Why Another Podcast?

For those who haven’t heard, I (with my collaborator Rob Baier) started the #DebateMath Podcast recently. The first episode two episodes are scheduled to air at the end of January 2022!

I wanted to take a moment to share some of the thinking behind the creation of this podcast, explore why we really wanted and needed to do this. There are four main reasons behind this podcast.

  1. We need more than a speech (sometimes). I love attending conferences and listening to speakers, but so often we just hear one side of an idea. Sometimes, I want (and need) to hear multiple view points on an idea to really understand it, to think about the implications and consequences. I don’t want teachers to jump on a new trend just because one speaker sounded sexy. I want teachers to hear multiple perspectives as they ponder what is best for their classroom and make even more informed decisions. Additionally, when we feel strongly about an idea, in this current climate, it is important to be prepared for push-back. Hearing two sides in a debate can help us prepare for the counterarguments we may get from colleagues, parents, admin, or students. Also, through the activity of debate, listeners and speakers are engaged because there’s a “competitive” aspect to it. You want to know who will “win” or be most convincing. So you can’t help but lean in a little more.
  2. We must explore nuance. I see so many ideas passed around on social media, and initially they sound good. However, we don’t always have the time or space to think through all the consequences, to think through all the nuance. I think we need to take more good ideas with a grain of salt. Also, when something is working well for someone else, there are so many things which led up to that new pedagogical move, that new technique or activity, that could be easily overlooked. We need to unpack all the layers about what made this particular idea work at this time and place, and how would it work for us in our own, unique (different) setting.
  3. The voice can provide more colors of tone. In listening to podcasts and recording our own, I’ve seen how much more can be expressed in the tone of an argument through the voice than in writing. There are good ideas shared in writing both in blogs and on social media, but without the tone (serious, sarcastic, or whatever), the message can easily get confused. Listening to another human explain in the audio (plus there will be YouTube videos of the episodes as well) adds so much more detail and clarity. I’ve read about how humans have evolved to think socially, and listening to a podcast provides a version of this.
  4. Podcasts are powerful. I listen to so many podcasts, when I’m driving to/from work, when I’m on the treadmill at the gym, when I’m alone at home and cleaning up, etc. It’s a great way for me to listen and think while doing simple tasks. We can listen to them in so many places and learn so much along the way. And podcasts can be as long as they need to be. We are not making TV shows that all have to be a certain length. Some of our debates may only be 20 minutes, some may be longer. Rob and I have been doing our best to give enough time to flesh out the sides of a debate, but not to let it go on too long. Additionally, podcasts can stay relevant. We are recording them once a month, and they can reflect current trends and topics that are up for debate!

Our goal is to release one episode a month. Rob and I are doing this in our spare time as educators, and we want to take time and care with each episode to give everyone something great to listen to. It is a lot of work to coordinate people/teams on two sides of a debate, give them time and help to prepare for the debate, and then actually do a live recording. But we enjoy it! It just will take us time to do one episode at at time.

I have to note that it has been a pleasure having chats with educators who are potential guests and asking them what they are passionate about. It has been wonderful to hear topics come up that I never thought some people were deeply pondering. And I’m excited that these passions can be shared with listeners on the podcast.

We are always looking for new guests and topics, too. We are slow moving, but anyone can suggest a name or topic idea on the website: debatemath.com.

I am so looking forward to putting this out in the world soon. I hope my fellow math educators will enjoy. Hope you subscribe ASAP on Apple Podcasts, Spotify, or wherever you listen to podcasts so you don’t miss one episode! And follow the Twitter handle @DebateMathPod and #DebateMathPod so you can get all the updates, join in the follow up conversations, and vote for your “winner” of each debate!

ps. A special “Episode 0” will be the first one, airing January 20, 2022. Enjoy!

pps. When I listen to podcasts, I usually play them at 1.25 or 1.5 speed to get through them a little faster. I’m not sure how ours will sound sped up, but it’s an option for those who are short on time!

CMC 2021

Waking up at the start of Day 2 of CMC – South 2021 and my mind is abuzz with so many great ideas I learned yesterday. I need to write some things down. I was first so excited/overwhelmed just to finally be in person with wonderful math teacher people again. It was emotional for me. I loved it! Some sessions that stick out right now:

1) Mike Flynn’s session on being a teacher and advocate really stands out the most. I think a lot about the current political climate we are in and all the school board meetings in the news right now with lots of aggressive and angry speakers. How can we, as teachers, be effective advocates for our students, our curriculum, etc?

Mike talked about the roles in advocacy (Agitator, Innovator, Orchestrator) and how each one is important. He stressed how we need to have a team (an advocacy team) that works together. The superhero mentality (one person making all the difference) does not lead to change. He also talked about the mix of advocacy and inquiry. There need to be times that advocates listen, observe, ask questions. We need to hear what people are thinking and build relationship (just like in our classrooms!).

