Midterm grades: what are they good for?

When I taught at Emory, there was no official policy or procedure for posting “midterm grades.” In fact, I had forgotten midterm grades were even a thing until just this past week. At my new job at the University of Kentucky, we must officially post each student’s midterm letter grade to the university grading system by a specific date. After several years of not assigning midterm letter grades, I initially didn’t see the point of this, but now I do: students in general are bad at understanding how they’re performing in a class, and their midterm grades help improve their perception. It’s the wakeup call some of them need to realize they’re in trouble.

Let me be clear, my students have every resource at hand to understand how they’re doing in my class at all times.

• Whenever they complete any online worksheet, homework assignment, quiz, or exam, they see their grade for it immediately.
• I keep all their grades up to date in the Canvas gradebook, which also calculates their current class grade.
• The syllabus describes the letter-grade breakdown, which is the usual one: 90% and above is an A, 80% and above is a B, and so one.

After each quiz and exam this semester, I’ve admonished any students who did poorly to talk to me about how they can improve going forward, and how they cannot expect to wait until mid-November to turn their grade around. After the first exam, I instructed every student who failed (25 of my 200 students) to come to me in office hours to talk about how to improve their study habits, and I reminded them of this instruction every day for a week and a half. Only two students came. It seems some students are in denial about how poorly they’re doing in the class, or aren’t willing or able to make the effort to change.

Yesterday, I sent out my students’ official midterm letter grades (along with a percentage grade), and I’ve since been inundated with emails from students who are worried about their performance and their final grades. Even though the students have always been able to see their percentage grade in the class and know what letter that equates to, it seems there’s something especially motivating about seeing a single letter assigned on the university’s grading system. A big fat F—well, UK uses E instead of F—on their gradebook was the jolt some students needed to reach out to me finally and ask for help. I now have several appointments set up over the next few days to help these students come up with a study plan and start to raise their grades.

I wonder if it might be worthwhile to send out letter grades even more frequently, or if this would dull the effect of receiving the midterm letter grade halfway through the semester. What I plan to do now at least is track the students who reach out to me versus those who don’t, and see if the help I offer them leads to improved grades between now and the end of the semester.

Project NExT: Final Takeaways

The pace of Project NExT finally caught up with me, and I wasn’t able to write a post about Day 4. Today I’ll write this post about the final two days of the program. This is maybe for the best, since I attended a mini-course that spanned both Thursday and Friday, so if I’d written this Thursday evening, I wouldn’t have had the whole picture. In fact, let’s say I planned not to write anything until today.

My main final takeaway from Project NExt is that mastery-based grading (aka standards-based grading) can help measure how many core skills students have truly mastered in your class; further, this method can reduce test anxiety for students, promotes a growth mindset, and gives students more agency over the learning process.

In a nutshell, mastery-based grading is a grading structure where students must pass each exam component completely (i.e. they must have mastered it) to get credit for it–there is no partial credit. Students who fail certain problems on an exam, however, may re-attempt similar problems on future exams until they have mastered the concept. Final grades in the class are determined by how many concepts a student has mastered. For example, if I were teaching a Calc I class with mastery-based grading, every exam would have a problem regarding the evaluation of a limit. Once a student successful answers this problem, they’ve shown mastery of the topic and never have to answer that problem-type again.

There are many benefits to mastery-based grading. First, exams aren’t as high-stakes for the students, since they know ahead of time that they can re-attempt any problem-type they miss (except on the final exam, of course). They also know exactly what types of problems to expect on each exam, since this list is provided ahead of time. Both these things help students feel less anxious about exams. Second, this philosophy emphasizes improvement and learning rather than perfection. It helps reinforce a growth mindset, telling students that it’s okay to make mistakes and try again, that this is part of the learning process. Third, students have more control over what they choose to study and answer an a given exam. Using the list of problem-types (that is, learning objectives), students can choose to focus their studying on whichever combination of topics they prefer or whichever ones will lead to their final grade goal. While an A student will need to pass almost all learning objectives, a C student will really have some choice here, and will feel empowered to pick which objectives to master.

There are some obvious drawbacks to master-based grading, though. We must write more exam questions, because each topic is tested on every exam, and we cannot repeat the some exact problem each time. There is also more grading to do, since students will attempt more problems per exam as the semester proceeds (though grading each problem should be faster, since no partial credit is given). Students may feel uncertain about how this grading system works or feel anxious about not having a percentage grade throughout the class.

