Reflecting on yearly goals & moving forward

I've set yearly goals a number of times, but feel like this is really the first year that I went back and reread my plans and made adjustments to actually try to reach them before the end of the year. Partly, this was due to blogging about them, as well as knowing that someone was going to check in with me in October to see how I was doing. Because the online accountability didn't end up being as ongoing as I thought it would be, I want to consciously reflect on my goals and think through what I want to continue doing or change for the rest of the year.

Recap of goals for this year (full post here):

Grading/feedback

• Give back quizzes with feedback only, share the grade later.
• I've been doing this and it's going well. Keep it up! Need to give more time in class to process and correct assessments, discuss with me and others.
• Students must correct original quiz and demonstrate evidence of work/learning done in order to reassess. They may not reassess on the same day. 
• I have not been as strict as I should be on this one. I basically tell them they need to do this, but don't actually check very rigorously. I have way fewer students reassessing this year though so it's not been a lot of work to manage. 
• Teach students how to use feedback effectively. 
• Sort of doing this... in-person meetings with students have been instrumental in my giving of feedback this year. I did a few peer feedback assignments and written reflections responding to my feedback, but need to do it more. 
• Have students self-assess their practices via a portfolio.
• Haven't done this yet, but plan to at the end of the semester. Need to check in with @crstn85, who's having her students self-assess their work on the mathematical practices and provide evidence for each one. 
• Get homework and projects graded more quickly.
• In-person meetings, describe here, have been really helpful with this. I discuss students' recent work with them in person so I am staying pretty current with grading. Definitely way more current with projects, which languished on my desk reproachfully for interminable periods of time last year.

Class culture

• Continue using Visible Random Groupings and whiteboarding.
• Done and done. I merged this with a goal for strengthening student-student relationships by having each daily random group go around answering a question (such as: if you could go anywhere in the world, where would you go? what is your spirit animal? what has been your favorite class this year? when were you last a good friend to someone? what is your favorite movie? etc) Students often give ideas for questions - it's been a great way to get to know students and to build trust and relationships in the classroom.
• Continue my policy of having students volunteer to participate in class discussions, with the caveat that each person must participate at least once or they must start the next day's discussion. In addition, when groups report out, I can call on any member of the group.
• Yep, still doing this and still prefer this over calling on random students. Everyone has to participate at some point, but they can choose when and how, which I think is important for communicating my beliefs about classroom culture and that contributions should be about learning, not for punitive or classroom management purposes.
• Introduce talking points and exploratory talk ideas into class discussions and teach language of argumentation and mathematical discourse.
• Uh, this one I've definitely failed to implement and not sure that I have the bandwidth to add another thing into the mix right now. I'll look over the list before the start of the next semester and plan out some actual activities rather than the vague idea to do this somehow somewhere.
• Continue assigning bi-weekly reflections; include metacognitive questions as well as questions about mathematics and thinking routines/student learning. 
• I have not been assigning longer metacognitive reflections as often as I did last year, but including a few reflection questions on most daily assignments. Definitely doing more questions that ask students to restate concepts in their own words, look for connections, and explain errors or discrepancies more frequently. I haven't really addressed the issue of thinking routines and how they are impacting student learning, other than asking students to reflect on their learning for bigger projects/write-ups. Need to balance the benefits of reflection with students not feeling like they are reflecting all. the. time.

Homework

• Be more intentional about homework: assign fewer problems, spiral it in more intentional ways, and always provide answers in advance and worked solutions from students' own work after we discuss homework in class. Organize homework into Review, Reflect, and Reach. 
• I'm much, much happier with how homework is going this year. I am lagging assignments, which allows class to be more flexible and not feel like it needs to cover a certain amount in order for homework to make sense. There is also more time for students to process and make connections in class before they are asked to do independent work on the topic. I am also seeing more retention since every assignment includes review that intentionally brings in topics that are related or that I think students could use more time with. I would like to assign fewer problems as we are still sometimes spending a lot of time going over questions, but they are good problems to be discussing.
• Ask students to give themselves feedback on their homework. Continue assigning a catch-up day every 2 weeks.
• I have had students give themselves feedback once, need to do it again. I have not had any catch-up days this year because students are able to complete assignments by the original due date since the work load has been more manageable and because they are getting feedback in person. Should definitely have at least one or few times that I ask students to pick an earlier assignment and revise it though.
• I will continue having students turn in pictures of their homework digitally while keeping an organized notebook. 
• I need to put more effort/teaching into students' notetaking. I would like students to take notes and write down questions/ideas/connections. This is especially important since we don't have textbooks so notes are students' only record of what we are learning. I also want to have students write down questions, conjectures, tests, and conclusions more formally - it will be really helpful to show models of actual student work so that students have a better idea of what this looks like. I am giving students time at the end of class to write down summary statements and examples, but need to do this at a few intervals instead of just at the end and give students more tips and feedback on how to do this better. I am providing a rubric for all assignments, but it would be great to have links in the rubric to samples of student work so that students have more specific models. I also need to provide students with models of what formal write-ups look like as well as what an "exemplary" (which is our highest level in the standards based grading rubric) level for each mathematical practice looks like. 

