This term, I have been mixing in some old Geometry lesson material into my current Geometry class, now that we're no longer doing purely Exeter problems. It has been so much fun!!! My kids are delighted every time I throw in something that I had used and liked in El Salvador. It makes me very happy to observe.

The last couple of days, we have been doing a fun little similarity activity on the Geoboard, that I had made back then and then revised for this year. See it here. I am using it to introduce similarity, as a lead in to special right triangles and right-triangle trigonometry. The kids are having so much fun with rubber bands that they don't realize I am sneaking in significant learning.

I still like the Exeter problems for their incredible richness, but balancing them out with other modes of learning is the way to go, I think!

PS. This has been a good teaching week. Two of my low-confidence students who have been working their BUTTS OFF for weeks each got an 100% on their requiz. HOT DAMN! I'm so, so proud!

# I Hope This Old Train Breaks Down...

## Friday, December 12, 2014

## Friday, October 31, 2014

### How it Must Feel

Imagine yourself as a student who has always had a hard time with math. Honestly, you are not even sure why there are sometimes multiple variables in the same formula. When the teacher starts a new topic, you may need to see it 4 different ways and have it be repeated a lot of times in order to become somewhat proficient at the skills involved, even though the general idea is sometimes (not always) accessible to you. When other people discuss mathematics in groups, the speed at which they are discussing the ideas often flies right over your head, but you feel embarrassed to ask them to explain every part to you, because other people all seem to understand it fine. So, you do ask some questions during class, enough to make some progress on the work, and then you wait until you can find the teacher to ask for more help. You can only find the teacher to meet outside of class about once a week, and by then you already have more questions accumulated than can be answered in one session, so that although you think the concept is probably important, you just want to know the bottom line of how to get through the various problems you are going to see on the quizzes. So, the quiz comes and goes and you feel defeated by your score, even though you are working hard everyday in class and your teacher can also see that, too. You go home with the quiz and you don't want to even look at it for now, even though you know that you probably should start thinking about the re-quiz and that the teacher will probably start something new either tomorrow or the next day, and that the cycle will likely repeat itself.

What do you do?

In every class I teach, there are some kids whom I meet outside of class on a weekly basis. For many of them, that is enough. For some, that isn't. I love the idea of heterogeneous grouping, because I believe in all the things that other math educators believe, which is that it promotes safer learning environments, teaches diverse students to work together, promotes growth mindset, etc. But, at the end of the day, I don't know what to do in a practical sense to help these kids who experience those cycles of disappointment in every unit. I try to vary up what I do to help them build the conceptual understanding of underlying topics (ie. Math Talk, problem-based learning, visual representations), but realistically I have to balance too much going-back-to-basics against the majority of the class's need to develop other, more sophisticated, skills and concepts. Open-ended problems sound awesome in theory, but in reality we still need coherent conceptual and skills development the majority of the time.

So, what do you do besides trying to be empathetic?

I don't have an answer right now. In the past, I have had great success in homogeneous grouping at helping the slower-paced classes build confidence and feel successful with a smaller set of topics, but I find that goal very challenging/elusive when those lower-confidence students are situated in an environment that is perhaps just faster-paced than they can handle. I am fortunate that most of my students give me 100%. I have no doubt about that. But, how can I grade them all on an absolute scale based on what they know, if they are all starting off at very different places along the spectrum of prior algebra experience? And, more so than grading, how can I serve them all?

Those are open questions in my mind. Would love if you could chime in to enlighten me in your thoughts about this.

## Thursday, October 2, 2014

### What to Do When You Are Busy

Today, one of the things I read over the summer came back to me. The thing I read was written by a therapist, regarding his patients who are stressed and unhappy because they feel like they're spread too thin among work, family, friends, etc. The therapist's advice is that you cannot make more minutes in a day, but you can increase the

