Function Yoga!

I wanted to take a moment to shout out two of my colleagues (Julian Rojas & Gemma Oliver) for adding a twist to the function dance activity that I really enjoyed. Some of you have done the function dance before, where students use their arms (and sometimes legs) to make the shape of a parent graph, like in this diagram:

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Then students practice transforming functions by moving around as the function would (For instance y=|x| and y=|x-2| could be two dance moves. The first requires students to put their arms in the air in a V-shape like the absolute value graph. The second tells you to keep the shape but to step to the right 2 spaces.)

My colleagues did a cool twist on this by changing it from function dancing to function yoga!

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Students acted out the same routine, but in a slower stretch-y way. It was so great and more manageable. My quick thoughts on this:

  • The slowness allowed students think time to slowly move from shape to shape (less rushed than in some of the dance versions)
  • The movement and stretching was great for the class.
  • This could easily be a daily warm-up for the rest of the week (or the year!) to get students up and moving at the start of class while also reviewing math concepts.


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DebateMath: Unit Circle Trig

I’ve had a few requests for more examples of #debatemath problems/questions. So from time to time, I’m going to add a few screen shots of things I’ve done around a topic in past years. Here are four examples from a Pre-Calc unit on Unit Circle trig, radians, etc.

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Clothesline Math!

Sometimes you learn about a teaching tool or technique at just the right time. That’s what Clothesline Math was to me in the past week. I had known about Chris Shore’s work with #ClotheslineMath (think open number line) for a while now, but I only knew the basic number line stuff he had started with. I happened to attend his session at the RSBCMTA #FallforMath conference, learning how many new ways he has taken this concept (Clothesline Math grew up!), and it was just what I need in both of my classes this year (PreCalc and Calculus) at that moment.


For Pre-Calc:

I was just about to start exploring the Unit Circle with my students. I know they always struggle with radians and making sense of the fractions. So, I started with a basic fraction Clothesline math for 5-ish minutes at the end of class one day:


It was amazing. This short number line actually took more than 5 minutes, with rich discussion among groups. It was amazing how many strong math students in Pre-Calculus wanted to say 1/2, 1/3 and 1/4 were equally spaced (probably due to the 2,3,4 in the denominators).

The next step was to make a Double Clothesline (!!!). We started making the top one in degrees, going from 0 to 360. Students had to put the other “common angles” on the number line proportionately. Then, we started a second number line below that in radians, going from 0 to 2pi. We talked as a class that 180 degrees would be where pi (or 1pi) would go, and then students had to figure out the rest using fraction reasoning. Here’s what it looked like by the end:


We did not finish because I did not realize how long it would take students. So we came back to this and re-did both number lines again at the start of next class. Worth. It.

In the following days, I have never had students so solid at reasoning through what fraction of pi each of the angles is. The manipulation of the clothesline, the time to really reason through it on their own, and starting with a horizontal number line before moving the fraction reasoning to the circle all contributed to making this a worthwhile use of time. I will never teach the unit circle/radians again without Clothesline Math!

For Calculus:

We had just started limits, and I spend the entire first day just having students make tables to see what the y-values are approaching on the table. So if we are talking about the limit as x approaches 4, for instance, students would make a table with an x-column including numbers like 3.8, 3.9, 3.99, 3.99 as well as 4.1, 4.01, 4.001, etc. I really want to emphasize how we are “squeezing” in around a number.

Students do pretty well with these tables. However, many always struggle when we have a problem where we are finding the limit as x approaches 0. Students usually make a table with 0.1, 0.01, 0.001 just fine, but on the other side of 0, they choose -0.9, -0.99, -0.999, etc. They get so into the habit of .001s and .999s with other numbers, they struggle with things reversing in the negative numbers, and being especially unique around 0.

