My Discovery-Based Math Elective: Introductions (Part 1)

In my short teaching career so far, I have been privileged in my exposure to teaching high school math electives. Given significant curricular and pedagogical liberty, I decided this year to make a pretty radical pedagogical shift in how I teach the class. It went pretty well, and my goal is to share the curriculum I developed, and reflect on structure and experience.

I'll reflect on the different components of the course over the course of a few blog posts, and will link them below. Providing this kind of Table of Contents will help me remember what all the components of the course are. I will link the posts below as I write them and might add things as they come up.

Table of Contents

1. Introductions (this post)
2. The Purpose of the Elective
3. A Typical Week
4. Practice Standards
5. Grading and Feedback
6. Circulation Notes Tracker
7. Compiling PSets
8. End-of-PSet Discussions
9. Exhibitions
10. Modifying for Distance Learning
11. Different Things Become Difficult
12. Pedagogy of the PSet
13. Architecture of a PSet
14. Content Threads
15. Impact on My Teaching

Before writing, let me share some relevant context:

• How is the curriculum packaged?
• The curriculum is packaged in a series of 26 problem sets (PSets). Here they are. They are a living document, and I'm always tweaking them. Check them out! Feel free to leave a comment on anything you have questions about, or if you notice an error.
• Full disclosure, however, they are imperfect, and I can't promise they are error free.
• There is also a set of unit plans to go with the PSets. Here they are.
• Same as above!
• If you have a question about any of the materials contained in this curriculum, leave a comment, or hit me up on Twitter @BearStMichael
• Where did you get the curriculum?
• James Tanton's Exploding Dots and Without Words, PROMYS for Teachers, PCMI, PlayWithYourMath.com, Joey Kelly, the #MTBoS, and a million other places. I haven't done a great job of citing sources, and it is an objective of mine to do so in a way that is helpful, honest, and honorable. If you see any problem that should have a source, feel free to leave a comment direction on the material, and I work on adding that citation.
• How did this course come to exist in your school?
• For a few years, my school required all juniors to take an SAT Math prep. They had scheduled me to teach a couple of sections. I asked if I could instead teach a pure math elective. Based on the meh experience that our school had with widespread mandated SAT prep (shocking), and the already virtually non-existent options for student choice in the classes they took, my school took a chance on letting me teach a portion of this course, as an optional alternative to SAT prep. I initially modeled the course pretty directly off of the course I'd seen Joey Kelly teach when I was his student teacher.
• The year after, we had built some momentum in favor of math electives and student choice, and my department was able to expand this course into a year-long course, broken into two independent semesters.
• What is your school's demographics and background?
• I teach in Boston Public Schools. You can see my school's data here, and it's pretty accurate, though an obviously incomplete picture of the school. But the major points are:

What I Tried During Distance Learning: Communicating With Students (Part 3)

I'm a reflective writer--it helps me organize and think through things. My administration asked me to do a writeup of the things I tried and did during our time of distance learning, to include as a part of my professional evaluation. So I figured I'd try to pivot it, and turn it into something that's useful for my practice and continued development. I hope this is helpful for other people too.

• Part 1: Necessary Changes
• Part 2: The Classwork
• Part 3: Communicating with Students (this post)
• Part 4: Questions I Still Have

Being systematic in my communication

• One of my first instincts with distance learning, in the middle of a pandemic, was that the goal and focus of our education should change in response. My working theory was that the thing that kids are going to experience, learn, and remember from this time is not math content. The thing kids are going to learn about is self-direction, independent learning, and communicating digitally.
• That being the assumption, I reallocated a ton of my teacher energy away from curriculum to building a strong and consistent structure for communication and connection with students. In doing so, I focused on coaching students on things like goal-setting, managing longer-term assignments, accessing resources independently, and keeping abreast of the constantly changing landscape of participating in school. That is to say, I focused on building student cognitive and emotional capacity.
• This was one of the most complicated new systems I had to work out. I never had to do anything like it before. And I do not like talking on the phone. But I knew it was important to do.
• Initially, I said I was going to call up each of my 80 kids at least once a week, and then expect to text/call follow up with all of them again a second time, later in the week. I used Google Voice, and a spreadsheet I had developed to track my communication. The calling each kid, over the course of two days, was somewhat manageable, if exhausting. I immediately discovered that calling/texting each kid twice a week was unmanageable. Then after about six weeks, I dialed it back a little bit, to be a bit more strategic with my energies, in the name of sustainability. And here is the system I have developed.

A tiered system of contact

• The big idea I had to eventually figure out, was that I should have a clear tiered system of communication. At the bottom, I should have a communication method that is low demand, to capture most students. Then as we proceed gradually down the spectrum, the communication attempts are harder to ignore, ending with direct phone calls to parents and referrals to the school's student support team (SST). I also had to figure a timeline that seemed reasonable to match with this system.
• Below, I have represented this tiered system, along with a timeline. This system is based on the goal of weekly two-way communication between me and the student, where two weeks is too long to go without any prior communication. The tiers are:
• Text student --> call student --> email student --> email student, cc'ing guardians --> text guardians --> call guardians --> loop in SST.