*quality*of your minutes in the day. If you're thinking of work when you are at home, and thinking of home when you're at work, then you're not making anyone feel valued and therefore the quality of your minutes spent with them is low. This way, everyone around you will feel dissatisfied, and you'll feel unhappy as a result. The wisdom he shares is to focus on making the people you are presently with feel like you are 110% present. Be proactive about it, instead of trying to coast by with little engagement. (In terms of active engagement, he distinguishes between your son telling you about his upcoming soccer game, and you remembering it yourself and mentioning, without being prompted, how much you already look forward to it. In both cases, your time commitment is the same -- you're still going to watch his game -- but just by being a little bit more thoughtful, the reception of your time spent will be very different.)
I thought of this today because I noticed that I have been working a lot of extra hours. But, when I meet with my students outside of class, it's not helpful for me to think even fleetingly about the many things that I still need to do that day. Instead, being fully present and taking an extra 10 minutes to let the kid slowly organize their binder or to tell me about their week before we start to look at where they need help in math, is so valuable. It slows me down and it slows them down, and it allows me to make them feel important before, during, and after our meeting. Maybe I cannot increase the amount of time that I have for them on a regular basis (and maybe I cannot solve all of their math troubles in one meeting), but I can make each encounter outside of class more meaningful by just taking a little extra time.

So, here is a reminder to myself to keep building the quality of my minutes in a day, especially as it gets really busy.

## Saturday, September 20, 2014

### The Start to a Great Year!

It has been 2.5 weeks since we got back to school! During this time, because of various back-to-school grade-level bonding activities, we only had 2 good weeks' worth of classes in each grade. Still, I am feeling somewhat settled into my classes, having learned all the kids' names, and am starting to figure out who needs help and to meet regularly with some students outside of class. So, things are good!

In addition to revising curricula, I am trying a lot of new classroom procedures this year. At first I thought these changes wouldn't be manageable, but so far, although I am working a fair amount, my work time has been very productive! I can clearly see the impact of various small changes on my classes, and that leaves me to spend my energy where it counts the most (ie. trying to tackle the big challenge of having many more high-need students this year).

Here are some things that I am trying this year:

*

**Assigning group roles for each group member:**The one benefit that has stood out the most to me is the role of the recorder. The group's facilitator and time-keeper help me ensure that every student in the group tries the problems independently and jots down their individual work, but at the end of the activity/discussion, I only collect 1 copy of the worksheet and notes (from the recorder) to make sure the answers they went over are in fact correct. I also ask the recorder to write down additional items as per their discussion, such as a description for the algebra process used and definitions for key terms. I then correct just that one copy (from a group of 3 or 4), and then make copies for the other group members to keep. This way, everyone has the conceptual explanations, to help them remember what was discussed in the group besides just the answers they have on their paper. It also cuts down on the amount of grading I have to do, while being able to give written feedback to the entire group. Playing the role of recorder has also been very motivating for the kids who are the weakest performers, stay actively engaged. They know that their role is highly relied upon, so there is some unsaid peer pressure for them to make sure that they understand the process enough to write down important pieces of it on paper. Besides the recorder, I also have a facilitator, a time-keeper, and a questioner in each group. (The basic construct comes from Complex Instruction, but I am not sure if I am following their protocol exactly.)
*

**Green, yellow, red stickers:**In lieu of grades, this year I have been, thus far, giving feedback in terms of colored stickers. Eventually we will have quizzes that will count for points, but in the mean time, "mock quizzes" are just about 3 questions I write on the board, and I collect them to see how the kids are doing (and to take notes on who needs reinforcement on what), and we discuss the answers as a class. The next day, I return the mock quizzes with a colored sticker to show whether each student needs to keep working on that concept. It's de-coupled from grades, but my secret hope is that kids will just*want*to get green stickers. And, so far, that is pretty true!! I have been using this colored-sticker system on the collected discussion notes (from recorders), mock quizzes, and conceptual "check-in" homework assigned individually. The kids who got red stickers from me on a quiz or an important assignment have approached me to automatically resubmit the assignment after revision, to ask for help, or have been very happy when I proactively approached them to set up a time to meet outside of class. They treat the red stickers very seriously, which is great. It's such a visual way to alert them that there is a gap. The fact that the assignments are de-coupled from grades makes it possible for me to focus on meeting with them for their learning, rather than the conversation being about grades. It also makes it possible for me to have multiple "mock quizzes" in the same week without it being stressful for the kids. So far, I love it! I got a pack of round stickers from a local drug store's stationary/school-supplies section, and it already has all the colors I needed. One of the science teachers started using the same system with his students after hearing me talk about it, and he thinks it's much more clear than giving kids check plus, check, and check minus. Eventually, my goal is to have the kids self-assess via a sticker when they turn in certain assignments. And then I'd give a second sticker upon returning it, to confirm or correct their self-assessment.
*

