We all know the user manual we get when we buy a new appliance or piece of technology. It comes with some information on how it works best and what things to avoid.

So why don’t we do something similar for ourselves?! To help in building relationships and clarifying some of our personal preferences, I want my teachers to take the time to reflect on how they work best (as well as their pet peeves). This manual is a chance to bring up personal preferences and routines so that we know how best to work with each other, possibly preventing some future conflicts. It is a place for some unfiltered brainstorming on how we work best. For instance:

- What is the best time of day to drop by your office?
- What is your preferred method of communication?
- Is there a routine you have during a part of the day?

As an example, for me, having quiet time in the early morning is really important. Before first period is a time where I focus and organize myself for my day, take one last look over my lesson plans, etc., and I prefer that time to be uninterrupted. So I can communicate in my own user manual that the mornings are not a good time to “stop by.” If there is a concern you have first thing in the morning, send me and email about it or to schedule a time to talk.

Here is the link to what I came up with: Teacher User Manual. I’d love to hear thoughts and suggestions for it before I roll it out!

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**How**** does your class move the needle on what your kids think about the ***doing*** of math, or what ***counts*** as math, or what math ***feels*** like, or ***who*** can do math?**

The short answer is: debate. I have spent a lot of time developing structures and strategies to make my classroom a place where students can discuss and debate math!

Those who know me, know I love to talk strategies for debate in the classroom. However, my struggle in writing this post is to be clear that in my classroom, debate is more than the claim/warrant structure I share in 30-60min sessions at conferences. For me, it is a real and important part of the fabric of my classroom culture. I want students to know that anyone can talk and debate math. I want them to know that doing math can involve opinions, and I want them to feel that math can be messy, argumentative and beautiful.

As a former speech and debate student-turned-coach, I saw the power of both public speaking and creating debatable arguments for students. Furthermore, in the subject of mathematics, a subject solidly grounded in reasoning, proof and argumentation, I saw a place to connect debate strategies to content learning. So, I have tried to create a classroom where students do the talking, explaining their thinking and creating opinions to open-ended questions. It is my goal that at least once every school day, students have an opportunity to debate and discuss *something* (though I can’t think of a day where it only happened once)!

My delight is seeing students struggle as their mindset shifts ** from** the idea that math is where you (a) learn a new formula/concept and then (b) solve a worksheet of similar problems

*What is the best method for solving this problem?**Which piece of information is the least important here?**How will you most affectively improve this work?*

FYI: More about my debate structure can be found on my blog here, here and here (among other posts)!

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As always, I was overwhelmed by great ideas. Here are some things I want to work on this year:

- Assessing with Desmos (a la Julie and Jonathan)–remember to press pause so they can not go back and change answers.
- Connecting with other math teachers and coaches about leadership in Heather’s session.
- Writing in math class! I want each student to have a math journal that stays in the classroom and they write in once a week. Thanks to Cindy at St. Ignatius!
- Need to check out Games for Young Minds.
- Remember: you are not an imposter. We are all teacher leaders! Let’s build up! (Thanks, Julie)

So looking forward to TMC19 in Berkeley, CA next July!!

]]>So, I’m doing formal observations as needed for the teachers in my department. At my school, that’s once every 3 years. But what about all the time in between? How can I really help my colleagues learn and grow in genuine ways? How do we even get to the point where we can be aware of areas to focus on? There’s so much out there to do, how do I help each individual find the right tools for growth?

Then I listened to this great podcast episode (that is also blogged here) from The Cult of Pedagogy about various types of PD. Choice #3 in the blog post inspired me to do my own choice/bingo board of options for PD. Below is what I came up with, a PD Bingo board of options that teachers in my department can do next year. The goal is that each month (just for next year) they will choose a different option. Each of these options can be very minimal in time commitment (but teachers are also welcome to run with it as they please!).

The instructions will be for teachers to work on a different square each month, providing a variety of experiences throughout the year. The goal is that through this variety they can find both (1) topics they want to focus on for growth and (2) the method(s) of coaching they most prefer. I want my colleagues to see that coaching and PD does not just have to be formal or informal observations. I want them to see a range of options that I consider PD or personal growth.

