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.!

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!

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.

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!