One Win from 2020: Goal-less problems

February 24, 2021

Here is one thing that I have been happy with this year. I decided to give the students goal-less problems for quizzes this year. Here how the process would go in a simplified Calculus I class with the following three learning outcomes.

  1. I can take derivatives of polynomials.
  2. I can determine antiderivatives of polynomials.
  3. I can solve definite integrals using u-substitution.

A quiz would then look something like this: “A car is traveling with velocity t^2 kilometers per second.”

Note that there is no question here. With a goal-less problem, the students need to supply both the question an the solution. So a student might do the following:

“Question: What is the acceleration of the car at time t=1? Answer: Acceleration is the derivative of velocity, so we can compute the derivative to get 2t, which evaluates to 2(1)=2 kilometers per seconds-squared when t=1.”

The student would then submit this to a Canvas assignment that corresponds to the first learning outcome (“derivatives of polynomials”).

But the student could write a second question/answer combination as follows:

“Question: How far does the car travel from t=0 to t=1? Answer: We can find the displacement by determining \int_0^1 t^2 \, dt = \frac{1}{3}t^3 \mid_0^1 =\frac{1}{3} kilometers.”

The student could then submit this to a Canvas assignment that corresponds to the second learning outcome (“antiderivatives of polynomials”). Since this situation doesn’t lend itself to u-substitution, the student could submit question/answer combinations to up to two of the three learning outcomes.

Here is why I am happy I am doing this: one of my unspoken goals (unspoken to myself, even, even though it makes total sense) is that students should not only know how to use calculus, but they should also know when to use calculus. The fact that my students really struggle (in fact, I had to add an entry into my FAQ page about this) suggests that I shouldn’t assume that they automatically learn this without being asked to do something like this. The fact that students don’t struggle with this after the first week tells me that this is useful.

Note that students don’t struggle coming up with questions for material that we learn in the third week of class. This suggests to me that the reason why they are able to develop questions after the first week is the fact that they are being told that knowing when to use calculus is important—it isn’t just that they understand the material from the first week better. They really seem to be telling themselves that part of learning material is learning when to apply it.

I am going to add this to my tool belt. I could see this being done in a bunch of courses. I could imagine giving the students a group in abstract algebra. The students could write questions like “What is the order?” or “Is this abelian?” My colleague has done similar things in linear algebra: he will give them a linear transformation. The students might then determine the kernel, the image, the eigenvalues/vectors, whether it is one-to-one/onto, etc.

I think that I ultimately like this because it breaks the input-output response cycle. The students have to think more—just a bit more—for themselves, but in a way that is not too burdensome.

FAQ Page

February 17, 2021

A big way I have improved my teaching is to provide more documentation. In particular, I have continued to provide detailed Specifications on how the grading will go, and I am now creating serious FAQ pages on Canvas (my LMS).

In particular, one issue with grading via some sort of Mastery Grading is that students aren’t used to computing their grades. I have taken care of this below by include screenshots that instruct students who to compute their grades. Here is the process I used to make them.

  • I change “Test Students” grade in Canvas to be an example of a particular grade.
  • I go into Student View on Canvas, and navigate to my grade.
  • I scroll to the bottom.
  • I open a PDF of the syllabus and shrink it so that it doesn’t cover any of Canvas’s grade information.
  • I take a screenshot.
  • I open the screenshot in gimp, and I draw in to direct students to the relevant information.

This whole process probably took 30 minutes.

Other features that seem to be helpful: I created Canvas Assignment groupings that exactly match how students will be graded, and I put “THIS WILL BE USED TO COMPUTE YOUR FINAL GRADE” in the name of the grouping.

None of this is novel or hard, but it seems to have helped the students. Plus, I can easily answer questions by directing them to the FAQ page of Specifications, saving me time in writing emails/Piazza messages.

A sample FAQ page is below.


Q:  Who belongs in this course?

A:  You!  If you are in this class, you belong here.  If you think that you don’t, please talk to me.  However, I am planning that you will be successful in this course, and I willing to help you with this.

By the way:  I fully expect that everyone else will treat you like you belong in this class, too.  Likewise, I expect that you will treat other people like they belong in the course.

Q:  What will a typical day of class look like.

A:  Very roughly, we will:

  • Spend an hour learning the new material.
  • Take a break.
  • Spend an hour reviewing old material.
  • Take a break.
  • Spend an hour working on projects.

Q:  What will my homework usually be like?

A:  Again, VERY roughly, you will work for about six hours.  Most students will roughly do the following:

  • Spend two hours learning the new material via the Tutorials.
  • Spend two hours on Pacework.
  • Spend two hours on DoMs (Quizzes and Projects).