New goals:

• Take pictures during class of work that is going on the whiteboards and post these - still need to figure out an organized way to put up this work so that students can find it easily if they need to review a specific topic and don't remember the date on which we worked on it. I currently provide a list of electronic resources for each topic we are studying - perhaps I can make a second column of links to pictures of our in-class work on that topic. I still need to think about what makes sense here - a chronological order or a topic-based order. One idea is to assign each topic to a student and put them in charge of writing a summary of the topic, including examples, and uploading this to a class online textbook.
• Keep better track of students' progress. I have been doing a lot of discussion and in-person meeting, but not a lot has been in writing or communicated more formally. I am now starting to write down a summary of our in-person meeting with specific objectives listed, such as "review binomial distribution model and reassess on it next week" and then emailing the list of feedback and to-dos to the student. I should make sure to loop in the advisor when appropriate as well.
• I need to have more regular meetings with my co-teachers. I have not been doing a great job of planning together or watching others on my team teach. I wanted to do some co-teaching and lesson study, but in the frenzy of the year, this has sort of fallen by the wayside. I'd like to recommit some time & energy to this in preparation for the second semester to figure out a regular plan for making this happen.
• I need to figure out better ways to support struggling students in my classes. I have been more focused this year on providing sufficient rigor and challenge to students who learn quickly and want to explore more topics, but haven't been addressing as much the needs of students who need more time and to sit with concepts for longer and from more perspectives in order to feel more comfortable with the material. I need to structure classwork and homework occasionally to allow for more differentiation and review/reach, as needed.

Student meetings for formative assessment

This year is going by in a blur, but I did finally squeeze out a few minutes to blog about something that I've tried this year that I'm really liking. Based on a description in the book, "Creating Cultures of Thinking," I implemented something that the book calls Individual Feedback Sessions (I just call them "regularly scheduled meetings," but sure, pick the fanciest name you think you can get away with), which were created by a teacher profiled in the book. His explanation of this strategy is here, but the gist of it is:

1. Schedule 20 minute bi-weekly meeting with each student (he recommended meeting with 2 students at the same time, which I would do if I had a more typical class load). I meet with students before school, during lunch, during their free periods (if they overlap with mine), during tutorial, and after school.
2. Discuss work that the student has turned in during the last two weeks (since the last time you've seen them). Go through their work with them in detail, giving verbal feedback. Either you or the student records a summary of the feedback (I put comments in the online gradebook and ask students to also write some notes in their notebook).
3. Discuss the student's overall progress in the class, how they're doing incorporating feedback from previous sessions, and ask for questions about the content and for feedback on what you can do to support them better. Follow up on any issues that have come up in class with that student.

I have a schedule of meetings posted on the class page so students know when we're supposed to meet. For students who need a bit of an extra reminder, I have an automated email that emails them the day before our scheduled meeting. It is more work. I do have a much busier schedule during the school day as a result with almost no "free" periods. BUT, I do virtually no grading at home (other than quizzes, which I like to grade more quickly than a 2 week window would allow) AND I feel like students understand my expectations, really hear my feedback, and develop stronger self-advocacy and ownership of their learning. 

Grading papers at home by myself vs. interacting with an actual student about their actual work...
You do the math

Doing this has made it possible for me to assign better problems. I have been using IMP Problem of the Week tasks in my teaching for a long time, but every year, after grading a few write-ups, I became quickly overwhelmed by the sheer volume of grading (how the heck do English teachers do it??) and stopped assigning them. This year, I have already assigned four Problem of the Week write-ups for every class. And graded them all with students. And seen tremendous growth in their ability to describe their process and reasoning. Yes, I do think it's a valuable skill for students to learn to interpret written feedback from teachers, and I plan to do some of this in the second semester (perhaps, I provide written feedback first, then we meet and student explains how they understood it). But in terms of actually understanding and learning from feedback, in person conversations are waaaaaay more effective. Especially when I'm asking students to do mathematical work that they are not used to doing (writing about process, explaining reasoning, providing evidence, using formal notation and clearly annotating work, etc). Overall, I do think it takes more time, but it's much more fun for me and results in more learning for students so I think it's worth it. 

