Notre Dame MathEd

Preparing for the Job Market

2008-01-21T13:58:00.002-05:00

Here are some pictures of the employment center at the joint meetings in San Diego in Jan 2008. Matt Rissler and Sara Quinn will describe these pictures and some of the ins and outs of job interviews at the joint meetings.

Mathed Events 2007-08

2008-01-14T16:04:00.005-05:00

The schedule for the teaching seminar for Spring 2008 is available in the course syllabus. It is summarized below (links will be added as they become available):

15 Jan 2008 - Information Session
22 Jan 2008 - Preparing for the job market
29 Jan 2008 - Understanding the student perspective
05 Feb 2008 - Lecturing
12 Feb 2008 - Learning from our students
19 Feb 2008 - Lecturing anecdotes
26 Feb 2008 - Discussion of Mock Lectures
4 Mar 2008 Spring Break
11 Mar 2008 - Mock Lecture - Jian Ge (moderator: Jeffrey Diller)
18 Mar 2008 - Mock Lecture - Jacob Carson (moderator: Brian Hall)
25 Mar 2008 - Mock Lecture - Giorgios Poulios (moderator: Frederico Xavier)
01 Apr 2008 - Mock Lecture - Yen-Chang Huang (moderator: Qing Han)
08 Apr 2008 - Constructing Short Quizzes
15 Apr 2008 - The Challenge of Diversity in Teaching and Learning
22 Apr 2008 - Mock Lecture - Stephen Flood (moderator: Arthur Lim)
29 Apr 2008 - Planning Meeting

Teaching Seminar Syllabus - Spr 08

2008-01-14T15:49:00.000-05:00

The syllabus and course information for the Mathematics Teaching Seminar are now available. In addition to the .pdf file, a .tex file is available primarily for use as a template for syllabus creation for our syllabus writing workshop and as a starting point for writing a syllabus for your course.

Syllabus .pdf
Syllabus .tex

Area and the Definite Integral

2007-04-30T13:45:00.000-04:00

Richard Gejji presented a mock lecture on the relationship between area under a curve and the definite integral. The lecture was motivated by the following "simple" goal:

Given a continuous curve y=f(x), approximate the area under the curve between x=a annd x=b.

For a general function, it is not obvious how to do this. To proceed, we need to generate a more intuitive notion of Area. The area is the "amount of space" a two dimensional object takes up.

In the case of f(x)=l, it is easy to calculate, because the area under the curve is a rectangle.

For f(x)=1-|x-1|, we have a triangle.

But if the area under the curve isn't a shape whose area we already know how to compute, what do we do? The general idea is as follows:

Step 1: Divide up the domain (x-values) of the area into n pieces.

Step 2: Estimate the area of the strips of this region with rectangles.

Q: (from class) What rectangles? (Speaker has not drawn any rectangles on the board.)

A: We'll see in a minute.

Step 3: Add together the areas of each of the rectangles to get an estimate for the total area.

Richard pointed out that there are many way to estimate the area of a strip by a rectangle. Without much explanation, he mentions right-endpoint, left-endpoint and mid-point methods.

Again the time for commentary was limited since the speaker shared time with another mock lecturer. The primary comment on the lecture was that the pace was too slow. However, it was noted that this particular lecture is one of the harder lectures to give in Calculus I. Suggestions were as follows:

Try an explicit example with specific rectangles.
Breaking down the process into steps was a good idea.
Too much time spent writing down complete sentences. Write down keywords instead.
Prepare a handout to help save time, or prepare slides/computer graphics (eg here).
Talk more about why the words estimate and approximate make sense here. Is it obvious that the Riemann Sum should converge to the actual area when n goes to infinity.

Integration By Parts

2007-04-30T12:46:00.000-04:00

Sujin Khomrutai presented a mock lecture on Integration by Parts. He began by reminding us of the tools we have learned so far for integrating functions; namely, definite integrals, riemann sums, indefinte integrals and the fundamental theorem of calculus.

Then he presented the following example:

Then he noted that

But now we can integrate both sides to find the integral of our original function.