• Note that the first two steps happen on the same day. I would text each kid in the morning, the following message:
• "How's your week going? I'd like to check in today. Unless I hear from you by then, we'll plan on our regular phone call check in at 2:40. Looking for an update on how you're doing and how [week's assignment] is going. Talk to you soon!"
• Towards the end of the school year, especially if it was a "light" week, or if it was a particularly exhausting week for the kids or I, I might change it up by sending either of the following messages:
• If we had solid two-way contact last week:
• "We can check in over text this week. I want to check in about how you're doing and how [week's assignment] is going."
• If I hadn't heard from them last week:
• "I'd like to check in today. Unless I hear from you by then, we'll plan on our regular phone call check in at 2:40. Looking for an update on how you're doing and how [week's assignment] is going. Talk to you soon!"
• The first option usually got a bunch of good responses. Kids were motivated to respond, because they don't like talking on the phone #GenZ. It also allows them to communicate in a way that fits their schedule a bit more. And it lets them plan their responses a little more, if need be.
• Offering the first option (text w/ no expectation of call) was also good for me, because it dramatically reduced the number of students I needed to call, reducing demand on my end. On the flip side, however, it did lead to a slight increase in the number of students I'd go more than a week without hearing from. So maybe a more appropriate use system would be to only extend this option to students who have demonstrated multiple consecutive weeks of successful communication via text, without me needing to call or email them or their guardians.
• All students had a pre-scheduled time that I called them--same time each week. I also did my calls on the same day each week. Initially that was Monday, but I switched that to Thursdays. This was because students typically hadn't started work for the week by Monday, so they didn't have a ton of questions yet. I also wanted to give a kid a few more days to reach out to me on their own, through office hours or something--which would then count as their point of contact for the week. If I could pick again, I might go for Wednesday next time, because it still gives kids a couple days for us to work with, in terms of planning for completing work later in the week. 

Tracking contact

• I wanted to pair this tiered communication system with a tracker that allows me to quickly update a kid's status, filter and sort for different kids with different statuses, all while putting any contact info I may need at the ready (namely phone numbers, emails, and preferred contact info). Some may ask, "Don't you have a student information system, that has all this?" To which I respond, "Yes," as well as, "LOL." Cumbersome at best, for this purpose.
• I made a spreadsheet that does this pretty well. It has adapted and grown over time, so this is the most advanced, cleanest version. Here is a blank template. Note the communication template. It has a drop-down menu with statuses for students that align to the communication system outlined above. Throw in a couple filters, and you're in business.
• Column H indicated the default mode for how I reach out to those students:
• Text: they've effectively communicated with me for multiple consecutive weeks via text, and so are extended the option to check in via text, w/o expectation of phone call later that day should they not respond
• Call: call unless they text back before scheduled time
• Hangouts: for students who don't use phones, I taught them how to use Google Hangouts. It's much quicker/easier than email
• Email: for the very few students who didn't want to use Hangouts, for whatever reason. This wasn't typically effective.
• Note, this was made to be useful for teachers and their work. Throughout all of this, downtown, administration, SST, and guidance were all constantly hitting us with different versions of different spreadsheets, the same spreadsheets, all requiring of teachers different levels of fluency with spreadsheets. I believe that it would be possible to give a school full of 50 teachers this kind of tracker, train them in how to use this, and then have a connected "admin view" that collects all of this information, and complies it. I'd be interested in hearing if any of your schools have done something like that.
• My school instituted daily "office hours," which were times where teachers were technically supposed to make themselves available for students. I was always available for students who needed help then, but I was also pretty much available whenever any students needed a quick support-session in general. Combined with the already existing regularity of my weekly communication routine, the office hours were pretty unused. So the existence of office hours didn't impact students accessing real-time support from me.

The next post in this series is Part 4: Questions I Still Have.

What I Tried During Distance Learning: Necessary Changes (Part 1)

I'm a reflective writer--it helps me organize and think through things. My administration asked me to do a writeup of the things I tried and did during our time of distance learning, to include as a part of my professional evaluation. So I figured I'd try to pivot it, and turn it into something that's useful for my practice and continued development. I hope this is helpful for other people too.