As the semester winds down, I am starting to ask students to give feedback to their own work and to the work of their peers so this is definitely not my only model for feedback. And as always, I'm curious to learn more about others' approaches and ideas on this.

Back to School Night

Aaaaaaand, it's a wrap.

Back to school night felt much more chill this year than it has in recent memory. Probably because I talked a lot less and tried to run it more similarly to how I run class with students. We started with a brief intro of me because parents are really curious about that stuff, but then jumped right into a problem I've liked for a long time.



I asked parents to make a guess. Crickets. I made a guess, and that helped break the ice. Once the gridded rectangles and scissors came out, parents got into it. They were finding volumes, realizing that the cut corner had to be removed from both sides, looking for patterns, talking to each other, asking if the corner length had to be an integer, and in general, being great students. When we discussed the need to be convincing, it made sense that a general rule could help with that.

Dum, dum, dum... enter desmos:



We talked about different approaches to this problem and why it makes sense to talk, collaborate, and learn from each other in math class. I used the problem as a way to describe my most common structure for class (problem posing --> intuition --> strategies, collaboration, checking for reasonability, changing ideas --> class consensus --> formalization, showing other approaches --> application to new problem space) and made the pitch that learning that happens in this order is far superior to just skipping straight ahead to the formalization step before anyone's hands have gotten dirty, both in terms of engagement and in the depth and quality of learning that's going to take place.

We talked a bit about content and the sequence of math courses at the school since it's a weird one.

And finally, my favorite slide: what I need from parents (inspired by @fawnpnguyen's back to school night slides, available here)


I'm not sure if this what parents wanted or expected, but it was fun for me! Wish there was a way to get quick feedback from my audience, but formative assessment is severely lacking in the back to school night business. It certainly beats going through a list of content objectives and the grading policy (ahem, what I used to do).

First two weeks of school

It's been a really fun first two weeks of the school year. Yes, exhausting as well, but super exhilarating and exciting too. This year, I started the year a bit differently, focusing more on how I wanted students to work together and think mathematically than on specific content. Because it was the first year I was doing this, I could do the exact same problems with all of my classes. Here's how it went:

Day 1:

• Students came in and saw a seating chart with randomly assigned groups of 3 or 4 and were directed to one of the vertical whiteboards. I wanted to establish this as the norm from the get go.
• Students filled out Google form describing a class in the past they've enjoyed, a class they have not enjoyed, questions that they have about this class, and questions that they have about me. I used the last two prompts as ways to discuss my expectations and structure for the class and to start building some personal relationships.
• Students worked individually for a few minutes and then discussed this problem, which I stole from IMP Year 2. Our first unit will be Statistics for all classes so I thought it would be good to do a fun, but challenging problems, that related to probabilities and ways of counting events.
• Homework was to fill out a Google form asking them about themselves and to keep working on the problem above.

Day 2:

• Students were grouped randomly anew and shared their work on the Tying the Knots problem. We spent the last half of class with group presentations sharing out their progress and practicing how to present and interact with presenters.
• Homework was to write a reflection on themselves as a learner and to start writing up the process and solution for the Tying the Knots problem (I used the Problem of the Week standard categories).

Day 3:

• New random groups, and I used one of @sophgermain's activities for helping students get to know each other. Nothing huge, kids just shared one thing they did over the weekend with their group.
• New problem! This one was incredibly fun. I originally thought that we would take it into proof by induction, but after input from @woutgeo and @hpicciotto decided to stick with a more intuitive visualization of the sequence.
• I basically let students work in their groups without too much guidance from me. Most realized it gave the Fibonacci sequence pretty quickly, but were not able to explain why. Many tried to develop a closed form rule, without much success (surprise, that's actually pretty hard to do). Most groups started trying the extensions, but didn't get super far. I stopped the class a few times and asked various students to explain their group's work. One of my classes this year doesn't have as many whiteboards as I'm used to having, but our desks can be written on so I'm going with that for now.
• Homework was a reflection on their process and feelings when working on these problems and presenting/watching presentations.

Day 4:

• New groups and I answered some more questions about the class and about me. I continued to have them share out a few personal tidbits in their groups as they are still very much getting to know each other (especially the freshmen). Today's questions were about favorite ice cream flavors and favorite movie.
• This was a slightly more structured day. I pushed students to be able to explain why the pattern that was produced matched the Fibonacci sequence. It was helpful to project pictures of their written work and explanations and use that to get more precise and tight in our language. I felt okay adding on to their explanations as needed since there were more extensions to explore (2 by 2 by n case and 3 by n case).
• Homework was to work on the two extensions and to start an integrated review problem set.

