Posts Tagged ‘homework’

Assessment Idea for Calculus I: Near Final Draft

August 18, 2014

Sorry about the two month hiatus—Dana Ernst sucked me into a great research project about games with finite groups.

I previously wrote about my plan for calculus I. Basically, it is this:

  1. I give the students a list of learning goals. These are much finer than I have done in the past, which means that there are many more of them.
  2. I give students quizzes in class.
  3. For each quiz question, the student solves the problem as best as she can.
  4. Here is the important part: after solving the problem, the student reviews her work and determines which learning goals she has met.
  5. She indicates exactly where she met each learning goal. If she does not claim a learning goal, she does not get credit for the learning goal.

This basic idea has not changed; I have decided to go for this to see how it works. I have made a couple of changes since last time, though:

  1. I change my learning goals (see below for a list).
  2. I am only requiring that they demonstrate mastery of each learning goal four times, rather than the six that I previously had. There just is not enough time to assess that much, considering that I try to give my students at least twice as many attempts as is required. I am able to cut from six to four by scaling down homework: I previously required at least three demonstrations on a quiz and up to three demonstrations on homework, but I have changed this to requiring at least three demonstrations on a quiz and up to one demonstration on homework.
  3. I change my quiz template to include a margin on the left side. This is where students will write their code for each achieved learning goal. They then need to circle exactly where the learning goal is met, and connect that circle to the code. This should make the quizzes easier to grade and easier to read (less messy). I think that I am not going to require that this be done in a different colored pen, either.

I think that is mainly it. I have included drafts of my learning goals and syllabus (sorry for being three weeks late on this, Robert) below. Please see my previous post to get an idea of what students will do with their quizzes.

As always: feedback is welcome.

View this document on Scribd
View this document on Scribd

Assessment Idea for Calculus I: Feedback desperately wanted!

June 25, 2014

I am planning an overhaul of Calculus I for the fall. I used a combination of Peer Instruction and student presentations in Fall 2012, and I was not completely happy with it.

So I am starting from scratch. I am following the backwards design approach, and I feel like I am close to being done with my list of goals for the students. Here is my draft of learning goals, sorted by the letter grades they are associated with:

View this document on Scribd

I previously had lists of “topics” (essentially “Problem Types”). These lists had 10–20 items, and tended to be broad (e.g. “Limits,” “Symbolic derivatives,” “Finding and classifying extrema”). This list will give me (and, I hope, the students) more detailed feedback on what they know.

This differs from how I did things in the past, in that I used to list “learning goals” as very broad topics (so they weren’t learning goals at all, but rather “topics” or “types of problem”). Students would then need to demonstrate their ability to do these goals on label-less quizzes.

The process would be this:

  1. A student does a homework problem or quiz problem.
  2. The student then “tags” every instance of where she provided evidence of a learning goal.
  3. The student hands in the problem.
  4. The grader grades it in the following way: the grader scans for the tags. If the tags correspond to correct, relevant work AND if the tag points to the specific relevant part of the solution, the students gets credit for demonstrating that she understands that learning goal. Otherwise, no.
  5. Repeat for each tag.
  6. Students need to demonstrate understanding/mastery/whatever for every learning goal n times throughout the semester.

Below are three examples of how this might be done on a quiz. The first example is work by an exemplary student: the student would get credit for every tag here (In all three of the examples, the blue ink represents the student work and the red ink indicates the tag).

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The second example has the same work and the same tags, but the student would not get credit due to lack of specificity; the student should have pointed out exactly where each learning goal was demonstrated.

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The third example (like the first) was tagged correctly. However, there are mistakes and omissions. In the third example, the student failed to claim credit for the “FToCI” and the “Sum/Difference Rule for Integrals.” Because of this, the student would not get credit for these two goals (even though the student did them; the point is to get students reflecting on what they did).

Additionally, the student incorrectly took the “antiderivative of the polynomial,” which caused the entire solution to the “problem of motion” to be wrong. Again, the student would not get credit for these two goals.

However, the student does correctly indicate that they know “when to use an integral,” could apply the “Constant Multiple Rule for integrals,” and “wrote in complete sentences.” The student would get credit for these three.

