Students Figure Out Which Standards They Meet

April 16, 2014

I am starting to think about planning for Calculus I for next year, and there is an idea I would like to try: I want to stop labelling problems according to the corresponding standard, and put the burden on the student to determine which standards they met. I have tried this before (as have other people), but I would implement it different from how I did it last time.

So each quiz would go like this: I give them several (unlabelled) quiz problems. The students do what they can. When they are done, they submit their work. However, when they submit, we make some sort of a copy (perhaps a paper copy, perhaps just take a picture with the smart phone), and then the student takes one copy home.

At home, the student tries to figure out which standards she met on the quiz. For each standard, she writes up an argument as to why she met that standard. Specificity is key—the student would need to explicitly say where and how she met the standard. She submits this at the next class period, and this is graded as I usually do.

Here are the things I like about this idea:

  1. Students have to reflect on their work in order to get credit. This could lead to higher quality writing.
  2. Students would have to take ownership of their learning. They need to be aware of the standards they are missing, and make a concerted attempt to learn it well enough to be able to apply it on a quiz (including recognizing where it makes sense to apply it).
  3. Students can solve problems any way they like. As long as they can solve the problem using a standard, it counts. For instance, a linear algebra student might get “eigenvalue” and “determinant” credit for finding the eigenvalues of a matrix.
  4. Students are forced to really think about what the standards are and mean. There could be metacognitive benefits.
  5. I can ask more synthesis questions on quizzes; I do not need to isolate ideas for each question.
  6. Students no longer get the hint that the label provides (if the quiz question is labelled as corresponding to the “Tangent line” standard, then the student has a pretty good idea that he should find a tangent line at some point).
  7. It might give me room to have more standards (and more specific standards of the “I can do this” variety, rather than standards that are really topics, as in “Tangent lines.” David Clark encouraged me to make this transition last weekend).

Here are some potential problems:

  1. If the problems are too synthesis-y, then students won’t be able to do very many on each quiz. This might be fine, but it would be bad for a student who gets stuck and does not know where to start (on the other hand, maybe it would help teach students to start with something?).
  2. Students may try to shoehorn standards where they do not belong. This is what I would do if I were missing a small subset of standards.
  3. I am not certain I can write quiz problems that will give everyone the opportunities they need at the end of the semester. Students need different things, so I would have to have a lot of questions (note: this actually doesn’t need to be any different than how it is now; I can just provide straightforward, say, “Tangent lines” problems to quizzes if I need to. So this actually isn’t much of a problem).
  4. It forces students to be aware of what they have not yet demonstrated; this might be asking too much of some first-years.

I am on the fence about this, although I would really like to try it. Perhaps I could do both: keep the old way (with the labels) and do the new way. I could make that work.

What am I missing? What other advantages, disadvantages, and difficulties would this have?

A Google Appointments Replacement

April 9, 2014

I am planning on doing some oral exams in my classes, and I previously used Google Appointments to do this. Unfortunately, Google Appointments is no longer available.

Instead, I plan on taking Jack Dougherty’s advice and use to schedule the oral exams. Dougherty gives a very detailed explanation on how to use this service in the previous link. I will write about how it works when the time comes.

(Disclaimer: I have not been contacted by YouCanBook.Me, and I have no affiliation with them).

Office Hours Again

April 3, 2014

I wrote about office hours three years ago, and I have noticed that my office hours less attended than my colleagues’ (some of them, anyway). I used to have packed office hours, but that slowed to a trickle a couple of years ago.

This concerns me a bit. While I am happy that students might be learning on their own, I have somehow internalized the message that “being a good professor means having a lot of students at your office hours.”

But then I learned of something that might make me feel better. I had the pleasure of meeting Andy Rundquist (and Matt Wiebold) for lunch last week, and he commented that has not had many students in office hours recently, either. We talked briefly about why this might be. Here are some possibilities:

  1. I am somehow intimidating, and students do not want to come to my office hours. Or, even if I am not intimidating, I am sending some message that students are not welcome.
  2. Neither Andy nor I collect homework that is graded for accuracy.
  3. Both Andy and I use something akin to Standards-Based Grading.

