Posts Tagged ‘goals’

Learning Outcomes for Calculus II

July 15, 2020

I have one rule in life: when Robert Talbert issues me a personal challenge, I respond. I have created learning outcomes for my Calculus II course, which are below.

There are a couple of things to note. First, this might seem to imply that I have thrown out the work I did on Dee Fink’s Significant Learning planning. This is not the case, though. Doing Fink’s exercise helped prep me for these learning outcomes. More importantly, Fink’s exercise was focused on “What do you want your students to remember several years from now?” I clearly could not assess such long-term goals in Fall 2020, so I needed to approximate the goals from the Fink exercise to something that I can assess this semester. This is the result.

Second, I intend to focus much more on applications than I normally do. I will couch differential equations in terms of Covid 19 modeling, and I will do much of integration in terms of probability (improper integrals and double integrals) and applied work/density problems. Series are then motivated by trying to solve differential equations and integrals that have no “nice” solution. I additionally will focus on successive approximations and estimating error—starting with a rough solution, and working toward a better one.

Third, we are on the block system this year, so I am intensively teaching this course over the course of four weeks. The advice I have gotten is that you must cut content in such situations, which is why I am missing your favorite convergence test for series (although there are some that are hidden in other outcomes).

I haven’t thought much about my assessments yet, so the implied modules below (Integration, Optimization, Series, Approximation, and Error) might change. In particular, I suspect that I might have several projects, and the modules might change to match the needs of the project rather than the more conceptual categorization below. I don’t know for sure, though.

Below are my learning outcomes for Calculus II. I welcome all comments, compliments, and criticisms.

  • Group I: I can use integrals to solve authentic real-life application problems.
    • I1: I can solve separable differential equations.
    • I2: I can compute integrals using integration by parts.
    • I3: I can solve real-world problems by slicing and integrating.
    • I4: I can compute improper integrals.
    • I5: I evaluate a double integral over a general region.
  • Group O: I can find optimal solutions to multivariable functions.
    • O1: I can optimize a 3D function.
  • Group S: I can determine whether series, including power series, converge.
    • S1: I can define what it means for a series to converge.
    • S2: I can show con(di)vergence of a series using the Direct Comparison Test.
    • S3: I can show con(di)vergence of a series using the Integral Test
    • S4: I can find the interval and radius of convergence of a power series.
    • S5: I can prove a sequence converges using an \epsilon-$N$ argument.
  • Group A: I can find good approximations to functions, values of integrals and solutions to differential equations.
    • A1: I can approximate the value of an integral using the Trapezoidal Rule, Midpoint Rule, or Simpson’s Rule.
    • A2: I can approximate the value of a solution to a differential equation using Euler’s Method.
    • A3: I can approximate a function with a Taylor polynomial
  • Group E: I can bound the error associated with the above approximations.
    • E1: I can bound the error when using the Trapezoidal Rule, Midpoint Rule, or Simpson’s Rule.
    • E2: I can bound the error when using Euler’s Method.
    • E3: I can bound the error of a Taylor polynomial using Taylor’s Theorem.

Honoring All Parts of the Mathematical Process

October 30, 2019

I went to a sectional MAA meeting 1.5 weeks ago, and I saw a fantastic talk by Aaron Wangberg. He basically said that there are about four parts to doing mathematics:

  1. Experimenting and developing intuition
  2. Making definitions
  3. Making conjectures
  4. Proving and problem solving

(These four may not be exactly the four that Aaron had, but they are close enough for this post). Aaron went on to say that we spend almost all of our time in the four area—proving and problem solving—in a typical math course. He thinks that we should work on all four aspects with students.

(A quick aside: I have always felt a bit uneasy about Moore Method-style notes like those found on They are great, but Aaron gave me a way to put my finger on one of the things that I don’t like about them: they definitely do not provide space to make definitions.)

In many ways, this is not new. However, he put things in such a way that organized my thinking to the point where I think that I can start doing this (which I am sure is not being captured in this blog post). I am newly inspired, and I am trying to figure out how to start doing this justice in my courses.

The other motivator for me is my success in my capstone class this semester. Briefly, I am giving them open problems to work on. They are doing amazingly well. They are going through all four of the parts of doing mathematics listed above, and it is awesome to watch. I am so happy with this course (and so are the students, based on the feedback I have gotten, which is from nearly everyone in the course).

