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!

Digesting NCTM

Just had an amazing 4 days being steeped in the world of math education at NCTM. On the plane ride home last night, I went through my notes from each of the sessions I attended and my own (by the way, I think that the most learning that I had this week was from planning my session with the inimitable @fnoschese and @_mattowen_ - there's nothing like preparing a presentation with thoughtful colleagues for elevating your own understanding of your practice) to try to congeal and connect all the various thoughts that I had in my head this week. Here are my conclusions and to-do's, the big takeaways being:

1. A curriculum is not a series of tasks, projects, and activities, no matter how open or interesting. It is a cohesive progression with clearly defined goals that needs to spiral within a given year and progress from one year to the next. Everything else will be piecemeal until we create a common understanding of our curriculum progression and look at it as a whole. On the other hand, it was nice to realize that we are already doing so many of the individual best practices I heard about at NCTM and just need to pull it all together.


2. Real-time professional collaboration is where it's at. In several of the sessions, it was evident how powerful lesson study, teacher time-outs, and opportunities to team teach and reflect on each others' practice in a supportive, nonjudgemental way can be. We are already working on a mentor program for new math teachers, but this reminded me of the need for this for all teachers. As wonderful as Twitter is as a resource, it doesn't replace working with your colleagues to move your school forward.

Specific notes from my plane ride brain dump (please feel free to stop reading, this is just for my personal recording and accountability):

More scaffolding for big projects

• Too low and too high guess to start
• Make a plan
• Work independently for 5 minutes
• Share with others in class
• Amend plan or make a new one, reflect on why the original one didn't work out
• Some time outside of class
• More time in class and check-ins through the process, not just following up with students after the deadline
• Required revision for at least one project (will connect to portfolio project and end-of-year defense of work)

Integrate projects more into course structure

• Follow up in class to share strategies and connections to content
• Activity or project can serve as launching point for several other problems, can be lynchpin for entire unit or subunit - use it to build cohesion and add more continuity and coherence into the unit
• Revisit same project or task at the end of the unit or do a similar one to reflect on progress

More frequent feedback on practices

• Update homework spreadsheet every week
• Students track own content scores (I can still use Active Grade to track it officially)
• Track class discourse and participation (from Carmel Schettino's handout)
• Individual meetings every 2 weeks for ongoing feedback and more back-and-forth rather than one direction for feedback
• Have students explictly reflect on practices as part of biweekly reflection on progress in course
• Have students rate themselves on practices and cite specific evidence for each (need to get a link to Carmel's handout for this)
• Get more frequent feedback from students as to what is working and what needs to be tweaked from my end. Be more open and inviting of feedback, solicit negative as well as positive feedback.

Professional collaboration

• Buddy up with a teacher to team teach one lesson per week in one person's class, switch off week to week; use teacher time outs during class
• Organize department-wide lesson study to plan together and revise - could this be done with teachers from other schools?
• Organize next year's schedule to have some same-level classes scheduled at the same time - can double up the two classes and two teachers once per week
• Continue K-12 strand work through the summer and next year to build better cohesion between courses

Improve questions and conversations

• Include "I learned..." and "I wonder..." either as exit ticket (digital) or as homework
• Take more time for labs - white boarding and debriefing are crucial, have students reflect on each others' work, discuss meaning and context, summarize as a class, create a space where summaries from one day to the next can be saved and seen
• Build on lab as a way to start a unit and investigate a new topic, should serve as launchpad for following activities (similar to opening project or task)
• Make labs more open
• Start by showing something and asking kids what is interesting, what we could measure 
• Identify a relationship to measure, ask kids to define variables (creating a model is a key part of modeling in addition to manipulating a given model)
• Have each student make their own data table and sketch in their notebook, each one answers questions in notebook before creating group whiteboard
• Don't tell them how to figure out the relationship always; start with more scaffolds: telling them to graph by hand and find equation by hand first, then show them Desmos, Excel, graphing calculators, then show regression models and let them choose how to represent (can require at least two representations or whatever makes sense)