I never realized it before, but my conversation with Andy makes me wonder if SBG and/or a No Homework Policy might naturally lead to a decrease in students coming to office hours.

For instance, I have found that while I have a smaller quantity of students in my office hours, I typically have a much higher quality interaction during the office hours. Students tend to come with specific questions about why they are stuck on a problem, or (better yet) specific questions about something they are just curious about. I remember this happening a lot less previously. Before, it seemed like there were mainly requests that I do homework problems (or problems similar to homework problems). So it seems like the No Homework policy got rid of students coming to office hours for the sole purpose of finishing busy work (I think this is a good thing).

[Edit 10:38 pm CDT: This is not just a matter of "the course is easier because there is no homework," which was my first thought of how to explain this. The students have closed notes quizzes on the SBG topics, so students still need to understand the material; they just demonstrate it on quizzes rather than on homework, which is harder to do.]

A plausible explanation for why SBG might lead to fewer students attending office hours is that students are being supported just enough to learn independently. When I used a Traditional Grading scheme, it likely was not clear what the most important ideas of the course were. I could see a student wanting more guidance if every detail in the course seems as important as every other detail (it probably did not help that I would typically respond with “Everything” when students asked what they should be studying for an exam). My hypothesis is that SBG gives students just enough guidance that they can determine what to study on their own.

This is a balancing act, of course: I do think that most everything that I do in class is important, and that students should know it. However, I would be willing to sacrifice students learning some of the course topics if it resulted in students learning the most important topics more deeply and becoming more independent learners. So I hope that this is what is happening.

Have other people noticed that office hour attendance is correlated with how you structure class? Can anyone think of any other explanation for the change in office hour attendance?

Reducing Stress with Tickler Files

March 25, 2014

I have been annoyed a bit this week with other faculty members who do not reply to email. My best guess is that these faculty members just lose track of the things they need to do (like respond to my email). So as a public service, I have decided to write about the single best thing that has helped me keep track of what I need to do: tickler file.

I learned about the tickler File from David Allen’s Getting Things Done.. A tickler file is essentially a system where you can write notes to your future self. Here are the basics:

  1. A tickler File consists of 43 folders labelled 1–31 and January–August (31+12 months=43).
  2. The folders labelled 1–31 correspond to the next month. So if today is March 25th and I know that I need to call Buffy on March 27th, I simply write “Call Buffy” (ideally with her phone number) on a sheet of scratch paper and put it in the folder labelled “27.”
  3. If I need to call Buffy on June 6th (and today is March 25th), then I write “Call Buffy on June 6th” and put it in the folder labelled “June.”
  4. I check the folder every day; today (March 25th), I checked the “25″ folder; tomorrow I will check the “26″ folder.
  5. On the first of April, I will check the “April” folder and distribute the notes to the appropriate 1–31 folders. I do a similar thing on the first of every month.

That is basically it. It is not complex, but it prevents me from forgetting about things; more importantly, it reduces my stress, because I know that I will not forget things.

Finally, I use this for things other than simple tasks like “Call Buffy.” I use it to remind myself to start working on a paper, to remember whose birthday it is, and to let me know what I need to do to prepare for class that day. It allows me to put off non-urgent decisions without worrying that I will forget about them.

My wife has started using my tickler file. She asks me to remind her of things that she needs to do if they are more than a month away.

This is a low-cost, high-reward system. You should start using a tickler file if you haven’t already.

Eliminate Meetings

March 20, 2014

+Vincent Knight posted this article on meetings from

I am mainly posting this so that I can find it later. I will be department chair in a couple of years, and I could be a committee chair sooner than that. I would like to remember this general philosophy.

I don’t want to go to the extremes that this paper does. For instance, “Meet only to support a decision that has already been made; do not use the meetings to make decisions” seems like a bad idea when the department is deciding whom to hire for a tenure-track job. However, I have found that many of my meetings could be replaced be electronic communication.