2) Dr. Cathery Yeh gave a talk titled “Mathematics as a Human Right,” which immediately drew me in. Early on, she had a slide that said “All students should have access to rigorous mathematical learning that respects and honors their identities and ways of knowing.” Boom.

On top of that, one big take away was that Dr. Yeh asks students:

  • What are your access needs?
  • What supports your learning?

Then she asked us to take time and write or discuss with a neighbor about our access needs (us adults!). I really struggled with this, as my table talked about, because I don’t think anyone had ever asked me this. In the American culture of “power through” and “figure it out” for yourself, I’ve never been asked or took the time to think about what my access needs are for various things. It lead to a great discussion with the teachers I was chatting with!

Debate Questions on Assessments

Since debate and argumentation are a regular part of my math classes, I also find it important to incorporate them into assessments. That is, I want students to create arguments on tests and quizzes. I’ve played with a few varieties over the years, but what I use most often (and what my students have come to expect) is a question like the following on every test.

This question is from a test early in the year in PreCalc. I don’t intend these questions to be too involved in the beginning of the year. I mostly want to see if they can give a decent argument, and I spend a lot effort giving written feedback on their arguments. My goal is to make clear what I expect on future tests (when I will be a little pickier in my grading).

Question B is one of my favorites to give feedback on and talk about with the whole class afterward. It is pretty clear to all students that the angle drawn is way more than 200 degrees. However, this is where I really get to talk about a quality and convincing argument.

Three typical pitfalls include:

  • Assuming – every year I get students who say that the pictured angle is 300 degrees because that is roughly what matches a point on their unit circle diagram. These students are only considering the key angles that our class has memorized coordinates for (every 30 or 45 degrees). My comments include the question: what if this is 302 degrees? They are correct that the angle is not 200, but their argument is not necessarily a true statement.
  • Vagueness – other students correctly say the angle is not 200 degrees, but then say vague ideas, presumably to have something written but possibly unsure how to be convincing. This includes statements like: This angle is not 200 and my warrant is that it is big and probably bigger than 200. A student who says this might have been able to give a convincing argument but was never pushed or had it made clear what was expected. This is where I like to talk with the whole class about precision.
  • Missing Connections – This last group is the one I really want to talk about with the whole class. This includes students who might say something like: This angle is not 200 and my warrant is that it is in the 4th quadrant. This student is SO CLOSE to having a solid argument (and at the beginning of the year I may give them credit, but with lots of comments). They are just missing a connection. I ask what about the 4th quadrant makes that angle not 200?

There are definitely some students who give a good argument from the start, and I show those to the whole class as well. I really like to show when someone says something like

My claim is that angle is not 200 degrees and my warrant is that it is in the 4th quadrant and angles in that quadrant are between 270 and 360 degrees.

This gives me a great contrast to show the students who have a missing connection to see what exactly they were missing.

*One more note about these argument questions is that they really help emphasize that there is not just one correct way to answer a problem. You don’t have to talk about the 4th quadrant to be correct. Other students might say:

  • That angle is not 200 degrees and my warrant is that 200 degrees is in the third quadrant and this angle is not.
  • That angle is not 200 degrees and my warrant is that it is greater than 270 degrees.

And this is just the start. I love building from here!

Igniting a Softer Side of Math

I’ve been privileged to be invited to give some Ignite talks (5 min talks with 20 powerpoint slides automatically advancing every 15 seconds) at conferences in the past two years. I have recordings of both, and I wanted to put them here to refer back to. Both are part of my journey into seeing math as more than a place where we focus on right answers, where we embrace ambiguity and the human side of learning math.

The first one “Math for Healing” was from late 2019.

It was given at the NorthWest Math Conference and then at CMC South (where the recording was made).

The second is “Non-Binary Math” from early 2021. It was given at two virtual NCTM conferences.

In Praise of Warm Ups

I haven’t blogged much at all in this year of remote teaching, but I’m getting back at it now!

I’ve done a lot of work with teachers in the past few years, and one thing that keeps coming up is how much the warm up activity can be a game changer for classes.

As teachers, we can easily feel so overwhelmed with all the content we *must* teach, all that we have to somehow squeeze into one school year, that it can be really difficult to think about the other things you want to focus on. This includes the standards of math practice (persevering, problem solving!), number sense and estimation activities (Clothesline math, Estimation180), and stats and data exploration (What’s Going On in This Graph?), not to mention just having time to Play With Your Math. And of course, there’s always a need to find time to DebateMath!

So how do we fit it all in? How do we help develop mathematical and problem solving skills? How do we make time to help students see that math is more than this year’s curriculum?

