Math as a Tool

I got into a spirited discussion with Karim a few days ago about his desire for math to be an instrument to look with as much as an object to look at, which he wrote about in this blog post. Karim's concern that too many activities billed as applications of mathematics are actually structured to develop conceptual understanding rather than be a true application with a primary purpose of understanding something about the world resonated with me. However, I took some issue with his proposed solution: math teachers taking applications on more fully than they currently do.

We had a long chat over Twitter about it, in which I argued that perhaps Math teachers aren't in the best position to fully develop authentic applications and investigations of the world in which math is a tool. This does NOT mean that I don't think Math teachers should only teach concepts and never delve into applications. Of course, Math is both a subject onto itself and a tool for better understanding the world. And of course, for all students, understanding and engaging in its use as a tool makes Math more relevant and is a vital part of their education. My argument is primarily that when we shove all math applications into Math class and ask Math teachers to shoulder that full load, that inevitably means teaching less math and very likely, also results in these applications being less authentic and deep than they can be. My counter-solution is that more applications should be happening cross-curricularly in order to harness the expertise of multiple teachers and approach real world applications in the interdisciplinary way they are actually approached in the real world.

For example, I think teaching a lesson on wage inequality using math to analyze and form a quantitative basis for the discussion is awesome. However, the discussion that I am going to facilitate as a Math teacher in Math class is not going to be as deep as the discussion that an economics teacher would be able to lead on this topic. It's not because I don't care about wage inequality, but because my area of expertise is mathematics and their area of expertise is economics. They're going to have a rich understanding of historical trends and societal pressures and opposing views on this topic that even if I were to spend significant time prepping (keeping in mind that I have three preps every day and want to do application problems from a variety of fields and disciplines in each of them), I would not be able to achieve. Imagine how much more powerful this same lesson would be if we spent a Math class learning different ways linear models allow us to find "break-even" points for situations and then students went next door to Economics/History/Civics class and looked at how these models have been or could be applied to look at wage debates in our country. If we go even further outside the standard school model of siloed subjects, the Economics/History/Civics teacher and I can join forces and teach a lesson together in which the math and its application are interwoven.

It's not that I don't want math to be applied. It's that I want to see math applied deeply, across various subjects, as much as possible, as a joint project between disciplines rather than a few question prompts crammed in at the end of a math lesson. I want to harness my strengths as a teacher of math in its pure form, as well as a tool that is uniquely powerful exactly because it's so abstract and generalizable, rather than dilute what I am able to accomplish by trying to do it all. Why do applications of Math have to be taught during Math class?

If your answer to that question is: "because teachers from other disciplines won't do it," I think that accepting that would be a huge fail on our part as Math teachers. Here are some concrete things that I think would help if you are a math teacher:

- Ask your school's science, history, economics, psychology, etc teachers what topics they are teaching in the next month and if they would like you to come visit their class and co-teach a lesson to include math related to this topic.
- Ask other teachers at your school if they would consider creating a joint assignment that students would turn in for both classes (or turn in one part to one teacher and the other part to the other teacher) that would allow for a more in depth investigation.
- Are there classes that all students at a particular grade at your school take or a field trip that they all go on? That might be a great starting point for a cross-curricular project that involves Math and one other discipline.

Here are some examples of cross-disciplinary application projects I have liked:

- Students in a History class taken by all 10th graders were analyzing racial relationships in colonial times. They read an article called, "Social Dimensions of Race: Mexico City, 1753," which looked at how perceived racial differences were the basic criteria for social differentiation and employment in Mexico City in the 1750s. In my Math class, we used the data in the article to run a chi-square test of independence to see the level of independence between race and employment. Students then came back to History class the next day to discuss the ramifications of this analysis.

- Students in a Math 3 class created original art using Desmos and a variety of functions and conic sections, which they also worked on during their art class and which they had to analyze from an artistic as well as a mathematical perspective. 