Still the benefits seem to outweigh the risks, and I am excited to try this method in some class this school year! There are many online resources to help you design a mastery-based course, and you should really use these your first time rather than trying to start from scratch. Some of the resources given by our mini-course leader Rachel Weir are here, here, here, and here.

Here are some final tidbits I want to remember from my first week of Project NExT:

1. Jamboard is a neat Google tool for collaborative brainstorming online. I will try to use this in my remote (and even in-person?) classes.
2. Announcing our goals publicly helps with accountability and ultimately helps us achieve those goals. I will write a blog post this month about my goals for the coming year(s) as I start my career at a new institution.
3. I really loved getting to know so many passionate teachers at Project NExT this week. I wish I could have met them in person so we could have become better friends, but I know we’ll meet someday. I’m going to make an active effort to stay in touch with some of them this year so I can continue to build my professional support network.

I’m looking forward to the winter 2021 session of Project NExT! Here’s hoping it’ll be in person in Washington, D.C.!

Project NExT: Day 3 Takeaways

From day 3 of Project NExT, I really want to remember the classroom Rites of Passage shared by Uri Treisman and Erica Winterer. They shared some techniques they’ve used to welcome new students into the mathematical community and to set up class norms.

One suggestion I loved is that they try to learn every student’s name before the first day of class. They use flashcards and the students’ university photos to study the names (and other info, like where they’re from). Can you imagine the look on a student’s face when you call on them by name on the first day of class? This technique makes students feel cared about and important, and lets them know they’ll be missed if they’re not in class. Caveat: if we’re going to bother learning names, let’s learn to pronounce them correctly. If we can learn to pronounce “Daenerys Targaryen,” we can learn to pronounce any name.

Another suggestion was to introduce students to their “mathematical geneology.” This means showing them the long line of mathematicians who have contributed to where we are now, and telling them, “You are next.” It’s important to show a diverse list of mathematicians, not just Europeans, but this isn’t hard to do. Let students know that what they’re learning is the culmination of thousands of years of work and that it’s not done yet. Someone has to take up the mantle and keep going, and we hope it will be them.

Yesterday I wrote about honoring hard work rather than natural ability. Treisman and Winterer touched on this with a third suggestion from their classes: have students research a recent MacArthur Fellow and learn about their work and history. Treisman himself is a MacArthur Fellow. What students generally find is that the MacArthur Fellows aren’t natural-born geniuses (despite the name “MacArthur Genius Grant”), but rather they’re people who work very hard on something they’re very passionate about. We need to dispel the myth that certain people are “geniuses” and everyone else need not bother; any of us can be successful in our fields through hard work and passion.

It’s worth noting that much of Treisman’s and Winterer’s presentation was actually video clips of their former students describing the class. These videos were made two years after the students finished the class. I was amazed at how much the students remembered and how fondly they spoke of the class. We should all hope to make such a strong positive impression on our students.

Some other tidbits I want to remember from day 3:

1. Classtime is precious, so use it all. I often spend a few minutes at the start of each class going over announcements and reminders, but these might be better saved for a Canvas announcement, or just written on the board. Let’s use those minutes for learning.
2. CalcPlot3D is an amazing app that helps with visualizing 3D shapes, curves, planes, etc. I can’t wait to use this in Calc 3 (or any other calc class) in the future.
3. In future Math Circle lessons, start each class with the Math Circle Pledge. The pledge basically says, “If I already know the answer, I will not spoil it for other people. Instead, I’ll try to find the answer in a new way.” Some students need this reminder every time.

I’m looking forward to learning even more this week!

Project NExT: Day 2 Takeaways

My favorite lesson during day 2 of Project NExT was on how to use tactile visualization techniques in class. Using manipulatives and movement in class helps students both learn and retain content more easily.

One manipulative we used today was homemade Play-Do (which was easy, cheap, and fun to make ahead of time!). We used the Play-Do to create models of different solids of revolution. I have always struggled with helping students visualize solids of revolution, and my attempts at 3D drawings on the whiteboard need improvement. Rather than using computer software to sketch the shapes of the students, students can practice making the shapes themselves, which is a more memorable lesson; further, by using a piece of dental floss, students can cut their models in half to see the shapes of the cross-sections, which will help them better understand the disc/washer method.

