# 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 Success

I have spent more time lately pondering ways I hope to improve in subsequent semesters. Since my improvement list seems to be countably infinite, I thought it would  be good for my psyche to also come up with things that went really well this semester. So here are my Top 5.

1. I learned more names than ever.

After some conversations on Twitter in early January, I made the goal of learning as many student names as possible this semester. I started with the mini-goal of learning 100% of student names in my Calculus II course (enrollment: about 30). It took a couple of weeks, but I managed it! I think that it made a big difference & I’m going to challenge myself to learn 100% of next semester’s student names.

2. I tried something new & adapted to its challenges.

I implemented a standards-based approach in my Calculus II section. While there were a few speed-bumps, overall I thought it was successful. I am still formulating who I am & who I want to be as an educator, and I think my SBG approach is more aligned with my pedagogical goals than other things I have done in the past. I am also pleased that my students were willing to embark on the challenge with me and that we were able to have some honest reflection about the process of teaching & learning, outside of the context of our Calculus II content.

3. I worked to develop a coach mentality.
I have had many conversations with students about how it isn’t good for them to tie their emotional well-being to their performance on a math test. I really don’t want them to feel badly about themselves if they do badly on an exam. Instead, I want their exam grade to reflect their understanding of the material and I want them to take that information & use it to help their learning process. (Part of what SBG allowed me to do was make more clear what exactly a learning target is & what it means to master a concept.) I think I did a better job offering my students support and guidance than in the past.

Also, I think I was better at taking my own advice. While I don’t want them to be emotionally upset about their mathematical performance, I struggle with rating my skills as an educator as a function of student exam grades.  This is not a good idea! The point for all of us in my classroom should be to work toward gradual improvement over time & if we’re doing that, we need to be less harsh to ourselves. I think planting the idea “You are the coach & your role is not to score the points for the players during the game” in my own head helped me deal with class performance stress better.

4. I worked toward inspiring curiosity about math, outside of any particular course topic.
About once a week I took about ten minutes of class time to introduce cool/interesting/bizarre math ideas to my students. Several students became great question-askers: They came up with really thought-provoking questions. And I tried to bite my tongue and not provide the answers.

Some of the things we talked about were: the Banach-Tarski paradox; the Cantor set; the Hilbert Hotel; the Numberphile video (about the sum of all positive numbers being -1/12); and a crazy Slinky video showing in slow-motion what happens when you drop a Slinky.

These things were not necessarily related to what we were covering in class. But one of the things I feel it’s my job to do is inspire wonder & show how mathematics is really beautiful. I give myself an “A” for this task this semester.

5. Success with ongoing lactation
I am a nursing mom of a 9-month-old. Maintaining lactation while working full-time is a serious challenge. It’s exhausting. It takes a lot of time I could be doing other things. It takes extra calories (I’m always starving!). It required me to overcome humiliation on several occasions when people used keys to enter into my locked, private office — without first knocking — to find me pumping at my desk. (I cried every time.) It also took countless hours in front of my kitchen sink washing parts and bottles, getting ready for the cycle again tomorrow. And I honestly cannot recall the last time I slept more than three hours in a row (but it was more than 11-months ago).

A student of mine wrote me a really lovely letter last week. One of the things she mentioned was that I had inspired her with how I have balanced my work life and my family life so well, and she hopes to be just as successful at it in the years to come. (The letter is in my top desk drawer and really meant a lot to me.) It was really satisfying to be appreciated in this way, especially since I struggle with feeling imbalanced between work & family, and I regularly ponder giving up my career entirely just to have more time with my kids.

But it’s students like her that remind me that I cannot leave this career path because I feel somehow responsible for displaying that it is possible to be a mom-wife-mathematician-professor and enjoy life at the same time. I did not have enough role models like this when I was on the student side of the desk & I want to help change the demographics in that direction.

# 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!

THE BACKGROUND
Back in March, I wrote a post called “Want Some Free Red Pens?” on my dream for digital exam grading. In my ideal world, I’d remove all the paper from my office entirely. Having only digital copies of exams would be splendid since I could get a lovely potted plant to put in place of my institutional-looking filing cabinet. Last semester, I did accomplish my goal of grading an entire set of exams without using any non-digital ink. Now I finally have the time to tell you how it went.

The exam was for our “Introductory Calculus” (MATH 120) course. It was the third exam of the semester and I had about 30 students enrolled. I gave the same exam I would have otherwise — it wasn’t an online test. If you’re really interested, you can find a copy of the test here. I photocopied it like usual, and my students took it like usual. I did choose 1-sided copies over my usual preference for double-sided to help with the scanning task.