- Students in Math 2 who were studying histograms, box plots, measures of center and variation, and outliers picked topics of interest in a country in which their world language was spoken and used the Gapminder global data set to analyze this topic over time in that country. They then wrote a paper and presented their findings to their world language class. 

My argument is that projects like these are inherently more relevant, authentic, and motivating to students than any applications I could find and facilitate on my own. And then I can put more of my time into teaching pure math ;)

Reflections

My school is committed to having students reflect on their learning, both in terms of math-specific development and student habits*. The research is pretty strong that reflecting on learning is a huge component of solidifying understanding. As John Dewey wrote, “We do not learn from experience... we learn from reflecting on experience.” Reflection as a skill is something that we intentionally cultivate and assess, but I am always working on making it a more integrated component of my classes and something that students value and appreciate.


Here are some ways that I've worked on doing this over the past few years:

Start of year reflections: establishing relational aspects of class and setting goals

We spend the first two weeks of each course working on open problems and having students read, watch, and discuss ideas that we think are important to setting the tone for the year, establishing classroom norms, and getting buy-in for learning through problem-solving

• Students respond to readings about mathematical practices and/or habits of mind:
• Reflection questions on first chapter of Make It Stick
• Reflection questions on "How to Learn Math" video from Youcubed and excerpt from Mathematical Mindsets
• Math autobiography
• Students reflect on their history as students and mathematicians and set goals for their work in this class

Reflections that emphasize content: after each lesson/assignment and after taking an assessment in order to correct course

We want students actively thinking about their progress in the course, returning to their goals, reflecting on their learning, and fine-tuning strategies in order to make progress.

• At the start of most classes, students summarize the main topics from the last class and homework assignment and reflect on their understanding through this Desmos Activity Builder.
• After most assessments, students reflect on their work in the class, both in terms of content learned and the development of their mathematical practices and student habits
• Typical post-assessment reflection; We realized that having students reflect on ALL mathematical practices was a bit much for a single reflection and tried to pull out a specific practice to reflect on  
• Correct and reflect on errors; plan what to do next

Reflections that emphasize practices and habits of learning: projects, homework, note-taking

• Students reflect on their learning and process on all projects and investigations 
• Menu of write-up reflection options (scroll to the bottom)
• Not strictly reflection, but we are starting to use peer feedback to have students develop specific mathematical practices
• Peer feedback on homework
• Reflection on note-taking
• Peer feedback on note-taking

Things I still need to work on/think about

The reflections were mostly created based on perceived need and don't necessarily spiral and build on each other as clearly as they could. I'd love to spend time going through the prompts and making them more specific - thinking about which mathematical practices should be cultivated at the start of the year, which ones later on, and which ones should be spiraled back to at later times. This would also help make the reflections shorter and more specific, encouraging deeper and more thorough responses. 

I'd love to hear about others' experiences with reflections so please comment or tweet at me with questions or feedback.

*Rubric of mathematical practices and habits of learning

Formative Feedback

I've been thinking a lot about feedback lately. It started with this tweet:

@MrAbend I would be really curious as to what the Math learning/cognitive science research says about error types. Does @mpershan know?

— Anna Blinstein (@Borschtwithanna) April 15, 2017

@mpershan was kind enough to respond with an email and sent me down a rabbit hole of articles and blog posts about the usefulness of feedback. Since Michael was the inspiration for this journey, it's only fitting that I try to imitate his style of writing out loud to try to organize my thoughts on this topic (sorry, Michael - reading this back after I've finished the blog post has shown me that you are inimitable. Also, I should probably avoid writing blog posts at 11 pm in the future).

The central question we discussed was: what is the purpose of feedback? Clearly, it is only useful if it changes a person's thinking. Does pointing out a mistake do this? Does categorizing the mistake do this? Does indicating a student's level of understanding of a topic ala Standards-Based-Grading do this? Do questions do this better than statements? Do students need to reflect on the feedback or do another problem related to the feedback received or implement it in some other way in order to get more benefit from it? Written vs. oral? Immediate vs. delayed?

