# Reboot of my list of standards

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

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 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

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.

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.

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.

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.

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.

# Some Thoughts for My Students

I spent some time over the last several days trying to track down documentation about SBG/SBL. I wanted to find something to pass along to my students to address some of their questions or concerns, like, “What’s this SBG thing?” or “How will this work in our course?” or “How is this going to be beneficial?”

Thankfully, Joshua Bowman came to the rescue and sent me something he gives out to his students. It addressed some of his students’ frequently asked questions and it was a great launchpad to write my own. I’ll post it below. I kept stole his format and questions, but re-wrote (most of?) the answers as the apply to my own course.

You are probably accustomed to the following system: You do an assignment (like for homework, a quiz, or a test) and give it to your instructor to grade. After grading, it is returned to you with a score like “14/15” or “93%”. In our course, I won’t keep track of how you do on particular assignments; instead, I will keep track of how well you master specific mathematical tasks or concepts that are called standards. Once I see your work, my goal is to give you meaningful feedback: I want my feedback to tell you what you have mastered, what you should practice, and how what you have mastered relates to the goals of our course.

There are three major advantages to this system:

• First, it rewards mastery instead of a “hunt for partial-credit” strategy. On an assignment with five problems, I believe it is better to do three problems extremely well (and leave two problems blank) than to just write stuff down on every page hoping you’ll earn enough points.
• Second, I hope that it will allow you to see how to improve your knowledge of our course material. This system will allow us to track what topics you understand well, and also what topics you should spend more time working on. This way, if you seek additional help, you will know exactly what you need help with! Since your grade on a standard is not a fixed number — it changes over time — it is always advantageous to go back and fill in any gaps in your knowledge.
• Third, it allows us to be clear about what the expectations of the course are (namely, demonstrating an understanding of topics in Calculus II) and how well you are meeting (or exceeding!) those expectations.

How will I know how well I did on a test?

Each assignment will probably look similar to those you have seen in prior courses. When I return them to you, you will be provided with a rubric. The rubric will give you two kinds of information. First, it will outline what standards correspond to each problem you solved. Second, it will outline the level of mastery you demonstrated on that problem, using a scale of 0-4. Apart from the rubric, my hope is to offer additional feedback on your solutions that will help you toward your goal of continued mastery.

How do I know which standards will be tested?

On each quiz, you can expect to see material we covered in the previous week. However, as you know, mathematics tends to build on itself. So although maybe we didn’t talk about the Quotient Rule last week, you will probably still have to know how to use it this week! Before each test, I will provide a list of all of the standards the test will cover. Since our course is cumulative, although a particular test might focus on recent standards, you might encounter problems that require knowledge of previous standards from earlier in our semester — or even prior mathematics courses.

How often will each standard be assessed?

It will depend on the particular standard. Standards that appear early in our course will be assessed multiple times, since we will be using them (either implicitly or explicitly) to solve problems later on. Toward the end of our course, you might only encounter a particular standard once or twice.

Why can my score on a standard go down?

It’s important that your score shows your current level of mastery. Your score on a standard may go down because you’ve forgotten some of the material, or you were unable to apply earlier techniques in solving problems later on.

In addition, some of our standards are quite broad: For instance, one of them deals with “techniques of integration.” We will see many of these techniques in our course. So your score may go down if you show mastery of the earlier techniques, but aren’t comfortable with techniques that show up later on.

How can I raise my score on a standard?

There are two ways to have a score on a standard raised.

First, you can wait for that standard to be re-assessed later on. For example, some standards assessed on quiz questions will be re-assessed on test problems. Especially early in the course, when there will be many opportunities to reassess standards, this may be the easiest way to raise your scores.

Second, it will be possible to “retest” a particular standard by making an appointment to meet with me. At this meeting, you will demonstrate your understanding by trying new problems and then answering questions I pose to you. You can make appointments to retest up to two standards each week. You choose which standards you would like to retest and when. You can retest any given standard more than once, as long as you only retest up to two each week. Each “retest” will take 10-15 minutes. Please request an appointment for re-assessment at least one class day in advance; this will allow me to prepare materials for you. You can request an appointment simply by e-mailing me and letting me know which standard you have chosen.

How many times can I ask for a standard to be reassessed?

You can ask for any standard to be reassessed as many times as you want, subject to the limitation that you may only retest two standards each week. If you require multiple attempts on a particular standard, I might ask you to work on some additional problems first (potentially with my help) so we can clear up any knowledge gap more quickly.

Our final exam will be cumulative and will have problems reflecting standards we have encountered throughout the course. Not every standard will be directly assessed on the final exam (after all, we don’t want to make it too lengthy!). Also, by the nature of final exams, you cannot re-assess any standard after the final exam. Your course score on each standard will be decided as follows:

• If a standard does not appear on the final exam, your course score for that standard will be your score as of Reading Day. For Spring 2014, the date is Thursday, April 24th.
• If a standard does appear on the final exam, your course score for that standard will be the average of [your score as of Reading Day] and [your score for that standard on the final exam].