My hope is that after a year of this, each teacher can set more specific individual goals and a specific method (or two) of working toward that goal. Then, instead of doing such a variety, we can focus on the best method for that teacher…though we can always mix it up again as wanted!

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**3.12 Monday** – **The Pi Chain**

During math classes the previous Friday and Monday, students decorated a link (a rectangular piece of construction paper) that represented one of the digits of pi. Each class was assigned a section of digits and made a chain for their section, connecting each student’s link.

The links were also color coded by digit.

Then, during Monday’s lunch period, we had a (not-so-ceremonial) hanging of the Pi Chain, wrapping it around all the math classrooms.

**3.13 Tuesday – The Scavenger Hunt**

Teams of 3-5 students signed up in advance for a scavenger hunt that would happen during Tuesday’s lunch. Our department worked in advance to create a series of clues that were puzzles, cyphers, etc, leading students around campus to various rooms to get the next clue. At the end of the clues, they were directed back to my classroom, where they had to put their clues together to figure out the code to the lock box, opening the locks (escape room style).

Teams had to come up with team names, and the winning teams were named “The Inte-girls” and “Dobby’s Sock.” The prizes were math-y tshirts, candy and these amazing ruler-slap-bracelets I recently learned about!

**3.14 Wednesday – Pi Day**

Due to various other events (like the student walk out) on Wednesday 3/14/18, we were lower-key on the actual pi day. In my classes, students wrote Pi-Kus (see the link for more info). Teachers and students wore math-y shirts, and during lunch, students were selling pies as a fundraiser!

Additionally, over all 3 days, we had tables set up during lunch periods with (student run) math puzzles, games, tangrams, rubik’s cubes, etc. Students were free to stop by and just play with math-y challenges. Teachers (not in math) dropped by too!

We are already talking about how to grow this for next year!!

]]>I wanted to give a quick thank you to all the people who attended my debate session! A copy of my slides are here: CMC Conference Slides! It was the first time I did the talk in a large room, a less intimate setting, and (though it all went well!), I’m starting to think about how I would re-work it in the future for a large space. I’m learning that I’m a teacher/presenter/adult that enjoys small group settings.

After a wonderful lunch with some of my favorite (and new!) math teacher friends, I made my way to Robert’s talk. If you haven’t seen his Open Middle problems or heard him talk, definitely check them out! I had a great time working alongside Eric Martin as we created our own open middle problems. A (messy) vision of what I did is below:

There’s something interesting that really stood out to me about these problems (and the presentation on a macro level)–I felt a similarity between the session I attended and the one I led. Creating these open middle problems and creating debate-able questions are two different styles/methods with similar goals. For one, we both want to help teachers see ways to improve their questioning. My style is to foster more discussion and debate through creating debate-able questions, and Open Middle is re-working questions in its way to challenge students and increase discussion/engagement around problems. We’re both deepening students’ understanding of the math and awareness of misconceptions. We’re both trying to increase student engagement and to allow for various methods. It’s just the details in how we do it that is different.

Of course there are plenty of ways to contrast the two, but sitting in Robert’s talk, I noticed how similar even the structure and style of the presentation was. We both take teachers through this journey of *doing* the kind of problems we want them to do, *creating *problems by sharing a method of developing problems similar to what we just did and *sharing resources* to extend their exploration and learning in what we are each doing. (Granted, Robert is waaaaaaay ahead of me on collecting and collating resources, but I’m just saying we have a similar flow.)

It’s really a feeling, and I still struggling to put the connection into words.

]]>However, I really wanted to add both more fun activities to the rest of my units AND have more of a continuous through-line. So, I tried to include a Barbie-centered activity in my introduction to derivatives, including average velocity vs. instantaneous velocity, etc. I came up with Barbie walking the runway!

Page 2:

It worked really well! They had fun, and we talked a lot about slope=velocity. The last page had some questions more textbook-y, asking them to apply the ideas in new problems:

Email me if you want the full doc!

]]>- Day 1 & 2: Finding limits by tables
- Day 3: Finding limits on a graph
- Day 4: Finding limits using algebra

The first two days, we just spent finding limits using tables (on Desmos and in the graphing calculator). I started with the “buffering” idea I mentioned here. Then, we just made lots of tables, explicitly talking about what numbers you would plug in to get “really close” to a certain x-value. Below is a screen shot of the worksheets I used. Notice that we got into infinite limits without talking about them as anything different.

