Friday, October 19, 2018

Connecting Math and CS with probability game simulations

One of my goals for this school year was to build out a few interesting and relevant projects into the 7th grade curriculum, which seemed a bit dry and skill-focused. One area that seemed to beg for an application project was the first unit on Data and Probability. Since one of my other goals was to incorporate more computer science into my classes, it was a no brainer. Developing a cross-over computer science project for this grade level proved to be a bit tricky because students are all over in terms of their experience with programming - we have students who have been coding for years as well as new students who have never coded anything before. I tried to develop a project that would differentiate appropriately and allow students to either explore the CS or the Math parts in greater depth, depending on their interest in and experience with programming.

Here is the project description. You'll notice that I created three distinct strands with different goals and let students select the one that was most appropriate and interesting for them. I was also lucky that the computer science teacher was able to come to my classes for some of the time that students worked on this project. Having many intermediate checkpoints for students to submit pieces of the project was very helpful here in ensuring that I could identify those who were behind or struggling and work with them during class.

Things that I would still like to build out:

  • A more robust peer editing process -- I'd like students to be able to present their optimal winning strategy to peers and get critical feedback on how convincing their reasoning is that they would be able to incorporate into their final draft
  • A revised rubric to make it more concise
  • Move some pieces of this project out to computer science class - this definitely took up quite a bit of time, especially because I felt that most or all of the coding work should happen during class where students would have support
  • A clearer division between group and individual aspects - this is always a challenge for me when designing group projects in terms of maximizing student learning and individual accountability. Students seemed to work well together during class, but this isn't an explicit part of the project currently. 
  • Some sort of final presentation - for projects like this, I think that having the final product on display or presented to others creates a much more authentic need for clarity and functionality. I haven't figured out a good way to do that for this project. Should students do a gallery walk of projects within the class? Can this be presented or shared with students in other classes somehow? What about with parents?
  • Other connections - is this something that can connect to students' work from previous years so that it feels less like a stand-alone project and more like a continuation of ongoing work and thinking? Are there other aspects of this project that can connect to other disciplines, like writing? Can we build on this in future years of either computer science or math curriculum?

Thursday, September 27, 2018

Feedback and communicating with families

A goal I wanted to work on this school year is more systematic feedback on mathematical practices as well as better communication with families about what students were working on and their progress. I also wanted to do it in a way that didn't emphasize grading and evaluation and kept the student at the center of setting goals, reflecting on progress, and owning the process.

This blog post had a great suggestion for using Google forms to have students reflect each week and have those reflections emailed to parents. The prompts asked students to describe what they learned that week and how they feel about the class. To be honest, the directions for setting up the emailing were a bit too complicated for me and involved using Add-Ons that our tech administrator wasn't too jazzed about, so I did it in a way that seemed more simple and worked well for me. I'll summarize the deets below, but wanted to first say that I've done this twice now (students are reflecting every other week) and have gotten very positive responses from parents. It takes a lot less time than emailing individual parents, and I think it makes a big difference for parents to hear about progress in their children's own voice.

I changed the questions to be a bit more focused on goal-setting and learning. The questions I'm asking are:

  1. What have you learned in the last two weeks? Be as specific as you can - feel free to look through your notebook.
  2. How do you feel about your learning of this material, both from class work and homework? (3 = I can teach it to someone else; 2 = I understand it pretty well, but have some questions; 1 = I am very confused and/or have a lot of questions)
  3. How do you feel about your class engagement and work? Have you been engaged and focused? Have you worked productively with a variety of classmates? Have you been a respectful skeptical peer and asked for feedback on your thinking?
  4. How do you feel about your homework effort? Did you allocate time well during the week? Pick problems at a good challenge level? Stick with hard problems? Try different things? Ask questions? Make corrections during class?
  5. What was your goal/next steps the last time you reflected? Did you make progress towards this goal? Why or why not?
  6. What are your next steps? What should you keep doing during class and at home? What should you do differently? Do you need to follow up with your teacher?
To clarify, students have a lot of choice in their homework each week - they have an hour to spend on a problem set that has questions at different levels of challenge and depth so I find it helpful for them to reflect on their choices and make changes, if needed. They also have a single assignment due at the end of each week so they should be thinking about how to best allocate their time during the week to avoid leaving it for the last minute.

