Category Archives: Uncategorized

Building Arguments with Probability and the Clips App

I don't like projects for assessment. I do like in class projects for the purposes of fostering discussion and other forms of interactions. I decided to put together something fun to build time into the unit while students developed their skills in applying binomial probability. From student feedback, they actually said it was fun, so this wasn't just hopeful thinking (this time). This also had the added value of giving students a change to work on Common Core mathematical practice standard 3: Construct viable arguments and critique the reasoning of others.

I gave pairs of groups of students a statement. The center paragraph was the same for both - a statement about probabilities. The paragraphs preceding and following that were different - conflicting contexts for each statement. Here's an example.

I ended up writing four sets of situations to make sure that each class had at least two groups working on the same probability statement, but different arguments.

I asked students to do calculations and write a 100 word abstract stating their argument. After learning that the Clips app, recently released by Apple, made for a really easy way for students creatively describe and document their thinking, I also asked students to create a two minute video documenting the situation and their argument. You can see a selection of the video results below.

Students were really challenged to search for the calculations and results that supported their arguments. Some reported that they felt dishonest doing so.

You can check out all four sets of scenarios and the rubric I used here. The students said that working in teams and working through this task was enjoyable and actually reinforced their understanding of how to use binomial probability. As with a previous unit, this project was graded for completion, not for a grade, a fact I stated up front. So far, the students haven't actually said this was a problem for them, and the quality of what they produced didn't seem to suffer much.

An Experiment: Swapping Numerical Grades for Skill-Levels and Emoji

I decided to try something different for my pre-Calculus class for the past three weeks. There was a mix of factors that led me to do this when I did:

  • The quarter ended one week, with spring break beginning at the end of the next. Not a great time to start a full unit.
  • I knew I wanted to include some conic sections content in the course since it appears on the SAT II, and since the graphs appear in IB and AP questions. Some familiarity might be useful. In addition, conic sections also appear as plus standards within CCSS.
  • The topic provides a really interesting opportunity to connect the worlds of geometry and algebra. Much of this connection, historically, is wrapped up in algebraic derivations. I wanted to use technology to do much of the heavy lifting here.
  • Students were exhibiting pretty high levels of stress around school in general, and I wanted to provide a bit of a break from that.
  • We are not in a hurry in this class.

Before I share the details of what I did, I have to share the other side to this. A long time ago, I was intrigued by the conversation started around the Twitter hashtag #emojigrading, a conversational fire stoked by Jon Smith, among many others. I like the idea of using emoji to communicate, particularly given my frustrations over the past year on how communication of grades as numbers distort their meaning and imply precision that doesn't exist. Emoji can be used communicate quickly, but can't be averaged.

I was also very pleased to find out that PowerSchool comments can contain emoji, and will display them correctly based on the operating system being used.

So here's the idea I pitched to students:

  • Unit 7 standards on conic sections would not be assessed with numerical grades, ever. As a result, these grades would not affect their numerical average.
  • We would still have standards quizzes and a unit exam, but instead of grades of 6, 8, and 10, there would be some other designation that students could help select. I would grade the quizzes and give feedback during the class, as with the rest of the units this year.
  • Questions related to Unit 7 would still appear on the final exam for the semester, where scores will be point based.

I also let students submit some examples of an appropriate scale. Here's what I settled on based on their recommendations:

I also asked them for their feedback before this all began. Here's what they said:

  • Positive Feedback:
    • Fourteen students made some mention of a reduction in stress or pressure. Some also mentioned the benefits of the grade being less specific being a good thing.
    • Three students talked about being able to focus more on learning as a result. Note that since I already use a standards based grading system, my students are pretty aware of how much I value learning being reflected in the grade book.
  • Constructive Feedback:
    • Students were concerned about their own motivation about studying or reassessing knowing that the grades would not be part of the numerical average.
    • Some students were concerned about not having knowledge about where they are relative to the boundaries of the grades. Note: I don't see this by itself as a bad thing, but perhaps as the start of a different conversation. Instead of how to raise my grade, it becomes how I develop the skills needed to reach a higher level.
    • There were also mentions of 'objectivity' and how I would measure their performance relative to standards. I explained during class that I would probably do what I always do: calculate scores on individual standards, and use those scores to inform my decisions on standards levels. I was careful to explain that I wasn't going to change how I generate the standards scores (which students have previously agreed are fair) but how I communicate them.

