U Substitution

Just last week I gave the lecture for "u substitution" to our class of first year calculus I students. In contrast, the the lecture that I gave on u substitution was a bit easier to present than the lecture for the Mean Value Theorem. By nature the two lectures are dissimilar in the sense that one is a technique for evaluating integrals and the other is something of a theoretical tool which does not get used much in a first year calculus class. Not only was u substitution easier to present but it was also much easier to prepare. I do feel, however, as someone who is new to teaching, that there are aspects of my style that need some improvement. While Barry did reassure me that my lecture on u substitution was indeed "very well done," I want to highlight some characteristics of a spring semester course in calculus I at UMM through my experiences teaching.

I'm not sure if I have made mention of the differences in mathematical maturity between students who take calculus I in the fall as opposed to those who take it in the spring, but in order to give some direction to this blog I will say that students who take calc in the fall are typically stronger than those who take it in the spring (at least this is true here at UMM). Since our calculus class is taking place in the spring, I have had the chance to witness and experience some of the difficulties that some of the students in a spring semester of calculus face. While a large part of the students can succeed, success is somewhat contingent on the presentation of the material. Barry was well aware that he need to treat our spring semester class just slightly different (with respect to how he presented the material) and he made sure I understood this. While I was aware of the students abilities, I did not quite catch on to how Barry has managed to polish his teaching style so that these students could improve in such a way that would prepare most of them to succeed in a section of calculus II.

I believe the way that this happens begins with how Barry lectures. For example, when I presented the material on u substitution I hurried through some of the tedious algebraic steps, believing that the students would follow what I was doing. And for some of the students did follow, and I feel that most who didn't follow right away could have followed if I had quickly jotted down the of the rule of exponents that we made use of. This may have taken just 10 seconds and would have ensured that all the students knew what I was doing. While Barry felt strongly about my presentation on u substitution, he made note of the fact that I should have made quick mention of the rule of exponents that I was using. While this is just a small detail that would have made my lecture more self contained, such details would benefit some of the students in our calculus I course.

For the most part much of the feedback that I received from Barry pertained to small details that, as Barry said, "Come with experience." It is my impression that Improving on these small yet important parts of my recent lectures would greatly improve the clarity and allow students to better grasp the material presented.

Preparing a Lecture

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I was given the opportunity to give our calculus class a lecture on the mean value theorem. I'd like to share the process of preparing this lecture.

As part of class assignments I've given in class presentations, most of which usually included a power point presentation. I knew teaching a small class of first semester calculus students the mean value theorem wouldn't exactly merit the same type of preparation as I had done previously for class presentations. And so with some experienced guidance from Barry I began to prepare. I started to prepare almost a week before the day of the lecture. For a while I thought I might be going overboard by giving myself that much time to prepare, however, I found that plenty of time to think about the mean value theorem beneficial. I found myself explaining to a fictional audience related theorems, such as Fermat's theorem, and thought about how clear my thoughts would have been had they been vocalized to a real audience. Eventually I started to plan what exactly I wanted to say in class with the hope that the evolution of the lecture would give way to a clear understanding.

After deciding on what I wanted to say and what examples I wanted to mention, I only needed to internalize the order in which I would present the ideas and the formal statement of the mean value theorem. I wrote down in a spare notebook the things I write on the board. All the details of what I would say to clarify the intuition behind the theorem had already been established from my pseudo-lectures in front of the fictional audience I had been practicing in front of for the past week.

When it came time to lecture I hardly needed my notebook, except occasionally to verify what I was writing on the board agreed with what I had in my notebook. I definitely had things I could improve on, such as my use of the whiteboard. While I enjoyed explaining the ideas behind the mean value theorem, Barry informed me that a bit more enthusiasm would improve the overall presentation.

Hopefully when I give the next lecture on the substitution rule for integration I can come through with a bit more charm and organization on the whiteboard. We'll see how it goes the second time around...    

A Successful Calculus Course

Sometime ago Barry and I talked about what it means to have a successful calculus class. We mentioned things like grades and appropriate test/quiz rigor, both of which naturally relate; if we gave our students easy tests, we would likely see a high class average. However, much more is considered when talking about the success of a course in calculus. If half of the original enrollment of students drop before the end of the semester then this may be reason for a very serious assessment of the course as a whole.

