Building Community through Debate!

I’m writing this in the summer of 2020, knowing that my upcoming school year will start remotely and pondering ways to make sure community and relationships are leading our work in math class.

My current plan is to give students a “non-mathy” debate question to do in their breakout rooms every single day, before they work together on the math problems for the day. I want students to have a chance to talk and connect, as some of them might not know their classmates well, and I want to normalize having fun/being silly at times. Connection will be so important. So I’m thinking that every time we go off into breakout rooms for a significant span of time (10-20+ mins) to work on problems, I will instruct them to first have everyone share a response to the debate prompt I give them. Then they can transition to the math work.

Here are some fun “non-mathy” prompts I might use. This is by no means an exhaustive list. Shoutouts to Claire, Patricia, and Karla for helping add to this list!

  • What is the best movie/TV show to watch right now?
  • What is the strangest thing one of your family members did this week?
  • What is the tastiest meal you have had at home?
  • Who was the best middle school teacher?
  • What is the worst freeway in Southern California?
  • What is the best sports team?
  • If you had unlimited funds, what would be the best place to visit that you’ve never been to?
  • What are the best pizza toppings?
  • Where is the best place to get coffee?
  • Should we allow electric scooters on the sidewalks?

The prompts above are more open-ended. Claire, Patricia and Karla also shared some two-sided debates they had with their students, such as:

  • Twizzlers vs. Red Vines
  • Vans vs. Nike
  • Cheddar vs. White Cheddar (mac and cheese)
  • Mountains vs. Ocean (most relaxing? most fun?)
  • Starbucks vs. Coffee Bean

Students should respond using the “my claim is…my warrant is…” debate prompt.

I’m looking forward to joining different breakout rooms and getting a taste of their personalities through these small, silly debate moments.

Levels of Convincing

When we think about teaching students to be convincing, we sometimes hear about these three levels:

  • Convince yourself
  • Convince a friend
  • Convince a skeptic

Additionally, I think a lot about the three types of justification from Thinking Mathematically:

  • Appeal to authority
  • Justification by example
  • Generalizable arguments

To me, the first two justifications are what I see often from students that “convince yourself,” while generalizable arguments are more for convincing others.

I want to talk a little more about these ideas and levels–how my students and I develop these ideas through debate structure. Let’s take an example statement I gave to my students last year.

True or False: The sum of two linear functions will always be a linear function.

I gave this statement to my students as a bonus (not worth any credit) question at the end of a quiz. Some students gave an example or two and said it was true. Others jumped into writing a convincing paragraph (presumably without even trying an example). Yet others said it was false and moved on. Below are some of the highlights of our discussion about how to be convincing.

Level 1: Convincing yourself.

When I am first introducing debate structures to my students, a general basic level of convincing–convince yourself–is acceptable for warrants. Early on, I will accept most any warrant that is relevant. In the starting weeks of the school year, my goal is more for students to find their voice and feel included than critiquing the quality of their arguments. 

That said, as time goes on, I push for stronger arguments. When I gave the question about the sum of two linear functions, I was ready to talk to students about being more convincing. After I handed back the quiz, we talked about/looked at examples that either had some sort of appeal to higher authority 

“Mr. Luz said it was true.” or

“It is impossible to get a quadratic.”

or were a justification by (one) example.

“2x+3 + 4x+5 = 6x+8”

Clearly, each of these students had convinced themselves in the moment, and perhaps did not think we had to go further. Up until now, these were acceptable answers in our debates. Since we had not talked much about a quality justification yet, I shared these examples as a foundation step. It is important to first take a moment to convince yourself.  

However, from now on, convincing yourself is like a rough draft; it is the work you might do on a scrap paper to start yourself on the path of justification. I compared this first step to creating a quick outline for an English or History essay. Convincing yourself is an important organizational step before jumping into writing a paragraph justification, but it is not, in itself, enough.

Level 2: Convincing others.

I talked to students about the next step: to imagine a friend. Yes, actually picture someone you are talking to, and write an example and some sentences that you think would convince them that you are correct. To highlight some examples of this, I shared that one student had written:

“(x+1) + (x+2) is linear because the x-value is not squared. Since you are not multiplying, the x-values are not squared, or cubed, etc. So it will be linear.”

