Dandy Candies + a Spreadsheet!

I know it’s probably an unpopular opinion, but I really love spreadsheets. The way you can set them up to adjust one value and have everything change, or manipulate a formula to adjust a fairly large (but manageable) set of data, really makes the calculations much more interesting and less tedious, and lets one start to make sense of some nice patterns. Spreadsheets are also (at least currently) still a part of the professional world, and properly writing an expression in the formula bar is a good and basic introduction to the syntax of programming languages. I don’t do it as often as I would like, but when the opportunity to incorporate a spreadsheet into a lesson presents itself as a good tool to simplify the calculations and get at the interesting math and conclusions, I jump at the chance. Chocolate Candies

Dan Meyer’s 3 act lesson, Dandy Candies, is an excellent way to explore basic surface area and volume comparisons. DandyCandies 4 PackagesBut what if we take it a step further?

In case you aren’t familiar with the premise, the lesson starts with a short video of 24 candies being packaged into various boxes, all with integer dimensions and a volume of 24 cubic candy units (where 1 candy = 1 cubic candy unit). Students then need to calculate the surface areas of the boxes and the length of ribbon required. But that’s about where the original problem ends. And this is where mine begins.

Students develop formulas for both surface area and ribbon length, and then create a spreadsheet in which they enter various dimensions for length, width, and height. They can then play around with different combinations to try to find some patterns that minimize both surface area and ribbon length. The next step, of course, is to minimize the actual cost. Students then must research (or be given) costs for cardboard and ribbon. For simplicity sake, we assume that 1 cubic candy unit is 1 cubic inch, meaning that we are looking for square inches of cardboard and inches of ribbon.

DandyCandiesThis gives another opportunity for some more formulas to find cost of cardboard for each configuration, cost of ribbon for each configuration, and their sum, the total cost of packaging for each box. I have a screenshot of one part of a student spreadsheet, to give an idea of what this (basically) looks like. We used Google Sheets, but this can be done in any other spreadsheet.

As a result of the extra spreadsheet work, along with some additional Gene-Wilder-as-Willy-Wonka-in-Willy-Wonka-The-Chocolate-Factoryresearch time, we now take about two full class periods on this assignment, but the amount of practice that students get with spreadsheets, spatial thinking, and applying a variety of skills makes this extra time well spent. Plus, I show a few clips from Charlie and the Chocolate Factory just to remind my students of the genius of Gene Wilder.

Why Debate in Math Class?

I have found my niche for this part of my career – I’m a math teacher and I love to make my students debate. Make no mistake – it takes some time to do a debate in class, it takes preparation, and it takes up a class period that could be used for instruction or assessment or a project or an exploration or a computer lab, which are the things that take place most days in most math classes around the country.

In the (no longer very new) Common Core Standards, Math Practice 3 (MP3) is one of the 8 Standards for Mathematical Practice, which are standards that cut across grade/content levels. MP3 states that students should be able to “Construct viable arguments and critique the reasoning of others.” A debate is defined by Oxford Dictionaries as “A formal discussion on a particular matter in a public meeting or legislative assembly, in which opposing arguments are put forward and which usually ends with a vote.” It is easy to see how well these can fit together. A student debate about a mathematical proposition is one in which they put forward arguments in favor of or opposition to the proposition. Through this process, they are given opportunities to further develop and synthesize ideas and create examples to support their arguments, which requires them to apply ideas that they can understand (and remember). In sum, it covers all the bands of Bloom’s taxonomy (which has its own controversy, acknowledged, but that’s another conversation).


For these reasons, I have found a debate to be an excellent way to review a unit after it has finished. For example, after covering solutions of linear equations in my Algebra 1 classes, we had a debate in which students argued that either elimination or substitution was the best approach to take in order to solve a linear system. I did not create an experiment that is worthy of peer review, but as a case study, in my two classes, assessment scores increased on average from 2.4 on substitution and 2.1 on elimination to 3.6 on substitution and 3.4 on elimination. The pass rates for substitution increased from 67% to 95%, and on elimination increased from 38% to 90%. (My assessments are SBG assessments scored on a scale from 0 to 4. These results cannot only explained by the debate, as students also got the feedback from their first assessments, and not all students retook their assessments. But it is some nice anecdotal evidence.)

