Thursday, July 30, 2015

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!!


  • 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. 

  • 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.
  • 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.

Tuesday, July 28, 2015

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.

Thursday, July 16, 2015

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!