THE PROCESS

1. Write, photocopy, proctor, collect exam. Alphabetize exams by student lastname and remove staple.
2. Scan exams to PDF files using department’s Xerox machine; export as e-mail attachment to myself.
3. Use husband’s perl script to “pull apart” multi-exam PDF file into 7-page segments. Rename files “lastname-exam3.pdf”. Transfer each file to iPad and open in GoodNotes.
5. Use LaTeX’s “pdfpages” package to combine each annotated exam with a very thorough “Solution Key” (with comments, hints, suggestions, etc) at the end. Send each student an e-mail containing their exam’s feedback with the Solution Key & notification that official exam grade is available on LMS. [This was done to avoid FERPA issues about sending graded assignments, or grades themselves, over e-mail.]
6. Save un-graded exams in my filing cabinet in case any student wants to pick theirs up. (As it turned out, no one did.)

THE GOOD THINGS
Here are the things I did like:

• No crayon marks! No spilled orange juice! No paper shuffling! No page flipping! No running out of ink! Grading at home with a toddler is a tedious process, but being able to get in eight minutes of grading while also providing parental supervision was fantastic.
• Forced Solutions. By giving every student a full Solution Key, I was able to write things like “See Remark on page 5” instead of re-writing the same paragraph of comments over and over again. Also, I didn’t have to feel guilty about printing thirty copies of said Solution Key, and I knew each and every student had been given the chance to see the solutions. (Usually, I upload the Solution Key to our LMS, but not every student bothers reading it, which is weird.)
• Grading was Fast! During the “active grading” phase, I think it went faster than grading on paper. I didn’t have to spend time turning pages. I could Copy-and-Paste similar remarks from one test onto a different test. Because I didn’t need as much physical desk space to spread out, I was able to get in five minutes of grading here, four minutes of grading there, and so forth, so I think I was able to return the exams sooner than I would have otherwise.

THINGS NEEDING IMPROVEMENT

• Hello, Copy Room. With about thirty students and a 7-page exam, the scanning task involved around 200 pages. It turns out that our Xerox machine does not like it when you ask it to scan anywhere near this many pages at once. After trying to scan 8 exams at once (56 pages), the Xerox’s “brain” would get hung up mid-process and a machine reboot was necessary. After this happened twice, I realized that I could only really scan 28-pages at once. So I set up four exams, pressed “SCAN”, and waited three minutes; lather, rinse, repeat. Four exams taking three scanning minutes meant about half an hour in the Copy Room I would have liked to spend elsewhere. (Thankfully, this wasn’t a total time loss since I could work on other tasks while the copy machine whirred.)

A colleague let me know that elsewhere on campus, there exists a better copy machine that could handle this type of task more easily. But, accounting for walking to-and-from time, I am not sure this would have taken any less than thirty minutes anyhow.

• Returning Exams. It had been my plan to use the LMS’s “Dropbox” functionality to return the exams. Unfortunately, I lost over an hour of my life trying to get this to work — without any success whatsoever. We use a Desire2Learn product, and after consulting back-and-forth with my Instructional Technologist, we concluded that you cannot return graded work unless a student has submitted ungraded work first.

In other words, there is no way for me to return a PDF file to a student unless and until they have uploaded a (potentially blank) PDF file to me. So, basically, there is a way to “reply” to an uploaded student document, but there is no way for me to “send” a student an uploaded document first.

• Big File Sizes. One has to be careful about writing too many GoodNotes comments. GoodNotes didn’t do a great job of compressing the PDF file size, and our LMS refused to allow me to send any file over 2MB in size as an e-mail attachment. Some of the exams were over this limit (too many comments) and others weren’t. To be fair, I am not sure if this is more annoying because of GoodNotes or more annoying because of our LMS. I also don’t know if GoodNotes has gotten better at saving from a GoodNotes document to an annotated PDF and keeping the file size smaller.

CONCLUSION

In the end, I don’t know if I’ll try this process again anytime soon. The biggest time drainers were the Xerox scanning & learning what didn’t work. If I were to do this again, I might investigate a better scanning technology. I would certainly ask my students to submit a blank PDF file to the LMS Dropbox, so I could “grade it” and instead return to them their graded test papers. My students really liked having a digital copy of their tests — it meant that when final exam week rolled around, they didn’t have to dig through their course materials to find their test. So, maybe I will revisit this idea sometime in the future? I’ll let you know if I do.

# The Pregnant Mathematician Continues Teaching, Probably

Recent Events
During the month of June, I’m teaching our “Introductory Calculus (Math 120)” course. We meet five days a week for 145 minutes, with a total of twenty class sessions running June 5th through July 2nd. Our final exam will be held on Wednesday, July 3rd. A few days ago, my very compassionate students expressed interest in knowing the probability their final exam would be cancelled due to an unexpectedly early birth of my daughter. I’m sure their inquiry was based solely on concerns for our health and not at all from them hoping to escape the intellectual challenge of a cumulative final exam!

