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.

Collaboration

Today, my 7th graders worked on a great activity from nrich.maths.org that combined practice with the distributive property (ostensibly, the content we are learning) with some very important aspects of groupwork that I wanted to highlight and discuss. Thanks to @Veganmathbeagle for tweeting it out a few days ago.

The activity provides 16 cards in which there are 4 sets of 4 equivalent expressions. The four members of a group start out with 4 random cards and the task ends when every member of the group has 4 equivalent cards. Key rules: no talking or non-verbal communication of any sort AND you cannot take a card from anyone else, only give one of your cards to someone. Each member of the group must have at least 2 cards at any time. If there is an extra person in a group, he or she acts as an observer to the process and takes notes on the ways in which the group members helped each other.

The expressions in the activity - the link above has them in an easy, printable version

This was challenging for my students both from a content perspective and due to the emphasis on collaboration. It was amazing to watch how well some groups gelled and how others were brought to a standstill by a disengaged student.

Comments from my students (roughly paraphrased) when I asked them to reflect on what made this task hard:

"If one person wasn't trying, the whole group got stuck."

"You couldn't do the work for anyone else."

"Some of them were hard and I just wanted to do the easy ones that I knew I could get and leave the hard ones for someone else. But sometimes, everyone left the hard ones for someone else and there was no someone else."

"It made me do more work than I usually do because my group was depending on me."

These are real issues that happen in groups, but are often concealed because other members do pick up the slack. They are really hard to solve in most situations because we do want students discussing and creating a single group product, which means that students who choose to do the bare minimum often can do so. Of course, I do try to build in individual accountability into group tasks, asking a random member of the group to explain the group's work or asking an individual follow-up question that each person must answer on their own. I have done "group quizzes" in order to give feedback to students on their collaborative skills. But this was definitely the most aware and open that I've ever seen my students in discussing the disparity in the level of effort that often takes place when working in groups. I'm hoping that in future tasks, we can refer back to this activity and students will have a better sense of their need to work with more parity and engagement. If you know of any other activities or ways to improve individual accountability in group tasks, please do share.

Some ways that I modified the activity: half-way through, I allowed students to use scratch paper. This reduced the cognitive load a great deal and allowed them to work more productively. In one class that was really struggling, I allowed the groups to talk to each other for a few minutes at the end. Different groups may need more or less of the restrictions in order to create the right level of challenge.

Sequences and Series and Differentiation

Things are moving right along in my 10th grade classes. We wrapped up the Stats unit with some really fun individual research projects in which students created a question about our school community that they wanted to answer, collected data, and performed either chi square or z-tests to answer their questions. I was really, really happy with the level of work students put into their projects and how much ownership they took over their learning.

Here is a picture of the summary slides I asked them to create to summarize their research questions and conclusions. It was really nice to be able to display the results of our labors to the school community.

We started working with sequences and series. This is a relatively short unit and I am pretty happy with the unit projects, which were due last week. Students needed to create their own visual pattern, write recursive and closed form rules for the pattern and its differences and sums, and try to prove one of their formulas using induction. That last part proved really hard for just about everyone. Maybe it's because I haven't really taught proof by induction before, but it was just a painful slog for everyone involved. I have no idea how to teach it in a constructivist fashion as it seems so far removed from the way that most students would approach a proof.

The other challenging part of this unit for me has been appropriate differentiation. For several students, writing rules and finding patterns seemed intuitive and they flew through classwork problems, while others have really struggled and I could tell they needed more support. Most of what we do in class is groupwork based, which has its advantages and disadvantages in terms of supporting struggling students. They can get help and work with peers, but they can also chill on the sidelines and rely on others to do most of the work. I do call on random group members to explain the group's work, but this isn't the same as actually doing the group's work. There is also a big discrepancy between students who are seeking me out for extra help outside of class and those who are avoiding me. Spoiler alert: it's not the students who really need the help who seek it out, for the most part. 

When I teach middle school students, I feel comfortable emailing home or just telling a student that they are required to work with me during lunch or before/after school. For high school students though, it feels overly babyish to do this. I want them to have independence and learn to reflect on their understanding and ask for help. Conferences were a great time for me to communicate to students what I would like to see them doing differently, but the challenge now is to find the time to follow up with individual students and remind them of the commitments they made in their conferences. It's a tough balance between giving them freedom to make their own choices and mistakes and also coaching them in how to learn from those choices and mistakes. One thing that I would like to do is to meet with each student one-on-one right after Thanksgiving break to discuss their progress. As always, finding the time to do this is a challenge.