Students' work on explaining the derivation of the recursive formula

Some work from the first day on developing a closed form. I was not sure as to whether I should discourage students from going in this direction as finding the closed form rule is extremely challenging. 

More fun student work at the beginning of the exploration.

P.S. @daveinstpaul shared a great follow-up programming project in which students need to write a program to generate all of the possible ways to tile a 2 by n rectangle and then extend it to an m by n rectangle. I'm going to check in with one of the programming teachers tomorrow to see if this might make sense as a posible extension in her class.

P.P.S. A new teacher who I think is going to be amazing visited my class today and I got completely turned around in what I was saying and did not do a great job of moderating the discussion. It's been too long since anyone has observed me, and I just didn't feel comfortable with the kids yet to laugh it off so awkwardness ensued. Bah. We need to be visiting each other's classrooms much more frequently.

Goals for 2015-2016

I find these goal-setting types of posts pretty helpful. Ideally, I will actually go back to this and reread it at some point into the school year. Maybe when we do the #1TMCthing check-in at the end of October? I just reread some of the goals that I had set in previous years and it's pretty hit and miss in terms of how much I actually accomplished. My main concern is that I have too many goals and it's maybe unrealistic that I will be able to accomplish all, but what's that terrible saying that I've seen in a bunch of classrooms that doesn't actually make any sense? Oh yes...

That literally makes no sense. The stars are farther.


Anyway, without further ado, my goals for the 2015-2016 school year!!

Grading/feedback

• Give back quizzes with feedback only, share the grade after class (we will have an SBG online gradebook this year that will hopefully make that easier), as described in this post by @mythagon.
• Students must correct original quiz and demonstrate evidence of work/learning done in order to reassess. They may not reassess on the same day. Still debating whether I want to put a limit on the number of attempts. I thought about making all of the attempts count (with a weight making the later attempts count more), but decided that this is not in the spirit of SBG.
• Teach students how to use feedback effectively. This was from @pegcagle's session at #TMC15 and is my official #1TMCthing that I have publically promised to follow up on this year. One way to do this is to give back feedback mixed up and not attached to assignments and have students in each group try to figure out which feedback should go to which person. Another idea that I want to try is to have students exchange papers and coach each other on what to do with the feedback they received. Last year, I did a bit of peer feedback prior to turning work in to me, with some success. I need to make this a more consistent part of my class.
• Incorporate homework and classwork into students' self-assessment of their practices. Last year, I asked students to do this for projects, but I would really like a portfolio each unit with self-assessment that is more global and includes homework and classwork with linked examples of their work as evidence. This idea is based on a post by @jacehan.
• I really need to get homework and projects graded more quickly so that students are getting feedback at a time when it's still useful to them. I got bogged down with grading big time last year, and I need to be better about staying on top of it. Update: after reading "Creating Cultures of Thinking" a few weeks ago, I would like to set up individual meeting times with each of my students outside of class every 2 weeks (we'll see if the schedule supports this) in order to discuss their progress in the class and go over their projects and reflections with them. One of the ideas in the book that I found really fascinating was that instead of thinking about time as the most limiting constraint, as we usually do, we can instead think about energy... what feels energizing and what feels draining. It might make sense to change something that takes less time, but is draining with something that takes longer, but energizes you. The example given was grading papers at home, which can be exchanged for in-person meetings with students with the paper graded in real time and written feedback accompanied by in-person interaction. I would like to try this model, knowing that it will mean trading off some time, but hopefully, will feel less painful than grading projects late at night.

Class culture

• Continue using daily random groupings and whiteboarding, as described by @AlexOverwijk to increase student participation and engagement. Based on a Twitter conversation with @fnoschese about gender balance in groups, in which he discussed the research that groups in which there are as many or more girls as boys have higher performance outcomes for girls than groups in which there are fewer girls than boys (single gender groups are okay), I will be tweaking the gender ratios in my random groups to help make them more balanced, if needed. But not always. It depends. Basically, it's on my radar, but I'm not 100% sure that gender always trumps other status issues and I really do believe in the overall benefit of visibly random groups.
• Continue my policy of having students volunteer to participate in class discussions, with the caveat that each person must participate at least once or they must start the next day's discussion. In addition, when groups report out, I can call on any member of the group.
• Introduce talking points and exploratory talk ideas into class discussions, as described by @cheesemonkeysf here and as described in a similar Visible Thinking routine called Micro Lab. Teach language of argumentation and mathematical discourse, as described in the Claim-Support-Question Visible Thinking routine.
• Continue assigning bi-weekly reflections as a way for students to reflect on their learning and also to build community and feelings of connection. Move from reflections that are only about learning and affect to reflections that also dig deeper into mathematical concepts and connections. Incorporate reflection questions into daily homework or exit tickets to have more formal processing opportunities and feedback on thinking routines and classroom structures and how they are impacting student learning. 
• Build student-student connections to strengthen class culture. I really enjoyed the activities that @sophgermain demonstrated in her session on restorative justice, which facilitate student-student connections in the classroom. Some examples:
• There are two circles of students facing each other, a question is asked and each person speaks for 1 minute on that topic (examples of questions and topics below), then one circle rotates to a new partner and another question/topic is asked.
• Making one circle for the class and popcorn or going around the whole circle sharing on a particular topic or to give appreciation to someone.
• Have students write their names on a piece of paper, then distribute randomly and each student must write something nice anonymously about the person whose paper they received. Can repeat this multiple times and then return to the original person.
• Ask students to share at the end of a task who was helpful to their learning and how. 

