Getting students to dig deeper into rich problems

So I was going to participate in the #MTBoS Challenges, but then, life happened. I did write a blog post responding to the first challenge, and even though I'm not participating in the full scope of challenges, I'd like to post what I can. So here is what I wrote in response to the question "What is one of your favorite open-ended/rich problems?  How do you use it in your classroom?"

Even though we were directed to write about a favorite rich problem, I’m going to write about a problem that is the most recent one that I’ve done with a class because I’m having some issues with the way it’s worked out and would love some feedback on how to make it better.

The problem that I recently gave my 2-year Algebra students (these are 8^th graders in the second year of a 2-year course that covers a standard Algebra 1 curriculum) was the first one in the Integrated Math Program, Year 1 book, called Broken Eggs. This was students’ first problem of the week, in which they are to write up their problem-solving process and justify their thinking and solution, if they find one. In this problem, you are told that a number of eggs when put into groups of 2, 3, 4, 5, or 6 always had one egg left over, but fit perfectly into groups of 7. Students were asked to determine whether there was only one unique solution or whether there were many possible solutions, and if so, how they were connected to each other.

Full problem:

Broken Eggs POW by Anna

I gave students 20 minutes in class to start working on the problem (5 minutes alone, then 15 minutes with their group members) and a week and a half to complete their write-ups. I also met with students individually who were struggling.

I thought that the problem was a good one with which to start as it could be approached from a variety of angles and would encourage for the looking of patterns. However, the results were pretty disappointing. Most kids were only able to find the first solution and did so using brute force (writing out the multiples of seven and testing each one to see how it divided by 2, 3, 4, 5, and 6). Quite a few kids just looked at numbers that weren’t evenly divisible rather than looking for a specific remainder. Almost no one found any other solutions and not a single student found a pattern between the solutions. Almost no students even attempted to find one. So the problem just turned into one that required some organization to keep track of things, but almost no algebraic thinking. So basically, the result was a lot of annoying calculations with little payoff.

I am trying to think about what I could have done differently to encourage students to keep going and to notice patterns that would make their work easier. Having students share strategies maybe would have helped to disseminate some of the shortcuts that a few of the students discovered, but not ones that no one figured out. I think that part of the tension for me is that I want open problems to really be about students’ thinking and approaches, but also be a learning opportunity that stretches them past their current abilities and into something more advanced, and I don’t know how to do that without giving hints or telling kids to change their approach. Basically, I want them to learn and be stretched mathematically, but have it be organic and come as an extension of their own thinking rather than a top-down approach where I direct them.

Part of the issue also is that kids are mostly to work on problems of the week outside of class so I’m not getting much of a chance to see their thinking before they turn them in to me. So another change that I’m thinking of doing is having students turn in a “rough draft” that I can give them feedback on or that we can confer about in person before they complete their write-up. There is a thin line between pushing a student’s thinking and directing it onto a predetermined path that takes away from the problem’s openness and richness, and I am still navigating how to do this in an optimal way. Suggestions welcome!

Whiteboarding wins and fails

Now that whiteboarding (my posts on it here, here, and here) has become a full-fledged part of my classroom, I wanted to reflect on how it's been going. So first, the successes...

Whiteboarding wins:

• Student cooperation and engagement in group activities has been kicked up a notch. I thought that I was doing fine with structuring group work and encouraging student engagement and participation, but whiteboarding has really increased cooperation and discourse between students. There's something about having one person write for everyone that creates more investment in the process and final product. A white board also encourages larger and clearer writing, making it easier for everyone in the group to see and participate.
• All students have to demonstrate understanding and it's harder to hide or have one person do all of the work. When groups present or when I go around the room and talk to groups, I pick students at random to explain their group's thinking. Rotating the marker between group members also help keep everyone involved. 
• Kids love to get picked to show their work and work harder to make their thinking more explicit and clear.
• It results in greater perseverance and willingness to try crazy things - something about the ease of erasing makes kids more willing to take risks and get messy with a problem.
• Kids get used to asking questions of others' work and disagreeing with their process politely, and I am seeing this carry over into non-whiteboarding work.
• Students really like it, and it lends itself to other activities & structures that allow it to continue feeling fresh and exciting.

