Tag Archives: calculus

Course Currency Model

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This semester (Fall 2017), I’m teaching Math 120: Introductory Calculus for the first time in a while. I’ve been debating introducing a standards-based assessment (SBG) system in the course but decided against it this semester. One of the things I really like about my prior SBG experiences is that SBG allows students flexibility when they need or want it. For example, in the past, I’ve allowed students to re-try quizzes during office hours to demonstrate a higher level of mastery on course topics. I have been trying to find a way to allow more flexibility in my Calculus course in a way that limits the amount of time and work it requires on my end.

I’ve decided to create a course currency system that I’m calling Calculus Tokens. I think the idea of “class tokens” came about from the “specifications grading” community and I’m not even sure where I first heard of this idea. Each calculus student in my course will begin the semester with 10 Calculus Tokens. These tokens can be redeemed, as needed, for a variety of things, like getting an extension on an online homework assignment, making up a quiz due to absence, or even re-trying a quiz in my office to improve a student’s score. Additionally, tokens can be earned by completing extra online assignments or by completing problems on my Study Guides before each test. If students have a balance of 8 or more Tokens at the end of the semester, they will earn a small (1% or 2%) grade boost on their score on the final exam.

I’m hoping this system works. My goals are:

  1. Allow students flexibility when they miss assignments due to absence;
  2. Allow students a re-assessment procedure for bringing up quiz grades;
  3. Allow students who need or want an extension on the homework a method of doing so that is transparent and fair to everyone.
  4. Let students have more ownership for the course, in the sense that I can be flexible in the direction that benefits them the most (and it doesn’t have to be the same for every student).

I’m not sure about the details of my implementation. The cost of making up a quiz, whether due to absence or just to re-assess, is 3 Tokens. The cost of getting a homework extension is 2 Tokens. Once students reach 0 Tokens, they can’t redeem any more for additional reasons.

I’ll let you know how it goes!

Documents related to SBG

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This afternoon I’ll be presenting about standards based grading as part of Teaching, Learning and Technology‘s “Faculty Showcase.” I’ll be giving a similar talk at an upcoming conference. In case you’re interested, here are some documents related to my presentations:

A lot of my FAQ document was borrowed from Joshua Bowman (@Thalesdisciple). This semester, I didn’t actually give my students the FAQ document — It turned out that after three semesters of SBG, my explanation to students about how our grading system works & why I think it’s a good idea has gotten a lot better.

Actually, that point speaks to one of the great things I’ve gotten out of using SBG: Implementing my system forced me to give deep consideration to exactly what mathematical content I want my students to get out of the course. Instead of debating if homework should count 10% or 12% of the overall grade, or what I should do if a student misses a quiz for an undocumented reason, or other administrative policies like those, the SBG system made my entire course planning process focus on the math stuff I want to teach and assess — instead of worrying about policies unrelated to mathematics (compliance with the rules, attendance, percentage breakdowns, etc).

Two Upcoming Talks on Standards Based Grading

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In the next month or so, I’ll be giving two talks on my implementation of standards based grading. (Okay, if you want to be really precise, that should say that I’m giving the same talk twice.) The first will be hosted by our “Teaching, Learning, and Technology” (@TLTCofC) division as part of their events for “Assessment Week”, and it will be on Wednesday, April 1st at 2pm. The second will be at SOCAMATYC  — the South Carolina Mathematical Association of Two-Year Colleges Annual Conference. They haven’t finalized their schedule yet, but the conference runs Friday 4/17 through Saturday 4/18. Thanks go to Frank Monterisi (@frank314) for letting me know about this opportunity.

Here’s a blurb about my talk:

In this presentation, we will give an overview of standards based grading (SBG) including helpful answers to questions of the form “What?”, “Why?” and “How?”. While an implementation specific to Calculus II will be discussed, the method outlined could be applied to courses in any discipline. If you’ve ever wondered about alternatives to traditional grading and how to avoid hearing the question, “What percent do I need to make on the final exam to get an 82% in the class?” then this is a great place to start.

Once I have put together my slides, I’m hoping to upload them here, along with some updated SBG documentation from my Calculus II course, like my current list of standards and the information provided to students about how the grading system works.