That should be enough of a reason to include debates in math classes, but I’ve skipped over the most important reason. It’s a lot of fun! Students get to collaborate, research, and then they are encouraged to argue, with some extra points (not extra credit of course) on the line. WODBCirclesBlankThere are a variety of formats that we can use for debates, and they can range from a short “Which One Doesn’t Belong” warm up activity to a formal debate, and everything in between. Students are able to understand the protocols fairly quickly, and get used to preparing for debates because they could come up at any time. In fact they will sometimes ask if we can debate a topic if it isn’t immediately settled in class. (Case in point- a student asked if parallelograms could be defined as a subset of trapezoids. In other words, are trapezoids defined as having at least one pair of parallel sides, or exactly one pair of parallel sides? I realized that I didn’t have a definite answer, and that it isn’t a settled definition, and lo and behold, students begged for a debate, which we did a week later.)

Why Debate NCTM2017

Are you sold on debates yet? For more information, you can take a look at my Google Slides document from my presentation at the NCTM 2017 Annual Conference in San Antonio.

Math Forum Debate at NCTM 2017

This year, I was fortunate to have my first proposal to speak at the NCTM Annual Conference accepted. Even more exciting, I was invited by the Math Forum to do a presentation on debates, which then turned into an actual debate. Two friends of mine from #MTBoS, Anna Blinstein (@borschtwithanna) and Mishaal Surti (@MrSurti), agreed to have a semi-formal debate, which went over very well (despite the small audience). It’s been a week now, so it seems like a good time to reflect on this experience.

For the Math Forum debate, I have a vision of it growing into an annual math-ed celebrity debate event. Where did I get that idea? It probably goes back several years, and originated with Professor Colin Adams (whose website was last updated in 2008, but looks the same as it did during my senior year in college in 1995). I saw him more than 20 years ago at an MAA conference, in a performance as his alter-ego  Mel Slugbate, in a talk about how to cheat your way to the knot merit badge. Always creative, he wrote an excellent book on knot theory, and later debated his colleague, Thomas Garrity, on the topics of “Pi vs. e” and “Integral vs. Derivative”. (On a side note, he received his Ph.D. in math from University of Wisconsin at Madison, where mathematician, author, and NCTM 2017 opening keynote speaker Jordan Ellenberg currently teaches.)  I’m sure that hearing about and then seeing videos of those debates first put the idea of debates in math class in my mind. It’s definitely what I’d love to see annually if I can find a way to do it. There are numerous topics that can be debated in the areas of math and math education. Once I put the word out and reached out to a few people, both Anna and Mishaal graciously accepted the challenge. And now we had to decide on a topic, as well as the format.

The format was easy – I suggested some guidelines for a semi-formal debate (opening statements, closing statements, and how questions would be answered). Both Anna and Mishaal were agreeable, and that was that. The topic of debate was not too difficult of a decision; I suggested a few topics in math, and a few topics in math education, and both Mishaal and Anna were pretty excited to debate the topic of traditional sequencing of secondary math courses (Algebra 1, Geometry, Algebra 2) vs. an integrated math curriculum (which is used, basically, everywhere else in the world). And here was the problem that ended up not being a problem: both Anna and Mishaal are fans of integrated math, and so one would have to debate a side that they opposed. In the world of debate, of course, this isn’t uncommon. (I often tell the story of how I was required to take the side of allowing teenage tobacco use for a high school debate. I won the debate, making the argument that teenage smoking should be required. Not my proudest moment…or was it?)


All three of us were quite busy with, you know, all the things that come up in the lives of teachers, so our plans for great preparation were not completely fulfilled. I did create a document of research sources, and a slide show (with very few graphics), and my own list of questions for each side. Anna and Mishaal downplayed how much they prepared, but their performances during the debate were outstanding – arguments and rebuttals were well thought out, they listened to each other and responded with eloquence and intention, and they both injected humor at appropriate times.

If you’d like to see the video, it’s on YouTube (complete with quickly added end credits, Creative Commons licensed music, and minimal editing).

