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9/1/2021

I recently hosted a seminar at the University of Arizona. The topic was on "Reigniting Student Interest in Mathematics", and I was going to talk about a two-pronged approach to accomplish this: fixing grades and fixing the curriculum.

Over the first four hour session, we ended up primarily talking about grades and grading. When I asked the teachers present what they like least about teaching, these were the responses:

• you’re never done

• mandatory teaching meetings

• oppressive structures in the system (including grading)

• assessment, procedural nature

• grading

• rushing to complete curriculum

• do what’s best but only do it this way 

• learning names every year

While I knew I couldn't address all of these things, several of them got at the idea that the way school is done is simply not ideal. I hope none of my superiors are reading this right now, but during the last school year (2020-2021 - 75% entirely remote learning, 25% "hybrid" due to COVID-19 pandemic) and the following summer, I decided that I wasn't just going to keep doing things a certain way because that is the way they had always been done. I decided that if I started doing things how I felt was best and someone told me that I couldn't, that I would just not listen to them. Of course, I was pretty sure this wouldn't happen since I knew that if I was going to do something different then it would be something I really believed in, and it is super easy to argue your point when you know you're right.

Today on September 1st, all of my classes have taken their first summative assessments and received their scores. Many students have begun the retake process, but not enough. I think my next step will be to figure out how to REQUIRE that students do a retake if they scored below a certain mark. The grading scale I adopted this year is different than what students are used to. If a student does pretty poorly on a test and does not demonstrate an acceptable amount of knowledge on any topic, then it is technically possible for them to have a low C in the class. Students are used to thinking that a C grade is "OK". In this case it is not. I may need to tweak how the letter grades are assigned in the gradebook. Before my next post I hope to have another summative assessment data point. I will also be doing some data analysis, such as running correlations between number of missing assignments and success on tests.

9/14/2021

Last Saturday, September 11, I presented the 2nd part of the HEMS-HS seminar. This was a much more practical session where we did many math activities. We still talked about the grading tweaks and attendees offered some interesting insights on that too.

Activity 1: Measure the height of the Mathematics Building on the University of Arizona campus.

I gave attendees the following materials and asked them to start by thinking about the problem and formulating a plan:

• ~10ft of string

• A compass

• A 25ft tape measurer

• A straw

• A few washers

This is an activity I normally do in my classes after we learn about the Law of Sines and how you can sight two different angles of elevation to the top of some thing (building, mountain, etc) separated by a known distance. I give students actual inclinometers, which is just a protractor with a straw attached to it and a string that hangs down weighted by some paperclips or washers. For this activity, I wanted to see what the attendees would come up with for measuring so I gave them less direction and less obvious materials. I had imagined they would use the compass to construct their own protractors (using angle bisectors and overlapping circles and stuff, or they would draw a semicircle and use the string to divide the arc up into radians and fractions of radians (also creating a protractor).

Instead, with the less direction, the groups came up with all kinds of interesting ideas. One group thought they would measure shadows and use similar triangles. That was really smart but it was about noon and the shadows were not obvious. Another group (or maybe it was the same group?) decided to measure a single brick, then count the number of bricks to the top of the building. This was a fairly high building and not all of it's height is in bricks, so there would need to be a fair amount of generalizing and approximation. Another strategy was to use purely right angle trigonometry by using the string to measure a straight and level line off the side of the building to the end of it's shadow (again not ideal), and they achieved getting this level string by putting the string through the straw and adjusting the string angle until the straw didn't move. I thought that was pretty genius.

The other things we did in the session ran pretty long so we didn't really get to conclude the activity, but that was besides the point.

We did and talked about a lot of other things during this session but I don't know how useful talking about them here would be.

If you're interested, I've zipped up all the resources I used for both seminar sessions and put them here:

https://drive.google.com/file/d/17ShyIBUqijrMlwKrspXUFFK0N8-DcPvj/view?usp=sharing

9/17/2021

Here is some data analysis I shared with my students recently (below).

After getting grades updated after all classes had taken a second summative assessment, I ran a correlation between the number of missing assignments students had and their overall grade in the class so far. I would expect a negative correlation between these two variables (as number of missing assignments increases, overall grade in the class will decrease). The results confirmed my expectations. Rather than take this as some kind of proof that my grading system isn't working (students are not finding the intrinsic motivation to do the assignments when they are not getting points that count towards their grade), I have instead chosen to share this information with them. I am showing them that this is proof that there is a relationship between them not doing assignments and not doing well in the class. I hope this confirms their own suspicions and helps them take some responsibility for doing the amount of work they personally need to do learn the material.

This led me to zoom in a bit and look at some trends I don't think I would be able to suss out using standard excel formulas. Specifically I was interested in how many students had a "large" number of missing assignments but were still doing "well" in the class. I defined a large number of assignments as greater than a third of the total assignments for the class, and doing well as greater than or equal to the class average.

• P3 College Algebra: 6 students (18.2% of the class)

• P4 College Algebra: 11 students (32.4% of the class)

• P5 Honors Trig: 12 students (31.6% of the class)

• P6 Honors Trig: 11 students (33.3% of the class)

I need to look at this data for the opposite thing also: students who have done all the work but are doing poorly in the class.

Assessment

Over the past couple of weeks I have been wracking my brain about how to handle the summative portfolio assessment I have planned for the semester. Although I let students know about the portfolio "being a thing" on the first day of school, a combination of me not following up, regularly checking in, and lots of student schedule changes led me to observe that not all students were being diligent about this. I read Sarr Sackstein's Hacking Assessment for some inspiration and direction about how to fix this. Before the fall break I let students know what things should be in their portfolios up until this point, and now, after fall break, we are in the midst of a period during which I am having short individual conferences with students where I check in on their portfolio progress and address any concerns they are having. I am also asking them to basically give me ideas about the things that interest them that involve math, so that I might be able to incorporate those things into a project or just generally into the curriculum in the future. After this week I think students will have a much better idea of the expectations for this, so hopefully things go more smoothly from here on out.

General Observations

Students are not taking advantage of my retake policy nearly as much as I thought they would. This worries me because at this point there are A LOT of students who really ought to be retaking sections of their tests and they are not. The only reasons I can imagine they are not doing that is because A) they don't think they need to and/or B) they are just waiting until the last minute. Neither of these is good for me. I will probably need to encourage them to get busy with that sooner than later.

Making whiteboards and markers super ubiquitous in the room has had upsides but also a big downside. Students seem to generally feel safer taking risks with problems, and I am able to walk around and sketch out some advice really conveniently, and both of these are good things. Some students enjoy writing on the whiteboards so much that they end up not getting anything down in their notes permanently, which I think is a bad thing. There is also just a TON of doodling happening. Although I thoroughly enjoy looking at the doodles and usually encourage creativity in the classroom, they are also wasting my damn markers and clearly distracted if they have time to doodle that much. IDK.

12/02/2021 - Progress on Portfolios and the End of Semester

Full Disclosure: I started writing this post/initially felt motivated to write it on December 2nd, in part because it is a cool palindromic date, in part because my "capstone" portfolio assignment thing had been fully assigned and distributed, and in part because I had already met with an amazing former mentor about modifications I would make to it for next semester (shout out the OG Erik Yoder).

Well, it's now December 27th. So yeah. At least I have more things to talk about now since the semester is wrapped up and I can give some final thoughts. I still think 12022021 is a mega cool date so I'm leaving it. This will be my only post for the month anyway.

12022021!!!!

Before I dive into the content part of this post, I wanted to share a link. I have a Google Pixel 6 phone and consequently (and through using most of their services) Google knows way too much about me. Recently they fed me this link from The Washington Post. I'd encourage you to read through it.

Grade kids’ homework: A plan to judge them less is a bad idea - The Washington Post 

(*-massive grain of salt addressed at the end of this section) 

Something you should know about me by now, from reading this page or if you know me in person, is that I try to keep an open mind when reading through educational commentary. Despite being what I would consider a "heavily researched" field, I feel it is important to keep an open mind about education because at the end of the day, I think everyone can agree that a one-size-fits-all approach usually doesn't work, and the field itself, as a product of research and the students themselves, is ever-evolving. I did the same while reading this post, but if you've read anything on this website you'll know that I disagree with a lot of what is in it.

I found the comment section far more interesting; it ranged from people posting concise agreements with the content of the article, to those offering polite alternate opinions supported by research. It struck me that the article itself does not have any research attached to it other than anecdotes from teachers in various districts and quotes from a letter allegedly sent by Teachers at Arlington, Virginia's Wakefield High School.

I will confess to not knowing this for sure, but my understanding is that this "not grading homework" idea does not mean that students are not given homework, or indeed that it is not graded, but that it simply is not given a grade value in the gradebook that effects a students' final grade in the class. At least I hope this is the case. The article does not seem to grasp this, and further argues that the practice and repetition afforded by homework is crucial for students' success on summative assessments. Opinions in the comments further allege that such practice assignments and the strict deadlines they come with are critical preparation for success in the real world.

I agree with the idea that practice and repetition is a critical part of the learning process and have told my students this repeatedly throughout the semester this year. The thing, though, is that not every student requires the same amount of practice. In an efficiently-run class, it is possible that many students do not require practice outside of what has been provided during the class. Others might need to do an entire assigned problem set. I contest that the self-awareness skills developed by students when discovering for themselves exactly how much practice is sufficient for them is a far more valuable lesson than any individual concept we teach them, and is a better way to prepare them for success in the real world than requiring them to complete a prescribed amount of work in a given time period.

*- The article focuses on the Arlington Public Schools district in Virginia, which is a very high-performing district. I teach at University High School in Tucson, AZ, which is routinely ranked in the top 20 of high-schools nationally. My belief is that this teaching strategy greatly enhances equity and would work well in a school that "underperforms" and has a greater population of students from historically underserved communities, but I definitely have no experience with this and frankly don't know where to turn to find out if this is true. This is just me checking my privilege.

--

Dear students who read this: hi! So proud of you!

Portfolio Summative Assessment

In my last update I mentioned portfolios and how I felt like until then I had just kind of said "here" and didn't really have a goal in mind for them. I always thought that the portfolio would somehow evolve into a summative assessment, but I wasn't sure how. I set out to fix that, and through A LOT of work and research, came up with the document below. This is a "Portfolio Reflection Activity", wherein I ask students to answer an assortment of questions.

I really recommend that you read through this whole document in order to gain a better understanding of what it represents/accomplishes, but here is a super quick breakdown of the why behind this:

• I want students to be reflective of their learning process.

• I want feedback from my students about how this whole thing went for them.

• I want there to be some form of summative assessment of the actual content from the class (I'll elaborate on this below)

• I want there to be some de-facto document that I can show to any of their future teachers who want to know what "so and so" knows about the content from my class. (Isn't that A LOT more useful information than a simple letter grade??)

Briefly about bullet point 3: You might ask, "why not just give a final exam then?". I have always hated final exams for a few reasons. First, they are often treated as a very high-stakes test that can absolutely ruin a student's grade in the class due to their weighting. Even if they are not scored in this way, that is still unfortunately the association students make with the words "Final Exam". Second, students just end up cramming for such a test anyway. If the majority of your students are just living test-to-test in your class and forgetting previous material shortly after being tested on it, a final exam isn't really different. (Note: I know that students do this in my class still. Trying to undo the task/reward brainwashing of school is tough.) Lastly, most students always just do poorly on a final exam anyway, so you kind of feel obligated to apply a curve, at which point, what do the results of the final exam even tell you?

Alternatively, my intention with the content-based questions in this assignment is not for students to recall answers in a high-pressure situation, but rather to go back and make sure they understand the associated concepts and provide thoughtful answers. There were reflection questions built-in to see what they thought about their weaknesses and their overall experience. From a teacher perspective the advantages (in my mind) were:

• This is a lot more meaningful information for me.

• This is a more meaningful task for students than a final exam.

• This is just a much more useful learning artifact that I can share with other teachers if needed.

So... how did it go?