In a way, it feels a little strange to prepare a talk about standards based grading when I feel like the relative newbie to this topic. My entire system came about after many conversations and interactions with fellow educators on Twitter, and I am still indebted to them for all of their helpful support and guidance. In particular, I couldn’t have gotten my course running smoothly without inspiration from Frank Noschese (@fnoschese) and Joshua Bowman (@thalesdisciple). A quick google search just told me that Joshua gave a similar talk about his transition to SBG; I stumbled on his slides here.

Reflection on Standards-Based Calculus

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Our semester is wrapping up and we only have one more class meeting day after Thanksgiving. I’ve been teaching two sections of “Calculus II” using my standards-based grading system that I’ve mentioned before. I think I made several improvements this semester and I wanted to share them, along with a couple of things I’m still contemplating. But first, here are things I thought went well:

  • I really liked having my standards organized by Big Questions. This is probably something I could have implemented outside of my grading system. Somehow writing and organizing my list of standards gave me the motivation and time and priority to think about the take-aways I wanted my students to get from our course.
  • Last Spring, I had approximately 18 standards, meaning about one per week. They were large learning targets. Take, for example, the “Techniques of Integration” standard that encompassed a couple of weeks of class time spent talking about integration by parts, by trigonometric identities, by trigonometric substitution, by partial fractions, and so on. This semester, I wanted more standards that were more specific. I hit my goal of 30 standards for the semester and this number worked well. On the one hand, the standards were specific enough that students could focus on just one idea at a time. On the other hand, there weren’t an unreasonable number for me to assess. Roughly they correlated to one standard per textbook section, spanning about 1.5 classes per idea.
  • Originally, I had a “policy of replacement” where a score would be updated each time a problem was attempted. In some cases, this seemed to harsh, since prior good work was “erased” easily. In some cases, this seemed to lenient, because sometimes an easy problem would earn a high score, but replace more thoughtful work on a harder problem. This semester scores were defined as the average of the scores from the last two attempts. This also made picking problems for re-assessments easier on me since I wasn’t as concerned about having them all be exactly the same difficulty. It also means that a score of 4 means a student demonstrated a strong level of mastery on two problems of a particular type, and that seems to work well.
  • I limited the number of re-assessments to one re-assessment topic per week. For example, if a student were struggling with Taylor Polynomials, they could come in throughout the week and try re-assessments. In some cases, they would just solve one problem. In other cases, they might solve four or five problems, each time getting a little more of the correct solution. Previously I let them do 2 standards per week but I found two problems with this: First, some students would just always pick their lowest two scores and try them, without really ever focusing on a single idea and working toward mastering it. Second, having multiple re-assessments on multiple topics times multiple students meant my grading workload was higher. So, one per weeks seems like a more manageable number for them to work on and it makes my grading workload lighter. Lastly, since we had 30 standards (but only 16 weeks) this policy pushed them to demonstrate mastery on in-class assignments (quizzes, exams) without just punting them to re-assess in my office later on.

Two things I don’t have data on yet:

  1. This semester I tried assigning online homework, with the homework contributing 5% to the overall course grade. I found assigning just textbook problems (and not grading them) did not work well. Perhaps I was not very good at motivating students to solve more problems on their own? I haven’t taken a detailed look at homework scores compared with course standing, so I am not sure if homework correlated with success on in-class assignments or not. I also feel a bit “icky” about assigning and grading homework, given some of the research I’ve seen.
  2. The other change I made was I separated “during semester scores” from “final exam scores.” So 70% of course grades will come from a letter grade assigned based on the scores on standards that were accumulated during the semester and 25% of course grades will come from a letter grade assigned based on scores on standards that will be accumulated on the final exam. This breakdown was in response to some conversations with students from last semester who felt that the old policy (“average of semester score and final exam score”) was too strict. We will see how this works out and if there is much movement in pre-final letter grades to post-final letter grades.

I’m teaching calculus II again in Spring 2015 and I plan to continue using this system. I am still entirely undecided about trying it in Pre-Calculus. I have several worries about trying it in that course.