Another student wrote: 

“My claim is this is true and my warrant is because you are adding the values together. So you will still end up with a linear function. Ex: (3x+4) + (3x+4) = 6x + 8.”

One of my classes really took to this as a good way to remember this level of convincing. Picture a friend. Give them an example and an explanation (or in debate terms, a claim and a warrant) that you think would convince them. Many students had already shared arguments in our debate activities that could count as convincing a friend, but I wanted to be clear that this should be the base goal for us from now on.

I thought it was a good goal for the class to move toward this routine for Level 2 convincing, thinking about what to say to truly convince your friend. For some, it was going to be a good jump in growth for their mathematical thinking. For others, it would help solidify writing a basic justification, one that would get them some “credit” on future problems. 

A few others were ready to talk about the next level, but I talked to them in individual feedback at this point. I wanted to save a discussion on Level 3 for the next quiz, leading us into a discussion of proof and how to convince a skeptic.

Level 3: Convince a skeptic.

For me, level three is where a rigorous justification or proof is needed. If I have planted the seeds for debate throughout the starting months of the school year–through Soapbox Debates, Circle Debates, using Debate Cards, and putting good debate questions (sometimes for no credit) on quizzes, students are primed to talk about a solid mathematical argument (or proof) later in the semester. What’s more, it feels natural to talk about mathematical proof. It’s not like we’re introducing this idea of argumentation for the first time as we start a unit on proof. It’s a skill we’ve been building up regularly, a skill key to being a mathematician.

I’m not going to go into proof right now, many have published great ideas and research better than I could do (and this post is already long!). The only thing I can add at this moment is when it comes to being thoroughly convincing, enough to convince a skeptic, I tell students to think of a counter-example a skeptic might try to come up with (like what about if 0 is involved? Or fractions?) and include how that is not a counter-argument in your explanation. 


Further reading (partly so I can find these links in the future again):

Robert Kaplinsky wrote an interesting blog on Levels of Convincing.

Jo Boaler wrote a good article Prove It To Me.

NCTM has a great article on “Promoting Mathematical Argumentation” too, but you need a membership to access it. (March 2016, Vol. 22, Issue 7) 

An Exponential Debate!

As I have done in the past several years, I ended my recent unit on exponential functions and logarithms with a debate that involved exponential growth. The debate (featured in my book Up for Debate!), centers around the question:

Which is the greater threat: Overpopulation or Outbreak?

Students work in groups to look at data on the population of the earth in years and the spread of Ebola, in months. They use the data to create exponential functions to model the exponential growth (if unimpeded) of each, and then they use those models to make predictions about overpopulation and outbreaks. Each group then provides a summary what they think is the bigger threat with the class.

I was so excited to be doing this activity again this year, as we are in the middle of an outbreak (Covid-19), and students have a more tangible sense of the spread and the consequences.

What I find so interesting is:

In past years students almost always said that overpopulation is the bigger concern. They stated that an outbreak can always be controlled by modern medicine and quarantining and other measures, but they argued that our population will continue to grow and there isn’t an easy way to slow that.

However, this year, the majority of the groups cited an outbreak as the bigger threat. They cited closures and death rates and stay-at-home orders, as they’re experiencing these first-hand.

It was clear the current climate weighs heavy on their minds, and I look forward to debriefing with them further next class!

Teaching Online

With school closing, we had to suddenly jump into online learning with just a few days’ notice. Like many of you, this is not familiar territory for me, but I wanted to share a few ideas I’ve learned/been thinking about in the past week.

Mainly, I keep hearing over and over again from those who have more experience with this that we need to take things slower and more relaxed. In a transitional time like this, we need to take it easy on ourselves and our students. We need to model calm. We need to be ok teaching a little less and giving students more time to complete less work. As I expressed to my department colleagues in our switch to online learning, my two goals for our math classes online are:

  1. To have students flex their math muscles – This can take many forms–maybe we pause the curriculum for the first few days and play with Desmos Marbleslides (check out the Marbleslides Challenge!) or check out What’s Going On in This Graph? I do want students to do some mathematical thinking each day (or even every other day) to provide some routine, some academic struggle, and some focus.
  2. To give students a sense of community – I’m guessing many of our students are feeling bored and/or isolated. I imagine they will be excited to see smiling faces (ours and their classmates) as well as just to hear from us. So having ways to meet over video, or have groups work on a project, or just check in over email, can help keep students connected to the community of their school.