Feedback while kids are working in class:

This is the type of feedback I think I know how to do the best. When kids are working on a task, either on their own or with someone else, I am usually able to ask questions, point out features of their work, or connect them with other students' thinking so they can make progress, identify and correct errors, and clarify their own ideas. The one blind spot that I think I still have in this area is when a student thinks about a problem in a way that is really, really different from methods I understand or have seen and thought about before. This doesn't happen very often, but when it does, I'm really stumped. I can help them verify that their answer is incorrect. I can ignore their method and show them a way to think about the problem correctly or point them to another student in the class with a different approach. But if I don't understand it, I can't help them resolve the cognitive dissonance of their incorrect approach, which means that my work is not complete.

But in general, this is the type of feedback that seems to pay the most dividends. The kid is right there with their work, we can have a conversation, I can see if they are able to implement my feedback and give more or of a different kind, as needed, or ask them to work on a related problem. This is really the best case scenario in feedback world for me.

Feedback on homework:

Things start to get real hazy real quick when I'm looking at a kid's work outside of class and my feedback is now provided in written form or via a conversation with them the next day. Will they have time/inclination to do anything about my feedback? Without the option of a conversation, I have to make a guess, which I suspect is often not great, about their thinking and the amount/level of information to provide back and how to do that in a way that opens thinking rather than closes it. Honestly, I don't have any evidence that students get a ton out of the written feedback on their homework assignments. I've thought about building in class time to have students read the feedback on their assignment from the previous day and do something with it (since homework is turned in digitally and feedback is provided digitally, I have no idea how thoroughly students would be reading my feedback otherwise), but it seems like I could just use this time to talk to students of concern about their work or have the class do a problem related to an issue that I saw on many papers. We already go over homework questions in class before it's turned in and the answers are provided in advance, so presumably, they know if they are understanding the material. If I'm very concerned, I would rather email a student or talk to them in class or ask them to work with me outside of class. Spending lots of time writing comments and then flinging them into a black hole of ??? doesn't seem like the best use of my limited time. But not providing feedback on homework also seems wrong. So I'm at a bit of an impasse here. I've moved some of my homework grading (especially for bigger projects) to in-person conversation and in an ideal world, I would be able to do that for all of my grading, but time with students is a precious commodity.

Feedback on assessments:

This type of written feedback seems to go better than homework. I think that there are a few components that have made it more successful:

1. Students perceive assessments to be more summative and take feedback on them more seriously. They know it's a check of their understanding that will more directly be reflected in their grade (grades as motivation.... laaaaaame, but I'm not sure how to get around this... I have to produce some sort of grade at the end, and I like homework to be purely for feedback so that leaves assessments for grading). As a result, they read comments more carefully and are more motivated to figure out their mistakes and learn from them so that they can show more understanding on the reassessment.


2. I separate the feedback and grading parts to help students focus more on the feedback initially. When I grade assessments, I only write comments/questions (and try not to say too much since I know I'll be there in person to continue the conversation). I record their SBG grades on the assessment in the online gradebook only a day or so later, based on the research that showed that when students receive written comments and a grade on an assessment, they basically ignore the comments and only look at the grade, and that this is not helpful for learning. Getting back their assessments with comments only helps to keep the conversations on concepts and learning only, not on grades, as well as encourages students to work together with less comparison to others. 


3. We spend class time correcting quizzes, usually in groups that are either assigned randomly or by common error types. The quiz corrections are an assignment that is collected, they are not for "earning back points" (I don't actually understand what that means), but they are required in order to reassess. I ask students to analyze their error (did they misunderstand an aspect of the concept? execute a procedure incorrectly? make a careless mechanical error?), as well as redo the problems on which they made errors. Based on my thinking around this issue, going forward, I'd like them to also state what they plan to do to make progress on the issue identified. Michael seemed to think that identifying the type of error is not particularly helpful to students, but I think that when followed up with a "next step," it is maybe more useful?