In my talks with Barry I've learned that many things contribute to how a course is graded as a success. If the number of students in a course who drop, withdraw, earn an incomplete, or earn a letter grade of D lower is adequately low then such a course may be called a success. The natural question that arises is the meaning of adequately low. From what I have seen in our section of calculus here at Morris, and from what I've heard about larger sections of calculus at other institutions, adequately low can vary substantially. For our particular section of calculus Barry and I aim for most students not to withdraw or drop for reasons of inadequate preparation, however, there are instances in which students are simply not ready to study calculus. For such students we hope that they first realize on their own that they are not well prepared, which seems to be the case for a large majority of students. For those who fail to see their lack of preparation, Barry sends emails asking to see students of this type in his office. Being able to inform one of our students, sooner rather than later, of the trouble they may face by continuing the course and earning a D or lower is seen by us as a success if the student does drop. While it is unfortunate, the student is given the opportunity for better preparation and a better experience with calculus by enrolling in a later semester.

It seems crucial to the overall progress of the class to be able to identify struggling students and provide them with appropriate advice. While this was not something I anticipated as a UTOP TA I see every need to assess these issues. Naturally, good preparation and organization on the part of the teacher is also a vital aspect to any successful class in calculus, but I hoped to shed some light on a "darker" side to being responsible for a class of calculus students.      

Roles as a UTOP Teaching Assistant in First Year Calculus

Something that has only briefly been mentioned (if at all) is my role as a teaching assistant under the UTOP program. Unlike the responsibilities that math TAs are usually given as undergraduate student workers here at University of Minnesota Morris, which mostly includes grading homework and quizzes, I, as a UTOP TA, get to construct the homework and quizzes and lecture.

At the beginning of the semester Barry and I met to discuss what my responsibilities should be as a more involved TA. It was decided that I would be solely responsible for constructing the online homework (Webwork). I also help in the construction in both quizzes and tests. Since being a TA still means that grading is involved, I do grade the quizzes while Barry grades the tests. Being more involved means that I attend three of the four classes every week to assist in answering questions, however, the most exciting part of being a more involved TA is having the chance to actually lecture. At one or two points in the semester I will be responsible for presenting the material to the class in full-length 65-minute lectures. While this is a bit intimidating I have always enjoyed helping others in leaning mathematics, so the chance to lecture in a first-year calculus course is very exciting for me.

Indeed, my role as an undergraduate TA is well beyond that of a typical undergrad TA. In completing the online homework, students have every chance to email me directly about problems they are having, and answering these questions is something math TAs here a UMM don't get a chance to do. My more personal involvement in the class allows me to help Barry in teaching and assisting the students while experiencing what it takes to be a great teacher.

Activities in Functional Notation

One of the first topics we covered in our Calculus class was functional notation. Since understanding the definition of the derivative requires the understanding of functional notation, Barry and I tried to make it clear that functional notation is important and will come up again and again as the course progresses. We quickly learned that the students were not understanding how to work with functional notation and one day in Barry's office we came up with a little in-class group activity which is the topic of this entry.

In an attempt to resolve the issues students were having with functional notation, Barry and I worked on a short in-class activity that was meant for students to understand the algebraic and graphical aspects of functional notation. Our hope was to witness a class-wide epiphany! This never actually happened the way we visualized it happening when we were constructing the activity in Barry's office. However, despite the lack of large scale epiphany, a reasonable number of students walked away with at least an inkling of what they had previously been doing wrong with functional notation. A few more students completely understood the point of the activity. While we may have not made our point clear to every student, it is our hope that students will help each other outside of class.

While working on this activity I've learned that just rephrasing something does not make it clear for everyone. Sometimes when we as teacher/tutors/TAs work with a group of students who struggle with the same problem we hope that what we have to say is understandable and clear. However, being so accustomed to concepts in mathematics sometime gets in the way of conveying the math and other means of explanation are sought out.


The Syllabus

Perhaps one of the first first things that should be discussed (it typically being the first thing students want to see) is the syllabus. I will describe what went into the syllabus for our section of Calculus I and why we choose to keep or loose certain parts of the initial proposed syllabus. I also want to compare what went into the syllabus with what should be in a syllabus (according to Thomas W. Rishel -- author of "Teaching First A Guide for New Mathematicians").

