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