I understand that these are senior math majors, so I would need to scaffold things a lot more for, say, my probability and statistics course (next semester) or courses for elementary education majors (next year). However, I am going to plan on doing this.

Here is my one barrier: I know that this is going to require me to grade very differently. I know that my grading system (and courses in general) have gotten too rigid, so I need to figure out how to grade in a way that (1) I feel good about and (2) allows for stuff like “developing intuition” to count.

Let me know if you have ideas!

Problem Solving for the Liberal Arts

March 7, 2014

I taught a “Math for Liberal Arts” course last semester based on Pólya-type problem solving. I want to change some things the next time I teach it, and I should write it down before I forget it.

Just to remind you (and also me, actually), I will list the major points about the course structure. I have two more-detailed posts here and here.

But here is the short version:

  1. I taught the students the problem solving process, including some carefully-chosen heuristics (solve an easier problem first, find an invariant, etc). We spent most every Monday and Friday working on two new problems for students to solve (Wednesdays were quizzes or review). I (mostly) carefully chose these problems so that they could be solved by applying the heuristics we had already discussed.
  2. If a student solved a problem, she could sign up to present the solution in class. If we all agreed it was correct, the problem was closed and no one else could get credit for it. If multiple people signed up to present the same problem on the same day, I would randomly select one person to present, while the other people handed in written solutions of the problem. Everyone with a correct solution got full credit for the problem.
  3. Once a problem was presented correctly, it was eligible to go on a quiz. So the quizzes consisted entirely of problems students have already seen solutions to. Once a student gave a correct solution on a quiz, he never had to answer that question again.
  4. Regardless of whether a student found a solution to a problem, the student could submit a Problem Report on that problem. The idea was to describe their problem solving process and mine out instances of good habits of mind to present as evidence for a higher grade (see this for more detail).
  5. The grading scheme is basically this: a student got a C for the semester if she did well on the habits of mind in the Problem Reports; a student got a B for the semester if she additionally could reproduce solutions she had already seen (i.e. “did well on the quizzes”); a student got an A for the semester if she additionally could create solutions to problems she had never seen before (i.e. “correctly presented many of the problems from the course”).

Here are a couple of examples of problems I gave the students:

  1. How many zeroes appear at the end of 100!, where 100! is the product all of the integers between 1 and 100 inclusive?
  2. A dragon has 100 heads. A knight can cut off exactly 15, 17, 20 or 5 heads with one blow of his sword. In each of these cases, respectively 24, 2, 14, or 17 new heads grow on its shoulders. If all heads are cut off, the dragon dies. Can the dragon ever die?
  3. What is the last digit in the following product? (2^1)(2^2)(2^3)(2^4)\ldots(2^{201})(2^{202})(2^{203})?
  4. An enormous 5 \times 5 checkerboard is painted on the floor and there is a student standing on each square. When the command is given each student moves to a square that is diagonally adjacent to their square. Then it is possible that some squares are empty and some squares have more than one student. Find the smallest number of empty squares.
  5. Suppose you are in a strange part of the world where everyone either always tells the truth (a Truthie) or always lies (a Liar). Two inhabitants, A and B, are sitting together. A says, “Either I am a Liar or else B is a Truthie.” What can you conclude?

The last type of “Truthie/Liar” problem is a standard one in logic, and I started including a lot of them at the end of the semester. This was both because students really enjoyed them and the students needed a lot of help getting the Perspectives habit of mind. Students had a very difficult time figuring out what this even means, and I need to do a better job helping them understand it in future semesters.

One consequence of including so many Truthie/Liar questions is that I would like to add a heuristic to the class list: “Break the problem into cases.”

One other thing that I would change about the course is the quiz structure. What I did was to pull problems that had been previously solved by members of the class. Instead, I would like to find 15 or 20 problems, present them myself to help teach/emphasize/remind students about heuristics and the problem solving process, and use these on the quizzes. This would solve a couple of problems:

  1. I had three sections, so I had to keep track of three sets of quiz questions. This way, I would only have one set.
  2. This would give students more time to digest all of the solutions. As I did it, students may have only had two weeks to learn a solution that was presented toward the end of the semester. If I control the quiz questions, I could pace them so that the last one is solved for them by mid-semester, giving them at least half of a semester to learn the solutions for the quiz problems.
  3. Similarly, I can raise the expectations for how many solutions they learn if they all have at least half of a semester to learn them. Depending on the problems I choose, I think that I could realistically expect a B-student to know all of the solutions.
  4. Perhaps most importantly, some solutions are more instructive and valuable than others. I would be able to show them solutions that can be modified to solve other problems.