One thing that I’ve noticed is that, as moderator and presentation operator, I personally find it difficult to pay as close attention and take notes during the debate as I would like. As a result, I general take video of all of the debates in my classes so I can watch them afterwards. It’s also really great to have some evidence of some of the very insightful and funny things that students say.

I’ll post again with a summary of my debate talk, but right now I want to put this dream out there. Would you like to see a debate at a future math conference? Maybe a celebrity that also happens to be a mathematician debate? (I was thinking maybe Danika McKellar vs. John Urschel would be a fun one to do). What about author mathematicians? (Simon Singh vs. Keith Devlin, perhaps?) Or maybe it would be more meaningful with celebrities of #MTBoS. Perhaps…even you? Tell Suzanne and Annie and Max at Math Forum that you know you missed a great show and would like to see a debate next year! Tell NCTM you want this to happen. And tell me if you want to take part. I could always use a debater, a researcher, another camera operator or two, a video editor, a stage manager, and someone to do something about my hair and wardrobe. Say…anyone want to write a grant to fund all this?

My First #ObserveMe Experience

First things first. It can be intimidating to have someone observe you. At least, when I first started teaching, I felt like any observation was a judgement (even when I was told it wasn’t). So for all you new teachers out there, if I just told you that it can be a rewarding and fruitful experience, would you believe me? No? Well, let me tell you what I got out of my first #ObserveMe experience. If you aren’t familiar with the #ObserveMe movement, it was popularized by Robert Kaplinsky as a way to help teachers take ownership to improve their practice. When he put out his call to action last August, I excitedly got on board, posted a sign outside of my classroom, announced that my door was open at several staff meetings, and then…nothing. To be fair, I’m in a tiny school, where there’s only one other math teacher, and non-math teachers probably felt that they wouldn’t get much out of observing someone outside their subject area. And this year, I couldn’t model observing (which had been one of my goals) because I have six classes this year and no prep periods. With that class load, I definitely did not want to write any more sub plans than necessary.

So summer turned to fall, which turned to winter, which turned to spring, and still no luck. And then I realized that I could go outside of my own school. I mentioned to a few people who I’d like to observe other teachers during my spring break (since I wasn’t going anywhere and was planning to work through it anyway). Then I mentioned it to Robert when I ran into him at the NCTM annual conference in San Antonio last week. And that did it – now I had to follow through! And that’s when I took to Twitter:

Soon, I got a response from Paul Jorgens, an 8th grade teacher in nearby Palo Alto. A few messages back and forth, and I was set to visit him to observe his Algebra 1 class. Paul got one of the Desmos fellowships that I’m so jealous of, and it sounds like it’s a pretty amazing experience.  Maybe I’ll be able to commit to it in 2018-2019 (assuming they have it again and I get accepted). He is also an experienced teacher who has been at the same middle school for longer than I’ve been teaching.

I arrived at the front office of his school, and then got a few minutes to chat with him in the staff lounge. When I asked him about what he wanted me to focus on during my observation, he mentioned the work of Schoenfeld, who researches out of UC Berkeley. He described the Teaching for Robust Understanding (TRU) Framework, something that wasn’t familiar to me, but which I’m eager to investigate. In his classroom, he handed me a paper with the table below, and suggested that I choose one area for my focus today. SchoenfeldObserveStudentEyes

I decided to focus on the “Equitable Access to Mathematics” row, which is similar to one of my own personal goals for this year. From a student’s perspective, “Do I get to participate in meaningful mathematical learning? Can I hide or be ignored?” Such important questions, and really, all of these questions are very student centered, and something I plan to bring back to my own practice, and to share with my school.

Then, we went to Paul’s classroom. Paul team-teaches this particular class with another teacher, Brian (whose last name I didn’t catch). Today’s lesson was a Desmos activity about exponential functions, and because Paul is a Desmos fellow, he had access to some interesting new features that I’m excited to try out when they get released to the masses.