• Part 1: Necessary Changes (this post)
• Part 2: The Classwork
• Part 3: Communicating with Students
• Part 4: Questions I Still Have

Necessary Changes

• In the classroom, we all have a collection of classroom routines and structures that are always working with. We then add in a "toolkit" of instructional routines and moves that we access every day, choosing whatever the right tool is for the job that day. Having a bigger, deeper toolkit, full of routines and moves you've thought deeply about, is helpful as a teacher.
• And this is good! Kids know that any given day is going to be different. And what's happening between their classes is going to be different. They have a schema for how days and classes can be different, because they've been doing this for years. Most of my 9th graders have been in school for almost a decade. They've developed a level of capacity and understanding, so they can roll with things changing quickly and often--they expect it. And even if something doesn't make sense one day, there are 30 other kids buzzing around doing the same thing, providing quick feedback and direction. Worst case scenario, they've got 179 other days that year.
• But when we are teaching things online, for the first time, most students don't have a decade of online learning experiences to provide a schema for what's going on. So every new routine and structure is VERY NEW, and as such has a much higher cognitive demand. Layer upon that the demand of simply navigating computers in general. Then seat that kid in the middle of a global health crisis, an economic implosion, and a nation's renewed public attention on anti-black racism. The bank of cognitive and emotional capacity is constantly being withdrawn from, simply trying to figure out what is going on. This leaves little cognitive space to dedicate to doing math.
• That is not to say that these are all excuses for not providing students with a rigorous and worthwhile education. This is simply evidence that to the greatest extent possible, in addition to build academic capacity, our schoolwork should do two things:
• build student cognitive and emotional capacity
• minimize unnecessary cognitive and emotional demand, while focusing it on necessary things (relevant and rigorous content)
• Yes, our school work would be doing this all the time. That's the goal, anyways. But on the spectrum between "don't do these at all" and "do these all the time," the context demands that we move in the direction on increased sensitivity.

Asynchronous > Synchronous

• Synchronous learning puts a lot of pressure on those students who can't dedicate their entire day to the schedules set by their teachers. And especially if your school is one of the majority schools in the country, and doesn't have a particularly coordinated or united instructional or administrative leadership, then your kids probably have anywhere from 4-7 different adults trying to schedule times to meet with students.
• And their means of communication are probably just as all over the place. Some are scheduling things on Google Calendar, some are emailing, some are posting in their online classrooms, some are texting/calling. So teachers can't just shrug and say, "Well I emailed them my office hours, so..." Or worse, "Within the first week of distance learning, every kid at least logged in to my Google Classroom. So they have no excuse!"
• I'm not saying that synchronous learning is never right. But based on what I know about my school, my kids, my context, and me...I'm putting all my chips in asynchronous learning this round.

The next post in this series is Part 2: The Classwork.

What I Tried During Distance Learning: Questions I Still Have (Part 4, final)

I'm a reflective writer--it helps me organize and think through things. My administration asked me to do a writeup of the things I tried and did during our time of distance learning, to include as a part of my professional evaluation. So I figured I'd try to pivot it, and turn it into something that's useful for my practice and continued development. I hope this is helpful for other people too.

• Part 1: Necessary Changes
• Part 2: The Classwork
• Part 3: Communicating with Students
• Part 4: Questions I Still Have (this post)

Helping students get familiar with "platforms"

• Whatever platforms I end up picking, there is a ton of fluency that students are going to need to develop, with respect to the platform itself.
• Tired: written instructions, possibly with screenshots
• Wired: video walkthroughs from the student perspectives
• Inspired: scavenger hunts
• Right away, I knew I was going to need to be able to provide screenshots of the student perspective when providing instructions that included navigation. I've also started including some kind of video instructions with many of my big assignments and updates, and putting those at the head of written instructions.
• I attended a Desmos PD last summer (pretty chill). One of the things they had us do on the first day (dedicated to the Desmos Calculator), was this scavenger hunt. It's just a list of challenges of increasing sophistication. The challenges are reasonable interesting puzzles on their own, and include a solution key. But they are an example of how to problematize fluency with a platform, making it a more authentic context for learning and exploration. I could make a scavenger hunt for Slides, Docs, Classroom, Gmail, Khan Academy, etc.
• I also would just be generally interested in getting familiar myself with more, different, effective platforms. Forward any reccs you have to me on Twitter @bearstmichael.

Choice boards

• There is a "board" (usually a slide), where there are a bunch of options for students to do different things. Often, they'll click on a link on that first page, which will take them to a document (or other slide) where it has the assignment or whatever.
• One of my more creative colleagues has a funny way of talking about
• Mustard: students must do everything in this section
• Mayo: students may do some items from this section, if they want
• Pickles: students pick which one (or more) of the items in this section they want to do
• Providing multiple options for students to pick from is generally a good move in any context. And for distance learning, it's especially powerful if the options are meaningful accommodations for students trying to manage learning from home, in the middle of a pandemic.
• What are some meaningful choices we can give kids? Record a video vs. write something vs. make a presentation? What do we need to do to make sure that students know how to use and explore the board, so they understand all the options available to them, at which point they can make an informed decision that actually feels like a real choice.
• Even though they're opting into these options, it's still possible for the variety of platforms, and their relative novelty, to be overwhelming for students. So we still want to be judicious with what platforms our options engage. Maybe some of our options can be "learn about this new platform"?

