Mental Math


Continuing the process of letting some ideas breath on the blog this summer. Here’s another.

It’s a simple activity for those few unexpected extra minutes near the end of the period…or if I just want to hit them with some quick mental stimulation. I picked it up from my fourth grade teacher: mental math.

I simply call out a sequence of operations with a pause between each operation. For example, I might say “2 plus 5 (pause)…times 8 (pause)…minus 10 (pause)…divided 13…what’s the answer?

Students can’t allowed to say anything out loud and, obviously, any electronic device is prohibited. I don’t require that everyone plays (most do). The students must wait until I say ‘What’s the answer?’ before raising their hands. I call on a different student each time and if that person’s answer is wrong, someone else gets a chance. Because it’s a terribly simple idea, it’s always engaging. The trick is to make it challenging to the point where they get hooked and want more.

Some tidbits: I’ll usually start with one that’s pretty straightforward with long pauses – especially at the beginning of the year. Things get interesting when I start to call out the operations lightning fast or the sequence contains something like 15 operations. Make things fun by using numbers in the millions – or even billions. Also, depending on the class, the level of the math can vary from basic arithmetic to roots and exponents to evaluating trig functions. It’s endless. And fun.

TMC16 Reflections

Twitter Math Camp 2016 has officially come and gone. Its impact will be felt for years to come.

This is the type professional development that stays with you. Some of the best minds in mathematics education from all over the world come together and, for three four and a half days, form a mosaic that’s impossible to duplicate. And at the center of it all is a deep-rooted passion for mathematics, pedagogy, and improvement. What happens at TMC changes careers.

This year was no different.

The keynotes by Sara, Jose, Tracy, and Dylan set the tone. Their themes of evangelism, social justice, K-16 collaboration, and deliberate practice all struck chords because they are all things that I’ve been thinking about lately. I found myself constantly referring back to them throughout the conference.

The morning sessions provided a calm, relaxing space to reflect deeply on a specific aspect of my practice: questioning. I rediscovered the power that a question holds – especially when it comes from your student.

The My Favorite and afternoon sessions added some valuable new tools to my toolbox. I am now fully equipped to jump off some cliffs take plenty of risks next year.

I can’t go a word further without thanking Math for America for supporting my TMC16 journey through their Impact Grant. Once again, they have demonstrated why they are the best thing to ever happen to my teaching career.

Everything below is not intended to be summary of TMC16. Instead, it is a collection of personal discoveries and relationships that I made over the course of those four and a half wonderful days in Minneapolis, Minnesota. There are many.


Day 0 (Desmos Preconference) – Friday, July 15, 2016

  • Damn. Dan Meyer is tall.
  • The collaboration document containing all pertinent links.
  • Desmos is now accessible for visually impaired users. This is huge. Desmos understands that as teachers and students become more and more dependent on their product, the calculator needs higher levels of accessibility to avoid disappointing the masses. Kudos to them for continually aiming to please their users.
  • I don’t sit and play with Desmos enough. This day gave me the opportunity to muck around via the scavenger hunt. Two standouts: learning how to use lists and regressions with draggable points. Good stuff.
  • Sara Vanderwerf challenged us all to be evangelists. This was pure inspiration. She’s all passion…one of those educators that I aspire to be. Takeaways: a) find your evangelical goal, b) read The Art of Evangelism by Guy Kawasaki, c) have students use the Desmos app in class (a lot) – especially at the beginning of the year…equity and access, and d) minimize visual noise in the classroom (read more about her math wall of shame). Notes.
  • I met Abby Rosa, who teaches at a correctional facility. This rocked my world. I had so many questions for her.
  • I chatted with Paul Kelley who is on the board of directors at NCTM. Pretty cool.
  • We can now create marbleslides and card sorts ourselves in Desmos. The collective uproar on this was deafening. I tinkered and created a card sort on relations.
  • Desmos socks. Finally.