I asked an additional question about what their parents would think about the change. My plan was to send out an email to all parents informing them of the specifics of the change, and I wanted students to think proactively about how their parents would respond. Their response in general: "They won't care much." This was surprising to me.

So I proceeded with the unit. I used a mix of direct instruction, some Trello style lists of tasks from textbooks, websites, and Desmos, and lots of circulating and helping students individually where they needed it. I tried to keep the only major change to this unit to be the communication of the scores through the grade book using the emoji and verbal designation of beginner, intermediate, expert. As I also said earlier, I gave skills quizzes throughout.

The unit exam was a series of medium level questions that I wanted to use to gauge where students were when everything was together. As with my other units, I gave a review class after the spring break where students could work on their own and in groups, asking questions where they needed it. Anecdotally, the class was as focused and productive as for any other unit this year.

I was able to ask one group some questions about this after their unit test, and here's how they responded:

The fact that the stress level was the same, if not less, was good to see. The effort level did drop in the case of a couple of students here, but for the most part, there isn't any major change. This class as a whole values working independently, so I'm not surprised that none reported working harder during this unit.

I also asked them to give me general feedback about the no-numerical-grades policy. Some of them deleted their responses before I could take a look, but here's some of what they shared:

    • Three students confirmed a lower stress level. One student explained that since there was no numerical grade, she "...couldn't force/motivate [her]self to study."
    • Five students said the change made little to no difference to them. One student summed it up nicely: "It wasn't much different than the numerical grades, but it definitely wasn't worse."
    • One student said this: "The emojis seemed abstract so I wasn't as sure of where I was within the unit compared to numbers." This is one of a couple of the students that had concerns about knowing how to move from one level to the next, so the unit didn't change this particular student's mind.


  • This was a really thought-provoking exercise. A move away from numerical grades is a compelling proposition, but a frequent argument against it is that grades motivate students. By no means have I disproven this fact in the results of my small case study. If a move like this can have a minimal effect on motivation, and students get the feedback they need to improve, it offers an opportunity for considering similar experiments in my other classes.

    There are a couple questions I still have on this. Will students choose to reassess on the learning standards from unit 7, given that they won't change the numerical average when we return to numerical grades for unit 8? The second involves the longer term retention of this material. How will students do on these questions when they appear on the final exam?

    I'll return to this when I have more answers.


SBG and Leveling Up - Part 2: Machine Learning

In my 100-point scale series last June, I wrote about how our system does a pretty cruddy job of classifying students based on raw point percentages. In a later post in that series, I proposed that machine learning might serve as a way to make sense of our intuition around student achievement levels and help provide insight into refining a rubric to better reflect a student's ability.

In I last post, I wrote about my desire to become more methodical about my process of deciding how a student moves from one standard level to the next. I typically know what I'm looking for when I see it. Observing students and their skill levels relative to a given set of tasks is often required to identify the level of a student students. Defining the characteristics of different levels is crucial to communicating those levels to students and parents, and for being consistent among different groups. This is precisely what we intend to do when we define a rubric or grading scale.

I need help relating my observations of different factors to a numerical scale. I want students to know clearly what they might expect to get in a given session. I want them to understand my expectations of what is necessary to go from a level 6 to a level 8. I don't believe I have the ability to design a simple grid rubric that describes all of this to them though. I could try, sure, but why not use some computational thinking to do the pattern finding for me?

In my last post, I detailed some elements that I typically consider in assigning a level to a student: previously recorded level, question difficulty, number of conceptual errors, and numbers of algebraic, and arithmetic errors. I had the goal of creating a system that lets me go through the following process:

  • I am presented with a series of scenarios with different initial scores, arithmetic errors, conceptual errors, and so on.
  • I decide what new numerical level I think is appropriate given this information. I enter that into the system.
  • The system uses these examples to make predictions for what score it thinks I will give a different set of parameters. I can choose to agree, or assign a different level.
  • With sufficient training, the computer should be able to agree with my assessment a majority of the time.

After a lot of trial and error, more learning about React, and figuring out how to use a different machine learning library than I used previously, I was able to piece together a working prototype.

You can play with my implementation yourself by visiting the CodePen that I used to write this. The first ten suggested scores are generated by increasing the input score by one, but the next ten use the neural network to generate the suggested scores.

In my next post in this series, I'll discuss the methodology I followed for training this neural network and how I've been sharing the results with my students.

Standards Based Grading and Leveling Up

I've been really happy since joining the SBG fan club a few years ago.