4. I think that more students actually know what they should do to make progress with assessment feedback. They've done a lot of work with the concepts being assessed. They can talk to peers to understand other approaches, they can talk to me, we can schedule a meeting outside of class to work together, they can refer to online resources organized by content topic to review a concept or procedure, they can do practice problems from homework assignments and previous reviews related to this concept so the feedback is both more closely connected to a concrete goal and to ways of reaching that goal. 

So my main questions right now are:

1. How can I make feedback on homework more useful in helping students change their thinking?
2. Are there ways to improve both my in-class and assessment feedback?
3. How can I move more of my feedback to conversation and away from enigmatic notes that try to strike just the right balance of tantalizing hint/information-giving and hook to motivate kids to want to look at their homework again and rethink their approach, but that mostly get ignored or scanned quickly and not attended to? Did I mention that writing tons of feedback on homework assignments takes a lot of time???
4. Are there aspects of feedback that I'm not considering?

Using Canvas to coordinate written feedback

Yesterday, @druinok asked for suggestions on providing more written feedback to many students quickly.

I wish I had the time to give my students more written feedback without a grade attached. How do you fit it all in?

— Druin (@druinok) April 15, 2017

It turns out that we both use Canvas, an LMS, at our schools. There are a lot of features about Canvas that are not the most user-friendly, but in terms of giving feedback to students quickly and easily, it's been really helpful. Here's how I use it:

1. All work that I collect from students passes through Canvas, including work that is not graded, but is just for feedback. I create an assignment with either a link to a pdf for problem sets or to a Google doc for projects/written reflections
   
   
   

2. Students complete the work either in their Math notebook or in a Google doc (for projects/written reflections). But all work is submitted digitally. The student's view has an electronic submission button. Most of my students have the Canvas app on their phones and can submit by snapping a photo of their written work. Those that don't have the app take a picture with their computer camera and submit it via the web version of Canvas. I remind students to submit their homework to Canvas after we go over homework questions in class.
   
   
   




3. In grading mode, I see the photo each student took, mark the assignment Complete or Incomplete, and type written feedback. If the assignment is graded, I indicate their level of proficiency and sometimes comment on the specific objectives graded (we use Standards Based Grading so students don't see points, only learning objectives and levels of proficiency).
   

4. The system is quick - I can go through all the work submitted by students, type or copy and paste comments, and click on levels of proficiency if I'm grading the assignment. Students see the feedback on individual assignments and can also look at feedback from past work, chronologically or organized by learning objective.
   
   
   
5. I love the fact that students have access to all of their Math work in their notebook at all times - there's no longer the loss of time in turning it in, waiting for me to write feedback, and then getting it back, accompanied by the inevitable loss of someone's work and of me lugging piles of papers back and forth from school. There's no longer a question of whether something was turned in or what the feedback on that work was. We can both always easily see a chronological record of the feedback given over the course of the year and track progress. If I ever create a portfolio system for summative assessment, all of the student's submitted work is already digital and organized.
   
   
6. The one drawback that I wish Canvas provided is the ability to annotate directly on student work. If I want to draw a student's attention to a particular problem, I have to write a note that says, "In question #2, look at..." instead of just circling question #2 on their paper. When students upload their files in pdf format, Canvas has an internal marking system that activates and allows you to annotate, highlight, and type directly on the page. But for most students, this would add an extra step of converting their picture to a pdf and uploading it in that format, and I would rather make homework submission as simple as possible. So for now, this is my system.

Task Makeover

We've all been there... you find a task that seems awesome. You start reading it and you get excited. There are so many different strategies that can be tried. There's a visual and algebraic aspect to it and a chance to try specific examples, make generalizations and predictions, test them, and build and justify a model. You spend a bunch of time exploring the different paths you think students might take, how you're going to give them feedback and what you'll assess with this task, how it fits with the rest of your curriculum, how you'll structure individual work time, collaboration and class discussion, how you think the lesson will flow, and how much time you plan to give to each component. Mostly, you're excited because you think it will be engaging and fun for students and will also bring up really interesting and important math ideas and practices. You introduce the task in class, eyes aglow with that special teacher light reserved for days like this, rubbing your hands in anticipation for the awesomeness about to unfold.