Postscript: Here are some links to some older blog posts about my SBG Calculus adventure:

https://blogs.charleston.edu/owensks/2014/10/09/big-questions/

https://blogs.charleston.edu/owensks/2014/08/18/list-reboot/

https://blogs.charleston.edu/owensks/2014/02/25/on-the-purpose-of-examinations/

https://blogs.charleston.edu/owensks/2014/01/09/sbg-faq/

My Feedback Experiment

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I’m trying something new in my exam grading process. I’ve been wondering for a while what sorts of comments to write on student solutions(1). My goals:

  • Convey to student what level of mastery their solution has demonstrated about the assorted topics
  • Convey to the student where any mistakes or errors were made
  • Avoid spending hours upon hours grading exams and/or leaving lengthy comments
  • Show positive & supportive sentiment

Recently, I’ve been writing questions instead of comments on exams. Some examples:

  • Rather than writing, “You forgot to use the quotient rule here” I’ll write instead, “How do we differentiate a quotient?”
  • Rather than correcting an antiderivative miscalculation, I’ll write instead, “Is the derivative of your answer equivalent to the integrand?”
  • Rather than fixing an arithmetical error, I’ll write instead, “Are you sure this should be 81?”
  • Rather than writing, “Your formula is wrong,” I’ll write instead, “What happens if we plug in x=4 on both sides?”

What I Wonder: I don’t know if what I’m trying is a good idea, a bad idea, or just totally crazy. When grading exams, I wonder (1) how to communicate mathematical corrections to a student and (2) how to be supportive of the emotions surrounding test-taking. I had a number of conversations this week with mathematicians about times they had really miserable experiences with feedback they got from their own math professors “back in the day.”

[It’s also interesting to me that we all seem to have stories of the form “The time my math professor made me cry was…”]

I really do not want to make any of my students cry about a calculus exam. I really do want to say helpful, supportive, thought-creating comments that help them move forward mathematically. I am trying to figure out how to balance both of these things. How do I say, “Your solution is wrong, but I really believe in you & your ability to mastery this material! Keep trying!”

Footnote (1): I’m completely ignoring what I know about the feedback & learning cycle. In particular, I am taking huge latitude here and ignoring the fact I know the research says I should be doing more formative assessment and less summative assessment. I’m also ignoring that I’m probably not going to be successful at my current quest because research has shown that regardless of what comments we give students, if we give them a grade at the same time, they tend to ignore the comments anyway. In fact, I’m going to ignore this so loudly, I won’t even track down the links to the ed journal articles I’ve read about this very thing.

Postscript. I went and hunted down some links to stuff I know about feedback. It all started when I asked something on Twitter:

Some stuff I learned:

and also, http://blog.mathed.net/2011/08/rysk-butlers-effects-on-intrinsic.html has a summary of Butler’s Effects on Intrinsic Motivation and Performance (1986) and Task-Involving and Ego-Involving Properties of Evaluation (1987).

Escaping the Lectureculture

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For years now I’ve been a reader of Robert Talbert‘s column Casting Out Nines hosted by The Chronicle of Higher Education. Last week he wrote a post (“Is lecture really the thing that needs fixing?“) that gave me a lot to chew on. Here’s where I find myself today:

  1. Lectureculture is a set of machinery that self-replicates and it has political, social, psychological, instructional, and institutional components. It is pervasive and I find it in the world all around me, and some of the cultural natives don’t even recognize its existence.
  2. When I run a course, my #1 goal is to help learners move from being introduced to a concept to understanding and displaying mastery of the concept. Lecture is not the most effective way to help learners*.
  3. If I do nothing but lecture in my classes, I am helping sustain lectureculture and I am not helping my learners toward mastery the best I can, in violation of my #1 goal.

My plan of action: I’m teaching “Calculus II” again this semester. Although I’m using a standards-based approach, I must fess up that last semester nearly all of our class time was devoted to either lecture or assessment.

I am a lectureculture native and it is hard for me to let go. But I have come up with two ways I want to add non-lecture content delivery this semester (that don’t involve me tossing out all of my old materials).

First, I plan to continue last semester’s “Madness Mondays.” On those days, I introduced my students to ideas not necessarily tied to our course. I wanted to pick topics that I thought would inspire curiosity or happy befuddlement in my students, so they would walk away wanting to know more about what they had heard. (Examples: The Cantor set. Hilbert’s Hotel. Countably infinite vs uncountably infinite). I hoped to approach these ideas using a type of moderated discussion, letting the students ask questions to each other and talk about what was perplexing, interesting, fascinating, confusing, etc.