Well, mostly really well. Students who took the assignment seriously generally gave me some incredible responses. Students who blew it off gave me some worthless responses. There were a handful of students who just didn't do it, while other students clearly cheated, which was particularly frustrating for me. Very quickly about the latter two groups: I was never unclear about this being a mandatory and significant assignment, and I am sooo upset that students continue to find ways to cheat in this class after I feel I have put a lot of effort into providing every opportunity for success and implementing a very lenient grading scale.

What qualified as a good response? I will post links to a few exceptional ones below, but generally a good response included the following things:

• Thorough responses to questions written in their own words.

• Insightful comments in the reflection and feedback sections.

That's it really. Here are some great ones (any identifying information has been removed (I think? If not, sorry about it.)):

Good Portfolio Reflection Activity Examples [12/2/22-Link temporarily removed for reasons. Email me if you need it.]

What made a response bad? Well, as previously mentioned, it was really easy to tell when there was little effort put into this assignment. That was clear when responses to certain questions were "I don't know" or "I don't remember". There were also a fair number of students who just chose to copy/paste answers off the internet to questions like "What is a radian?" or "What is an extraneous solution?". This is obvious when there are a bunch of students who answer the question in the same exact way using words they don't understand.

Overall I thought it went well. I learned a few things from this that will lead to changes I will make for next semester and I'll discuss those in the section below called "Changes for Next Time".

Nuts and Bolts

For any teachers reading this that are thinking about implementing something like this for their own classes, here is how I did it. I'm not saying this is perfect but it worked. First of all, feel free to steal anything I've posted here or if you need other resources just email me (thomas.gribble@tusd1.org or togribble@gmail.com).

The first thing I did was distribute the blank document to students via an assignment on Microsoft Teams and had them read through it during class. I gave them 10-15 minutes to do this and walked around the room to make sure they were actually doing what I asked. I took any questions after that. I then explained the whole thing to them again just to make sure everyone was on the same page about the expectations and the due date. The benefit of using Microsoft Teams here was that students could click on the assignment and edit their own copy of the document right there. This proved to be too difficult for some students, which led to me having to deal with annoying technical issues later on for them, but for the most part this was a really convenient way to do it.

Next, I made a schedule for individual 5-minute conferences with every student. I did this over the course of four class sessions for each class. Those four class sessions would normally be devoted to review and the final exam, so it's not like this took more time than a final exam would have taken. I made the schedule by exporting student lists from our SIS into excel and using =RAND() and sorting by the random number. If you don't know how to do this you're tripping. I filled in schedule slots on another sheet of the excel document that I had pre-populated based on the scheduled class times. I also included a buffer before meeting with students each class period to get my wits about me.

Before meeting with any students, on the due date for the assignment I went through all of the submissions and made a note about if they had been completed and a very rough idea of the thoroughness. This took about 45 seconds per student so it ended up taking about 2.5 hours with breaks on the due date. I felt like I needed to do this for it to be fair for students who were scheduled to meet on the first day vs. those who were scheduled for later days. Since I believe I gave more than enough time to complete this assignment (about 1.5 weeks total), I wanted everyone to have the same amount of time. Many students still turned this in very late, but looking at the submissions on the due date allowed me to consider that when meeting with them.

The next thing I did was create a Microsoft Form where I would enter in the student's name and their period, and after hitting submit it would take me to a page where I would evaluate them based on the rubric that I had copy pasted right into the form. This allowed me to see each rubric item right in front of me each time, and when all was said and done I had all the information in a single spreadsheet that I could sort by period.

Here is a link to duplicate this form for your own use: https://forms.office.com/Pages/ShareFormPage.aspx?id=4FBwvMxLCUiSReqLZQhIZSl3obk0smFIrCau1PPBYgVUOTM3RVg4RjdMTUI1STdCNEpUVk9XVjZUUi4u&sharetoken=YIo4c2nmZTjVgocUPbpk

Here is a link you can use to enter random info to see how it works: https://forms.office.com/r/4DCS2Hvd5v

Changes for Next Time

First of all I took a lot of inspiration from Susan Blum's chapter in Ungrading: Why Rating Students Undermines Learning (and What to Do Instead) while coming up with this. While I was working on it I had a lot of questions about efficacy and reached out to a few people for advice. Unfortunately a lot of those people, while brilliant, don't have much experience with this type of thing. The one person I really wanted to meet with was unable to due to a COVID scare, so I had to go with my gut and the feedback I got from colleagues. I eventually was able to meet with this person and he gave me incredibly wonderful feedback that will lead me to change a few things about this for next semester. I will also tweak a couple of things based on the submissions I got and my discussions with students. Here are those changes:

1. More clarity with expectations. I learned that simply saying responses to questions should be "thorough" is not clear enough. Students wanted clarity on how long their responses should be. I will probably tweak this language to talk about how the only requirement is that the question is fully answered and if you can do that in a few sentences then that is great, but if you feel like you need more of a paragraph, then that is great too. I do believe showing exemplar submissions from this semester will help clear up expectations as well.

2. Some requirement that student's share this (Portfolio+Reflection) with somebody (parent, teacher, counselor, someone else). This would be to increase the value for the student- it gives them more bang for the buck and hopefully increases buy-in. In my 2nd year of teaching I was asked to create a portfolio and my mentor teacher (the person who I mention above!) told me they would like to share it with other people, so I made sure to do a really good job on it.

3. Tweak reflection questions to include something about how to improve. This will tie in to...

4. Include an element to this at the beginning of the semester where students set goals for themselves in the class. This will include things like grade goals.

5. Add to the self-evaluation section somewhere for students to tabulate how many poor test scores they initially received and how many test sections they did retakes for

6. Include a regular self-assessment/reflection element to the class about time spent and number of assignments done.

7. Possibly include a checklist in the final reflection activity and ask students to tell what the data shows.

Final Thoughts

Honestly this has all been pretty exhausting. I think overall it has been an improvement over previous years, and I also have to consider things through the lens of "students just went through a year of online learning and are still going through a traumatic life experience amidst a global pandemic". There have probably been too many uncontrolled variables in this experiment for the results to be reliable, but I am choosing to turn a blind eye to that possibility because I really have put in what I believe to be an honest effort here, and I have pretty high standards for myself. 

After briefly meeting with my colleagues who had common preps this semester and did not do what I am doing with grades (they chose to go a more traditional route and homework was required and graded, though not heavily weighted), it seems like our distribution of grades was fairly similar in the end. I don't know exactly what this means because I think what grades represent can be very inconsistent across teachers, but I thought I would mention this anyway.

If you are a reader of this blog I hope that you agree with me that I have been transparent about this process and have done the due diligence.

Next semester is going to be extremely challenging for me on a professional and personal level, and I am interested to see how different changes will effect the outcome. I am hopeful that I a lot of the work I have put into figuring this out this semester will allow me to do many things next semester more efficiently and ultimately save some time. Maybe I won't have to work 60 hours per week again next semester??

Thanks for reading.

1/20/2022 - Semester 2 Tweaks

This will be a quick post about how I am implementing some of the tweaks mentioned in the previous post now that the 2nd Semester has started, as well as some other little changes in the classroom that I think will improve the overall student experience. Maybe.

Grade Accountability Sheet

On the first day of this semester I gave students this Semester Grade Accountability Sheet. The goal of this was for students to set a target. I might have mentioned this in my manifesto on grading, but for the most part, students can hit a target that they can see and that isn't moving. If the student has also self-identified that target, I think they will also be more intrinsically motivated to hit it. After students identify the grade they would like to get for the semester, there are a couple of different things that they will use the sheet for throughout the semester. First, after every test, they will record any sections on which they scored beneath their grade goal. By keeping a list of these sections, it is my hope that students will take more accountability for then retaking these sections in an attempt to get closer to (or hopefully surpass) their grade goal. At the end of the semester, students will be able to tally the number of sections of tests they scored beneath their grade goal on, along with the total number of sections they retook during the semester, and hopefully those numbers will be equal. If not, then there is clear objective reason why they may not have met their grade goal for the semester that they can take responsibility for and reflect on.

Second, at each grading period (4 throughout the semester), students will complete a brief reflection on the back of the sheet. This will include a quick check-in on their current grade vs. their grade goal, a tally of the number of retakes they took, and some reflection on the relative effort they've put into the class up until that point. I am also asking them to write about what is going well for them or not going well for them. Ultimately I hope they reflect on their learning process much more frequently than this, but at least I am providing them class time to do this 4 times per semester.

Grading Scale

I decided to tweak the grading scale a bit to address some of the things I observed. It is now this:

• A: 4-3.5

• B: 3.49-2.75

• C: 2.74-2

• D: 1.99-1

• F: 0.99-0

This change, like a few of the others I am making this semester, is meant to get students to want to retake test sections. I tell them repeatedly that they really shouldn't be OK with a score of 2 on a test section. However, then they get that 2 and see a "C" in the gradebook (pesky gradebook) and feel OK about it. If making a "2" more precarious will get them to go back and reinforce their learning and retake a test section to improve, then that is a good result in my opinion. I'm happy if every student gets an A. They probably don't believe that, but it's true.

Realizations

One of the guest teachers that covered for me during my absence is also someone who has been a terrific colleague and mentor throughout my educational career. For one reason or another, I always get a bit down in the dumps at the end of the school year, and my usual reaction is to desperately reach out to people for validation. I know this isn't healthy and also pretty childish, but luckily most of the people I reach out to recognize that and play along anyway by telling me how great I am. This helps a little, but what helps A LOT is when someone sits me down and spits some facts at me.

The aforementioned colleague and mentor did just that for me this year. She has seen me put unrealistic amounts of effort into things for close to 6 years now, then consistently be hyper-critical of myself or question if what I am doing is the right thing. She happened to be on campus during the last week of school covering for math teacher who was taking a group of students to a national competition in Iowa, and I asked her if she would come take a look at the end-of-semester summative and cumulative assignment I gave to students and let me know what she thought. She spent a few minutes reading through it and basically said she thinks all tests should be written in this kind of format if we really want to know if students are reaching the educational objectives we say we want them to reach. (Note: this is a whole other thread and maybe something I will expand on later during the summer and even dare to implement next year.) I then shared with her some of the things I thought went wrong it this year, including a fair number of very inadequate submissions and a bit of cheating.

She told me a lot of things that I will attempt to summarize here:

• In response to me saying I wasn't sure if all my effort was worth it: Years ago she had a colleague that all the students loved. He was a younger teacher and relatable, and it always seemed like he was really enjoying his work and wasn't stressed at all. Upon further investigation, she found out that all he would do in class was have students read out of the textbook for their lessons. I don't remember what the homework and testing situation was, but the reason the students loved him was because his class was insanely easy. Meanwhile, her colleagues and her were busting their asses to provide their students with every chance to succeed and in many cases the students either did not like them because they were making the class "hard" by enforcing rigor, or really appreciated all the extra effort. I had a couple of takeaways from this: students can like you if your class is super easy and you basically give them a free period, or they can like you because they know you care and are trying very hard to provide them with the best learning experience possible even if your class is a bit more difficult; and at the end of the day, does it really make that big of a difference? I put a lot of stock into the "idea" that I might make a big difference in the lives/educational careers of a handful of students each year. In the grand scheme of things, was that really that big of a deal when I have 150+ students?

• In response to me sharing with her that there were a lot of students who turned in really poor work and others that clearly cheated: There are always going to be students who don't care. Regardless of the style you teach or kind of tests you give, some students will always do very poorly, and not because they haven't been able to learn the material, but because they just don't care. Some of those same students will also cheat, because they just do not care about the learning and they have always been told that good grades = success, so if they have not put any effort into learning, there is no way for them to get a good grade without cheating. She reminded me that when I gave traditional final exams, there would inevitably be students who did extremely poorly. She reminded me that no matter how much work you put in to something to make it meaningful for students, some of them just won't care and will not try or cheat. If you tried really hard to "fix" those students over the course of the year and it still happens, that's not on you and you shouldn't feel bad about it.

