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. 

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 :)

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

Math 1 Problem Solving Unit Day 1

Math 1 Problem Solving Unit Day 2

Math 1 Problem Solving Unit Day 3

Individual/Group/Class Norms Revised

In my previous post, I wrote about my updated group norms. But then, I got some great feedback in the form of comments, a few Twitter conversations, and a post by Sarah on her updated groupwork norms... damn you, MTBoS with your feedback, always making me want to change stuff to make it better.

I decided to break up my groupwork norms into three components:

1. Individual accountability
2. Group accountability
3. Class accountability

Here is the poster for each set of norms.

Individual Norms

Group Norms

Class Norms

Finally, I made a poster for the green/yellow/red cup strategy Avery uses in his middle school classes. I went back and forth a bunch of times to see if this was perhaps not going to work in high school, especially if students are usually working on whiteboards around the room rather than sitting at a desk, and if there was maybe a way to do this electronically, but eventually, decided to just do it the same way that Avery does it and then make changes, if needed.

The idea is that each group starts with a stack of three cups, with green on top, yellow in the middle, and red on the bottom.

If the group feels stuck or confused, they should move the green cup to the bottom of the stack. The yellow cup is a sign to the group to discuss their confusion together and try to get themselves unstuck using the various strategies we've discussed or by checking in with other groups. 

If they have discussed and tried to get themselves unstuck, but were not successful, then they come up with a single group question that every member of the group needs to be able to articulate, and can switch their cups so that the red one is on top, at which point a teacher will come over and ask a random member of the group what their question is.

Avery's selling points for me were:

• The yellow cup is an important step to prompt students to reflect on their confusion as a team and get better at the "unsticking" process that is such an important part of productive struggle.
• There is a clear visual for the teacher in scanning the room where groups are at and which might need attention soon (currently at yellow).
• You can hear cups switching so without even looking, have a sense of group need.
• The proportion of the time that various cups are on top gives you valuable information regarding the challenge level of the task you've given students that day. Ideally, cups are changing back and forth between green and yellow as groups become puzzled and then figure things out on their own.

Here's a poster summarizing this for students:

Cups

As always, feedback and suggestions for improvement are welcome!

Formalizing Routines

In my last post, I blogged about #TMC16 and how excited I was to take what I learned in @davidwees's workshop on instructional routines and apply it to what I do most in my class, which is guided investigations (aka problem sets that scaffold instruction) and open investigations, which are more focused on exploring connections and representations of student thinking. I've taken a first stab at writing out the steps and teacher moves involved in both types of investigations (links below), including writing prompts for students and class norms. The class norms were especially tricky to nail down because I've been thinking all summer about how to marry the norms that I learned in Complex Instruction, which are all about valuing different types of participation and making the group a cohesive and supportive unit, with what I'm seeing as emerging from the research on Visibly Random Groupings, which values flow and makes the entire class a unit of idea exchange and interdependence. Complex instruction often has assigned roles within the group and clear instructions on establishing a "group question" before a teacher can be called over for help. By contrast, in a VRG class structure, students are encouraged to share ideas with and ask for help from anyone in the class. Groups change daily and roles are eschewed in favor of flow of ideas and vertical whiteboards that encourage easy participation and engagement.

My attempt to merge these two cooperative structures (as well as my other goals for students) has resulted in the following group norms:

Group Norms

I am going to continue randomly assigning students to groups when working on problem sets or open investigations and avoid assigning roles. There will probably be one day every week or two when students are grouped homogeneously based on their self-assessment of their needs (more structure/support/direct instruction, same level (stay with guided inquiry), explore independently). I have to think about tweaks to the group norms that need to happen on those days.

I also wrote out the protocol for when a group can ask me for help. They need to first attempt the strategies posted in the classroom for getting unstuck (listed below), look around to see what other groups are doing and send a representative to another group to discuss and share ideas, and if they're still stuck, to formulate a single question to ask me... aka a group question. I should be able to ask anyone in the group what their question is and be assured that it was indeed a group decision to get help.

If your group is stuck:

I will try to remember to write another post discussing the various reflection prompts and closing questions that I've adapted, but here are the links to the two routines, which have all of the prompts I've thought of so far.

Guided Inquiry Routine

Open Investigation Routine

Feedback is super appreciated! These are still very much in the planning stages, but it's been immensely helpful to write out and formalize the routines that I normally use in my classes. My goal is to work on making these better this year, both in my classes and in those of my colleagues, through lesson study focused specifically on routines.