Cue clothesline math! After this first day of tables, we started the next day with a short clothesline math activity. Students put some whole numbers (0,2,3,6) on a number line. Then organized the same fractions that I had used in PreCalc (1/2, 1/3, 1/4). Lastly, I asked them to put the following on a number line:

0, 1, -1, 0.1, 0.5, 0.9, -0.9, -0.4, -0.001

They got it. They just needed the time to refresh their understanding of the number line and strengthen their number sense. I also think some of them just needed permission to draw a number line anytime they want in the future. (This activity helped make drawing a number line seem not-so-juvenile/really important.)

Some quick tips:

  • I actually started clothesline math in each class with a “easy” set of numbers: 0,2,3,6. It was a great “easy” way to introduce it and help students understand the idea of scale/proportionality. Plenty of students struggled at first with just these numbers.
  • I made the numbers on the clothesline by folding over index cards. Quick and easy.
  • Each round (each set of numbers) I had one group come to the front (a different group each round) to put the index card numbers on the clothesline. The other groups were all working on small white boards, drawing a number line and discussing with their group where the numbers went. Thanks to Chris Shore for this pro-tip! (See example below)



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Summer Reading

I had a quite relaxing summer this year with not many plans (no traveling for me). So, I took the opportunity to read a few books on my list. Below is a picture of them all (the last two were re-reads for me).


A few quick take-aways:

Stop Talking, Start Influencing (the third book in the top row) had the biggest impact on how I deliver information, both in the classroom and in presentations. It is full of brain science on how to make messages easy to understand and easy to stick. (I considered it a nice follow up to Make it Stick!)

Grading for Equity was an easy read that I couldn’t put down. If you’ve ever considered or tried Standards Based Grading or other alternative ways of grading, this book will probably resonate with you. It addresses the biases in grading and grade books and how we can work to be more equitable.

Crucial Conversations was a great read about how to have difficult conversations with other adults (colleagues, administrators, partners). I really appreciated the “crucial” part of the title, as the author pointed out that these tough conversations need to happen (don’t avoid them) because they are crucial to healthy relationships.

And I recommend Lani Horn’s Motivated to teachers so often that I thought I would re-read it more carefully!

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Making Math Emotional

This post is my response to the prompt for The Virtual Conference on Humanizing Mathematics:

How do you highlight that the doing of mathematics is a human endeavor?

Math was a pretty bland subject for me in school (K-12). It wasn’t until I got into more abstract math (Calculus and beyond) that I had some really engaging teachers and classes that blew my mind and got me excited to do mathematics!

I now try to spread that love of mathematics with my students (grades 6-12). I also want to make math feel more personal and alive. I want it to feel like an emotional, human endeavor. (Think of how emotional students can get about a novel or piece of music. Why can’t the same be true in math?!)

The two big parts of my classroom that help with this are:

  1. #DebateMath! Those of you who know me, know I LOVE to debate in math class. You can find more about math debate routines on my website (and in my upcoming book Up for Debate! this November!!). What I love about having a question (or several) that involves students debating each day is that doing math becomes subjective. Doing math in my classroom is not focused on the objectivity of answer-getting. Instead, students discuss and debate their methods, their reasoning and favorite parts. Students are talking about math, sharing opinions and ideas, and they often develop an emotional attachment to the math they are doing. Whether or not the math we are doing has a “real-world” context, students are continuously able to find personalization in the math that they are doing.
  2. Journals! (lovingly borrowed from Cindy Reagan) I have long wanted to have more opportunities for students to write about math, but I struggled to find the right fit for myself and my students. Then, I was blown away by Cindy Reagan’s wonderful presentation at TMC18. Through her presentation and blog, I was empowered to start journals in math class. In the past year, journals have significantly changed the way my students interact with math. Though they initially struggled/resisted journaling in math class, SO MANY students grew to LOVE journaling. It was a chance for them to look back at their work in the past week or two and talk about their thoughts and feelings. Students quickly seemed to form emotional connections to certain problems and methods. I gained new insights into my students’ thoughts and reasoning, and they got to privately share some of their insights/hopes/frustrations to me in writing.