• In response to seeing how stressed out I was over everything at the end of the semester: What is important to me? I am a new father and here I was stressing out over a handful of students who I work with at my job. The stress I was feeling at the end of the semester was like a parasite eating away at my time and the amount of attention I could give to things that are genuinely more important than my job: my physical and mental health, and my family. I have always been someone who has attached a lot of their self-worth to performing well at work. Even when I worked at Target and a pizza place, I always tried to do my best because when I felt like I was doing a good job, it made me feel like I was a good, productive person. She was not saying that my mindset on this needs to change, more that it is already abundantly clear to everyone watching (except me) that I am trying to do a good job.

• In response to me asking if this all seemed like a good idea to her and if I should keep doing it: Not only did she say yes, but she suggested that it might be something worth expanding. She said that maybe part of the reason some of the responses were so poor is because it was the first (second, really) time many of the students were being asked to provide thoughtful responses on a math test, rather than just reproducing solutions to problems they've seen before. If they had to respond to similar prompts throughout the semester, not only would they be familiar with the types of questions and the expectations, but their responses would probably improve in quality as well.

• Lastly, on the topic of gradebook category weighting and making it 100% assessment-based in order to make grading as equitable as possible: Equity is a hard problem to solve. It is admirable to drastically change policies in the pursuit of equity, but there is a reason that homework has been a requirement for students for decades - their brains are still developing and they may not all be able to get to the point where they discover the amount of work that they need to do in order to effectively learn the material and then self-motivate to actually do that work in the absence of an incentive. I've always maintained that practice is an essential part of the learning process but that the amount of practice required can vary drastically from student to student. Due to students developing at different rates, is it actually inequitable to expect all of them to "figure it out" when it comes to the amount of work they need to do? I don't know how to solve this problem either, but it is something I will think about over the summer. Some students actually suggested possible solutions to this in the feedback section of the final assignment I gave, so I will review those suggestions as well. (They included things like having one assignment per unit being mandatory, or weighting assignments in the gradebook at a very small non-zero amount to at least give some incentive while still making it possible for a student who does no or almost no homework to get an "A" in the class.)

Shout outs to Deb Pettit.

Final Takeaways

I don't think it would be fair to call this year's experiment anything other than successful. Even if the outcomes for all students weren't great, it would still have been a success just from the amount of information and experience gained. To be clear, I think the outcomes for students were predominantly good. The students who were going to do well regardless of the class structure still did well, students who potentially would have struggled a lot in other classes due to mandatory work loads were given the opportunity to do well and in most cases did, and there wasn't much of a difference for the students who were probably going to do poorly regardless, except for that they could have done well if they wanted to because all the chances were there.

Another thing that enforces my belief that this was successful was the consistent feedback from students. At worst the feedback was only ever lukewarm, and the main idea behind that feedback was concerning not getting any points for doing homework for one of two reasons: it removed their motivation to do the homework and so they just didn't even though they knew they should have, and there wasn't any grade cushion in case they didn't do well on a test. I've already discussed, at length, why I don't think that second point is valid (both mathematically and philosophically). The first point is potentially valid and, as mentioned previously in this post, is something I will put some thought into addressing. Overwhelmingly, however, the feedback was positive. The main things that students seemed to appreciate this year were the generous retake policy, not having to be stressed about getting math homework done in light of the huge workloads from other classes (especially AP classes) and extracurriculars, the understanding that these changes were implemented not only to improve equity but also to help their personal mental health (many comments about how they could tell that I actually cared about them as people and that they hope more teachers will "follow my lead" in that way), and various comments about specific things that I did or said in class.

The last thing I need to say in this post is thank you. I don't know if anyone actually reads this stuff I write (and I would be very happy to hear from you if you do), but having this blog throughout the year has held me accountable for following up on the things I have done in the classroom. It's almost like a lab report in science. You can do the experiment and see what happens, but the analysis and reflection and summary parts in the lab report are the things that really make sure you learn from the results.

3/30/23 11/30/23 12/3/23 - Assessment, New Thoughts on Grading, and Artificial Intelligence

My toxic trait is that every school year I get obsessed with one particular thing about my teaching practice and then work way too hard to adjust it and make it better. Sometimes good things come out of this, but it also leads to a lot of stress and sometimes friction with colleagues (and my wife, who very generously allows me to have time to do said work). Last year's obsession was equitable grading practices, and this year's obsession is assessment.

I've been thinking of assessment this year from 3 sides:

1. Do the assessments I give really show me if students know the material in a meaningful way?

2. Do the assessments I give even assess the things I really want students to know/leave my class with?

3. How can I do less grading?

I've really struggled this year because while I know none of those three questions are easy to answer, I'm pretty sure they do have answers. Let's look at each of them.

Do the assessments I give really show me if students know the material in a meaningful way?

I think there is a consensus among teachers in general, but especially math teachers, that we don't just want students to know a bunch of facts. What we DO want students to know is how to apply the breadth of knowledge they have ideally learned to novel problems. What that essentially means in math is that we don't want students to simply be able to mimic what we do or memorize how to solve a specific type of problem, we instead want them to be able to be presented with a completely new problem and apply the different rules and processes and logic associated with a topic in order to come up with a solution.

Unfortunately when we give a student a problem like "Solve the following quadratic equation by completing the square 3x^2-6x+2=0", we aren't always going to know if a student understands the significance of completing the square not only as a method for solving a quadratic equation, but for changing the form of a quadratic equation and that it is really just the quadratic formula in disguise. A student answering that question really only tells us that a student can complete the square for a quadratic with a leading coefficient greater than 1, and when the coefficient of the linear term is even. Even then, we don't know if the knowledge the student applied to solve that problem is lasting just by looking at the one success alone.

You might be thinking something like, "Well, isn't it always kind of the case that we don't know if students are going to really remember anything from our class?" Yes, but is it supposed to be like that? If you're a teacher, you've probably experienced the disappointment from realizing that students have crammed for a test in your class and within a week or so have forgotten a lot of the material. This is often evident in math classes where material builds on itself; A student that could add fractions last week according to their test suddenly doesn't understand what a lowest common denominator is this week.

This issue probably needs to be addressed in two ways: get students out of the cram->perform->forget cycle first and give tests that demand the kind of deep knowledge that isn't likely to be forgotten quickly (alternatively, spiral curriculum in such a way that forgetting material immediately after a test will prevent future success).

The first side of this is ideally one of the main benefits of equitable grading practices. According to what we hope equitable grading practices accomplish, if students know they will have opportunities to retake tests and that grades are not being lorded over them as instruments of discipline, they will not feel pressured to cram for an exam they know they are not ready for, and instead focus on building the knowledge that will allow them to be successful in the future. So far in my experience with this I have had a non-zero amount of success, but it hasn't caught on as much as I wish it would. The highly competitive students that I have are still subject to traditional grading practices in other classes, so it is hard for them to change out of that mode when they get to my class. It could also be the case that I haven't done a good enough job of convincing them that I really truly care a lot more about their learning than I do what grade they get. They care more about what grade they get than about their learning, and to some extent it will probably just stay that way until there is institutional change.

The second side of this is the one that requires the kind of time and effort I am short of these days. In my College Algebra class, I have a student who suffered a significant TBI during the previous school year and since then has not been able to experience success on any form of traditional assessment. I have spent hours trying to devise alternate assessments for her for each of the units. My goal is to present her with a set of questions or prompts that require her to have a comprehensive understanding of the pertinent topics, and to give her ample time to demonstrate that knowledge. It has been a tremendous learning experience for me because once she submits her responses I can see if a question did what I wanted it to, and occasionally it does not. To refine these kinds of assessments, which would be called product-based assessments, and the corresponding rubric for each of them, requires a lot of work and trial and error. The other downside is that grading this style of assessment can also be more time consuming, which is something I am not interested in.

Ultimately I guess the direction I want to move to is one where every student takes the aforementioned style of assessment, as I believe that type of thing is a better way to determine if a student has a comprehensive understanding of the material than the traditional assessments I usually give. With time I am sure I can do this for all units in the classes I teach, but I worry about the grading that will come along with it. More on that later.

Do the assessments I give even assess the things I really want students to know/leave my class with?

In many cases we care more about the process than the destination, because "rational thinking and problem solving literacy" is the overarching goal of a high school math program. Of course we want students to get the correct answers to problems we give them, but I think it is fair to say that seeing a student really wrestle with a problem, take risks and make positive progress towards a solution is a more gratifying and meaningful observation than simply seeing the correct answer in the box at the end.

My school hangs its hat on how well it prepares students for college. Indeed, the metrics traditionally associated with that - number of AP courses offered, AP tests taken, and the % of students who enroll in college - suggest that we rank in the top ~25 schools in the country in that regard. But what does "College Prep" really mean? I am about to write a few paragraphs about something that I think I have some idea about, but I could be off the mark. It's a blog that serves as a brain dump for me, what do you want?

Reflecting on my own college experience, I can think of two ways my high school prepared me (or didn't prepare me):

1. An amount of rigor or at least volume of coursework that meant I either didn't have to take certain requisite classes in college (AP), or that I was seeing a lot of material for the second time which made those classes very easy/manageable (Honors, electives).

2. Developing a strong sense of personal responsibility for my own learning, which included organization, time management, communication, and good study skills/habits.

Don't get me wrong, I think both of these items are important, especially in age where college courses/credits and just the cost of living in general are so outrageously expensive (2023), but in my opinion the 2nd one is more significant. If I am being honest, I don't think my high school did a particularly great job of either of these things. That is a bit tough for me to admit because as a high school teacher now, I tend to look back at my high school experience and teachers with rose-tinted glasses. I also recognize, just in the moment while writing this, that the extent to which my high school experience prepared me for college was partially my responsibility; there were some AP courses I chose not to take (English, Chemistry, Government), and I was not immune to procrastination and typical teenage distractions.

I might have mentioned this on the blog before, but as a high school math teacher I can tell you that somewhere in the ballpark of 80% of the material I am required to teach is either really boring, not that important, or both. I have come to love math and think it is a worthwhile subject to study and invest time into. However, I would never claim to students that their success in my class (if that measures their understanding of the material) is going to be a meaningful factor in how their life plays out. I think some teachers, and this is just my observation, hold their subject in such high regard that they tend to lose sight of the fact that a large portion of their students don't actually care about the material and are living complicated and often very difficult lives. For example, the fact that Andrew Jackson served two terms as the President of the United States, that he was the 7th president, and that he was elected in 1828 is a piece of information that I can honestly say is completely meaningless to me. The only reason I even know it is that I recently watched a YouTube video where the presenter called Euler's Number, e, the Andrew Jackson Number, because it is 2.71828...

I bring this up here because I think a lot of teachers focus in on the first item, commonly at the expense of the second. It is a great thing when teachers are passionate about their subject area and its natural that they would want students to feel the same. I recently thought my students were going to be awestruck by how cool it is to discover that multiplying a complex number by another one has the effect of rotating it on the complex plane, but when I looked at their faces I remembered that 95% of them just want to get through the day and might also think I am a big dumb idiot (which is fair, to be honest).

So teachers naturally lean more towards the first item than the second. And? Well, the second one is really important, and we don't, at least most of us don't, regularly hold students accountable (through assessment and policies/procedures) for that. We might regularly remind them of these things, some of us every day, but in a world where school has become so transactional - you do this thing and you get this thing in return (a grade) - students are going to focus more of their effort on what earns them points.

Do I want students to learn the material in my class, especially the 20% or so of stuff that is actually kind of important? Yes, absolutely. But what I really want students to learn from their experience in my class is the importance of taking responsibility for their own learning, which involves managing their time effectively and learning what studying techniques or habits are most effective for them, among other things. If a kid knows every bit of math from my precalculus class they will be able to go into Calculus with a great footing, but if they don't have the abovementioned skills they are susceptible to not being able to overcome challenges when faced with difficult new material. On the other hand, if a student has built the kind of study habits and perseverance I want them to leave my class with, while still having a decent foundation, they'll be able to figure out that difficult new material.

I don't think any of this is controversial. We had a professional development towards the end of last school year where the presenter tasked each department to represent the ideal graduate on a big poster, including all the qualities we would hope to instill in them during their time in our program. Our department's result is below. Notice how it doesn't say take a derivative, establish a trigonometric identity, or complete the square anywhere on it.

12/4/23 - Everything is a lie.