Except that it doesn't. At all. Kids seem confused. Then, frustrated. Heads start to go down onto desks along with pencils. Silent think/work time becomes sad, frustrated time, then out-loud complaining time as you slowly realize that this task is bombing and how. The kids you'd especially hoped and planned to engage, the ones who only sometimes engage, are the first ones to go. You try to rally the troops, but it's a lost cause, and you end the class demoralized and humbled. X years into this thing and every day still has the potential for catastrophe and epic failing (you may or may not be exaggerating the dramatic nature of the experience, most kids probably shrugged their shoulders and went on with their lives, but it was a hard 30 minutes for me, dammit!)

Where do you go for solace and a sympathetic ear? To the Math Twitter-Blog-o-Sphere, of course!

That feeling when you spend hours planning an awesome investigation for students and they hate it... pic.twitter.com/0xg8yIPcrh

— Anna Blinstein (@Borschtwithanna) March 21, 2017

Thanks to the advice and thoughtful questions of all you fine folks, I was able to reflect on the task design and recognize that the sheer wordiness and immediate jumping into very abstract ideas was a huge turn-off for many students.

Students had been doing so well with open investigations that even though it had been a little while since we had done one, I had completely abandoned the normal structures that coax kids who are not super sold on this Math thing just yet to try things, engage, take guesses, get a foot in the door, and progress towards increasing abstraction and formality at their own pace.

Fortunately, I had another class the next day with which I had planned to try this task. Back to the drawing board.

I started with a story. It's my birthday, but I'm really, really obsessed with all things square. My entire party has a square theme. Of course, I demand a square cake and that all pieces served to guests are perfect squares too. I can have my own party and eat the entire square cake myself. I can be a bit more generous and have a party for 4 since I can cut the cake into 4 perfect square slices. I can be even more generous and throw a party for 7 by cutting the cake further. 

This was a natural segue to asking students what they noticed and wondered, which brought out all the key features of the problem that in the earlier version were laid out in many, many words. Namely - is it possible to throw a party of any size if the slices must be square (but don't need to be of equal size)?

Students immediately had gut reactions and strong opinions. Some were ready to look for patterns right away, but for most, an opening question of: can you make another arrangement that we don't already have on the board? sent them on their way. 

Students quickly determined that they could make 1, 4, 9, 16, 25, etc pieces and that they could always add three to the number of pieces by cutting one of them into 4 and add 8 to the number of pieces by cutting them into 9.

There was another breakthrough when a student presented a convincing case for 6 pieces (as well as 11) and others realized that they could always add 5 more pieces by putting another layer on the outside.

This was the class I'd been hoping and prepping for when I found this task. Engaged, arguing, changing their minds, kids working past the end of class and needing to figure this out. 

Take-aways for me? Don't assume that kids have graduated past scaffolds that help them get started and build up to abstraction. If you're going to take them away, be aware and think deeply about how to do that carefully and thoughtfully. It's hard to reclaim a class that's lost its confidence so pay attention to this part.

I went back to the first class and tried again. With the reformulated question, things went much better. One of the students who had struggled the most the first day came up with a great organizational chart (that she said was inspired by Pascal's triangle) for tracking possible party sizes. 

She did have to amend it when another student came up with the 6 and 11 square versions since her version only assumed you could have perfect squares and add either 3 or 8.

Next step for both classes - helping them transform the patterns and ideas they have into more formal written explanations and justifications.

More ideas on working with students who really, really don't like mathematical exploration

As I've blogged before, the area in which our program has perhaps received the most criticism is in the challenge that open tasks, labs, mathematical explorations, and group problem solving pose for students who crave a more structured, algorithmic, and predictable approach. I met with a student (new to me this semester) last week who told me that she was incredibly frustrated with her current Math class (I am the teacher) because in her prior Math class, homework was 1 through whatever odd and both homework and quizzes were repeat versions of what the teacher had shown students in class. She had found this prior class soothing and comfortable and was an excellent student in this environment, whereas now, she felt that every facet of class was constantly asking her to figure out problems she hadn't seen before and she never knew if she really understood or felt like she was on solid, comfortable ground. She was worried that her confidence was slipping and that she wasn't learning as well as she had in the more traditional environment.