Also recently, my husband and I went on an “Expectant Parents Tour” of the hospital where I will be delivering. Our son was born in November 2010, but at a different hospital; this time, I will be delivering at a hospital that was still under construction then! Since they have recently opened their doors, their NICU [Neonatal Intensive Care Unit] is still a “Level 1” facility. This means that they are certified to care for healthy infants born at 36 weeks or later (measured from LMP) or 34 weeks gestational age (i.e., since conception). Infants who are born with complications, or who are born before 36 weeks, are usually transferred to another local hospital. It turns out that reaching “Week 36” of my pregnancy and having lots of calculus final exams to grade will coincide perfectly.

Some Probability
While on our tour, I started wondering about the chance that I would go into labor early enough that it would affect my summer class. Not only would this be an unfortunate inconvenience for my students, it would also mean that I would probably have to deliver at a different hospital since I wouldn’t be at 36 weeks yet. What’s the chance this happens?

First, I should point out that I’ve had a relatively uneventful pregnancy — thankfully, both my unborn daughter and I have been in excellent health (if not a little grumpy from having to share the same circulatory system and oxygen supply). Second, my son was born within a few days of his due date, a little on the early side. Third, I’m not carrying multiples, nor am I expecting any major complications as my due date approaches. So we will just assume that this is an average pregnancy as far as the medical issues are concerned.

One of the things I talked about in my original “Pregnant Mathematician” post was how due dates are calculated using Naegele’s rule. Also, there was a rather large (n=427,582) study done in Norway [See Duration of human singleton pregnancy—a population-based study, Bergsjφ P, Denman DW, Hoffman HJ, Meirik O.] that found the mean gestational length for singleton pregnancies was 281 days, with a standard deviation of 13 days.

Let’s assume a mean of 281 days and a standard deviation of 13 days. What’s the chance a woman goes into labor 251 days or earlier (corresponding to 36 weeks)? Notice that 251 is about 281+(-2.31)*13, so giving birth prior to 36 weeks means you’re about 2.31 standard deviations away from the mean. By the Empirical Rule, I know this would be quite rare: There’s less than a 2.5% chance!

We can use Wolfram|Alpha to compute the exact probability. Our input is the command “CDF[NormalDistribution[mean, stdev], X]”; in this case, we are taking mean=281, stdev=13, and X=251. Wolfram|Alpha returns a result of 0.0105081, meaning there is about a 1.1% probability that I will give birth early enough to impact my current students.

It’s Gonna Be How Hot?!!

According to tomorrow’s weather forecast, the heat index here will reach 110 degrees tomorrow. This is miserably hot for everyone, but especially if you’re pregnant. Although my actual due “date” is in the first few days of August, I’m sincerely hoping that the birth occurs sometime in July. August heat in South Carolina is no joke! Wolfram|Alpha comforted me with its computation that there’s about a 41% chance I will give birth before I have the opportunity to be pregnant in August.

I’m also going to assume my chances of a July delivery are even higher than this, since human gestation doesn’t exactly follow a normal distribution. While a measurable percentage of moms give birth two or more weeks early, nearly none give birth two or more weeks late. By that point, OBs usually induce labor because of declining amounts of amniotic fluid and concerns for the health of the newborn. I’m going to take this and a “fingers crossed” approach and assume that a July birthday is at least 50% possible.

[There’s a name for such a distribution; I thought it was a truncated normal distribution, but that doesn’t seem to be quite right. The statistician who told me the term isn’t in his office at present! Anyone know what it’s called?]

Postscript: Dear Students,
Regardless of when I deliver, I can assure you that your calculus final examination will occur as scheduled. I know lots and lots and lots and lots of people who really enjoy torturing unsuspecting college students with tough calculus exams, and it would be easy for me to cajole one of those people into proctoring your test! So, don’t fear: You will certainly have the opportunity to demonstrate all the calculus you have learned this summer & feel proud of your scholastic achievement upon completing our course — including its final exam. 🙂

# Digital Plan for Digital Action

It turns out that several people had some great suggestions about my wish for digital exam grading. I’ve decided to attempt it for my next Calculus exam, scheduled for Tuesday, March 26th. Here’s an outline of the plan:

1. Photocopy exams single-sided and unstapled. Place a copy of each exam into an empty file folder.
2. Subject unsuspecting Calculus students to grueling exam on these topics: Related Rates; Linear Approximation; Mean Value Theorem; Derivatives and Graphs.
3. Alphabetize exams as they are turned in according to course roster. For absent students, place blank exam where theirs should be.
4. Use department copy machine to scan all ~350 pages to a single PDF file and send it to me via e-mail.
5. Thank my husband profusely for writing pdftk bash script that will take the single PDF file and break it apart, at every ~9th page, and rename the files according to last name (keeping alphabetical order in place). If this works, I should end up with 36 PDF files where each student has a file called “Owens-Calculus-Exam3.pdf” or something similar.
6. Create Dropbox folders for the ungraded exam PDFs and the graded exam PDFs. Use GoodNotes to grade the exams on my iPad. Export the finished product back to Dropbox.