**Green, yellow, red cups:**I know this is not news for other teachers, but I wanted to try the green, yellow, red cups in groups as a passive indicator to me whether the students need help. Sometimes, I think as a teacher, it's hard for me to tell when students are positively frustrated or negatively frustrated by a task. The cups are a clear way to indicate to me whether I should intervene. My secret hope in introducing this was that if they looked around the room and the other students all have green cups, maybe the group that was already about to give up would push themselves to persevere just a little more. Also, because they could only indicate green, yellow, or red to me as a group, they would be forced to communicate amongst themselves before they reached out to me for help on a problem. I have thus far only used this in my lower-grade classes (Algebra 2 and Geometry), because I am not sure whether it's a bit too cheesy for my 11th- and 12th-graders. But, I may roll it out to them eventually.
*

**Self-reporting math efforts outside of class:**I feel strongly that as juniors and seniors in my Calculus class, the students need to be doing self-directed learning outside of class. Practically, that means that I don't assign problems everyday, because I want them to use their at-home time to make flashcards, concept outlines, re-do tricky problems, do new problems on their own, see me for help, etc. It does*not*mean that on those days when I don't assign specific problems, they shouldn't be working on math at home!!!! But, I am also a realist; I know that most teens will do the minimum unless there is some visibility into their action. So, I made a Google Form that collects data about what the students are doing. (*You can click on it. This is a copy of the link I sent out to the kids, so even if you fill it out, it won't mess up their data.*) I created a bit.ly link to the form I made for my students, and I asked them to fill it out every time that they do math outside of class. I stated that, as "homework", I expect that they're doing/logging 20 minutes of math outside of class, 5 times a week. Of course, they can clump the times if that is not possible, but if they are doing 100 minutes of math once a week, that is probably something I should talk to them about. Anyhow, I think the information collected this way will be a great resource to allow me to have productive learning dialogue with my students, while encouraging them to be self-directed learners, asking me questions like, "What else can I be doing with my studying time?" I've only rolled this out on Thursday, and already I can see some data being logged by some students, with comments on what they might need from me as next steps. I'm very excited about this!!!!! If this is successful, I will extend it to my Algebra 2 class. Throughout the term, it'll be a very valuable resource for me in terms of giving them specific feedback on their learning strategies. It'll also make writing narrative report cards a breeze at the end of the term.
So far, these are the "systematic" changes that I am trying to make. I have already noticed a tremendous difference in my classroom culture this year, in terms of how equitably and actively students participate, and how positive they are. I will observe a bit more and write a big post about that, maybe next week!

## Thursday, September 4, 2014

### Experimenting with Structured Group Work

I am trying to change my classes this year drastically in the way that I handle group work! I want to be intentional, intentional, intentional.

So far, it has been one day and that one day has been excellent. I decided that for Day 1, I would talk to the kids about inquiry-based learning vs. direct instruction, and then pass out a task that is accessible for everyone to start. All they had time for on Day 1 was the individual thinking time, which I am going to build into every group activity this year.

In Geometry and Algebra 2, I did Mark Driscoll's folding task on Day 1, but emphasized that the content of the task is not nearly as important as the students practicing the expectations for group work. In Calculus, I made a custom sequence from Desmo's Function Carnival that consisted only of the parachute height vs. time, followed by misconception analysis, and parachute vertical velocity vs. time, with subsequent misconception analysis. The students in all classes then were asked to go home and finish the rest of their individual thinking.

Tomorrow, when we come back, we will do verrrry structured group work, and I will ask the students to do first one round of just making observations with no comments from their peers (Round 1). Then, they will go around and ask clarifying questions or challenge each other to justify their thinking (Round 2). Then, the group will engage in an open discussion while the recorder continues to take careful notes, to turn in later and to distribute to the group. Afterwards, they will summarize the findings and record what questions they still have as a group. To help the Calculus students focus in on the misconceptions I saw today, I will give them this handout as generated from their parachute graphs, and ask them to brainstorm as many observations as possible for each graph before we discuss as a whole class. The recorder of each group will record all the accuracies and inaccuracies that they notice about each graph.

Here is some language I will offer the groups tomorrow to help them with their structured discussions:

Middle of discussion (Round 2):

"I was confused about how to _______________"

So far, it has been one day and that one day has been excellent. I decided that for Day 1, I would talk to the kids about inquiry-based learning vs. direct instruction, and then pass out a task that is accessible for everyone to start. All they had time for on Day 1 was the individual thinking time, which I am going to build into every group activity this year.