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I like this method over my previous method because (1) I can have finer grained standards and (2) students will not only “do,” but also reflect on what they did. I do not like this method because it is more cumbersome than other grading schemes.

My current idea (after talking a lot to my wife and Robert Campbell, and then stealing an idea from David Clark) is to require that each student show that he/she can do each learning goal six times, but up to three of them can be done on homework (so at least three have to be done on quizzes). I usually have not assigned any homework, save for the practice that students need to do to do well on the quizzes. This is a change in policy that (1) frees up some class time, (2) helps train the students on how to think about what the learning goals mean, (3) force some extra review of the material, (4) provide an additional opportunity to collaborate with other students, and (5) provide an opportunity for students to practice quiz-type problems.

My basic idea is that I will ask harder questions on the homework, but grade it more leniently (which implies that I will ask easier questions on the quizzes, but grade it more strictly).

I have been relying solely on quizzes for the past several years, so grading homework will be something that I haven’t done for a while. I initially planned on only allowing quizzes for this system, too, but it seemed like things would be overwhelming for everyone: we would likely have daily quizzes (rather than maybe twice per week); I would likely not give class time to “tag” quizzes, so students would do this at home (creating a logical nightmare); I would probably have to spend a lot more time coaching students on how to tag (whereas they now get to practice it on the homework with other people).

Let’s end this post, Rundquist-style, with some starters for you.

  1. This is an awesome idea because …
  2. This is a terrible idea because …
  3. This is a good idea, but not worth the effort because …
  4. This is not workable as it is, but it would be if you changed …
  5. Homework is a terrible idea because …
  6. You are missing this learning goal …
  7. My name is TJ, and you are missing this process goal …

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?

How much homework is enough?

March 2, 2010

As a tenure-track professor, my colleagues visit my class every semester to better get to know me. Aside from the fact that this is an evaluation with professional consequences for me, I find this to be a great practice when done correctly—in fact, I wish that tenured professors would make a habit of visiting each other’s classes. It is easy to become too familiar with one’s own teaching, and it is good to periodically have an outsider question your choices.

I recently had a visitor, and he was concerned about my homework assignments. It was a blessing that we had this conversation, because it reminded me about why I make some of the decisions that I do; I have internalized these, but it is good to make them explicit from time to time.

  1. I believe that any sort of upper-level mathematics that is worth doing is worth taking your time on. Upper-level mathematics is about ideas and problem-solving, not computations and memorization (for the most part). I believe that a well-written homework assignment should require that the student take time to complete it; it should require stops and starts, dead-ends, and sometimes flashes of inspiration. These problems cannot be churned out, factory-like, on a schedule. Because of this, I eschew the practice of giving proofs after every class period; rather, I prefer to give students a “cycle” (6 days—recall that my college/s is/are weird) so they can dwell, fail, try again, and ultimately succeed.
  2. I believe that the teacher’s job is to make him/herself expendable. I used to revel in the fact that my students would come to my office hours on a daily basis for help. However, I no longer think this is in the best interest of the student. While it does make me feel good to feel needed, I have (in the past) had a tendency to set up co-dependent relationships with my students, where the students never feel like they can do the mathematics outside of my presence. This, in my opinion, is the opposite of education, and I have been improving on this by leaps and bounds over the past eight years. My latest tool is cooperative learning, which has been considered a great success by my students. A second success is giving my students more flexibility in when the homework is done; again, I have found that giving students homework every cycle creates less dependence than making homework due every lecture—even if the amount of homework is the same in both cases.
  3. I believe in quality over quantity, if I must choose. I prefer to give fewer homework problems while expecting a higher quality (and giving a higher quality of feedback) than more homework problems of lower quality.
  4. I believe that mathematics homework is not just about mathematics; rather, this is an opportunity to improve writing skills, which will ultimately be used more than the mathematic skills for 99% of my students.

One of my main points is that I do not think that having homework due every lecture meshes with my values, but this does not mean that I do assign homework that is due the next lecture period. Rather, I reserve these homework questions for simple computational problems or problems that apply a definition.

