One of my recent favorite reads was Vanessa Vakharia’s Math Therapy, and I can’t recommend it enough! What’s more, I thought it pairs nicely with DebateMath to create more engaging and welcoming classrooms.
Vanessa hit me hard right from the start, sharing how serious math trauma can be. What’s more, that trauma is often holding students back from trying new things and taking risks. As she says: “We can’t expect our students to take risks or exhibit vulnerability when they are protecting…an open wound” (p.11).
Vanessa doesn’t just talk about helpings students with math anxiety and math trauma, she offers clear, concrete activities you can do immediately. These include both short and long options, including some of my old favorites like name tents and Math-ographies.
But what stood out to me the most is the parallels to DebateMath. When I read statements like:
“Mythbust the idea that math is just about getting the right answer” (p.72)
“We rarely ask our students to actually reflect or feel in math class” (p.80)
“Tell kids that you are just as interested in their process as you are in the final result” (p.212)
I kept thinking about the messages I want students to understand when I ask them to debate in math class. I want it to be be more about the process and their thinking. I want them to be creative and share their unique way of looking at things. I have less of an emphasis on the answer, and more of an emphasis on celebrating ideas and welcoming new ones.
There’s so much I could say about this great book, but I’ll let you discover all its hidden gems yourself! And feel free to message me. I’d love to hear your thoughts!
I’ve spent a lot of time in classrooms lately: observing as well as co-teaching and modeling lessons. One thing that I’m noticing really helps a lesson flow and stand out to students is having a clear objective in mind. Now let me be clear: I’m not a fan of teachers having to write their objective on the board every morning (like SWBAT) because of some admin requirement. In fact, I prefer to never write an objective for students to see, because it can often ruin the “surprise” of the lesson.
What I’m talking about is the teachers having a clear objective in mind, and being able to state it clearly. Let me share two examples to explain what I’m getting at.
A Struggle
Last school year, I was working with some middle school teachers to plan a lesson on slope. Together, we spent an hour planning a lesson that we were then going to go teach. However, in that hour we were not able to finish planning a lesson (and we went into teaching a lesson that was only about 80% planned!). It was a little stressful, but upon reflection, I realized the issue: We did not have a clear objective.
We started the day by talking about our goals for the lesson, and we even stated an objective. However, it was too vague. I can’t remember the exact wording, but we said something like Students will be able to understand slope. With this vague objective, planning was more of a struggle than we expected it to be. We spent a lot of time asking questions like
Do we teach rise over run?
Do we teach the slope formula?
Do we include negative slopes?
Do we want to include real world examples to highlight the use of slope?
Do we look at how any two points on the same line can will you the same slope?
Do we look at the formula y=mx+b?
etc
We spent so much time continually asking what we needed to include that we hardly had time to plan what it is the students will actually do in the lesson.
If we had one, clear, specific objective in mind, it would have helped guide our planning, AND it would have helped me focus what questions I ask students in the moment during the lesson. It also would have helped me consolidate things at the end with more clarity.
If, for instance, we had stated an objective of calculating slope by counting the rise over run (as perhaps a first introduction to slope), we would have had that clarity in our planning. That is NOT to say that we couldn’t have a problem that includes negative slope or that we couldn’t tie it into the slope formula or y=mx+b. We also could have gone on a tangent (planned or not) about, say, simplifying fractions. Other things are possible, and sometimes things come up even if you didn’t plan for it. That is all good. The point I’m making is that having one clear goal for the lesson gives a driving direction, an objective we can (hopefully) achieve, in addition to other things that happen in the lesson.
A Success Story
This current school year I planned and taught a lesson on proofs with a group of teachers that went really well. A big part of that was the specificity of the objective. In this class, students had already seen proofs for a day or two. As we started our planning time, we talked about several options for lesson goals, and we settled on this: Students will understand that the order of the steps in a proof has a logical flow. That is, the order of the steps usually matters. You can change the order of some pieces, but not others, as the flow has to have a logical progression.