#TMC16 Recap

I certainly didn’t need another TMC conference to remind myself of the power and specialness of the #MTBoS, but it didn’t hurt either. I think that this year, I needed the professional rejuvenation of seeing everyone in person just a bit more than usual after a difficult year. So my first and foremost task in this blog post is to convey my gratitude to everyone at the conference who made time for me to vent, invited me out to dinner, came to and participated in my session, gave me amazing advice, and was a source of support. It’s amazingly powerful to have a common starting point with a group of people, to pick up where we left off, either from last year, at other conferences, or Internet conversations. There’s no need to explain, to get our bearings, to wade through small talk and establish common ground and trust – these people get me, we’re on the same team, and it’s effin’ going places. There were so many friends with whom I didn’t have a chance to reconnect, unfortunately, but huge shout outs to @davidwees, @park_star, @jaz_math, @MrJanesMath, @AlexOverwijk, @normabgordon, @JamiDanielle, and @crstn85 for helping me unpack the challenges of this year and get excited about the year to come.

Even though the power of the TMC for me is in the relationships, not the workshops, I have to give huge props to @davidwees for completely challenging my notions of professional development for teachers. I have always been of the camp that the less structure, the better, and that our work together should be about discussing best practices, research, and ideas. Let individual teachers determine how to tweak and interpret these ideas for their particular classrooms, on which they are the experts. It turns out, tightly structured practice with specific strategies, seeing and debriefing them from different angles, and trying to do them yourself while getting feedback is pretty damn powerful. I can’t do a three-day morning session justice here, but check out the materials available about two of their instructional routines here, and I really hope that there are videos available for people who were not able to attend since this is really the kind of thing that needs to be experienced to be understood.

On a meta level, I was surprised by how much practice and feedback we (who were mostly very experienced teachers) needed to get halfway decent at a single routine. We saw it demonstrated several times and debriefed these examples, which is generally as good as it gets in PD. But once we tried implementing it ourselves, new issues and questions surfaced and were addressed. David’s focus on improving teaching practice, not individual teachers, was instrumental here to allow us to be vulnerable and know that we were working together to learn how to do something new and not to be judged (either positively or negatively) by our colleagues. Going to internalize the heck out of this and bring this thinking to my work with other teachers in my department and in the PD that I lead.

I feel like I learned several key points about instructional routines themselves from David and his awesome colleagues (yay Jasper and Kaitlin!):

• Having a good routine frees us to dig deep into the learning, which is extremely counterintuitive for me as my tendency was to assume that structure is limiting and restricting. Nope! Having good guidelines (which can certainly be tweaked for tasks that would benefit from that) means that our cognitive energy is directed solely at learning and the intellectual work at hand rather than trying to figure out expectations and how that learning should proceed. The routine allows students to do more challenging work. It is nothing like the mindless-recipe-following that I imagined.
• A quick pace and keeping the focus on what we can learn about patterns/shortcuts/whatever the purpose of the routine is makes the activity engaging and mediates students’ relationship with math in a way that honors their thinking, but pushes them to go deeper, see connections, and understand other methods and representations. This allows all students, not just those who intuitively know how to learn math, to access the learning.
• A routine is not the time to teach something new. Its purpose is to explore students’ thinking, bring out and connect their ideas, and help them represent and synthesize key mathematical ideas. I can certainly “teach” something in a more traditional sense after the routine, but genuinely focusing it on student thinking keeps it engaging and powerful.

Finally (longest blog post everrrrrrr, bless you if you’ve stuck with me for this long), I loved David’s suggestion of using instructional routines as a focal point for lesson study rather than the traditional use of lesson study to develop a content topic. Different teachers working together can develop different lessons around the same instructional routine and then observe each other, give feedback, and improve the routine in their work together, improving their teaching along the way and creating programmatic change, not change on the individual level.

I am planning on working out my own instructional routines for doing guided and open investigations (similar to Exeter problems and Interactive Math Program tasks) since these form the backbone of my class, but could see separate routines being developed for Open Middle problems (paging @math8_teacher), Would You Rather, Estimation180, or Which One Doesn’t Belong. Contemplate then Calculate was basically created with @fawnpnguyen’s VisualPatterns in mind and it feels like @ddmeyer has already created a 3Act routine, so those two are all set. Any others?