I make a new version of the form every two weeks and the responses feed into a spreadsheet. I also ask for their name so I can sort the responses alphabetically. I add a column at the end where I add any additional notes I want to share with the family. Usually, it's things like, "This is a great goal. It sounds like X is ready to try some harder problems on the homework next week." I have a list of parent emails that I can then paste in as well as two somewhat fancy things that make the whole system work (not that fancy in actuality, but let me get excited here for a sec). The first one is a cell that combines all of the student responses in one place for ease of emailing. 

The code to make that magic happen is: 

CHAR(10) just creates a line break between responses. The & symbol concatenates responses so that they appear next to the question. Otherwise, it just pulls the responses into a single cell. Drag down the formula to have this for all of the students. Then, add another column to the right that will track whether an email has been sent (this is useful if some students are absent and do this later so you end up running the email script multiple times and don't want to resend the emails that already sent).

When you're done, you have a spreadsheet that looks like this:

(your email sent column will initially be blank)

Okay, this is where the fun really begins. Under Tools, select Script Editor. I found a script for emailing from a spreadsheet and amended it to email two addresses. You can use it too. Ta da.

The code in that link is:

// This constant is written in column C for rows for which an email
// has been sent successfully.

 * Sends non-duplicate emails with data from the current spreadsheet.
function sendEmails2() {
  var sheet = SpreadsheetApp.getActiveSheet();
  var startRow = 1; // First row of data to process
  var numRows = 28; // Number of rows to process
  // Fetch the range of cells desired
  var dataRange = sheet.getRange(startRow, 1, numRows, 4);
  // Fetch values for each column in the Range.
  var data = dataRange.getValues();
  for (var i = 0; i < data.length; ++i) {
    var row = data[i];
    var emailAddress1 = row[0]; // First column
    var emailAddress2 = row[1]; // Second column
    var message = row[2]; // Third column
    var emailSent = row[3]; // Fourth column
    if (emailSent != EMAIL_SENT) { // Prevents sending duplicates
      var subject = 'Bi-Weekly Math Update';
      MailApp.sendEmail(emailAddress1, subject, message);
      MailApp.sendEmail(emailAddress2, subject, message);
      sheet.getRange(startRow + i, 4).setValue(EMAIL_SENT);
      // Make sure the cell is updated right away in case the script is interrupted

Notice that my script currently starts on the first row and processes 28 rows (I piloted this in two sections only). You might have more students so will need to process a larger number of rows. You do need to make sure you don't go too far and get to an empty row. The script doesn't like it when there's no data in a cell it's calling up. By the way, when parents respond to this script-generated email, their response goes directly to my regular school email address because Google is magical.

How do students have access to all of their reflections, you ask? I went a bit Google spreadsheet happy and added a tab to my master grading spreadsheet that pulls in the reflection responses for each student using the IMPORTRANGE function. Each student then has their own spreadsheet that pulls in just their reflection responses (as well as feedback on content learning goals). There is now a chain of Google sheets happily talking to each other and emailing parents every two weeks. What a world.

Saturday, August 25, 2018

Culture of Mathematics

This post is part of the Virtual Conference on Mathematical Flavors, and is part of a group thinking about different cultures within mathematics, and how those relate to teaching. Our group draws its initial inspiration from writing by mathematicians that describe different camps and cultures -- from problem solvers and theoristsmusicians and artistsexplorers, alchemists and wrestlers, to "makers of patterns." Are each of these cultures represented in the math curriculum? Do different teachers emphasize different aspects of mathematics? Are all of these ways of thinking about math useful when thinking about teaching, or are some of them harmful? These are the sorts of questions our group is asking. 

