MAT313 Linear Algebra is an advanced mathematics course. Individual professors will design their own syllabi and final exam. The course and major outcomes will be assessed on the final exam. A sample syllabus of this course as taught in 2014 is available here and covers both theory and applications.
Topics:
1. Solving systems of linear equations, dimensions of solutions sets
2. Determinants, Inverses, eigenvalues and eigenvectors of matrices
3. Vector spaces, Linear maps, Linear Independance of Vectors
Math Major Outcomes incorporated into MAT313:
A. Perform numeric and symbolic computations
B. Construct and apply symbolic and graphical representations of functions
C. Model real-life problems mathematically
E. State and apply mathematical definitions and theorems
F. Prove fundamental theorems
Fall 2017 Assessment Report:
The results of the final exam from the assessment coordinator's Fall 2017 section of MAT 313 were the basis of the assessment. The class consisted of 37 students, 28 of whom were majoring in Computer Science. The following shows, for each problem from the final exam, the number of students that demonstrated mastery (i.e. lost no more than one point) on that problem.
Problem #1: 22
Problem #2: 18
Problem #3: 11
Problem #4: 12
Problem #5: 5
The Math Major Outcomes (MMO) reflected in the final exam were A (problems 1-5), B (problems 1 & 5), and E (problems 1-5). The most glaring result was the paucity of correct solutions to problem #5 (which had to do with orthogonal complements and projections). The associated material was covered during the last two weeks of the course, which is one reason why the students did so poorly. Another reason may be that the instructor made handing-in the last two homework assignments optional (a mistake he will not repeat!). The results on Problem 3, which consisted of three statements that students had to judge as true or false (with justification), were also telling, though not surprising. MAT 313 is often the course in which students are asked to write proofs for the first time. To solve part (a) of Problem 3 the students had to make a short argument invoking two major theorems from the course; two solve parts (b) and (c) they needed to recognize that the given statements were false and to construct valid counterexamples. Since writing such proofs is usually the most difficult part of the course, it is somewhat encouraging that almost a third of the class showed mastery in Problem 3.
MMO F was not reflected in the final exam, but was so on a number of the homework assignments. The next time MAT 313 is used for assessment, MMO F will be included by having the students complete 4 or 5 proof-writing assignments (with one re-write allowed on each).
MMO D (Use technology appropriately...) is not a required part of MAT 313, though this semester Maple problems were included on most homework assignments. Overall the students did quite well with the Maple problems (not surprisingly perhaps since many of them were CS majors).
MMO C was only present tangentially in MAT 313 this semester. Linear algebra has many real-world applications, but it is not easy to find the time to discuss them in any substantial way.
In Fall 2013, this course was assessed using two Fall 2014 final exams of a similar nature designed by the professors teaching the course, each following their own syllabus and using a different textbook. Both courses covered the following list of topics and tested them on the final. The finals were not identical but tested each of these topics:
Topics:
1. Solving systems of linear equations, dimensions of solutions sets
2. Determinants, Inverses, eigenvalues and eigenvectors of matrices
3. Vector spaces, Linear maps, Linear Independance of Vectors
The course outcomes were very vague and should perhaps be changed to the topics above:
1. Learn to prove theorems: Students should be able to observe connections between different topics in the course description above, especially systems of equations, matrices and vector spaces.
2. Learn to perform calculations: Students must learn how to perform calculations in different settings. They should be comfortable doing algebra on matrices, solving systems of linear equations and working with vectors.
Math Major Outcomes incorporated into MAT313:
A. Perform numeric and symbolic computations
B. Construct and apply symbolic and graphical representations of functions
C. Model real-life problems mathematically
E. State and apply mathematical definitions and theorems
F. Prove fundamental theorems
Spreadsheet showing performance of students on each topic on the finals in both sections of the class.
Analysis: Observe that all the topics were covered well and that students performed well on topics taught later in the course, like eigenvalues and eigenvectors.
The students had excellent performance in solving systems, finding inverses and determinants. In both sections the students have learned the key topics involving calculations from the course which are applied in future courses. The questions with disturbing results all involved theoretical proofs. The topic of vector spaces, which involves proofs, had strong performance by only about 1/3 of the passing students while 2/3 performed poorly on the this questions on the exam (note that in both sections all students either performed well on the proof or failed to complete the proof). Students also had difficulty with proofs of linear independance although the performance there was a little better in both sections with some students managing half the students. One section also tested proving the distributive property for nxn matrices (allowing partial credit for a proof in the 2x2 case) with similarly poor performances. This is the first course in the math major which involves proofs on an exam and so it is perhaps good news that 1/3 of the passing students have caught on to the idea of proof. Keep in mind also that the majority of the students in the class are computer science majors.