Google teacher training

• My district is big on the GSuite. I've never taught in any other district, so I don't know if it's universal. But it's probably pretty universal, right? Regardless, Google has a series of online PDs about how to use their stuff. I've heard the PD is decent. Especially if Google is a big part of my vision for an online class, it could be useful?

Student-student collaboration

• Student-student collaboration is probably the number one focus of my math class (importantly, above the math, even). It was a swift and certain casualty of my commitment to asynchronous learning. This has been the biggest impact on my pedagogy. Which sucks. Student-student connection is so incredibly good for improving the socioemotional experience of learning, while also significantly improving the academic rigor and relevance as well.
• In the online classes I've taken, the teachers have had us do forum posts, and then required us to respond to 2-3 other students' responses, as a part of the assignment. This was always more annoying than educational, and I think it's because it's such an inauthentic practice. It's not useful! Collaboration for its own sake has questionable value (I think). But what are the applications, what is the classwork, for which collaboration is an authentic support?
• I also don't just want to make every assignment a group assignment, because that's exhausting. I want students to be able to collaborate in groups, on work that is ultimately their own individual classwork.
• As far as digital collaboration, having students draw on AwwApp.com is pretty good, b/c they can draw with each other. It's a little awkward, but if you play around with it, even the free version is pretty effective. I understand that a best practice for synchronous collaboration is having students collaborate on Google Slides, instead of Docs. This adds a ton more flexibility, in terms of the kinds of objects they can interact with. It also avoids some of the awkwardness that comes with multiple people editing the same Google Doc.
• Unfortunately the drawing "scribble" tool in Slides is super inadequate, because it requires you to click the button after each stroke.
• There would also need to be a TON of support in teaching kids how to do all the cool things you can do with slides. You can get really creative with all the flexibility it allows, but that takes training. And I don't just mean changing font, size, and type. I'm talking images, GIFs, tables, shapes, hyperlinks, comments, etc. These tools push Slides into the realm of feasible math platforms.

Technology literacy

• Even as our students are born into a digital age, not all of them are as fluent as their teachers think. I've had to show multiple students how to compose an email. I have had to teach two different students how to minimize a window. Raise your hand if you have watch a student use caps lock to capitalize the first letter of a sentence. Even now, I still get emails from students where the whole body of the email is written in the subject line.
• None of these are intended to shame any kid or adult who does these things. It's meant to communicate the negligence we have demonstrated by assuming too much about our students' computer skills, and so failed to develop them.
• This doesn't even include the many many computer functionalities that can tremendously improve student capacity to work on computers. Pinning tabs. Keeping an organized Google Drive. Keyboard shortcuts for efficiency. Screenshots. What is the best way to do this? How do we roll this out over the course of the year, on a need-to-know basis, so as not to overwhelm?
• Surely some of these platforms are within the domain of all students' classes, right? Especially in the context of distance learning, Gmail and the whole GSuite have got to be worth explicitly teaching students how to use. Cross-disciplinary work is tough though. Maybe different content teams at schools take responsibility for using different common platforms extra, and including development on how to use them well? For example, all ELA teachers teach kids how to use emails. Science teachers teach Google Slides. Math teachers teach kids how to take pictures, screenshots, and edit them, and upload them on different platforms.

Building community

• One of my colleagues (who has really been leading the charge on creatively adapting to distance learning), would post a daily question designed to get kids logged into the Classroom, and responding to her and each other. She would post questions in Google Classroom, intentionally chosen to get kids to log into the classroom. This marginally boosted engagement on actually classwork--the question got kids in the door, so to speak. It also generally improved the flow of "student gets an email about a thing in Google Classroom --> then goes to Classroom to do it right then."
• Splitting up all your students into sections of 20-30 students is a constraint forced upon us by budgets and schedules. Is it still useful during virtual learning? Or is it better to have smaller "teams," where students are in groups of like 7-10? Maybe smaller groups are easier to build community? Maybe we can take advantage of things like group competitions and stuff to build community within smaller groups?
• What about competition in general? Personally, I'm a very competitive person, and it has pushed me very hard in my academics and extracurriculars. But it has also filled me with tons of stress and anxiety about school. I generally try to avoid it in the classroom at basically every opportunity. But I also know how competition with other groups does a lot to promote in-group cohesion. How can I engage that positive aspect, while disrupting the negative ones? I've got a pretty clear philosophy on where that belongs in a regular classroom....but how does that change since we're online, where motivating connection is more important than ever?
• How do I want to balance a student's desire for, and right to, privacy, while at the same time, trying to push them a little out of their comfort zone? I mean, next year I'm teaching 9th graders, in a 9-12. So this is going to be kids' first times in the whole school. They won't have many pre-existing friendships with other students, or relationships with teachers. What can we do then?