Homework

• Be more intentional about homework. I blogged about this here already, but I'd like to assign fewer homework problems, spiral it in more intentional ways, and always provide answers in advance and worked solutions from students' own work after we discuss homework in class. Homework will be organized into three sections: Review (questions/problems relating to old material), Reflect (processing questions relating to new material, connections between content), and Reach (deeper/harder questions from mostly old material).
• From @bowmanimal's post on homework, I'm going to ask students to give themselves feedback on their homework and emphasize it as a learning tool. I normally tell students to limit themselves to 45 minutes per assignment and then provide a day every 2 weeks that is a designated catch-up day when they are expected to go back to old assignments that were not finished and put more work into them. I will do this more explicitly this year.
• I am still playing around with how I want to handle homework questions. I tried using a Google form, as described by @z_cress, last year, and it was a bit hit or miss. Quite a few students simply didn't do it and of those that did, many filled it out too late for it to be useful in my planning of the next day. In @pegcagle's workshop, she discussed using entrance tickets as a way to assess which homework questions are worthy of discussion and to anchor that day's learning, with questions like, "Which homework problem was the hardest for you? Which homework problem was the most interesting?" I definitely like giving groups a few minutes to discuss homework questions together and asking students to present solutions to problems that many are wondering about or that are especially important, but am still working on how to do this efficiently and in a way that maximizes everyone's learning. I definitely treat homework as a vital part of class, with questions that bring out connections between topics and that preview new content or skills that will be useful in that day's learning.
• I will continue having students turn in pictures of their homework digitally while keeping an organized notebook. I really liked that students had access to all of their work and that I didn't have to track papers last year.

Other

• I would like to emphasize organization a bit more. Ideally (if I can make myself do this consistently), I would like students to create a table of contents at the front and number pages in their notebook. I was reminded of Magdalene Lampert's structures for student notebook writing in @sgnagni's post describing the sections that she had students create (Date, Problem of the Day, Experiments, and Reasoning). For my high school students, I am thinking something like:
• Date
• Questions being investigated
• Tasks/Mathematical Work
• Summary and Reasons
• I would like to continue giving students 5 or so minutes at the end of each class to organize their written work and complete the summary section of their notes. I started doing this halfway through the year last year and students felt that this was very helpful in solidifying their learning, especially if much of their work had been done on whiteboards and they wanted a record of their thinking for that day.
• I will spend more time planning tasks in anticipating student responses and how I will treat them. I did too much of this on the fly this past year, and while much of the time, it went okay, I am starting to see the benefit of spending more time on advance planning. I would also like to provide rubrics for all projects in advance. I did this sometimes and always got positive feedback when I was able to do it.

Modeling tasks: what if there is no third act?

As part of a Modeling with Mathematics workshop that I ran for teachers with @zmill415 a few weeks ago, I had the chance to play with a few different formats for modeling activities and reflect on the types of thinking and representation that were done. We introduced a number of Three Act tasks, which the teachers really enjoyed. Generally, Three Act tasks have a specific question that students are trying to answer by creating a model... it's in the very structure of the format, in which the third act is the "reveal." I've always felt some tension between this format and the idea that the focus should be on process and not solution. Yes, we're valuing different paths and approaches, but ultimately, the one that gets us to the right answer is the one that's going to feel most rewarding for students. So for one of the modeling tasks with which we had teachers engage (a slinky lab in which we investigate the relationship between the weight attached to a slinkie and its length), I specifically did not ask the teachers any questions or set up a conflict that needed resolution. We simply discussed what they noticed and wondered about a slinky and then selected the relationship between weight and length as one that would be investigated in more detail. I was really struck by the richness of the conversations and representations that came out of this activity compared to other modeling tasks we had done together. The goal was to understand and represent the relationship and look for interesting connections between various representations, and the teachers really dug into this question.