My solution is to use the warm up time for this. I take 5-7mins (sometimes a little more or a little less) at the start of each class to do something that is outside the curriculum. Once a week, we notice and wonder at a NYTimes graph. Once a week we have a short debate or solve a math riddle. Each day, we start by seeing math as interesting, playful, and/or relevant. We might start an interesting puzzle or discussion that we can’t finish, but the rest is left for students to explore as they want to. Math might spill over into their lunch or family time later that day or another.

Not only does it get students wanting to get to class on time and get started, but it provides a joyful moment to start the class. It also shows students that math is not just about learning to use the quadratic formula. Students always write on their end of year surveys that those 5mins of “outside the box” math really changed the way they see math class. They see math as interesting and important.

And as a bonus, I see the students being more resilient and playful in the rest of the class. When they hit challenges in the curriculum, they approach them as puzzles. “Let’s see what we can figure out,” is a phrase I often hear.

“Rumors” Virtual Style

This is a guest blog, written by my colleague Gemma Oliver.

While preparing for the beginning of this school year, I was struggling to incorporate ways to get to know my students, let them get to know each other, AND leave enough time in our schedule to get through the material for the semester. So much to do and so little time! Giving space for the students to share out about themselves and bring their personalities into the classroom was not something I was willing to sacrifice. In the end, I chose to adapt an activity that I have seen Chris use on campus in the past called Rumors. I was so happy with how well it translated into the remote world!

On campus, the game goes a little like this:

  • Everyone is asked to write down a response to 2-3 prompts on a notecard (Rose/Thorn, burning questions, etc.)
  • Then, everyone is asked to stand up, walk a few steps, and find a partner.
  • Once everyone has a partner, they have two minutes to both share out their responses and swap cards.
  • Once the two minutes are up, they will have to take their first partner’s card and find a new partner.
  • For the second round, they will each share out the responses of their previous partner instead of their own and, again, switch cards. This could continue on for a few more rounds.

One of the main reasons I like this game is because it allows students time at the beginning to think about what they want to share out and prompts them to make a “cue card” for themselves. This provides great structure and processing time for students who are more hesitant to talk in class and students who would otherwise talk too much and take time away from others. Being that Zoom already makes it more difficult to speak up in class, I wanted to provide plenty of structure at the beginning so that we could get to know each other and begin building the rapport necessary to comfortably engage through Zoom.

Here’s how I formatted the game in the remote setting:

  • As students were joining the call, I had a slide screen shared with the prompts for them to respond to.
  • When they were ready, I sent them into two-person breakout rooms with the instructions to each share out their responses and take some quick notes on what the other person’s responses were. (I used timed breakout rooms and gave them three minutes instead of two to account for the extra time needed to write down some notes.)
  • When the three minutes were up, the breakout rooms closed and I randomly assigned them to new breakout rooms immediately after.  
  • After three rounds, I brought them all back to the main room and asked them to write at least one thing that they had learned about someone else. As their comments were coming in, I read some aloud and oohed and aahed at the glorious facts I was learning about them! *This part was especially important to include since we lose the ability to “eavesdrop” on multiple conversations when we use Zoom. Without this, a lot of the information would have been lost.*

So! There are a couple of things that I loved about this. Hearing students tell me about themselves is great, but hearing them tell me exciting things about each other? That was some real heartwarming stuff! You could tell in the chat how excited they were to tell me about their classmates. On the same note, they seemed more comfortable adding something to the chat when it wasn’t something about themselves. This did a great job of taking some pressure off of them on the already stressful first day. I was also able to learn a TON of things about my students in about 10 minutes. I love how this activity transitioned from one on one conversations to a whole class experience. It allowed us to have more personal connections while also giving us a chance to learn about our entire class without taking up too much precious class time.

Although there is still plenty more for me to learn about my students, I think that this activity more than served its purpose of breaking the ice and forming some connections amongst students. In our current state, connections amongst students is about the biggest win I can think of!

Huge thanks to Gemma Oliver for sharing this!!!

Debating Math Remotely

I have been struggling to imagine ways to have rich math discourse and debate as my school plans to start the year remotely. I’m actually happy to start remotely, rather than in a socially-distanced classroom, because I don’t know how to have discourse with people 6ft apart?!

During classes, I think some of the time we can use routines similar to when we are in person, and other times we will try out new ways of interacting with technology. Below are a few ideas on my mind that I hope to try out. I welcome any other ideas!!

*Some details: I’m working with Zoom. All students have a laptop provided by the school. We will have 90min blocks every other day for our classes (only half the classes meet each day).