In Geometry and Algebra 2, I did Mark Driscoll's folding task on Day 1, but emphasized that the content of the task is not nearly as important as the students practicing the expectations for group work. In Calculus, I made a custom sequence from Desmo's Function Carnival that consisted only of the parachute height vs. time, followed by misconception analysis, and parachute vertical velocity vs. time, with subsequent misconception analysis. The students in all classes then were asked to go home and finish the rest of their individual thinking.

Tomorrow, when we come back, we will do verrrry structured group work, and I will ask the students to do first one round of just making observations with no comments from their peers (Round 1). Then, they will go around and ask clarifying questions or challenge each other to justify their thinking (Round 2). Then, the group will engage in an open discussion while the recorder continues to take careful notes, to turn in later and to distribute to the group. Afterwards, they will summarize the findings and record what questions they still have as a group. To help the Calculus students focus in on the misconceptions I saw today, I will give them this handout as generated from their parachute graphs, and ask them to brainstorm as many observations as possible for each graph before we discuss as a whole class. The recorder of each group will record all the accuracies and inaccuracies that they notice about each graph.

Here is some language I will offer the groups tomorrow to help them with their structured discussions:

Beginning of discussion (Round 1):

"What have you tried so far?"

"I noticed that ______________"

"I tried __________ and found that to be (un)helpful,
because _____________"

"I was confused about how to _______________"

"Are you saying that _______________?"

"Can you explain why you think ___________ ?"

"I don’t get that.
Can you explain it in another way?"

"If we changed __________, then what would the result look
like?"

End of discussion:

"In conclusion, we agreed that ______________"

"We found it hard to agree on _____________"

"As a group, we still have trouble understanding ____________"

I am really excited about this! Having structured group work is allowing me to slow down the pace at the start of the year and to emphasize quality over quantity of work done, in order to set the right tone for the rest of the year. I plan to have the kids stay in these groups for about a week, and then we'll discuss what have been the most helpful parts of the structure, and then switch into new groups. Wish me luck!!!

## Friday, August 1, 2014

### One Resource a (Week)Day #19: Using Desmos in Calculus

I am starting to focus in to think about what I want to do with my classes during the first week of school. For Calculus, I think the choice is obvious. I should start them off by playing with Function Carnival, over at Desmos.com!! (For those of you who, like I, did not attend Twitter Math Camp and have missed the demo / all the blogosphere buzz, you should sign up for an account at teacher.desmos.com to play with their interactive applets immediately. The interface is thoughtful and it allows you to collect data regarding what your students are creating as graphs and how they analyze other people's errors, and you can use it at the start of a unit to build understanding from recognizing and addressing misconceptions.) I will be starting them off easy, on Day 1, by playing with the regular algebra version to warm up. And then, on Day 2, plunge them right into predicting and testing velocity function graphs in the Function Carnival, in order to jump-start their interest in thinking about rates of change.

To reiterate things that others have said, I really like the ability to both toggle through individual students' responses and to look at the whole class's responses at once. To start with this activity at the very beginning of the year can help me to set the tone that it is okay to experiment and to make mistakes, because that is how we learn math.

Yay Desmos! This is really a fantastic tool that incorporates real mathematical instruction,

To reiterate things that others have said, I really like the ability to both toggle through individual students' responses and to look at the whole class's responses at once. To start with this activity at the very beginning of the year can help me to set the tone that it is okay to experiment and to make mistakes, because that is how we learn math.

Yay Desmos! This is really a fantastic tool that incorporates real mathematical instruction,

*not*by replacing the whole-group experience but to reinforce it by giving each student a structured think-time before the whole-group discussion. I really hope that this is the future of digital instruction!## Thursday, July 31, 2014

### Thinking About Calculus

If you teach Calculus in high school, you probably have a similar observation as mine that the student population is highly heterogeneous (a fact that is perhaps surprising to outsiders). Many students feel pressure from their parents and peers (ie. social pressure) to stay on the Calculus track, so within the same group they can span truly a wide range, from those who really could have benefited from another year of Precalculus-type skills-building, to those who are somewhat okay with procedures but who really need more experience/exposure with integrated problem-solving (something like a "Math 4" course that involves college-level math language), to those who are truly ready -- and eager -- to tackle new topics, both conceptually and algebraically.

