Sunday, March 29, 2020

Remote Teaching

As I end the first full week of remote teaching, I wanted to quickly jot down some of the things that I've tried this week and what is and isn't working for me. First, a bit of context:

- I'm in San Francisco and teach Math to 7th and 8th graders at a K-12 independent school
- We are 1-to-1 with laptops in the middle and high school and have reached out to students with internet connection issues to help them access remote classes
- We don't have textbooks or give grades, which means that we need to think differently about structures and motivation for kids to engage in remote learning
- Two weeks ago, school rolled out a Remote Learning Plan that involves synchronous video classes over Zoom for most core classes, along with assignments and drop-in hours, but no synchronous class, for P.E., music, art, and electives. We are on an A/B block schedule and classes have been shortened from 75 to 60 minutes so kids get a 20-minute break in between, but otherwise, have 3-4 remote classes four days a week, and then a day to work on homework, projects, work from their asynchronous classes, and meet with teachers one-on-one.

Some things that I've found really helpful so far:

  • A class webpage... I use this to structure what we're going to do during each remote class. It was super, duper easy to make a webpage using Google Sites. Mine is not fancy, but has a tab for the day's agenda, another one with assignments students complete outside of class (which are also posted to Google Classroom), and a final tab that organizes class notes and online resources for each topic, since we don't have a physical textbook.

Here's last Wednesday's agenda for 8th grade:

Kids can use this to get to the different parts of that day's lesson, which is especially helpful if they lose their connection and have to reconnect to the Zoom class or have bad audio.
  • Structures that break up the class into different types of learning environments seem to be going well. We usually start with a warm-up problem that is linked in Google Slides that we either discuss as a class or in breakout rooms. Then, I have each breakout room work on problems together in a slide on Jamboard, a virtual whiteboard. 
Here's a whiteboard from one breakout room in a 7th grade class from last week:

Some kids are writing using their mouse and trackpad, some kids are using text, others are doing work in a notebook and uploading photos of their work. Having the virtual whiteboard means that I know which groups are struggling and can be more efficient in which breakout rooms I visit while they work together, as well as identifying responses I want to highlight when we debrief as a whole class.

The last part of class is usually a Desmos activity I can use to see how individual kids are doing with the topic and decide on whether I want to see any for one-on-one time. I'm mostly taking Illustrative Mathematics lessons and converting them into Desmos Activities or creating quick exit tickets that I can have students complete in the last five minutes of class.
  • Using Google Classroom to get homework out and collect written work is going really well. Students are submitting photos or scans of their handwritten work on homework assignments and the Classroom app for iPad lets me annotate submissions so that I can give them written feedback on their work. This is another important avenue for me to see how individual kids are doing and to identify kids I want to follow up with one-on-one. That might mean asking a student to see me during office hours or recording and sending them a short video addressing something I am seeing in their homework. I just figured out that my iPad has its own screen recording function so I don't need to download and use a different app, I can just jot and talk through a quick note in Notability, upload it to Google Drive, and send the student a link.

Things I still want to work on as California will remain "sheltered in place" for at least for the next month:
  • One-on-one meetings with every student; I have been mostly using office hours as an optional drop-in time for students, although I have reached out to those that I can see are struggling, but I want to schedule a one-on-one meeting with each and every student in the next few weeks to check in and find out how things are going for them and to look over their work together. I'm finding that middle school students are just not great at taking in written feedback over the computer and knowing what to do with that information or what their next step should be. 
  • More variety in the activities I am doing with students during class; right now, they seem to like the mix of class discussion, breakout rooms, virtual whiteboarding, and Desmos activities, but I imagine that these will start to get old soon. These are pretty similar to activities we did in class when regular school was in session, but we also did collaborative projects, labs where students gathered and modeled data, student presentations, and explorations with manipulatives, which are all missing from remote Math class. I'd love ideas of what others are doing that might create more balance and variety in working with students remotely. 
  • Ways to check in with every student during class; I really miss the ability to walk around the room, scan what students are doing, and ask quick questions to probe their thinking or identify cool ideas that I could then ask them to share with the class. It's really hard to informally see the work of individual students and and talk to them without drawing a lot of attention to it in a video meeting while still keeping an eye on the rest of the class. 
  • I want to build more community and connections between students because I know this is something they're really missing in this remote learning space and is one of the main reasons that kids are excited to go to school and learn. Initially, I focused on structures that would help with content and organization because that's how I deal with stressful situations, but now that classes are running pretty smoothly and we have a system, I want to develop more ways for us to be human and connected together. 