GMail Keyboard Shortcuts

March 13, 2014

I know that I am really late on this, but I started using the GMail keyboard shortcuts a couple of weeks ago. I don’t know if this is true, but it seems like I get through my email significantly faster than I used to.

Here are the only ones I regularly use so far. I really only use two labels—”Inbox” and “Action”—and some of the commands I only use in one label. I basically follow Merlin Mann’s advice about processing email.

  • Type “g” followed by “i” to go to the Inbox.
  • Type “g” followed by “l” to access a dropdown list of labels. I then press the down arrow until Action is selected.
  • In the Inbox, type “o” to open the first message.
  • In the Inbox, type “l” to access the Label menu and hit return when the Action label is highlighted. I use this when I know that I need to return to the email after I am done processing my inbox.
  • In the Inbox, type “{” to archive the current message and move to the next message. I use this when I am done with that email, either because I put an Action label on it already or because there is no action needed. (Edit on April 9, 2014: I have found that typing “[” accomplishes the same thing; this means that you do not need to press SHIFT, and you can use the same keystrokes in Inbox and Action).
  • In the Inbox, type “#” to delete a message that I do not want to save. I then press “o” again to access the next message.
  • Within the Action label, type “[” to remove the label from the current message and move to the next message. I use this when I am done with the required action for that email.
  • In any context, type “c” to compose a new message.
  • When composing a new message, type “control-shift-c” to include addresses as cc; type “control-shift-b” to include addresses as bcc.
  • Type “control-enter” to send a message.

It only took me about a day to become fluent with these eleven commands, and now I almost never have to use the mouse in GMail.

Does anyone else have any keyboard shortcuts that they use regularly in GMail?

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?

Linear Algebra Grading Plan

February 28, 2014

I described my structure for my linear algebra class last week. This week, I will describe my grading system.

First, recall that this is typically a sophomore-level class (although 9 of my 16 students are first-years). It is typically the first mathematics course that a mathematics student takes after calculus II. The purpose is to learn about linear algebra—including abstract vector spaces—and gently introduce the students to proofs.

The grading system hinges on what I want the grades of A, B, and C to mean. My linear algebra class does not vary from my usual interpretation: To get a C, you must demonstrate conceptual understanding of the material. To get a B, you must demonstrate both conceptual understanding and computational proficiency. To get an A, you must additionally demonstrate an ability to learn independently.

Below, I describe the graded portions of the class, followed by a set of rules that determines the semester grade for each student. I reserve the right to raise a student’s grade from what this rubric says, but I will never lower it. The rest of this post is taken directly from my syllabus (modulo some formatting and possible editorial comments); I would appreciate any thoughts that you have on how to improve this.

Here we go; it may be helpful to see my structuring if you have questions about anything.


Each time you present a Presentation Problem, I will make a note of it. It is better to correctly do a problem than to incorrectly do a problem, and it is better to correctly do a difficult problem than it is to do a relatively easy problem. Both the quality and quantity of the Presentations you give will be considered in determining your semester grade.

Part of the goal of this class is to help you hone your mathematical judgment. Because of this, I will limit the number of presentations that I confirm are correct (I will confirm 2 per day for the first three classes, 1 per day for the enxt three classes, and then 5 for the entirety of the remaining part of the semester). Please note that I will still do something during the semester to correct any misconceptions the class has—I just won’t necessarily do it immediately following the presentation unless the majority of the class wants to spend a Confirmation.

Giving a Presentation can never hurt your grade; the worst it can do is to fail to help your grade.

Daily Homework

You will be expected to submit you work for the Routine Problems and Presentation Problems at the end of each Monday and Friday during the first half of the semester. These will be graded on completion: as long as you make an honest effort at solving every Routine and Presentation Problem due that day, you will receive full credit. You are not expected to use \LaTeX for Daily Homework.

Since these are due at the end of the class, you may write on the homework with a marker (I will supply them, although I would appreciate it if any student brought his/her own). Your grade for Daily Homework will be based on the work you came to class with—the work in marker will not be graded.

