Yesterday, I wrote the quiz using the same mindset I would use if I had been giving a graded quiz. As I gave this quiz to two of my classes today, the whole experience felt different. A few kids needed to leave class a bit early; not a problem. They could leave the work they had done with me, and take a blank copy of the quiz with them to finish the last two problems, to bring to class on Monday for me. Kids who typically have extra time on tests could use that time, or not, as they chose. I felt like I was helping them, and they seemed eager for the opportunity for true feedback.

Some students finished their quiz in 15 minutes, while others stayed for 45 minutes. I checked in with each kid as they turned in their work. Quite a few students said they felt good about the quiz. At least six other students asked to schedule a time to meet with me, knowing we would be having a graded quiz in about a week. Overall, I think I just liked the feel of the whole process. Students didn’t get stressed out. Instead, they had an opportunity to test themselves, see how it felt, and then gain feedback from me. I suppose it would be even better if they got that feedback before they left the classroom but I don’t see how to do that without using significant technology.

I would like to keep thinking about ways that I can view my role as “serving” my students. What else do they need that I’m not currently providing for them? What do your students need?

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It is part way through exam week as Semester One draws to a close, and I realize that many of my “New Year’s Resolutions” from this fall have faded from my mind. I am about to pick up another class and want to spend the time to get to know them so we can work well together in the second half of the year. As my mind goes there, I realize that I have not done some of the things I wanted to do this year. In September (and even earlier), I was planning to find ways to make my classroom feel more like an elementary school classroom. I tried a few days with “centers” in my Calculus class, but my activities were too long and made the rotation idea less realistic. Although I was initially enthused, I didn’t work at it hard enough to make it viable. Now, I’ve just stopped.

I have three classes with very different personalities. One small group is extremely quiet. What changes do I need to make so I hear their voices more often, and they get what they need out of class? Another group is boisterous, eager to jump up and do problems at the board. However, two or three students sometimes get left out of that vibe. How can I reclaim some class time to make sure everyone has their “moment”? My last section is my favorite. I know you’re not supposed to have a favorite, but I do. I’m crediting some of the time I put into the class at the start of the year to build the environment we want in the room. However, a lot of it is pure luck of the draw with who is in the class and how they interact. It’s a variety of freshmen, sophomores and juniors, all of whom are curious. They have just submitted their second big project of the year, and have had two real opportunities to explore an idea of interest to them. We have had multiple days where I have had what felt like the perfect class with people all over the place, all working on different things in slightly different ways, and me interacting with a few of them in significant mathematical conversations. I want to make opportunities for more of that to happen.

I know I’m tired, and that it’s a cold, rainy day in January. However, I want to find the energy to direct to each of my classes (including the new one) so that each student has the best experience in the second half of the year as possible. But first, I need to grade some exam projects.

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In this mindset, I decided to attend any session that interested me. I was awe-inspired from the start, as I listened to Wanda Hill as she received the Ruzicka Compass Award for work as an educator. Hearing her story about helping minority students gain access to boarding schools in New England, beginning in the early 1980’s. She was incredibly brave in her work, and was vocal about what needed to change. For example, she would not send any of “her” students to a school where students on financial aid were required to work on campus. I recommend looking up her full story; it’s a fascinating one.

Charles Best was the official keynote speaker, the founder of donorschoose.org. He was suitably stunned to have to follow the impressive Wanda Hill onto the stage. Nonetheless, he told an incredibly compelling history of donorschoose.org, how it works, and where he thinks it could lead. It was a great start to the day.

As I moved into an initial session, I went to hear Andrew Watson (@Andrew WatsonTTB) speak about Long-Term Memory Creation. Although nothing he said was new to me, the ideas and research have solidified in my mind in a way they clearly had not been before. Andrew spoke about interleaving, the benefits of blank page review, and how to provide “desirable difficulties.” He was incredibly concise as he spoke, explaining the extensive research in this area. Perhaps because my mind wasn’t racing with what I needed to plan for class tomorrow, and what was in my bag waiting to be graded, I could finally process what he was saying.