This sort of trick is one that we can generalize. The general form is called integration by parts. The integration by parts formula is as follows:

Q: (from class) Can't we just write:

A: If we write that down, try differentiating both sides. Both sides should have the same derivative if they are equal. But they don't.

Q: (from class) Can we switch f and g?

A: We will look at some examples. We can switch f and g, but it may not be useful to do so.

Ex: Let's rework the original example with the integration by parts formula:

If we switch f and g, the expression gets more complicated, rather than easier to integrate.

Ex 2:

Ex 3:

Q: (from class) How do I choose f & g?

A: After trying a few examples, you will be able to see the pattern.

Q: (from class) Will g'(x)=1 work for any function?

A: You can always try it, but it may not help.

Time for comment was limited due to the need to fit two mock lectures into one meeting. Suggestions for Sujin included the following:

Clearly state the objective and motivation for the topic. In this case, we need to generate a method of integration that allows us to integrate t cos t.
Give some general rules for picking f and g. Three cases in particular: polynomial * ln, polynomial* trig, ln * trig.
If students don't respond/ask questions, that doesn't mean that they understand.

Writing a Teaching Philosophy

2007-04-06T15:37:00.001-04:00

Alex Hahn led a discussion/workshop on writing a teaching philosophy as part of the process of seeeking an academic job. The discussion was organized by an article written by Helen G. Grundman titled "Writing a Teaching Philosophy Statement."

Prior to embarking on a discussion of this article Prof. Hahn offered some directional comments about the exercise of writing a teaching philosophy.

1. This is a process of reflecting on what we are doing when we are teaching mathematics.
2. "Don't be a slave of your inbox." (Wisdom he passed along from a former student who is now a life coach.)
3. The teaching statement should be less about us and our actions in the classroom and more about the students, what they learn and how we contribute to their learning.

If our goal in writing a teaching statement is to focus on our students, thinking about who are students are is critically important. Prof Hahn encouraged us to think of students in a terminal course. What are their concerns? How will they need mathematics in the future? What kinds of activities will motivate them and help them engage the subject matter most effectively?

For example, "terminal" students (students in a terminal course) may be concerned about the course in so much as it is a hoop they MUST jump in order to graduate. "Terminal" students in the College of Arts and Letters may not feel like mathematics is a "useful" skill for them; but they may need to speak quantitatively in mattters of public policy or otherwise. If we combine these two, an effective sort of activity for these students might be a technical report. Such a report might appeal to Arts and Letters students since it is verbal/linguistic and it would prepare the same students to speak on quantitative matters in public policy.

Finally, Prof. Hahn admonished us to never say the following in a teaching statement: "My students don't have the mental capacity to understand what I am delivering." This seems obvious, but Prof Hahn has seen this type of statement in the past.

The remainder of the discussion (which far exceeded the time allowed) focused on following the activities suggested by Grundman in the suggested article. For several minutes we composed our responses to exercises 1 and 2. Then we wrote our responses on the board and discussed the merits of the responses.

The two responses that generated the most discussion were:

1) I want to teach mathematics, because I want to help students learn to think and organize their thoughts.
2) I want to teach mathematics, because I want to share its beauty, profundity and fun.

In the spirit of reflecting on what we do when we teach mathematics, the discussion of 1) diverged somewhat to the discussion of whether or not mathematics instruction generally achieves the aim of helping students learn to think and organize their thoughts. The underlying point was that if you want to claim that 1) is your motivation, presumably you would have to demonstrate that you teach courses in such a way that students do learn to think and organize their thoughts, or at least that they could do so.

While none of us left the discussion with a teaching statement written, the workshop and the article together put us in a good position to compose strong teaching statements.

Truth Tables

2007-04-06T15:18:00.000-04:00

Demirhan Tunc presented a mock lecture on Truth Tables as a lecture from the Beginning Logic course. Most of the audience was unfamiliar with this particular course. As a result, the analysis of this mock lecture was more vague than usual.

He began with a summary of where they (would have) left off in the previous lecture. This included a summary of the impact of logical operators on truth.

Regarding 3., A. Pilkington asked what happens if both G and H are True. D. Tunc replied that at least one of those must be True.