I would also change one detail of the Problem Reports. I required at least three in each category to be eligible for a C, six for a B, and nine for an A. I think that three was too low, so I would probably change it to 5 for a C, 5 for a B, and 10 for an A.

Finally, I spent too much of the class letting the students freely try to solve problems. I need to figure out how to incorporate more instruction into these. For instance, I could charge each team trying out an assigned heuristic on a problem, let them work, and then have the teams report how they worked to apply the heuristic. This would regularly review the heuristics and help the students get in the habit of using them (I think that most students did not consciously use them).

Does anyone else have any ideas about any of this—particularly concerning the previous paragraph?

Campus Famous

October 5, 2013

Here is a problem that I have. Or, maybe, here is something that is true about me that I wish were not true: I want to be famous.

I actually don’t want to be an actual celebrity, but I want to be well-known on my campus; I want to be campus famous (it is worth knowing that I work at a liberal arts school with roughly 350 faculty members).

In and of itself, this is not bad. In fact, it may be good. This stems from the fact that I feel a strong sense of community and I want to nurture it. To do this, I want to build bonds with a lot of people. But even with people I don’t have a bond with, I want them to know who I am. This is useful, since I feel like I have skills that other people might find useful (just as I have people on campus that I seek out when I have a problem that needs to be solved).

But the problem comes in in actually becoming campus famous. The easiest way to do this to do a lot of service. This is because it is very easy to get to work with people outside of my department in doing service, but very difficult to do by teaching or research.

Now I actually enjoy doing some amount of service, I indubitably do my share of service, and I think that much of it is worthwhile. The problem comes from the fact that it is very tempting to keep increasing the amount of service I do. The more service I do, the more people I meet. The more people I meet, the more relationships I build on campus. The more relationships I build on campus, the better the campus is and the happier I am at work.

Part of this is that I want to be a good employee, and doing service is part of that. I also take pride in my school and want to see it reach its potential. I should do enough service to help make these two things happen. But I should not do more than that simply because I want to be campus famous; that is just ego. Doing service just to feed my ego does not align with my goals. I need to be aware of this when I make decisions.

Finally, here is some news that is less related than it initially seems: I recently agreed to be on a campus-wide committee to assess our “Common Curriculum.” But I thought of writing this post before I was even offered a place on this committee, and so I was very mindful of my desire to become campus famous when I accepted. Also, I think that this meets my goal of “Continue to try to improve my teaching.” Having goals is an essential part of good teaching, and assessing them is also important. This is a couple steps removed from how I normally think about my classroom, but I think that I should learn about what we, as a college, are trying to teach our students (and how well we are doing it).

So I am pretty sure that this isn’t just me trying to be campus famous.

My goals worked; I said “no.”

April 18, 2013

I previously listed some goals/guiding principles for my career. Here they are:

  • I want to continue to improve my teaching.
  • I want to do research in finite group theory.
  • I want to help K-12 teachers improve their teaching. This is mostly done by teaching pre-service courses, but this could be met in other ways.
  • I want to help close the gender gap in mathematics.

I have two bits of news related to these:

First, I think that I can now provide more detail for the second goal. Part of the goal will be to focus on undergraduate research. By the end of the week, I will have been to four thesis defenses in ten days (I was an advisor for two of them and a reader for the other two). I thoroughly enjoy this process, and it is something that benefits the college and the students. I think that “undergraduate research” might be the focus of my upcoming sabbatical.

Second, these goals are working. I previously wrote, “This means that my default answer should be ‘no’ to any other requests of my time” (aside from these goals), and I used this list to figure out that I should say “no” to a request. I was given the opportunity to volunteer for a service job last week. There are several reasons why I wanted to volunteer for the job, but it did not meet the criteria on this list (and I already had enough other service jobs to fulfill the requirements for my job). So I said “no” (in the sense that I did not volunteer).