I found out recently that at least one person (besides me) has read this blog. I won't get too much into the details, but my blog was cited in a document that was given to me to look at. I still don't know exactly how to feel about that, but I think it would be a good time to remind any current or future readers that this blog is really only a way for me to get the things I'm thinking about all the time out of my head. I guess if people are interested in my thoughts then that's good, but I'm not doing any official academic research and I'm not writing peer-reviewed journal articles here. As such, please don't treat these words as gospel. I certainly think I'm on the right track, but, as I'm fond of saying, at the end of the day none of us really know what we're doing.

If you've read my other posts on this blog, a common thing I mention is how education is always changing because the students are always changing. When we try to resist that change, bad things typically happen (or at least no progress is made). One thing that never changes, however, is that there is always some "new hotness" in the education field. A buzzword, a pedagogy, some research that promises to be a revolution in the way we teach young people. If you compare a modern classroom to one from 20 years ago, fundamentally not that much has changed. There is usually a lot more technology, but the average classroom then looks a lot like the average classroom now. I can say this with confidence since I was in high school 20 years ago (sidenote: Wow. 20 years!? Hello darkness my old friend...), but I expect if you go back another 20 years more this would still be largely true. This means that the vast majority of those buzzwords didn't quite deliver on their promise. That's not to say that no positive progress has been made, just that we've not seen the kind of paradigm shift that these things often claim to represent.

As is tradition, before this school year (23-24) even had a chance to begin, emails started rolling across my desk about John Hattie and Visible Learning. I'll just put a link to this criticism of his work here, offer a very brief summary, and then attempt to explain why I think we can never trust the latest trends in education. So basically, attempting to use generalizations across different populations and arbitrarily defining a baseline completely invalidates the lion's share of the strong effect sizes that Hattie found in some of the 135 interventions he saw in the meta-analyses that form his data. Also basically, statistics is an incredibly complicated branch of mathematics and most people that don't know how to do it screw it up. This is partly why I made disclaimers all over the place when I presented my own data and analyses in previous blog posts.

I think the underlying reason the education book has not been re-written by this constant stream of new ideas is that the way we define success is inconsistent. Are grades how we should be defining success? Let's say a study says that yes, grades are the best measure of success. See above for my thoughts on grades and how there is no consensus on what each letter means. College acceptance rates? Scores on standardized tests? All of these things are complicated, inconsistent, and have a lot of well-recognized problems. Personally I think a good measure of success these days is just getting through high school alive with a set of skills that enable you to adapt to and overcome the challenges you will face in the next phase of life. I wouldn't know where to start when it comes to measuring that, and I certainly wouldn't want to make it the basis behind a entire book series worth of "research".

Last year, Peter Liljedahl's Building Thinking Classrooms was the new hotness. I don't want to be overly-critical of the work because I have tried many of the strategies highlighted in the toolkit the book presents and many of them seem to be having a positive impact on my class. However, despite being based from his own research, Liljedahl's analyses have a lot of the same shortcomings as Hattie's. Students are always changing. Classroom environments (including resources, socioeconomic status of students, number of students, etc.) can vary wildly from country to country, state to state, district to district, school to school - hell, even room to room at the same school sometimes. What we consider success and how we measure whatever that is can change and be inconsistent in the same way. Attempting to generalize all that data and claim that a thing will for sure work seems a bit silly to me.

So what are we to do? Well, there is only one person (sometimes maybe two people I guess?) in each classroom that knows for sure if something is working or isn't working. It's OK for people to get excited about their own research, write books, and push their own methods. Most everyone in education has the best interests of students in mind (there are definitely exceptions to this, hence the use of the word "most"), so it is hard to fault people for trying. However, at the end of the day, the classroom teacher needs to be the one to give things an honest try, see if they work, and adjust as necessary.

As someone who regularly tries new things in the classroom, analyzes data (as a non-statistician), solicits student feedback, and makes adjustments fairly often, I can tell you that this process can be very productive and rewarding. The problem is that it is also enormously time-consuming. For decades society has pushed more and more of its obligations onto the education system. That increase has not been met with commensurate increases in funding, so the education system exists in a constant state of financial strain. Not having enough personnel results in larger class sizes, which results in more grading/feedback workload for teachers, which results in less time to implement and develop new ideas. Not paying teachers, who are highly-trained, highly-qualified professionals, many with advanced degrees in their content area or in education, enough money for them to support a family on single income does not motivate them to spend time outside of school to do these things either.

One last note on this: I think people often overlook that last point, that classroom teachers are highly-trained and highly-qualified. With few exceptions, teachers go through exhaustive teacher training programs where they learn about educational psychology, best pedagogical practices, laws surrounding education, and tons of supervised real-world training in the classroom with real students. Here is a list of people who have not done this (at least not to the same extent):

• Elected officials

• Administrators

• Parents (Even if those parents are themselves professionals (e.g. lawyers, doctors, or even college professors (college professors typically do not have formal teacher training or certifications)))

I don't mean to suggest that all of these groups need to stay out of all classroom decisions, just that as teachers are the group best suited for making those classroom decisions, the others are best suited for other roles. Elected officials set policies and put laws in place to make sure all students have access to a free public education. Administrators are best suited to taking care of the day to day operation of the school and handling the administrative duties that they are specially trained for. Parents are best suited to support their students at home and support the school's mission through parent associations. Sure everyone has a say in every other thing, but a well-oiled educational machine is one where each group mostly stays in their own lane. Getting on a soapbox and telling teachers that they need to be doing this or that when you don't have expertise or even any training in the matter is a surefire way to lose their attention and their respect.

**THIS POST IS A WORK IN PROGRESS** (12/7/2023)

3/1/2024 - Can't Stop Us Now

The 3rd quarter just ended, and there were more D's and F's on student report cards for my class than ever (except for the online learning year). Over the last 5 days or so I've been wondering why. The following is a post with a bunch of maybe connected thoughts that I hope will lead me to acceptance.

If anything, the changes I've made to my class, from instruction to grading policy, have raised the overall "average grade" amongst my students. So what's different about this quarter/semester? A few things. First, students are really not doing their homework this semester. I don't want to go through the hassle of pulling numbers, but the lack of doing Check Your Understanding Problems is leading students to get less practice than they need. Compounding this problem is a conscious shift in my instructional practices where I provide students with more opportunity to practice, but give them more autonomy over that practice. This is just a change from, "OK, now do this problem" with me walking around the classroom to, "Hey here are a handful of problems, now get into these groups and work on them together at the whiteboards" with my guidance and groups collaborating with other groups.

I've read the research and seen promising results with my own implementation of the "Building Thinking Classrooms" instructional techniques, but there are enough of my students who resist this process - refusing to engage in a task for long enough for it to be meaningful then never end up getting any practice - for it to feel like herding cats at times. These students who don't end up getting practice and often wander around the room interrupting the work of others inevitably do poorly on assessments, but that doesn't fully explain the lower grades. In my older/other way of doing instruction, the "OK, now do this problem" approach, mostly those same students end up stalling worse than Joseph to the point where they never actually engage their mind with the problem and just write down whatever ends up getting written down on the board (which at times is a complete solution, at times is a shorthand solution, at times is just an answer, and still other times is just nothing). These students end up with mostly-blank pages of notes with beautiful headings and fully written-out definitions, or a word problem copied down verbatim with no solution whatsoever. Anyway, there's a lot of ∩ between these groups.

Maybe the rest is explained by my decision at the beginning of the semester to make thinking and genuine understanding a requisite for success in my class. Stating this is also me admitting that that hasn't always been the case in my class. The students I work with are all incredibly capable. Unfortunately many of them decide to not take advantage of their capabilities, or use them in a different ways, and instead do everything in their ability to game the system. I don't blame them for doing this, because the system emphasizes grades and GPA above all else, so they should do exactly what they need to do to maximize their grades and GPA and nothing else. If that means gaming the system then that's what they often do. And the thing is, just like with everything else, they're incredibly capable at it. Many students have done well in my class simply by memorizing large amounts of material in advance of a test, doing well or well enough on the test, then forgetting the material. They can do this in my class for a number of reasons, such as my grading scale and the relative weight (or non-weight) of the final, and are mostly penalized by the nature of material building on itself over the year (typically their scores on tests decrease over the semester, but they still average out to good or good enough).

I really don't like this. This isn't me being self-important or wanting my students to believe math is more important than their other classes - I just want them to learn in my class. What I want them to learn most isn't even the content, it's the resiliency, perseverance, and self-motivation that comes from solving problems for the purpose of solving problems. I want them to be able to problem solve so they can solve problems, and so that they know when they are faced with challenges that they have the ability to work hard and come out on top. I don't know if this is sounds grandiose, because I mostly just talk to myself when I'm thinking about these things and I think everything I say sounds pretty reasonable, but I don't think I'm asking for too much here. I just want my students to try. I want them to try, not give up, experience failure, and experience success. Learning is a guaranteed byproduct of that process. Students who refuse to try or give up at the first sign of adversity or failure are going to do poorly in my class. This is something I can assure through the type and number of assessments I give, along with the contents of those assessments.

Does this really explain the lower overall average grades in my classes this semester? I think so, but to be honest, I don't know. There may be other compounding factors, like how most of the students who are doing poorly in my class also took Algebra 1 during the online learning year, and so this year's class is the spike in learning loss due to insert reason here (ineffective instruction, existential dread over the pandemic, lack of social interaction, pervasiveness of blatant cheating, etc.) coming home to roost. I don't know.

What I do know is that I like the direction things are going. If there are a few grade casualties along the way, I think I'm ok with that. As a colleague of mine is wont to say, "Sucks to suck."

3/9/2024 - MEAD and Authentic Assessment

An email came through in October 2023 while I was isolating in the guest bedroom of our house, desperately trying not to give my wife and son COVID. It was a call for presenters at the upcoming MEAD conference. MEAD stands for Mathematics Educators Appreciation Day, and it consists of a conference for Arizona educators hosted by the University of Arizona's Center for Recruitment and Retention of Math Teachers (CRR) at a high school that is across the street from the university. Over the years I have thoroughly enjoyed attending MEAD and learning from the presenters and keynote speakers, but never really felt like taking on the responsibility of presenting myself. When I read the aforementioned email, I had nothing but time on my hands. "Why not?" I thought. Have I mentioned that I'm an idiot?

The session I volunteered to present would be titled "Authentic Assessment in the Modern Mathematics Classroom", and the description was, "As our instructional strategies evolve to meet the needs of an ever-changing group of learners, we must also adjust our approach to assessment. This session aims to provide discussion and examples of how to authentically assess student learning in the age of AI and collaborative classrooms."

The idea was that the math education community as a whole was being swept away in the Building Thinking Classrooms hype (justifiably), but that the way we traditionally assess students in math actually does not require a whole lot of thinking. While sitting there sick in bed I thought, "I have a lot of good ideas already, I'm reading and thinking a lot about this stuff, and I'll be a pro/expert by the time the conference comes around in late January." I swear, it's like I temporarily forgot I have a toddler.

Anyway, long story short, I procrastinated on this a lot and by the time I knew the session proposal had been accepted I had done next to nothing. At that point I was incredibly busy with other things and ended up only really being able to prepare for the session sparingly over the two weekends that preceded it. Here's the gist of what I came up with:

1. The way we do things now (when it comes to assessment) is sub-optimal. We do it this way because we've always done it this way, and in many (if not most) cases we are asking students to replicate/mimic working out math problems that they have seen before. So...

2. This does not align with the "thinking task" emphasis that is currently sweeping the math education field. Also...

3. When we measure student success by test scores (standardized tests) and reward them with grades which then have an outsized influence on their educational and scholarship opportunities, we are sending the message that performance on tests and getting good grades is what we value the most, when really...

4. We all desire school to be a place where students go to learn, and we want to measure their success at school by their willingness and ability to learn along with the outcomes of their learning experiences. So, in practice,...

5. We need to provide them with many learning opportunities and many and novel ways for them to demonstrate that learning. We need to evaluate the things we are most interested in (evidence of thinking, perseverance, creativity, etc.) and provide meaningful and actionable feedback on those things. And, as a bonus,...

