Reaction Time & Web Data Collection

If you put out an open call through email to complete a task for nothing in return, it might make sense not to expect much. I tried to make it as simple as possible to gather some reaction time data for my IB Mathematics SL class to analyze. My goal for each class has been to get an interesting data set each time and see what students can make out of it. After several hours of having this open, I had a really nice set of data to give the class.

I know my social networks are connections between some phenomenal people. That said, I didn't know that the interest in trying this out would be so substantial, and in several cases, get people to try multiple times to get their own best time. In less than a week, I've collected more than 1,000 responses to my request to click a button:
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I coded this pretty quickly and left out the error correction I would have included given the number of people that did this. I've been told that between phones, tablets, desktops, laptops, and even SmartBoards, there have been many different use cases for times ranging from hundredths of a second to more than five minutes - clearly an indication that this badly needs to be tweaked and fixed. That said, I am eager to share the results with the community that helped me out, along with the rest of the world. A histogram:

There's nothing surprising here to report on a first look. It is clear that my lazy use of jQuery to handle the click event made for a prominent second peak at around 0.75 seconds for those tapping on a screen rather than clicking. Some anecdotal reporting from Facebook confirmed this might have been the explanation. The rest of the random data outside of the reasonable range is nothing more than poorly coding the user experience on my part. Sorry, folks.

This isn't the first time I've done a data collection task involving clicking a button - far from it. It's amazing what can be collected with a simple task and little entry cost, even when it's a mathematical one. One of the things I wonder about these days is which tools are needed to make it easy for anyone (including students) to build a collection system like this and investigate something of personal importance. This has become much easier with tools such as Google Docs, but it isn't easy to get a clean interface that strips away the surrounding material to make the content the focus. For all I know, there may already be a solution out there. I'd love to hear about it if you know.


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Maintaining Sanity, Reviewing Priorities

I've had a really busy year. I've always said at the start of the school year that I'm going to say 'no' more frequently in as politely a way as possible. I've said I'd be more honest about priorities. Instead of spending time writing code for something that might be really cool as part of a lesson next week, I need to get tests graded today. I've had more preps this year than ever before. I have big scale planning to do relative to my IB classes and their two year sequence of lessons, labs, and assessments. In a small school like ours, it's difficult to avoid being on multiple committees that all want to meet on the same day.

Probably the hardest part has been figuring out what my true classroom priorities are. I'd love to look at every student's homework, but I don't have time. I'd love to make videos of all of my direct instruction, but I don't have time. I'd love to curate a full collection of existing resources for every learning standard in my courses, but despite designing my own system to do this, I haven't had time.

Over the course of the year, however, I've found that the set of goals I have for every class can be boiled down to three big ones:

Give short SBG assessments as frequently as possible.

These need to be looked at and given back in the course of a class period, or they lose their effectiveness for students and for my own course correction when needed.

Provide more time for students to work during class. Use the remaining time to give direct instruction only as needed, and only to those that really need it.

Time I spend talking is unnecessary for the students who get concepts, and doesn't help the students that do not. If I'm going to spend time doing this, it needs to be worth it. This also means that I may not know what we need to review until during the class, so forget having full detailed lesson plans created a week at a time. I think I've accepted that I'm better at correcting errors along the way than I am at creating a solid, clear presentation of material from start to finish, at least given time constraints.

It has been more efficient for me to give students a set of problems and see how they approach them than tell them what to do from the start. There are all sorts of reasons why this is also educationally better for everyone involved.

Focus planning time on creating or finding interesting mathematical tasks, not on presentation.

I've always thought this, but a tweet from Michael Pershan made it really clear:

What I teach comes from the learning standards that I either create or am given. Maximizing opportunities for students to do the heavy cognitive lifting also maximizes the time these ideas spend simmering in their heads. This rarely occurs as a result of a solid presentation of material. It doesn't necessarily (or even usually) happen by watching a perfect video crafted by an expert. When you have a variety of mental situations in which to place your students and see how they react, you understand their needs and can provide support only when necessary. Anything can be turned into a puzzle. Finding the way to do that pays significant dividends over spending an extra ten minutes perfecting a video.