Day 1 – Saturday, July 16, 2016

  • Started off by scouring the area for a decent breakfast spot. I hate hotels without complimentary morning grub. It’s a crime.
  • I chose my morning session because I’ll be facilitating a book club in the fall with Math for America and wanted to get my feet wet. The group only consists of four of us. Norma Gordon is the facilitator. The focus of the day was A More Beautiful Question by Warren Berger, which I have read before. We spent 45 minutes reading the book followed by discourse around inquiry and why it’s so vital to student achievement. Takeaways: Get students to ask the questions, ask students questions that Google can’t answer, that students are the “research and development division” of the human species, game-changing ideas stem from a cycle of What?, What if?, and Why? questions, and teaching inquiry through experience and not merely because the teacher says to. Notes.
  • I spent the majority of lunch with Tina Cardone, who is also part of the book club morning sessions. I’ve known of her for a long time, so getting the opportunity to have an extended conversation and get to know her better was a real treat. We talked about our schools, why we teach where we do, the transparency that’s needed with students with special needs, and our mutual love for root beer. She even introduced me to Pokemon!
  • My Favorites. Jonathan Claydon’s Varisty Math is outstanding. He’s turned learning mathematics into a brand. And he’s marketing it. Next year, I’d love to start off with stickers/patches and possibly lead up t-shirts at some point. Brilliant stuff here from Jonathan. Also, I should read into Ms. Pacman transformations from Robert Kaplinsky and other teachers that have used it. Notes.
  • Jose Vilson‘s keynote was full of candid talk (video). After my recent post of my own inspired by his book, I was excited to hear him speak. He message was clear: I need you. His call to the whole of MTBoS was direct. We need to place more of a focus on social justice and equity for the students that need it most. We need to go beyond mathematics. No matter where we teach, your students are my students and mine are yours. He fielded questions and challenged us all to deliberately seek to address issues of race and social class. Notes.
  • Next up was the Talking Points session with Elizabeth Statmore. I think she’s my new hero. Backed by research, she demonstrated why quality exploratory talk amongst students is the number one predictor of effective group work. This is especially pertinent to me because of my goals for group work. Talking Points are a simple way to promote flexible thinking, vulnerability, and listening. Here are some examples. I also loved Reader’s Theatre…it is a fun way to introduce it (or any other structure) in a way that is friendly and inviting. Maybe I should read the book this summer? Also, Glen Waddell mentioned Open Middle problems, which I had never heard of. Notes.
  • Julie Wright closed the day with her talk on using feedback quizzes to promote low-stakes learning. She described her system of providing detailed, written, unscored assessments. I pondered this idea of (efficient) feedback last year. It involves scanning students exams as a single PDF, inserting typed feedback (no scores), and handing them back to students for revision. Feedback is colored-coded for the teacher, but students only see black. All students receive equal amounts of feedback. Notes.


Day 2 – Sunday, July 17, 2016

  • Cereal in my room. A solid decision.
  • My favorites. David Sabol, a Clevelander (!), used some pretty cool data to illustrate his usage of maps and voronoi diagrams to promote engagement, modeling, and low-floor, high-ceiling mathematics. Anna Blinstein highlighted her commitment to flipped assessment (feedback meetings) with students, which really interested me because I considered (private) the same thing last February. She wants to maximize the time spends on feedback and conferencing with individual, pairs, or triads of students is her way of doing this. I see value here and want to pick her brain more about her system. Notes.
  • Day 2 of the book club kicked off with an hour of reading A More Beautiful Question followed by using the Interview Design and Dialog protocol. I felt good about the protocol. Takeaways from the day’s reading: how student-generated questions from underserved youth can evoke social justice, why the “professionalism” of asking questions serves to maintain the status quo of those in authority, and the potential impact of a “Be a Skeptic” theme for my classroom next year. Notes.
  • I spent lunch getting to know Sandy Ketterling, a.k.a. the queen of baking. She teaches on an Indian reservation. Sweet lady. Go warriors!
  • My favorites. Sam Shah again amazed me with his work on Connie Haugneland made my weekend by sharing her uplifting story about sponsoring a student and building a school in Rwanda. Her experiences have inspired her to move to Rwanda and dive even deeper into this work. Connie, thank you for embracing a cause that many of us distance ourselves from. Notes.
  • I caught up with Andy Pethan. A cool dude. Down-to-earth. Relatable. Smart. Shame there was no frisbee this go around.
  • In her keynote, Tracy Zager did a lot to open up the collaboration pathways between elementary and secondary mathematics teachers (video). Her passion called on us to do a better job of bridging the divide that exists between us. Interestingly, I’ve recently given thought to this. We have so much to learn from one another and she provided several examples of mathematical pedagogy that prove that we are stronger together. When we make our learning and vulnerabilities public, amazing things happen. Let’s be intentional and move beyond the self-imposed boundaries we call grade levels. And please, don’t call elementary mathematics the “basics.” Notes.
  • Dylan Kane and Nicole Hansen supplemented the New Visions course I’m taking this summer by exploring mathematical structure, Mathematical Practice Standard 7. I’ve long admired Dylan as a humble, well-articulated, knowledgable source on teaching and Nicole proved likewise. Through this session I realized the need for a routine (ahem…New Visions) to surface structure for kids and help them to leverage it. The shift from procedural to structural thinking has caused me to rethink so much of mathematics and how I teach it. The goal is getting kids thinking flexibly using structure within a problem and between problems. I feel the earth moving on this one. Oh yeah. Big improvements on the way. Notes.
  • The day wrapped up with Jonathan Claydon and his unorthodox approach of implementing curriculum. He is so damn inspiring. Last week, by happenstance, I stumbled upon his post that motivated the session, so I had a pretty good idea what was on tap. He shared how and why he decided to hack the traditional curriculum to pieces. His centers his planning on skills rather than content. This allows for more continuity throughout the year and less disjointed learning. It’s a really interesting structure that I’m looking to adopt. The inception button and marble races were also outstanding takeaways. Notes.