6. These are things that humans can do but robots cannot (at least yet). In a way-zoomed out view of education, hopefully we're taking a path that diverges from the one where artificial intelligences can competently do all the things we request of students.

With all this in mind we discussed what an "authentic assessment" looks like. This is what the group came up with: Authentic assessment gives meaningful information about student understanding of concept; Involves genuine application of conceptual and procedural knowledge, perhaps novel; Has an element of student buy-in; Gives information about how well a student can do the things we want them to be able to do; Values problem solving skills and justification; Requires deep procedural knowledge.

I provided two examples of assessments I intended to give to my students later that school year, one about roller coasters, and the other about an evil mechanical arm. I can't imagine a world that I won't write something more detailed about the evil mechanical arm problem in the future, but I don't think it fits in this post.

The final element of changing assessment is changing grading. The google search term "alternate grading" was a real game changer for me in my reading, but to give you a tip of the iceberg and a single good, simple, clearly-communicated example of something we can do instead of traditional grading, https://rtalbert.org/specs-grading-emrf/. 

After the MEAD conference, I also presented this material in slightly modified format to a group of math professors at the university in a math instructors colloquium. That also went pretty well.

At this point I should remind the reader that I am privileged to teach high-achieving students and have some freedoms in the classroom that not all other teachers have. "GRIBBLE", I hear you saying. "But how on earth does this actually happen in the classroom and what are the results?"

I have so much to tell you.

5/28/2024 - School Year

What I did, how it went, what I learned and what to change for next time

As I've mentioned several times at this point, my primary focus/obsession during the 23-24 school year was on assessment. For as long as I've had the bandwidth to think about it critically, I've known that the "traditional" way of assessing students in math is bad. It is bad because even if we don't mean it to, it perpetuates the study-test-forget game that students are used to playing in their math classes (and other classes), and the kinds of questions we ask are not really requiring them to think and problem solve. Some might disagree with this and that's OK, but I would say two things about that.

First, if we are asking students to solve problems on tests that are phrased and solved similarly to ones they've seen before, then students who are unprepared for the test may end up having a genuine, authentic problem solving experience, while students who have studied for the test or have paid attention in class are ultimately just mimicking the process for solving a problem they've seen before. Second, I know that Bloom's Taxonomy exists, and that when we ask some of the higher level questions which loosely correspond to DOK 3 and DOK 4 type questions, we are often (not always) asking our students to engage in genuine thinking and problem solving. That's good! The problem is that, if we're being honest, none of us really do that enough. I don't blame us for that.

Teaching is hard. In fact, I think for most people it is an impossible job to do. The only way that teaching as a profession exists is because there are actual superheroes walking around who have the powers of extreme patience and time management in addition to being absolute subject matter experts. (I know that I am calling myself a superhero here, but to be fair I am typically very self-deprecating on this blog and I would even say that I am terrible at time management so I'm like only kind of a superhero (like Batman (no shade on Batman, I LOVE Batman, but my dude doesn't have any actual super powers, does he? (Genuine question, e-mail me answers thomas.gribble@tusd1.org))).

As such, we can only do our best at any given time, and sometimes we are forced to make strategic decisions that make our lives easier in one area at the expense of what we know is the absolute best thing to do pedagogically. For me, and I'd imagine this is true for >50% of teachers, at least I hope, that means I give more cookie-cutter style questions on assessments because students tend to do better on those, which makes them easier and faster to grade, which allows me to spend more time doing other things that I need to do to not drown.

So, I guess one way to talk about the big thing I did this year is simply to say that I made a conscious decision to NOT do that. Often that meant that certain other parts of my teaching practice were sacrificed because I spent so much time on assessment, but that was a price I was willing to pay to make meaningful strides towards figuring out how to assess students authentically.

Nuts and Bolts

You know the feeling you get when you assemble a piece of furniture and there are parts leftover? It's like simultaneously: fear, "what the hell did I do wrong!? Is it going to fall apart?"; defeat, "oh my god I have no idea where these are supposed to go and I'm too exhausted to do it over again."; and disappointment, "well, I'm an idiot, oh well." Turns out they usually just give you some extra parts in case you lose some in the assembly process.

For me, the process of trying to figure out assessment this year was like putting together a piece of furniture without an instruction manual and no certainty about what the thing was actually supposed to be in the first place. Forget extra parts, I started out not knowing for sure what I was even building. There were a few things I knew, and at the risk of sounding like a broken record, they were:

-Traditional math assessments (or at least what had been traditional for me and as far as I know all of my colleagues) was mostly testing students on their ability to replicate processes they had already seen before.

-We talk a lot about how one of the things we want most from our students is the willingness and ability to dive in to a messy problem, persevere through the solving process, and come out on the other end having learned something. We typically are not formally assessing that "grit".

-I wanted to provide my students with novel problem solving opportunities, aligned with the learning objectives/standards of the class as much as possible (without sacrificing authenticity), and I wanted to do it is as much as I possibly could.

As you can see, not a super well-defined gameplan. So I started reading. I re-read parts of books that I'd read before, I ordered some new books from amazon, I scoured the web for people who were thinking similar thoughts (there were many), and I started to come up with a blueprint for how I was going to do this.

After MONTHS of trying to figure things out, some of this time happening after the school year had already started, here was my starting point:

1. Think deeply about the kinds of question that require students to apply what they have learned in ways that they could understand but that they hadn't seen before.

2. Leverage my engineering background and general inability to not wonder why certain things are the way they are (the way they are designed, organized, built...) to dream up some actual real-world questions for students to consider and struggle with.

3. Condition students during class sessions to be comfortable with this type of learning/thinking experience.

4. Grade/evaluate student submissions in a way that values the process, effort, and perseverance more heavily than a correct answer.

5. Allow students the opportunity to demonstrate their knowledge on different topics multiple times (whether through retakes/reassessments or "spiral quizzes")

6. Require students to engage in self-reflection regularly throughout the term.

I don't want to lay out every little tiny thing I did or tried to do in an effort to accomplish all of these things, because that would take forever, so I'll just write a little about some of it.

First, I've done a lot of this work already. Throughout my career I have written unique problems here and there that fulfill item #1 on the above list. Although my file system isn't perfect, it is good enough that I could find and compile most of those problems, which was a nice jumping off point. I also have a fair amount of experience under my belt when it comes to generating relatable real-world problems about things students have seen, heard about, or maybe even are actually interested in. For the past 5 years, excluding the tragic online COVID year (20-21), I have been dabbling in an assortment of different grading practices that have all been moving towards what I describe above, so making this final* leap wouldn't be too difficult. (*Spoiler for future post: turns out it was not the final leap.) And lastly, I already had systems in place to help students take responsibility for the learning, including a self-reflection structure and a self-evaluation portion of the semester finals that I give.

Despite having a bit of a head start, things were far from perfect at the beginning of the year. While they measurably improved throughout the year, they never got anywhere near perfect.

How it Went

I don't think it is fair to me, my students, or the end-result of the class to call my efforts this year anything other than a success. There is no such thing as perfect. There were always going to be hiccups, fails, and upset/disgruntled/discouraged students at the end of the year. Part of what I have been learning and remembering very recently is that no one thing is going to work for every teacher and every institution with every group of students. To be successful we simply need to: do our best, ask thoughtful and critical questions of ourselves and our practice, work within the boundaries that we're presented with and occasionally push against them if it is what is best for students, and involve students in the process. I am confident that I did all of that, so I am confident that the year was successful. That being said...

Wow, assessment is a really difficult subject to figure out. Here are some obvious and maybe some not-so-obvious reasons why:

-There are handful established cadences for assessment in a math class that are hard to move away from (unit tests, midterms->finals, etc.)

-Writing more meaningful assessments is more difficult.

-Grading more meaningful assessments can be more difficult.

-Training students to be confident with the kinds of questions seen on more meaningful assessments is not something we (teachers, the education system) are used to doing.

-Tests tend to be very impactful on students' grades (if you have read my previous blogarithms and grading policy page then you know in my class they are really the only impactful thing (this is changing)).

-Students are conditioned to take tests and have a sense of when they should be based on the amount of material that has been covered, and any deviation from that can be scary.

-Similarly, students are skeptical of teachers making major changes to structures that have worked for them (the students and the teacher) in the past, and often need persuading or coaxing to give it an honest attempt.

-School districts (at least mine) would really like grades to be updated frequently, and for those grades to mean anything there ought to be data points in the more heavily-weighted categories so that the grade is more mathematically representative of the student's standing in the class (as opposed to a handful of smaller assignments that do not have a lot of grade inertia).

Writing that last one made me physically ill. I've known this for a while, have been writing about it for a while, have made significant changes to address this over the last 5 years, but only recently have been convinced to tackle it head on - grades have completely taken over the educational system and have informed almost everything that is currently bad about schooling in the United States (and I assume many other countries but I can't speak much to that). However, this is subject of my next blog post - let's get back to how the things I tried this last year did or didn't work.

In my grading policy manifesto I restate what Feldman writes in Grading for Equity about how grades should be "mathematically accurate". To me this means that we need to adjust for the outlier skewing that occurs in a traditional grading systems, but also that we should strive to make the grades we do give as representative of a student's understanding of the material as we possibly can (this is different than grading only assessments). To that end, I think the grades I gave this year are the most accurate I've ever given. The reason I think this is that in total, the work that I graded - students' answers on the assessments I gave - was more representative of their true understanding of the material than it has been in the past. This is because of the simple change I mentioned above, which was to give less cookie-cutter problems on assessments. HOWEVER, there are a few things that I know I could have done better. I didn't see myself making these mistakes in the moment, but upon reflection they seem pretty obvious.

First, I was not consistent about assessment over the course of the year. My explanation for this was that I was trying a lot of new things and wanted to know how they would work, but that is not an excuse. I should have been more consistent and communicated changes and other things a bit better with my students. For example, the first tests I gave in the class were pretty traditional. This is partly due to there being more parity between the different teachers that teach this class at the beginning when we teach a mostly review/diagnostic unit, and partly because I was treading water at the beginning of the school year and couldn't devise a more meaningful assessment for the first few units. In my class I usually give review sheets/study guides to students before traditional assessments, and many of my students understandably use this as their primary preparation tool (disappointingly, many of them use it as literally their only preparation tool, which is something I am going to try to address in the sweeping changes I'll be making next year). Although I told my students at the beginning of the year that tests in my class would focus more on conceptual understanding than simply replicating the solutions to problems similar to one's they'd seen before, the class didn't start out that way. Then, when we were going to have a more "conceptual" assessment (one that fits the description I talked about above), while I did tell them this was going to happen, I probably did not communicate what that actually meant as well as I should have. The first time this happened, the test they took did not actually look a lot different from the very normal one they had previously taken, but the questions did require more critical thinking and there were not similar questions (like for like with different numbers) given on the review sheet or study guide. There were a lot of questions where as long as the student had a good conceptual grasp of the concept, the process for solving them was relatively straightforward. Students in my class were more used to studying a range of different problem types and memorizing how they were solved (so, solution over process over concept). And that's also mostly if not all my fault, which leads me to another thing I didn't do that well.

I did not give students enough experience in class working through these "different" problem types. I allot more than enough time in my class for practice, but often that practice is in the form of thinly-sliced, ever-more-difficult versions of similar problems. Basically the problems that I gave most frequently as practice did not require students to apply the concepts in ways other than how they would apply them on the type of cookie-cutter problems I've described. As a result, they didn't have much of a reason to think it would be prudent to practice the skill I desired them to have, and even if they did, they wouldn't have had much of a library of problems to work through (because even thought they exist, they don't exist in large quantity and they are certainly not in the textbook). I think naively my hope was that by implementing some of the Building Thinking Classrooms pedagogy/toolkits into our routine that students would naturally be inclined to make sure they had the concepts down (knowing the justification for why certain things are done in the problem solving process). I think that did happen in my class, just not to the extent I had hoped, and so students were typically not as prepared as I assumed they were. To be fair, I did work on this consistently throughout the school year (cueing students to understand why rather than just how).