Modeling teachers reflecting on their experience with the slinky lab

More views of their whiteboards

Obviously, I'm not saying that we should be throwing away the Three Act framework. Clearly, tasks with a fun and engaging hook that sets up a conflict or ones where a student might engage quickly with a guess are popular with students and provide excellent "needs" that prompt the development and application of mathematical tools. But, I do think that there is a place for tasks or activities in which the goal is just to tinker and think about how something works, where there is no resolution or ultimate reveal. I also make the bold claim that thinking of modeling only or primarily within the Three Act structure does a disservice to students in its focus on getting the right answer, especially if the purpose is to more closely mimic how actual mathematics is done.

I was recently reading a biography of Terry Tao, a leading mathematician and Fields medalist, and was struck by his description of doing mathematics:

From "The Singular Mind of Terry Tao" by Gareth Cook for the New York Times

His analogy to doing mathematics as being similar to being a jazz musician really struck me, as well as the notion that mathematicians are not handed problems to solve. I don't disregard the ability of Three Act tasks to hook and engage students, but I do hope that there is balance with the types of activities and tasks we ask students to do, and that there is also inclusion of more open investigations, an opening of our students' minds to curiosity and wonder about how things work and how we might describe and understand them better using mathematics, not just because it might give us correct answers, but because it's interesting and engaging to try to know and understand something beautiful.

Projects: theory versus practice

This is the first of a few blog posts that I want to write as reflections on this year. One of the topics that I'm thinking about and would love to get feedback on is the use of independent projects in teaching math. I understand that projects aren't a well-defined entity, but are somewhere on the spectrum past tasks (aka problems) in terms of self-direction, depth, and length of time.

Here's a little visual of how I imagine traditional problems compare to tasks to projects:

I am much more comfortable operating in the "task" or problem space: there are a limited number of routes and answers and they can more obviously be connected to specific content that I would like students to learn. We can have productive discussions and students can learn from other approaches and how they relate to their own. 

Projects are a more challenging space for me to navigate. I feel like I won't predict all the different possible things that students might choose to investigate, and I'm not always sure what they're going to learn or how it's going to connect to the content of the course or to what other students are doing. There is also greater potential that some students will go off in totally unproductive directions or into a space neither of us understands, and I will not know how to help them make sense of what they uncover.

Students are preparing to launch M&Ms using a launcher they designed after creating a mathematical model for the launch in a unit on quadratic functions. This project was actually pretty structured, so probably not technically a project.

I am also generally unclear how tightly projects should connect to course content. I often use tasks as a way for students to explore and learn specific mathematics, but because projects are more individualized and go farther and deeper, students engaging in them will tend to learn different things and often not ones within the narrowly defined boundary of the course. I have seen teachers basically separate the course into two clear portions: content and projects, which are done separately and often have little to do with each other. Personally, I am not a fan of this because it tends to push content instruction to more traditional methods as a time saver and students see two conflicting ways of doing math. But trying to make projects an integrated part of the curriculum means that I can't always use ones that are really open and need to have a sense of each student's direction and how it connects to the course.

This is a pattern created by a student for a project in the series and sequences unit. Some students got a ton out of this project, others created simple linear patterns and went on with their day. How do I make sure that everyone is appropriately challenged with open projects?

The other aspect of projects that has been hard for me to manage is time. My students can't just be assigned a project and complete it individually (I guess if that were the case, the project is probably not sufficiently challenging). They need time to figure out how to explore, to get feedback on their ideas and attempts to communicate those ideas, and to get unstuck and make sense of what they find. It is really hard to find projects that will appropriately challenge all of my students and then make enough time and provide the right amount of support to have them be valuable, positive experiences that promote learning and self-confidence. Additionally, they take much more time to grade and provide useful feedback on because students have gone off in totally different directions and learned varying amounts and types of math. How can I use a common rubric? How can I assess a student's work that seems so much less productive than another student's, but perhaps, the first student actually worked harder and learned more? 

Solar cooker project completed by students in @michaelpeller's class. Hopefully next year, this will be a collaborative project between Math and Engineering.

Clearly, I think they are worthwhile to do, but there are many things that can go wrong and I need to really think harder about the purpose of projects and the optimal frequency for doing them in my classes. Ideally, the math teachers in my school would discuss this as a department and come to a consensus on these questions and create a project thread that went through all of our courses and spiraled and built on work from prior years. In an ideal world, projects would also cross disciplines and leverage students' strengths and interests.