I do “blank page” review often, but not often enough. That’s an easy change to make. However, I do not interleave on a daily basis. I think about the concept regularly, and try to think about overlaps between units over the course of the year. This is not enough. The yearly approach isn’t a bad idea; it’s just nowhere near enough. I need to mix up problems every time students are doing them, so they aren’t in Section 4.1, needing related rates, and therefore expecting to use that skill. This is not a hard change to make because I don’t use textbooks in many of my classes. However, I need to make a commitment to changing the type of homework I assign, in particular. My students will learn more by doing 3 or 4 different types of problems than doing 4 identical ones.

I am eager to commit to making these changes! Thanks, #TABS2016.

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Joanne is taking what is essentially an Algebra I class at her college. She has attempted three other math classes and has either dropped them or failed them, and she needs to pass one math class to earn her degree. Her advisor suggested she try a self-paced course, which happened to also be computer-based, using the ALEKS program.

As we began to work on expanding polynomials, I shared a few techniques that seemed to help, at least at that moment. Based on some of the comments she made, I realized that she didn’t really understand what an expression like “2y^6″ meant. If students don’t understand the building blocks, and why they are useful, how can we expect them to make sense of multiplying one polynomial by another? Our assumptions can be risky, and mostly they are risky for the students in our classroom.

The other thing that has stuck with me that Joanne mentioned was about her former math teacher. She sees her high school teacher at her job, and that teacher has chided her about her math struggles, saying” Joanne, you know this stuff! We did all of this together in class.” Again, this is an issue of assumptions (which include mine as well). What did Joanne understand when she studied these ideas the first time? If she did really comprehend all aspects of her work at the time, what makes it so hard the second time around? If nothing else, it points out how rarely anyone is being asked to use some of the skills we emphasize in high school. How can we make the work that we do more relevant?

All of these ideas coalesced in my mind this weekend. I was out in the city, and overheard two people talking about the incredible sales going on. One person was accurately estimating what the sale price of something would be if it were 40% off. They talked through their technique, and reached a correct answer. Then, the person turned to their friend, joking and saying, “Math! It’s finally helping me!” This makes me sad. I’m not quite sure what to do with that feeling, beyond keeping it in my mind every day.

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Overall, these visits have been really enjoyable. Visiting the art class was probably my favorite, but I did realize that even just spending 20 minutes in a class gives me plenty to observe and a lot to think about. Here are a few of my takeaways after being in others’ classes:

- It’s invigorating and refreshing to watch my colleagues teach, regardless of discipline.
- I can easily empathize with students who might have an entire class day with teachers talking at them. I also find myself thinking about the student who is less inclined to participate in class discussions. That person could spend the whole day not speaking, and not engaging, depending on the activities planned in each class. I need to make sure my class engages all of my students at least some of the time.
- A one-on-one private conversation with a student is often the best first step to take to understand someone. I found myself giving the advice on multiple occasions to talk directly with the student whose actions are frustrating to you; step away from the heat of the moment, and ask some questions about why they are acting the way that they do. Then, be prepared to listen. How does that kid who is dominating the class discussion want you to notify them that they need to take a break? Empower the student to navigate whatever situation you are in, and work on a solution together.
- Returning student work promptly is always a good idea, and will always be appreciated.
- Teachers doing all or most of the talking does not result in a good class experience.
- Teachers need to be paying attention all of the time as to what is going on in their classroom. They won’t see everything but they will see a lot if they look. This makes me of a think of recent post from @dcox21 on When the Activity Isn’t Enough. He wrote about watching students actively doing a Marbleslides activity on Desmos, and observing what we can learn about how they approach the challenge of finding the correct equation. We all need to keep watching, and keep learning.

I’m sure there are many more items to add to this list, but these are my highlights for now. When can you find 20 minutes to visit someone else’s class? If you can make this happen, post something you learned or were reminded about by conducting that visit.