-For more info on debate in math class check, sign up for my monthly newsletter here and check out the PBS video of my classroom!


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The “Most Important” Project

I have been struggling to come up with an engaging summary project for my Calculus and PreCalculus classes. I’ve tried a few things in the recent years, but had not yet found just the right fit. Then I stumbled across this tweet a few months ago:

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Larissa shared a link to her version of the project. And now, I have my own version: link.

I got a copy of Margaret Wise Brown’s The Important Book on Then, to introduce students to the project, we read the book in what my students called “kindergarten style,” meaning we passed the book around, each student read one page and then showed the associated picture(s) to the class before passing it on. I must say the book is adorable, and my students enjoyed the 5 minutes we spent reading it together. Each page describes a different object, such as:

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I then told the students that we were going to make a book of the important topics from this school year. Each student can choose to work solo or with a partner, and each student or pair will produce two pages of the book: one page will be an important topic from this past year, one will be an important topic we did not get to (polar coordinates, matrices, etc).  I wanted students to both summarize an important topic from the year and learn some math on their own!

Additionally, on the back of each page, I asked students to create, solve and explain one math problem for each of their two topics. I asked them to choose a problem carefully, as we want a challenging problem that will point out different key components of the topic.

The students have just started turning them in. I will show some examples here soon. I’m excited to put them together to make a summary booklet that we can all look through as our closing activity for the year!


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NCTM SD 2019 Reflection

In early April, I spent a week in San Diego for the NCSM and NCTM conferences. It’s taken me some time to gather my thoughts in writing, but here are my big takeaways from the learning!

1. I want to video (myself and others) teaching..a lot!

Dr. Ilana Horn was one of the first speakers I saw. I loved hearing about her work with MfA LA around growth for experienced math teachers. She talked about how a teacher would pose a question, that teacher’s classroom would be videotaped, and Dr. Horn’s team would share specific video clips that are related to the question asked. I especially liked the idea of targeted video study–just looking at short clips that are directly related to a teacher’s guiding question.

I felt a similar message in part of Steve Leinwand’s talk on PD. He talked about how teachers can find PD unhelpful or not useful. And among the strategies he talked about, videotaping was one.

The one-two punch of two great math leaders expressing the importance of videotaping our teaching really stuck with me and make me want to incorporate that as a regular habit for our department next year!

2. Creating inclusive math spaces is imperative. 

It really felt that NCTM was challenging us all to think deeply about diversity, equity and inclusion in the math classroom this year. I started NCTM with a pre-conference day on social justice math. We talked openly about deficit based thinking and explored the types of problems that embrace diversity and social justice issues in math class.

Two wonderful educators of color gave the opening and closing keynotes. I was blown away by all they said. Additionally, Chrissy Newell gave a great ShadowCon talk about gender diversity in mathematics, focusing on her (and her daughter’s!) #mathgals project. Dr. Talithia Williams ended the event with her powerful personal stories about being a woman of color and pursuing a career (and advanced degree) in mathematics. I can’t wait to read “Power in Number.”

3. Students (and adults) need play in math.

Two Chrises–Christopher Danielson and Chris Nho made me think deeply about the power of play in math. Christopher Danielson led on session on categorizing hexagons. My table had such a fun time coming up with names and rules, a great way to develop geometric definitions with playfulness. Additionally, Chris Nho challenged us to think: if adults continue to read outside of school with book clubs, why don’t we continue math with “Problem Clubs.” Why are kids the only ones allowed to have fun?

4. #DebateMath is everywhere!

Tweeting was abounding before and after my session on #debatemath! We were trending on Twitter!


Twitter hype aside, I really did see debate math everywhere. So many sessions talked about the need for student discourse and/or the importance of developing argumentation. Others talked about the need for clear routines for students to think critically and debate. I went to an excellent presentation by Mario Valdez where he brought his students (how cool!) to show his routines for his 5th grade students to explore and discuss challenging math problems.


Thanks to everyone who came out to the #debate math session!

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