With this objective in mind, we planned a solid lesson that had a good flow to it. As one of the activities, we cut out the steps of a finished proof, mixed up the order and asked pairs to put it back into order. We also planned an Exit Ticket that checked in on students’ understanding of this objective. AND during the lesson, because I had this clear objective in mind, I could ask students (whether in their groups or as a whole class) questions like: Is it ok if I switch these two steps? What about these two? Students were able to talk about the logic of the proof and the connections between steps in a wonderful way.
Again, this does not mean other things didn’t happen. We talked about the “Given” and reviewed Triangle Congruence Theorems, discussing which one might be useful in a particular problem. Near the end of the lesson, students wrote their own proof (or two) from scratch. We will still reinforcing the ideas and skills behind proofs. However, that clear objective gave me a clear focus to keep coming back to!
In Conclusion
Having a clear objective gave a clear point to the lesson, a driving force. We didn’t need to write it down for students, and we could also go on tangents and explore some other things along the way. However, that one clear goal (One Goal to Rule Them All!) really helped focus our planning AND made it easier for us to question student thinking during the lesson, bringing out the main idea strongly.
In the past 20 years, I’ve taught some version of a Pre-Calculus course for most of those years. And the Unit Circle has become MY FAVORITE topic to teach. I wanted to share one of my favorite activities that I built for the Unit Circle unit–coordinates at certain points on the circle. If you want to know more about this unit, email me!
Background: Let me first say that this is after we’ve already talked about angles (negative degrees, more than 360 degrees) and radians (as another way to measure angles). You can read more about one of my favorite radian discovery activities in this previous blog post.
Once we’ve had some lessons to talk about measuring angles in radians and degrees, I introduce the unit circle definition: just that it is a circle whose radius is one unit in length. Then I give them a full page unit circle on a grid that has gridlines every 0.1 units.
Step 1: I ask students to share with their partner what they notice and wonder on this circle (for just a minute or two).
**Below is the circle. I zoomed in on the top half so you can see some of the details better. At the bottom of this post is the full circle.
Step 2: I ask groups to work on a set of questions/tasks.
If all the sectors around the circle above are of equal size, how many degrees are in each angle?
Label the angles marked on the Unit Circle (on the lines) in degrees (from 0 to 360).
If all the sectors around the circle above are of equal size, how many radians are in each angle?
Label the angles marked on the Unit Circle (on the lines) in radians (from 0 to 2pi). Simplify all fractions.
Use the graph paper the circle is on to estimate (to two decimal places) the coordinates at each of the angles drawn. Write the coordinates outside the Unit Circle.
Question #5 is THE KEY question here. The first four parts keep students reviewing angles in degrees and radians, but #5 is taking it forward. And without me having to tell them students will be able to state a LOT of great facts about the Unit Circle, for example:
“I notice that all the coordinates are decimals less than 1”
“I notice that the coordinates at 45 degrees have the same x and y value” (about 0.7)
“I notice that the y-value of 30 degrees seems to be exactly 0.5”
“I notice that the coordinates of 30 and 60 are basically the same but x and y are switched”
“I notice that the coordinates in Quadrant 2 are the exact same numbers but the x is negative”
Basically, students say all the patterns we want them to understand about the coordinates, AND it really sinks in that these coordinates are all small numbers less than 1 (and greater than -1).
Step 3: Is to have students share out so they can hear these noticings from each other. After this activity students tend to have a more concrete grasp of the unit circle, its patterns, and the magnitude of the coordinates.
Now, we can find some of the exact coordinates in Quadrant 1, and students fly through filling out the rest of the circle. What’s more, I NEVER have to again remind them of the patterns in the circle or why a certain sin/cos answer is negative.
*Below is the full circle if you want to use it! If you want this and any of my other Unit Circle resources, just email me and we can connect!
I know that Standard for Math Practice #6 calls for students to “Attend to Precision,” but what if we got it wrong by making it a practice standard? I think we go too far with this statement. I worry that our over-emphasis on precision is detrimental to student learning. I’m concerned we are being too rigid and exclusionary with this focus on precision.
