Monday, February 13, 2017

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

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

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

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

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

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

Tuesday, January 10, 2017

Why might students be motivated in math class?

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

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

  1. Patterns and beauty inherent in mathematical structures

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

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

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

    These students seem to be predominantly motivated by positive interactions with others, whether that's the teacher or their peers. Classroom structures that increase conversations and collaboration between students and that make students feel known and connected to others have been helpful in motivating this group in my experience, as well as putting more of an emphasis on my relationship with them. 
Obviously, most students are some mix of these categories, but for many in my experience, one is more dominant. I think that a classroom that tries to balance between these different student needs will likely result in broader student success than one that caters to only one type. I would love pushback on my preliminary and perhaps too simplified analysis. Are there any categories you see being useful for thinking about student motivation? What other tools and strategies have you used to help students foster their curiosity and interest about math and motivation to exert effort towards the class?

Friday, January 6, 2017

#MtbosBlogsplosion - My Favorites

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



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


Goals for second semester

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

Changes for my personal teaching:

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

Big Picture Curriculum:

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

Professional Development:

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

Wednesday, December 14, 2016

Lessons from this year - supporting struggling students

One of the big issues that's been emerging for me this year is how to support struggling students. As a school, we've made a commitment not to track and to differentiate instruction so that students from a variety of backgrounds can be supported and pursue their interests fully. The desire to provide challenging content to all students is one I very much support and know the research backs it up. The problem, of course, is that if students are grouped heterogeneously, but the content is the same as what would be taught in an honors section, students who have not been successful in Math in the past do not magically overcome those challenges. What does end up happening is that half the class is frustrated and feels like the pace of instruction is too slow and the other half of the class has their preexisting images of themselves as unsuccessful Math students confirmed.

I have also been quite surprised to see that it's often students who are struggling who give pushback to teaching methods that emphasize choice, group work, student-constructed knowledge, and open problems. They feel unsuccessful with these teaching styles and crave direct instruction, structure, and concrete, repetitive problems. These students (and their families) have been asking for textbooks, lecture, and an explicit curricular progression in which students are walked through algorithms and given lots of practice. In teaching these students, when I see how much more scaffolding they need to successfully mediate their relationship with mathematics, I understand their perspective and needs much better than I did before. Their gaps are often not in prior knowledge (although that's there too), but in how to learn Math. As a department, I think that we've done a great job of building a rigorous and interesting curriculum that works well for successful Math students who jump into open problems, ask questions, tinker and test, iterate, confer with peers, look for connections and patterns, reflect on their understanding, and figure out what they do and don't know independently. When they lack some of these skills, they are receptive to feedback and observation of peers who model them. We have not yet, however, figured out how to teach all of these skills while simultaneously asking students who don't yet have them to grapple with difficult mathematics in an environment that requires these skills to be successful in that work.

One solution to this issue is to give the students what they want: a choice between a track of open/challenging/problem-based math and a track of traditional/lecture-based math. For many reasons, this is not a solution that I can get behind. Perhaps I'm wrong, but I have not seen incontrovertible evidence that there are some students who just can't learn Math without lecture and repeated drill. If we really think this, we are basically saying that these students can't learn Math and let's just teach them how to regurgitate some procedures so they can get by on their standardized tests. I would have a very hard time supporting a bifurcated system like this.

Other ideas I have had that might help this issue are:

  • Provide an extra Math class for struggling students that would focus on just content or just mathematical practices/habits; either make this optional or required
  • Provide a summer bridge program for students who we worry might struggle in our program, focusing on building up their ability to learn and mathematical practices
  • Start the year with work on mathematical habits and ways of learning Math with little to no focus on content for all classes. 
  • Work on improving our curriculum so that it incorporates more of the principles of Complex Instruction and can highlight students' strengths.
I would love to hear from others who have grappled with this issue and ways that they and their schools have approached it, either successful or not. 

Thursday, October 20, 2016

Starting Lesson Study and Update on Classes

This isn't going to be a very coherent post, but I need to get our current work into writing to better organize my thoughts. Here are the projects that are in progress right now:


  • Lesson Study
    • We've broken up our Upper School Math teachers into several groups of 3-4 teachers who teach across different grade levels. We considered doing a more traditional lesson study in which teachers plan a lesson centered around specific content, but decided to focus our efforts on developing our practice around a particular instructional routine that would be relevant across many grades and that would help us fine tune a specific pedagogical approach and learn from colleagues with whom we rarely get to work.
    • Everyone read this article from KQED to orient themselves to the lesson study process in advance of our first meeting.
    • Each group selected an instructional routine to plan out, teach, and refine this cycle. The routines selected were:
      • Differentiation for students who learn at different paces
      • Guided investigation
      • Students giving feedback to each other

    • In the next meeting, groups will plan a specific lesson around their instructional routine based on the first teacher who will be modeling it and decide on an observation time and what to look for when observing
    • I'm super excited for this initiative to be gaining traction! We got some time to work on this while students were taking the PSAT or doing other activities, but I'm worried that if we don't get specific time off to work on this, people will become significantly less enthusiastic. 