Thank you for reflecting with me! If you have any questions, comments, or thoughts related to this series "What I tried During Distance Learning," please feel free to @ me, or message me, on Twitter @bearstmichael

What I Tried During Distance Learning: The Classwork (Part 2)

I'm a reflective writer--it helps me organize and think through things. My administration asked me to do a writeup of the things I tried and did during our time of distance learning, to include as a part of my professional evaluation. So I figured I'd try to pivot it, and turn it into something that's useful for my practice and continued development. I hope this is helpful for other people too.

• Part 1: Necessary Changes
• Part 2: The Classwork (this post)
• Part 3: Communicating with Students
• Part 4: Questions I Still Have

Fewer, more versatile platforms

• Consider:
• Desmos is great for interactive lessons.
• Google Slides is great for group collaboration and presentations.
• Google Forms is great for regular, qualitative, in depth responses from students
• EdPuzzle is great for making interactive videos
• Khan Academy is great for developing procedural fluency with frequence immediate evaluative feedback
• Uploading pictures of handwritten work is great for math that can't easily or naturally be done on the computer
• AwwApp is great for online collaboration
• The Questions feature in Google Classroom is great for quick, public questions, where students and respond to each other
• CK12, DeltaMath, are probably great for different things
• But every new platform is a new investment of precious energy and time. So I wanted to remain as strategic as possible. I chose a few: Khan Academy, Desmos Activity Builders, and Google Classroom.

Khan Academy

• Why I chose it:
• It's very quick and easy to assign a bunch of pretty specific chunks of work.
• The videos, hints, and teacher dashboard are robust and about as good as it gets for what it is.
• Many kids are already pretty familiar with it.
• How I used it:
• Assign a big chunk of work that I am expecting kids to take a couple months to finish.
• I assign it all at once, so that different students can work at different paces.
• Through weekly check-ins, the majority of support I provide is in goal-setting and managing the big assignment in general.
• How I wish I'd used it:
• Only assign the problems, not the notes or videos. This reduces the "clutter" of the "things I need to do" part of the student-facing portal. The videos and notes are all right there if they need them. I also don't want a kid to feel like they have to watch a video if they don't need to.
• Assign things a week at a time. Lots of students had no idea how to manage such a big project. Some did, and it was cool, but most didn't. And the conversation, "Last week you said you'd do two hours of Khan, and you didn't do any? What do you think got in the way of that?" gets really old and discouraging, really quickly. This also allows me to easily change tack as the class unfolds over time.
• Have all the topics in Khan posted, linked, and shared, so that if a student wanted to jump ahead, they could.
• Provide some instructional videos for how to use Khan. I underestimated the difficulty of this. There were a bunch of onboarding, and ongoing, difficulties that could have been cleared up and prevented. It would have been effective to provide a walkthrough of the student-facing navigation.
• Have some differentiated pathways. Through a combination of teacher recommendation, and student self-selection, provide ~3 "pathways" of topics on Khan for students to pick. These pathways would be mostly differentiated based on readiness level. Ideally, adjacent pathways would overlap in about half their topics. It's also 
• The biggest way to differentiate Khan work is to simply assign different work--which can fine, if done appropriately. Some students need a review of earlier procedures before learning new procedures. Fortunately, this is the minority of what math is, as a whole, which is why tracking is largely unnecessary. But it is literally 100% of Khan. It's not particularly flexible for students with different levels of readiness.
• Questions I still have about Khan:
• How can I make it so that students can do more to pick the Khan they want to do? Like, we figure out what their goal is, and then they go through and pick the Khan Academy topics from the courses they want to do on their own? This would be a super amazing skills/disposition for their long-term education. Just an an example, my significant other basically used Khan to put herself through most of pre-med on her own. What a tremendous resource! I'd love to learn more about how to help students use it better.
• I'm not excited about how narrowly defined success is on Khan Academy, and the impact it has on differentiation. But for what it is (structured support for procedural fluency for accuracy), it's totally solid. Is it possible to use as like...20% of a kid's experience? Just a supplement? How little can I use it, while still having it be a meaningful thing to do on the side for students?
• What are the other things kids need to learn, in order for Khan to fulfill it's potential as a self-paced, adaptive, opportunity for students to take some real agency in their learning?

