Tag Archives: precalculus

Standards-Based Grading in Fall 2018

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An Overview of My Semester

This semester, I’ll be teaching three different courses:

  • Pre-Calculus Mathematics (MATH 111), a course designed to review algebra and trigonometry for students who plan to take our (scientific) calculus sequence;
  • College Algebra (MATH 101), a course designed to cover algebra and function basics for students who will continue on either to MATH 111 (pre-calculus with trigonometry) or MATH 105 (business calculus); and
  • “Applications of Mathematics Across the Curriculum with Technology” (SMFT 516), a graduate-level course designed for in-service science & math teachers who are working toward an interdisciplinary M.Ed.

Due to enrollment challenges, my schedule for courses shifted in late July, so I spent a while during the summer trying to ditch my old plans for the semester and start over. Although I was planning to attend Mathfest in Denver, CO and give a talk in the session on #MasteryGrading based on my years of experience implementing standards-based grading in my courses, I must admit that before my trip I had no plans to use SBG in any of my courses this semester.

But then… UGH!!! INSPIRATION FROM PEOPLE. AND TOO MANY GOOD IDEAS.

So I attended every talk in the #MasteryGrading session at Mathfest. And wow, I got a ton of great ideas from all of the talks [stay tuned for future blog post] and, on a personal level, I really enjoyed our conversations, meals, and hang-outs outside of the session itself. (Thanks, y’all!)

Unfortunately a couple of days into Mathfest I realized I just couldn’t go back to traditional grading, so I threw out all my traditional plans for the semester and committed to myself that I would implement SBG/SBSG/MasteryGrading in 100% of my courses this semester.

Did I mention that I got home from Mathfest only 15 days in advance of my semester start?

The Nuts and Bolts of Fall 2018: SBG PreCalculus and SBG College Algebra

Rachel Weir, of Allegheny College, is maintaining a repository of course documents for secondary Mathematics courses that are using Standards-based grading, Specifications-based grading, or Mastery-Based Grading: Rachel’s SBG Repository

Both my Pre-Calculus and College Algebra courses are using the exact same setup. It’s very similar to Tom Mahoney ‘s (@MathProfTom) approach in his College Algebra courses. Here is the basic setup:

  1. I have written 25 standards for each course.
  2. Every time a student completes a problem on a standard, I will assess the solution using a “SGN Rubric” (see below). This assigns either 0 points, 1 point, or 2 points to each attempt.
  3. A student’s score on a standard is the average of their best two attempts.
  4. A student earns total points out of 50 possible (25 standards*2 points max). Together with work in an online homework system, this converts to a usual letter grade*.

For example, for any given standard, I will track a student’s progress as something like “0,1,2,1,1,1,0,2” and this student will earn a 2. After two perfectly correct solutions, the student isn’t required to answer problems on that topic again.

*My department requires a departmental-wide final exam that is graded using a partial credit, percentage system, and this exam must be worth at least 25% of each student’s course grade. So the actual grade computation is (75% performance on standards)+(25% final exam performance).

Links to Possibly Useful Things

Here are some links that I’ve freely distributed to my students. Perhaps reading them will shine some light on how I explained this system to them. Also, there are more details about the “SGN Rubric” I mentioned above and explanation about online homework & how it fits in.

Things To Do Later

I haven’t mentioned that third class (“Applications of Mathematics Across the Curriculum with Technology”). It runs double speed for half the time, during our Express II semester, and it doesn’t start until October. I want this course to be a project-based course, so I’m going to figure out some way to introduce specifications grading into my design. Robert Talbert (@RobertTalbert) has written extensively about his use of specs-grading and it’s my plan to steal as many ideas from his MTH 350 F18 Syllabus as I can. Our courses are very different, but he has so many clever ideas for his course skeleton. Once I write my syllabus, I’ll tell you about it.

Student Assessment of Learning Gains

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The College of Charleston has recently moved to a paperless, online-only course instruction evaluation system. The obvious benefit of the new system is that instructors are not required to use class time for student evaluations, and no students are required to shuffle sealed envelopes from one building to another once the evaluations are complete. I’m a big proponent of technology-enhanced learning and while I appreciate the time (and environmental) savings of the new system, I find myself frustrated with it. One problem is that every semester, there are problems with a very low response rate. Any of our Math 104 (“Elementary Statistics”) students can tell you about the issues with a voluntary response sample.

But the low response rate isn’t my main problem with the evaluations. In an ideal world, the course evaluations would provide statistically meaningful data that is useful in helping me guide course design, structure, and content. Unfortunately, the evaluations don’t do this. For example, one question asks students to rate (using a Likert scale) the statement, “The instructor showed enthusiasm for teaching the subject.” Yes, I am enthusiastic in my classroom (both about teaching and about mathematics), and I am happy that my students notice and enjoy my enthusiasm. But this doesn’t help me teach the course better. I would prefer student feedback on statements like, “In this course I learned to work cooperatively with my peers to learn mathematical concepts.”

Overall, my issue with the evaluations is that the questions posed are teacher-centered instead of learner-centered. Example: Rate the statement “Overall this instructor is an effective teacher.” This statement removes the student’s responsibility for their own learning. Compare with the following: Rate the statement “Overall in this course I developed skills as an effective learner.” The biggest goal I have in a mathematics course is to provide students with problem solving skills that they can use beyond my classroom. If a professor often gives a fantastic lecture, then that’s great; but that may not be helpful to students five years from now. Instead I hope to give students skills, practice, and experience in critical thinking, problem solving, complex reasoning, etc. Rating whether or not they’ve learned these skills is more important than rating “Overall, the required textbook was useful.”