Thursday, January 16, 2020

Proof in a non-Geometry classroom

A beef that I have long had with the standard math curriculum is that for many students who don't take college math courses, proof is a weird one-off that you do in a Geometry class for one year, writing algorithmically in two-columns full of acronyms. You know it's coming, but certainly aren't expected to do anything about it until you get to Geometry, and then, just as enigmatically as it appeared, it vanishes from the curriculum again. Truly, a mystery.

I began to rethink my own views on proof and how its definition might be broadened to make it a more regular component of ALL math classrooms when I read Avery's blog post on redefining proof several years ago. More recently, I've been thinking a lot about how to help students become more rigorous and formal in their proof writing, while still treating their informal reasoning and ideas as valid and interesting in their own right, not just as a stepping stone to "more correct" proof.

Teaching an integrated math course to 8th graders this year gave me some unique opportunities to play around with proof and formalization. The year starts with a unit centered on the Pythagorean Theorem and culminates with a project in which students choose a proof of the theorem (we used a collection of proofs at Cut the Knot) to study and then present their proof of choice to peers. There's a great variety of proofs there in terms of complexity and usage of algebra/geometry/trig/similar triangle concepts and focusing on analysis of existing proofs emphasizes what it means to understand a proof and be convinced by its reasoning. We also had a great class discussion about the difference between an example or demonstration and a proof, looking at several videos "proving" the Pythagorean Theorem that really only showed it to be true for a specific instance.

Image result for prove pythagorean theorem
Look, it works!

That was pretty great, but there was nothing in the curriculum that built on this for the rest of the year so I decided that halfway through the next unit, we would do an exploration of polygon areas that would lead students to Pick's Theorem where they could start to write their own proofs, but I really struggled with structuring it in a way that honored students' existing reasoning, but also pushed them to formalize more and consider proof techniques, such as casework. If you haven't seen Pick's Theorem, it's a neat little formula that connects the area of a polygon whose vertices are points on a lattice to the number of lattice dots on the perimeter of the polygon and the number of lattice dots in interior of the polygon. 

The structure I eventually created seemed okay, but definitely produced mixed results. All students were able to come up with a conjecture for the area of a lattice polygon, but even with my hints (which I thought were maybe too helpful), virtually no students were able to make progress on proving the theorem on their own.

Questions I now have:
  • Was it too big of a jump to go from analyzing proofs to having students write their own proof, even with lots of hints?
  • Is this problem perhaps not the right one for first proof writing?
  • What other problems or structures could I have used to transition more effectively towards proof-writing that still build on students' original reasoning and perspectives?
  • Would students have benefited from writing a proof (or the start of a proof) together as a class first?
  • Where should I go next to develop students' formal proof-writing skills?

I do have some thoughts on these, but would love to get more input and ideas from the community.

Tuesday, February 19, 2019

Flexible groupings

I wanted to respond to @MarkChubb3's tweet (below) in more characters than Twitter would allow.

Mark asks some great questions about groupings, specifically focusing on issues of equity, identity, and experience vs achievement gaps. I would highly recommend going to read his post, where he details his recommendation for a progression between Tier 1 (all students), Tier 2 (small groups), and Tier 3 (individual students) instruction.

These are questions near and dear to my heart. My school, like many, has tried a variety of approaches to groupings over the years. As a school that targets gifted learners whose needs were not being met, we see huge ranges in students' experience in Math that make purely heterogeneous groupings challenging. However, as Mark points out, the danger of groupings, especially ones that are static, holistic, and that lead to diverging learning and opportunities, are manifold. This year, we have piloted a model in our 5th and 6th grade Math classes that has tried to walk the fine line between the benefits of some groupings, while trying to avoid some of the negative effects of groupings discussed above.

Flexible groupings model

Our model relies on the fact that for 5th and 6th grade, students have Math at one of two distinct times every day they meet. That means that there are several same grade Math classes scheduled at the same time. The other component of our schedule that makes flexible groupings easier is that these Math classes are happening in classrooms next to each other and two of the rooms share a retractable wall that can be moved out of the way to create a large shared space.