Desmos Activity Builders (ABs)

• In addition to the big, long-term assignment of working through the Khan Academy I assigned, I assigned weekly Desmos Activity Builders, as "Review/Previews," to borrow a term from CPM. These were meant to review the major work of the course. They started out more as mixed-spaced practice, with a handful of different math topics, and then over time I opted more into the pre-built Desmos ABs, which are less mixed, and more focused on individual topics.
• Active learning that's not assessment
• Khan Academy basically has two modalities:
• "Take Notes": students as mostly-passive receptor during videos or notes passages.
• "Performance": students being quizzed on content they ostensibly should be able to do, after the videos/notes. Note, even if they aren't called "quizzes," and they're just "practice problems," they're basically the same thing.
• But there is another major modality that should make up the vast majority of a student's work: exploration, attempts, guessing/checking, looking for mistakes, tracking what other people are trying, and literally every other math-doing verb. Desmos does a better job of this. It's actually one of their design principles. [ensure students are doing a variety of verbs.] This made it feel like the natural supplement to Khan Academy.
• Automatic feedback
• Dan Meyer and the Desmos team have shared a lot of reasoning, and concrete examples about the difference between evaluative and interpretive feedback. One of many incredibly influential concepts I've learned from them. Go check it out, if you're not already familiar. I've tried to incorporate a variety of automatic feedback mechanisms into the activities, because prompt (not always immediate) feedback is helpful for student learning.
• Below is an example of how I tried to design for interpretive feedback in the ABs I made. It doesn't immediately tell them what is right/wrong, but it signals that something is wrong. It also provides a model for thinking about the problem in a different way (graphically, as opposed to algebraically).

via GIPHY

• On the final slide of the AB, I also included a slide outlining which problems they got "probably right" or if they needed to "keep trying." Here's an example of what that might look like:

• Students seemed to value this slide. In opted to put it on the final slide to make it intentionally a bit of a hassle, hopefully motivating students to rely on the interpretive feedback provided directly on the slide. But I also know how some students appreciate some direct feedback, especially if they're too lost to be able to make enough use of the interpretive feedback.
• Share Classmate's Response
• Goal 1: Increase opportunities for students to meaningfully communicate with other students.
• Every time a student clicks "Share with Class," they are given a few other student responses. Importantly, not all student responses are definitely correct, or easy to understand. But students have a natural desire to see what other students said, to see if it validates their own response. Students are motivated to try to make sense of what other people are saying. This, in and of itself is an educational experience.
• Goal 2: Increase visibility of other students doing the same work.
• In the context of asynchronous learning, students don't have nearly as much interaction with each other, as a result of their common work in the class. But seeing other students' names, and seeing that they're doing the same work, can hopefully make students feel more like they are in community with other students.
• It was difficult to manage wanting students to look at other students' work, while also not wanting them to see all the work done for them, spoiling the problem. Over time, I got better at figuring out how to ask helpful "follow up" questions, the responses of which provide some general feedback, without spoiling most of the work on the slide.
• Focus of the work
• Lessons: These are ABs focused on a single topic, to some depth. For these, I typically used a relevant pre-made Desmos AB that I liked. Or tweaked one so that it fit my class better. Here's an example of the Desmos classic Match My Line, edited with love by me.
• Review/Preview: Here is an example of a Review/Preview AB I made from scratch. For reasons outlined above, most RPs followed this same structure:
• Private check in (shared only w/ me)
• Public check in (shared w/ class)
• Hints: a link to a helpful video, article, or example related to each problem
• Notice/Wonder
• ~6 problems, one problem per slide
• Correctness checker: indicating whether all the check-able work was correct.
• Questions I still have about Desmos:
• Most Desmos ABs that are single lessons are mostly designed to be executed in a class, with other kids, and other teachers. They are meant to facilitate classroom learning. How does the design of a "lesson" AB have to be different, in the absence of a teach? Of other students?

Problem Sets

• Problem Sets (PSets) were the primary mode of classwork for my elective, whereas my Math 1 class was Khan + Desmos. I wrote a bunch about what they are here. In particular, I wrote about how I used them during distance learning here. The two things I really liked about them were their natural capacity for offering choice as well as a reasonable system for feedback and revision.

The next post in this series is Part 3: Communicating with Students.

My Discovery-Based Math Elective: Impact on My Teaching (Part 15, final)

This is part 15, of a 15-part series of posts detailing how I developed and piloted a discovery-based high school math elective. The first, introductory, blog post for this series can be found here [Introductions]. The goal of this post is to discuss how teaching this elective course impacted parts of my development as a teacher. It also re-identifies some of the biggest contributors to the development of this curriculum.