... and now for some failures...

Whiteboarding fails:

• Annoyingly, kids are still uptight about making mistakes and having errors publicly attributed to them. Even after playing the Mistake Game, kids were still getting upset and unhappy with each other when unintentional mistakes would get exposed in front of the class. I had one incident where a group of boys exploded with accusations about whose fault the mistake was and had to talk to them out in the hallway about their behavior. Kids still find it embarrassing to be wrong, which I am finding frustrating. Like, how many times do I have to say, "This is the awesomest, most interesting mistake ever! We are learning so much more from it than we would from the "correct" approach!" before it sinks in? When I would ask groups to present, kids would still anxiously ask "but is it right or wrong?" Right now, I'm working around this by putting up boards with mistakes and having the class give feedback without attributing from which group the mistake came, but I'd like to get to a place where students are actually comfortable with their errors and aren't embarrassed by having them publicly discussed.
• Kids are still kind of bad at asking questions rather than just pointing out the mistakes. Or their questions are really just statements with a lilt at the end: "Should you get 8 when you square 4?" 
• The boards are sometimes hard to display if the group wrote too small, and the whiteboard is too big to stick under the document camera. My workaround has been to take a photo of the whiteboard and send it to the projector where I can zoom in on their work. Hopefully, their writing will get better as we continue to use whiteboarding regularly.
• Students would like to have notes they can take home, and whiteboarding does not lend itself to that. I thought Bowman's post on giving students time at the end of a period spent whiteboarding to reflect and write down some notes was really great, but it's hard to find the time for this what with going over homework, whiteboarding, and student explanations of their work, all of which must fit into a 45 minute period, 4 days per week. I've thought about taking photos of their whiteboards and putting them on our class webpage and perhaps asking them to do some reflecting and summarizing for homework, but haven't actually followed up on this.

Overall, I am really happy with how whiteboarding has gone this year!!

Thoughts on homework

Julie put out a call for posts about homework differentiation, which came at a perfect time for me since this is something I've been thinking about recently. I'll get to differentiation in a sec, but first, some general issues that I'm having with homework:

It has sort of taken over my teaching life. We spend a lot of time on it in class, and I spend a lot of time on it outside of class. I am not sure if this is good or bad. The way that I do homework, students have all of the answers and must check their assignment and identify problems they have done incorrectly or that they don't know how to do. In class, students discuss their questions with each other in groups for about 5 minutes, then we make a list of questions to go over as a class. Students volunteer to present their solutions to the identified questions and answer any follow up questions from classmates. Those who got those questions wrong or didn't complete them are expected to make corrections to their work. The whole thing takes about 20 minutes, on average, which is almost half of the 45 minute class period. For my end, I look at each problem for each kid and write feedback on each paper. If there are errors or incomplete work or misunderstandings demonstrated, credit is reduced and students are expected to correct those assignments and turn them back in to me. Once the assignment meets my criteria for "goodness," full credit is given. Students are also required to write up corrections and explanations of their errors for tests and complete a journal entry reflecting on their progress each chapter. For my four classes, which have four homework assignments each week, the whole grading process takes me on average about 2 hours every day. That's a lot of time!!!

On the plus side, I think doing problems independently, discussing their thinking with other students, presenting their work and having to make sure that it adequately communicates their process, and getting regular, frequent feedback from me are all really helpful for their learning of both the math content and how to be a better student. On the minus side, it means that there isn't much time left for group explorations or as thorough grounding of each lesson as I would like, both in terms of how much class time is available and how much time I have to plan lessons outside of class.

I use this system not because I think it's super duper awesome, but because I haven't seen any other way that accomplishes the student learning piece better. A few things that have helped me be slightly more efficient:

1. If I see a bunch of errors, say three or more, I stop grading the paper and write a note to the student reminding them that they need to check their answers and make corrections, either on their own or with help from me. The assignment doesn't get any credit until that happens. I don't want them relying on me to check their answers for them, and I want them to come in and get help.
2. As the year continues, I start restricting the amount of time given before corrections can be turned back in to me. Eventually, they are not allowed to correct any more and must come in and get help before turning in the work in the first place. This helps them transition to their future math classes, none of which allow corrections on homework assignments. 
3. I try not to assign too many problems and assign ones that I think are meaningful and helpful to students' learning. Kids do see the connection between putting effort into their homework and their learning and progress in the class.