I stumbled across this research summary from years ago (dated 1988) that noted a shift in college Calculus curricula away from conceptual understanding and toward procedural manipulation. Even though the article is dated, I find myself torn in the middle of a similar tug-of-war in my own thinking of Calculus. Fortunately, I teach a non-AP class, but my juniors (within a class mixed with seniors) do largely feed into another year of Multivariable Calculus after my class (combined with students from another class), so I have to take that into consideration in organizing my course. But, the statement from the article that we're trying to make the course "be all things to all people" really resonates with me. In the same class, I am supposed (and trying!) to differentiate so much as to help all the kids who need an extra year of algebra practice; to help all the kids who need an extra year of problem-solving; to cover all the major Calculus concepts;

*and*to develop all of their skills as learners and to nurture their mathematical practices (all of which, mind you, take time). It was really half a miracle that many of the kids ended up enjoying and appreciating the experience of my first year of trial-by-fire to accomplish ALL of this at a new school; the task set for us can be daunting, frustrating, or exciting, depending on how you choose to look at it.
One of the things that I did this year which actually helped a lot with teaching a heterogeneous group was to make sure that I made time for projects in this class. The projects gave the kids time to self-differentiate. Many of the kids kept working on their projects until the last minute, throwing in extra features or sometimes even working on it after it was due and presented. The next year, I will keep most of the projects but re-arrange the pacing so that they are better timed and hopefully a little less stressful for the kids.

Projects I did:

* Economics mini-project (but this one I will switch out for something better, more individualized)

* Graphical organizer showing connections between all the learned skills

* Rollercoaster design using piecewise functions

* Function pictures including shaded definite integrals (calculated via GeoGebra but also by hand)

* 3D-modeling using vases that they created and volumes of revolution / scaling up to find and test against real volumes

I also did a group quiz this year on the minimization of coordinate distances (to help review the distance formula) and on related rates, which was assigned to be completed mostly outside of class. It was such a great experience for me and most of the kids, that I'll definitely have to find some opportunity to repeat it. I decided that next year, I'll have to bump up the Function Pictures project to the first grading period, to help the kids review functional forms and inverses. That way, when we revisit their projects to fill in the integrals later on in the year, their focus will be more on the integrals and less on the outlines.

Since I need to replace the economics mini-project, I am going to try doing a sustainability project this year, that involves some regression and rate-analysis at the start of the year, and then have an individual component where the kids look for something else (an interesting data set) that does not have a time domain, to extend their understanding of rates beyond the time domain.

Another choice that really helped me with approaching the heterogeneity was teaching Calculus in reverse. First, we did a lot of graph sketching and graphical analysis by calculator. This evened out the playing field because the kids who were weak with prerequisite algebra skills could still access and feel successful immediately about the new concepts. Then, we learned differential Calculus skills via exploration, which helped the kids build some of those valuable mathematical practices and to build their confidence in playing around with math. I took our time on this part of the course, to go through old algebra skills and to practice things that were challenging as they came up. The really intuitive students started during this unit to peek ahead on their own at how to procedurally un-do differentiation. After the derivative skills, I took a significant amount of time to work on related rates with the kids, to give everyone some quality time with problem-solving. (It wasn't nearly enough time though. Obviously, you can never spend enough time on teaching and practicing problem-solving.) Later, through explorations, the students were able to gather most of the core concepts about integrals. We then used projects to reinforce their algebra skills. Limits came last in my course (because it is the most abstract, and I think it made the most sense to introduce it last as a way to

*prove*the things we thought were true), and at that point, it was a really nice tie-up of the whole year, reinforcing the definitions we had learned throughout the year regarding derivatives and anti-derivatives by proving them via limits.
Even though I have been thoughtful with my course, I wish I could feel more certain that I am making the right choices for my class. For everything that I decided to spend more time on, it was a choice to leave something out. I tried to keep the class fluid, so that if an interesting question came up during a project with a bunch of kids, I expanded that during the next class to go into the relevant material, even if it's not part of the classic Calculus 1 curriculum. (For example, my students really wanted to know how to integrate circular areas on their projects, so we did an example together and about a third of the class then followed suit to use trig-substitution to help them on their projects.) Never in my class did I feel like I had wasted time, but I couldn't shake the feeling that Calculus 1, the way that I was teaching the class, could easily have spanned 1.5 years.

What do you think? What choices have you had to make for your own Calculus classes? Do you think they are worth it?

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