I am excited to write a post as part of a group of bloggers thinking about the tension between problem solving and theoretical understanding, among other tensions. Moreover, the benefit of procrastinating and getting terribly behind is that I get to read and respond to some of the other blogs written as part of this group. Michael's post, in which he discusses the reasons that he has moved away from problem solving as a classroom focus, was one that really struck me and prompted me to want to respond. I think that he makes some excellent points about wanting to move away from answer getting as an inherently inequitable and exclusionary practice in which some students race ahead while others are left behind. It's a great read, and I highly recommend you pause here and read his post in full. 

The main place where I found myself disagreeing was in the setup, in which problem-solving is positioned diametrically opposed to theory-building, and the two trade off against each other. This, to me, seems like a confusing and artificial construction... both are just questions that we are posing about the world, where perhaps problem-solving takes the form of slightly more specific questions and theory-building is what we call questions that are more general. Joshua Bowman calls out this false dichotomy in his post as well, adding it to the list of polarities like applied vs. theoretical and individual vs. communal and urging for math teachers to value both types of thinking because we just don't know what's going to motivate or interest a particular student and the more variety and ways there are to be hooked into mathematical thinking, the better. 

I would say that as teachers, we can't help but be biased towards ways of thinking that are aligned to how we ourselves think and what we value. When I first started teaching, I was very much tapping into my own personal experiences as a math student - the complete disconnect I had felt from math as an intellectual discipline in high school and why I fell in love with math as an undergraduate, thinking for the first time about real (to me) mathematical questions that sparked my curiosity and wonder and ideas that blew my mind and made me want to learn more. I posed problems to my high school students in the way that I would have wanted them posed to me. There were some kids who came along for the ride, but there were also definitely some who were left behind because I was not speaking their language.

Joshua's conscious choice to provide students with many options and potential hooks is a way to move away from this form of me-centered teaching, which can be such a natural trap. He chooses to be agnostic and let students construct knowledge in the way that works for them. I find it interesting that Michael is perhaps doing the same thing, but in a way that purposefully deemphasizes problem-solving because it is such a dominant paradigm in mathematics so that students are exposed to other ways of doing math. The sentiment behind these teacher decisions definitely resonates for me, and I think should be central in teacher preparation and planning for courses - what values are you emphasizing in your classroom structures, teacher moves, and curriculum? 

I have certainly seen problem-solving play out in the same troubling ways that Michael referenced in his post - primarily when I have attended math team practice and felt the anxiety I often feel in these types of hyper-competitive-speed-based-publicly-exposed environments. But for me, it isn't problem-solving that's the culprit, but the types of problems that have been posed, the environment in which they are done, and their purpose. For example, I attended PCMI last summer - this is a place where math teachers are solving problems together for hours every day. There is a huge amount of variety in mathematical background knowledge, experience with math teaching, and familiarity with the PCMI style. Yet norms are set and problem sets written in such a way that connections, representations, deep and novel ways of thinking and analyzing, and thoughtful questions are what is valued, resulting in a community that while not quite a mathematical utopia, is pretty damn close. Good problems + clear norms + teacher moves to support norms = learning that aligns to the values of the program and access and motivation for many students.

In my own teaching, I have moved towards student-posed questions and projects as something that more closely matches my values in teaching and moves away from my subjective opinion of what is interesting towards my students' perspectives and interests. I value good problem posing as an opportunity to both pique interest, stimulate thinking, and help students better understand what makes for a good problem so they can move on from problems posed by me to problems they pose themselves. It's much less important to me if the questions they ask are specific (problem-solving) or more general (theory-building) - it's in the asking of questions and seeking to understand and construct the world around them that I see the purpose of my teaching.

Sunday, June 3, 2018

End of year celebration of knowledge

Dan Meyer started a discussion on Twitter recently about the unnecessary stress that final exams cause for students at the end of the year, questioning how much insight they really give into student learning. It’s been a helpful reminder that while I definitely agree that high-stakes final exams are terrible, I really don’t have a great system yet for wrapping up the year.