Day 3 – Monday, July 18, 2016

  • A breakfast disappointment. Although, there was fresh fruit was involved – so it wasn’t a total meltdown.
  • My favorites. Joel Bezaire discussed an incredibly simple game that he uses with his students. I want to try this. It’s straightforward, but can get deceptively challenging. Sort of reminds of a Sudoku or KenKen. He has a website dedicated to the game and an archive of versions he’s already created. Gregory Taylor blew everyone away with his math song about the cubic formula based on a hymn from Sister Act. And Edmund Harris delighted us all by revealing that his new coloring book, Visions of the Universe, will debut on November 29, 2016. To ice the cake, there will be four lesson plans written by MTBoS teachers (Sam is one) that will align with the two coloring books. Notes.
  • The final day of the book club opened up with an informal discussion about protocols that digressed into teaching strategies. Tina showed us a “step-by-step” activity similar to this one, but in her case students fold the paper backwards to hide all previous work before passing it left or right. I feel desk surfaces are underutilized spaces, and Norma made a great point that whatever is attached shouldn’t be static because, over time, students will simply ignore it. Shop ticket holders were brought up as a possible solution. Takeaways from the day’s reading/discussion: stay away from answers – live in the questions, we must stop doing and knowing in order to ask, we know far less than our intuition leads us to believe, ask why five times, and Tina’s idea of having a reflection question based on the day of the week (e.g. Tuesday: What’s one good thing that happened to you in the past week?). A possible MTBoS book exchange was also brought up, which I love. Notes.
  • My Favorites. Denis Sheeran discussed his initiative I See Math, which encourages us to step back, simplify our approach, let student misconceptions come forward, and frame real-world situations with mathematics. It’s a structure consisting of three slides: title, image, and question. He also showed us two cool games to play with Google Maps: Geoguessr (old) and SmartyPins (new), both of which are really fun. Notes.
  • Dylan Kane was the last keynote of the conference. Dylan detailed his first few years teaching by admitting that he tirelessly searched for and tried out so many resources he found through MTBoS. What he learned was that despite being extremely dedicated to improvement, he wasn’t deliberate. He constantly referenced the elements of deliberate practice as outlined in Peak by Anders Ericsson and Robert Pool (a superb book): getting out of your comfort zone, being specific, including feedback, and being goal-oriented. His inspiration seems to stem from asking the question, “What would MTBoS do?” when stuck between a rock and a hard place. He mentioned that we can meaningfully improve our practice by about 10% each year. Call to action: How will I distribute that 10% and how will I be more purposeful about it? Notes.
  • I missed out on the Assessing What We Value (Experience-First Mathematics) afternoon session. I want to read up on this.
  • The awesome Alex Overwijk was up next with Go With the Flow. His speciality is accomplishing flow in the classroom and references the work of Peter Liljedahl with vertical non-permanent surfaces (VNPS) and visible random groupings (VRG) to accomplish this. He recommended the book Flow by Mihaly Csikszentmihalyi and an article by Peter. He proceeded to demo the strategy with the taxman problem which left me in awe. Critical note: The brilliance behind this strategy goes far beyond using vertical whiteboards. The majority of the impact lies in how one facilitates student work and keeps them in an optimal state of engagement. He challenged us to give students less and allow learning to happen naturally. Now I know what all the hype is about with VNPS. At some point, I must try this next year. Notes.
  • Minnesota-based teacher Annie Perkins brought folks together in her flex session to talk about how she introduces students to mathematicians that look more like them in terms of ethnicity, LGBT, etc. Here is an example. It’s a form of social justice that she calls “The Mathematicians Project.” She uses Wikipedia to research mathematicians and dedicates 5-10 minutes each Friday to present one mathematician to her students. She polls her students to determine who to present. The impact on her students has been remarkable. She is looking to post all the mathematicians she presents in her classroom and even see if she can get some living ones to pay a visit to her students. The close of the session included a discussion about how to advance social justice within our classrooms and the MTBoS. The hashtag #sjmath was mentioned and Radical Math was shared for mathematics-based, social justice-themed resources. Sheila Orr and Wendy Menard are also great ambassadors of this work.  Notes.