• Work on including more open questions
• Embed review content into applications or new contexts
• Ask students how the problem might be changed to make it easier? Harder?
• Ask questions in which students have choice a la Marian Small: "Make two quadratic functions with intercepts at -1 and 5" instead of, "find the intercepts of this function." Then you can discuss the characteristics of all the functions students generated.
• Spiral up investigations and tasks to remove scaffolding as the year progresses, should end with investigation of their own design (progressively more complex from one year to the next)
• Include "Would you rather..." and "Which one doesn't belong?" and all the other techniques mentioned by Geoff Krall to open up tasks

Next year plans

• Summer math class for incoming 9th to fill in gaps in content and practices
• Require graph notebook (binder? digital?) - decide as a discipline what we want for students; it could be different year to year, but should be a cohesive progression
• This will tie into portfolio project - digital might make sense if kids are taking pictures and turning in all assignments digitally
• Look at the progression of our math courses: how are we spiraling content and practices year to year? Can we build on projects/mathematical spaces as students develop a more sophisticated understanding of content?
• Look for better projects and tasks to build more coherent progression within the year and between years (investigate Carmel's materials, 3 act tasks, IMP books, Geoff Krall's materials, Robert Kaplinsky's materials, list of labs from Casey Rutherford)
• Coordinate more with other disciplines; goal is at least one collaborative project with each discipline per year
• Look into a capstone project connected to grade trip; 9th grade Peru trip can connect to statistics and data analysis, 10th grade Costa Rica trip can connect to modeling

Books to read

• Good Questions: Great Ways to Differentiate Mathematics Instruction, Marian Small
• Art of Problem Solving series
• How to Solve It, Polya
• Fostering Geometric Thinking, Mark Driscoll
• Mathematics Formative Assessment, Keeley
• Investigate Geogebra, Python (may need online class), TI-Nspire, Sketch Explorer, Mathematica, Wolfram, new programming project from Bootstrap
• Look through CME Project integrated series for possible adoption

Introducing logarithms

Having not taught logarithms for about 10 years, I looked through several blog posts detailing how to best introduce this topic to students, given how much less experience they've had with logs versus other types of functions. The main idea that I wanted to reinforce for students is that logs are defined as an inverse of exponential functions and to continue connecting them back to features of those functions, which they have had much more experience manipulating, graphing, and applying. As usual, Mimi had a lesson that resonated with me. I liked her formulation of a log equation as a form of a question.

From her blog:

But I wanted to add more emphasis on its derivation as an inverse. So I stole liberally from the CME Algebra 2 lesson on introducing logarithms. We started with a discussion of one-to-one functions (I was surprised to hear that they had never heard of this term before) to know if we could even go into the land of inverses. Then, once we had established that exponential functions were indeed one-to-one and did a quick summary of inverse functions, with which they were already familiar, I had them create a table of value for the function $y=2^{x}$ and then create a table for the inverse function, which we were going to call $L_{2}(x)$. We then started playing around with this function... What would it look like when graphed? Why? Is it also one-to-one? Why or why not? Can we evaluate it for certain values of x? Are there any restrictions on its domain or range? Why?

I really liked how this lesson went. It seemed to build a good deal of intuition for logarithmic functions that I felt had been missing for me when I'd taught it in the past. There was a feeling of exploring a new function, but not one that had been plopped down from the sky and was completely mysterious. There was definitely enough newness to keep it interesting, but not so much that it was overwhelming. I liked how the CME lesson didn't even call the function a logarithm, just L. The less bogging down in new terminology, the better, I say.

I showed students how to evaluate logarithms of any base using technology (both Wolfram-Alpha and Desmos are friendly to different bases) and then gave them the base change formula so that they could solve application problems now rather than waiting until we had developed more of the properties and derived the formula. We'll derive it soon, but I think that it will be more motivating now that they know why it's useful and I wanted them to be able to apply logarithms on day 1 without fending off tons of properties coming at them from left and right.