I have so many questions about how to do this... what are good starting projects? How can I get better at supporting and providing feedback on projects? How do I help students learn to better manage their time and work more independently/cooperatively on projects? If you have had some success with projects, please share any resources, both print and digital, that you have found helpful. If this is something that you'd like to work on next year, let me know and we can trial and error together!

Quadratic Functions project

Thanks to @SweenWSweens and his M&M Catapult Project (explained here and here and updated here), we are ending the quadratic functions unit in my 10th grade class with a bang. Well, a whooooosh, but you know what I mean. Kids had a ton of fun with this activity and it gave great practice for writing and solving quadratic equations. The basic idea (but really, you should check out Sean's posts) is that groups launch an M&M and measure the horizontal distance traveled and approximate the vertical distance traveled by using the time, putting this together to create a quadratic function that models this relationship. Then, they apply this model when the launcher is placed a given height above the ground to figure out where to place a target.

First of all, Sean was super helpful, walking me through the lab and giving me great tips on how to adapt it for my students. Love that #MTBoS. My project description and follow-up questions are here.

Here are the changes that I made to Sean's awesome plans and why:

1. I let kids build their own launchers. I shared Sean's basic design (pictured at right), but let them tweak it or do their own thing altogether. It actually took kids only about 20 minutes using our engineering lab, which had all of the supplies already, except for clothespins, as opposed to the few hours I would have spent making all of them and then dealing with kid complaints that their launcher wasn't good. Next time, I will do this again, but will also share Sean's updated design, which I did not see in time (below).

Here is what my kids built (most just did the basic design, a few went nuts and did their own thing):







2. Little direction was provided about lab technique or how to find the equation relating the height vs. horizontal distance. We did discuss the equation relating vertical distance traveled and falling time, but next year, I will do a better job of integrating this concept into earlier problems so that students can generate this idea themselves. What I liked as a result of giving less structure:
• Students incorporated other topics, which I did not anticipate. A few groups did statistical analysis to look for outlier data, which was awesome since that was a concept learned way back in September. Others compared lab protocols from different science classes and their applicability to this project.
• There was much more variety in approaches, which allowed for richer discussions within and between groups and more connections made. Some groups used the vertex, some used intercepts, and others used quadratic regression on desmos to generate equations. There was likewise diversity in how to change the model to incorporate the new starting height for the final launch. 
• The intellectual rigor was higher - students had to figure out what to do and then for their write-up, remember and reflect on their approach.
3. I used some class time after the activity for groups to whiteboard their approaches and then share out with the class and get feedback on their thinking. I also used 15 minutes the day that the write-ups were due for students to peer edit each other's work. The goal was to have more cross-pollination of ideas and connections made, as well as a chance to justify their own and critique each other's reasoning. I'm hoping that this also helped to produce higher quality final products and deeper understanding. Next year, I hope to run a more structured peer-editing process with specific questions for students to address.
4. More individual accountability - students were asked to divvy up points to their group members and describe each person's contributions as well as complete individual follow-up questions. I need to think about this more to see if I think this overall contributed to students' learning and experience with this project and helped or hurt their collaboration.

And now, more pictures!!

Building the launchers:

Gathering data:

Final launch day:

A few student whiteboards:

Once again, huge thanks and shoutout to Sean for creating this!! it ended up being a great project for this unit. Students had a blast, but were also appropriately challenged. 

Feedback from students:

My issue with hints

Yesterday, @mpershan asked for feedback on his Shadowcon talk regarding the usefulness of hints.

Would love to hear from you, *especially* if you disagree. An invitation to comment on my #shadowcon15 talk. http://t.co/lCzGmtgXWI

— Michael Pershan (@mpershan) April 29, 2015

His contention is that the pedagogy of hints for 9 - 12 math teaching is not very developed and could be improved by thinking about the context, reasons, and specificity of the hints.

As much as I agree that hints could be improved by these things, I also have a lot of discomfort around hints in general. Too often, I find that they funnel student thinking in a predetermined direction... as in, the student is stuck, and the teacher is trying to direct them onto a path that they think is productive by using hints, but it's a predetermined path and therefore removes a lot of the exploration that one would presumably want a student doing in solving this problem. Michael argued that this was only true for bad hints, that good hints should not simplify the problem or do the heavy lifting for the student or close off avenues of thoughts and overspecify a direction. I'm still not sure if we're arguing semantics or if we genuinely have different views on whether hints are good or bad and thought it might be helpful to look at specific examples.

Random exhibit A from a recent assignment:

I gave my students a problem of the week from the IMP curriculum in which you are told that there are five bales of hay, but that instead of being weighed individually, they were weighed in all possible combos of two. We know all of these dual weights, but would like to know how much each individual bale weighs.