We proceeded to compute the truth value of:

given that P is False and Q is True. To motivate today's topic, he suggested that we try to analyze this expression all possible truth values of P and Q. Then he drew the following table on the board (minus the truth values):

S. Broad asked how to know how to break down the expression into smaller pieces (i.e. where to break it apart and in what order to do so). D. Tunc replied that we have already covered the propositional calculus and its order of operations.

We filled in this truth table together. Then truth tables for the following expressions were drawn:

It was noted that we have drawn three types of truth tables: one with T's and F's in the result column, one with all T's in the result column, and one with all F's in the result column. Following this, we developed the classification of formulas into three types: Tautologous (Example 2), Inconsistent (Example 3) and Contingent (Example 1).

Finally, a truth table for the following statement was drawn:

This truth table had eight rows. We discussed briefly how the values of P, Q and R could be assigned to guarantee all possibilities were considered. D. Tunc presented the method of alternating the truth values: first 4 T's, 4 F's, then alternating 2 T's and 2 F's, then alternating 1 T and 1 F.

Many questions were asked about how students could learn to populate these values.

Following the mock lecture, the analysis focused on two areas:

1. How can students determine all possible combinations of truth values?
2. Are students going to need something concrete to anchor the symbolic formalism of truth table analysis?

Suggestions included the following:

1. Draw horizontal lines to keep truth tables from "leaning."
2. Mention the use of binary trees for determining truth value combinations or at least demonstrate that the alternation method achieves all possible combinations of truth values.
3. Even though students should be comfortable with manipulation of formulae at this stage in the course, a concrete example of how truth tables analyze all possible combinations of truth values seems reasonable.
4. Draw truth tables for all basic logical operators on the board and demonstrate that more complicated truth tables are all computed from the basic truth tables.

Overall the lecture was quite competent and showed great promise.

Taylor Polynomials

2007-03-22T14:04:00.000-04:00

Don Brower presented a mock lecture on Taylor Polynomials. To introduce the notion, he asked the question: "Is it possible to approximate a function with polynomials?"

To develop the way in which this might be done, he drew the graph of f(x)=e^x and drew our attention to the point x=1. Then he drew the line y=e and asserted that it approximated f near x=1, but qualified that by saying that it wasn't a very good approximation.

Next he asked the class to recall the notion of linear approximation from the prior semester. We computed the tangent line to the graph of f(x)=e^x to get y=f(1)+f'(1)(x-1)=e+e(x-1). From Calc A/I, we understand that this is an approximation for y=f(x).

At this stage, Don asked if we could find a function that would match up to the second derivative which would be a polynomial of degree 2. He suggested that we try the following:

y=f(1)+f'(1)(x-1)+f''(1)(x-1)² (1)

The problem with this is that y''(1)=2f''(1). But we can correct that by dividing by 2, to get:

y=f(1)+f'(1)(x-1)+f''(1)(x-1)²/2 (2)

At this point A. Pilkington asked whether or not (1) had the same first derivative as our original function. Don responded that yes it did, but it wasn't a suitable candidate (for an approximation) because the 2nd order derivative did not match.

Don suggested that we could keep going and match the first n derivatives of f. He wrote down the following general formula for the nth degree Taylor Polynomial at c.

P_n(x)=f(c)+f'(c)(x-c)+...+f⁽ⁿ⁾(c)/n! (x-c)ⁿ (3)

B. Hall asked what Don meant when he said that he would divide by n! accounting for the factorial when taking the derivative. Don then began computing derivatives of P_n to show that the kth derivative of the Taylor polynomial would eliminate terms of degree < k because they didn't have enough factors of x-c and it would eliminate terms of degree > k because there would be remaining factors of x-c which would vanish when x=c. Furthermore, the surviving term of degree k, would have been differentiated k times producing a factor of k!.

After some time, Don was asked to give an example of a Taylor polynomial approximating a function. Don showed that the 3rd order Taylor polynomial for f(x)=2x³-x²-2x+1 recovered the original polynomial, hence being a very good approximation for f(x). At this point, the mock lecture was concluded.

The following question was raised as a footnote to the lecture:

If I want to approximate e^x, how do I choose c? It seemed that the intent of the question was to require a discussion of what is meant by the phrase "an approximation of f near c".