The next day, we talked only about finding limits on graphs. I gave lots of graphs, and we found the limit by reading what the graph is approaching on each side.

The last day we spent some time using “algebra tricks” to also find limits.

My goal was that by the test, students could find limits using whatever method(s) they prefer, having a deep intuitive understanding of what a limit is. And the scores were so much better this year!

As an additional test of how it worked, the day before the test, I gave the following as a warm up problem. I’ve done it every year, and this is the first year students could do it on their own, without any help! They did a great job!

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The day started with a great call to action from Chris Shore, focusing us on the Common Core Standards for Math Practice.

He also demonstrated the difference between traditional quizzing and some variation, but more on that later…

Chris’s key note focused our sessions around the SMPs in a good way. I gave a session on debate in the math class. Then, inspired by Tracy Zager’s TMC16 talk, I went to an elementary school teacher’s session on classroom discourse. It was a great session led by Mary Vongsavanh, and it got me thinking more about classroom discussion structures. Something she said that really struck me was “if want want them to do x, give them that experience”…as well as “The more you talk about something, the deeper your understanding.”

One take away is an extension of a Falsification activity I already do. Her example is below. Based on the picture, what is the rule for what fractions go inside the circle? You can “test out” the rule when you think you know it by naming a fraction that you believe goes in (or out of) the circle, and the teacher will confirm.

I have lots more to say about discussion and assessments and all that, but I’ll get more into that later. Thanks RSBCMTA for a great conference!

]]>Now, let me say that I avoided it for many years because it felt to me to be a bit too contrived or too geared towards younger students. However, I’ve come around on my thinking, and I use it in all my classes, 7th grade through Calculus, and I LOVE it.

Part of the reason I love it is that it forces me to thoroughly flesh out the new vocab word, for the benefit of my students.

Another reason I like it is that I’ve found two ways of structuring how we fill in the boxes (see below).

Let me just add that I hand out to students a page where this box covers half a page (pictured to the right). I usually have two boxes on the front and one or two on the back, depending on how many important definitions I plan to go over in the span of a class or two.

I then also display one box on the board when we are filling it out in class.

Now, on to the two ways I work through the boxes:

**Style 1:**

A nice, straightforward way to use the model is to follow it in a U-shape. It flows like this:

- Start in the top left corner: I give a formal definition.
- Move to the bottom left corner: as a class, let’s come up with some examples.
- Move to the bottom right corner: with a partner, come up with some non-examples (which I may have some students put on the board).
- End in the top right corner: As a class, we add any notes that came up during our examples of things we want to remember. I may also come back to this box in the following day or two to add more notes if necessary.

Here’s an example we did in my Integrated Math 3 (Algebra 2) class.

We filled in the examples with tables and sequences that had a quadratic pattern. The “Essential Characteristics” popped up as an animation after we filled in the bottom boxes. We later went back to fill in “Essential Characteristics” with notes like *forms a parabola when graphed*.

**Style 2:**

My newer use of the box (inspired by one of my awesome colleagues at TMC17) forces students to create good definitions based on examples. I fill out the bottom boxes and they come up with a definition that is “unbreakable.” It flows like this:

- Start in the bottom left and bottom right: Prepared examples or on the spot examples. Have students look at those, consider.
- Move to the top left box: have students pair share a good definition (that is unbreakable) and then discuss as a class. The teacher tries to “break” the definitions students come up with using examples and non-examples (possibly the ones already there), until the class comes to a solid definition.
- End in the top right box: fill in notes about anything important that came up in the “trying to make a definition” process. I may also come back to this box in the following day or two to add more notes if necessary.

This time, the thing I start with looks like this:

Here, students started to say things like “it goes through the vertex.” However, my horizontal and diagonal lines go through the vertex, too. So I was able to “break” that definition. Students love playing this “game” of making a solid definition, and in this class they eventually honed in on the point that it had to vertical and cut the parabola “in half.” The fact that it went through the vertex was a “cool side effect” as one student said, and that went in our Essential Characteristics box.

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