If we meet these two goals in each of our math “lessons,” then we will have done our job for the day.

Additionally, this could be the time to try to change a few things, get creative. My middle school colleague Jill created a multiple choice check in/math scavenger hunt with slides (like the two below). Students will work out the question, go around their house to find the object that matches the answer they chose, and hold it up to the camera on Zoom.

Screen Shot 2020-03-18 at 8.04.37 AM

I love that Jill has included an element of #MathMovement!

The number one question I keep hearing is: how do we do a major assessment? This I’m still pondering. I’m hoping to give some “quizzes” in the coming days and come back to share out how/what worked/didn’t work…

Stay safe, calm and connected everyone.

Explore Math with Prompts – Semester 1

A few years ago, I learned about Sam Shah’s wonderful Explore Math project. (Sam wrote a great blog post on it.) The goal was to have students explore some math outside the curriculum so they could begin to see some of what math could be, not just the content standard they have to learn this year. I think it is SO important for students to see that math is more than graphing a line or factoring expressions. Using Sam’s template, my students had a menu of options to choose from (reading an article from the NYTimes about math, exploring Visual Patterns, and many, many more). For each Explore Math assignment, students would choose one option, learn something about math, and then write a paragraph reflection on it.

I wanted to keep it manageable and low-stakes. I had them do about one a month (really about 3 a semester). And most students would get 90% or higher just for writing a decent paragraph. I really liked that it was low-stakes, and because there was not pressure to “perform” in some way, students had fun with it. Many students would talk about it in their end of year surveys as the time when they “actually had fun with math” or learned that “math could be cool.”

While I like the choice option, this year, my goal was to narrow the focus a little for each assignment. I’m so happy with how it went! Below is what I did for Semester 1.

*One caveat: students were able to substitute the focused assignment for anything from the full menu if they really wanted to. So there still was some choice, though nearly every student stuck to the focused assignment.

Assignment #1: Who is a mathematician?

To start the year, I really wanted to challenge students’ pre-conceived notion about who can and does “do math.” Are “math people” only those who make a life-long commitment to the study of math? I think not! We are all mathematicians in our own ways! So inspired by Annie PerkinsMathematician Project (mathematicians are not just old, dead white dudes), I asked students to find a mathematician to learn a little about. Their ultimate task was to write one paragraph about what makes this mathematician interesting and include a picture. I asked about what makes the mathematician interesting specifically to avoid a paragraph of random facts about the date of birth, family facts, etc. I wanted students to get to the heart/excitement of who that person was.

I asked students to get approval from me on the person they chose, so that I could make sure no two students from the same section researched the same mathematician. What I was pleasantly surprised by is that, in addition to many wonderful historical mathematicians to choose from (I shared this website as a resource), students started asking if they could interview a family member (who perhaps worked in a STEM field), a favorite STEM teacher of theirs on campus, or a friend of theirs who they thought was a mathematician (perhaps someone in an AP math class).

What resulted was a wonderful wall display of mathematicians who were female, people of color, currently working in the field, teachers on campus and students. We had this wonderful collection of photos and “interesting paragraphs” on the wall near the pencil sharpener that students could look at anytime they walked by.


And thus we are able to constantly talk about and remind ourselves that everyone can be a mathematician. That math isn’t mystical. It is something we all do!


Assignment #2: What makes math cool?

I wanted student to look around and find a concept (even if they don’t fully understand it) that shows some cool math. My instructions were as follows:

“OK, so you learned about a mathematician. Now what? Now it’s your turn to explore some cool math! What’s a golygon? Have you heard of sexy primes? The Hilbert problems? Fractals? The different sizes of infinity? Or do you have a cool math challenge you want to solve and discuss?