But really, this is a crazy inefficient system, especially in terms of my time use. I don't know about everyone else, but grading homework for me is somewhere up there between having hot pokers shoved into my eyes and being trapped in an airless mine shaft. It's seriously obnoxious. But I hate the idea of just grading based on completion because then there's no feedback to the student and much less of a connection between homework and learning. So, I'm stuck with my current system unless someone has any suggestions.

Okay, on to differentiation!!

So I don't currently actively differentiate a lot of homework, other than test corrections and journal entries. There is a good deal of differentiation that happens as a natural outcome of how I structure homework though because students who didn't get full credit on assignments have to correct them, which means that they are addressing their own specific misunderstandings. Also, students who score less than 70% on a test are required to complete extra work as well as meet with me.

Here are some other ideas of things I'd like to try:

1. After a quiz, breaking up the class into groups based on the types of mistakes they made. Each group will have all the kids that need to work on a particular concept together. Then, I can work with each group to reteach/address misunderstandings on that topic and their homework will be problems targeting that specific concept. The kids that rocked the quiz will work together on a challenge problem. Their homework will be to complete it and do a write-up of their process.
2. In my regular-paced class, which is completing Algebra 1 over two years, we often spend two days on a learning target. For the first day, I'd like to assign everyone the same basic homework assignment. Then, for the second night's homework assignment, students will have the option of either doing more basic practice OR to do a more challenging, shorter set.
3. I loved this post by Bob Lochel that I recently read about differentiating homework assignments in which different problems are worth different point amounts (depending on difficulty) and the students complete enough problems of their choice to earn the assigned total point value.

Looking forward to reading everyone else's differentiation strategies and ideas.

Technology in the MS Classroom #msSunFun

See all the other #msSunFun submissions for this week

iPad apps (I have an iPad, my students do not):

• Doceri: I just started using this last week after reading about it on @danbowdoin's blog. So far, I've made a few video examples for students to watch as review and for a sub to show when I was away on a field trip one day. It is incredibly easy to use, which is a huge selling point for me. You hit the record button and write with a stylus or finger while commenting on what you're writing. Hit stop when you're done and upload the whole thing to YouTube. Done and done. We are slowly, but surely, moving to 1-to-1 for students, so this would be a cool app for students to use to create videos for each other.
• AirPlay: This allows me to mirror my iPad screen on my projector via an Apple TV device. I'm actually still waiting for the Apple TV to be installed in my room (any day now, IS...), but I've played around with this before asking for my own and it's awesome. You have the ability to project stuff wirelessly from anywhere in the room instead of being tethered to where the computer is plugged in. Although, I just saw that you can do this type of iPad mirroring without the Apple TV. If someone does this, let me know in the comments because I'm thinking of trying it also while I wait for that Apple TV to get installed.

Classroom Management Technology:

• Edmodo: I'm using it for the first time this year, and it's so much better than my old class webpage for providing kids with easy access to class materials and encouraging interaction outside of class. Students often post questions (sometimes, they take a picture of their work using their phone or iPad and post that too) and answer each other's questions. They have arguments about homework problems (whose answer is right??) and post interesting math questions they are wondering about. So, so cool.
• Google docs: I use these to survey students anonymously through Google Forms, get input on what music they want to listen to in class, and have them write and share reflections and conference preparation documents with me. I also use it to share documents with other teachers (for example, the other Algebra 1 teacher and I have a shared spreadsheet we use to plan out the unit), such as notes from meetings and committee reports.