Day 4 – Tuesday, July 19, 2016

  • I ate the hotel breakfast!
  • My favorites. Amy Zimmer shared a really cool ice breaker that focuses on the collective interests of the members of the group. What stood out to me was how she highlights the process the groups use arrive at their choices. This turns into a great class discussion about group chemistry and teamwork. Max shared his a congruent triangles example of his work on ARCs, each of which consist of NCTM resources pieced together in single coherent resource. He’s going to share those with me, which I’d like to play around with. Glenn Waddell received a standing ovation for sharing how he has faced his fears…and how it has completely changed his life (video). Megan testified on how a boring moment at a PD can lead to presenting at a State Fair with the likes of Christopher Danielson. I appreciate her humility. And what an interesting exploration she developed! Hannah Mesick gave me a really good idea about displaying birthdays as functions (video). So intuitive. Notes.



Contemplate then Calculate

I’m taking a Structured Inquiry course this summer offered by New Visions for Public Schools. As part of the course, we are required to design a Contemplate then Calculate instructional routine and rehearse it with the class. David Wees and Kaitlin Ruggiero have both done some inspiring work with #CthenC routines and much of my motivation comes from them.

If you’re unaware of Contemplate then Calculate, the goal is to get students thinking about mathematical structure. The routine leverages student observations, honors their approaches, and highlights reflections on their own thinking. There is a public collection already created by David, Kaitlin, and the New Visions team.

Here was my task.


  • I initially planned for this to come at the end of an exponentials unit, but after some discussion, I think it would fit much better at the beginning of the unit – or at least before we discuss exponential equations. One thing I’m great at is having students get overly consumed with procedures, so my hope is to get them thinking instead about the structure of the equation and the relationships that exist within it before any mention of the term “exponential equation.” I want their mathematical insights to drive how we solve these equations…and the entire curriculum.
  • To this end, I thought of four strategies that students may use, ranging from guess and check to equating exponents (excluding the one involving logarithms since the task would be presented before they appear). Interestingly, guess and check was not an approach that was adopted much during rehearsal (possibly because they were all math teachers).
  • Since my phrasing of the number of solutions was ambiguous, Robin mentioned that students could use a graphical representation to address this. They could reason that only one solution exists because f(x) = 10^(x+5) -1 is one-to-one.
  • I also considered how to represent the equation. Specifically, I thought about 100 = 10^(x-5),  10^(x-5) = 100, 100 – 10^(x-5) = 0, or using any other set of constants in place of 99 and 1. All of these alternatives have consequences for how students may approach the problem.
  • During the rehearsal, I did a fairly poor job at annotating, which is a critical aspect of the activity because it models student thinking for the rest of the class. Although I anticipated all the strategies while planning, I rushed myself during the process and the result was unclear and unorganized.
  • The reflection prompts should be tailored to the goals of the activity. The more specific they are, the more beneficial the reflection.
  • I should omit the question (e.g. “find all values of x”) during the flash of the task. This move opens things up for more diverse observations and student thinking.
  • An interesting extension was asking what if the 1 wasn’t there? How would this impact our strategy? This is a nice prelude to logarithms.
  • Designing this activity made me think about mathematical structure like I never have before. Often times I take student thinking related to structure for granted. This activity helped me better value student understandings of mathematical structure – and how to leverage those understandings to enhance learning.
  • One of the most beautiful aspects of this routine is that it goes beyond mere discovery learning. The goal isn’t for students to end up at the same strategy. The idea is to develop fluency and flexibility in their numeracy. It echoes the theme from my current book, Building Powerful Numeracy for Middle and High School Students by Pam Weber Harris.
  • All of my planning materials can be found here, which include the anticipated strategies students will use, my planned annotations, the annotations I actually did during the rehearsal, and the slides.



