Tag Archives: calculus

Some Thoughts for My Students

Posted on by
2

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.

Introduction to Standards-Based Grading

How is standards-based grading different from traditional grading?

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.

What about the final exam?

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?

Your midterm grade and course grade will be the usual sorts of letter grades you are accustomed to. Here is how I will convert your mastery of the course standards into letter grades:

  • 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

Posted on by
10

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!

 

Digital Grading Follow-Up

Posted on by
4

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.
  4. Correct each exam, save graded copy as “lastname-exam3-done.pdf”, compile exam grades, and upload grades onto our LMS.
  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.

Even More Weather Data

Posted on by
3

I’ve had fun over the last few days chatting with colleagues, friends, and family about the March-related weather in Charleston. See my previous blog posts to find out the background of this information. Here I’ll outline two new developments that came up today. Again, all of this pertains only to the month of March and only in Charleston SC.

(1) When do we ever use calculus, anyway?

In an e-mail yesterday, Dan Jarratt remarked that he was surprised by the result that the today-to-tomorrow temperature change had an average (mean) of +0.037 degrees Fahrenheit. In other words, given today’s high temperature, on average we expect it’ll be about 0.04 degrees warmer tomorrow. This isn’t a very big difference, as Dan remarked; it was lower than he thought it would be. This made me wonder if I too thought this was a small temperature change.

I would guess that the hottest month in Charleston is August. (That is, August is the month that has the highest average temperature.) Also, I would guess that the coldest month is six months from August, so that would mean February. Assuming that the temperature shifts in a sinusoidal fashion, we’d get a nice sine function with a period of 12 months; a local maximum in August; and a local minimum in February. This led to the following question, which I asked my Calculus students today:

When is the temperature in Charleston increasing most rapidly?

First, we had some discussion on how we could rephrase this question into one about calculus. If the temperature is increasing most rapidly, that would mean that the slope of the tangent line is its largest; this would occur halfway between February and August. We agreed that this would be the month of May. Let T(x) be the temperature at time x. Graphically, if the temperature is increasing the most rapidly, then this is where T'(x) has a local maximum, so T”(x) changes sign from positive to negative. In calculus, we call this a point of inflection: a place on the graph where the tangent line increases most rapidly (should such a place exist). Alternatively, it is where the graph changes concavity — in this case, from being concave up to concave down.

(2) What would a numerical simulation tell us?
Both my College of Charleston colleague Jason Howell and Dartmouth professor François G. Dorais suggested my predictive model wasn’t great, and

Thankfully, Jason was willing to help write some code toward this goal. (His code is given below, written for MATLAB, in case you’re interested!) He gathered historical averages for high temperatures for each date in Charleston, restricted to the month of March, from weather.com. The averages were computed using data from 1893 through 2013. Given that the average temperature change was +0.037F and the standard deviation of this temperature change was 8.39, we can run a number of trials to answer the question

How many days in March can we expect to be at or below their historical average temperature?

Jason’s code simulated one million different March months, given a starting temperature on March 1st of 68 degrees F. Here’s a histogram of results:

resultsOf course, March 2013 hasn’t finished quite yet. But this histogram does tell us that if we end up with 23 or 24 days with an “at or below average” temperature, this isn’t exceedingly rare — or it isn’t as uncommon as I had thought it would be.

Here’s a graph of the daily average temperatures (based on the same historical data):

historical

Jason’s Code:

function y=weatherexp(start_temp, num_trials)

%set historical averages, from weather.com
hist_avgs = [62 62 62 63 63 63 63 63 64 64 64 64 64 …
65 65 65 65 65 66 66 66 66 67 67 67 67 67 68 …
68 68 68];
%initialization
num_days=length(hist_avgs);
temp_diff = zeros(size(hist_avgs));
num_days_below = zeros(1,num_trials);
%parameters for normally distributed daily temperature changes
%from months of March from 1893 to 2012
stdev = 8.39;
avg = 0.037;
%loop over trials
for j=1:num_trials
%set temperature for start of a simulated month/week/etc.
curr_temp = start_temp;
for i=1:num_days
%get temperature change
temp_change = avg+stdev*randn(1,1);
%new temp
curr_temp = curr_temp + temp_change;
%how far from average?
temp_diff(i) = curr_temp – hist_avgs(i);
end
%count number of days below average
num_days_below(j)=sum(temp_diff<0);
%temp_diff
end
%histogram of the num_days_below data
figure
hist(num_days_below,[0:31])
y = num_days_below;

Digital Plan for Digital Action

Posted on by
8

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.
  7. Disseminate graded exams and grades to students.

It’s likely my first attempt at this will take longer than nondigital grading. One of the things I will have to do as I go is come up with “Correction JPGs” for those errors that happen most frequently and store them somewhere on Dropbox. I think these should be easy to add to each exam using the “import JPG” feature of GoodNotes. Usually I estimate that grading will take no longer than 10 minutes per exam. For my 36 calculus students, this means regular grading should take me about six hours. Hopefully this digital grading effort won’t take too much longer than this.

For Step 7, I also need to find out about FERPA. Provided I have a “sign for consent” on my exam header page, is that enough for it to be okay for me to e-mail each student her graded exam? Alternatively, is there a way using our Desire2Learn-Dropbox (on our Learning Management System) to return the exams to the students in some easy way?

Wish me luck!

Project Based Learning

Posted on by
1

Our classes for the Fall 2012 semester start today. Thankfully, my teaching schedule doesn’t include Tuesdays, so I don’t start until tomorrow! I’m hoping to use Tuesdays this semester to work on several other projects, including adding more blog postings. Wish me luck.

This semester I’ll be teaching two sections (Section 05 and 17) of our Pre-Calculus class (Math 111) and one section (Section 05) of our Calculus I class (Math 120). Each class meets for 50-minutes per day on Mondays, Wednesdays, and Fridays, and an additional 75-minutes on Thursdays. The longer meetings on Thursdays will be useful in my current quest to incorporate Project Based Learning (“PBL”) into my classes.

I’ve begun the task of designing “Lab assignments” for students to work on, in small groups, during our Thursday meetings. Ideally they would be assignments that require no pre-lecture and ask the students to draw from their course content knowledge to form connections between ideas. By working together in a group, the students could collaborate (hopefully allowing for some peer instruction), ask questions, have a discussion, and digest what we’ve talked about during our other class meetings. According to my calendar, the students will have ten lab assignments over the course of the semester.

Yesterday I began working on the third lab assignment for Calculus. The topics covered earlier that week will be limits at infinity; asymptotic behavior; and continuity. I found an activity called “Carousel Game” from the NCTM‘s Illuminations series and modified it for my class. Here’s a brief overview of this lab:

  • Topic: Graphing rational functions
  • Goal: To correctly determine the equation that corresponds to the problem situation or graph
  • Technology Required: None allowed!
  • Warm-up: Vocabulary assessment, including: asymptote,  rational function, exponential function, end behavior, domain, range
  • Activity: Students will use a description or a graph to find the equation for twelve functions
  • Assessment: After finding the functions, students will find domain, range, vertical asymptotes, horizontal asymptotes, and all intercepts. This will be turned in and graded.

I also uploaded a copy of the lab instructions to my public Dropbox. If you are interested in seeing the entire lab, check it out here: http://dl.dropbox.com/u/59433434/120-Lab2.pdf. (Notice that it’s 120-Lab2, even though I mentioned before it is really our third lab — I start numbering things with zero.)

I’m hoping to reuse this activity in Pre-Calculus later in the semester, once we cover material about rational functions.