Group Norms

Working collaboratively is an incredibly important skill, and increasingly so. So in addition to teaching math, I’ve been working on strategies for teaching communication and collaboration skills.

I got a jump on this during bonding week this year. Every year we divide the school into small groups of maybe 5 or 6 and task them with creating boats. We have this rotation – year 1 these boats are strictly cardboard and duct tape, year 2 strictly trash bags and duct tape, and year 3 cardboard, trash bags, and duct tape. It’s year 3 but this picture is from last year:

boats for blog

Chaos, right? But super fun chaos, and we get more buy-in than you’d expect. Even still, we had noticed this tendency for boat group communication to be subpar. The groups would usually defer to the ideas of whoever was loudest, kids sometimes got mad that their idea didn’t feel listened to, and without a teacher consistently in the room (which would ruin it in other ways), it was totally OK to not participate for kids who didn’t feel like it.

So this year we had them do limited-resource challenges in small groups, and then reflect on these group dynamics upon entering their boat groups. We orchestrated the groups so that nobody was in a group with someone they just worked with, so they could talk openly about their experience. I wrote up a set of reflection questions:

Communication Discussion Questions:
1. What was the most challenging part of communicating with your previous group?
2. Did you feel that your ideas were listened to?
3. Did you always feel comfortable sharing your ideas?
4. Did you feel that you listened to other ideas? Why/ why not?
5. What qualities in someone else make them easy to communicate/collaborate with?
6. What qualities in someone else make them difficult to communicate/collaborate with?
7. Do you find it helpful when someone steps up to be the leader, or does that make you more reluctant to share?
8. What can you do as a group member to make sure nobody is dominating the discussion?
9. What can you do as a group member to make sure quieter people are joining in?
10. What can you do as a group member to make the decision-making process easier?

So now in my math 2 class we’ve started our first group task, the rectangle pattern task I discovered last year. Before breaking into groups we went over the group norms, based on concepts we talked about in my college classes and the book Designing Groupwork by Elizabeth Cohen. Group norms:

For a productive group…
1) Stay in your group
2) Ask the other group members first
3) Everyone is accountable for what the group is doing
4) You are responsible to ask for and offer help

I like the last one best although the first one is a close second.

Best question ever: “wait – are we doing this to practice working in groups or to learn the math?” Answer – both!!! I think I exclaimed that with three exclamation points in class as well.

Tomorrow we finish the task in math 2 and launch the vector fields task in math 4.


Intro to Math Week Year 3: Arts Integration Edition

Just finished my last academic day of the first academic week of the school year. As it has been for the last couple of years, this week is “intro to math week,” and this year I focused specifically on arts integration, keeping the mathematical standards of practice front and center.

On day 1 we spent some time on the syllabus, focusing on class procedures and previewing the upcoming units. Thanks to keeping this blog I have plenty of pictures of previous classes doing the work they’ll be doing, so it’s not just a line about functions or data or whatnot on a sheet of paper. Then I had them each take a Math Attitude Survey. Somewhat unsurprisingly, most of the new ones had some pretty negative associations with math. Kids also weirdly LOVE sharing bad experiences in math classrooms, so it’s a nice first day icebreaker. I took everyone’s three words and put them into a word cloud:

math word cloud 2015

I’ve always been too scared to re-administer the survey at the end of the year, but I think I should do it, or at least have them give me three words again.

The next day was all about arts integration. I had them re-examine the mathematical practices and compare them to the National Core Arts Standards, since I had personally noticed so many great links. Some of them were a little unsure at first, or had trouble understanding what individual standards meant, but they didn’t shy away from asking each other or me for clarification, and the resulting conversations were pretty cool.

art and math

Then we watched this TED talk by Daina Taimina, or at least most of it – it’s super long [for my kids’ attention spans & non-university math levels] and the important parts are at the beginning and end. I love this talk so much though. It inspires me to see her construct her own understanding, especially given her initial troubles with the subject, I enjoy seeing stereotypes smashed, and is the perfect example of arts integration. The art form makes the mathematics more concrete and comprehensible, and the mathematics provides structure and context for some interesting-looking pieces.

Typically I have some major emphasis on group norms, but I didn’t want to insert something else random into this week, so instead today we just had an extended journal reflection and then dove right into our first topics. I’ll teach them the group norms as we start doing our first group tasks. I realized earlier that I’ve never actually blogged the group norms, so I’m working on a separate post about that.

So far things are going well BUT hey it’s only the first week.