Before you rush to email me to tell me that part of math is getting the right answer, let me tell you about some recent experiences I’ve had as a learner. Those that know me know that I have been trying to learn French through classes and apps for the past few years. This past year, I decided to really dig in. I wanted to finally reach a level of conversational fluency with French. But I hit several roadblocks:
Typos. I’m regularly marked wrong (losing points) for every typo in my written work. For instance, I might forget the s at the end of a verb (like parler) in the second person conjugation (Tu parles). Sometimes, this is just a careless mistake. I do understand the conjugations of regular verbs. However, careless or not, I JUST DON’T CARE when it comes to my learning. I want to be able to speak and understand conversations in French, and in that language, parle and parles sound exactly the same. So my goal of communication is met the minute I speak it, whether or not I remember the s. Constantly being corrected over minor mistakes (often just careless) frustrates me and makes me want to stop learning. It makes me feel excluded and not welcome to continue the learning.
THE Way. Sometimes, depending on the teacher or the app that I’m using, I am reprimanded for not saying it the way they want me to. I may not remember or fully understand the most elegant way to say something. So I may take a circuitous route to communicate the idea. I may need some gestures to aid in my communication too. It’s not always the prettiest, but if I were hanging out at a cafe in Paris, I would be able to get my point across. However, in “learning” French, I’m constantly getting the message (however conscious or not the teacher is of this) that there is only one correct way to say something. I need to learn the way. It feels very restrictive and alienating. I feel like I’m memorizing full sentences in isolation just so I can repeat back what is expected, rather than learning the words and concepts more generally.
Details. When I write French, I often write without adding in accent marks. My defense: I’ve lived for 40 plus years writing in English where there are almost no accent marks, and on top of that, my computer and certain programs can make it difficult to add in accents. So I just skip them. However, this means I’m “not good at French” to instructors. No matter how well I can pronounce the words, spell the words, or put a grammatically correct sentence together, the whole thing is marked wrong if I forget that little flair above some letters. I understand that precise French would include these details, but I have no desire to be a French scholar. Again, I want to chat up some friends at dinner in Quebec, not write the next great French novel. My lack of elegance and the constant reminders about them send me the message that French is only for some privileged elite group. It is not welcoming, and it indicates that French is only for those who want to dedicate themselves to perfection.
So here’s the dilemma, my goal is to be able to speak and listen to others in French. However, my language classes and apps require that I speak, write, hear, read, spell, conjugate, live and die with precision. I want to be a functioning adult in the French world mais quand suis-je assez bien? And the message I keep getting over and over is that “you are not good enough” or “you do not belong.”
So I wonder and worry: how many students have we alienated every time we correct an error (taking off a point for each tiny error)? Or when have we sent the message that math is only for those who are “perfect” at it? How do we engage students and work with them to establish goals that meet their plans?
I want to live in a mathematically literate world, one where no one would dare say “I’m not good at math.” But at the same time not everyone has to be an engineer or physicist. How do we balance improving numeracy and fluency with also helping students to feel included?
I recently added some Venn Diagram puzzles I’ve made with some teachers this year for Geometry in this post. I wanted to add some Algebra ones here! Though most were created with certain answers in mind, we welcome creative ideas from students. So there may be more than one answer!
Some are good for vocab.
Some involve solving for x first.
Or looking at graphs.
Some are focused on what process you might prefer to use…
We made some “challenging” ones that may not always have answers for every question mark.
I’d love to hear what you think or how it goes if you try one with your students. Feel free to share any you make!
Back in 2020, I wrote about some Venn Diagram debate questions that I had made for student discussion. I’ve finally had the time and space this year to play with this more and make more!
I’ve been fortunate to work with many teachers across the US (and Canada) in consulting role, and sometimes I’ve worked with teachers who wanted to create some Venn Diagram questions. Below are a few of the ones we came up with for Geometry topics.
Some of these are open-ended, with more than one possible answer. Some might only have one. They were great to use as a warm-up thinking problem (maybe have students discuss with a partner) at the start of class. They were also great as part of a problem set (for class or homework), to give students a challenge to think more deeply about vocabulary and categories…and to be creative!
