Reflecting on the arc of my curriculum

This is only my second year teaching 6th and 7th grade math. I work at a school with a lot of students who score a level 3 or 4 on the state exam; for example, most of the students I teach who struggled in my class last year, still got a 3 or 4! So there are different challenges to teaching these students – I need way more extensions than I’m used to needing. I don’t need to scaffold as much (and in fact, I need to consider carefully whether I’m over-scaffolding and underestimating what they can do on their own).

Additionally, since this is only my second year teaching these two grades, I’m not as familiar with the curriculum: both the content in these two grades and the arc or story of the math. I am still figuring out the big ideas, and how best to build them together. I have a 3 – 5 year “plan” or expectation. Last year, I was just merely considering the topics and skills from a stand-point of making sure that I knew all of the topics AND that I knew the skills myself. Unfortunately, this meant that last year, I didn’t necessarily do my best teaching. Some topics were taught in ways that were more true to my pedagogical beliefs, while others were delivered in a way more congruent with my colleague’s beliefs, because I was using their curriculum to do my instruction.

This year, I’ve started to make changes. I’ve considered how I want to build the arc of the various units I am teaching, and I’m thinking about better problems to use with my students. I’m thinking about how to find the rich tasks for students to engage in, as well as the different instructional models I can leverage for different types of tasks. In the course of participating in my MfA PLTs and other PDs, I’m trying out various instructional routines and best practices – everything from the routines for reasoning from Amy Lucenta and Grace Kelmanik to number strings from Kara Imm and Pamela Weber Harris. I tried out an “important stuff, neat stuff, tough stuff,” style packet from PCMI. I have been using some of the content from CMP3, as well as some of the content from, openupresources from Illustrative Mathematics. I’ve continued to use some of the resources from my colleagues (especially in the first seventh grade unit on Probability, since it’s drawn from Engage NY in a very specific manner – though I confess that’s currently my least favorite unit because I feel the least ownership and understanding and enjoyment of it).

I am still tweaking the arcs of my curriculum, and considering how and when to cover the various content areas. I’m considering how to pull the various resources together to create a coherent set of units and to address the major skills and avenues for thinking. I’m considering how to build a mathematical community of learners and how to strengthen my students’ mathematical curiosity. I’m enjoying this work, but I do sometimes have to take the “long vision” – I’m not going to perfect my curriculum overnight. I am still making sense of the big ideas and mapping the topics that I have to teach to the fundamental layers of mathematics.

This week in particular, I spent a lot of time using the Illustrative Mathematics 6th grade unit 4 on division of fractions and thinking about which parts were necessary for my students and my classroom, and which things were unfeasible. I’m excited to hear the conversations my students are having, and the ways in which they’re developing their understanding. Earlier today, I had one student confess to the whole class (after multiplying by the reciprocal), “I know to do that, but I don’t know why.” We talked about making sense of the algorithm, and connected it to the diagram another student had drawn – suddenly, it became clear that the reciprocal was the number of that fraction in one whole – and multiplying it by how many wholes you had would get you the total number of that fraction in the dividend. We haven’t done every experience in the Illustrative Mathematics curriculum (and we’re going out of order because my school has a particular arc to our units), but it was such a rich conversation. I’ll write more about it in a separate post. My students have been making sense of models.

Meanwhile, in 7th grade, I’ve been using the routines for reasoning to develop my students’ understanding of how to write equivalent expressions. We did three Contemplate then Calculates to develop my students’ ability to chunk diagrams and see how to find the tenth, hundredth, and nth stages. Today, we launched a new routine: Connecting Representations. I was surprised by the results of it – and again, I’ll make a separate post about this.

The other thing I’ve been doing is collecting more and more student work and analyzing their thinking. In teaching a rational number operations unit in 7th grade, I made a close analysis of the kinds of mistakes the students were making on the quizzes. This will help me both this year, in teaching some of those same skills to my 6th graders as well as going back and doing targeted instruction for students who still made certain mistakes. It will help me even more next year, in planning specific activities to help my students unpack certain common misconceptions and develop a deeper understanding.

Finally, I’ve been thinking about ways to engage my students. I don’t want any of my students to be bored, I want to manage the range of learners in my classroom, and I want to develop a mathematical community more. I have begun to notice that there are fewer and fewer hands going up in my classes (especially in the 7th grade!), and I constantly hear from the same students. I am seeking new routines to change up who the talkers are. I don’t want to do random cold-calling, but I think in January, I might ask students to fill out a survey/reflection about their class participation and to start tracking which students are raising their hands and/or being called on/sharing.

The other thing I want to work on is the small group and partner conversations being more equitable. I use visibly random groupings (students get a card each day to find their new seat), but I’ve begun to notice that when I approach a random group and ask them “What have you discussed?” during a turn-and-talk, it is almost always a boy who jumps in to answer first. And if two students both start talking simultaneously, the boy often talks over the girl to continue and the girl usually backs off to allow the boy to finish his idea. I’ve started naming students and making eye contact with the girls to be conscious of hearing more of the girls in my math class. Probably another post on that at some point.