How will my final grade be computed from my scores?

• In order to guarantee a grade of A, you should attain 4s (or 5s) on 85% of course standards and have no scores below 3.
• In order to guarantee a grade of B, you should attain 3s on 85% of course standards and have no scores below 2.
• In order to guarantee a grade of C, you should attain 2s on at least 85% of course standards.

Plus and minus grades will be given based on how closely your performance is to a full letter grade. (For example, if you earn 3s on only 80% of course standards, and 2s on the other 20% of course standards, a grade of “B-” may be more appropriate than a grade of “B.”)

If I don’t like this method of grading, can I tell you about it?

Please! This is my first time using standards-based grading, and there are bound to be hiccups. However, I truly believe it will provide more helpful feedback and give you a better chance to prove your mastery of the material, so I ask that you at least give it a try, even if it seems strange at first.

If I have questions about how I’m doing in the class, can I ask you about it?

Absolutely! One drawback of this system of assessment is that you may have questions about your performance in the class. If you have questions or concerns about this, feel free to come talk with me and I will try my best to give you an accurate picture of your progress with our course material.

# An Adventure in Standards Based Calculus

Today was the first day of our new semester. This spring, I’ll be teaching two sections of “Calculus I” and one section of “Calculus II.” I feel like “Calculus I” is basically on autopilot; I’ve taught the class every semester for the last couple years and so I’m very comfortable with the course content. But this will be my first time teaching “Calculus II” in many years. (I think the last time I taught it was 2006 or so, at the University of South Carolina, using an entirely different textbook.) I’ve decided that I want to try something different & I am embarking on my first attempt at Standards Based Grading (SBG) — or as someone suggested today on twitter, maybe Standards Based Learning (SBL) is more appropriate?

Why Am I Doing This?
For the last few years, I’ve noticed a few things about traditional grading (TG) that I did not like. One thing that has bothered me is that a student can go the entire semester without ever solving a problem 100% correctly, yet still do very well in the course. For example, it is entirely possible to earn a “B+” grade, by performing pretty well on everything, but never really and truly mastering a single topic or problem type. I hope that Standards Based Grading helps me motivate my students to really try to master specific sorts of problems, rather than try to bounce around, hoping they can earn enough “partial credit” points to propel them to success. Really, I want to reward a student who gets four problems absolutely correct (and skips two problems) more than a student who just writes jumbled stuff down on every page. I think SBG will allow me to do this.

Another (related) thing that has bothered me: The point of calculus class is not to earn as many points as possible, doing the least effort possible. I will admit that I have used a TG scheme for years and years; I have no idea how many college-level courses I’ve taught. And I am pretty sure that I can look at a calculus quiz question, assign it a score between 0 and 10, and accurately give a number close to what my colleagues would give for that same problem. We might all agree, “Okay, this solution is worth 7 out of 10 points for these reasons.” But I think this gives the students the idea that the reason they should study is to earn points on the quiz — after all, 9 points is better than 7 points! Instead, I think the reason they should study is to understand the material deeper than they presently do now, and I think by assigning X points out of 100 sends them the wrong message.

Something that has really bothered me recently is that when a student is struggling with the course, I am never entirely sure what to tell them. I look up their grades in my gradebook; I see that they have an average of 62%; and then I try to give them advice. But what advice should I give? The 62% in my gradebook does not tell me very much: I do not know if this student is struggling because they need more practice in trigonometry. Or maybe they were doing very well, but bombed our last test because they got some bad news the night before. Or maybe they got L’Hopital’s Rule confused with the Quotient Rule. I want to be able to tell a student exactly what they can do to improve their understanding. By tracking each student’s mastery of particular standards, if a student comes to my office for extra help, I can tell that student, “Okay, it looks like you need extra help with [insert specific topic].”

Lastly, I would like to give students more low-stakes feedback about their understanding: That is, feedback without the worry that it will negatively affect their grade in the class. I will be giving a weekly quiz, and I will grade it, offer feedback, and return it to my students; then (eventually) their score on that standard can be replaced with a newer [hopefully better!] score. I will constantly replace their previous score on a standard with their current score on a standard. This way, if they are really struggling with (say) Taylor polynomials, I can communicate this to them early, they can seek extra help and resources, and then they can be re-assessed without penalty for their original lack of understanding.

What Worries Me?
I have lots of different things worrying me about this system! For example, since this is my first time teaching Calculus II in many years, I don’t know all the “common pitfalls” that my students will encounter, so I don’t feel like I’m going to see them coming until they’re already here. Also, I am worried that students will struggle to understand this method of assessment & won’t really “get it” about how they are doing in the course — or won’t take the opportunity to re-assess when they need it. Lastly, despite reading online that “before a course begins, start by making a list of what you want them to master (a.k.a, the standards)” I was unable to do this. I have the first half (or so), but I don’t know how good they are. Am I being too vague? Am I being too specific? Do I have too many? Too few? How difficult will they be to assess?

Some Resources
In my own course planning, here are links to resources I found helpful:

Wish me luck!