Comments and suggestions were as follows:

(1) This was a difficult topic on which to give a mock lecture.
(2) This might have been a situation in which computer graphics could have added the discussion.
(3) There was something of a disconnect between the stated goal of the lecture and the discussion that followed (i.e. how does this exercise of getting polynomials to match the derivatives of a function at a point relate to approximating f?).
(4) Motivate the need for approximation. In fairness, Don did say that approximating by polynomials would simplify computations and other operations like integration and differentiation. But the comment was a bit of a sidebar, in the sense that no examples demonstrated it.
(5) Try to think of a numerical application of Taylor polynomials such as approximating 26^1/2.
(6) Don said that he wished he had been more in touch with the geometric development of the Taylor polynomials as an approximation technique. In other words, a better explanation of how adding a parabola to a line would improve the approximation of a function might have been useful.

The discussion that followed the mock lecture raised many questions as well, owing to the fact that these notions are slippery for first-year undergrads. While there were a couple holes in the lecture related to motivation and the geometric viewpoint, the intent and preparation of the lecture were certainly in the right direction.

Cramer’s Rule

2007-03-15T08:37:00.000-04:00

Gun Sunyeekhan presented a portion of lecture for the Linear Algebra and Differential Equations class on Cramer’s rule. He began by recalling some definition from the previous lesson.

Recall: A is an invertible n×n matrix if and only if for each b in Rⁿ, Ax=b has a unique solution.

A is invertible if there exists B an n×n matrix such that BA=I and AB=I. We write B=A^-1.

Theorem (Cramer’s Rule)
Let A be an invertible n×n matrix, for each b in Rⁿ, the unique solution x of Ax=b has entries given by:

where A_i(b) is obtained from A by replacing the i^th column by b i.e. if

then

(A. Pilkington asked why we need Cramer’s rule when we can just use the inverse of A to the solution of Ax=b. Gun pointed out that Cramer’s rule gives the formula for the inverse of A.)

Next Gun proceeds to prove Cramer’s rule. This is followed by an example where Cramer’s rule is used to solve a system of equations:

Example: Use Cramer’s rule to solve the system:

Solution: View the linear system as

(A. Pilkington asked why not make x=[x₁ x₂ x₃]? Gun pointed out that matrices with incompatible sizes could not be multiplied together.)

So

We also have

hence det A₁(b)=29. Similarly, det A₂(b)=5 and det A₃(b)=-4.

Therefore the unique solution of the system is

Here are some comments made by the audience:
(1) Audience pointed out that Cramer’s rule is useful when vector b has lots of zeroes and A is a complicated matrix with no-trivial entries.
(2) A. Pilkington anticipated that engineers could have trouble understanding the statements of Cramer’s rule; whether vector b is a fixed vector or not.
(3) A. Lim suggested that speak could use a 2×2 system of equation to verify the statement of Cramer’s rule and clarify what the statement says and make it less intimidating to students. This could address the issue pointed out in (2).
(4) Audience discusses how exam problems could be asked to test Cramer’s rule and calculator issues. A. Lim suggested using matrices with variable terms and not all numbers.
(5) An audience member pointed out that b can be chosen to be a column of matrix A so that det A₁(b), say, is zero and point out that Cramer’s rule could still be applied. Compare with det A=0.

Generally, the presentation was great; beautiful board work and handwriting.

- Posted by Arthur Lim

The Derivative and the Tangent Line Problem

2007-03-06T11:23:00.000-05:00

Removed at request of presenter.

General comments. Usually, the faculty guest is selected due to a great familiarity with teaching a specific course. The graduate students are interested in the subtleties of teaching each specific course:

student demographic (major, math experience, etc.),
student abilities (what can we assume they know?),
philosophical approach to course content (conceptual vs. quantitative, etc.)
troublesome course material,
good and bad textbooks (if there is a choice),
appropriate balance between theory and applications, etc,
comparison with teaching other related courses.

Invited faculty members are welcome to distribute this wisdom either by direct discussion of these matters, or by responding to graduate student questions. In either case, it is a “roundtable” discussion format (i.e. the students and faculty guest sit in a big circle and talk).