Lots of students were confused and stuck. Here were some things that I did not say (although I really, really wanted to) because I think of these hints as being too helpful and pushing kids in a certain direction in their problem solving.

• How many times does each bale of hay come up in all the weighings?
• What is the total weight of all of the combos? Why might this be helpful?
• How can you represent this using equations?
• Can you organize the combos in order of weight?
• Are there any combos the weights of which we can figure out? Any that we cannot?
• Can you make equations to represent what you know?
• Can you make a table to organize what you know?
• Can you make an easier version of this problem?
• I see that you have four equations, but five unknowns. How do you think that will play out in trying to solve this problem?

Here are some things that I did say:

• What have you tried?
• Have you talked to anyone else in the class?
• Where are you stuck? How do you know that what you're doing isn't working?
• What information would be helpful to get unstuck?
• What things do you think that you know? What don't you know?
• How are you organizing your thinking? 
• How are you representing your understanding of this problem?
• Are you making any assumptions? Which ones? How will you know if your assumptions are correct?
• How will the person reading this understand what you did?
• What are strategies that might be helpful here that you haven't tried yet?
• You are making a lot of progress! Read through what you have already and see if you can restate it in a different way.

I make a distinction between teaching a specific procedure or specific content when you would want to channel students' thinking perhaps more narrowly - there may be multiple paths, but not an infinite number of them, and it is likely important that students understand which paths are more efficient under what circumstances and how they connect to each other - versus when you are asking students to work on a more open problem in which they are meant to develop problem-solving and sense-making. It seems like half the purpose of open problems are for students to come up with different approaches, persevere past sticking points, learn to think flexibly and independently, and make sense of unknown situations. And yes, that almost requires that they be stuck and frustrated for parts of it. If a problem can be solved by a student easily and without any false starts, then it's not much of an open problem. To me, hints like the ones I listed in the first section decrease this cognitive load significantly. I want students coming up with those ideas, not following mine.

I am trying not to get bogged down in the word "hint," but it just has this connotation of "I have the right answer in my head, but you can't figure out what it is so let me make it a bit easier for you to get it." If we redefine "hint" to also include questions or statements that push the student to think more deeply and develop their own internal resources rather than as a way to make the process smoother for them by external means, then I think that I can get behind good hints. 

Reflecting on homework

Last week, I posted on Twitter asking for help with homework structure and routines to limit the amount of time we were spending in class going over questions, and boy, did I get some great responses.

Hey #MTBoS, teach me how to go over hw. I give answer keys, but it still takes like half the period to answer questions & make connections
— Anna Blinstein (@Borschtwithanna) April 24, 2015

First of all, I really appreciated everyone who took the time to give ideas and feedback. This is what makes the #MTBoS so amazing. I've gone through all of the suggestions to see which ones make sense for me and my classes, and here is my attempt to summarize and make a plan for myself:

1. Spiral homework so that it lags classwork (great idea from @hpicciotto, @cheesemonkeysf, and @pegcagle) - questions that relate to classwork from a few days ago give students time to process more deeply, have metacognition about their learning, and make stronger connections to the material. I would like to structure my homework into Review (questions that relate to content from a few days ago), Reflect (processing current content and making connections), and Reach (more challenging problems and questions to preview upcoming content) sections.
2. Provide answer keys in advance (I try to do this, but it doesn't always happen due to time constraints with developing an "emerging" curriculum... I need to remember that when I don't do it, it means I lose a ton of class time) and upload detailed solutions after we have gone over assignments.
3. Remind students that they can ask each other questions on Google classroom. I used to use a more bloggy class blog so students naturally commented and discussed online, but after switching to Google classroom, it has totally faded as a tool for student discussion outside of class. I'm hoping that with some reminders and maybe an assignment to comment or respond to a comment, I can jumpstart this type of interaction.
4. After students finish homework, they give feedback regarding their understanding. @z_cress shared an awesome Google Form for doing this, which looks like this and will help me have a better idea in advance of how much time homework will need to take and how much support students need with this content:





5. Start class by having students work in groups to ask each other questions and clarify problems for a few minutes; at the same time, ask various students to put up specific problems that many are confused on or that will be useful to discuss as a class on whiteboards. I still need to think through this a bit more - do I want everyone putting up work and circulating around the room and discussing (as suggested by @dandersod here) or more focused on working in groups and having only a few group questions put up on the board? I would like to play around with these and see what works for me. 

A few other blog posts on how others are handling homework:

• Henri's explanation of lagging homework
• E's plan for homework for next year
• Daniel's switch from homework to exit tickets

If you have other suggestions or blog posts to share about homework structures, I'd love to see 'em!