Sometimes, I have trouble wrapping up my thoughts in a post like this, because I don’t have the answer to the questions I pose. Continue the conversation with me on twitter at @MrKitMath.


Fractions, Decimals & Mixed Numbers, Oh My! #MTBoS HELP!

When I taught 8th grade math, my primary content focus was to master linear relationships. I learned everything I could about slope and y-intercept. I expanded into systems of equations. I discovered a variety of techniques to use to solve problems and many ways of teaching students how to do it. I learned multiple models and I got satisfied with some of them.

When I switched to 6th and 7th grade last year, I knew integer operations were something I was going to need to learn how to teach well. I did some research, and I settled into using a combination of the patterns for multiplying and dividing (based on CMP3’s accentuate the negative), and using the hot air balloon game with addition and subtraction (and even with absolute value and comparing numbers to some degree). I think I do a pretty decent job of teaching it (though obviously, it’s not perfect and some students struggle with some of the concepts, even by the end – especially with subtraction!).

Now I’m recognizing the next content area I need to focus on mastering: fraction/decimal/mixed number operations. Although the focus in 6th grade is on fraction division and on all decimal operations, I also think it’s vital for the students to master the other operations as well – and in my seventh grade class, we are focusing on the operations as well, especially with mixed numbers and combining decimals with mixed numbers (and including negatives!). So I need to find classes, resources, etc. to deal with those ideas next.

What are your favorite resources for multiplying mixed numbers? Dividing fractions and mixed numbers? For addition and subtraction with mixed numbers (especially negatives!). For dealing with decimals in general?

#MfAMT2 Become the Subject (A Pedagogical Use of Self)

Tonight, I had the honor of giving a TED-style talk at the Math for America annual event, Master Teachers on Teaching (Affectionately known as MT-squared). This year’s theme was Truth Matters: Trust, Lies, and Logic in the STEM classroom. This theme really inspired me to share my evolution as a teacher, so I wrote a proposal, and was one of eight teachers accepted to give a speech. I remember bringing my rough draft to the first session of our mini-course to prepare for giving the speech, and feeling like I had finished my speech and just needed to shorten it. After getting some feedback, I realized that I had somehow written 5 different speeches overlapping each other and I needed to pull out one speech to deliver. And I only had ten minutes (my original draft was more like a 25 minute speech!).

I ultimately selected my notion of a pedagogical use of self to focus on, and I wrote this speech. I’m sharing the text of what was written here as well, but it’s not a perfect transcript. MfA asked us to memorize our speeches, and while I did a pretty good job, I had a bit of stage fright in the beginning, and literally forgot my lines! I had to check my script to find my place. Luckily, once I told my story about Bubbe’s bagels, I got more confident, and I delivered the rest of my speech with fidelity. I even got comfortable enough to ad-lib a comment “Thanks Kid,” – which for those of you who know how awkward I can be, especially with improv and delivering speeches, you can understand why this was such an achievement.

My next steps with the idea of pedagogical use of self is that Kara Imm and I are collaborating on writing an article for the NCTM middle school teacher journal. Hopefully, they’ll accept our article!

The text of my speech (as written):

Become the Subject

Kit Golan, Master Teacher

Good evening. I’m delighted to talk with you tonight. First, let me share a dirty little secret: math class is not as fun as science class! Controversial, I know. I realized this as a first-year teacher when I taught both. The problem is I’m a math teacher, and passionate about it!

Those of you who are science teachers likely enjoy designing classroom experiments that facilitate discussions by unpacking student observations. I remember one such experiment, my first-year teaching, when I put food coloring in cups of cold and hot water.   My 6th graders watched, fascinated, as the color spread much more quickly in the hot water.  In an animated discussion, we made connections to brewing tea in hot water, and ultimately spoke about the impact of temperature on the speed of molecules. My students were excited and talked enthusiastically about the experiment.

Later that day, I taught those same students a math lesson. They showed much less interest in adding fractions with unlike denominators. Class became a battle: I spent all my energy trying to get students to do the math, while they spent all their energy trying to distract me and avoid doing math. “What are you doing this weekend, Mr. G?” they asked. “Do you have a girlfriend?” “Do you have any kids?” When asked a personal question, I stonewalled with, “That’s off topic” or “That’s irrelevant.” I thought I was keeping them on task. But what I didn’t realize was that some students were genuinely curious about who I was and wanted to connect with me, and I was rebuffing their efforts!

This leads me to the challenge we face: “How do we provide students with classroom experiences that promote rich discussions and engage their curiosity in math class?” My principal advised, “Just make your lessons more engaging.” “But how?” I asked. I hadn’t learned how to do that in grad school! Meanwhile, my students seemed overly curious about me and uninterested in the math. Then, during my second year of teaching, I had an epiphany and realized that I could use one problem to solve the other!