Organization Technology:

• Evernote: Started using this over the summer, and it's helped me be much more organized this year. Because I have it on my phone, iPad, and laptop and all the accounts are synced, I can track cool teaching ideas from twitter or blogs, information about students (including the missing homework form idea I stole from @approx_normal - I take a picture of each one and stick that in the note for that student), textbook sign-out sheets, and notes from meetings/conferences, not to mention personal stuff, like receipts, restaurants, books/articles I want to read, and vacation planning materials.
• Hackpad: I blogged about this app earlier this year. I use it similarly to google docs, but it has the added feature that the name of the person writing shows up next to their text so it's really handy when you want to know which person wrote which part of a document.
• Google calendar: Our school recently moved to Google Apps for Education, so everyone has a school google account. We use shared google calendars to keep track of shared events (field trips, meetings, and advisory activities) and large tests and projects to help make sure that we're not over scheduling kids or putting too many assessments on one day. I also need this for my own personal organization to keep from imploding. All events in my life go here and my husband and I can see & edit each other's calendars to keep from double-booking ourselves or our kid.

Default mode

This week was a little disappointing. I had been very excited about how the year had started - I put in a lot of work this summer to revise the first big chapter (exponent properties and polynomial operations) to make it more constructivist and engaging. There was lots of group work (including using my new big whiteboards!!), writing and reflection questions, and classes that built from exploration --> sharing out --> a teeny bit of summary from me to pull it all together. It felt organic and like the ideas were coming from the students themselves. Almost all of the students did a really great job pulling everything together for the chapter test - the average was much higher than last year.

Working polynomial problems in their groups

I had students give me anonymous feedback via a google form towards the end of the chapter (yay for actually doing one of my goals for the year!) and was happy to see how positive students were feeling about the class and their understanding of the material. My favorite quotes:

"I think that we should keep spending a lot of time going over homework and learning by making mistakes in the notes."

"I like how you have us interacting with others with our groups to solve problems."

"I liked how last year we consistently did notes, like a pattern, and at the beginning of this year, it was hard to adjust to what we do now. But now I realize that I like figuring things out in my own ways, not just following the repetitive steps on the notes."

Given how well things were going last chapter, I was surprised with how meh this week has gone. Classes felt boring and I noticed that I was doing lots and lots of the talking and that students were passively completing problems with little engagement and interaction. Then I gave a short & sweet quiz on Friday on some early factoring concepts (factoring by GCF and factoring by grouping). Holy cow. So awful. So so so so awful. It's like they learned almost nothing this week. I can count on one hand the number of students who did even remotely well on this quiz. Thinking things through, I've realized that I've completely reverted to my default mode of teaching - here students, I'm going to work through some example problems, and you just follow along. Now, you try some and I will help you if you get stuck. Oh look, class is over. Let's do that again tomorrow. Ughhhhh!! 

This quiz was a good wake-up call for me that old habits die hard and that I need to be vigilant and keep the big picture in mind for how I want class to go and what I need to do to make sure that it's student-centered and engaging and that students are actually learning and not just following along mindlessly. My plan for Monday is to provide one actual incorrect approach for each problem from the quiz (one per group) and have students analyze the error and explain what went wrong and how to do it correctly to the class. I also want to talk about study strategies because I definitely got the sense that few students actually reviewed for this quiz or did so in an effective manner. I made this handout to help them think through studying for math quizzes and tests:

Study for Math

Then, it's back to the drawing board for me to plan the rest of the week's lessons with what I've learned this past week in mind. It's nice knowing that every day, I get a fresh start and a chance to get things right.

Writing and reflection

One of the goals that I set for myself this year was to make writing a more regular part of my class, rather than the add-on journal entries I've had students write the past few years. Kids had been resistant to these (a few refused to do them at all) and I felt like I wasn't seeing much improvement as the year went on. Those who were reflective and took the assignments seriously got something out of it, but lots of kids did a crappy job, took a 1/3 or 2/3 score and moved on with their lives.

So this year, I started the first real unit (after reviewing last year's Algebra 1A material) with worksheets that kids started on in class and that had more problems and a reflection piece at the end for them to complete at home. Here's one (adapted very closely from CME Project Algebra):

Ch. 7 Day 1 12-13

I used the same writing prompt each day:

How well did you understand today’s lesson? Use one or more of the following prompts to help you answer this question (write at least a few sentences, include at least one example).

a. One thing that I understand really well from this lesson is…

b. One thing that I didn’t understand at first from this lesson, but now do understand is...

c. One thing that is still confusing to me from this lesson is...

d. Something that I’m wondering about that is related to this lesson is…

Some positives: 

• Every kid is responding to these. Maybe because it's the last question on the assignment and they've already put in all the rest of the work, or because it is an almost daily component of their work and thus normalized, but I'm having much less resistance to writing this year.
• I feel like I'm getting a better understanding of kids' misconceptions and questions. Yes, there are some who always say "I understand everything. Here's a trivial example." But lots of kids are taking the time to write about a problem type they don't understand or a question they have about the topic that wasn't addressed in class. 