As I've gained experience, I've been able to hone my definitions of what it means to be a six, eight, or ten. Much of what happens when students sign up to do a reassessment is based on applying my experience to evaluating individual students against these definitions. I give a student a problem or two, ask him or her to talk to me about it, and based on the overall interaction, I decide where students are on that scale.

And yet, with all of that experience, I still sometimes fear that I might not be as consistent as I think I am. I've wondered if my mood, fatigue level, the time of day affect my assessment of that level. From a more cynical perspective, I also really really hope that past experiences with a given student, gender, nationality, and other characteristics don't enter into the process. I don't know how I would measure the effect of all of these to confirm these are not significant effects, if they exist at all. I don't think I fully trust myself to be truly unbiased, as well intentioned and unbiased as I might try to be or think I am.

Before the winter break, I came up with a new way to look at the problem. If I can define what demonstrated characteristics should matter for assessing a student's level, and test myself to decide how I would respond to different arrangements of those characteristics, I might have a way to better define this for myself, and more importantly, communicate those to my students.

I determined the following to be the parameters I use to decide where a student is on my scale based on a given reassessment session:

  1. A student's previously assessed level. This is an indicator of past performance. With measurement error and a whole host of other factors affecting the connection between this level and where a student actually is at any given time, I don't think this is necessarily the most important. It is, in reality, information that I use to decide what type of question to give a student, and as such, is usually my starting point.
  2. The difficulty of the question(s). A student that really struggled on the first assessment is not going to get a high level synthesis question. A student at the upper end of the scale is going to get a question that requires transfer and understanding. I think this is probably the most obvious out of the factors I'm listing here.
  3. Conceptual errors made by the student during the reassessment. In the context of the previous two, this is key in whether a student should (or should not) advance. Is a conceptual error in the context of basic skills the same as one of application of those skills? These apply differently at a level six versus a level eight. I know this effect when I see it and feel pretty confident in my ability to identify one or more of these errors.
  4. Arithmetic/Sign errors and Algebraic errors. I consider these separately when I look at a student's work. Using a calculator appropriately to check arithmetic is something students should be able to do. Deciding to do this when calculations don't make sense is a sign of a more skilled student in comparison to one that does not. Observing these errors is routinely something I identify as a barrier to advancement, but not necessarily in decreasing a student's level.

There are, of course, other factors to consider. I decided to settle on the ones mentioned above for the next steps of my winter break project.

I'll share how I moved forward on this in my next post in the series.

Unit Circle Practice (#TeachersCoding)

I've always wanted a simple interface to help my students practice the unit circle. I've found Quizlet sites that help with this, as well as the occasional Khan Academy exercise that approaches what I want. The big issue I find with most of these is that the interface and the questions ask much more than what I'm looking for. I want a simple flashcard-like situation with no bells and whistles that gets my students the repetition and opportunity to think through the functions with feedback.

Over the winter break, I decided I needed to build the resource I had in mind. Here's the result:

The live site can be accessed here:

This is essentially a digital version of a set of flash cards, but they never stop. The angles rotate around the unit circle and the trigonometric function used is randomized. Since I am holding my PreCalculus students responsible for the reciprocal functions, but my IB students don't need them, I added the ability to flip those on and off.

I decided to do this on CodePen in case you want to look under the hood to see how it works. The editor view that contains my code is here. Let me know if you use it for something useful.

Releasing Today: States-n-Plates

I'm excited to share States-n-Plates , a project I built with Dan Meyer.

Dan proposed the idea for this activity a while ago with his typically high level of excitement about activities that provoke interesting and productive classroom conversation. This time, however, it wasn't about mathematics. I was looking for a bigger scale project to help me develop my ReactJS skills, so I took it on. Dan was patient enough to let me hack away at the project in this context. Though I could have certainly done it more quickly using jQuery or another framework, I wanted to try building this project in a particular way.


  • I wanted to be able to play the game myself when I was done. Hard coding everything into a series of HTML pages would have likely resulted in my seeing each plate and the answer over the many times I reloaded during development. By abstracting the behavior of the game to be automated for each group of license plates, I saw most of the plates for the first time during testing.
  • I wanted to experiment with a drag and drop library for React as an exercise for use in future experiments.
  • I also wanted to have a slightly different UI behavior for the desktop and mobile versions. This functionality came from Bootstrap. This led to a bit of wonkiness on small phone displays, but larger tablets work great using touch, and the desktop version works well using drag and drop.
  • I also wanted to experiment with modularity of both files and React component JSX files. I used Webpack. I don't understand Webpack.