My initial internal reaction was to try to convince her that my pedagogy was sound, that it would indeed be better for her long term to struggle and make sense of novel situations, apply and stretch herself, learn how to tinker and problem solve rather than regurgitate algorithms repeatedly, but I felt that this would be minimizing her experience and negating her sense of her learning and mathematical identity. She had clearly stated that things make sense to her after she is given a method and does a lot of similar problems - only then does she believe that she is able to generalize and form an underlying concept. This isn't how our program is designed and I absolutely believe that it is better for most students to experiment and play first, forming conjectures and identifying patterns before coming to or seeing more formal methods (if needed), but maybe it's not better for her. At the very least, if she is convinced that this is the wrong way for her to learn, then it will be very difficult for her to interpret her experience otherwise, thus creating a self-perpetuating cycle. 

So I'm trying something new, and I'm not sure how well it's going to work. Every week, I'm going to email her a list of concepts that we will be working on next week, along with resources either in the textbook or online for her to see these concepts explained and practice problems for her to work on. A preview, if you will. Class will then not be a time for her to explore and invent, like it is for other students, but a time for her to generalize and prove the patterns that have already been revealed and practiced. In exchange, she has agreed that in a few weeks, she will again try exploring a new topic and be open to coaching by me in order to also get better at this way of learning. 

I'm hoping that by engaging in good faith, I am able to bridge the divide in expectations and meet this student at her current level of need and that she is able to grow over time in the mathematical habits of mind that I believe are just as important as, if not more than, content knowledge. It is certainly possible that she will continue preferring doing math in predictable and routine ways, following a pattern shown to her by someone else, on mathematical autopilot. I really hope that I can convince her that she can be successful and that it's worthwhile to engage in math in a different way than she has in the past. But it's okay if that's not where she is right now. I have a whole semester to build a relationship of trust and forment and celebrate moments of mathematical success for her.

Have you had students who actively and eloquently resisted your view of math or ways of teaching? What are some ways that you've made progress over time in their willingness to go there with you? Are there students who never changed their minds? Any and all advice welcome, as always :)

Why might students be motivated in math class?

At the end of the first semester, as part of students' self-evaluations, I asked them to reflect on their habits of learning, including curiosity and passion, asking, "Do you do work just to get it done? Do you cultivate your mathematical strengths and interests? How motivated/passionate are you and how might you improve here?" I received some pretty interesting responses to this series of questions, many of which boiled down to: I've never been that interested in or motivated by math and I don't know where to start to develop this.

In my reflection on this reflection, I came up with four main categories from my experience that describe why students have been interested in or motivated to study and learn mathematics.

1. Patterns and beauty inherent in mathematical structures

Some students are intrigued by looking for, identifying, and explaining patterns; others enjoy the beauty inherent in visual representations of mathematical objects and relationships. These students appreciate a teacher who encourages and rewards their curiosity, but overall, require the least amount of effort on the teacher's part to motivate and support since they're often speaking the same language as the teacher already.

2. Applications between mathematics and the real world

Other students I have taught were less interested in math in and of itself, but did find the idea of math as a tool to understand, explain, and predict the real world motivating. These were often students with an existing interest in science or social science who saw the usefulness of math in their respective fields of interest. Interesting projects were obvious choices in hooking and motivating these students, as well as a greater emphasis on practice and application than on derivation or justification. 

3. Being a good student

   This third category of student is one that is invested in an image of themselves as a good student. They care about doing well and meeting their goals and are motivated by seeing their progress, exerting effort and seeing it pay off, as well as specific feedback on how to improve and clear objectives for the course. 