I am pleased that I will be able to spend more time next year on my career goals.

Students are Gaming Your System

February 18, 2013

There was an amusing story recently about some clever students who found a way to get an A on the final without doing any work.

The professor’s policy was that all exam scores are re-normed so that the highest score on an exam becomes the new “100%” (I have a serious issue with his statement that this system is the “most predictable and consistent way” of comparing students’ work to their peers, since I think that students should judged on the basis of their knowledge of the material and NOT by comparing them to their peers. But that is a topic for a different post). His students recently got together and decided that none of them would take the final. Thus, the highest grade was 0%, and everyone got an A.

The professor did give everyone an A on the final exam, but later said: “I have changed my grading scheme to include ‘everybody has 0 points means that everybody gets 0 percent, and I also added a clause stating that I reserve the right to give everybody 0 percent if I get the impression that the students are trying to ‘game’ the system again.”

Here is the thing: students are always trying to game the system; this is because they are largely rational people. Moreover, professors mostly want students to game the system.

For example: I know of many professors who count homework as maybe 5-10% of a student’s final grade. The most common reason I hear from giving a low, but non-zero, weight to homework is “I want to make the students do the homework.” Translated: “I want my students to ‘game’ the system by doing the homework, whether they learn from it or not.” This is also true of attendance policies, participation policies, or really anything that has points attached to it.

What we (thoughtful people, at least) are really interested in is “student learning.” This is difficult to measure, so we use a proxy—”points”—to measure it. But then we fall subject to Campbell’s Law, where we confuse the proxy with the real thing.

Thus, the phrase “I reserve the right to give everybody 0 percent if I get the impression that the students are trying to ‘game’ the system again” really means
“I reserve the right to give everybody 0 percent if I get the impression that the students are trying to ‘game’ the system again in a way that I do not approve of.”

Moreover, the professor is not being clear about what are allowable ways of gaming and what aren’t. This conjures the memory of the famous case of cheating in Central Florida. The students found a test bank available online and studied from it. To me, this sounds like a completely reasonable way to study, and—to the best of my knowledge—this was not explicitly prohibited by the instructor.

The Central Florida was asking the students to ‘game’ his system by performing well on the exam. He was not clear about the allowable ways to ‘game’ the system, but he expected them to know what was allowable and what was not. This seems very unreasonable to me (there are some things that I think that we mostly agree on. It is not allowable to ‘game’ the system by writing down exactly what your neighbor wrote down, for instance. But I don’t think the Central Florida example is such a culturally agreed upon situation).

This is one of the many reasons why I switched to Standards-Based Grading: the proxies are at the very least less familiar, and most likely better associated with our ultimate goal of “student learning.” My proxies are not points, but rather “the number of times you demonstrated that you can do a particular type of problem to me.” It is tough to ‘game’ this system in a way that I am not in favor of, since most of the ‘gaming’ involves learning something well enough to convince me that you understand it.

It is still possible to game the system, though. For instance, students can demonstrate understanding through quizzes, and they can game the system by copying down their neighbors’ answers. But most of the examples of gaming SBG that I can think of falls into the “everyone knows that you are not supposed to do that type of gaming” category.

But the main point is: let’s not pretend that we don’t want students to game the system in certain ways. Let’s remember that the system is not what is important, and we must not lose sight of the reason why there is a system in the first place.

Career Goals

February 8, 2013

I have been thinking recently about how I would like my career to be. I am trying to narrow my focus so that I can figure out what is important (and then say “no” to things that get in the way of the important thing). Then I had a serendipitous conversation with my wife last night. She was making a list of “values” for herself. This list will help her make decisions about what she should and should not do in the future.

I realized that this is exactly what I have been considering recently, only I have been thinking about my “career values” rather than “life values” (this is not a perfect match—the word “goals” applies much more to my example than hers).

So here is what I currently have for a list of values. These are the things that I want to focus on for my career, and I will work to minimize things that do not belong on this list. This is also only a draft—I will be preparing my tenure file this summer, and I will do some deeper thinking about this then.