The other thing that I caught myself doing multiple times throughout the school year but still did anyway, was completely change the "look and feel" of the actual assessment documents and in some cases the system I used to grade them. A small example of this was that I removed the space for students to write their names and instead I assigned them a randomized number. This was done in an attempt to minimize any biases I might have had towards different students. I had done this before in my career and I think on the surface it is a good idea. The problem is that I didn't tell them about it before hand so a lot of them were taken aback right at the start of the test and kind of needed a minute to get back on track. Another problem was that then on the next test I didn't do that, and then I did it again on the one after that. The reason for the inconsistency on that middle test was just that I forgot to do it before making the copies. However, this gave students whiplash and not knowing what to expect, structurally, on an assessment is not something that sets you up for success. 

A larger example of this was during the 2nd semester when I made two quite significant changes on a single assessment that happened to be on polynomial and rational functions. Not only was the test a more conceptual one, but I A) decided that I would try something different for the retake/revision process and B) tried to use a completely different grading system for it (an EMRF-style rubric (Exceeds expectations, meets expectations, revision required, non-assessable)).

I want to go on a quick stroll off-topic here and write a little about these two things because I think they are really good ideas. I don't know if I will be doing them again next year just yet, but that is not because they aren't good ideas, just that I'm not sure if what I am doing next year will really require this kind of thing. 

First, the revision process. For the past 5 years in my class, "the retake process" has been basically this:

1. Receive feedback/marks on original test and decide if you want to retake any parts of the original test to improve score.

2. Fill out a "Test Wrapper" that includes 7 steps wherein students identify what items they did poorly on, reflect on why they might have done poorly, re-learn or re-familiarize themselves with the material, perform test corrections, write a statement about what they can do differently on the next attempt to try to improve, and then schedule a time to sit a retake. (Said retake does not include any limitations to how well you can do on it and there are no restrictions to who can do retakes.)

I think this is drastically better than just handing students another version of a test on a specified date, but especially during the first part of this last school year I grew increasingly unsatisfied with it for a few reasons. Students were still obviously trying to "game" the retake system by completing the test wrapper quickly and using reference materials (another student's test or photomath), and were not actually making a meaningful effort to relearn the material. My suspicion, which I'm pretty sure is correct, is that they just wanted to intensely review one very specific type of problem and essentially memorize exactly how to do it, because they thought they would get a nearly-identical problem on the retake. This is, as I've highlighted before, not learning, but mimicking. The reality is that the problems they were going to receive on the retake were not nearly-identical, but still assessed the same conceptual knowledge. So if a student developed a deep understanding of the mathematical relationships at play in a certain concept, they would be successful on the retake regardless. This was always my intention with the Test Wrapper thing, but in practice it didn't work out that way - the understandable desire for students to take shortcuts to earn a better grade (because they have 5 other classes, limited time in the day, and have been trained for double-digit years that grades are the only thing that matter) was frequently too much of a factor.

The problem was not in the procedure or the document itself, and I repeatedly lectured my students on the importance of gaining those strong conceptual/general understandings of the topics rather than trying to memorize how to do certain problems. As mentioned, my belief is that students were just trying to game the system. If a system is gameable, students will game it. They are gamers. They have been trained to play the game. We can't blame them for that.

So, what I tried to do later in the school year in the abovementioned unit on polynomial and rational functions, was create an un-gameable retake system. It worked like this:

1. Students took the original assessment.

2. I marked the original assessment based on the EMRF/N rubric that I linked to above (I altered it slightly to look more like this: https://rtalbert.org/emrn/ I told students about this rubric well before giving them the test and provided it for them on the actual test as a reference). Their tests were also returned with various usual "feedback" notes like check marks, x's, questions, etc. that attempt to sus out where they went wrong on a problem without giving away exactly what they did wrong.

3. Students made a note of what they received (E,M,R,N) on each of the 4 sections of the assessment, and I collected the original assessment back.

4. Students decided if they wanted to improve upon the EMRN mark they received. Note that at this point students were not given any information about how individual EMRN marks correlated to the numerical scoring system we usually use in class. So when students were making the choice to improve or not, they were not necessarily doing so out of a motivation to get a specific numerical score (which they could then use to estimate how it would impact their grade in the class (or not estimate because they could just use the stupid calculator built-in to the student information system and know exactly what their grade would be in the class OMG this is such a terrible feature don't even get me started)).

5. I provided a "Revision Ticket" to each student. This was basically a very-targeted study guide/review worksheet that asked questions that required a strong general understanding of the basic concepts needed to be successful on the original assessment. The questions themselves were not exactly like the questions on the assessment. They were much simpler, but if a student answered them correctly and knew why their answers were correct, they could apply that understanding to the questions on the original assessment.

6. Students would show this to me to ensure that their answers were correct, I would ask some probing questions to determine if they really understood, and then I would hand them back their original assessment so that they could perform revisions in a different-colored pen or pencil. They could use the revision ticket they completed while they completed the revisions. This took place either during class in a planned, dedicated time, or outside of class during a conference or lunch period.

7. After they had revised their original assessment, I would go back and look at the new work and re-mark on the EMRN rubric, which I eventually translated into a numerical score that went in the gradebook.

Something I really liked about this was that students were still hopeless if they did not take the time to actually understand the concepts that were being assessed. Simply a complete and correct revision ticket was unhelpful if there was not genuine understanding, and since a successfully completed revision ticket was meant to represent genuine understanding, I had no problem with allowing students to use it as a reference when they did their revisions. Another thing I liked was that since I re-collected the original assessments from the students, they could not closely examine the specific problems I gave and memorize how to solve them. Instead, they were forced to re-visit the underlying material, gain solid, comprehensive understanding, and then only after showing me that new understanding would they be able to revise the original problems.

Something I didn't like was that it was more work for me. I also didn't like the 10 million questions from students about what an E, M, R, and N meant in terms of points. I intended the actual revisions themselves to be done very quickly, because my thought was that if students understood the concepts in the way I hoped they would after completing the revision ticket, applying those concepts to the original questions would be easy. While that did happen for a lot of students, it was still the case that students were spending way too much time on the revision. I suppose I could have crafted the special simple conceptual questions a little more optimally, but I suspect that many of the students who took a long time on their revisions were still trying to take shortcuts. I know that is not true of all the students who took a long time on their revisions because I watched them throughout the process and math is just really difficult for them, and I am very proud of those students.

OK, so that wasn't very quick, but the other thing I meant to "quickly" talk about was the EMRN rubric itself. My next blog post is going to be a hit piece on grades, among other things, but this EMRN thing is one of many ways to make the idea of evaluating students more meaningful. One way it is more meaningful is just in what the letters themselves represent. Like, "E" means something. It means that the work is Exemplary/Excellent, and is further Elaborated on in the rubric itself. Reader, what does an "A" mean? Absolutely aNothing aSpecific. Does a "B" mean barebones? Basic? "C" means caca? Another way it is more meaningful is that there are just two binary choices made in assigning each letter. The first question the person reviewing the work asks themselves is if the work demonstrates thorough understanding of the concepts/does it meet expectations of the assignment. After that you ask yourself a follow up question to determine which specific letter is assigned to the work. Since everything is clearly laid out in the rubric, it is usually quite clear why a specific letter was given. Another way it is more meaningful is that it is intended for growth. An R or an N should signal to the student that they need to revise or re-do the assignment in whichever way the assignment parameters permit. This is because the R and N actually mean something. If a student recieves a D or an F on an assignment, there is no implication that goes along with it that they should try again. They just got a D or an F and that's it. Sucks to suck I guess?

6/26/2024 - What if we... (incomplete 7/22/2024) (completed 12/28/24)

I read a book this summer called Undoing the Grade: Why We Grade, and How to Stop by Jesse Stommel. I've read some of Stommel's work before in Blum's Ungrading, which I read a few years ago concurrently with Feldman's Grading for Equity, and I'm a big fan. Something that's been hard for me to gauge since embarking on my journey towards creating a better learning experience for my students is if I have been radicalized, and if, because of that, whenever I read anything about "progressive" education, I agree with it so much due to confirmation bias. Stommel has a way of laying things out clearly, presenting the common questions/criticisms of the "ungrading" movement, and letting the reader decide for themselves if they agree. I agree. I strongly recommend all teachers and administrators read this book. There is no prerequisite for reading it, and it's pretty short. A lot of what I am going to write about in this blog post is informed by or comes directly from it.

I am going to try something with my class this coming year that is pretty radical. As of writing this right now on June 30, 2024, I'm not completely sure I'll be allowed to do this (I will probably do it anyway). If there is something that I am going to try that I might not be allowed to do, I figure I should front-load a fair amount of justification before I even say what it is. So here that goes.

One of the most-referenced pieces of writing on the topic of grades is The Case Against Grades by Alfie Kohn. You should read it. Here is a quick taste (and another one after it):

"...when students from elementary school to college who are led to focus on grades are compared with those who aren’t, the results support three robust conclusions:

*  Grades tend to diminish students’ interest in whatever they’re learning.  A “grading orientation” and a “learning orientation” have been shown to be inversely related and, as far as I can tell, every study that has ever investigated the impact on intrinsic motivation of receiving grades (or instructions that emphasize the importance of getting good grades) has found a negative effect.

*  Grades create a preference for the easiest possible task.  Impress upon students that what they’re doing will count toward their grade, and their response will likely be to avoid taking any unnecessary intellectual risks.  They’ll choose a shorter book, or a project on a familiar topic, in order to minimize the chance of doing poorly — not because they’re “unmotivated” but because they’re rational.  They’re responding to adults who, by telling them the goal is to get a good mark, have sent the message that success matters more than learning.

*  Grades tend to reduce the quality of students’ thinking.  They may skim books for what they’ll “need to know.” They’re less likely to wonder, say, “How can we be sure that’s true?” than to ask “Is this going to be on the test?”  In one experiment, students told they’d be graded on how well they learned a social studies lesson had more trouble understanding the main point of the text than did students who were told that no grades would be involved.  Even on a measure of rote recall, the graded group remembered fewer facts a week later (Grolnick and Ryan, 1987).

If you are a teacher, think about if any of this is true for your students. Was it true for you as a student? Repeated research on the topic of grading backs these findings.

I like the whole work, but the last two paragraphs are a highlight:

Indeed, research suggests that the common tendency of students to focus on grades doesn’t reflect an innate predilection or a “learning style” to be accommodated; rather, it’s due to having been led for years to work for grades.  In one study (Butler, 1992), some students were encouraged to think about how well they performed at a creative task while others were just invited to be imaginative.  Each student was then taken to a room that contained a pile of pictures that other people had drawn in response to the same instructions.  It also contained some information that told them how to figure out their “creativity score.” Sure enough, the children who were told to think about their performance now wanted to know how they had done relative to their peers; those who had been allowed to become immersed in the task were more interested in seeing what their peers had done.

Grades don’t prepare children for the “real world” — unless one has in mind a world where interest in learning and quality of thinking are unimportant.  Nor are grades a necessary part of schooling, any more than paddling or taking extended dictation could be described that way.  Still, it takes courage to do right by kids in an era when the quantitative matters more than the qualitative, when meeting (someone else’s) standards counts for more than exploring ideas, and when anything “rigorous” is automatically assumed to be valuable.  We have to be willing to challenge the conventional wisdom, which in this case means asking not how to improve grades but how to jettison them once and for all.

So this is kind of the basis for why grades are bad, but I've said before in this blogarithm that research in education is incredibly difficult to conduct well due to the subjects (children) - the ethics of doing research with children and the ever-changing nature of them.

^^I wrote the above in June/July 2024 and the rest of this post is being written in late December 2024.

