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. 

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.
var EMAIL_SENT = 'EMAIL_SENT';

/**
 * 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
      SpreadsheetApp.flush();
    }
  }
}

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 ;)