Another suggestion I loved from today’s lesson is about graph transformations. As the instructor, create an x– and y-axis on the floor of your classroom, and have each student stand somewhere on the “plane” of the floor. Call a student’s original location (x, f(x)). Now apply graph transformations to your class. For example, for f(x)+1, they each take a step forward. For f(x+1), they each go one step to the left. For -f(x), they go to the opposite side of the x-axis, on the other side of the room. You can even compose multiple transformations! It may be helpful for students to compare their starting and ending positions with those of their nearby classmates, to understand how an entire function would be transformed. I like to think of this activity as a Transformation Line Dance.

Because of today’s lesson on tactile learning, I want to spend more time thinking about how to implement methods like these into my fall classes. Remote learning provides additional challenges here (e.g. I can only use materials that every student certainly has at home, like paper, but not necessarily Play-Do), but I still want to make it work.

Here are a few other ideas I want to remember from today:

1. “Inquiry-oriented learning” is bottom-up; it starts with a topic but lets students guide the questions asked and the methods used. “Inquire-based learning” is top-down; the lesson is guided by predetermined questions that encourage the students to think in a certain way.
2. To create high morale in my classes/department, I should remember to recognize hard work, not natural ability. This can look like giving completion-based homework grades instead of correctness-based, or honoring all students who participated in a completion rather than just those who scored well.
3. Dave Kung (based on his connections to the Potsdam Program) recommends a “No Criticism Zone” in the office. This is a department policy of never criticizing or complaining about students in any public space where you could be overheard. After some time of purposefully hiding these rant sessions in private offices, he’s found that faculty tend to stop criticizing students altogether. This leads to improved attitudes toward students in general.

After two days of Project NExT, I’m starting to feel overloaded with information, but that’s just another reason for me to stay dedicated to recording my thoughts here each night.

Project NExT: Day 1 Takeaways

The main point I want to remember from my first day of Project NExT is focus not on teaching math but rather on teaching students. I want to put the emphasis on people, not content.

In practice, this means abandoning lecture as the primary mode of instruction (and indeed, all the research on math education supports active learning as the more effective and equitable technique when compared to lecturing). This means providing individual instruction and support to students when possible. This means instilling in my students the skills of high-level thinking and problem-solving, rather than rote memorization and algorithm-repetition.

When we focus on teaching students, we are more equipped to recognize and address the problems our students are having. It is easy for an instructor to claim, “I taught the material, so I don’t know why some of my students don’t understand.” Rather we should ask ourselves, “Did I teach my students?” and if many students seem to be struggling, then we need to reflect on how change that. By talking with our students, we can learn about their struggles and how better to help them learn. To be in this mindset, we must be focused on the people, not the material.

Teaching students also means being cognizant of the inequities in our classrooms and working to remedy them. An instructor can ignore all the realities of our world–the biases, systemic obstacles, and individual hurdles–and give a well-delivered lecture focused on the math. This method serves to perpetuate these problems and the inequalities they create. An instructor who’s focused on teaching students must confront these problems in order for their student to succeed. This means being culturally responsive in our lessons and setting class norms that create an inclusive environment.

Some other tidbits I want to remember from today:

1. Tactile learning helps students remember lessons better. For example, using pieces of string to model curves and their derivatives is more memorable than simply drawing the curves and their derivatives on paper.
2. While I prefer to keep politics out of the classroom, human rights and equity are not inherently political. For example, I do not consider “Black lives matter” to be a political statement, and it’s not a phrase I should avoid using in class.
3. Give students the chance to convince each other of their answers after a round of polling (clickers), then let them answer the poll again. Students will learn to explain and defend their thinking and will also practice changing their stance based on new information.

I had a great first day at Project NExT and am excited to see what the week brings!

Calculus memes

As a fitting end to a Very Online semester, I invited my students to create and share memes about our calculus class. Some of the memes are about calculus, but most are about learning online. Below I’ve shared some of my favorites.

Seeing these memes reminds me how funny and creative my students are. I feel like I have missed out on getting to know this class, first because I was traveling a lot for job interviews, and then because in-person classes ended. Relationships with students are much harder to cultivate online. I’m hoping to connect more with my students in the fall, even if classes are held online. I will spend the summer learning about ways to encourage a close class culture virtually.

By Julian R.

By Nimra A.

By Hansa V.

By Dilara S.

By Sophia C.

By Cassie S.

By Erin C.

By Sammy J.

By Leida C.

By Jt R.

By Zeyad H. Explanation: I use a deck of Harry Potter playing cards to sort my students into groups in class

By Shiva D. He meant to write “bats an eye”

By Stephanie R.