I am super, super, super excited about this as creating routines is something that I think I’m decent at (as opposed to creating creative problems/lessons, which I’m absolutely not) and it’s something that can fundamentally improve our practice in a way that sharing one-off “clever lessons” doesn’t (thanks Dylan… such a valuable point). I’ve already committed our department to doing lesson study this year, and we’re in the midst of building a new program and integrating a number of teachers new to the school so this couldn’t have come at a better time. If there are other teachers out there who would like to give feedback on my ideas or collaborate on creating and tweaking instructional routines around guided/open investigations, I would love to hear from you.

So there you have it… my #1TMCthing is to develop two instructional routines and get as many people as I can to collaborate and give me feedback. Ready, set, go!

Co-teaching and the start of semester 2

Welp, it's been a month or so since we've been in the full swing of things, and "intense" would be the understatement of the year. As a department, we rolled out a co-teaching model for almost all of the core Math courses this semester, which has created many new opportunities and also, challenges. Here's a quick summary of how we're approaching co-teaching, pros, cons, and next steps/thoughts/questions.

Summary of approach:
The core Math courses we offer follow a Math 1 --> Math 2 --> Math 3 sequence, after which many students will take Calculus (as a new school, we have not yet firmed up all the elective options we would like to eventually offer students concurrently or after the core sequence, but that's a whole other blog post). The way that co-teaching is working this semester is that two classes of the same level are scheduled at the same time in two rooms that have a divider down the middle that can be lowered or raised. There are about 26 - 36 students in each co-taught class with two Math teachers. Each of the rooms has a projector and an Apple TV so that a single computer can project to both classrooms, if desired.

The two or three teachers who teach the same prep meet during a joint planning period for two long blocks and one short block each week. Most of us have two preps, so are attending multiple co-teaching meetings. We also try to carve out a few hours once a month or so for all of the co-teaching teachers to meet and discuss how co-teaching is going overall.

Pros:
Many more grouping options are now possible than were available in a traditional classroom setting. We have had some success with differentiated groupings based on support level needed (either student-selected or teacher-assigned) and would like to explore other groupings (theoretical vs. applied, review vs. extension, project topic selected, or just interest in a topic... for example, we recently had a class where students could either be audience members for fellow students presenting on their projects or help generate a proof for an earlier topic we hadn't had time to delve into in depth). We like the idea that students can potentially learn in different ways from different teachers and have more options for who to see outside of class for help as well as choice in their groupings. Peer-peer interactions can also be more rich when there are more options of students with whom to interact and there can be a cohort rather than one or two students with a particular need. Curricula are also much more aligned than they were when classes were taught individually since we're planning so much together.

Cons:
Mainly, the cons are in figuring out how to make the best use of all of the new choices available to us and time to do all of this well. All of the pros above in terms of coordinated planning and differentiated groupings can only be realized with very thoughtful and intentional implementation, which translates into lots of time together... time discussing content/process goals, creating lesson plans, reflecting on those plans, thinking about what work various students did that day and how they received feedback on that work, and how all of these things will inform the next day/week/month.

Next steps/thoughts/questions:

There are so many things left to work out. A big one are the different responsibilities that co-teachers now have and that need to be partitioned: Who will do what during class? Who will take the lead on a particular lesson or do all lessons need to be jointly planned? Who will be responsible for giving feedback to which students and then communicating to the other teacher those student needs and what has been done so far? Who will help to make sure that the administrative stuff necessary for running a class is being done? How will we reflect on co-teaching as a practice systematically in order to learn from this semester and make it better for next year? How will co-teachers interact with each other during class, communicate about changes to the lesson plan, or give each other feedback about the lesson/teaching? And where is all of this time going to magically come from?

One area of huge support has been the hiring of an assistant teacher into our program a few weeks ago. She has already made a huge difference in the level of overwhelmingness we were feeling initially and has been instrumental in being an extra pair of eyes in the classroom, both for student learning and feedback on the lesson, as well as in helping with some of the grading and administrative work. I now understand why co-teaching often involves removing a prep or class from each teacher involved - there is so much more to do now. Trying to do it in addition to full loads for both co-teachers was perhaps a bit rash, and we're so glad to have the extra help now.

Next steps are to make progress in answering some of the questions above, as well as in continuing to play with the various possible groupings to try to harness more of co-teaching's power. It will also be important to get more student input into the process - how do they see this improving their learning and what ideas do they have for creative/new ways of organizing a classroom?

I would love any feedback, suggestions of literature that might be helpful for us, or questions!