This being the first time I had to help construct and critique a syllabus for a first semester Calculus section, I had expectations of simply stating the crucial components of the class: grading scale, required texts, office hours, etc. As a student I mostly cared for grading scales and for the percentage of my final grade for which homework, tests, and quizzes counted. After reading Rishel's book I understood the need to have a more complete syllabus. Rishel describes the syllabus as a contract between the professor and the students. I very much agree with this! I, as a student have always looked at a syllabus as what was expected from me and from the professor. Rishel continues to describe a syllabus as something that should be more than just a summary of the course.

As for the syllabus created for Barry McQuarrie's Calculus class, much of what Rishel said should be in a syllabus was in the syllabus (and then some!). The syllabus that Barry and I decided on actually came from a past spring semester Calculus I course, and so most (if not all) of the necessary components were in place. This was nice, since that left us with mulling over whether or not we should do applied projects and in-class homework presentations. In retrospect, the completeness and thoroughness of the syllabus is far beyond what Rishel ever advised in his book (which makes sense since Rishel's audience is probably comprised of new first-time teachers). Rishel's recommendations for what should be in a syllabus merely scratches the surface of what is in the syllabus for our Calculus I course.

I planned on describing the syllabus, but instead I find it more useful to simply provide a link: syllabus.

The second week of class I found myself needing to look at the syllabus to remind a student (and myself) of the agreement dealing with late homework. Despite having read the syllabus several times over just a week before I felt the need to keep in mind my part of the "contract."
Having such a complete account of what is expected from both sides of the classroom is not only helpful to students, but also to TA's such as myself. While our syllabus is rather lengthy, it does not set an example of what should be in every course syllabus. Obviously each class varies from subject to subject and from teacher to teacher, and each syllabus should vary accordingly. 


The First Week

As someone who has never crossed that threshold (from learning to teaching) there is a lot to be learned. Before the first week of class Barry and I met to discuss the syllabus and the components of Calculus I that we found conducive to learning and understanding the concepts of Calculus I. Such things as applied projects, homework, quizzes, tests, and journals we found to be essential parts to a good Calculus class.
Class journals are one thing I have never had to do for a math class, however, I completely agree with their use in Calculus. The journals ensure that students communicate with the professor, and this can be extremely valuable for both the student and professor. Student feedback is essential to gauging how well the students are grasping the class material, and journals are a source for immediate student feedback. -- Many classes (especial math and science classes) can benefit from journals.

The first week of class I spent "evaluating" Barry as he lectured. Having never taught a class before, I observed the way in which Barry spoke and wrote on the board (simple things). You can't learn from a professor who stands in the way of what he or she is writing on the board. Beyond the simple things I listened to the ways Barry would explain what he had written.

During our meetings outside of class, Barry and I discussed grading in more detail. We talked about the importance of consistent fair grading. I have graded homework for other classes before, but something I've learned already is how to grade consistently. Grading the same problem for each quiz before moving on to the next, for example, can help to be more consistent. And better yet, to go back and review each graded problem for each student to ensure that it was roughly graded with the same rigor for each student is more though. Barry mentioned how he will sort graded papers in stacks according to the letter grade, and how he will return to all the B papers, for example, and make sure each paper in that stack truly deserves a B.

We decided that as the class TA I would be responsible for constructing quizzes homework sets. Constructing three sets of homework on Webwork and one quiz, I've begun to understand the process of finding good problems for students. A good problem can show how well a class or individual understands or does not understand something. A good problem can tell a professor that either a class or student is slacking or, more likely, if every student is struggling with the same problem, that the professor is not communicating as well as he or she thought during lecture. This is another type of feedback that I learned is very crucial for a professor to pick up on. Barry has mentioned how he has learned to explain certain topic better because of such problems. Hopefully by the end of the semester I can get a little better at finding good problems!

For now we are working on finalizing the first exam. Again, we are looking for good problems that cover the material discussed during lecture, and problems that will ensure good feedback.


The UTOP Begins

David successfully applied for a University Teaching Opportunities Program grant for spring 2010. We decided it might be useful to blog our thoughts as a means to document the process.

There are two courses we will focus on, Survey of Math and Calculus I. Calculus I is the standard calculus class taken by a variety of majors, and typically has students enrolled with a wide variety of mathematical background and ability (from PSEO high school students to seniors meeting a major requirement). Survey of Math is a general education M/SR course that was created to meet standards for teaching licensure in the state of Minnesota. While Survey of Math is open to all students enrollment is predominantly elementary education majors.

For the first ten weeks or so of the semester, we will be working on Calculus I. For the last few weeks, we will turn our attention to Survey of Math.

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