Not only did he use Desmos, he started the Desmos activity with a “Which One Doesn’t Belong” activity. Already using two of my own favorite things to do in class. But of course, I did have a focus for my observation, and it’s important to remember that even with an amazing lesson plan, it’s really the implementation of the lesson that dictates its success. A great lesson in the hands of an unprepared teacher can go awry in the same way that a poorly designed lesson generally cannot be saved by even the greatest of teachers without abandoning the lesson plan.

In this class, all the students knew what they needed to be doing, and were well engaged. Each student had their own Chromebook, and technology problems didn’t deter anyone. A few Chromebooks had to be traded in for new ones, and there was some sharing, but there was no real disruption. There were a number of things that Paul said which made it clear that all students got to participate. I heard a lot of “Tell me more about what you said” or “Tell me more about what said.” Paul also gave students time to reflect quietly and then write down thoughts before he called on students. This meant that students could not hide, but they had the opportunity to prepare which makes participation less intimidating for those introverts out there. (Adding more time for students to write before answering is something I really need to work on in my class.)

I also really liked the technique of not asking “What do you think?”, but rather asking “Who heard something in their group that made sense? What did you hear?”. This is one of my favorite strategies in my classes to prevent the students who always know the answer from answering, and to give a chance for a student who maybe feels like they aren’t an expert the opportunity to evaluate what they heard from classmates and to share that information. It’s a great way to give everyone an opportunity to have a voice.

One other area that Paul mentioned he was working on was to give a stronger voice to the students from disenfranchised backgrounds, and so he called on those students a lot more often. I wondered if it was perhaps too often, at the expense of other students. There were definitely a few students that neither Paul nor Brian interacted with directly (that I saw), which is understandable in a class of 32. You probably can’t chat meaningfully with each of them in a 60 minute class period. However, how can you be sure that the same student(s) are not being forgotten or left out of the conversation?

Paul did say he’d rather err on the side of working with underrepresented students too often rather than not often enough, which I agree with, but I wonder if there’s a way to mix things up a little more. Perhaps with a random walk from table to table to check in with students? With the random variable groups that are created through cards at the beginning of class, students are never in the same groups two days in a row, which is great. I do wonder, though, if focusing on a group at a time, and within that group, a student from a disenfranchised background, may be a little more balanced, while still giving weight to those students who may be underrepresented in higher math classes later on.

One of my favorite moments came towards the end of the class, when Paul put up a long list of equations that students had written to try to model a particular exponential function to go through two points (I believe). He told students how much he was “excited to look at all of these equations <they> found to fit those points. Should there just be one equation? Could all of these equations match that graph?” What a great way to validate the variety of thought that students used to come up with their ideas, and to give them a sense to critique each other and think more deeply mathematically.

I was so thankful for this opportunity that Paul gave me to observe his class. I’m looking forward to setting up a time for him to #ObserveMe in the coming weeks, to give me some feedback on some of the areas that I’ve been working on in terms of student engagement and how they participate.

Again, I absolutely encourage you to get someone to observe you. Another math teacher at your school, another teacher of any discipline, a teacher outside of your school or district, and even a teacher outside of the grade levels that you teach. There are so many things we can learn from these observations, in a completely constructive and positive manner. My recommendation is to stick to one or two very specific and measureable goals, just like the kinds of goals we try to give our students. How do you want to improve upon or build up your teaching practice, and how can you know that it’s happening? Both getting feedback from another teacher and observing another teacher to give them feedback are meaningful and inspiring ways to continue your growth as a teacher.

Cited Work:

Schoenfeld, A. H., and the Teaching for Robust Understanding Project. (2016). The Teaching for Robust Understanding (TRU) observation guide for mathematics: A tool for teachers, coaches, administrators, and professional learning communities. Berkeley, CA: Graduate School of Education, University of California, Berkeley. Retrieved from: http://map.mathshell.org/.

How I Do Honors

In response to a tweet about differentiation, I connected it to my honors class. Benjamin Leis asked if I had any posts, and I realized I had been meaning to write one. So here it is.


I teach at a small independent school, and we don’t have tracks of classes. We don’t have A.P. classes. What we do have is in-class differentiation. We have the opportunity for students to take classes at what we call “Skills” level, for students who are taking the class purely for exposure to the material. This is often used for students who have some significant challenges with success in a class, due to learning disabilities or lack of exposure to prerequisite concepts and skills. The class does not count as a college prep class. Often, a student will take a class at skills level one year, then repeat it the next year as a regular college prep class, with greater success due to the extra exposure to the ideas.