Of course, figuring out how students have grown academically or intellectually is difficult. In this semester’s Precalculus classes, I’m working together with another instructor on designing course content. One of the things we decided to do was to use something similar to the Student Assessment of their Learning Gains (SALG) tool in an attempt to gather data on student progress through the course. Initially, the students are asked to take a benchmark SALG survey and they will repeat a similar survey two to three times throughout this semester. We are hoping to gather meaningful data on the growth of their skills by tracking things like whether they are in the habit of “using systematic reasoning in the approach to problems” or “using a critical approach to analyze arguments in daily life.” Hopefully this data will prove useful as we continue to tweak the course moving forward.

Rational Power Functions

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One of the topics our PreCalculus syllabus (Math 111) covers is “Rational Power Functions.” Since functions like f(x)=x^n are called power functions, the rational power functions would be those of the form f(x)= x^{p/q} (where p/q \in \mathbb{Q} ). Unfortunately, this topic isn’t covered in our textbook (Zill’s “Essentials of Precalculus with Calculus Previews“).

Tom Kunkle, on his Math111 Homepage, provides his students a nice summary of how these functions behave: See ratpowfunc.pdf. When thinking about this week’s Lab Assignment, my goal was to give students practice on drawing graphs of functions like

y = -(x-5)^{4/3}+1
Lab Description
I created nine cards, which I’m calling “Function Cards.” Each card is a half-page (printed on heavy-duty card stock). On one side, the card has a Graph of some translated rational power function. On the other side, the card has an Equation of a translated rational power function. However, the equation and the graph are not the same function!

Here are the directions I provided to students:

  1. When you get your Function Card, flip it to the Equation side. Make a note of the card number. Write down the equation.
  2. As a group, work together to sketch a graph of the equation. You may use the Rational Power Function handout (if you like). You may NOT use a calculator. Work together! Your sketch should clearly display & label all intercepts, any cusp points, any vertical asymptotes, any locations where the tangent line is vertical, and the parent function.
  3. Once your group has agreed upon what you think the graph looks like, draw your sketch on the Answer Sheet, along with the information listed in Step 2. Then go around to the other groups. Find the Function Card with the graph that matches your sketch. Ask nicely, then take it from that group.
  4. Return to Step 1.

The students will have the entire class period to create graphs of each of the nine Functions. Since they will track down the graph most closely matching theirs, they will have a chance to check their answers. Even once they see the answer (i.e., the graph), they are still asked to do some thinking since they have to provide information about the graph’s important features.

Hopefully this process will work. The only thing that concerns me is how long it will take them. I am really terrible at gauging how long it takes students to think about things. My best guess was that it would take about 4 minutes to sketch a quick graph, and then another 3 minutes to write down its important features. So, (7 minutes)x(9 Function Cards) should be a little over an hour.

I picked the nine functions from Tom’s “Homework Handout” on this Section, available here. If the weather complies, maybe we can do this activity outside. The only thing better than math class is math class in the sunshine.

Friendship Policy

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I have my first course meetings this morning. Right now I’m enjoying a one-hour break between classes in what will become my Office Hours once students figure out what Office Hours are for. I thought I’d take the time to write about an important topic I covered during today’s PreCalculus class.

A Very Important Course Policy:

One of the notable policies I have on my syllabus is called my Friendship Policy: Students in my courses are required to make two friends from class. For those of you who, like me, haven’t been a college student in a number of years, this policy may seem very silly and totally unnecessary! However, the policy has an important function at fixing a “problem” I noticed a few semesters ago.

Before class, I would find students sitting on benches in the hallway for several minutes waiting for the previous class to end. There would be, say, ten or twelve students all from the same course, standing in the same hallway, and it was library silent. No person was talking to any other person! Instead, every single one of them was texting someone on their phone, checking Facebook on their iPad, playing a game on their laptop, etc. Eventually they would all enter the same classroom and continue their technologically dependent anti-social activities.

When I pointed this out to my students, they had never noticed this phenomenon and they didn’t understand why I thought it was weird!

“Back in my day,” says the professor…

There were no cell phones. In order to fill the awkward silence, students in my classes would talk to each other, real-time, face-to-face. Sure, we would talk about course-related things like homework or exam studying, but we would also talk about social activities or sporting events or movies or whatever. This is how we made new friends.

I realize that students in my class have lots of friends. (Otherwise, who would they be constantly texting?) But I still have not figured out how they make new friends. Hence the birth of my Friendship Policy:

Friendship Policy:

You are required to make friends with students in this class. If you are absent from class, your friends will be very happy to lend you their notes to copy! In fact, I think cooperative learning is so important I am going to leave blank space on this syllabus for you to write down the names of two of your class friends and their contact information.

After explaining all this to the students, they usually look at me with confused faces until I say something along the lines of, “Friendship Time: Commence!” and then stare at my wristwatch expectantly. Within seconds, the room explodes in conversation. Occasionally, I have to nudge some of the shy students in the right direction.

Results and Analysis

After several classes over several semesters, this policy seems to make a big difference. First, no one sits before class in techno-quiet. They talk to each other, get to know each other, and occasionally I have caught them teaching each other how to do math problems. Second, I no longer get e-mails asking, “What did you cover in class yesterday?” Third, I learn a lot from my students by participating in before class conversations. For example, in this morning’s class, one student is here on a golf scholarship from Sweden! (How awesome is that!)

I still have two more classes this morning. We’ll see how those groups take to forced friendship-making time.