mathematical language.

Outcome 3

Produce rigorous proofs of results that arise in the context of abstract algebra. 

Outcome 6

Write solutions to problems and proofs of results that meet rigorous standards based on content, organization, coherence, logical arguments, and style.

Evaluation

Two assignments were used in this assessment. The first is a month-long group project, the second is the final exam. A description of the project is attached. The final exam was a 110 minute cumulative evaluation to be completed without use of any reference materials or computational aids. It consisted of three types of activities: fill in the blank definitions of key concepts, true or false statement identification, and theorems to be proved. 

Results

Of the four group projects submitted, two indicated acceptable performance on all of the outcomes, one indicated marginal performance on all outcomes, and one indicated unacceptable performance on all outcomes.

Group Projects Exceptional Acceptable Marginal Unacceptable
LO 1 0 2 1 1
LO 3 0 2 1 1
LO 6 0 2 1 1

 

Of the twelve exams, one indicated excellent performance on all outcomes, four demonstrated acceptable performance on all outcomes, six demonstrated marginal performance on all outcomes, and one demonstrated unacceptable performance on all outcomes. Of the marginal and unacceptable performances, one exam was by non-mathematics major. In total, this course did not meet the threshold. 

Final Exams Exceptional Acceptable Marginal Unacceptable
LO 1 1 4 6 1
LO 3 1 4 6 1
LO 6 1 4 6 1

Recommendations

The assignments indicate that the program has not supported the majority of students in developing the level of understanding necessary to meet these learning outcomes. Many students have not integrated material from M 242 (Methods of Proof) or M 333 (Linear Algebra) into their skill set. In particular, many students display discomfort with the notion of a mathematical definition, have difficulty in identifying the parts of a proof by induction, and have not internalized concepts such as injectivity or linearity. Another problematic point is the difference between a proof of a theorem and an example confirming the validity of the theorem. We suggest that efforts be directed to the improvement of student learning in M 242 and M 333, the prerequisites to this course. 

Program Learning Outcomes

LO# Students should demonstrate the ability to:
1. Effectively communicate mathematical ideas by precisely formulating them in proper mathematical language (M 333, M 383, M 384, M 431).
2. Produce rigorous proofs of results that arise in the context of real analysis (M 383, M 384).
3. Produce rigorous proofs of results that arise in the context of abstract algebra (M 431).
4. Produce rigorous proofs of results that arise in the context of linear algebra (M 333).
5. Construct direct, indirect, and proofs by induction and determine the appropriateness of each type in a particular setting. Analyze and critique proofs with respect to logic and correctness. (M 333, M 383, M 384)
6. Write solutions to problems and proofs of results that meet rigorous standards based on content, organization, coherence, logical arguments, and style. (M333, M 383, M 384, M 431) 

Curriculum Map and Assessment Schedule

Course 1 2 3 4 5 6 Assessment Schedule
M333 X     X X X Even fall semesters
M431 X   X     X Odd spring semesters
M383 X X     X X Odd fall semesters
M384 X X     X X Even spring semesters

Rubric

LO  Unacceptable Marginal Acceptable Excellent
1 Terms are used improperly or key definitions are missing. Terms are properly used, but logical flow and organization do not lead to clear communication. Terms are properly used, and flow is logical, though organization lacks attention to detail that would lead to a clearly communicated result. The work is fully correct and complete with relevant terms properly employed and shows that work, and mathematical concepts are well-organized into a logical sequence.
2-5 The work is not logical and complete because either terms are used improperly or key ideas are missing or organization is unlikely to result in a correct proof even if a few more ideas are inserted. The work is not correct and complete because key ideas are missing, but the terms are properly used and the work shows a type of organization that might work if the right ideas were inserted in the proper places. Also, the work is "marginal" if most of it leads toward a correct proof, but a false statement is inserted.  The work is almost correct with relevant terms used and ideas that work, but not well-organized; for example, with some steps out of order, or with something relatively minor. The work is fully correct and complete with relevant terms properly employed and shows that work, and steps well organized into a logical sequence. 
6 If the work is not correct and complete because either there are fundamental gaps in understanding the underlying mathematical methods or there are two or more significant errors in the computations. The work is not correct and complete because a significant component of the analysis is missing or incorrect, but most of the components are present. The work is almost correct with the appropriate methods employed but with a minor error or misunderstanding one part of the computations. The work is fully correct and complete and displays full understanding of the appropriate mathematical methods. 

Threshold

At least half of the majors in each of the courses are assessed as "excellent" or "acceptable" for all the learning outcomes. 

 

PDF of M333 Assessment Report