This will vary by night, though.  Sometimes Tutorials will go by quickly and you will spend a lot more time on Pacework.  Other times, the Tutorials will take longer, and you might spend less time on the Quiz.

Q:  How will I communicate with my classmates if we aren’t in the same room?

A:  Most students will use a combination of Zoom, Canvas Collaborations (on the left-menu of Canvas, which is very similar to Google Docs), and Piazza.

Q:  What are the expectations for using Zoom?

A:  You will be using Zoom a lot.  The basic flow is this:  when we are in a large group setting, you should have your camera off.  When you are in a breakout session, I highly recommend that you have the camera on. 

I will mostly take questions via the comments in Zoom (feel free to only send the question to me).  The reason for this is that I want to make sure that everyone has a chance to ask questions (some students are much more likely to type questions than verbally ask questions).

Q:  How should I submit my Tutorials?

A:  Most students will take a picture of their work done on scratch paper.  This does not need to be fancy.  Try to make it a bit readable, but don’t stress out too much over that, either.  There are no Specifications, and the main goal of uploading your work to Canvas is so that you can see the solutions.

Q: How do I sign up for to do the Pacework?

Go to (Links to an external site.) and get a a free account.  Once you are there, look for Course ID:  XXXXX; the enrollment key is xxxxxxx.

Q:  Where do I go if I need help setting up technology?

A:  Our IT Department is well-trained on how to support you.  You can email them at [email address] or call them at [phone number]

Q:  What are the expectations for working online?

A:  First and foremost, act according to the Benedictine values.  If you are kind and listen, that will solve most problems.  Also, though, be brave:  ask questions that you have; both you and the class will benefit.

Q:  I am really stressed out about this class/CSBSJU/life in general.  What should I do?

Start by contacting me via Piazza (make it private to me).   I might be able to respond to you there, set up a Zoom call, or refer you to someone who can better help you.

Q:  How do I use Piazza?

A:  I am about to show you a video (a friend of mine actually made it).  His school uses “Blackboard” instead of Canvas, so just think “Canvas” whenever he says “Blackboard.”  Here is the video.   (Links to an external site.) Here is another video (Links to an external site.), although the first minute on signing up may not be relevant, since Piazza is integrated into Canvas.

Q:  How do I use Perusall?

A:  Here is a video.   (Links to an external site.)

Q:  I didn’t get credit for an assignment.  What can I do?

A:  Generally speaking, you didn’t get credit because you did not fulfill all of the Specifications for the assignment.  You can usually resubmit these assignments.  If you failed to meet a Specification because you did not correctly do the calculus we are learning this semester, you will not be charged a token to resubmit; if you failed to meet a Specification for some other reason (e.g. you submitted the assignment late, you didn’t write in complete sentences, etc), then you will need to spend a token to re-submit.  

Q:  Can I change the information given in a Quiz?

A:  You can add to the information—it will often be useful to provide a context—but do not change the information.  There are several reasons for this.  First, changing the information might make the problem too easy or too difficult (part of my job is to find the Goldilocks problems for you).  Second, it makes things much tougher to grade, which means that it takes longer to get you feedback.

Q:  I think that I know how to solve Quiz problems, but I don’t know how to ask the questions.  What should I do?

A:  Go back and look at the appropriate Tutorials, Daily Worksheets, and textbook chapters for the Learning Outcome you want to write a question for.  See what types of questions they ask, and mimic them.

Q:  Can I submit an old Quiz to a new Learning Outcome?  That is:  if I take a Quiz and then learn about a different Learning Outcome a week later, can I re-do the old Quiz to demonstrate the new Learning Outcome?

A:  Yes.

Q:  I am submitting a Quiz to a DoM Assignment for a second time.  If I submit the second assignment, will you still be able to see the first assignment?

A:  The teaching staff is able to see all assignments that you have ever submitted.  Unfortunately, students can only see the most recent assignment (I don’t know why they do it this way).  I recommend that you save your submitted Quizzes to your computer for your records, but feel free to contact Bret if you want an old submission.  I recommend leaving a comment that there are multiple Quizzes to be graded so that we are less likely to miss the first Quiz.

Q:  What are the tokens?  Can I trade in unused tokens at the end of the semester to raise my grade?

A:  Tokens are an artificial currency to allow for the fact that sometimes life doesn’t go as planned.  They can be traded for extra attempts at an assignment.  You only start with a limited number (five) because they are not to be relied on, but rather you should take comfort that they are there if you get a flat tire/have a really busy week in your other class/simply forget to do an assignment.

