PD for recitation instructors with tie-ins to partner departments

#1

Many upper division students take linear algebra and differential equations from math departments. Sometimes the courses are separate, sometimes the two topics are linked. Often they are taught in multi-section format - Sometimes in lecture/recitation format, sometimes many smaller classes. Often they are required by many partner disciplines. Math faculty and recitation instructors may be unfamiliar with the applications. Many students from STEM majors want to see the applications. Thus the PD for lecturers and recitation instructors should involve faculty from partner disciplines who can introduce common misconceptions and explain sample problems for math people who may be unfamiliar.

Math people like generality and abstraction to facilitate a variety of applications. The folks I have talked to in STEM and Econ point out that their students are not familiar with the processes by which sample problems lead to general statements or the process by which abstract results can be specialized to applied problems.

Conversations will help as will concrete examples. Here is one from chemical engineering: Many situation are modeled as systems of linear equations. However the unknowns are usually denoted by letter suggesting the quantities they stand for and the coefficients may be called parameters not coefficients. The matrix of coefficients will likely not appear as doubly indexed letters Similarly the constants on the right-hand side may not appear as letters y with single subscripts. This difference of notational convention seems to be a major obstacle for the engineers. This is especially a problem for engineer who too an “integrated” ODE and Linear Alg course, where the emphasis may have been on single linear and non-linear ode’s rather than on systems of linear equations. CRAFTY publishes on these issues. Huge number of instructional staff in doctoral and masters universities may not be familiar with MAA at all or CRAFTY documents. Face to face PD discussions may be needed at least every other year to advance the trans-cultural understanding.

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#2

Important point! I agree completely that faculty from partner disciplines like economics and chemistry should be brought in to teach mathematicians the application-specific notations and topics, and help develop these applications to be taught in our linear algebra and ODE courses.

Thanks for pointing to the MAA CRAFTY materials. CRAFTY’s website is here, for readers not familiar with it: http://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/crafty

I love the idea of interdisciplinary professional development workshops and think it a (small) and wise investment for universities to make. Even a short 2-day workshop during the summer on campus can go a long way toward building these bridges. Pay each faculty member a small amount to attend. Have an IT person as part of the group to help with software sharing and teaching.

If anyone has specific resources of applications as these, please share!

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#3

I am an engineering professor and I believe that abstractions are important to develop the engineering mind. We just need to agree to meet in the middle. Only after the abstraction and some practice with some rote questions, do we want to see an application, and then see if students can apply it later to a different application. I do not want application and abstraction taught at the same time as it can overwhelm the brain. The arguments of a function should always be carried in ODEs as many students will do all sorts of crazy integrations and differentiations. I would like emphasis on numerical solution of ODEs by hand as well as programming languages as only then will students see merit in solving coupled ODEs and the need to put them in state variable form. Should linear algebra and ODEs taught at the same time, I do not think so. Linear algebra should be a pre-requisite/co-requisite to ODEs. We may as experts see merit of combining the two but let’s scaffold the material for the students.

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#4

The previous post discusses sequencing of topics and courses. I personally agree that linear algebra and ODE’s should be taught separately. However, what I understand from some engineering colleagues is that a BS in engineering already requires more credits than the “usual” 120 credits for a BA, so there is pressure from students and deans not to increase the number of three credit courses but to integrate the material into one 3 or 4 credit course.

Returning to the topic of PD and effective pedagogy, I agree with the previous post that students can benefit from spacing out the motivating examples, the generalizations or abstractions, and the applications. Many instructors feel time-pressure to cover these parts of the material “all at once” lest the students get confused by “covering” more than one topic per class period. By contrast, I have had some success with the following scheme across 3 class periods: (1) using a sample problem at the end of one class; (2) using the following class to do a different motivation for generalization and abstraction; (3) beginning the class after that with a “recap” which reviews in the context of a somewhat new application.

I hope to learn of successful professional development efforts for the course instructors and/or recitation instructors in both math and engineering who could benefit from understanding the motivating problems, the useful forms of generalization and abstraction, and ways in which students learn to apply those general or abstract results in unfamiliar cases. My department runs a “seminar in teaching math” which might include from time to time a speaker of panel from engineering on the ways of making “pure” and “applied” linear algebra or ODE more coherent.

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#5

I appreciate hearing from an engineer, pointing to the importance of “abstraction.” My own view is that linear algebra (abstract ideas and also computational and applied linear algebra) is they keystone of the undergraduate curriculum in mathematics (pure and applied). All departments should consider the role of this course (these courses) carefully, and make them top quality! Linear algebra serves as the entry point to the major for many majors (the place we attract majors) and simultaneously the final math course that many other scientists take.

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