As in my past collaborations with Dan, I learned to do a number of things I didn't think I could do. For example, I told Dan 'no' on the fading effect at one point, and then subsequently figured out how to make it happen through lots of searches, StackOverflow, and careful reading of the React documentation.

If you want to play with the code, the Github repository is at You don't need the big node_modules directory for this to work locally, but it is required if you want to change the bundle.js file.

I have more thoughts on the learning process I went through, but that will be shared soon. Have fun and share with your friends.

Computational Thinking and Spreadsheets, Teacher Edition (#TeachersCoding)

I ran a workshop last week giving some teachers ideas on how to use computational thinking to improve their workflow. I've written in the past about how spreadsheets can serve as a way to get students thinking like programmers, without the intimidation of a text-based development environment. I don't find teachers any different in this regard.

I spent the beginning of this workshop sharing a bit about my views on why teachers should develop their computational thinking skills. I then set them off to work through answering the following questions about each task in the video below:

  • What is the spreadsheet being programmed to do?
  • What commands are being used?
  • How would I use this in my own practice?

I'm reasonably sure that a majority of teachers have a spreadsheet somewhere that contains student data like the one in the video. My hope is that teachers that watch the video and see what I've done with this spreadsheet will have one of a few possible responses:

  • Wow, I do that by hand right now. Now I know there's an easier way that will save me time.
  • That isn't useful to me, but it does give me an idea of how to do some other task that involves iteration, sorting, or another task best suited for a computer.
  • I do that already. Is that computational thinking?

If I elicit any of these responses, and then get someone to then build a tool that is useful to him or her, I think I've done my job. Learning to code for its own sake isn't necessarily worth a teacher's valuable time. Outsourcing tasks that computers do best to a computer can free a teacher to have more time for those tasks that require the expertise, experience, and a personal touch that only a person can provide. If learning a bit of computational thinking can do that, doing so might be worth the time.

Please comment on the video or below to let me know what you think.

Let's Collect Some Data Together!

I've hacked together a data collection tool with the Desmos API for an activity tomorrow - I'd love if you could help me test it out.

Please visit and enter 3DR9 as the name of the data set. Then enter an ordered pair in the form (age, # of years teaching).


If everything works correctly, we'll be able to put together a data set across the world and see it graphed in real time.

I'll talk more about this soon - thanks in advance for helping out, folks.

Scaling Reassessments, Part 2

A quick comment before hitting the hay after another busy day: the reassessment system has hit it big in my new school.

Some facts to share:

  • In the month since my reassessment sign-up system went up, 87% of my students have done at least one self-initiated reassessment, 69% doing more than one. This is much more usage than my system has had, well, ever.
  • Last Friday was an all time high number of 53 reassessments over the course of a day. I will not be doing that again, ever.
  • Students are not hoarding their credits, they are actually using them. I've committed to expiring them if they go unused, and they will all be expired by the end of the quarter, which is essentially tomorrow.

I need to come up with some new systems to manage the volume. I'll likely limit the number of slots available in the morning, at lunch, and after school to encourage them to spread these out throughout the upcoming units instead of waiting, but more needs to be done. This is what I've been hoping for, and I need to capitalize on the enthusiasm students are showing for the system. Now I need to make it so I don't pull all my hair out in the process.

Moving to Vietnam

After a whirlwind tour visiting family, friends, and taking care of many more errands than in a typical summer vacation, my family and I arrived in Vietnam mid-July. The 27 hours of travel went far more smoothly and quickly than expected. This was at least partly due to the fact that the under-filled coach cabin yielded our now eight-month old daughter her own seat.

All of this was a big step toward the next stage of my teaching career: I've joined the high school faculty at the Saigon South International School, located in District 7 of Ho Chi Minh City. This past week, I started my year teaching two sections of the first year of IB Mathematics SL, two sections of pre-Calculus, and a section of Algebra 2 & trigonometry. If you've heard me discuss my teaching load at my previous school, you'll know that this is half the number of preps, and one more open block in my schedule than I've had for the past six years. I've been amazed by my colleagues and their range of international experiences, both in and out of my department. The energy to try new things and a drive to challenge my teaching practices are both part of the culture here, and it's very exciting to be on this team for the new year.

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I'll continue to write on this blog, which has often played second fiddle to other obligations in the past couple of years. My hope is to reflect more regularly as part of an effort to do fewer things, but with greater focus. I hope you'll continue to join me.