Recommendations: It seems that our students have shown they can handle advanced topics involving calculations including eigenvalues and eigenvectors, so that these topics should definitely always form part of the linear algebra course. Neither textbook or syllabus appeared significantly better and so either could be followed in future courses.
In Spring 2012, this course was assessed using the Spring 2012 final exam designed by the professor teaching the course following his own syllabus.
MAT313 Course Outcomes (on Spring 2012 syllabus):
1. Learn to prove theorems. Students should be able to observe connections between different topics in the course description above, especially systems of equations, matrices and vector spaces.
2. Learn to perform calculations. Students must learn how to perform calculations in different settings. They should be comfortable doing algebra on matrices, solving systems of linear equations and working with vectors.
Math Major Outcomes incorporated into MAT313:
A. Perform numeric and symbolic computations
B. Construct and apply symbolic and graphical representations of functions
C. Model real-life problems mathematically
E. State and apply mathematical definitions and theorems
F. Prove fundamental theorems
Data: spreadsheet
Comments before data analysis: Due to a lack of funding, this class consisted of two sections of Linear Algebra merged into a single section. Although the department was given some money to hire a grader, no one could be hired due to the excessive paperwork required for a new hire. Perhaps in the future, graders might be hired as consultants. We should also avoid having extra large sessions of courses unless we are allowed to hire a teaching assistant to meet with the students in smaller groups. In the end, many students dropped this course, 16 failed the final and 24 passed the final. The data analysis is conducted on the passing students as usual, since we are assessing the level of knowledge achieved by those students who completed the course successfully. However it should
be noted that most of the students who failed the final did learn topics from the earlier portion of the course. They might have passed if a teaching assistant had been provided.
It is also the opinion of the assessment ambassador that the course outcomes for this course are too vague and that instead one should view the list of topics on the syllabus as the true course outcomes. There is also a concern that the listed topics in this course as taught in Spring 2012 are not as extensive as they should be. Every problem in the final tested both of the official course outcomes. Every problem in the final tested major outcomes A, B and E. None of the problems on the final tested major outcomes C or F. So in order to conduct an analysis of the course, we analyze the performance of students on the various topics as described in the spreadsheet below.
Data Analysis: spreadsheet
Conclusions: We are pleased to report that 87% of passing students were able to turn a system of equations into a matrix equation, invert the matrix and solve the system (with 50% completing this multistep process perfectly). 83% of the passing students were able to find a plane spanning three points and produce a line which does not intersect that plane (with 45% completing this perfectly). 71% of the passing students were able to complete a problem concerning the spans of vectors (29% perfectly). Only 33% of the passing students were able to determine if three points lied on the same line (25% perfectly). However 54% of the passing students were able to turn a linear system of equations into a matrix equation and determine the determinant of the matrix (42% perfectly). It appears that the students who passed the class have a reasonably strong understanding of four of the topics covered in this very basic linear algebra class. It is unknown if the students learned to write proofs or have an understanding of vector spaces as an abstract concept. In the future, this course should be taught with a smaller class so that the more intensively difficult abstract proving related concepts can be taught with lots of attention to each student as an individual. The assessment ambassador would also suggest that the course outcomes be restated more precisely as follows:
1. Solving linear systems and inverting matrices (a,b,c)
2. Computations involving vector spaces and linear transformations (a,b)
3. Proofs involving vector spaces and linear transformations (e,f,g).
4. Determinants, Eigenvalues and Eigenvectors (a,b,e)
See that here it is also recommended that we add Major Outcome G (Construct and present a rigorous mathematical argument) to this course. Possible proofs that could be on a final exam need not be difficult (e.g. prove that the kernel of a linear map is a vector space, or prove sum of two linear maps is a linear map).
Departmental Action: Starting Fall 2012, the class has been offered in smaller sections as it had in the past. In Spring 2013 an ad hoc committee has been formed to consider the possibility of a uniform syllabus in linear algebra.