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!

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!

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!

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.

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.

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 Perkins‘ Mathematician 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?

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:

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!

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.

 

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)

 

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!

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:

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 Amazon.com. 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:

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!