I think that it was also helpful that while we were working with exponential functions and equations, I would routinely throw in questions that required figuring out the value of the exponent, which students were forced to solve via guess and check and every time, I would mention that soon, we will be able to find these values more precisely with logs. Again, I like the idea of removing the mystery and foreignness of a new concept as much as possible... previewing and embedding it within more familiar concepts makes it less scary and connects it to prior knowledge when it is finally encountered.

Here's my actual handout:

Day 5 - Logarithms

I do regret that I didn't do a project in this unit, but we've been doing tons of projects and at this point, content coverage needs to take a bit of a priority. Also, assigning projects requires me to grade the previously assigned projects and well, yeah, let's just say that I'm a teensy bit behind.

Update:

Just saw that Henry Picciotto wrote a recent blog post explaining his approach to teaching logarithms as super-scientific notation. I would like to rework my lesson plan to include both approaches for next time.

SBG: Assessing Mathematical Practices

In an earlier post, I wrote about the challenges of giving meaningful feedback without using grades as motivation. But now, I am thinking about the challenges when grades are part of the picture. Over the summer, we put together a set of mathematical practices (a mix of aspects unique to our school, Common Core, Park School's Mathematical Habits of Mind, and work done by Avery (his post on Mathematical Habits is here), who teaches 5th and 6th grade Math at our school). The five main categories are

• Investigate, Explore, and Play
• Represent
• Reason
• Communicate
• Growth Mindset

Within each one are four sub-categories and four "levels:" Emerging, Developing, Strong, and Leading. 

Mathematical Practices 14-15

This template formed the backbone of my feedback this year. Every chapter has its own content objectives, but the practices continue the entire year and are consistently being used by all of the Upper School math teachers.

In order for this to be useful to students, most assignments had a content component, which was assessed separately, and a practices component. No assignment included all of the practices, but each one included at least a few. I would let students know in advance which practices I would be assessing with a particular assignment/project. Sometimes, there would be a self-assessment component first ("highlight the level you think you have demonstrated for each practice assessed on this assignment and give evidence for your conclusion."). They would get back a rubric with the appropriate cells highlighted, along with comments and suggestions for improvement.

Pros: very specific and detailed feedback, it was very clear to students how highly the practices were being valued, they understood them better as they continued to self-assess and get feedback on them over time, they began incorporating the language of the practices in their overall reflections on the course and in their work for the class, and they demonstrated progress and growth over time

Cons: there are so many sub-categories and so much detail that it took a while before students were really clear as to what each one meant, it took an almost unreasonably long period of time for me to do each assessment and justify each rating, compiling all of this in a non-formulaic way for a final semester grade was a Herculean effort that I don't know that I can ever undertake again, and the sheer volume of feedback that this resulted in for students, families, and advisors was overwhelming and therefore not practical in the long run

The main change that I think we will make for next year is to eliminate the sub-categories. They can be there in the background if we want to make reference to specific aspects of each practice, but always including each one is just too much. I would also like to build in more time for students to revise their work and make improvements on a project they've gotten back rather than waiting for the next project or paper in order to improve. I'm currently in discussions with other 10th grade teachers to use the last week of the school year as a time for students to put together a portfolio that will include one paper or project from Math, History, English, Science, and World Language from earlier in the year, but revised and improved to incorporate the feedback and learning that has taken place since then. I would love for revision and iteration to be a regular part of the learning cycle in all of our math classes and for feedback to be a step along the way, not the end.

I also need more regular ways to give feedback to students on practices that are not always assessed on projects, such as ones having to do with their growth mindset and collaboration and contribution towards class. There is so much already to plan, assess, and give feedback on that this one definitely slips through the cracks. But I keep reminding myself that if I want something to be a vital part of the class and for students to make progress on it, I need to regularly assess it, give feedback on it, provide explicit instruction on how to improve it, and ample opportunity to revise and iterate and apply it again and again. It makes sense to me that quality is way better than quantity here. Decreasing the number of practices, but assessing them more often and with depth, clear feedback, and explicit instruction and mentoring of students to move them along the spectrum is much better than spreading myself thin.