These are ideas that I have come to in my years of teaching, although I make no claim that these are optimal. I prefer weekly/cycle homework to daily homework for proofs, but I feel like I could easily change my thinking on this if I hear a good argument. I would appreciate suggestions and other people’s rationales in the comments.

On creating lies

February 3, 2010

I typically write my own homework packets, rather than just selecting problems out of the book. I have several reasons for this:

  1. Students are forced to use the space that I provide. Since there is a bigger problem students not using enough white space, I can make most students’ assignments neater by forcing enough white space.
  2. Students cannot simply look up the solution in the solution manual; they need to figure it out or speak with someone about it (note: I still have the flexibility to assign these problems if I want them to be able to see the solutions, and I usually have some problems of this sort).
  3. I am not limited to the problems of a particular textbook: I can borrow/steal problems from other sources, and I can write my own.

Today I am going to focus on writing homework problems. Many of my problems are of the “Prove or disprove” variety, as opposed to the “Prove” variety. I think that this is important, since it forces the student to decide if the proposition is true. In a larger context, this forces students to think critically, since they are not told to blindly believe that the proposition is true. This is a habit I hope students build in my classes—to evaluate whether a statement is believable.

I have found a challenge in this, though: it is somewhat difficult for me to create “Prove or disprove” statements that the students are supposed to disprove. I create some based on common student misconceptions. For example, I would include a “Prove or disprove: (a+b)2=a2+b2 for all real numbers a and b” if I were teaching algebra. However, it is generally difficult to create statements that “look” true, but are actually false. With luck, I will be proficient at this by the end of the semester.

Course Collaboration Project—Part 4 (Homework Policies)

January 2, 2010

With goals and content in mind, I can now focus on how to best get the students to learn the material. One aspect of this is homework.

This is a proof-based course. My theory is that there are three things that need to happen if you are going to learn how to successfully do proofs:

  1. You must read a lot of proofs.
  2. You must write a lot of proofs.
  3. You must analyze the proofs you read.

The third point will largely be done in class, since I do not think I can expect students to know how to analyze proofs. I have several ideas for formats that will allow the students to read and write a lot of proofs:

  1. I might have students evaluate their own homework. Students would give me a photocopy of their homework, but keep the original for themselves. I would create a solution key/rubric. They would use the rubric to evaluate the homework outside of class; perhaps students could comment on the “differences, omissions, and additions” of their proofs compared to mine, and comment on how important these differences/omissions/additions are. Students would email me their evaluation, noting the strengths and weakness of their proofs. I would spot-check their work by using the photocopied homework.
  2. I might allow students to resubmit unlimited attempts on homework problems to me. Problems would have two possible grades: “Near-perfect” and “Incomplete.” Students would resubmit until they received a grade of “Near-perfect.” I would provided detailed comments on their proofs to help them with the next draft.
  3. I might have students evaluate other students’ proofs as part of their homework. I would create a packet of 3-5 student attempts at proofs. Students would be expected to contribute to class discussions on the proofs.
  4. I might have “homework committees.” This idea comes from from Patrick Bahls. Here, a committee of 2-3 students would look 1-2 selected problems from the homework assignment. This committee would look at all of the student solutions that were submitted, categorize the different approaches that students used, and discuss the relative strengths, weaknesses, and validity of each approach. The committee would give a short summary of what students did in class.

I think a combination of these approaches would work well to get students to read, write, and analyze a variety of proofs. I am leaning toward a combination of the first three approaches. I am planning on giving 3-5 problems that the students will self-evaluate each “cycle” (6 school days=1 cycle). Students would additionally get 1-2 problems that students would be allowed to revise as many times as needed. I would use these revisable problems to create the packets for students to evaluate. On the fourth approach, I am in agreement with Patrick that the homework committees might create more overhead than I care to handle.

I am strongly considering following Patrick’s lead and teaching the class LaTeX. I would then require students to use LaTeX on the revisable homework, which would make their revisions easier.

The one point that have not settled on: I would like students to give presentations. I have not yet determine how this should relate to the homework. I welcome input on how I should organize the course—on the subject of presentations, or any other aspect of homework.