Your final grade for Math 120 will be based on the amount of mathematical knowledge you demonstrated by the end of the course.  As such, tokens can only purchase extra attempts at demonstrating what you know—they won’t raise your grade by themselves.  Any unused tokens at the end of the semester just evaporate and cannot be used for anything.

Q:  Is there a way for me to get more tokens during the semester?

A:  Yes—be the first person to let me know where I have made an error on Canvas.   You must be specific enough that I can easily find and correct the error.

Q:  I need to revise a Quiz or Project.  How long do I have to do this?

A:  If you got a grade of T, you have 24 hours (as stated in Specifications.pdf.  If you got a grade of R, you can submit it until the end of class on the last day of the block.

Q:  How do I figure out my grade?

A:  Do the following:

  1. Go to Canvas/Gradebook, and scroll all the way to the bottom.
  2. Open up the syllabus and scroll to Section 9.
  3. See how your scores in Canvas (marked USE TO DETERMINE YOUR FINAL GRADE) correspond to the grades in the syllabus.  Note that the percentages in Canvas do not matter at all—just look at the counts.

For example, suppose that your Canvas/Gradebook looks like this (note the syllabus in the upper right).


Try to determine the grade for this student.  Here are the important parts you need to consider (marked in green).


This student would get a B.  They do not have enough  Pacework Assignments (in red) and DoMs (in blue) to get an AB, but they have all of the other requirements for a B.


Similarly, try to figure out what grade this student gets.


This student gets a CD since they do not have enough Pacework assignments for a C.   This is unfortunate, since they could have gotten an A if they had kept up with the Pacework.


Federal Definition of a Credit Hour

February 10, 2021

Partially to communicate (and justify) my expectations for the workload for my classes, I have started referencing the federal definition of a credit hour. This basically means “three hours of work per week for every credit,” so a 4-credit course should be 12 hours worth of work each week (usually 3 hours in-class, 9 hours homework for me).

I am finding that students are indirectly communicating to me that this is a new workload for them—they suggest that they don’t normally have that much work in their classes.

I will admit that I used to think that two-hours-of-homework-for-every-hour-of-class was a rule of thumb, and I may not have given enough work to satisfy the federal definition in some of my previous courses. But I get the feeling that it is rare for a course to give this much homework.

What has been your experience?

Discrete Math Modeling Planning

February 3, 2021

I suddenly have to teach a modeling course. I am excited about this, but it is a lot of work—particular for someone who doesn’t know a lot about modeling.

There were two decisions I had to make right away.

  • Will this be a class on models, or will it be a class on modeling?
  • How active will this class be?

These were answered simultaneously, in some sense. First, some background.

A course on models teaches students a bunch of models (linear exponential, etc), and then has the students apply the models to different problems. A course on modeling starts with messy problems, and works from there to figure out how to make sense of it. Grading in a course on models is straightforward: you have a good idea what to expect from a student, since they will most likely be applying a linear model when you are working on the chapter on linear models. Grading in a course on modeling is surprising. The students might apply a variety of different models, and their assumptions might be very different.

Both courses have their place, but I think that a course on modeling is more appropriate in my liberal arts setting. So I am looking for messy, real-world problems that students can approach. One problem, which I am already using in Calculus II, is “Use mathematics to predict how many people in [insert geographic region here] will have Covid on June 30, 2021.” I also ask about cumulative cases. Questions like this will be at the heart of my course.

This also requires students to be very active, since this course is about the modeling process rather than mastering known modeling techniques. This aligns with my goal to up the levels of active learning in my class (which have been good for about a decade, but I can still fall into lecturing too much sometimes).

There is some trickiness here. How are students suppose to learn about what models are possible? What will the assignments look like? What will grading look like?

I will start to answer the first here: I am hoping to introduce models by some combination of

  • Working through some models together, with me making suggestions (and giving mini-lectures where appropriate) when students get stuck.
  • Having the students work in different teams to create models, and then having them share the results. They will likely do different things, which will help the different teams learn about new techniques (I will be coaching the individual teams to help them refine their ideas, of course).
  • Introducing models as they would be done in a course on models (I am not a purist). I will likely introduce several models in the first week of course without practicing them a lot. This will introduce them to some tools.

As always, I am looking for advice for this course—it you know of messy, real-world problems that would be good for a non-calculus-based modeling course, I would love to hear them!

Discrete Mathematical Modeling

January 27, 2021

Our semester has started now, and I am teaching on the block schedule again (three hours of class on MTRF, and I have one hour of office hours each of those days, too).