I guess this means I'm being courageous. This upcoming school year, I won't be giving grades. I will enter grades into the gradebook at the end of the semester as my school district requires, but I will not be determining those grades, I will not lord them over students, and, to the extent possible, I will try to convince my students to not think - at all - about their grade in math. I'm writing this in December 2024, having completed a full semester of the school year where I did, in fact, not give grades, but even before the semester and running this by my school's administration, I could foresee some of the skepticisms. I think those skepticisms can be grouped into two categories:

1. How will student grades be accurate?

2. If students are not getting a grade, will they even do anything?

If you expect me to just blast these two points with research-based arguments then you're partially correct, but it's worth understanding where they're coming from. It's kind of a good thing I couldn't finish this blog post before the semester, because now that it's over I can give actual answers to them. They are: 

1. They won't be that accurate.

2. Yes.

For all the reasons I've discussed on this blog and more, I'm not convinced that grades have ever been truly accurate. If you give every student the exact same assessments and mark every one of them with the exact same rubric, then you will have numerators and denominators that you can compare and then rank from greatest to least. This evaluation of a student doesn't consider individual differences. If for whatever reason a highly knowledgeable student is not able to convey the extent of their knowledge on the assessments you gave, does that mean they should get a lower grade in the class than a comparable student who can? Furthermore, do we always have full confidence that the assessments we give are written in a way where student responses are accurate measures of what they know? Are the questions confusingly worded? Does every teacher mark an arithmetic error off the same way? What about two arithmetic errors?

I have to believe most teachers know about these inherent weaknesses with the way we assess students. All of us (I'm sure this is true for teachers of every subject but I'm definitely thinking about math teachers here) have grappled with whether to take points off for an arithmetic error here or there, or to search for context in a student's work to see if this was a significant misunderstanding of a concept or just a silly mistake, and then trying to figure out the severity of that mistake. We've all probably played the game at the end of the semester where we looked at the grades that show up in the gradebook next to student names, then thought more holistically about that student and their effort in class over the term and made (or didn't make) tweaks somewhere to bump them up a letter (or not).

I'm a big believer in equitable grading practices, and in theory working towards making the grades in your class motivational, mathematically accurate, and free of bias are things all teachers should strive to do no matter what. I don't think any of that is controversial and I will tirelessly encourage all of my direct and indirect colleagues to pursue those goals. The problem that I see, as someone who at this point is so red-pilled by progressive literature on education that I might as well be Morpheus, is that grades themselves still stand in the way of what we want students to do in our classrooms: learn. Learning requires curiosity, risk-taking, and reflection. Grades squelch curiosity, discourage risk-taking, and ignore reflection. Even equitable grades.

Equitable grading seeks to remove the "stick" part of grades as a reinforcement system. Not giving grades removes the "carrot". So with no threat of bad grades and no promise of good grades, will students even do any work?

Before the semester I would say, "I hope so". I had to believe that students would do work, and when they did that work they would be doing it for no other reason than trying to learn the material. After the semester I can say that that was mostly true. Students definitely did do work, and I can be far more certain than I've ever been before that they did that work primarily because they were trying to learn the material.

Before the semester I was optimistic. After the semester I'm encouraged. That doesn't mean everything went well.

So...What if we just didn't give grades? Read about the semester on the next episode of Dragon Ball Z!

 1/5/2025 - Going Gradeless (S1 2024)

So just don't give grades right? How hard could it be?

Well, if you're a teacher reading this then you are either curious about how I managed to jump through all the hoops, or curious to see how badly everything went.

No matter where you work, there is likely some sort of institutional inertia behind grades and even prescribed grading systems. In some cases this isn't completely intentional - someone originally selected the "default" options for how to do grading and figured it would just be the easiest route for everyone. In others, it definitely is. The want to standardize assessment and grading across a department, school, district, or state in an attempt to justify funding can be strong.

I wouldn't say my situation is especially well-suited for making wholesale changes to my grading system. I work in a very large district in a state that competes for the bottom of the list in education on a yearly basis. However, I happen to teach at one of the top-ranked public high schools in the country and have been in my role for long enough and done a good enough job that for some reason people at least kind of trust me to make good decisions about how I do things in the classroom.

All this to say, when I broached the subject with my department chair and admin before the school year started, they at least agreed to hear me out. 

Before I became a teacher, a large part of my job co-owning a small business was basically sales. I had to do market research, come up with a product, build a compelling story around it, and do my best to tell that story to potential customers. It sucks that selling this idea of not giving grades felt a bit like old times for me, but I think it worked out. Here's what I did:

First, I did market research. I read countless blogs, many essays, a few research papers, and two books about alternative grading. I took notes on post-its, consolidated those notes a few times, distilled the main ideas down into a short list, and then came up with a product.

The product that I would use to not give grades would look like this:

• Develop a "Mastery Log" that students would use to keep track of their mastery level of each "topic" or "standard" that we covered in the semester.

• Develop an assessment structure where there is built-in overlap of concepts, which would constitute multiple mastery attempts at each concept.

• Have students routinely reflect on their learning process.

• Keep records of student performance and progress in parallel with students' own records

• Have students do a final reflection at the end of the semester that culminates with them assigning the letter grade that they feel is appropriate based on evidence from their mastery logs and reflections.

I need to give the administrators at my school a lot of credit for trusting me with this. As someone who has been engaging with these "progressive grading model" ideas for a long time, I am borderline if not totally radicalized against grades. At the same time, I recognize that you can't just not submit a grade for a student at the end of the semester, because that information is tied to credit status and GPA and so many other things. I hate it, but in the system my district has (for now, at least), grades are an absolutely necessary part of the equation. This is the world that admin exists in and they themselves kind of exist to make sure that everything is taken care of so that ultimately grades can be submitted. As I write that, it disgusts me that grades are basically the product/currency of school. This same system has used grades as sticks or carrots for so long that it is difficult to grasp that teachers teaching students can happen without grades (meaning, that learning can happen without grades). So, my pitch to them required that they place a lot of trust in me that I would get grades submitted, but more importantly, that none of this meant I was not going to be teaching and requiring that students engage with and learn the material.

Obviously admin agreed to let me do this or I wouldn't be writing this right now.

The Journey

I'm sure I've said this before, but even if I had an entire summer break to work on planning for every possible scenario, something would still go wrong that required some on-the-spot diagnosis and treatment. The reality is that I don't have the entire summer because I have a family and they take complete priority while I try to process the previous school year and prepare for the new one in the scraps of time that become available. This year I only had a solid week to try to prepare for the arrival of students. Unfortunately that week also includes a lot of mandatory pre-service meetings on campus and getting the physical learning space (classroom) set up. Also unfortunately I randomly sprained my ankle pretty badly which put a wrench in the gears.

So I'm just going to say this upfront, I wasn't able to get any of the things I needed to do done in a way that I was happy with. My classroom wasn't completely organized (still isn't as I write this in the final days of Winter break), I didn't have content prepared for the first couple of weeks of school (I have two new-to-me preps this year so that's a lot of work), and, most troublingly, I didn't have my mastery log and assessment system ready to go. This seems to be the new normal for me - going into things somewhat unprepared - but I definitely had more than concepts of a plan. As a result, there was a fair amount of developing and tweaking that took place during the first several weeks of the school year.

Here is an example of what we ended up with for the mastery log.

(note: I would just share this file but it's a page in a OneNote notebook so I don't think it shares nicely. It is not that difficult to replicate this by hand if you wanted to.)

What I was trying to accomplish with this was to give students somewhat of a map for what the concepts they were going to be learning were and then a way for them to record their performance on assessments of these concepts and reflect on those performances. One thing (of many) that was challenging for me with creating these was determining what to include in the list of concepts, what concepts may be better grouped, and what concepts may be better to exclude. Part of what made this difficult for me is that both of the classes I'm teaching this year are new to me (I have taught Algebra 1 before but it was fully-online during COVID, and I have taught Precalculus before but the AP curriculum is structured quite a lot differently than what I've done before), so I wasn't intimately familiar with what the most important concepts were and how they are meant to be presented. For example, I didn't have time to read way ahead into the AP Precalculus curriculum in the second half of Unit 1 where we discuss rational functions. Section 1.9 is about vertical asymptotes in rational functions, and section 1.10 is about holes in rational functions. These are quite similar ideas and could be consolidated into a single "topic" or concept called "Discontinuities of Rational Functions". The end result is that I think I ended up with far too many topics/concepts in these mastery logs. AP Precalculus ended with 41 topics assessed, and Honors Algebra 1 ended with 46. Ideally I would like these numbers to be less than 30 for an entire semester. I have read about different examples where the topics are more generally or broadly defined so there end up being somewhere between 10-15 for a semester, and that sounds lovely to me but I would need to read more about how nuance is handled in those cases.

Another thing that was challenging for me, which is also due to just not having the experience with the courses/curriculum and not having enough time to plan ahead, was that I was delayed in getting these to the students. In most cases they ended up having these to look at only after they were assessed on the material. This was still useful but negates a lot of the beneficial parts of having somewhat of a curriculum map for them to look at. There were other things that I know I did wrong with these mastery logs, such as starting out not including the "What success is" column, and I'll write about what specific tweaks I have in mind for mastery logs in a later section of this blog post.

So what information do students put in the mastery log? I'm going to give you a complicated and confusing answer to this question. I recognize that that is a really big problem. I am still thinking about how to address this problem and maybe by the time I'm done fleshing out this blog post I will have some ideas about that that I can write about. Anyway, consider the following pictures as I try to explain this.

 The first picture is an example of what a basic problem might look like on a quiz or a unit test. What you should notice about this is the [1.4A, 2.1B, 2.3] tag that is next to the problem. These numbers and letters communicate which "topics" are being assessed by this question. For reference:

• 1.4A Equations and Inequalities - Understand the concept of an equation or inequality

• 2.1B Algebraic Basics - Find solutions to equations.

• 2.3 Properties of Equality - Recognize and use the properties of equality in solving equations (specifically for this problem, the Zero Product Property).

The next picture is of an example of a student response (black) with my markings (red). [note: this is not actual student work, I just made something up]. What you should notice here is the nature of the marks and also what is missing: a score or letter of any kind indicating a final judgement of how the student did on the problem.

The final picture is an example of a slip of paper that a student would receive when the tests are handed back. For an entire test, this slip of paper would have an entry for each of the topics that have been tagged in the questions. (One per topic, not an entry per question. So, if there were several questions assessing the 2.1B topic, there would still only be one 2.1B entry space.)

When students get their tests back (along with the slip), I will briefly go over some of the common mistakes that I saw when going through all the tests and giving feedback. Students then use that time to go through all the feedback, make notes to themselves on the test and potentially correct missed questions, then fill out each entry on the slip with an E, M, R, or N according to the rubric the class adopted at the beginning of the semester. This is self-evaluation, and in most cases I have already recorded my own EMRN marks for their tests and entered them in Canvas. The students' self-evaluation marks would at some point get transferred into their mastery logs with notes and reflection. Some students did this right then and there (great!), others did it on the days that I had set aside for just that purpose (good!), others waited until the last minute to do it all at the end of the semester (bad!), and a handful just never ended up doing it (wtf!).

The process itself might not sound that confusing to any teachers reading this, but it is a bit of a logistical challenge and is asking students to do some things they have not been asked to do before: self-evaluate and look at a holistic performance (across entire assessment) rather than a granular one (getting individual questions right or wrong). I'm not going to pretend that this process went off without a hitch. Students definitely got more comfortable with this over the course of the semester, but I also improved the process a lot from where it started. As an example, introducing the slips mid-way through the semester helped students understand what marks they were meant to be putting, but at first I didn't have the descriptions of each topic listed on those slips, which meant they didn't automatically know what actual topic each topic number was.

For the make believe student in the example shown above, me, I would say they have a good understanding of inverse operations and just recognizing that the statement is an equation meant to be solved, but they are missing something when it comes to the Zero Product Property. So maybe like an M, M, R in the self-eval would be appropriate for this student.

Here's the main thing I would like anyone reading this to take away from the mastery log stuff:

• Emphasis on reflection

• Self-eval and reflection leads to metacognition and gives students a voice in how they are being "graded"

• No evaluative letters or numbers (A-F, numerators/denominators) appeared anywhere ever in my feedback. Students frequently asked "so what did I get?". Ha! Nothing! Look at the feedback! And they did. And they thought about it. And, hopefully, they made a note (either physically in the mastery log or mentally) about where they should work to improve. [note: I am certain that some students never did this last part, and as a result almost all of those students did quite poorly in the class.]