The task is usually to “fill in the blank” where you see a big red question mark.
Sometimes we had more than one red question mark.
Some were much more open to interpretation, which led to rich discussions.
If you know me, you know I LOVE puzzles and riddles. I’m a big escape room enthusiast. So I was delighted to find the following riddle in a book I was reading, which the author attributes the the mathematician Tartaglia! See if you can solve it! I left a hint…and then the solution.
THE RIDDLE: A man dies, leaving 17 camels to be divided among his three heirs, in the proportions 1/2, 1/3 and 1/9. How can this be done?
I had a lot of fun with this, trying to think of creative solutions. Just to be clear: we are not hurting any camels. That is, none of them are being cut up into fractional pieces.
As a HINT:
*SPOILER ALERT*
Something I noticed after playing around was that the sum of the fractions was almost, but not quite, equal to one whole. That is, 1/2 + 1/3 + 1/9 can be thought of as 9/18 + 6/18 + 2/18, which is equal to 17/18…not quite 1.
THE SOLUTION:
*SPOILER ALERT*
Are you sure you want to know it?
I’ll start from the hint…You may have noticed the fractions added up to 17 out of 18, and we have 17 camels…so imagine we borrow a camel for a moment. We now have 18 camels. We can then give away 1/2 (9 camels), 1/3 (6 camels), and 1/9 (2 camels) to the heirs. That’s a total of 17 camels. Every heir got their fraction, and we still have the borrowed camel that we can now return.
One of the biggest skills we can gain from debate is to become better listeners. People who do well in debates listen well to their opponents to truly understand them and respond accordingly. Teachers can help students if they really listen for what is happening in students’ minds. And all of us can connect better with people if we are good at truly listening. I want us to practice listening to learn.
Of course listening includes really trying to hear out the other person, paying attention to details, nodding along. However, I’ve learned a few things that may be a little counter-intuitive (at first). For instance:
Listening does not mean maintaining constant eye contact. Yes, some eye contact is nice, but staring at someone’s eyes and smiling the entire time they talk can be a bit artificial (if not creepy!). When you are comfortable and having a genuine conversation, your eyes will naturally wander. You’ll fidget. You’ll look inward and ponder. All of this is OK, and it has been shown to be helpful in communicating that you are listening. So, listening does not mean you have to be rigid.
Listening also means responding. Yes, there is a time to be quiet and listen, but if you are in a conversation with someone one-on-one, there needs to be some level of give and take. Have you heard of the 36 Questions for Increasing Closeness? In the studies they conducted, researchers found that having one person ask each of the 36 questions first (one at a time) and listening to the answers did not help people connect as well as having each person answer one question at a time. If one person was doing most of the talking and the other person was just listening the whole time, it didn’t necessarily lead to deeper connections. It can feel awkward. When you are sharing a story, say, you want the person listening to respond with their thoughts or their connection to your story. Their response shows that the other person was truly listening.
Listening includes asking for clarity. Similar to the point above, asking questions to clarify are a sign to your partner that you are listening. Something I’ve learned a lot from work with a crisis hotline is that asking for clarity is extremely important when anything is vague or confusing. When a teenager tells me they “hooked up” with someone, I need to check in with them to ask what that means. A term like this could mean anything from a kiss to more intimate moment. And the person talking usually appreciates questions of clarity because you are helping them to tell their story more clearly.
Listening can sway someone to change their mind. Sometimes, we hope to convince someone of an idea by “giving them all the fact.” Yet, as we all probably have experienced, this does not always work. Instead, we often just need to listen. As David McRaney says in his book How Minds Change: “If people feel heard, they further articulate their opinions and often begin to question them.“ We won’t necessarily change someone’s mind by dumping information on them. So much of what we humans believe is rooted is feelings and emotion. No amount of information can overcome that. However, listening to someone, hearing them out (and responding with your own stories), can be a powerful way to help someone think more deeply about a topic or re-think their stance.