  1. I want to continue to improve my teaching. This is the most obvious one, but it should be included.
  2. I want to do research in finite group theory. This is one of the two values that has the least direction. So far, I have kind of been all over the place in my research, and not in a good way. The only theme I have seen so far is “I seem to be interested in maximal subgroups.”
  3. I want to help K-12 teachers improve their teaching. This is mostly done by teaching pre-service courses, but this could be met in other ways.
  4. I want to help close the gender gap in mathematics. Kate Owens has been inspiring me on this lately, and it is the value that I have the very least direction on. I have also only begun to think about it. At some point, I am going to have to sit down and read a bunch of the literature.

This is my list for now. This means that my default answer should be “no” to any other requests of my time (in theory, at least). For instance, I had considered eventually running for Chair of the Faculty Senate, but faculty governance is much less important to me than these four things. So I will stop actively thinking about this, although I would be happy to be a senator again in the Faculty Senate (which requires a lot less time—I have to do some service, after all).

Has anyone else thought of their careers like this? Any advice?

Grading for Probability and Statistics

January 23, 2013

Here is what I came up with for grading my probability and statistics course. First, I came up with standards my students should know:

“Interpreting” standards (these correspond to expectations for a student who will earn a C for the course.

  1. Means, Medians, and Such
  2. Standard Deviation
  3. z-scores
  4. Correlation vs. Causation and Study Types
  5. Linear Regression and Correlation
  6. Simple Probability
  7. Confidence Intervals
  8. p-values
  9. Statistical Significance

“Creating” standards (these correspond to a “B” grade):

  1. Means, Medians, and Standard Deviations
  2. Probability
  3. Probability
  4. Probability
  5. Confidence Intervals
  6. z-scores, t-scores, and p-values
  7. z-scores, t-scores, and p-values

(I repeat some standards to give them higher weight).

Finally, I have “Advanced” standards (these correspond to an “A” grade):

  1. Sign Test
  2. Chi-Square Test

Here is how the grading works: students take quizzes. Each quiz question is tied to a standard. Here are examples of some quiz questions:

(Interpreting: Means, Medians, and Such) Suppose the mean salary at a company is $50,000 with a standard deviation of $8,000, and the median salary is $42,000. Suppose everyone gets a raise of $3,000. What is the best answer to the following question: what is the new mean salary at the company?

(Interpreting: Standard Deviation) Pick four whole numbers from 1, . . . , 9 such that the standard deviation is as large as possible (you are allowed to repeat numbers).

(Creating: Means, Medians, and Standard Deviations) Find the mean, median, and standard
deviation of the data set below. It must be clear how you arrived at the answer (i.e. reading the answer off of the calculator is not sufficient). Here are the numbers: 48, 51, 37, 23, 49.

Advanced standard questions will look similar to Creating questions.

At the end of the semester, for each standard, I count how many questions the students gets completely correct in each standard. If the number is at least 3 (for Creating and Advanced) or at least 4 (for Interpreting), the student is said to have “completed” that standard (the student may opt to stop doing those quiz questions once the student has “completed” the standard).

If a student has “completed” every standard within the Interpreting standards, we say the student has “completed” the Interpreting standards. Similarly with Creating and Advanced.

Here are the grading guidelines (an “AB” is our grade that is between an A and a B):

-A student gets at least a C for a semester grade if and only if the student “completes” the Interpreting standards and gets at least a CD on the final exam.
-A student gets at least a B for the semester grade if and only if the student “completes” the Interpreting and Creating standards and gets at least a BC on the final exam.
-A student gets an A for the semester grade if and only if the student “completes” all of the standards, gets at least an AB on the final exam, and completes a project.

The project will be to do some experiment or observational study that uses a z-test, t-test, chi-square test, or sign test. It can be on any topic they want, and they can choose to collect data or use existing data. The students will have a poster presentation at my school’s Scholarship and Creativity Day.

I would appreciate any feedback that you have, although we are 1.5 weeks into the semester, so I am unlikely to incorporate it.

Grading Epiphany (or, Bret was completely wrong about everything)

January 8, 2013

LED Light Bulb

I am teaching probability and statistics, a course for first-year students, for the first time this spring. I have been struggling with how to grade the students.

This course is unusual in that there is only a little mathematics in the course (we throw in all of the probability that we can, but it is still mainly statistics). This requires that I think like a statistician, which is new and somewhat painful.