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It was pure coincidence that we were about to begin an axiomatic trigonometry multi-day lesson, where students would be working together and I would not be a prominent voice in the room. These three days (especially the last two) made me so happy! On day one, the first few proofs to do were fairly straightforward, so the students were working quietly and diligently on their own. I didn’t like how it felt because I was hoping they would be talking to each other, and they weren’t doing that. However, I realized afterwards that they didn’t need that help yet; they were doing fine solo. Once they reached a harder proof, they immediately started talking. For most students, that began on Day 2 of our in-class work on these problems. Then, I started looking around (but forgot to take a few pictures) and realized I was seeing much more of the classroom that I hoped to have. Even then, I couldn’t observe it all because I was deep in conversation for about 10 minutes at a time with one student at a time. I did see students up at the board (of their own volition), talking through how to solve one problem. Others were huddled (standing) in a group of about 6 or 7 people, listening to one person explain how to think about another question. No one was using our two beanbags that day, but three girls were at the stand-up desk, and others moved around the room when they wanted to do so. I had music playing through the projector.

I was already excited before I walked in the door to Day 3 of this unit. In many ways, this was a repeat of Day 2, with new problems to consider. A few new things arose:

- My two students who were in the play and in the midst of run week apologized for not making it farther through the packet. As I had already said to them, this was an experience and not the type of assignment where every student needed to get to the last question. Whatever they could do in class would be the right amount. I have to believe that this shift of expectations helped to make their week more manageable. It also made me wonder if I could make this same statement about other sets of problems I ask kids to do. Is it ever about doing all of the problems?
- One of my freshman began to shine. I had seen his energy before, but others in the class didn’t yet know him well enough to see his “light.” As I looked over his shoulder, he definitely did not need my help. Instead, he wanted to show me the proof he wanted to type up for homework. They were going home that weekend with the assignment below:

*Using LaTeX, type up three of your Axiomatic Trig proofs. Include:*

*(1) a proof where you felt great about your final product, but you really struggled figuring out the proof*

*(2) a proof where you knew what to do, did it, and wrote a great proof*

*and *

*(3) a proof that showcases something else about this week for you. In your .tex file, be sure to explain why/how you chose the proofs that you did, and what your third proof represents for you, and why you selected it. How did this week go for you? As a student? As a learner? As a way to solidify prior trig knowledge? I really want to know what you think! This is a journal entry-type thing but your writing will be with your proofs a .tex’d document. *

This student was so proud of his work on a particular problem that he knew he wanted to use that proof as his “3” choice proof. It was fun to hear his exuberance and to be able to share that with him!

3. This same student had a chance, at my suggestion, to help two others in the class who hadn’t made it as far, and were struggling with one particular proof. I love seeing a freshmen teach a junior something, and have that be a real, fruitful conversation.

So – as I had these two great days, I want to try and help myself remember what led to this moment:

- letting go of being at the front of the room for any significant portion of class time
- encouraging students to help each other, particularly building connections across grade levels
- having students progress through material at their own pace, and only set the amount of total time spent on the unit (here, three days)
- recognizing that students working quietly and productively can also be useful

Mostly, how can I provide just enough of a structure that there is something to explore, and then get out of the way so the students can dive in?

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In some ways, those numbers represent what I want. An A- average seems great. Having two kids who still have some misconceptions about this topic still able to master 80% of the questions seems about right.

However, the numbers don’t tell the whole story. In looking at some of the student work on the test, here are a few samples that caught my eye:

This student circled this problem, and put a question mark at the top of his work. He included a clear calculation of the period of the function. Then, there are calculations all over the page (the student did not have a calculator during this assessment). What made this student doubt themselves on this problem? What part of the process felt “off” to them? I am eager to ask.