We started the year with all students who were taking Math at that time in the large shared space, working in random, heterogeneous groups that we mixed up every day. All three Math teachers for that grade were present so that students could get to know the entire teaching team and vice versa. The first unit for both of these grades focuses on Mathematical habits of mind so problems are rich and low-floor/high-ceiling and don't have specific content objectives. Students were able to work with a variety of peers and we were able to gather a great deal of data of how students approach new problems, collaborate and communicate, and write down and process their thinking. We then moved into content-based units, which all followed a similar pattern:

Key features of our model -- student-facing

  • Students start each new unit in random, heterogeneous small groups within a large shared space with multiple teachers, working on open tasks that introduce some of the new concepts.
  • At the same time, students complete a take-home pre-assessment that looks at their prerequisite knowledge, as well as knowledge of the concepts and skills to be taught in the upcoming unit.
  • Teachers create new groupings based on the pre-assessment and observational data of students' learning strengths and needs.
  • Students move into their new groups, which last for two-three weeks.
  • There is some student choice built into each grouping - as we review and prepare for an assessment, students select what and how they would like to review and are regrouped based on this choice. 
  • Students are assessed on their understanding of content for this grouping cycle and are given feedback and opportunity for individual revision, intervention, and/or extensions. 
  • These unit assessments and pre-assessments are used to drive groupings for the next cycle. 
  • Most homework assignments are the same for all groups and are differentiated by giving students choice over which problems to work on. 
  • Students regularly reflect on their needs, choice of homework problems, and how they are working in different groupings and settings.
  • Units that are more project-based (we have three large units like this during the year) are done entirely in heterogeneous groups.

Key features of our model -- teacher-facing
  • Teachers who share a common pool of students use a shared spreadsheet to track observations/feedback on classwork, homework, assessments, and projects. This is really important in order to know how students are doing as they move through different groups and work with different teachers. We spend time as a team discussing what we think is important to include in our notes and what our notes reveal about different students' needs.
  • Each grade level team meets twice per week to discuss lesson plans, how students will be grouped that week, students we are concerned or wondering about, how we're giving feedback, and all the other little things that need to be aligned when co-teaching. 

Benefits of flexible groupings
Students and families have been really positive about this implementation. It has many of the benefits of both heterogeneous and homogeneous groupings and has ameliorated a lot of the issues we have seen in the past with groups forming a fixed mindset about their abilities and trajectory. Especially in our project-based units, we see students working productively with a greater variety of peers and having a better understanding of themselves as learners. It supports a philosophy of meeting students where they are and the idea that different students have different mathematical strengths and areas that need more support.

Questions and next steps
I would like to see more heterogeneous mixing within a unit rather than just at the start when the unit is being launched. There should be other opportunities for rich, low-floor high-ceiling tasks that many different students can access and work on together. At the same time, I would also like to see more differentiated materials used for intervention/re-teaching/practice. We currently give students access to a spreadsheet of practice problems and guided notes, but don't have fine-tuned intervention and reteaching strategies or problem sets. I would also like to see more spiral review built in to the curriculum so that students who are still working on concepts can continue with the rest of the grade, but continue to revisit material. 

The biggest question that I have is whether this model can continue into the higher grades. In 7th and 8th grade, we have traditionally broken students up into two tracks that had different curricula and that resulted in different placement in high school math classes. Our high school math classes are not grouped or tracked, but students can start at different points in the sequence of classes, which is another way of trying to avoid homogeneous, fixed groups, but is very different and operates on the assumption that students of different grade levels can take Math together. This is really different from our middle school model, where the entire schedule for a student is driven by their grade level. I'm very curious to hear how others (both middle and high school) are solving this issue and whether anyone has been able to make flexible groupings work for higher grades.

11/22/19 update: I recently co-presented on how the flexible grouping model interfaces with a growth mindset at a conference with one of my coworkers. Here are the slides from this presentation.

Thursday, December 13, 2018

Differentiation and the limitations of groupwork

It's important in any profession to stay humble, but teaching has a way of reminding you of this in particularly in-your-face ways, I believe. This semester has really brought home this issue for me in the challenges presented by my upper school Math 3 class. The issues have been around productive groupwork, an area in which I have felt particularly strong and well-trained, so it was perhaps an especially humbling experience to see all of my strategies and approaches come crumbling down and leave me turning to my Twitter network and colleagues to find new ways of helping students work together and feel confident in their progress. I wanted to share and summarize here some of the issues I've worked through that might perhaps be helpful to others.

First, some background.

I have been incorporating the essential elements of a Thinking Classroom in all of my Math classes for the past few years, but most notably in my high school class, where the focus on content and pressures to teach to the test are greater. This year, just like last, I had students read and reflect on Thinking Classrooms and we discussed why most of our time together is spent working on problems in random groups, sharing out ideas and conclusions, and using these to synthesize and summarize learning from the bottom up rather than top down via teacher-led instruction. Students initially seemed bought in and supportive of this type of classroom environment. We set up class norms and discussed the use of group roles, how to step up/step back in group environments, and how to be a skeptical peer and give respectful pushback on ideas.