Honors is different. Since we are in California, and many of our students are applying to UC schools, we make sure our courses meet UC a-g requirements. Image result for UC a-gThis also means that only certain courses can be classified as honors level. Last summer, I prepared the paperwork and got approval for an Honors Precalculus course. So what does that mean?

In my classroom, any student in the class can take the class as an honors class. There are two ways that the class is different for them, but as long as they take on the commitment, they get the honors distinction at the end of the semester. No student is turned away, and all students are encouraged to take on the challenge.

The first way that the class is different is in the way assessments are assessed. I have adopted a version of standards based grading (SBG) in my class. Assessments are scored on a scale from 1 to 4. A score of 4 means that a student can solve problems and is able to apply a concept (or the “how” of the math). Honors students are assessed on a scale from 1 to 6, where a 5 or 6 demonstrates a deeper and more thorough understanding of how that concept works and fits in with other ideas in math (or the “why” of the math). Make no mistake – a student who isn’t taking honors can still receive scores of 5 or 6 (which will improve their semester grade, as they’ve gone above and beyond), but to receive the honors distinction, they’d have to also meet the second requirement.

For the second requirement, each student must complete a research project for each unit and then share their findings with the class in a short summary presentation. For example, when we did our logarithms unit, the honors project was to build a slide rule, and record a short video in which they demonstrated how the slide rule uses logarithms to make arithmetic calculations. When we did our trigonometry unit, students had to develop and explain a trigonometric proof to the class. For matrices, students compared and contrasted using different methods (Cramer’s Rule, Gauss-Jordan elimination, standard algebraic solving linear systems by substitution or elimination) to determine which methods were better, and under what conditions, when solving by hand, with a calculator, or with a computer.

In both requirements, the honors students are expected to demonstrate a more sophisticated and thorough understanding of the math concepts, and to work on how to effectively communicate those ideas. That isn’t to say that the non-honors students aren’t required to understand concepts and communicate, but rather that they do what we have time for in class, whereas the honors students must take some time outside of class to deepen their understanding. For example, in class, we covered the trigonometric identities, and proved a few of them to demonstrate that it could be done. Honors students were required to prove all of the main ones, and demonstrate an understanding of how to prove a challenging one on their own.

Again, there is no restriction on who can elect to take the course with the honors distinction. If a student chooses to do the work, they will get an “H” on their transcript. I would like to see more students of color attempt to take on the challenge, but I would like to see more students of color in my PreCalculus class as well. This is a goal that we have to work on earlier, but that’s a separate blog post.

Three Days of Amazing PD, Bookended by Travel Problems (That Weren’t That Bad)

Staying up all night in an airport really isn’t my thing, but neither is trying to sleep on these uncomfortable chairs with the background hum of vacuum cleaners and occasional conversations. This is as good a time as any, I suppose, to reflect, at least a little bit, on the NCTM conference this week in San Antonio.

Let me get the travel part out of the way. I had set my alarm for 5am on Wednesday morning, but like a small child on Christmas Eve, I really couldn’t sleep. I woke up at 4:50, and was out the door by 6. My flight wasn’t until 8, and it’s really only a 15 minute drive, but I was so excited. Three days with math teachers, with my people, really means a lot to me, and since I found the #MTBoS community, I found a truly international staff lounge where I could share (and steal) great ideas. Seeing these fellow math teachers that I admire and respect in person is invaluable. But more about them later – back to the travel. I met up with Nicola (the other math teacher at my school) at the San Jose airport, and we had no problems getting on that first flight. For the first time, I had TSA Pre-Check, which was a nice novelty. Such a privilege to keep my shoes and belt on.

When we arrived in L.A. for our connecting flight, we found it was delayed. Then it got delayed some more, and some more. We were originally supposed to arrive at 3:30 for an early evening keynote by the author Jordan Ellenberg who wrote the book How Not to Be Wrong. I got to see him last fall at a Stanford Public Math Lecture, and was excited to see him again, and this time to get him to sign my book. Well, interestingly enough, I did read an interesting chapter.