After years of being good, I have over-committed myself this semester. This is only partially my fault. In addition to my usual teaching and chair duties, I volunteered to lead two student research projects, which I love doing. I am also teaching one extra class that was canceled due to low enrollments—four students would have graduation issues if we didn’t do this, so I am happy to do this, too. We are also hiring, and I am chair of that committee; I love that work, too.

All of this would be manageable. However, I recently found out that I also need to teach a third course this semester, and that is what is putting me over the top. The teaching isn’t so much a problem, but rather I don’t have much time to plan it.

The course I am teaching is brand new (no one has taught it before), and it is called discrete mathematical modeling. While I am a bit stressed about finding time to plan it, I am really looking forward to it (I also know that it will get done. Because it has to get done.). We are going to talk about optimization, discrete dynamical systems, and modeling using probability, and the only prereqs are a good high school math background (in particular, calculus is not a prerequisite).

I am just starting to plan this. Here are my initial ideas prior to starting the proper way.

  • The focus will truly be on “modeling” rather than “models.” The latter is showing students a bunch of models other people have made (linear models, exponential models, dynamical systems, Markov chains, etc), and then having them apply them to situations. While there will be some of that, I want the course to be more about the process of creating the models. This is half-baked (the execution, that is, not the idea), but my hope is to provide students with semi-messy situations, have them come up with some sort of mathematical model, and then perhaps show them a standard model for that situation.
  • In-class will be 90% having students create models and compare the results. The remaining 10% are showing them “my” models (a standard way of modeling the situation).
  • Specifications grading seems appropriate here. I need to figure out a set of specifications for modeling reports.
  • Several places have suggested that it is a good idea to quiz students on the basics of the standard modeling techniques, since students otherwise will tend to use the same modeling method repeatedly.
  • I want to incorporate some game theory (model the Cold War!), but I am going to try to stop myself; we are in a condensed block schedule, and I don’t have much time to plan it. I need to keep this simple.
  • The main technology will be spreadsheets.

One last cool thing is that this course has a Justice theme, so roughly 25% of the class should be about Justice. This is a helpful constraint to narrow down my choices, and I am also happy to learn more about Justice. I have already benefited from the MAA’s book on social justice (paid for by my work on the AMS Math Reviews).

My main sources that I am going to draw from are:

More than usual, I welcome suggestions on anything related to modeling.

Students’ Time Management

December 17, 2020

I have had three students tell me the same thing, which I find endlessly intriguing. Here is the set-up: I taught two courses that met weekly this semester. For these classes, I had one due date per week where everything was due. For example, everything was due on Tuesdays at 1:30 pm for one of my classes (the class met on Wednesday). This was recommended to me as a best practice by several experts, since it provides a predictable structure.

It worked out well, but I had three students tell me something like this: “I have class every Tuesday until 1 pm, so I won’t have time to get everything done. Could I turn things in on Wednesdays before class instead so that I have time to finish the assignments?”

This confused me each time it happened, since the students basically had a full week to get things done. It turns out that the students did not realize that they would work on assignments before the deadline was approaching.

I am not writing this to condemn the students. In fact, one of the students who said this is among the best students at the school. Rather, I want to point out that students do not automatically know things like this; we need to explicitly teach them this. Indeed, my reply to each student was: “Why don’t you work on the assignments immediately after class. That way, you will have time to get help if you need it.” None of the students responded as if this was common sense; all of them responded as if this were a new idea.

I quickly and haphazardly put together a curriculum for my classes to help teach them these skills. Basically, I gave them a video each week on time management for students (also, how to study effectively). That is all I did. If I were a better teacher who didn’t just throw this in at the last minute, I would devote some class time to talking about these ideas. Maybe next semester.

Token Trouble

December 4, 2020

I am teaching Calculus II, which is going amazingly. I am teaching it three times this year (we are doing blocks). I am halfway through the course and loving it.

I created a bit of a problem for myself, though. I gave them 5 tokens to start the semester for being able to resubmit work, as usual. Their grades are based on their ability to demonstrate learning outcomes, also as usual, and the students have various Quizzes and Projects that they can do to demonstrate learning outcomes.

But it is possible that a student might miss some chances to demonstrate learning outcomes early on, and I wanted to create an alternate way for students to demonstrate outcomes. I decided that students could do an oral exam with me at the end of the semester, which is a solution I am mostly happy with. However, this is expensive for me, since it takes a lot of time.