Another problem:

For the first time this year, I did not have my students complete a single-sided worksheet with about 40 problems like this on it. Clearly, this student’s solution is not a particularly efficient one. There are some lengthy calculations present because the student only wanted to think about the problem in degrees. Also, it shows me someone who likely uses their calculator briefly in questions like this, not hesitating to multiply 23 by 180, and then divide by 4, and also by 360. Because they haven’t needed to be efficient in this process, there is not an inclination to simplify the problem before multiplying and dividing. How can I better build up my student’s number sense to avoid spending time on something like this on a future test? Or should I instead provide a more basic calculator to each kid in case their working strategy involves computations like this?

Here’s one more solution that was interesting to me:

This student’s work is interesting because he had part of his answer incorrect. However, I think it was only off because of a miscalculation of the period (or of the coefficient of x in the equation).

Mostly, I’m ending this unit with some questions for my students about their thought process. I’m eager to see what they wrote in their math journals this weekend about the experience, and to have them reflect on a problem (on the test or from another moment) that did not make sense to them right away, as part of their ongoing math portfolio. I’m trying hard to see my students, and realize that as I look harder, I will only have more questions for them.

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As I redid the problem myself prior to class, I found myself redrawing the diagram on a larger scale (which I find my students don’t often do), and I also knew it would be a question that not everyone could solve. As the class went over homework together in small groups, I asked four different sets of people to put their solution on the board. One group imposed a scale of 0 to 100 on the problem, placing point R at 0 and point S at 100. They then found the numerical placement of the other midpoints in order to find the result that WZ=3x. When they presented their ideas, they said they were thinking of 0-100%, which is what led to that scale.

A second group did something similar, but chose to place point R at 0 and point S at 8. They also reached the same solution, that WZ=3x, but all of the points they placed on their “ruler” were at integer values.

A third group did an identical scale, although one boy shared as the group presented that he had actually used 0 to 24 on his paper, not 0 to 8.

The fourth solution on the board was drawn by one person [let’s call him Steve] who had gone to the board early in the class so he could explain his work to his two partners, who hadn’t figure out the problem. Steve opted for a more algebraic path to a solution, but reached the same end result. He had included some statements which included defining some lengths as being 5/8 of other segments in the problem.

As Steve explained his solution to the class, I could see the confusion in other’s eyes as to how he had approached the problem. We worked through this solution together to make sure everyone understood Steve’s idea, and then paused.

I asked the class to consider the three different solutions we had up on the board (since two were the same). Was any path a better one than another? Are all of the solutions reasonable? What should we take away from having done this problem? One girl raised her hand right away and said, “I think we should all be using Steve’s method because it’s the hardest.”

As soon as the words were out of her mouth, I realized how much of the typical student experience with high school math could be summarized by that one line. To many students (and perhaps to many teachers too), the hardest path is the one we should be taking. I felt pained as I heard this girl say this out loud, but at the same time, I felt grateful that she was willing to put that thought out there. I immediately shared my feelings with the class, and our conversation continued, but I find myself still thinking about her comment now. I shared the thought right away with a few of my colleagues, almost in a joking way. However, it feels like there is so much more to draw out of the remark.

If most students think that they have a path to a solution, but they doubt themselves because it isn’t “hard enough”, then I have work to do as their teacher. If we all share solutions in class as described above, debrief, and students still leave the room thinking that the hardest path is the one they “should have” taken, then I have work to do too. Is there any way to combat this feeling in students of math? Why doesn’t an easier path seem plausible? What has happened in students’ prior math experiences to make them think that harder is better?

As I experiment with a variety of different classroom structures this year, I want to be sure to remember to find the moments to point out the multitude of solutions, and try to emphasize that the path that makes sense to each person has validity and does not have to be hard. I also want to try to work to change my own mindset. When I wrote my solution to this particular problem, I wrote this:

It didn’t even occur to me to place a ruler-type scale on top of this image to simplify the work. I think that is the math teacher in me, feeling like I should be able to express the situation algebraically, even there is a simpler path.

Overall, I’m grateful to my students for showing me an easier path to a solution, and eager for the year ahead to try and help all of my students understand that math does not need to involve choosing the hardest path.