Since we didn’t arrive until a few hours after his keynote ended, my book’s title page is still naked.

While waiting to eventually board our plane, I bumped into Jed, who I met last year at the #MTBoS booth at NCTM (I think?). I got to chat with him a bit about his talk on geometric transformations, which is his niche I think, the way that my niche is currently debates. Eventually, we did make it onto the plane, back off, and I took my first Uber ride with Nicola and Jed to attend the #MTBoS Game Night. Yes, it’s as geeky and as nerdy as it sounds – lots of math teachers, sitting around playing various games, including some fun Estimation 180 contests and a rock-paper-scissors tournament.

The following are the things that I attended. I’ll blog more about these after I get some sleep, which hopefully means later this week, but could mean the end of June. We’ll see.

  • Thursday morning: Math Debate with Mishal Surti and Anna Blinstein!
  • Thursday afternoon: Megan Schmidt – Statistics for Social Justice; John Urschel
  • Thursday night: ShadowCon; Mathalicious Party
  • Friday morning: Journal Writing; Josh Wilkerson – Service Learning; Mishal Surti – Depth of Knowledge; Jose Vilson – Math, Equity, What’s Not Adding Up; My Debates Talk
  • Friday night: Ignite; Desmos Trivia Night
  • Saturday morning: Chris Shore – Clothesline Math; Mark Couturier – Out of Classroom Experiences; Simon Singh – FLT to Simpson’s Math

Earlier today, I got a message that our flight to L.A. was going to be delayed, but no big deal – we’d still have an hour for our layover, and I’d still get home by 10 tonight. It also meant I got a chance to spend some more time with new and old friends from #MTBoS. Maybe we did an adult version of Which One Doesn’t Belong?

And then, another message, just as I was closing my tab. Our flight was delayed more. I met up with Nicola, and we got a cab to the airport to try to figure things out. However, our flight kept getting pushed back. We were told that we would receive a voucher in L.A., but we were considering just renting a car and driving back to the bay area. Sitting around, I found that some of the people waiting were also math teachers on their way back from NCTM…but not all. That got me thinking…how many of us on the plane would be math teachers? Seems like a good Estimation 180 task – how many passengers on the plane, and what proportion are math teachers? And who should show up, but Andrew Stadel, the developer of Estimation 180. We were chatting with a couple of other math teacher friends, waiting, and waiting, and waiting to board, and finally it happened!

On the plane we went! Seemed like a good time to pose that estimation question.

And so, after a fairly uneventful flight, we landed. Early! But that meant that our gate wasn’t ready. It took 40 minutes for us to arrive at our gate. Just before we pulled in, we did a big loop on the runway. I am pretty sure the pilot was just doing donuts.

So, we stumbled off of the plane, tired, and ready to grab our vouchers and go home. Except that the line for the Delta service desk was…about….40 people deep? That didn’t seem too bad, until we found that the people at the front of the line had been there for three hours with no resolution yet. We started exploring other options for hotels and car rentals and anything that would get us back earlier. Little by little, people in front of us gave up and got out of line, and soon we got to the front. (And by soon, I mean 2 hours). At this point, we’d tried calling hotels, but they were all booked. Also, I got a text that our flight tomorrow morning was delayed, from 8:10 to 11:30. Seemed strange, so I checked Google, and sure enough, there was a 3 hour and 20 minute delay. Then I checked the trusty Delta app, and it said the flight was on time. The gate agent also said the flight was on time. Not sure who to trust anymore, but since there was nothing more the gate agent could do except give us paper boarding passes and reassure us that we’d get compensated somehow eventually for our troubles, we went off to try to sleep or rest or something to pass the time.

And that’s where we are now…waiting for this last leg of the journey. I hope it’s swift, because I miss my wife and daughters and my bed and sleep. But really, I have to remind myself that as bad as it seems, it could be so much worse. There were people I met who had been stuck in the airport for three days, small children, persons with mobility based disabilities, and others who had every reason to be far more frustrated than I. Their difficulties don’t make mine any less, but they should change my perspective.