4. Relationships

   These students seem to be predominantly motivated by positive interactions with others, whether that's the teacher or their peers. Classroom structures that increase conversations and collaboration between students and that make students feel known and connected to others have been helpful in motivating this group in my experience, as well as putting more of an emphasis on my relationship with them. 

Obviously, most students are some mix of these categories, but for many in my experience, one is more dominant. I think that a classroom that tries to balance between these different student needs will likely result in broader student success than one that caters to only one type. I would love pushback on my preliminary and perhaps too simplified analysis. Are there any categories you see being useful for thinking about student motivation? What other tools and strategies have you used to help students foster their curiosity and interest about math and motivation to exert effort towards the class?

#MtbosBlogsplosion - My Favorites

Carl and Julie have kick-started a new blogging initiative, and the timing is perfect, as I'm trying to get myself blogging more often instead of waiting for An Amazing Inspiration. This week's theme is My Favorites, and I wanted to share a really helpful framing for peer editing created by my awesome colleague. We've been working on using peer feedback more productively this year, and her document (shared below) gives a good structure for students to reflect on and give feedback to their peers' write-ups and oh hey, they also learn a lot about what makes for a good write-up and use this understanding to do a better job themselves. Mandy has incorporated a peer feedback step for all write-ups, with that night's assignment for students to revise their own work. I would love to do more structured peer feedback in other components of the class, such as homework assignments, note-taking, and studying for assessments. The setup is very basic - students exchange papers, give each other feedback, get their peer's feedback back, and turn it in with their revised write-up, documenting any revisions that they made.



Here's my first draft for a homework feedback form. Would love any feedback and suggestions for improvement.

Goals for second semester

As I've been wrapping up grading from semester 1 and planning semester 2 for my classes, I'm realizing that I did not set goals at the start of this year the way that I have in the past. Better late than never!

Changes for my personal teaching:

• Get back to individual feedback meetings. I blogged about them here, but the general idea is that I set aside 20 or so minutes to meet with each student approximately every two weeks in order to sit down together and look over their work and have a feedback conversation. I've found these incredibly helpful for students to actually attend to my feedback, understand what I mean and why I think it's important, and explain their thinking to me. This year has been very tricky since the schedule was changed and students lost a floating free period that I used to be able to use for these meetings. I am recommitting to instituting them again, using class time, if needed. It's been the best way for me to get through grading big projects in a timely manner since it's actually fun and rewarding to sit and discuss students' work with them rather than grading on my own after a long day (since, let's face it, grading gets put off and off).
• Be more on top of students who are struggling. I am committing to looking at work that is turned in every week to check up on students who are missing work or need additional support. If anyone has a good system for keeping track of interactions/observations/progress for all students and how they make sure that no one is falling through the cracks, I'd love to chat.
• More nuanced and thoughtful reflection questions - I think that the balance of reflection vs. doing math has been better this year, but I'd like to focus the questions I ask students in order to hone in on specific mathematical practices rather than just general "what's going well? what do you need to work on?" type questions. I also want to bring back, "what's one good thing that happened this week?" - it was a great way to regularly check in and connect with students.
• Collaboration quizzes to give more direct feedback to students on their groupwork and engagement and help them internalize expectations more effectively.
• More peer feedback. I've started doing this more this year, and love how much motivation it creates for students to express themselves more clearly and justify their thinking. I'm hoping to use peer feedback this semester to help students get better at analyzing strategy, getting positive feedback for extensions they create, and to deepen their understanding of different approaches. One of the lesson study groups worked on peer feedback last semester and I'm really excited to learn from them. I would also like to use a Slack channel for classes so that students can discuss and share ideas outside of class more easily.
• Better differentiation. I'd like to meet with students to set individual goals and do more follow up to help them stay on track with these. I think that there's already a fair amount of choice in problem sets and homework assignments, but I'd like to do a better job of teaching students how to use those choices better. One way will be to have them reflect at the end of class on the type of work they need to do to follow up on that day's learning (review of prior concepts, practice, connections, and/or reach problems). I know that they are learning project management skills in their other classes, but in Math, the product is the process, which is more abstract and harder for them to track and plan. 
• Continue and get better at classroom routines that foster reflection and a clear arc from start to finish. 
• I have often used Desmos Activity Builder to start and end class, but would like to do this more consistently and help students get better at constructing meaning from problem-based lessons by selecting useful reflections and comments to share. I still have work to do on making sure that meaning and connection emerges from students' own thinking and not ignoring times when they don't emerge or simply telling students what they should have learned. One way is to do more planning of student responses and how to connect these and have the main ideas of the lesson emerge from them, sharing methods and responses that did not emerge as part of that process. 
• This also connects to better note-taking. I have given feedback to students once or twice on their note-taking and organization and definitely need to do this again. I haven't really figured out a solution for sharing board work and "notes" from class since I've emphasized process and individual needs. I do share presentations, if they were used, but those generally do not contain worked solutions. If anyone has good ideas on this, I'm all ears. 
• I would also like to do this on a unit-level rather than just lesson-by-lesson by using student-generated essential questions, concept maps, and study-guides more this semester. There is still a fair amount of tension between student-generated conclusions/connections and teacher-generated ones that are more "efficient" and feel more comfortable and structured for students, especially if they're oriented towards maximizing content acquisition. I am working to help students get better at this and at understanding why I think that it's important, both of which are necessary to get more buy-in for the process and rewards that actualize when students do more of this work. One way is to be more transparent about the structures that I'm using and why - I observed a teacher recently giving an intro to a lesson by explaining the groupwork structure that he would be using and what he hoped it would achieve, and I think that enlisting students as teammates in this process is hugely beneficial. 
• Continue the following changes I implemented last year:
• Each assessment includes reassessment of previous content
• Visibly Random Groupings (new groups daily) and whiteboarding
• Homework that's spiraled and includes Retention, Review, Reflect, and Reach sections; students self-select problems to do (should sometimes group students by homework problems completed the next day though)
• Students submit all work digitally, all feedback is recorded digitally in one place (online gradebook)