Several weeks into the semester, however, I started noticing a troubling pattern - some students were disengaging from their collaborative work and seemed very hesitant in sharing their thinking within either small groups or the larger class. Then, I started to hear two different complaints from students - some were feeling that their work during class was unproductive because their groups moved too fast and they were feeling increasingly anxious and uncertain about their mathematical understanding and abilities. They were feeling unprepared to do problems independently on homework assignments or assessments and wanted more teacher guidance and structure, as well as more opportunities to go at their own pace and understand ideas more fully. Other students raised the opposite issue - they felt that the pace of the class was too slow, that they were doing too many problems that they already knew how to do or could figure out quickly and wanted more challenging and deeper problems, both during class and on homework assignments.

When group work goes wrong:

In reflecting on these issues and why they were coming up this year, I realized that we had actually made quite a large change to the Math program without making any changes to our curriculum or pedagogy. This was the first year that we had decided to mix all grades taking a particular Math course - Math had been the only discipline at the Upper School in which students were separated out by grade level. In the past, 9th and 10th graders taking Math 3 (students who had accelerated the normal sequence) were in different sections from 11th graders taking the class (students who were on grade level in terms of their progress through the sequence). This year really was different in terms of prior math experiences, expectations, and desire for challenge/acceleration for students in the same class and the normal groupwork structures were not sufficient to bring together students with such varying backgrounds and approaches.

My next step was to look for feedback from colleagues as well as the Twitter math teacher community. Some suggestions that I implemented that seemed to make a difference:

  • Taking a break from random groups to help students regain their trust that the class would meet their needs; doing some work in pairs designed to foster productive collaboration; allowing students choice as to who to work with while also asking them to work with different students at times; being explicit when the goal of a task was to build collaborative skills
  • Structuring activities so there was time at the start for individual exploration before asking students to share their thinking with others thus giving more processing time for students who worked more slowly; circulating and helping some students get started; building more optional challenge into tasks for students who worked very quickly or who had already had prior experience with a topic; creating tasks that could be approached with a greater variety of methods and building more writing into tasks so that different ways of thinking mathematically could be valued
  • Meeting students where they were to regain trust and buy-in; this included at times splitting the class into two groups (students chose which group to join) - a more free-form exploratory group with more open and challenging problems and a more structured group where students would get some problems to activate prior knowledge and smaller, more concrete problems that would build over time to greater generalization and abstraction and more teacher guidance and reassurance that they were on the right track
  • Noticing struggling students' successes and highlighting them publicly; selecting which students would share their thinking to make sure that different voices could be heard over time
  • Make sure to leave time for synthesis and practice problems (at different levels) during class - this helped address student concerns that they were leaving class with lots of questions and feeling unsettled about the concepts they had explored that day
  • Giving students more feedback during class about their understanding of a topic rather than relying more heavily on groupwork and self-assessment for students to know how they were doing and what might be helpful next steps
  • Providing more problems at different levels and helping students navigate which problems might be more helpful for them to do during/after a particular lesson - here is an example of a tiered homework problem set.
  • Providing more textbook resources - explicitly linking textbook sections to problem sets for students who wanted more references and examples
We are considering sorting students for next semester by grade level to decrease the heterogeneity of classes - while these strategies have alleviated the issues significantly, it does seem that productive collaboration and exploration is challenged when students in a class are so spread out. Despite these strategies, for example, it usually doesn't make sense for students who have worked quickly and deeply and have figured out challenging extensions to share their ideas with the whole class, most of whom have not even tried these problems. As a result, the class sometimes lacks cohesion and feeling like a true community. Additionally, the amount of work required to run a class with this much differentiation is really, really high. I'm essentially designing at least two different classes, creating both lesson plans and homework assignments that can reach the full spread of student interest and background, and giving individual and frequent feedback to students or small groups of students into which the class has fragmented. This is not really tenable for the whole year given my other preps and teaching responsibilities. However, breaking up students by grade level seems to run counter to our values of equitable access to challenging mathematics for all students and means that Math classes are essentially different from all other classes at the Upper School.

I would welcome any feedback or suggestions that others have around this issue - what strategies have worked for you in working with very heterogeneous groups? 

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.