In our hyper-polarized world right now, I’m worried that everyone is more concerned with talking than listening. Good debaters know the power of listening, and I hope we all can work on listening better. Let’s be a great listener in our next conversation.
While I speak a lot about getting students to debate in math class, I recently had the opportunity to speak about what adults can learn from debate (at the OMCA 2024 Retreat!). It’s something I’ve been spending more and more time thinking about and working on. In our highly politicized environment, where everyone seems to be competing for a viral sound byte, I think we can learn a lot by slowing down and having deep nuanced conversations (debates!). Some of the things I’ve been talking about include:
1) Everyone’s voice matters.
In an argument, people may be talking over each other. Or one person could dominate the discussion. To avoid dominance or groupthink, we need to give everyone a chance to be heard. This is where debate comes in. Debate is a structured conversation, where each person has time to share their thoughts (like an opening statement). To avoid overlooking details or brushing someone’s ideas aside, we need to give everyone, especially those who may be shy or more hesitant to speak out, a chance to talk.
2) Disconfirmation can lead to truth more than confirmation.
It’s natural for us to side with someone who we agree with or who feels “right.” However, many topics have complexities and nuances that we can easily overlook. If we only listen to people we agree with, we may miss important details. Sometimes both sides have something important to add to the conversation, and if we aren’t open to learning how we might be wrong, we may miss things. The more we try to prove ourselves wrong, the more we’ll learn what is really “true.”
3) Our brains are efficient/lazy.
In, perhaps, an effort to conserve energy, our brains often look for the easiest or most efficient solution. This is where biases can affect our reasoning. We try to fit each new situation into a pattern. This is also why groups have a big influence on our thinking. It’s easier to go along with the crowd. It takes thoughtful, conscious effort to question and analyze our reactions. Listening to debates is where I force myself to hear out both sides and think critically about the topic.
In conclusion:
To paraphrase author/debater Bo Seo in his book Good Arguments: perhaps the opposite of bad arguments isn’t calm or peacefulness. Perhaps the opposite of bad arguments is good arguments. In other words, perhaps we need to lean into arguments more. We shouldn’t avoid difficult conversations. Rather, we should have structures in place to argue in a healthy way when there are disagreements.
That’s why I’m doing my part to train adults in having good arguments (like formal debates), where we can grow and learn together.
I only hope school and political leaders can someday model this…
I’ve had the privilege this past school year to visit and work with teachers and classrooms all around the U.S., and I have loved the times I’ve been able to model #DebateMath in the classroom. I wanted to share an amazing experience I had last semester. Some quick background: Using the Learning Lab model, I work with a group a teachers to plan a lesson that I teach the very next period. In several of these Learning Labs, a focus has been on using DebateMath to increase student discourse. Teachers have seen me present about DebateMath, but many are skeptical that it will work in their classroom with their students (which is why the Learning Labs can be so powerful!).
One teacher invited me to modify her lesson plan on GCF and LCM problems with her 6th graders. So we planned a lesson that involved DebateMath, and below are some highlights of the lesson.
1) First, I had to explain to students how to debate in math class. I introduced the Claim and Warrant sentence stem, and we started with something fun.
2) Then, students solved some problems involving GCF and LCM. However, instead of asking students to just solve the problem, we asked them whether it would be a GCF or LCM problem and to defend their answer!
3) Students were given time to practice a few problems, and we concluded with a debate on who was correct, using LCM or GCF:
Takeaways:
Students were highly engaged the entire time in solving the problems because they had to justify in the debate format.
After the lesson, the teacher shared that the FIRST student to start speaking had NEVER spoken up in class so far this school year!
Students did well on the exit ticket, showing solid knowledge of the difference in an LCM vs GCF problem.
Teachers who were in the room observing wanted to you the lesson as is for their own classes!
It only took us part of an period to plan and modify the lesson to include DebateMath. Teachers were excited by easily and smoothly it can be added to enhance a lesson and by how much it increased student discourse!
CLopen Mathdebater 