• This past year was my first time teaching this course in its entirety (modulo distance learning due to a pandemic). As I started to think about the structure of the class, I think I was so ready and excited about it, and I had laid so much conceptual groundwork in my own head, that it kind of just exploded and built itself. It really felt like the last five or six years of math, teaching, life, and professional development all kind of converged on this experiment of a course. It has been one of the most interesting and validating things I've done in my short career so far.
• To be clear, the opportunity to design and teach this course was an uncommon privilege to have as a public school teacher. I had an evaluator, math team, and school environment that was ready and willing to give me the chance to be a experimental and radical with this class. Especially as an early-career teacher, that trust, and the opportunity to try and make good on it, was an enormous investment that my school and professional learning community made in me, as a professional.
• And teaching this course, while also teaching my traditional CPM Integrated Math 1 class, has yielded very interesting comparisons. Even though it was still offered for graduation-worthy math credit, this class didn't need to be the class that prepped students for AP classes, or the state standardized tests. So I was freed a little bit to try things out. And I can say, unequivocally that it has developed me professionally. So by teaching this class, I am better able to teach my 9th graders Math 1, a class that is very much preparing students for for state standardized tests, and AP classes as well, hopefully.
• I would again like to direct my appreciation to those whose work in mathematics and teaching have done so much to make accessible a tremendous amount of math. James Tanton's Exploding Dots and Without Words I and II, PROMYS for Teachers, PCMI, PlayWithYourMath.com, Joey Kelly, Numberphile, my own school's math department, the #MTBoS, and a million other educators and resources.
• Joey Kelly in particular, as my mentor teacher and continued colleague/friend, has done an enormous amount of precedent setting, resource gathering, and modelling. I can't overstate how much he has contributed to the existence of this course, and how very much I appreciate his guidance, support, and collaboration.

Student Perception Surveys

Have you ever watched video of yourself teaching? It's painful. Even on days where everything's goes well, it's a pretty humbling experience. You watch yourself miss opportunities, spend too much/little time on things, and as clear as you think you're being when you give directions, it's incredible how easy it is to be super confusing. Even hearing a recording of your own voice is the worst thing. And this is the person in front of your students every day!

In the name of reflection, growth, and professional development, understanding how your students perceive you as their teacher and leader is important. Trying to see yourself the way that your students see you, is a humbling experience, and it forces you to reflect and be constructively critical of yourself and your practice. One way that I try to do this in my practice is by giving Student Perception Surveys.

What Are Student Perception Surveys
As a graduate student, we were asked to give a student perception survey. The survey was designed by Tripod Education Partners, and was intended collect data on how students perceived their teacher, and how the teacher/class made them feel. Some example questions are:

• My classmates behave the way my teacher wants them to.
• This class does not keep my attention--I get bored.
• My teacher takes the time to summarize what we learn each day.
• In this class, we learn a lot almost every day.
• My teacher seems to know if something is bothering me.

Students then respond on a scale from "totally untrue" to "totally true." The survey has some great design features, and some mediocre ones as well, which is what you'd expect from a private education business thing. But the spirit behind the survey is what matters. Just like you would collect data on which of your students can identify the roots of a polynomial, and adjust your practice to suit, you can collect data on how you make your students feel--because it matters.

I'm not making a pitch for this particular Tripod Survey they made us give. Tripod doesn't know I'm writing this post about their survey (I hope they don't mind), and I won't post my survey or results because it's technically a private product that someone has to pay for. IDK, maybe you can convince your admin/district to sponsor your school or district. Or you can make your own, or maybe find a free one online. If you do, let me know because I'd love to see what kind of items you put on your survey.

Example of How It Can Help--A Personal Case Study
Last year, I taught two sections of Integrated Math 1 for 9th graders, and two sections of a Discrete Math elective for juniors. As I was looking at the data after the first survey, I was particularly surprised by student responses to the statement: "My teacher seems to know if something is bothering me." Fewer than 50% of students agreed with this statement. I was discouraged by this response, because I try to put a lot of work and thought into making sure that my students feel emotionally attended to, and it was something I thought I did well. After seeing this data, however, I tried to incorporate more check-in questions on my Do Nows asking students how they're doing.

I also modified one of my quick check-in routines that I use. At some point during a class, when I'm scanning the room, or students are taking a quiz, or working independently, any time I see a student who has their head down (in a "sad" way), or who looks worried about something, or if they're even just "spacing out," I try to check in with them non-verbally. I do this by making eye contact with them, and then giving them a thumbs up, sideways, and down, non-verbally asking them how they're doing. This gives them a chance to simply respond with the appropriate thumbs-up/sideways/down.

Before, I considered the check-in itself to be the primary means of attending to the student. Letting them know that I see them, and want to see if they're okay, especially if it looks like they're struggling. In an effort to be more attentive to students who might be struggling, now if a student gives me anything but a thumbs-up in return, I'll go kneel next to them and check in with them privately, asking what's up and if there's anything I can do to help.

After these two modifications, I saw a moderate improvement in teacher-favorable responses, up from 49% to 55%. This is still much lower than I was hoping for, and this area continued to be a weakness for me after the third administration of the survey. Nevertheless, in general, the survey directed me towards a blind-spot I had, which I appreciated, even as it continued to challenge me.