Similar to In Person

  1. Whole Class Debates (Soapbox): Similar to the Soapbox debates I do in class, I can replicate this on Zoom, where one student at a time un-mutes themself and shares a claim and warrant. I will continue to use resources like What’s Going On in This Graph? as start-of-lesson debates. Because Zoom can be awkward to know when it is a good time to jump into a conversation, I will call on students one at a time.
  2. Small Group/Breakout Room Debates: I can easily send students into breakout rooms to discuss (probably in Soapbox Debate style) a given prompt or prompts. I can see them speaking up much more easily in groups of 3-4. I wrote a blog post a few weeks ago about how I plan to have breakout rooms always start with a short, “fun” debate.

Unique to Remote Teaching…

  1. Google Slides: I’m excited to use Google Slides (and Google Docs) as a space for students to record their responses/ideas. I attended a webinar lead by Mike Flynn where he talked about having one long google doc for an assignment, where each breakout group has one (or more) slides they fill out with their responses. I like this a lot because when in breakout rooms, students can’t hear anyone outside of their group, but in person, they can overhear some of the groups nearby. Google Slides allows students to peek at what other groups are doing for ideas and inspiration, a virtual way to “overhear” others.
  2. Kialo: Based on a recommendation from someone on Twitter, I’ve started exploring the website Kialo. It allows students/groups to make a nice tree diagram to organize arguments. I was thinking this could be another way for students in small breakout groups to record their ideas, their initial thoughts, and then talk together and decide what to share with the whole class. They can use the Kialo diagram of their ideas to explore the strongest argument to share with the class.
  3. Desmos (Activity Builder): Last spring when we went remote, I started using Desmos for some of my assessments, and it worked well. I like to put in boxes for students to explain their answer or create an argument (claim/warrant). I’m hoping to make a few Desmos ABs in the coming weeks for my classes to discuss and debate. It’s great that Desmos has the option of allowing students to see what others have written after they submit their answer for a question.

EDIT 8/19/20: Thanks to Anna Blinstein and Karla Doyle for reminding me of two more online resources, especially for asynchronous debates:

  1. Padlet: Students can use Padlet to leave a comment (or agument), like they would put post-its on a poster. They can also leave a response to another student’s post. This is a great, low-stress way to have students share out in class.
  2. Flipgrid: I only used Flipgrid once last semester, but it was a great way to have students record very short videos of themselves, sharing a response. My strategy was to have every student upload a video with their argument (claim/warrant) and then respond in video to one other student (preferably someone who did not have a reply yet).

Roles for Speaking…

One last facet I want to add to classroom discussions is exploring what “roles” students take on during a discussion. Inspired by my wonderful colleague Kathleen Niles, I want to talk openly with students about some of the ways people participate in a discussion. Students should still share out their arguments (claims/warrants) and respond to each other, but I want to put a name to some of the ways students speak up and have them explore their personal preferences.

  • Initiator – the person who starts a new thread of discussion. In a Soapbox Debate, it would be nearly everyone who has a unique opinion. In a larger debate, it would be anyone taking us in a new direction, not building on what has already been said.
  • Builder – the person who hears/reads an ideas and adds to that argument or line of thinking. This is someone who would say “I agree with…and I want to add…”
  • Disruptor – the person who (nicely) challenges an idea. I could see person asking questions such as: Will this always work? or Does that work for negative numbers? The Disruptor wouldn’t necessarily have to disagree; rather, this person can be pushing for clarification or evidence.
  • Connector – the person who hears/reads different ideas and shares ways to connect them.
  • Summarizer – the person who summarizes the main arguments we just heard. I’m thinking of having a different student assigned this role each time, notifying them ahead of time. It could be great after, say, a What’s Going On in This Graph? discussion to have one student summarize the main points (and maybe even enter them into the NYTimes comments section!!).

Inspired by my colleague, Kathleen Niles, I’m going to use a tracker like the one below. For the first two weeks of classes, I plan to just let students debate/discuss and to keep track of how they interact on this tracker. Then perhaps, students can reflecting on the list above and identify an area or two of strength. Later in the year, I can challenge them to try a different role.

Venn Diagram Debates

I was fortunate to attend part of the Boston Debate League’s week-long course on incorporating debate across subject areas this week. They have some wonderful resources, and I wanted to share a debatable idea I was inspired to create “by mistake.”

There was a question we were debating in our math teacher breakout room that said something like: Could one of the headings below be “quadrilaterals”?

My group assumed the blank area on the right contained shapes that had not been revealed to us. So we went about discussing the possible headings based on this assumption. We later learned that this was meant to be a complete diagram with nothing in the right hand area, but nevertheless, I could see fun debate questions coming from incomplete Venn Diagrams. So “by mistake,” I developed a new string of debatable prompts for warm-ups (or other places!).

For instance, for the picture below, I could ask students to debate something like:

One of these groups could be named “Quadrant 2” or

What is the best label for the overlapping area?

Similarly, my middle schoolers could debate a heading I give them (or they create on their own) for a diagram like this:

I’m excited to try out this new set of questions this year!

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)