We certainly don't want students feeling like this:

But what makes for a good alternative?

It seems challenging to balance the goal of ending the year with celebration and anticipation of more learning, while also gaining information about retention and content synthesis. I want students to end the year on a high note, feeling positive about their progress and provided with the opportunity to dig deeply into a particular topic, but it would also be great to be able to identify topics from the entire year that would benefit from review and work with them to do that.

In some ideal universe where time doesn't exist and Firefly is still on the air, I would be able to do both: a meaty project in which students can shine and review and an assessment of all of the things. However, even given this bounty of time, I'm not sure that a timed, paper and pencil, silent, individual assessment would really promote the most learning and information for me and students.

So I spent a bunch time the last few weeks reading up on various ideas and here is my current compilation.

  • A group whiteboard assessment that looks at problem solving and tying together big concepts from the year, something like what @AlexOverwijk does with his classes:
    This would require careful teacher observation to untangle individual understanding and contribution to the group product, but seems like a much closer fit to what students do in class every day and therefore a more accurate picture of their understanding, as well as obviously being less stressful.
  • An annotated portfolio of work throughout the term, which would require students to find evidence of learning for previous topics, identify important connections, revise work, and identify topics that need further attention themselves. I really like this option as it puts the student in the driver's seat. However, this would be fairly time-consuming and likely need students to have been tracking their work throughout the semester. It's something I'm strongly considering for next year. If you do this, I'd love more information - directions, rubrics, advice for someone who wants to try it. How do you make this work in large classes?
  • An oral final exam in which each student has a one-on-one interview and discusses their process and reasoning for one or two problems, which @JadeMohrWhite proposed:

    This seems great for digging deep into mathematical practices and student thinking, but would only give limited content knowledge information due to time constraints. Building in class time for every student to have a 20 minute interview or so also seems a bit daunting in the end-of-year crunch, but could potentially complement a final project or portfolio assignment, during which students are working relatively independently.
  • Final individual project and group presentation. This is the model I'm trying this year in one of my classes. Students selected a topic of personal interest to them that is related to the content in the course and did research and Math work related to this topic. They were then placed into groups based on some possible common threads between projects and created a presentation that highlighted their individual work AND the connections between them, as well as how what they learned related to their Math course this year. Detailed directions are here.

    I like how positive and forward-looking the projects have been this year - it does feel like a celebration and memorable opportunity for students to shine. However, because projects are typically looking at a single topic in a great deal of depth, this way of ending the year misses out on the whole cumulative, wrapping everything up feeling that I like to have. 
  • Bring back the final exam, but have it be extremely low stakes by focusing on retention, connections, and structured so that it can only help a student's grade, not hurt it. This is how I've done final exams before - as a final opportunity for a student to show understanding of a topic from a previous unit and a place to look at cumulative retention and synthesis. It's efficient and serves that purpose well, but isn't the kind of experience I want students to take away with them as their last memory of my class, so if I brought it back, I would definitely want to pair it with one of the above ideas.
  • Edited to add:

    Take-home final exams, as described by @benjamin_leis below, seem like another way to get more comprehensive information about content knowledge in a less-stressful setting. I like the idea of removing time pressure from the equation and letting students assess in a more comfortable and familiar setting where they can take breaks and dig deeper into problems. Again, because this more closely replicates the ways that students do math in my class during the year, it should be a better assessment of what they know. I also think questions on a take-home final should be more interesting and less routine than what I would ask on an in-class timed assessment. 

I would love to know of other ideas people have for alternatives to high-stakes final exams or any feedback on these still-cooking ones. Share them in the comments or send out a tweet.

Sunday, September 24, 2017

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 ;)

Sunday, July 9, 2017


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

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

Reflections that emphasize practices and habits of learning: projects, homework, 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.

Wednesday, April 19, 2017

Formative Feedback

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

@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?