Tiles, free art, and practice problems

I have a “BLOG POST” alarm on my phone that goes off every 2 weeks, which is not even all that frequent. But over the summer I just don’t have a whole ton to report.

I guess I’ve been writing a ton more practice problems in order to reach some summer financial goals. Being able to do math in order to do things like get my hotel for my friends’ wedding is actually so wonderful.

I also read lots of books about architectural drawing and have that project in the works. I want them to learn about line weights and 1-point perspective in order to draw a scale rendering of the farmhouse at school. Part of me thinks I should write a rubric now while I’m not so busy, but part of me feels like I should wait and see which kids are even in that class. We’ve definitely got a bunch of new ones I don’t know much about, which is exciting!

Did you hear that a new tessellating pentagon was discovered? We think of math as so closed and black/white right/wrong, and yet new discoveries are still being made of colorful exciting-looking things. I’ve been looking for examples of cool tilings in my everyday life. The best one was on the bathroom floor at the new Baked and Wired, where we have our bookclub meetings:

literal tiles

I see triangles, hexagons, pentagons, more triangles – this has so much potential.

I went to the Freer/Sackler galleries with my teacher friend Anna in search of some Islamic art, or some other examples of geometric tilings, but I didn’t really find what I was looking for. This bowl has some intricate patterns, but it’s not exactly a tessellation.

freer bowl

I’m trying to think of more ways I can take advantage of the Smithsonian in my quest to make my curriculum more arts-integrated. Sweet, free, wonderful Smithsonian.

Summer PD Part II

Flipping the calendar to August and seeing the words “retreat and teacher workdays” on the page was a little bit of a shock, I have to say. But I’m feeling more and more ready. I’ve been at work on my arts integration plan.

I found out about the the national core standards during a Kennedy Center workshop, and so I just read through them. This was extremely helpful to me because they bring up questions and concepts I would have never thought of. For example, the idea that students should be planning their project and choosing an appropriate medium makes a lot of sense, but it’s not something I’ve been having them do. If I introduce a math/art thing, I usually just tell them what to do and how to do it, then help them with the math if they need it.

The concept behind arts integration is that the students learn an art form and the math content in ways that are “mutually reinforcing.” The art standard that most closely mirrored a math standard was “Through experimentation, practice, and persistence, demonstrate acquisition of skills and knowledge in a chosen art form,” which could reinforce and be reinforced by “Make sense of problems and persevere in solving them.” I love the wording behind both of these because they don’t expect us to be talented or to find clarity instantly, they encourage us to keep working even if things aren’t coming naturally. That’s a huge issue in both math and art, the notion that it’s only for people with a natural aptitude.

The project that I’m most excited about trying next year is a rose curve/ op art/ colored pencil task. In the past I’ve just told students “a rose curve is in the form r = a sin (theta b)” and told them what a and b do. Then I’ve taught them the other types of polar graphs similarly, with notes. This year I want them to investigate using different coefficients to see what types of graphs are created, then generalize. Then I want them to choose two and color them either in black and white, or in colored pencils, and explain why they chose the graphs they chose. I made some examples and they look pretty neat.

rose color rose op

I just really want to make sure that when we’re doing this, they understand a) what polar graphs are, and the different categories they fall into b) what types of equations generate rose curves c) good colored-pencil techniques so that these actually look decent [my example could totally be better] d) what would create an appealing black-and-white “op-art” style piece [so why would I not have chosen the first curve to color in black and white]

Once we’ve got all that, then we can talk about fun things like converting between rectangular and polar. The company I’m tutoring with lets us write problem sets for $$$, and I just wrote a bunch of those problems. I hadn’t done that math in a while and it’s actually great.

In terms of other arts integration stuff, I still need a lot of help with my sine curves/ jazz unit, and I’m also developing a similar figures/ architectural drawing project, right now mainly by reading through all the architectural drawing books at the library. Stay tuned for whatever I end up with!