Second, I was really inspired by a recent video by Jo Boaler about “Number Talks” and I plan to try doing a weekly “Number Talk” (or something like it) with my calculus students.

My husband asked me why I wasn’t combining these things under one umbrella. To me, they hit two different–but equally important–goals for my course that can’t be found directly on our syllabus. They are

  1. I want my students to develop an appreciation for mathematics outside of what will show up on their next exam. I want them to be exposed to the kinds of questions mathematicians ask. I want them to practice the difficult skill of speaking with others about mathematical ideas.
  2. I want my students to become more fluent in numeration. I want my students to practice looking at the same problem from multiple perspectives. I want my students to see mathematics as a creative endeavor and get away from the idea that what mathematicians do is “apply a standard algorithm, proceed the same way, get the right answer.”

[Many of my digital colleagues seem to use some type of presentation requirement in their courses to get at item (1.) above. While I think that having students present math problems, solutions, ideas, etc. to each other would help develop this skill, and other skills too, I remember how terrified I was as an undergraduate at the thought of standing up in front of people and I don’t think I could impose those feelings on anyone in my classroom.]

Hopefully I will come up with other ways to push back against lectureculture in my classroom.

Footnote:
*As I was writing this post, the following MOOC announcement appeared in my Twitter feed & seemed quite apropos:

 

Reboot of my list of standards

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I’m about to start my second semester of using a standards-based approach in Calculus II. One of the things I wanted to change was my list of standards. Last semester, I ended up with about sixteen standards. When thinking about improvements for this semester, I wanted to pull apart my standards in a different way and I wanted to have more of them. Also, another big goal I have is to offer a broader picture of what calculus is really about. I’ve decided to re-categorize my (now) thirty standards under some Big Questions. Here’s what I have so far:

  • What background skills are important before we begin?
  • What kinds of applied problems can we solve using integration?
  • What techniques can we use to evaluate integrals?
  • How can we add infinitely many things together?
  • When and how can polynomials be used to approximate functions?
  • How can we model phenomena if we know how they change over time?
  • What can we say about the motion of objects moving in more than one dimension?*

Here’s a Dropbox link to my current standards list: m220-f2014-standards.pdf (Apologies if this link isn’t stable; this is a working document undergoing continual changes)

* Thanks to Joshua Bowman for help with this last one!

 

On Improvement

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Our semester is rapidly winding up. I have about eight more course meetings to tell my students the things I want them to know before our Final Exams. Just as they are starting to reflect on the material we covered this semester, I am also reflecting on the things we covered this semester & all the things I want to do better next time.

Things I want to improve:

  • I need to break apart some of my “Calculus II” learning standards. I didn’t have a complete list at the start of this term, and I realize now I wish I had made them smaller than I did. (I had been afraid of having too many, so I overcompensated.)
  • I need to come up with a good “Missed Exam” policy. Since I switched to standards-based grading, I’ve focused on the current value of a student’s score. As such, some students have missed (skipped?) entire exams and have wanted to make up the exams on a later date. This has been extremely difficult on my side of things, since it usually means writing an entirely different test for them, grading it at a different time, etc. I am philosophically stuck with what to do. On the one hand, I want a policy that says “You must take the exam on the specified date, unless truly unforeseeable circumstances beyond your control occur.” On the other hand, if my idea is their grade ought to reflect their mastery of course material, and not “mastery of this topic with a deadline of Wednesday,” I am not sure how to implement such a policy.
  • I need to come up with a good “Schedule of Expectations.” Some students have been consistently behind the course, in terms of what problems they are able to solve. To help students in the future, I think it would be good to have some kind of date-to-learning-target function that tells them, “You should master this learning target before this date.”
  • I need to make grading quickly a bigger priority. I know I have gotten behind schedule on various assignments this semester. This is always an issue. Things pop up, kids get sick, cars need maintenance, and somehow “grading assignments by the next class period” is one of the first things I let go of when life gets hectic. I want to hold myself to a higher standard about returning work quickly.
  • I need to have on hand problems for re-assessment, so if a student wants to re-assess a particular topic I don’t have to think up new problems on the fly.
  • I’ve implemented something I’m calling “Madness Mondays” in Calculus II. I’ve been taking class time to talk about mathematical things that aren’t directly related to what we’re talking about in class. For example, today I spent a while talking about the Hilbert Hotel. I’ve really enjoyed this part of the week. I have been impressed by the curiosity of my students. I’ve also been really pleased about how great they are at asking interesting questions. I think that talking to them about this random assortment of topics has helped them get away from the idea that “the point of math class is to solve problems and get the right answer.” Instead, I hope they now see that one major point of math class is to get them to think about mathematical ideas, outside of the context of any particular homework problem. But what I want to do is formulate a complete list of topics for Madness Monday, from which I can pull ideas in subsequent semesters.
  • I want to learn ALL of my students’ names. I’ve always struggled with this. I made this a priority this semester, and I have learned a higher percentage of names this term than ever previously. But I’d really like to get better & learn all of their names.