My solution: I will make it expensive for the students so they don’t rely on this. Students can spend two of their five tokens for one oral exam, which limits the amount of oral exams I would need to do.

At the same time, I essentially wrote a textbook for this class (I produced probably 200–300 pages for the students), and I have typos all over the place. Because I sometimes charge students tokens for typos, I figure that it should go the other way, too. Thus, I charged a bounty for any sort of error in the class, and I give students a token for each error they find in my documents.

Some students are awesome at this. One now has 15 tokens.

The problem: I might pay for this in a week when the students start asking me for more oral exams than I planned. The good news is that I should be giving away a lot fewer tokens next semester, since I am fixing the errors as they come in.

Last(?) Post on “Weekly” Time Blocking?

November 12, 2020

This is a continuation of the last two posts. One of the reasons I like having a weekly template for my time blocking is that I get a new task or fail to finish a task assigned for the day, I rarely reschedule it for a long time in the future. This new way allows me to type fewer keystrokes when scheduling it.

For instance, suppose that I wanted to schedule something for tomorrow. In the previous way, I would do the following:

  • Type :tabnew to open up the tickler file for the 13th of the month (today is the 12th) (really, I would type :tabnew 13<tab> to autofill this).
  • Type :5 and f\ to go to the correct line (Line 5) and the correct place for my tickler file (this won’t make a lot of sense without having seen the file, but not that I have four keystrokes).
  • Type the to-do item.
  • Later, when time blocking for the 13th, I then need to find the right place in the file and paste (shift-p) in the item (which is already in my to-do list, thanks to cron).

Compare this to the new way:

  • Type /FRI go down to the part of the file for FRIDAY.
  • I then need to find the right place in the file and paste in the item (which is already in my to-do list, thanks to cron).
  • Type the to-do item.

In the previous way, I have 13 extra keystrokes. This isn’t a lot, but it does add up, and it just seems like a lot more effort, since I have two files to think about and I have to have the item in three locations (tickler file, top of time block file, and then correct place in time block file).

I suppose that I mainly just like being more efficient. In itself, this won’t save me much time, but it all adds up.

“Weekly” Time Blocking Update

November 5, 2020

Lest you are concerned that I got on Cal Newport’s bad side by time blocking weekly instead of daily, I need to report that he is cool with what I do. At the 10 minute mark of Episode 38, Newport describes his weekly planning schedule. This is essentially what I meant by “weekly time blocking.”

So I was calling “weekly time blocking” what he calls “weekly planning,” and his name is more accurate.

I am sure you all are very relieved.

“Weekly” Timeblocking

October 29, 2020

I started a new way of time blocking, and I like it. The old way was this: at the end of every day, I would plan what I would do for the next work in 30 minute increments. It would probably take me 10–15 minutes each day, which dropped to 5–10 minutes after I automated much of it.

My improvement was to start doing 90% of my time blocking for the week on the previous Friday. I changed my automation to put a template for my week on my to-do list every Friday afternoon. At the end of each day, I then very slightly tweak the time blocking for the next day, but there usually isn’t much to do.

This new process has several advantages.

  • I can automate better. Previously, I basically had two scripts: one for Monday/Wednesday/Friday, and one for Tuesday/Thursday. This is because these days tend to have similar recurring events (class meetings, office hours, etc) in academics. However, Tuesdays and Thursdays, say, don’t have the exact same schedule, so I would have to manually adjust the template some each day (I could have had separate programs for each day, but I have an aversion to having too many files. When I want to change something, I want to change it in one file, if possible). Now, I have the template be exactly correct for each day’s recurring events.
  • I can more intentionally figure out when tasks will get done. In the previous daily scheme, many times I would get a one-off task that showed up on a day when I had no time to do it. Now, I can see all of those on Friday, and then I can put them in slots throughout the week accordingly.
  • It ultimately saves me time. Due to both the batching and the automation advantages, I went from spending maybe 45 minutes per week time blocking to about 25 minutes per week.
  • I like it better. The automation is key, but I also don’t like having a large task to do at the end of each day. I would rather carve out a bigger time once per week, and then have just 1–2 minutes of work at the end of each day.

Strangely enough, the day I decided to do this, Cal Newport said on his podcast that you shouldn’t time block by week. I think that he thinks that it is too much. It could be that my 1–2 minutes of time blocking at the end of the day would make him happy (maybe this is only “weekly” time blocking), it could be that his advice is just wrong for my situation, or it could be that I will come to regret this and wish I would have listened to him. For now, though, I think it is a good think for me.

What do you think: is it better to plan most of your day’s schedule the day before (or morning of) or the previous weekend?