Better Math Discussions

When I first started having students to present their work to the class, I quickly realized that wanting to have good student presentations was oh so removed from having good student presentations. Half the time, the work was hard to follow or the explanation didn't make a ton of sense to the rest of the class or the way they did the problem was so wrong as to not even really be useful for discussion or even when everything was great, the rest of the class wasn't remotely interested because they had just done the same problem in the same way so why did they need to hear another group explain what they had already done themselves? And kids were just not used to sitting through presentations done by other kids. They were restless, distracted, and lots of classroom management was required to get some semblance of presentation. Not to mention the fact that I don't think these presentations did much in terms of any sort of math learning. I think I was just doing them because I thought I was supposed to do so?

I loved "5 Practices for Orchestrating Productive Mathematics Discussions" when I first read it a few years ago. It was the most succinct and clear set of explanations for how to organize discussions around student work. The key idea for me was that it involved NOT calling on random groups to present and that it required giving students problems for which there are multiple approaches. The idea is that the teacher selects which work should be discussed and that there's a rhyme and reason to the work that's selected and the order in which it's presented.

Having an Apple TV and iPad in my classroom the past two years has also been very helpful for doing the selection of student work and sequencing it in an order that makes mathematical sense. Before the Apple TV, I would run around the room, taking notes and trying to then remember which group I wanted to go first and why. Also, even with my giant whiteboards, some students feel the need to write microscopically.

Now, I just take a picture of the various approaches that I want the class to discuss, put them in the order that I want using the Notability app (you could use any app that allows you to stick in pictures), and then projecting each picture in the correct order and big enough that even my tiny-handwriting students' work can be read.

My other breakthrough was that having problems with multiple approaches is key because it really is boring and pointless to hear presentations of the same thing you've already done. It's much more interesting to see and learn from different ways to the same problem. If the content doesn't lend itself to problems with multiple approaches, you can assign similar, but not the same, problems to different groups. It can make for a good progression when the problems build on each other or when each has only one element that has been tweaked.

I have also been working with students on making the presentations more useful. Instead of just talking about what they did from start to finish, every group after the first one is asked to explain what they did differently from the other groups and discuss the benefits and drawbacks of the approaches. At the end of presentations, I also ask students to summarize the different ways and discuss when we might use one versus another.

Another issue that I had is with putting up work that had mistakes in it. Students would freak and be horrified that their WRONG work was being shown to everyone.

It helped a lot when I stopped attributing which group the work with an error came from and emphasizing how useful the mistake was to our learning. It's February, and I'm finally getting to the point where students ask whether their mistake is interesting enough for showing to the class. Score.

The things I'm still struggling with for group presentations are how to talk less and make them talk more. I still find myself doing a lot of the "summarizing" and pointing out key aspects of the work to the class, and I want to push myself to turn that over to the students. I would also like to find more ways to increase the engagement of the students who are listening to the presenting group. Perhaps asking them to write a summary or some other response after presentations or require each group to ask a question of the presenters. Other ideas?

Student skills

I've been thinking a lot recently about the differences between successful and struggling students. It is obviously easier to be a successful student if the material comes easier to you, but there are plenty of students for whom the learning part is hard and yet who are quite successful and even more that are the opposite. There are a lot of specific student skills, such as organization, time management, ability to focus during instruction, etc that I think students at this age are aware they need in order to be successful. But there are also others that I think are less obvious to them: asking questions, knowing when you don't know, and the big one: persevering when something doesn't work at first. I feel like I need to start paying more attention to these intangibles and teach them to struggling students explicitly rather than being annoyed if they're not "motivated" to do well. Obviously, this presupposes that the material is being taught in a way that is accessible and engaging, but I think that while the teacher has an important role, the student's role is even more fundamental. A student who has these skills will likely do well regardless of the teacher while a student who doesn't will likely struggle no matter how talented the instruction.

So, some things that I've been trying to get at these "student skills":

1) Journal prompts asking students to reflect on how well they feel like they're doing with the current lesson, what questions they have, and how they're going to address their questions.

2) More journal prompts asking students to reflect more broadly on how they're studying and keeping themselves engaged during class.

3) One-on-one meetings with students where we set individual goals and talk about progress in meeting them. (I haven't done these as much as I'd like this year... this is a good reminder to start these back up.)

4) Emphasizing over and over again the importance of effort, perseverance, and asking questions and that this is what's under their control and should be the goal towards which they're working rather than an abstract "I want to get a good grade."

5) Distinguishing between general effort and reflective and directed effort... for example, homework credit is only given for assignments in which the answers have been checked and mistakes have been corrected (with ample opportunity and encouragement to ask questions in class), not just for attempting problems. Part of students' participation grade (10% of their  overall grade for my more-struggling classes) is predicated on asking thoughtful questions.

6) Focusing my comments on grade reports on progress made and effort expended and not just on results reached.

I'd love to hear of ways that others encourage students to push themselves and become better learners.

Some journal prompts:

Graphing quadratic functions foldable

Nothing too fancy or mind-shattering here, but perhaps someone may find this useful. Just a quick, little graphic organizer for the steps I want students to follow to make a nice graph of a quadratic function.

Graphing quadratic function foldable (updated) by Anna

You need to print both pages and copy from one-sided to two-sided so that you can fold over the top and make it look like this:

Students should cut each numbered flap and then fill in each step with an example problem. Here is what the final copy will look like:

Outside:

Inside:

Group success!!