Summer PD Part I

Summer has officially started, and here in DC its arrival is never subtle. I can’t walk three blocks without being drenched in sweat, my water bottle is never full enough, and honestly I am loving every minute of it. Last summer I felt like I didn’t have enough to do and got sad about it, but this summer I’m trying to take advantage of my unstructured time. In addition to doing significantly more yoga and binge-watching Orange is the New Black, I’m working on accomplishing these things:

1. Design a statistics class for seniors, based [loosely?] around the AP curriculum.
2. Learn statistics – I somehow never took a course
3. Re-design my classes for 9th, 10th, and 11th graders, because now that I’m teaching all 4 grades there is no way to put them on different tracks, so I’ll have to differentiate like crazy. Plus most of the fun visual geometry stuff was in math 1 which no longer exists, so I need to find places to put that in 2, 3, or 4, probably 4 which means I’ll have to significantly enrich it. Like uranium. 4. Write curriculum for Spanish that assumes that they were all with me last year, because most of them were, but that will accommodate any new students.
5. Design a service learning course that continues Hope’s original vision but that I can feel ownership over.
6. Seriously plan the **** out of everything because I am scheduled with zero planning time next school year. ZERO.
7. Create an arts integration plan based on the Kennedy Center workshops I attended last year.
8. General professional development

I’m mainly here to blog about the last one, but this made me realize I’ll have plenty to share throughout the summer. It’s also really sinking in that I will be at a serious loss of unstructured time in the near future.

Anyway, PD. In the past, we’ve all read the same book over the summer and then discussed it on the retreat. This year we couldn’t decide on a single book, so everyone was tasked with just going out and finding professional development opportunities and then coming back and sharing.

So far, I’ve read one book in the name of PD: Building a Better Teacher by Elizabeth Green. I linked to her “Why Do Americans Stink At Math?” article last summer, and some of that same content was in the book. A lot of the book actually centers around math education, and excitingly for me, around one teacher, Deborah Ball. This is so exciting for me because Deborah Ball is the dean of the School of Education at the University of Michigan [GO BLUE], so I have taken her class and feel well-versed in her philosophy of teaching. I can also hope that her greatness had even the slightest impact on my own teaching. The book focuses on her work, in conjunction with another Michigan teacher, to help new teachers create investigative, collaborative classroom environments like her own.

For a great example, watch this video of some elementary students having a spirited yet respectful debate about even and odd numbers.

The book also focuses on Doug Lemov, whose book Teach Like a Champion provides extremely practical classroom strategies, revealing the secrets to managing an effective learning environment. He takes what seems like a magically smooth classroom and boils it down to concrete actions that any teacher can learn to take. While I was doing my student teaching, my mom mailed me this book and I remember reading it on the elliptical after school, dreaming of my future orderly classroom full of excited learners. More on that later.

The two visions of a productive classroom – Ball’s lively class discussions and Lemov’s orderly SLANTing – seem somewhat at odds, but what they have in common, and what the entire point of the book is, is that teaching is a skill that can be taught, not just an ability that some are born with. The immediate question then becomes: if I went through Michigan’s superior teacher training program and read Teach Like a Champion cover to cover, why am I not the best teacher in the world?

I surround myself with very positive, supportive people who would answer that with “but I’m sure you ARE a great teacher,” and that’s nice, but in reality there is no way that I internalized every piece of effective pedagogy. There is a huge difference between being inspired by something you hear in class and writing it down, and knowing how to actually apply it in my own class.

My major takeaway from this book is that no matter how amazing teacher training is – and the book implies as I’ve always assumed that mine was extra good – these skills need to be reinforced. What I really want is someone in my classroom frequently, not in a frightening evaluative way, but in a helpful “are you doing what you learned was best?” questioning and guiding way. I’m sure that after 4 years in the classroom I’m doing a lot of things better, but I’ve also strayed from some original intentions and picked up some bad habits.

What I’ll Share at the retreat: a) They should all watch the video of the 3rd grade number debate, because it will challenge some assumptions about the level of discourse we can expect from our students, and it will give us a jumping-off point for explicitly teaching them to communicate and hear each others ideas. b) We need to revisit doing classroom observations and assessing ourselves. That’s going to be incredibly hard to schedule because we’ll be down a couple staff next year, but I think it’s important. I’m tired of people assuming that I’m doing amazingly in my classroom just because the kids aren’t unhappy, because I want to be better than that.

“Learn What You Missed Week” and MI stations

Greetings! I have recently returned from this year’s senior trip. As you can see from my wrist accessories, we went to the Rock and Roll Hall of Fame in Cleveland and the Franklin Institute in Philadelphia.


We can worry about how cool my tattoo is later. Anyway, before the trip happened we were back in class for 2 days after exams. I will be honest and say that I was not extremely in favor of this decision – I did not feel like I could come up with 2 days worth of engaging lessons on “going over the test.” Ick. Instead of trying to fix every mistake they made on the test and magically make them understand what they hadn’t just one week prior, I created an initially simple-seeming activity.