By Allison T.

By Taylor S.

By Gaby D.

By Katie K.

By me, Brad Elliott

Student-created video content

How do we encourage students to study for an at-home, open-book test?

I recently gave my first at-home Calculus exam. My students had 24 hours to complete their exam, and were allowed to use all class resources (books, notes, my lecture videos, etc.). I am realistic and know some of them probably worked together or used other online resources as well. My worry was that students would not study in advance, thinking there was no need. I wanted to incentivize them to look over the material and work practice problems before the actual test began, so that they would internalize more of what we’re studying.

So I asked my students to create videos of themselves solving practice problems and talking through the steps/reasoning. I provided a list of problems spanning all of this exam’s concepts. Each student who chose to participate (for extra credit on the exam) claimed a problem, recorded themselves solving it, and posted it to our Canvas discussion board. I viewed the videos for correctness (though many students checked their work with me before making their videos) and commented on any small errors they contained. 31 of my 39 students partook in this opportunity.

Since I had already curated this list of problems to choose from, I also encouraged all students to use the list as a study guide, pointing out that they could check their work by looking over the videos posted by their classmates. In class the day after the exam, I polled my students asking if they had viewed their classmates’ videos, and whether that was helpful. Over 60% of the students who answered the poll said they had viewed a classmate’s video, and of those, all of them said viewing it was helpful. They also all said that creating the videos was helpful for their studying.

I expected the video-creation to cause more issues than it did, but the students all seemed to figure out which technology worked best for them, and I did not put any stipulations on how they created/shared the video. I showed them how to make a video in Zoom (just by recording a meeting with no one else in it and using the Whiteboard) and how to share the video link. Some students used other apps on their computers or tablets to record themselves working. Some students even solved the problem on paper and recorded their work with their phone camera. For students without access to video technology, I allowed them just to solve their problem on paper and to share a picture of their work. All of my students have reported having access to a camera at least, so this solution seemed to work for everyone.

I would use this extra credit assignment again. I think it was a great way to get students engaged with a problem and helping each other. It also created a trove of study material for other students; now instead of 40 students emailing me to check their work on a review problem, I could just point them toward the videos made by their classmates. Ideally students would comment on each other’s videos with questions or suggestions, but this didn’t happen, despite my urging them to. I think in a class that started off online (and not a class that was forced online in March) with a clearer initial expectation of commenting, this could be changed.

I’d love to hear how other math instructors have encouraged students engagement online!

Setting Clear, High, and Uniform Expectations for Students

I was recently learning about culturally responsive teaching, which is a teaching practice that emphasizes the inclusion of students’ cultural references in all aspects of the classroom. Some examples of culturally responsive teaching techniques are student-centered instruction and reshaping the curriculum. One aspect of this pedagogy that I realized I was lapsing in was communication of high expectations.

It is my nature to praise a student when they solve a problem, and to empathize with them when they get one wrong, but I’ve learned that these behaviors can be a sign of having low expectations for a student, and students can pick up on this. When I lavish praise onto a student for correctly simplifying some fractions, I’m conveying implicitly that I didn’t believe they’d be able to do this (pre-requisite) task. Further, I’m behaving as if they’ve mastered the material at hand, which they haven’t yet. I shouldn’t imply through my reactions that they’ve accomplished what I set out for them to; I should only praise them when they’ve mastered that activity’s goal.

Similarly, when I empathize with their failures, or offer unsolicited help, I may be sending the message that I never believed they can solve the problem. Students who think their instructor doesn’t believe in them will have a harder time believing in themselves.

Instructors can also show implicit bias through our use of praise or sympathy. Instructors are more likely to convey low expectations to students from certain cultural groups. This teaches these students that we don’t expect them to accomplish as much as their peers. Often this occurs with students from under-represented groups, and these students will be less likely to continue in math if their instructors act as if they don’t belong, worsening the representation problem.

To counteract this problem, one of my goals for my new semester is to set clear, consistent, and high expectations for all of my students. I will try to reserve my praise for when students meet these goals, and not empathize with their failures so much that it makes them think I accept their failings as inevitable. I will only offer help when asked, after I’ve seen a student try without me first.

If you’re interested in reading more about inclusive pedagogy, here’s a really good long read.