Today had one of those class periods. You know the ones. Where everyone is working and discussing and arguing and engaged and I'm? I'm just there. Totally and completely unnecessary to the workings of my class. Amaze.

Today was the first day of learning to solve systems by substitution. We went through an example together using @cheesemonkeysf's patented "substitution by stars" method. Then, I passed out some problems for them to work through, telling them to work with their group members and only go to the next problem when everyone in the group was totally good with the previous one. There were some where students had to apply the distributive property and some where they had to isolate a variable first. We had not talked about these. I did not stop class to discuss and fix and give hints, just circulated the room and looked on. Not one kid asked me for help for the 20 minutes that they worked - they puzzled through and argued and made mistakes and erased them and made some more and eventually figured it out all on their own. Holy wow. You guys get it. We all want group work to go exactly like this, but it so rarely does.

But finally!! All of the working on norms and redirecting questions with, "Have you asked your group yet?" or "Is this a group question?" and biting my lip to avoid butting in have paid off and there was just this amazing energy and focus and I was so, so proud. Cause there aren't that many glorious classroom moments like this, and it's so easy to focus on all the ways that we aren't perfect and they aren't either. So I want to stop and acknowledge that today was exactly right and keep it in my memory bank for when I'm feeling frustrated and disgruntled. Hooray!!!!

Axis of symmetry for a quadratic function

My two-year Algebra course is just starting our study of quadratics. So far, we've looked at graphs and tables for quadratic functions and compared them to linear functions and talked about the symmetry of the shape and finding the axis of symmetry from the x-intercepts and finding the reflections of points across the axis of symmetry. I would like students to derive the formula for the axis of symmetry: for the quadratic function . The Exeter Math 1 curriculum does this by having students graph functions in the form , then generalizing the idea that one of the x-intercepts is zero and the other one is . I created a worksheet that extends on this idea using desmos to graph and having students also derive the formula for the axis of symmetry by averaging the two x-intercepts to get . Then, just like in the Exeter packet, I will have students compare graphs of  to graphs of  to realize that the axis of symmetry is not changed by adding a c value to the equation. I'm hoping that the logical progression of the graphs will make sense and that the questions are open enough for this to be genuinely based on student thinking and discovery, but with enough structure that students who are not necessarily used to deriving formulas will have something concrete to think about and answer. Any and all feedback welcome!!

Axis of symmetry by Anna

Baby steps

There was a good discussion recently on Twitter about complex tasks and why many teachers and students shy away from engaging with them or give up in frustration and return to low-level tasks.

Thoughts/critiques of this framework? pic.twitter.com/vsTdBKbOhC
— Geoff Krall (@emergentmath) November 20, 2013

I think that we can all come up with reasons why it's difficult for many teachers (including myself) to move out of their comfort zones and implement rich tasks in their classrooms. I am also interested in figuring out why students would resist complex tasks. @MathEdnet blogged about the various reasons that complex tasks can empower students by giving them more control and a voice as mathematicians and doers. The idea is that working with rich problems allows students to see their knowledge as valuable and themselves as active users of such knowledge. In implementing such tasks in the classroom, however, I have often seen student frustration and discomfort with the change in expectations from previous classes or from how the class had been functioning. This is sometimes especially true for students who care about their progress the most and who have certain ways of doing mathematics that have worked for them in the past that no longer work in a framework of complex problem solving. For these students, complex tasks appear confusing, unfamiliar, and an obstacle to their goal of doing well in the class. It can feel very frustrating to the teacher, especially if she hopes that implementing a complex task will increase student buy-in and engagement. Everybody is unhappy.

There are many ways of working on this issue, I think, and each is unique to the particular confluence of school, teacher, and group of students. Some teachers have big enough personalities that they can persuade students to trust them and step out of their comfort zones through sheer awesomeness.

Not a teacher, but would probably be an awesome one.

Teachers like me who have a hard time not being liked by our students and are not inspiring enough to get everyone to drink the Kool Aid come up with more gentle approaches. Baby steps, if you will. I have been working on a mix of traditional and complex instruction that takes students from the type of work that they're used to doing in math classes and gradually, inserts some open problems, starting with smaller tasks that are worked on in class and give students plenty of supports to hopefully build on more and more rich problems as students' comfort level increases.

I am, by no means, amazing at this. I definitely give tasks that are too open for students to handle and they freak out. Or alternatively, too many low-level tasks, which undo some of the work I've put into pushing them past that point. But this is the type of thing that is really, really hard to learn to do. Or, at least, it is for me. It's not something that is part of a graduate course or can be picked up by watching a lesson or two taught by a master teacher. And I have certainly never seen a pre-made curriculum that does this type of nuanced dance between what this particular group of students is comfortable doing and something that's just a bit outside of their comfort and ability zone so that they feel challenged and interested, but not overwhelmed and frustrated or bored and disengaged. So. My point. I did have one. I feel like lots of us on Twitter are stabbing away at this teaching thing, but with different tools, personalities, and kids. And it's easy to feel frustrated that I'm not doing amazing open tasks every day with my students or month-long cross-curricular projects that empower and engage them to the utmost.

Wait, this isn't what your classroom looks like every day?

But, I'm working just outside of my comfort zone and pushing my students to do the same. Baby steps. But progress, nonetheless. And I'm confident that y'all are doing the same, in your own way.

So coming back to the original question - perhaps what I'm hoping for is more recognition of baby steps and meeting people where they are, both teachers and students, to help them make small, but noticeable progress, as a way out of the cycle that @emergentmath described.

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!