The above isn’t a complete list. I always think of dozens of things I want to do better, so this is only a start.

The last thing I want to do better is I need to be less hard on myself. I think I am probably my worst critic. Often times I walk out of class kicking myself for messing up a problem, or for not explaining something the best way, or for not spending enough time on this or that, or … At the end of the day (semester?), I wish I could give myself a break. My goal should be gradual improvement over time, not 100% perfection in every class on every day in every semester and with every student.

On the purpose of examinations

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I have just finished the second round of mid-semester exams in my calculus courses. As I may have mentioned before, I’m teaching two sections of “Calculus I” using a traditional grading scheme and one section of “Calculus II” using standards based grading. Both courses encountered their tests last week and had them returned with feedback this week.

While grading, I mentioned the following on Twitter:

Joe Heafner then replied: 

 

Since then, I’ve spent quite a while thinking about my purpose of giving them exams. I have the sense that students think the purpose of exams, in way exams are traditionally graded, is to compile as many points as possible. Here is a common approach:

  1. Write something down for every problem, whether or not you know how to do it, because you might get partial credit points for having at least something right.
  2. If you know how to do a problem, do it as quickly as possible; so long as you get the right answer at the end, there isn’t a need to write clearly or keep track of notation or show your chain of thought. If the answer is “7” then so long as you have “7” on your paper at the end, you’ll get full credit.
  3. Once the exam is returned, look it over. Ask questions of the form, “Why did I lose three points on this problem?”

I really dislike every step of this approach, but I don’t think I have ever communicated this well to my students and I have not been able to reward a change in behavior away from this strategy when I’ve used traditional grading. On the other hand, with standards based grading, I’ve tried to put effort into motivating my students to follow these steps instead:

  1. Keep track about what you know how to do and where you still need work. If you encounter a problem that you aren’t confident you can solve, don’t worry about it until you have more practice, seek more guidance, or have more time to study that topic.
  2. If you know how to do a problem, show all of your work. This is your chance to show off! Write neatly, explain your reasoning, and demonstrate mastery of the process. Don’t fret if the answer is “7” and you got “8”; we are interested in conceptual understanding and will happily overlook inconsequential arithmetic errors.
  3. Once the exam is returned, look it over. Ask questions of the form, “I am missing something conceptual here. Can you help clear it up for me?” or “I wasn’t sure how to attempt this problem. After I have more time to work on it on my own, can we go over it together?“

I suspect that the two different grading schemes would result in very different types of solutions for process problems. One such problem on the “Calculus I” test stated, “Use the limit definition of the derivative to find the derivative of the specified function.” I was not very successful at getting my beginning calculus students to understand that I am more interested in the process they are using instead of just their final answer. Standards based grading has allowed me to have conversations during class about the reason we ask these types of problems and what constitutes a solution versus just an answer.

The purpose of giving exams in my courses is to allow the students the opportunity to communicate their level of mastery of the course material. I’m looking for their ability to demonstrate conceptual understanding and their fluency with the technical processes needed in various problem-solving situations.

I think that standards based grading has made it easier for me to explain this to my students. I don’t think my students see exams as an adversarial process where I am judging them or their abilities. It is my hope they see exams as an opportunity to show what they know, to discover what they still need to work on, and to give us both a clear picture of where we should go from here.