The breakthrough came after I had delivered a boring lesson on converting Celsius to Fahrenheit. My coach observed that while my students were plugging numbers into the formulas, they had no motivation—it was sheer drudgery. She suggested an alternative: What if I had launched the day’s lesson with a story? Suppose I said I’d gone to Canada, checked the weather before going outdoors and it was 20 degrees. I bundled up tight, expecting it to be bitterly cold, but when I got outside, I was sweating. What do you think happened? This would pique my students’ interest and elicit from them that I’d interpreted a temperature in Celsius as Fahrenheit. Now we had a REASON to learn how to convert temperatures – so we wouldn’t make the same mistake Mr. G made on his Canadian trip.

Using storytelling to hook students and help them learn math was a brand-new idea for me. Immediately, I found it highly effective. The first time I tried this, the kids were more attentive and enthusiastic than I had ever seen them. And so, I became a storytelling teacher. The purpose of my stories wasn’t to tell the literal truth about my experiences, but to cultivate rapport with my students and develop a reason for the mathematics. I leveraged student curiosity about me to engage them before they even realized we were solving math problems. Gradually, I evolved from fabricating stories to turning actual incidents in my life into math problems.

For example, last year, I launched a problem in class like this. “How many of you like bagels?” [Encourage audience to raise their hands]. Me too! I love bagels. We’re lucky we live in NYC, because we have the world’s best bagels. Sadly, my Bubbe doesn’t live in NYC; she lives in Connecticut, where they don’t have great bagels. So like any good grandson, I brought her New York bagels whenever I visited. One time I had to visit on a Monday, when my local bagel store is closed, so I couldn’t bring her any bagels! Let me tell you, my Bubbe never let me hear the end of that! Every subsequent visit, she would ask, “Did you bring the bagels this time?” “Can you believe he forgot the bagels?” So to make sure it would never happen again, I did a little bit of research and I found two other good bagel stores in my neighborhood: Bob’s Bagels and Tom’s Bagels. And both are open on Monday! Can you all help me figure out where I would get the better deal on bagels for my Bubbe?”

My students really got on board with this problem. As I monitored student discussion, I heard lots of conversations about their favorite bagel stores. Students analyzed each other’s methods of determining which store offered the better deal. They were seeing for themselves how mathematics is a powerful tool to solve problems in daily life.

At the end of the year, I asked my students to write a letter with any advice on how to be a better teacher and how they would like me to remember them. One student wrote, “Now, as for any advice, I liked how you would often turn your stories into math problems, it makes math more fun… I want you to remember me as… ‘the kid who reminded you that you forgot to bring your grandma bagels that one time.’” That story had stuck with him through subsequent units, and he recalled it as THE thing he wanted me to remember about him. Clearly, my stories resonated with him.

In building relationships with my students, I’m drawing on what I know about relationships in general. In psychotherapy, there is a term for using yourself to create empathy and a relationship with your patients: a therapeutic use of self. I propose that teaching needs to coin its own term: a pedagogical use of self. A pedagogical use of self is when you strategically embed yourself into the curriculum in stories that will captivate students and cultivate a community of mathematicians—or scientists—in your classroom. Such sharing about yourself will strengthen your relationships with your students. Your curriculum will come to life, your students will get to know you and you’ll draw them into your subject matter.

By using my life pedagogically, I model what mathematics can do for anyone and show students how math can be a tool to help us make sense of, explain, and evaluate our own lives. In eight years of teaching, I’ve evolved from following the lie “Don’t smile until Christmas” and worrying about staying on task, to spending the whole first day of school getting to know who my students are and introducing myself to them – everything from the dog I have to the absurd number of board games I own. By sharing my truths with the students, I build trust and inspire them to share their own truths with me. As a result, I am currently experiencing powerful, vibrant relationships with students and enjoying seeing some of them develop a passion for math.

I hope my evolution will inspire you to embrace a pedagogical use of self as a valuable tool in your own classrooms. Ask yourself, “What are your Bubbe’s bagels stories?” Thank you.

My speech can be viewed on twitter and facebook, by going to Math for America’s page and watching the live stream. I closed out the first half of the program, before intermission. On the twitter version, I get up to talk at 53 minutes.

Time Well Spent #MTBoS

I struggle with time management, especially when I’m lesson planning. I type this even as I should be lesson planning for tomorrow or grading! But I had this realization just now, as I was sorting through different ideas about how to teach tomorrow’s lesson. I was trying to create a lesson about divisibility rules (that could spring somewhat organically from this week’s lessons about GCF and LCM), and I realized that I could use the puzzle I have to launch a NEED to have divisibility rules. Then I found Prime Climb, and I realized that I could further using that to introduce the idea of prime factorization – which some of my students already know about, but this could deepen their understanding and connect the two.

So while I spent more than 30 minutes just thinking about the tweets I wrote a few weeks ago and reviewing the things people shared with me, I feel like I’ve made a much deeper investigation that starts with a question, gives motivation for discovering the rules, and connects them to the larger picture of the unit. All good things.

I guess my conclusion is that GOOD lesson planning requires more time! 😦 Too bad time is so finite!