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I also want to bring an idea I had last year back into my “start-of-the-year” mind. My Honors Precalculus class was in the midst of doing a long-term project, where they were working pairs. As I looked at each group, I could see the way that each person brought something to make that group even better. For example, there were two girls were exploring the odds of winning a tennis match if you win the first point. Three things stand out to me as I think back about this particular partnership. First of all, these two girls are (and were) powerhouses. They are going to change the world and do amazing things. I can see it in how they go after an idea, asking great questions, helping others, and not being afraid to be the “skeptic” in class, not letting some niggling concern go and going after a concept completely. Wow! Just thinking about what they may do someday gets me excited.

The second reason that this pairing stuck in my mind is because of how their project idea came to be. They had followed my initial brainstorming instructions and landed on this tennis idea (which they eventually pursued). I chatted with them briefly and tried to sway their work in another direction. In retrospect, I don’t know why I did that. Instead of letting them draw me in, and helping them find the path to the math they could explore, I tried to get them to pick another idea where the math was more immediately apparent to me. I don’t want to do that again, so that moment was a great turning point for me.

Lastly, and the reason I’m writing this blog post, this group was a team in the best sense of the word. One of them had concurrently taken Honors Statistics, so she could engage in a statistical analysis of the question they wanted to pursue. The other student was enrolled in Introductory Programming and was making great progress in that class. Simultaneously, these two students were able to do an incredibly thoughtful analysis of the tennis question. One wrote a simulation to run repeated random trials and investigate, while the other student wrote the statistical analysis (using R) to assess the question.

As I described this group working together, I found myself noting how they made such a great team. It let me think of each of the other partners in a similar fashion and see this everywhere. Another set of two boys together were extremely interested in data analysis, and both had strong skills using a spreadsheet. Their work together was improved by being a joint project. A third group benefitted from a Python coder in the midst.

The reason I write all of this is to remember that feeling. I want my class to feel like we are all on the same team. I’m glad one person is there because they know more about coding than I do; you are glad I’m here because I may be able to explain hard ideas better to you than the teacher can. As I think more about group norms and the start of the year, I know I want to make sure to be explicit about the team goal for all of us. As an entire class, we will be an impressive group. When we work in smaller teams, we still all have skills to be sharing out. In fact, this may be the most important thing to have students share about themselves early in the year. What are you good at? What can you do, but don’t really like to do? What do you think of as an area in which you’d like to improve?

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- I’m still intrigued by the basketball problem @carlolitwitter had us consider. I want to do more of the math. My pattern-seeking self wants to find more, and think about ways to display what I have found. I am again reminded that Pascal’s Triangle is everywhere…. It also made me think and reflect with @bkdidact about doing math with my department, with tasks that are accessible to all. Lastly, working on this problem brought my colleague Jeanne to my mind, and her love of her circle cevian problem as a way to warn students about thinking they see a pattern that doesn’t hold true. (Short version: draw a circle, adding points to the circumference. For each point added, connect that point to every other point and count how many distinct shapes/areas are created and look for a pattern in the results.)
- I am perhaps not surprised but a bit saddened to hear how many people aren’t happy at their current schools, regardless of their tenure at that institution. I continue to feel like that I like what I do and love the school at which I teach.
- I heard much frustration directed at the DOE.
- Talking with @alittlestats made me think again about the role that Statistics teachers, or at least the good ones, get put in. If you do something well, you will be asked to keep doing it. Teaching the same thing again and again can be fun but even the best teachers need to pause to reflect and rejuvenate. That said, my pause from teaching Calculus is ending and I cannot wait to dive back into that class this year.
- I enjoyed talking with @_b_p about joining a new school, the unexpected benefits of changing jobs and interviewing, parenting and math. It’s always great to meet more teachers!
- I think I’m most excited about @NicoraPlaca‘s Master Designer task to start off the school year and help develop group norms. Her task description is here. This idea has been in my mind for a long time, based on numerous science museums I’ve visited where two people sit across from each other and try to build the same structure based on verbal descriptions only (sight unseen). I cannot wait to do this to start the year in each of my classes!

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