  • Parent Math Night
    • Our team is working on developing an informational night to help parents better understand our program, available resources, and philosophy. It's just in the planning stages, but I think will be really helpful in getting on the same page with families. Right now, whatever information they receive when applying is the extent of it. 
    • This needs to be thoughtful and informative for parents while also clearly conveying our position and getting buy-in and understanding of the program. If you have any resources or ideas to share, would love to have them.

  • Math 1 is finishing our unit on Counting, Probability, and Sets, designing a game that has students analyzing probability and expected value to determine best strategy and fair outcomes. Next week, students will be playing each other's game and reflecting on what they've learned. This is a good opportunity to differentiate and identify gaps in understanding linear functions and algebraic manipulation skills as we prepare to move into a functions unit next.
    • We're starting each unit with a "preview" assignment to look at prior knowledge and pre-requisite skills and concepts so that those students who have gaps can be identified and given extra support. Here is the preview assignment for linear functions.
    • We're also starting the unit with another open investigation, this one more directly related to functions ("Cutting the Pie" task from IMP Year 1). I'm curious to see if students pursue a recursive or closed rule for this function. When we worked with Pascal's triangle patterns, students had a hard time moving from "each row is twice the previous row" to the rule f(x) = 2^(x-1).

  • Math 2 is still in the depths of statistics, working through the Central Limit Theorem and connecting probability and the normal and binomial distributions. I'm realizing how much better I understand the material in my third year of teaching it and how much less formulaic and prescriptive my teaching is now that I have deeper content knowledge in this mathematical space. I have known that strong content knowledge is necessary, but it's amazing how much more depth is needed if you want students to explore and create and test their own theories, both in terms of creating those scenarios and in guiding the discussions that ensue. Maybe it's true then that teachers have to first progress through traditional pedagogy as they build up their depth of content knowledge before they can start to incorporate more problem-based or project-based learning. Would love some pushback on this though :)

Thursday, September 1, 2016

Habits of Mind Unit - Math 1

We've had four whirlwind days of school so far - I'm really enjoying starting with a Habits of Mind unit in each of my classes as it means students are working on tasks and learning the routines of the class every time we meet and I am getting to know them and the flow of the new year.

In Math 1, we have been working on several different tasks, each of which is related to combinatorics, the first unit that we'll be officially starting next week. In each task, students start with an introductory question and then each group creates an extension to pursue next. The three tasks we've done so far are below. I'm still tweaking the fourth one and will post it when I'm done (hint: this is one of the things I need help deciding).

Task 1: How many paths from A to B if you can only travel down and to the right?
Extensions created by students: generalize for a grid of any size, allow travel up and to the left (without crossing over), allow traveling diagonally




Task 2: Consider a game in which you flip a coin four times. At the beginning of the game, your score is 0. Each time you get heads, you get a point. Each time you get tails, you lose a point. What are the different scores that are possible and how likely is each of these scores?
Extensions created by students: generalize for n flips, what about dice that have 4, 5, 6, etc sides?


Task 3: How many different monetary values can you make from these bills?
No extensions created yet, will have more time on this next week



Scroll down for the presentations from class for each of the investigations, which include slides about group/class norms.

Two big questions with which I'm wrestling in doing these tasks are:

  1. How much, if any, content teaching should there be? Students are practically begging for more efficient methods than just listing out all of the options, but should this unit really be about helping students get better at exploring their own thinking or is it better to teach some content while they're hooked and eager rather than coming back to it when it actually comes up in the unit? For those who incorporate student-driven investigations along with teacher-led instruction, when do you do the latter? 
  2. Relatedly, how much should I be pushing students to make the connections between these problems more explicit? I feel like I've been dropping some (subtle) hints and revisiting student work from previous problems in the hopes that some students will point out the underlying connections, but no such luck. Again, is it better to show these connections now, even if it means they will mostly be teacher-driven, or better to wait until later and let these problems simmer for a while longer?

My current thinking on these two questions is that I will require each student to work on generalizing one of the tasks and then have students present their generalizations and ask more explicitly about connections between them at that time. I have to now choose a fourth task that I hope will make the connection more obvious... suggestions? What are some tasks/problems you've liked for hooking students on combinations?


P.S. I am super happy with how group norms and vertical whiteboarding is going so far this year. Using the same routine with a new math task each day so far has created a really nice flow and students are interacting well and starting to independently leave their groups to find out what other groups are doing to bring those ideas back. It was definitely worth taking a few days out of the content rush to set things up.
Presentations from class