Your second task is to read about some math topic you never heard of before and write a short paragraph about it. Be sure to include visuals/examples! Alternatively, you may solve some math challenge we have introduced in class (or one you find on your own) and write a paragraph about it.”

Again, students submitted a paragraph and included some sort of visual. I was so delighted by what I saw. One student stumbled upon the problem of the Bridges of Königsberg, one of my favorite problems, and she started playing with it for fun. Another student taught herself (and me) what the math was in the movie Good Will Hunting. Being her favorite movie (and to her surprise, one of my favorites too!), she found math in the art world that she latched onto and ran with.


Assignment #3: How does your family do math?

For the last assignment of the semester (and I purposely gave this out before the Thanksgiving break in case students wanted to do it then!), I wanted students to bring the ideas that (1) they are mathematicians and (2) math can be cool to their family!  Here were my directions:

“Wow! You’ve explored some cool math! You learned about lots of cool mathematicians. One more thing to go…What is your family’s take on math?

Your third task is to do some math with your family (#FamilyMathNight). You can define your family however you want. You can do any kind of math you want. But you need to be explicit about it and take a photo! Think about all the ways you and your family already do math: calculating costs, counting items, using fractions in cooking, figuring out a percent of a cost (tip or sale), etc. Use something like this or a challenge problem or something original. You get to decide!”

Students again had to write one paragraph and submit a picture with it, but they could interpret the assignment in many ways. Some students chose to do a challenge problem we had done in class or that they found on their own with a sibling or parent. Some tried to teach their family a topic from their current course over Thanksgiving dinner!

This turned out to be my favorite of the three tasks, as I learned so much about students and their families, and I got to laugh and ponder with students as they turned in the assignment. Many students talked about how frustrated their parents got with what they thought was “simple math” and reflected on how they never realized how far they have come in math (these are my Calc and Pre-Calc classes). Some have gone further in math than their parents were ever able to or interested in. Some talked about how their parents seemed to have forgotten everything. I think it was shocking to many students how wide-ranging their parents’ reactions were. It built confidence in students to see how some adults in their lives were so math-averse, while (with a growth mindset and a new idea of who could be a mathematician) they didn’t see math as something to run away from.

Overall, this was a fascinating explore math adventure for me and my students. I’ve learned so much more about them and their families, and I’ve seen them open themselves up to learning math so much more. I’m excited to keep this up and am pondering what to do next semester…any ideas?

Debate IS Learning

I was astounded recently when the current Senate Majority Leader shared that his mind was already made up in the matter of impeachment, before the trial has even begun. (You can read more details here.) I don’t want to get too political in this post, except to say that the point of a trial is for the jury to hear both sides, possibly learn some new information, and then to make up their minds. So, it shocks me, as a life-long learner and student of debate, that a national leader outwardly admits to being fixed in his thinking.

Where is the growth mindset?

This is why I love debate and encourage my students to debate whenever we can. It’s not about winning and losing; it’s about hearing different opinions on an idea and learning. I fear we have become so polarized as a nation, that winning is more important than understanding. I got hooked on debate when I was in high school because in hearing a debate (as a competitor, audience member, or judge) I learned SO MUCH. That’s why I enjoy bringing debate into math class. I am a life-long learner, and I want my students to be life-long learners, too!

Debates can be exciting because there’s a competitive part to it, a facet of debate that keeps the audience’s attention probably more than a speech full of facts. I love listening to IQ2 Debates as a podcast for this reason. They debate so many topics that I know little about or have little interest in beforehand (Healthcare, China, Net Neutrality, College Athletics), but the debate format draws me in. I want to learn more and be convinced. I want to hear what each side has to say as their most persuasive arguments. I want to find out how the audience voted. I get sucked in and can’t help but learn about new topics.

Shame on our national leaders for not modeling an open mind to learning. Our students deserve better. They need to learn to have an open mind, that life is all about learning, that there are so many perspectives on every topic. This is why we MUST #debatemath!


For more info, check out my book, Up for Debate!

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:

Screen Shot 2019-11-13 at 7.45.02 AM.png

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!

Screen Shot 2019-11-13 at 7.47.24 AM.png

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.


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)



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!