For each of my classes I identified the three most-missed objectives. In case you are curious, for math 4 they were modular arithmetic, fractals, and summation notation (sigma). For math 3 they were complex number operations, data, and exponential growth and decay. Finally, for math 2 they were factoring, similar figures, and probability. Then for each of the 3 objectives I found or created (usually created) a task for each of the 9 intelligences in Gardner’s theory. Those are: interpersonal, intrapersonal, existential, naturalistic, logical/mathematical, visual/spatial, verbal/linguistic, musical, and kinesthetic.

So that’s 3 classes x 3 objectives x 9 intelligences = 81 tasks. Luckily the art objective for exponential growth also worked for fractals so actually just 80 but still! Luckily again, these went really well, and it was all worth it.


This was the kinesthetic data task, where they had to throw the paper glob a bunch of times and record the distance, then find the mean, median, etc. I like the looks of that full page of mathematical text – that is not the work of an un-engaged math learner. This also shows the bin of algebra tiles, which I’ve never seriously used but actually really like. I experimented with them for the kinesthetic factoring task, and I think they have potential.


I printed the tasks on colored paper and cut them out so that they could just select one and then grab it and do it. Can you slightly see that the big one has music notes on it? I think that particular music task was one of the ones that was a bit of a stretch actually – they had to look at the notes and graph the melody on a complex plane, which probably doesn’t have musical meaning. The best ever musical task was the musical data one. I drew inspiration from this & had them record data about their favorite songs’ danceability, valence, and speechiness.


There’s a lot going on in this picture. The picture of ghosts was for a task inspired by this which is very neat. The reason why it is ghosts is probably because of the manipulatives I created to practice multiplying complex numbers. Those are in the blue cup in the front of the picture. They are pretty much algebra tiles, only instead of “x” blocks there are positive and negative ghosts to represent i. The positive ghosts are smiling and the negative ones are frowning. It was cute, initially confusing, and potentially very effective.

I’m done teaching until the fall but I think I will still have plenty of things to write about. We shall see.

Group Exams and Assessment in General

Exams are over! As a teacher, I secretly love exams – I love grading in brightly-colored pens, noticing trends in student performance, and somewhat shamefully, I enjoy a few days of just sitting. It is a really interesting balance, though, of stressing the importance of demonstrating knowledge on a culminating assessment without heightening any students’ anxieties. All last week and now this week I’ve been on the verge of sending mixed messages, applauding students for doing well but reassuring ones who didn’t knock it out of the park that their grade won’t drop down catastrophically.

The rest of the year I give short, low-pressure assessments as soon as students feel they’ve mastered a particular objective. I think that making these as small a deal as possible helps get more accurate data – students aren’t nervous about their performance, they’re just demonstrating their knowledge.

I tried to make the exam pretty non-threatening as well, even though it inherently feels high-pressure. A big part of that is the group exam. This is usually a multi-part task with multiple entry points to maximize student participation. This year for math 4 I had them build three iterations of the Koch snowflake out of craft sticks. I was going to have them build 4 but we would NOT have had time, or probably space. Then they had to find the area of each iteration based on the patterns of exponentially-increasing triangles. It turned out very nice!

snowflake1 snowflake2

My favorite part about this was that they walked in for the exam to see all the tables pushed to the walls and just knew something was up. I’ll attach the text of the group exam as well: Math 4 Group Final 2015

I should note that I do a group final as well as a standard individual final because of the way exams are scheduled at my school – I have each class for 1.5 hours before lunch, then another 1.5 hours after lunch. I didn’t think that it would benefit anyone to write a 3-hour math test. These tasks could work in a non-assessment situation, too, though.

For math 2, we’ve been doing all this great work with polyhedra so I had them continue for the group exam. As a sidenote I’ve been very inspired by this myself and have been hard at work crocheting pentagons for a dodecahedron – I hope to post that at some point. This exam was centered around the tetrahemihexahedron, which I think is amazing. I had them cut out nets and fold them with basically zero guidance – they did an amazing job intuiting how these should be folded. Then I asked them to create toothpick-and-marshmallow structures with things in common with this concave shape, such as the cuboctahedron below with the same vertex figure, 2 squares and 2 triangles:


That’s actually one I made, theirs was kinda lopsided, but they tried SO HARD. Most of the time my philosophy is that trying hard is NOT the same as doing good work, but sometimes I let lopsided polyhedra and the like slide.

Here’s the full text of the task: MATH 2 GROUP EXAM final 2015

So I’m almost done in the classroom, but I still have enough odds and ends to be posting well into the summer. I’m actually about to head back to school now to see the seniors present their projects, which is always a really nice culmination of some interesting independent student work. Plus I made cookies!