Going back to these three questions has helped me move forward when I am overwhelmed. How might I assess students working independently? What do I really need to show them how to do? What can I have my students think about today that will build a need for content, allow them to engage in mathematical practice, or be genuinely interesting for them to ponder?

What are your priorities?


Filed under reflection, teaching philosophy

Students Coding Tilman's Art with HTML5

I'm a big fan of the Geometry Daily Tumblr. Tilman's minimalist geometry images are beautiful in their simplicity. I've always wondered about reproducing art in code as a vehicle for learning to code, and have had it on my list to do this myself using Tilman's work.

In my web programming class, where we are currently playing around with the HTML5 canvas and its drawing capabilities, this concept was a perfect opportunity to let students play around with the art form. They quickly observed the beauty of what can be created using these tools, and the power of doing so by studying someone that is great at it.

Here are some of the results of their sketches. Some did precise imitations, others did their own interpretations. Click on the image to see their code posted on JSFiddle.


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Circuits & Building Mental Models

I moved up my electric circuit unit this year for the senior physics class. Usually I put it after a full unit on waves, but after completing the waves unit with the IB students, I wasn't so pumped to go through it again from the beginning.

I began by having students try to generate the largest voltage they could from a set of batteries, motors, solar panels, lemons, and some other fun gadgets. That was a great way to spend a full 90 minute block. The next class, we played around with the PhET circuit simulator as I described in a previous post. The goal was to get them to have some intuition about circuits before we actually got down to analyzing it. Our conversation focused on batteries generally contributing energy to the circuit, and other circuit elements using that energy, leaving nothing behind at the negative terminal of the starting battery. Our working definitions for voltage, current, and resistance came out of the need for describing what was happening in the circuit.

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This started a pretty textbook version of the modeling process. I gave students some circuits, asked them to make a prediction for voltage/current, they made the predictions, and then tested them in the simulator. As needed, they made adjustments to their mental model to make it consistent across all examples.
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What interested me most about the results here was that the students put together a pretty solid mental model that centered on the voltage divider concept. This came out of their other assertion that current is the same for resistors in series.

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This led to the students tackling this problem on the second day of looking at circuits this way:
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In my AP Physics sequence, this is something I don't get to until Kirchoff's rules, so I was impressed with how nonchalantly they reported their answers after only a minute or so of thinking about the circuit.

On day 3, we went through an approximation of this lesson that I described in a previous post titled Starting at the end. We didn't get to the more complex circuits, but did get to the concept of parallel circuits.

On day 4, we spent a day getting our hands dirty building actual circuits, not with the simulator. The students had a good time piecing things together and seeing bulbs light up and make measurements with actual voltmeters and ammeters.

Today, on day 5, I was finally thinking I was going to teach them about equivalent resistance...but I hesitated. I was too scared that providing a formula would risk undermining all of the intuition they had developed.

The students worked through some Physics Bowl questions from a while back. Here's one:
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I noted down a student's explanation of why the answer was 36 volts, and another student's addition to explain why it had to be 42 volts:
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It then I threw this one at students:
Screen Shot 2015-05-13 at 9.57.47 AM I set the battery voltage to be 10 volts.

If my students had followed the sequence of physics lessons from the 2005 me, this would have been a piece of cake because they would have had the formula. Instead, they went through a nice sequence of stating what they knew and didn't know and making guesses. I suggested a spreadsheet as a way to keep track of those guesses and their reasoning in one place:

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We went through the spreadsheet cell by cell and decided on formulas to put in. In the end, they figured out that the final two currents had to be the same.

I did some guessing and checking following their monitoring of the values, and eventually ended up with the 100 ohm resistor having a voltage drop of 9.923 volts.

Only at this point (which was five minutes before the end of class) did I apply an equivalent resistance formula:
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It was a great moment to end on. My presentation of the equivalent resistance formula came out of a need, and for that reason, I was glad to provide it. I'm so happy I waited.