Question and answer. There is not a clear boundary between this section and the previous. It is meant to emphasize that some questions can be anticipated, for which comments will hopefully be prepared in advance while others cannot.

MathEd Tutorial Resources

2006-08-15T16:00:00.001-04:00

Tutorial group activities for the following courses are being collected for the following courses. The use of group activities in tutorials began in Fall 2004 for 10550/560 and in Fall 2005 for 10350/360.

MATH 10350 Activity Portfolio Fall 2005 Fall 2006 (.zip files)
MATH 10360 Activity Portfolio Spring 2007 (.zip files)
MATH 10550 Activity Portfolio Fall 2004 Fall 2005 (.zip files)
MATH 10560 Activity Portfolio

We also have access to some already TeXed lecture examples used by A. Lim in previous lectureships. These will become available at this site in the near future.

MathEd TA Training

2006-08-15T15:59:00.000-04:00

Prior to their first tutorial session, new TAs are required to attend TA training. There are two tracks for the TA Training which are aimed at the two course sequences every TA will most likely be assigned at some point: MATH 10350 and MATH 10550.

TAs whose first tutorials are in the spring semester are required to attend TA training in the Fall, since there is no TA training in the Spring.

Course Kickstart
New TAs and TAs for MATH 10350/550 develop/discuss the group activity for the first tutorial of the semester. Many of the basic questions involved running a tutorial are discussed also.

Follow-up Discussions
Faculty members present their experiences teaching courses that graduate students are likely to teach at Notre Dame.

MathEd Teaching Seminar Mock Lectures

2006-08-15T15:58:00.001-04:00

What is a mock lecture?

What is expected of the lecturer?

Mathematics graduate students prepare for teaching responsibilities – in part – by participating in the Teaching Seminar. Prior to teaching a course, a graduate student must deliver a 20 minute mock lecture in the seminar. Following the mock lecture, faculty and graduate students offer their comments and suggestions which will be posted on the MathEd webpage.

The mock lecture is called a mock lecture, because no undergraduate students are present to listen to it. Even so, the mock lecture should be prepared as though the audience was a class of undergraduates. In particular, the lecturer should select course material from an existing undergraduate course at Notre Dame. The lesson should be prepared as though it would be given as a lecture in the course.

If you are preparing a mock lecture, please consider completing the Lesson Plan template (.pdf, .tex). These lesson plans can be posted on the Teaching Seminar web page for seminar participants to view prior to your mock lecture. An example of a completed lesson plan is available (.pdf, .tex).

Following the mock lecture, it is recommended that speakers review the feedback from the other participants. Some follow-up questions are listed below to help you evaluate your performance.

In what ways was this lesson effective?
How did this lesson further the course goals?
How would you change this lesson if you teach it again?
Did your students find the lesson meaningful?

MathEd Teaching Seminar

2006-08-15T15:57:00.000-04:00

Q&A Panel
Past TAs present their impressions of TAing and respond to questions from new TAs.

Teaching and Learning
Faculty members present their experiences teaching courses that graduate students are likely to teach at Notre Dame.

Mock Lectures (click here for additional details)
This is required before students are allowed to teach a course at Notre Dame. Graduate students present lectures from a Notre Dame course of their choice. The "audience" is undergraduates who would typically enroll in the course. Faculty and graduate students comment of the mock lecture afterward.

Roundtable Discussion
Students and faculty discuss teaching philosophy and specific strategies for teaching mathematics.

The MathEd Mission

2006-08-15T15:56:00.000-04:00

The MathEd program and website work together to provide guidance and resources for graduate students in the University of Notre Dame's Department of Mathematics as they begin their careers as mathematics educators.

The program has two components: the Teaching Seminar (MATH 83990, a 1 credit course offered in the Spring) and the TA training (offered during the first two weeks of the Fall). The Seminar is a course required for all graduate students who will teach a course. The TA training prepares TAs for their duties as tutorial instructors. (The TA training is required for all new TAs.)

The website presents resources for teaching and TAing in the Mathematics Department. It also has information about the Teaching Seminar for faculty and graduate students. Please see the links on the left for details.