Lecturing vs Active Learning

I have just finished my first semester teaching an active learning class (of Calculus I). This means that every day when students came to class, I sorted them into groups of four and gave them worksheets to complete. The worksheets guided them through that day’s material, helping them discover the concept and practice examples. I would facilitate the worksheets by answering questions and discussing students’ solutions at the board. Having taught many lecture-style classes, I see how active learning has some pros and some cons by comparison.

Pro: Students get to know each other better. By working together every day, students are more likely to build partnerships and friendships, which not only helps them when they need someone to study with, but also enhances their college experience.

Con: Their mathematical writing is sloppier. I do very little writing at the board, so students don’t see many examples of how solutions should be written precisely. (If they check the posted worksheet solutions online after class, they’ll see typed versions of the work, but I know many students don’t do this.) Some students are mis-remembering notation from their high school math classes, and passing it along to their classmates.

Pro: They get to discover and develop most of the concepts themselves. The worksheets are designed to help students form an intuitive idea of the lesson (e.g. where the limit definition of the derivative come from) and then helps them prove it to themselves. This provides deeper learning than when they passively take notes, and is more likely to stick with them into the future.

Con: We cover less material in class. On the surface, we have covered the same number of sections in the textbook as when I teach a lecture-based class; however, students often have to finish the worksheets at home because there isn’t enough time in class. The “deeper learning” provided by the worksheets comes at a cost that the students have to pay with extra time after class. This time is in addition to what they spend doing homework and studying for assessments.

Pro: Students receive more consistent feedback on their work. Every day, I collect the worksheets, grade one of them from each group, and return them all by the next class period. This way, students are receiving feedback in class every day, either on their own worksheet or on the worksheet of a group member. (The worksheet answer within a group are very similar in general.) They also receive verbal feedback during class as they ask questions to me and their group members. This helps them to correct misunderstandings every day, not just after the weekly quiz.

Con: Administering the course takes up more time. This one surprised me. I expected active learning to be easier on my time than it has been. Before every class period, I have to compile the worksheet, work it myself, tweak any pieces I don’t like, and print it. After class, I have to grade the worksheets, enter grades online, and post the worksheet with solutions online. Each of these is a small task, but the time adds up quickly. It seems like, when I teach this active learning course again in the future, it will take almost as much time to administer each lesson as it does now; there are few tasks that don’t have to be repeated, and even the worksheets will need to be altered now that I have experience working with them once. I don’t mind the time commitment, but it could be prohibitive to someone teaching three or four courses this way.

Ultimately, I think a mix of lecturing and active-learning may be the best approach. I love to see my students working together and having those “Aha!” moments, but I also love explaining a difficult concept to them and pointing out the techniques that will help them solve a certain type of problem. Next time I teach Calc I, I’ll be excited to find the right balance between these two styles.

How to give a good math talk…

(…or a good talk of any kind.)

I recently read an excellent article called Giving Good Talks, by Satyan L. Devadoss, in the Notices of the AMS. The timing of the article was good, because just last week I gave an invited research talk at the University of South Carolina. I was able to use the advice given to improve my talk beforehand. The Devadoss makes many good points, and I recommend reading the article. Here, I’ll just list my take-aways.

• Tell a story. Do not deviate from that story, even if the deviation seems interesting to you. Stay focused! Your audience needs to be able to follow along without too much jumping around.
• Do the work for your audience. Distill major points down to their essence, so that they are clear and memorable. It’s okay to ignore subtleties and exceptional cases. Use intuition and images over precise definitions and explanations.
  • A note here: it is very tempting when creating a slideshow based on a research paper to cut-and-paste from the paper into the slides. Do not do this. Language that is appropriate for a paper is often too verbose for a slideshow. Your audience can’t go back and re-read definitions or mull over the statement of a theorem, so it needs to be presented simply and memorably the first time.
• Use your lecture slides wisely. Just because you can cram more info onto a slide, or run through many slides quickly, doesn’t mean you should. Less is more here (and pretty much everywhere else).
• The most important one: Speak slowly and end on time. If you’re running behind, do not just talk faster. Rather, leave out some stuff (in fact, decide beforehand what can be left out). Do no go over your time. DO NOT GO OVER YOUR TIME.

Most of these lessons apply to giving any type of talk, not just a math talk. Also, they apply when lecturing to a class. When I teach, I try to connect the lesson to what we’ve learned already and what we ultimately want to learn, so the lessons build a kind of story. I first present intuitive explanations and pictures (often through an example) before giving technical definitions and theorems. And of course, I try to speak and write slowly, remembering that my students are hearing this information for the first time and need time to process it.