Big Picture Curriculum:

• Decide on mathematical practices and habits that should be emphasized within a given year/semester/unit and link them to specific lessons and activities. I've been doing a much better job this year of giving students regular feedback on these, but haven't been very intentional about which habits will be emphasized when, noticing which ones students are making progress on and which ones need more work, and how (besides getting feedback) they might get better at them.
• Create more opportunities for interdisciplinary connections. I've put out some feelers to Science teachers and will do the same for Computer Science, English, and History to see where we can join forces and create projects that can support and enrich both disciplines.
• Formulate a more cohesive picture of our curriculum and mission so that our core sequence is less content-driven and so that we can explain to students and families why acceleration is not necessary or desirable. This will require a reducing/reworking of our acceleration pathways, enriching/differentiating core classes, and deciding how electives should support the overall program. 
• Start developing a portfolio assessment for one Math course. It might not be ready to go this semester, but if I can pilot a beta version in one class, I can work on tweaking/developing it more over the summer so that it's ready to go in more classes next year.
• Work on developing group assessments (and other differentiated assessments) for at least one unit in each class.

Professional Development:

• Continue lesson study this semester and figure out good systems for sharing the results that each group has found, both within the discipline team and with the school community more broadly. Possibly help other disciplines/divisions begin the lesson study process. Think about presenting about lesson study next year and the types of resources and supports teachers would need to get started with this.
• Figure out what I want to work on over the summer. Major contenders currently are:
• Attending PCMI
• Teaching at summer institutes for teachers
• Start compiling our existing curriculum into a more easily shared and edited form for students, families, and teachers
• Curriculum development for my school, focusing on alignment between courses, portfolio assessment, projects that connect to other disciplines and class trips, parent education, and developing new electives
• Summer math support for students who are doing independent work or working more directly on accelerating/remediating/enriching
• Coordinating with the middle school on curriculum, parent education, and development of mathematical practices