A Note on Overlapping Assessments

I'm going to use an Algebra 1 example here because probably everyone reading this blog can relate. At the beginning of an Algebra 1 course, something that is taught is inverse operations. If you are asked to solve a mathematical equation like 5x-3=7 for the variable x, the objective is to manipulate that equation to form an equivalent equation in the form of x=a, where a is the solution to the equation. You do this in adherence with the properties of equality (what you do to one side you have to do to the other, or keeping the scale balanced), and you employ inverse operations to achieve it. For example, in the given equation the 3 is being subtracted from the 5x. The inverse operation of subtraction is addition, so a good first step to would be to add 3 (inverse operation of subtracting 3) to both sides of the equation (Addition Property of Equality). This results in 5x=10. Now, the x is being multiplied by 5, so we should divide (inverse operation of multiplication) both sides of the equation by 5 (Division Property of Equality). The end result is x=2, which is an equivalent equation to the given one but in the desired form x=a.

So we teach that, we assess it, and then a couple of months later we are asking students to solve systems of equations by substitution and elimination. A student simply cannot solve a system of equations without knowing all of that stuff about inverse operations and properties of equality. Let's say a student was a slow starter and had some problems with the inverse operation or properties of equality stuff. (A common case is if a student divided both sides of the equation by 5 first (which is not incorrect), but neglected to divide the -3 by 5 as well (which leads to a wrong answer and indicates that a student is missing some knowledge about the Division Property of Equality or maybe the Distributive Property or whatever)). So they did "poorly" on the early assessment. Two months later they are successfully solving systems of equations using the elimination method. They clearly know all of this stuff now. Let's say they ace the assessment on systems of equations. The traditional system is going to give them some kind of average of the two assessments. That is just plain absurd.

This can also apply to shorter timeframes. Let's say there is a mid-unit quiz that a student does poorly on (somewhat expected). They receive feedback and note where they need to improve. A week or two later is a unit exam and they do much better, maybe because they have spent more time on the material, or maybe they also went back and filled in the gaps of their understanding of the things they missed on the quiz after reading the feedback. Either way, in a traditional system there is going to be some kind of average computed numerically between the two assessments.

Tagging each question with all the relevant topic numbers is a way to account for this growth over time that is not possible in a traditional system (but teachers do try to consider this with category weighting, lower point values for "quizzes" vs. "tests", etc.).

This semester I tried my best to make sure each standard/topic was assessed multiple times by default. That could mean they were directly assessed on a quiz proceeded by an exam, and then maybe again on an exploration, or indirectly on a future assessment that requires knowledge of that previous topic. I did not do a perfect job of this but I am happy with what I did and know I can improve on it fairly simply next semester.

Results

"OK cool. So at the end of the semester, the students just give themselves a grade and you put it into the report card and that's it?" Pretty much, ya.

"Won't everyone just give themselves an A?" Maybe, probably not.

"Grade inflation?" I do not care about this.

I have data to share now. First, though, let me quickly describe the process a little more. The last assessment of the semester gets completed with a good week-ish remaining. All of a student's assessments are meant to be stored in a folder in the filing cabinet in my room. Students will retrieve that folder that has all of their self-evaluation slips and all of my feedback. Some of them will have done a good job maintaining their mastery logs up to this point, others will have not. Either way, they spend class time going through all of their assessments and all of the feedback, and they populate their mastery logs. This can be a time consuming process that includes a lot of reflection. They are able to see their progress/growth over the course of the semester and just generally develop a sense of how well they did in the class in a way that is much more meaningful than a percentage or a letter. Since this pertains to a student's performance in the class, I'm not sure if I'm allowed to share an individual example of a completed mastery log on here. I will look into this and if I can, I will update this blog and include it just after these words right here.

So anyway, here is some data. 

I have a lot more data than this, more data than I know what to do with actually, but I read your mind a little earlier and this is probably what you wanted to know about how it all went.

The first graph titled Grade Distribution is just the distribution of letter grades that my students received. Periods 2 and 6 are Honors Algebra 1 (A1) and Periods 4 and 5 are AP Precalculus (PC). The second graph is a comparison between the overall grade distribution in my classes against the overall grade distribution of every other section of each class that was taught (thank you colleagues for this data!). The table shows the data on agreement vs. disagreement. The intent with this is to show how well student self-evaluations aligned with what I would probably give them if I was grading traditionally. In the table, "Higher" means they gave themselves a higher grade than I would have given them. I disaggregated the data to show Honors Algebra 1 (freshman) vs. AP Precalculus (mixed, but mostly upper-classmen) and male vs. female (according to what is listed on the student information system) because a colleague was curious about it and I thought it might be interesting.

I'm not entirely sure I know how to comment on the results I just presented. Are the findings good? Are they bad? Is this proof that all of this works? Is this evidence that the semester was a disaster? I think I have a good guess about the answers to all of these questions (YNYN), but ultimately that's not important. Here's why:

The data I presented above is entirely about grades. Grades are stupid. Grades harm students. The research on this is crystal clear. (By the way, I finished reading another book about all this recently and it was fantastic - I might write about some of what I read in a future blog post.)

There are really just two things that I care about when determining if this "experiment" was successful or not. The first is if my students were able to focus on learning the material for the sake of learning the material without stressing out too much about their grade in the class. This first thing is far and away the most important to me. The second is if my students actually learned the material in the absence of grades being an incentive (carrot) or threat (stick). This is only less important to me because I know the answer to this one throughout the semester when I give students feedback and discuss performance on assessments with other teachers who are teaching the same classes and are using more traditional grading models. With that in mind, here is a bit of additional data.

Q: Leave some feedback. (Example questions that you could answer while leaving feedback: How did the grading system work for you? Was it more or less stressful? Do you feel like you were able to focus more on learning? Was the material presented clearly? Do you understand the expectations of the class? Did you learn something that was particularly interesting to you? How are things going generally?)

Not all students answered, not all students who answered left meaningful feedback, and not all students who left meaningful feedback wrote about the grading system, which is the topic of this blog post; but what you see above is feedback from my students after they had entered their grades for the class and completed the aforementioned mastery log work. I truncated a lot of the responses to keep just the relevant information. You'll see that it is not all positive. However, I would say it is overwhelmingly positive. Several of the responses gave me goosebumps because it seems like the message of all this really hit home with a lot students. Here are some of the highlights for me:

• Thank you for this grading system - I can focus on the joy of learning instead of trying to aquire an intangible measure of intellegence by a mark of a letter. I feel that you believe in me and I appreciate the graciousness and patience you show me with my questions and the class. This is what I want true learning to look like - I like that I don't have to strive to prove my understanding by a letter grade; you acknowledge intelligence is multifaceted.

• For the feedback, I feel like this grading system did two things for me, which made my learning experience better in a lot of ways. First of all, I am super less stressed about tests, although they do account for my "grade", I feel like I have a breathing room for failure and learning, rather than some stress that something will literally affect me for the rest of my life/high school career. I feel like it made me perform better on tests/quizzes overall, as I can actually relax and make mistakes and learn from them. Second of all, I understood more of what the class expects and it's overall much much more engaging than all my other classes. I feel so much more free with this system of grading than I would with a normal class which forces me to make all-nighters a common occurrence. Finally, to end off this long Feedback, at first I didn't know how to feel about the system, but so far, it's worked in my favor rather than against, math is definitely a subject that benefits from this grading policy as I feel like a lot of people benefit from a lax policy to learn at their own pace and make mistakes that don't cost too much. It makes it much more enjoyable and easier to learn, as you aren't stressed about "I NEED to understand it NOW," rather its at a good pace so that people can catch up..

• I loved the grading system, it was much less stressful than the traditional grading systems. Usually when I take tests, I get nervous and it is much more high stakes and sometimes that inhibits me. In contrast, with this grading system it was way less stressful and I think it made my test scores improve even more than the traditional grading system. Also, I understood all the content very well because of the feedback I would receive and tests and it was less final than traditional tests, and I was able to actually learn from my mistakes. 

• I am truly pleased with the grading system in this class because it not only understands the pressure student's feel about their material letter grade and eliminates it but I do feel like it helps with retention of information. I like to think of myself as a person who is generally petty good with math but when I started this class I found myself struggling with the heavy concepts thrown at us. I used to think of math as only numbers but now concepts are incorporated. I feel this class has stretched my mind in math and that was only possible because of the way you teach us the material. You provide quality feedback on all our work and teach in a manner that is serious but still includes humor which makes the class fun and attention-grabbing resulting in a safe and effective learning environment. Thank you.  

By the way if any of my current students are reading this: If you responded to this part of the self-eval form in a meaningful way, thank you; if you didn't respond to this part of the self-eval form in a meaningful way, please do a better job of reading directions in the future; if you didn't even do the self-eval form, I really want to know why because I just don't understand.

Moving Forward

If you asked me if I thought the semester went well or not, I would say that yes, it went well. However, there were many things that I didn't like, some of which will need to be addressed via changes and others that I think will just improve naturally.

• Increased time. In my first couple of years of teaching I probably worked longer hours than this, but this work is more difficult than that work was and has taken up far too much of my time outside of school. I stay up late after putting Ollie down and doing chores, which ruins my sleep and is just generally unhealthy in every way. A few reasons why this last semester was so much work:

1. Creating a whole new set of assessment and evaluation instruments and revising them throughout the year, and the growing pains that come along with that.

2. Writing more meaningful feedback on assessments takes longer (though is much more useful and rewarding than just writing 1/2/3/4 on every problem).

3. Both Honors Algebra 1 and AP Precalculus are new classes for me so there is a lot of that "new teacher" type work that is just time consuming.

Most of that time consuming work for #1 is done, but there will still need to be some revisions. I don't know a way around #2, but I will say that keeping my own meticulous records in parallel with students is a particularly time-consuming part of this that I'm not convinced is necessary. Good for data and accountability but I just don't think it is completely necessary for carrying out my objectives for all of this. Also no way around #3. This is just an obnoxious fact of teaching when you're given a new prep (or even a partially-new prep). I just can't use existing materials that someone else made because I don't always know what they intended with certain things and there isn't room for me to insert some completely inappropriate humor. If I teach these classes again, however, this work will largely be done (always need to make scheduling adjustments and stuff, but yeah).

• Confusion about mastery logs and what to do with that (from students and myself). The mastery log thing was a learning process for sure. A few things I know need to happen with those that I started to work towards in the second half of the semester:

1. Reduce the number of standards/topics as mentioned previously in this post.

2. Do a better job of tagging problems for past standards (like the example mentioned above).

3. Streamline the data entry process by having students update it more regularly.

4. Make sure the students have the mastery log during the unit rather than only afterwards. (This will be difficult for me because I am disorganized and also I am still figuring out how all the content and sequencing works for both of the classes I'm teaching.)

• Student uncertainty about "standing". This was expressed in some of the feedback that I shared above, but also in-person and on other forms throughout the semester. Some students were not sure that they were on track or heading in the right direction at various points, and that was due to them not having a current grade of any kind as a benchmark. Before I say what I'm going to do about this I think it's important to mention that I think these students (who are amazing and fantastic) are lacking in confidence about what they know and don't know, and that this is all a product of the traditional grading system they've been working with their whole lives. They have ample data at all times to help them understand how they are doing in the class, but they are used to being given a single number or letter and for that to reassure them or to inform changes to their efforts. None of that is their fault but I hope I can help them learn these things. That being said, there are couple of things I think I can do to help in this regard.

1. Teacher-Student check-ins/conferences (at least) twice a semester (during progress grading periods). The first check-in can serve as a redirect if necessary, while the second is to make sure both parties are on the same page about how they are doing in the class.

2. Have some kind of "behind the scenes" translation scale between what students see on the student information system and what their approximate grade in the class would be. This would exist only for students who wanted to see it. I'm worried about this negating a lot of the intended purpose of not maintaining grades throughout the year for student, so at the very least I'll bring it up in discussion on one of the first days back to school.

End

I probably have a lot more I could write about on this post but it's way too long already. If you have any questions about any details I glossed over or are curious about anything else, please email me! thomas.gribble@tusd1.org or togribble@gmail.com. 

//2025 - Evil Arm (coming soon)