Submitting Daily Homework can never hurt your grade; the worst it can do is to fail to help your grade.

Portfolio Homework

Each Monday and Friday during the first half of the semester, I will denote 1—2 of the Presentation Problems to be Portfolio Homework. This is homework that you are to write up nice solutions on \LaTeX.

Each problem will receive one of two grades: Complete or Resubmit. I will read your solution until I find an error. Once I find an error—it could be mathematical, grammatical, etc—I will mark it with Resubmit (or some suitable shorthand) and stop reading it. You can resubmit the same problem multiple times without penalty; if you eventually get a grade of Complete, you have been 100% successful on the problem, whether it was on your first attempt or your twenty-first.

You can only submit up to two Portfolio Homework problems per class day. This means that you should start working on them immediately—you do not want to have a lot of problems left to do during the last week of the semester, since you will not be able to submit all of them.

Submitting Portfolio Homework can never hurt your grade; the worst it can do is to fail to help your grade.

Mind Maps

You will be required to keep a “mind map” of all of the ideas described by the Chapter titles and Sub-Chapter titles in the table of contents of the course notes. Do this online using, name it YOURNAME239MindMap, and share your work with me by using the “Share” button. You are to update this mind map weekly; I will check to make sure that it is up-to-date periodically, although I may not announce when I do so. (This is not in the syllabus: the whole point of the mind map is that I want students to start to intentionally make connections among the different topics..)

We will be having quizzes during the second half of the semester. Each question will be linked to a Learning Goal (and clearly labelled indicating which Goal it is attached to), and will be graded either as Acceptable (if there are no errors of any sort) or Not Acceptable (if an error exists).

You will need to get five problems correct for every Learning Goal to be considered to have successfully learned a particular Learning Goal. There will be a few Learning Goals (This is not in the syllabus: the topics are LU and QR factorization; the only time I am going to mention them at all in class is when I put them on the quiz. I will grade those quizzes, and perhaps answer some specific questions of students who come to office hours. But students really are expected to truly learn these topics on their own if they want an A.) that are not covered in class; these are only for students who are aiming to get an A in the class, and they are meant to be learned on your own.

Calculators will not be allowed on quizzes. You should also not need them.

Submitting a Quiz can never hurt your grade; the worst it can do is to fail to help your grade. Because of this, the course policy is that make-up quizzes will not be given; you should plan on “making up” the quiz by doing well on later quizzes.

The quizzes largely take the place of most examinations. We will, however, have a final exam. There will be two components: We will have an in-class final exam on Thursday, May 15th at 8:00 am. The location is our usual classroom. Also, there will be a take-home portion of the final exam will be assigned in the last week of classes to be handed in at the final exam.

For students who wish to get an A for the semester, there will be a brief oral examination. You will receive the topic prior to the oral examination. This oral examination can only be done once you have at least 4 correct answers for each of the quiz Learning Goals (so this does not need to be done during Finals Week).

You are encouraged, but not required to do a project for this course. These projects will be mini-research projects. Your job is to find a problem (I will provide some possible problems), try to solve the problem, create a poster for it, and be prepared to answer questions about the topic and poster. Toward the end of the semester, we will have a poster presentation session for the class. The poster presentation will likely occur on Scholarship and Creativity Day.

Department Colloquia

Part of being a mathematician is to listen to mathematics. Because of this, you will be expected to attend some number of the Mathematics Colloquium, which occurs most every other Thursday at 2:40 pm (if you have a scheduling conflict, please let me know).


Here is how your semester grade will be determined:

To get a C for the semester, you must:

  1. You successfully answered at least one Query
  2. You presented (perhaps unsuccessfully) at least a few times
  3. You received credit on Daily Homework on all but at most three attempts
  4. You have grades of Acceptable on all but at most two Portfolio Homework problems
  5. You were successful on all of the Quiz Learning Goals in the Conceptual Learning Goal section
  6. You maintained a mostly complete and mostly up-to-date mind map for the entire semester
  7. You got at least a CD on the final exam.