My hope for group work…and introverts

Lately I’ve been thinking about group work.

This past spring, I reached a point in a lesson where I wanted students to work together on a few tasks. Prior knowledge was there. Things were accessible. Heterogeneous groups abound. All the standard stuff. There was no reason why the kids shouldn’t have been good to go.

What did they do? They waited for me. They couldn’t get started without me but, even worse, they couldn’t even use one another to get the ball rolling. Despite being more than capable, they wanted me to feed them…again. I say again because this was a fairly common theme all year. It just took this particular lesson for it to hit me.

Realizing this, I didn’t want to lecture them on how I knew that together they could accomplish the tasks I set forth. So I sat on a desk and stared at them. The result was a bunch of concerned faces asking me why I was so quiet. I responded with silent eye contact to each and every one of them. It took a full three minutes of awkwardness before they pieced things together. Oh, he wants us to figure this stuff out. 

In the moment, I was really disappointed with them. I was borderline furious. I overplan my lessons, pour growth mindset into them all year, and live with a low floor and high ceiling. Yet why couldn’t they work together, independent of me?

It didn’t take long for me to realize that I was the culprit. This situation was a direct consequence of me neglecting to develop a culture of interdependency. Now that I think about it, my classes have been like this for years.

Next year I am determined to get out of the way. For everyone’s sake, my students must need me less. I want group work to be the norm. A successful mathematics class is dependent on communication and inquiry – both of these are byproducts of collaboration.

I’m still finalizing a structure, but thanks to a workshop by Phil Dituri, I have some tentative group norms that I’ll use next year.

  • If you have a question, ask your group before asking me.
  • If someone asks a question, do your best to help that person.
  • It is the responsibility of the group to ensure that each and every person in the group understands the task at hand.
  • If you finish and check your work, you should ask others in your group if they need help.
  • Discuss different answers and try to agree on one. You should be able to explain your group mates’ solutions as if they were your own.
  • No talking to other groups.

It will be challenging to develop these norms with students that may not understand how to work together effectively. There are strategies that will be helpful in the process, but I still may have to start with one or two at the beginning of the year and build on those.

I want my students to value collaboration and learning from one another, but there’s another aspect to this talk of group work that’s worth noting. It’s the societal belief that collaboration is the root of all things great and that everyone must collaborate in the same way. Susan Cain argues against this mantra in her book Quiet and I agree with her. She calls this the New Groupthink because it “…elevates teamwork above all else. It insists that creativity and intellectual achievement come from a gregarious place.”

This line of thinking is a huge disservice to our introverted students. I bring this up because of my own introverted tendencies, like preferring one-on-one conversations over group discussions, enjoying listening more than speaking, and leading in non-traditional ways. The need for some students to think and work in solitude is something I get. It’s how I’m most productive. Reading the book was like uncovering so much of myself.

Studies show that one third to one half of us are introverts. The class I mentioned was full of introverts and they needed a structure to work together. I hope that my system encourages collaboration and interdependency while addressing the needs of all my students, but especially my more introverted ones. My norms won’t be a saving grace since research advocates for other strategies to support introverts – such as small groups, individual think time, and supporting individual passions, all of which I could improve upon. But hopefully my group norms help to celebrate introversion and make it easier for students to rely on one another as opposed to me.



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