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) 

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.