      To get a B for the semester, you must:

      1. You successfully answered at least two Queries
      2. You many successful presentations
      3. You received credit on Daily Homework on all but at most two attempts
      4. You have grades of Acceptable on all Portfolio Homework problems
      5. You were successful on all of the Quiz Learning Goals in the Conceptual Learning Goal and Computation Learning Goal sections
      6. You maintained a complete and up-to-date mind map for the entire semester
      7. You attend at least 1 Mathematics Colloquium this semester
      8. You got at least a BC on the final exam.

      To get a A for the semester, you must:

      1. You successfully answered at least two Queries
      2. You many successful presentations of difficult problems
      3. You received credit on Daily Homework on all but at most two attempts
      4. You have grades of Acceptable on all Portfolio Homework problems
      5. You were successful on all of the Quiz Learning Goals
      6. You maintained a complete and up-to-date mind map for the entire semester
      7. You successfully complete an oral examination
      8. You successfully complete a project
      9. You attend at least 2 Mathematics Colloquia this semester
      10. You got at least a $AB$ on the final exam.

      I will also make a judgment call about the grades of AB, BC, D, and F.

Linear Algebra Class Structure

February 21, 2014

I was originally scheduled to teach abstract algebra this semester, but my section was cancelled due to low enrollment. Instead, I am teaching linear algebra, as we had higher-than-expected enrollment there.

The good news is that I can use the same basic course structure for linear algebra that I was planning to use for abstract algebra. The model is this:

  1. The semester is divided into two parts. The first part, from January 15th until March 31st, is where we learn the content. The second part is all review and assessment.
  2. For the first part, we do IBL-type presentations on Mondays and Fridays. Each day, we can do 4–6 presentations in 55 minutes. On Wednesdays, we review what we learned on the previous Monday and Friday. The reason why I chose Wednesday as the review day was so that students could have at least three nights to prepare for each presentation day.
  3. For the second half of the semester, we will alternate between assessment days and review days. Students will be able to choose what they want to review based on what they found most confusing from the first half of the semester AND from the recent assessments.

One advantage of having the Wednesdays saved for review is that I can use it for an emergency presentation day if a Monday or Friday class is cancelled; this has happened twice so far this semester, due to cold and snow (including today).

One problem that I have is that the course notes I wrote for linear algebra have 314 problems in them. Since I am compressing the presentation part into the first part of the semester AND only using Mondays and Fridays for presentations, I only have 20 presentations days for the 314 problems. This means that we need to average 16 problems per presentation day. I accomplish this by designating 6 problems as “Presentation Problems” (which will be presented, naturally), creating video solutions for another (roughly) 6 problems, and then leaving the remaining four-ish problems without solutions (these are mostly computational problems for which the students were given a video “template” on how to do the process).

It took a while to create the videos, but they are pretty much necessary for our course. This course serves as a very gentle “Introduction to Proofs” course, but the level of proof that is expected is of the “figure out how the proof follows directly from the definition” type. Since there are more complicated proofs that need to be done in the course, I would either need to lecture in class, have the students read the proofs from a textbook (which we don’t have), or create video lectures.

Also, given that we only have six Presentation Problems each day, I have developed a method of having the students volunteer for the problems that cuts down on the amount of work that I have assigning students to problems. My usual way of doing this is putting one essay quiz on Moodle that asks “Which problems would you like to present?” I still do this for my capstone course, in which we present 15 problems per day. For linear algebra, though, I put one quiz consisting of one multiple choice question for each problem that is to be presented. The students are given three choices: “I want to present this problem,” “I really want to present this problem,” and “I changed my mind—I no longer wish to present this problem” (a student who does not want to present does not need to complete the quiz for that particular question). I assign each question 10 points, 5 points, and 0 points, respectively. These points do not affect a student’s grade, but a there simply so I can look at the quiz summary to see each student’s preference quickly without much clicking. The drawback to this is that there is a lot more to do on Moodle (6 quizzes per day instead just one). However, I created all of the quizzes at the very beginning of the semester, and it didn’t actually take that long to do once I learned about the “duplicate” feature on Moodle.