Things that still need to be worked out:

• Getting kids to use the example as evidence for what they are saying in words. I want the response to be a coherent piece of writing with the math embedded in the words, not as an add-on because it's a requirement to include an example. 
• Having kids go deeper in their explanations, rather than just stating a procedure they used (or not explaining what they did at all). I want them to explain why their approach worked (if they're using prompts a and b) or where they got stuck (for prompt c). I would also like them to write more. I think that if I require at least a paragraph minimum, fewer kids (ahem, boys) would be tempted to just pick the easiest example from the notes and try to regurgitate it back to me in the fewest number of words possible.

Here's one of the better ones from last week because this student actually explained their example in detail. Again, I'd like them to go a bit deeper into the "why," but at this point in the year, I'll take it.

So, a few things that I know I need to do to promote better writing:

• Give specific feedback. I've been saying things like, "needs to be longer" or "explain your example," but I should really talk to kids and tell them more specifically what I want them to change.
• Show examples of strong math writing and have kids point out what the person has done well, in addition to things that they can still improve on. 
• Tell kids why it is that I'm having them do math writing. Perhaps it would be helpful to (in general terms) talk about the research on metacognition and learning. 
• Change up the writing prompts and have more writing responses to actual math problems. I just had kids do an investigation in class with the homework assignment being a writeup of the problem, their process, and solution, if any. Once I grade these, I will have a better sense of where they are at with their writing about math and where to go from here.

Any other suggestions??

Making Math Class Easier

One of the things that I've been thinking about lately is this idea of "making math class easier." @ddmeyer asked about people's opinions on foldables earlier today and linked to a blog post that stated, "I really wish I had math notes like these growing up…math would have been so much easier!" Now, I'm obviously not ripping on this woman. I'm sure that she's an awesome and caring teacher. It just really made me think about this idea of "easier" and whether this is what we should be aiming for in math education. I have seen lots of teachers teach "tricks" or "mnemonic devices" that are supposed to help students remember concepts and procedures. A colleague of mine tells her students about colored socks - she pulls two socks out of her drawer and if they are the same color, that's good, but if they are different colors, that's bad. This is supposed to help students remember that when signs are the same, the product is positive, but when signs are different, the product is negative. Most of us have probably seen the little mnemonic device for absolute value inequalities: greatOR and less thAND to "help" kids convert absolute value inequalities to and/or compound inequalities. There are a lot of these hanging around.

To me, foldables are in the same category of "cute" device that will help you remember something. Yes, kids love them. They are easy. All these devices are colorful or cute or make a little rhyme or whatever. But like it or not, I feel that there's a tension between this aspect of math and the side where kids are grappling with rich problems, constructing meaning, and having ownership of ideas. I understand that many teachers use these "shortcuts" as ways to summarize a concept or wrap up a discovery lesson. And hardly anyone gets away from teaching any shortcuts whatsoever. Maybe that's not the point. But I do think that we need to think critically about what we're doing when we emphasize these shortcuts. Even if it comes at the end of a deep, rich lesson, there's going to be a bunch of kids that are going to remember the "trick" superficially, and it's going to be what the lesson was all about in their mind. It's going to train them, to some extent, to expect tricks like that in the future and avoid the harder work of understanding and internalizing the concepts underlying the shortcut.

Yes, a hook can be powerful for motivating student engagement. But I think it can also be junk food that distracts us from the substantive meal, which is not as shiny or easy to digest at first glance. Let's look instead for ways to make the mathematics more profound, more apparent, and more rich for kids. It really doesn't need to be dressed up and tricked out because it's pretty darn awesome on its own.