It also makes designing a course more difficult. I have the basics of what I want to do, but—as mentioned above—how to grade the students is the most difficult part. I want to stick with an SBG approach, but I was not sure how to set up the standards.

In calculus last semester (and other courses in previous semesters), I had a general format of: “if you can do the basic skills from the course, you will get a C. To get a grade higher than C, you must demonstrated some conceptual understanding.”

I realized as I was brushing my teeth last night that this is completely and utterly backwards.

I want my C students to understand the concepts of the course, but not necessarily be able to do the computations and symbolic manipulations. My B students should, in addition to understanding the concepts, be able to do many of the computations and symbolic manipulations. My A students should, in addition to understanding the concepts, be able to do all of the computations AND demonstrate that they can do some self-guided work.

Here is my rationale for requiring understanding the concepts to get a C:

  1. I am convinced that the concepts are easier in most college-level mathematics courses—students are better at drawing tangent lines on graphs of functions than they are at finding the equations of tangent lines.
  2. The students who need the calculation and symbolic manipulation skills are the ones who are going to continue taking more mathematics (and related) courses. I am guessing that C students are less likely to continue taking these courses.
  3. Computers can now do much of the calculation and symbolic manipulation, although the user has to understand the concepts to correctly enter the information.
  4. Most importantly, the concepts are the most important part of the course! I want to explicitly encourage students to focus on the concepts—I don’t want, say, a calculus student to be able to get an A in the course by only having good algebra skills (a colleague yesterday complained to me about such students; I view this as a flaw in the grading system).

I am not happy about having this completely backwards, and I feel bad for my previous students. But I am happy that I now understand what I want.

The tough part is designing assessments that isolate concepts. But that is part of my job, and I find it fun to come up with such questions.

How can this be improved? Am I wrong about this?

(image is “LED Light Bulb” by flickr user Wade Brooks, Creative Commons License)

Should we fire bad teachers?

March 16, 2010

Newsweek ran a story last week titled Why we must fire bad teachers. It makes some good points, but also evades most of the conversation that needs to occur before real education reform occurs.

First, I agree that we need to fire bad teachers. I am generally pro-teacher, due to the fact that they have a difficult job that is extremely important. Most work long hours for a small pay check, and the work conditions can be pretty stressful. However, I am more pro-student than I am pro-teacher, and this is why I agree that we should fire bad teachers.

However, the article made no mention as to how to determine which teachers are “good” and which are “bad.” I spent three years evaluating college teachers, and I did not make much progress in determining what makes a good teacher—the best I could do is rely on a gut feeling, or to paraphrase Potter Stewart: “I know bad teaching when I see it.”

Some would argue that we should indirectly measure teacher effectiveness by using student test scores. However, I have seen very little evidence that these scores mean anything. This is largely because I rarely hear discussion about our goals. If we decide that our goal is for students to know as many facts and one-step algorithms as possible, I would believe that these test scores could accurately measure that. If our goal is to create students who are creative, caring, diligent, and thoughtful, I am very skeptical. In fact, there are trade-offs involved: the more time you spend on facts and simple algorithms, the less time you spend on having students be creative and thoughtful. If this is the case, using test scores to determine teachers’ competence levels could have the effect that we fire the good teachers who are nurturing creativity and thoughtfulness.

Until someone convinces me that we have a way of measuring our educational goals correctly (also, they should tell me what the goals are for their school district), I am going to be very wary of firing teachers.

The article also makes a huge assumption, which is: the educational system is mostly good. If this is the case, I would agree that we should focus our energy on only having good teachers. However, “good” means “good within the context of the current education system.” Education is currently much more about obedience and compliance than creativity and thoughtfulness. Again, we need to decide if these are the goals we want (note: you can probably tell, especially if you have read previous posts, is that I think that we should have different educational goals that what we are actually teaching).

Note that there is no reason to believe that the skills required of a teacher whose main goal is compliance would be the same as if the main goal were thoughtfulness (although I think that there is still a large amount of overlap). For this reason, I would suggest that we reform all of education first according to our actual goals, and then sort teachers into “good” and “bad” piles based on the criteria of the system we want rather than the system we currently have.

I appreciate this article. I wish that the mainstream media would discuss education more. However, articles like these serve more to preserve the status quo than to effect change, and I know of very few people who are satisfied with the state of public education.