We are just over halfway through the presentation days, and the class is going really well. I think that I have a remarkably good class, so I cannot really say how this class structure is working; I think that any class structure would work with this particular group of students. On the other hand, this shows that this class structure can work, given the right set of students.

Team Quizzes

February 14, 2014

Inspired by Eric Mazur (h/t Robert Talbert [Edit 2/15/2014: I also meant to credit Joss Ives, who intially planted this idea in my head a couple of years ago]), I decided to try team quizzes in my linear algebra class. Here is basically how it went:

The topic was “Subspaces.” I gave students 10 minutes to answer four multiple choice questions. Each of the four questions is about a subset W of \mathbb{R}^4. The students need to answer questions about the following four subsets:

  1. W=\{(a,b,c,d) \in \mathbb{R}^4 : abcd \geq 0\}
  2. W=\{(a,b,c,d) \in \mathbb{R}^4 : b=1\}
  3. W=\{(a,b,c,d) \in \mathbb{R}^4 : b is twice the sum of c+d\}
  4. W=\{(a,b,c,d) \in \mathbb{R}^4 : a+b > c+d\}

For each subset, students had to pick the best answer from the following list:

  1. W is a subspace.
  2. W is not a subspace, and the only one of the three axioms that fails is “W contains the zero element.”
  3. W is not a subspace, and the only one of the three axioms that fails is “W is closed under addition.”
  4. W is not a subspace, and the only one of the three axioms that fails is “W is closed under scalar multiplication.”
  5. W is not a subspace, and the only one of the three axioms that holds is “W contains the zero element.”
  6. W is not a subspace, and the only one of the three axioms that holds is “W is closed under addition.”
  7. W is not a subspace, and the only one of the three axioms that holds is “W is closed under scalar multiplication.”
  8. W is not a subspace, and none of the three axioms holds.

The students wrote their answers (just their choice, not an explanation) on two copies of the quiz. After the 10 minutes of individual work, students handed one of the copies of their answers to me and got in teams of four. The four students then repeated the same quiz collaboratively. I did this by putting the quiz on Moodle. Teams could keep answering until they got the right answer, although there is a penalty for each incorrect attempt.

For this quiz, a student received SBG credit for one “Linear Spaces and Subspaces” question if the student answered only one question incorrectly total between the individual and the team quiz. So a student who did perfectly on the individual quiz could have their team answer one questions incorrectly, a student who missed exactly one question on the individual quiz had to have a perfect team quiz score, and a student who missed two questions on the individual quiz did not receive credit.

Aside from the fact that I did not give the students enough time (alternatively, I gave the students too many questions), student reviews ranged from “this was good” to “this is freakin’ awesome.” No student said they did not like, and about a quarter of the class seemed desperate for more team quizzes.

It was a tiny bit tricky setting this up on Moodle. I probably forgot some details, but here are some things that I needed to do to get it to work:

  1. Make a regular Moodle quiz. This means that I had to create four separate questions, and then put all four questions on the quiz.
  2. Change “How quiz behaves” to “Adaptive mode.” This allows students to attempt the same question multiple times.
  3. Uncheck the box “Right Answer” under “Review Options” so that students are not shown the correct answer after each attempt (the second attempt becomes really easy if you were just told the answer).
  4. In each question, I think that I had to assign a penalty (0.1 works fine) to let me know how many attempts each team took. I think that I also changed from the default so that the answers were not shuffled.

I am going to try this again, but not until I finish the course content at the end of March (I race through the content so that I can have 1.5 months of review and assessment). This would be great to do for the entire class period, but I do not see how I can make it work reasonably well in less than 45 minutes (cutting back on the number of questions decreases the confidence that I have that a student understands, and also does not allow for students who do well on the individual portion to have a cushion on the team portion).

But this worked really well, and I am looking forward to building it into my courses next year.


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