Logistics
Things I Have Done In The Past

• Give the survey multiple times over the course of the year. I typically give the survey three times, once each in November, February, and May. This give students some time at the beginning of the year to cultivate some feelings about me, and gives me some time after each survey to reflect and try to improve things.
• Divorce the survey from big tests/grades. It's super tempting to give the survey after a big test, or at the end of each quarter. But this might bias the data by drawing a connection between how some test just went, or what letter you sent home on a report card. And conversely, it also increases the temptation for teachers to tweak grades or routines in an effort to inflate their own survey results.
• Make it a normal day. Like I mentioned above, if you want the survey to be an accurate and honest assessment, you don't want to make a big speech beforehand, or do it the day after a field trip, or some other random day.
• Make sure students know the responses are anonymous. Data could be biased in either direction if students thought you could see who responded what. Also, especially if you have smaller groups of students, don't play the game where you look at the data and try to guess whose responses were whose.
• Offer the survey in native languages. My first year teaching, I had a section where almost the entire class had moved to the country in the past year. I wanted to assess their perceptions of me, not their language. So I made translations of the survey. Yes, this might have introduced new biases due to imperfect translations, but I figured it was worth it.

Things I Haven't Done (But Want To This Year)

• Give the survey with a team of teachers. Find a team of like-minded colleagues in your building, and form a working group. You can all give the survey to your students around the same time. Then you can reflect on the data and experience together. You don't have to share all your data with each other--it is pretty sensitive and personal data. But you can if you feel comfortable--it's an intense exercise in professional vulnerability. Especially if the results hit a sensitive nerve, like mine did in the personal case study I gave above. You can use the data as a starting point for discussing problems of practice, or brainstorming ways to respond to the data. You can also help each other stay accountable to the efforts you make to respond to the data.
• Share some of the results with the students. It would probably be overwhelming to share all of the data with your students, but maybe you can share an interesting or challenging cluster of data, and ask students what they think. If you're feeling really stuck on how to respond, this can help you to brainstorm ideas. It also messages to the students that you are actually attending to the data, and care about what they have to say. If you do this, make sure it comes from a position of you accepting responsibility for this, and seeking their support and input. It must not be about you blaming them or trying to justify why their responses were not favorable to you.

Using This Survey in Teacher Evaluations
I am one of the advisors for my school's student government. Every year, students share that they are frustrated that they have little to no input on teacher evaluations. They are certainly the people most proximal and most affected by the quality of the teacher. And they often cite the fact that student evaluations are common at the college level. I always see their point, and every year respond with how complicated and litigious the teacher evaluation process is. So from an administrative perspective, making student perception surveys a mandatory part of the teacher evaluation process is basically impossible, and definitely problematic.

Regardless, I voluntarily share my survey data, and my reflections on it, with my administrator as a part of my annual professional evaluation. It makes for a great artifact, I gotta say. But, I also trust my administrator to look at the data holistically, in context, and with an eye for me trying to be reflective. I trust that my administrator will not use the data to penalize me for the areas in which I need to grow. I know that not all teachers are privileged to have this kind of evaluator. Share the data with those who you feel safe doing so.

It would be inappropriate for an administrator to evaluate a teacher, or for teachers to evaluate each other, based solely on the raw data from this kind of survey. All this data exists in a context from which it can/should not be extricated. As such, it would also be inappropriate to use this data to compare teachers with each other, even those with the same students. The degree to which an administrator attempts to do this, is the degree to which they incentivize a teacher to game the survey, and is the same degree to which they erase the value of the survey as a professional exercise.

Indeed, it is very common for teachers to avoid these kinds of surveys for evaluative purposes, because they are concerned that the surveys end up as more of a "popularity" or "likability" assessment, which are not necessarily meaningful assessments of the quality of a teacher. It is impossible to avoid this bias, true. Nevertheless, I claim that students are much better at divorcing their student experience from their personal experience than we give them credit for. So we can feel comfortable giving this kind of data some real professional credence.

Regardless, and this is more of a personal opinion that not all teachers have to hold, I DO think that it is important for teachers to have a modicum of accessibility, likability, and collegiality. Despite the popular belief that students like "easy" teachers, I do think that students also find it easy to "like" the teacher that teaches them respectfully and well. Kids WANT to learn. Kids WANT to work hard. And kinds appreciate teachers who help them to do that.

Long story short, yes, a student perception survey will be biased in some way by whether or not students "like" you. No, that does not invalidate the professional utility of the data. Yes, it does complicate it. And if you want to invite this data into your professional evaluation, that's your professional prerogative to do so, recognizing that the evaluative focus should be on your reflection on the data, and your response to it, instead of raw performance.


If you end up putting together a working group at your school to develop or use some kind of student perception survey, let me know! I'd love to hear how it goes, what your assessment looks like, and how you or your team reflect on, and respond to the data.

UPDATE (6/9/2020): I actually ended up forming a teacher working group based on these surveys. It was a ton of fun. Maybe one day I'll be able to reflect on our work, archiving it for posterity and the #MTBoS. If you're thinking of doing this and interested in accessing the experience of someone who has done it before, DM me or tweet at me @bearstmichael. I'd love to talk about it!