Postscript. One issue I need to work on in upcoming courses is motivating students toward mastery earlier. Some SBG students are a bit behind in showing mastery of the standards, and while this hasn’t been a problem for them yet, I expect they will begin struggling quite a bit very soon. Now that our course material is building on itself at a swift pace (integration by parts into improper integrals into the Integral Test for Convergence of Series) I worry that their [lack of] progress on our “integration by parts” standard will cause them difficulty keeping up with the course. My students have been great and I feel like this was my failure at putting together an accurate timeline of what they should know and when they should know it. I think maybe I focused too much on “you can always improve later” that the message “…but there’s no time like the present, so do it today!” was lost.

The Adventure Continues

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I am now four weeks into my adventure in standards based calculus for this semester’s “Calculus II” class. Over the last week, I’ve given the semester’s first round of exams, both in Calculus I (using a traditional grading method) and in Calculus II (using standards-based grading). All of my students have received back their exams with my feedback. In this post, I’m hoping to reflect on my experience with both sets of exams, and give an update on how things are going.

About the Exam Grading Experience
Something I’ve struggled with using a Traditional Grading [TG] system is how grading exams makes me feel. Sure, no one enjoys grading exams, but I’ve found it can be a really miserable experience. For instance, when I see a solution that has a bunch of algebraic errors, instead of noting, “This student needs more practice with algebra” I have thought, “I didn’t explain the algebra very well” or I wonder, “Should we have gone over more algebra review? Should I have assigned more homework problems on this topic?” etc.

A second thing that has bothered me is that while I can easily grade an “A” paper, and I can easily grade an “F” paper, it is somewhat time-consuming to assign grades to the in between cases. For example, on a problem graded out of 14 points, I have to make lots of decisions of the form “Is this solution worth 3/14, 4/14, or 5/14?” — and this feels really subjective. I also believe it sends the student the message “try to get as many points as you can” rather than “try to master this topic perfectly”

The last big thing that is bothersome about the TG system has to do with what happens after I hand back an exam. In the instance of a student who has done poorly, I have seen them stuff their graded exam into their bag, and it is never to be seen again. When students ask me questions about their exam, the questions are always of the form: “Can I have another point on this question?” or “Can I still make a B+ in the course?” This is unfortunate since I think better questions would be, “What idea am I missing that caused this error?” or “I missed a step in this line of reasoning, can you help me find where I went wrong?” or “How come 1/0 is undefined but 0/1 isn’t?”

Happily, my first round of SBG exams resolved both these issues. First, grading the exams was a lot easier on me, since I knew each student would have as many opportunities to re-demonstrate what they missed. Second, instead of figuring out if they “learned” 20% of the idea or 22% of the idea or 24% of the idea, I could simply suggest they practice more and try again later, so the exam grading process went much faster. Lastly, since I handed back the tests, the questions students have asked have been all about mathematical ideas, and not about trying to find the optimal point-getting strategy.

Beyond Grading
I’ve also gotten a lot of positive feedback from my SBG students. Several of them have mentioned that they appreciate having less pressure on exam and quiz days, since they know (a) their scores will be replaced later on and (b) they can bring up their scores at any time by doing a re-assessment. Also, I’m getting many more students during my office hours and I have a much better sense on where each student is with our material. This is great because I can offer better advice on how they can improve. I know that this student needs more practice on integration by parts, and another student is having troubles remembering all our trigonometric identities.

Quick Summary:

  • My SBG assessment is going faster than my TG assessment, (even though the number of problems I’m assessing per student has gone up substantially). The grading is much faster. Students want to learn how to do the problems after I hand them back, rather than just toss them out.
  • My SBG students seem happy with the way the course is going; many of them come to my office hours regularly and want to do more problems. They are asking better questions and no one has argued for more points or a better grade on anything.
  • I hope I am sending the important message that to be successful in mathematics, you have to get used to self-correcting. In other words, you don’t have to get a problem right the very first time; instead, the better skill is to have the patience and confidence to re-attack what you don’t know — even if learning it takes multiple attempts.
  • And now I really wish I had set up my Calculus I course with an SBG system, too.