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Online School Resumes with Meteor

As you may know, I've been teaching a web programming course this year. I wrote previously about the work we did at the beginning of the year making interactive websites using the Meteor framework. Since then, we've spent time exploring the use of templates, event handlers, databases, and routing to build single page applications.

The latest assignment I gave students was to create an online school resume site with a working guestbook. I frequently discuss the importance of having a positive digital footprint online, and one of the most beneficial ways of establishing this is through a site created to share their work. Students worked last week to complete this and submitted their projects. We've had connectivity issues to the Meteor servers from China from school. As a result, some students used Meteorpad, which unfortunately means their sites aren't permanent.

Those that were successful at deploying, however, have persistent guestbooks that anyone can visit and comment upon. Some students added secret pages or like buttons to show that they have learned how to use the reactive features of Meteor. The students were excited when I said I would post links on my blog and have given me permission to share. Here is the set of deployed sites:

Maria's Site
Dominick's Site
Tanay's Site
Luke's Site
Steven's Site
Tiffany's Site

I'm really proud of how far these students have come since the beginning of the year. They have accrued some bad habits of copying code and avoiding commenting their Javascript, but I take some responsibility for not holding them accountable for this. My goal was to have the focus of this course be on building and creating as the first priority, and the second on developing skills as programmers. As with many of the subjects I teach, helping students see the need for the basics is often most easily done with the end product in mind.

If anyone wants recommendations for a summer hire, let me know.

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Code for Teachers: What do you want to learn to build?

A conversation with Dan Anderson(@dandersod) this morning has pushed me to revisit a coding for teachers concept that I've nudged forward before, but haven't made happen to my liking yet. There's an amazing variety of coding materials and tutorials out there, but few that I've seen take the approach of helping teachers build immediately useful tools to improve their workflow.

 To be done right, this must acknowledge the fact that this valid sentiment is out there:

@cheesemonkeysf: @dandersod @emwdx @dcox21 But are you prepared for… the "coding-impaired"?

As any person that has dabbled in programming knows, there's always a non-trivial period of frustration and bug hunting that comes with writing code. This discomfort is a part of learning any new skill, of course. It's also easy to say that you aren't a code person, just as someone can say that he or she isn't a math person. What pushes us (and our students) through this label to learn anyway?

  • Minimal hand-waving about how it 'just works'
  • Experiences that demonstrate the power of a growth mindset
  • Concrete ideas first, abstraction later
  • Building the need for better tools
  • Maybe the most important: having the right people at your side

I want to work to make this happen. Consider this a pile of rocks marking the beginning of that trail.
How do we start? I see this as an opportunity to use computational thinking as a way to improve what we do in the classroom. This project should be built on improving workflow, with the design constraint that it needs to be accessible and as useful as possible. I also want to use a range of languages and structures - block programming, spreadsheet, Automator, everything is fair game.

I want to first crowdsource a list of tools that would be useful to learn to build. Let's not limit ourselves to things that are easy at this point - let's see what the community wants first. I've posted a document here:

What Do You Want To Learn To Build?

Go there and share your ideas. I don't want to wait any longer to start talking about what this could be.

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    IB Mathematics HL: Vectors & Planes

    There's nothing big to report here, but I did want to share a really successful approach I put together relating vectors and planes. This is a required topic for the IB HL Mathematics curriculum. All of the textbooks I looked in did a fairly theoretical analysis of Cartesian and vector forms for planes from the start. I wanted to present a lesson that gave students a bit more intuition about the concepts involved, and then get to the mathematical vocabulary when needed.

    Vectors and Planes

    These notes were created live during class using OneNote. I don't intend these notes to replace the textbook, but I do want them to serve as the 'residue of logic' that we used during the lesson so that students can go back and review them to remember the key ideas. I have a small group, so we can sit around a big table and work together. There's lots of conversation between us and between students when I set them loose to do an exercise.

    All of the students demonstrated good understanding throughout the lesson in the problems I gave. The students that did the homework immediately after the lesson did well on a subsequent quiz. The student that didn't, well, didn't. No surprise there.

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    A Small Change: Unit Circle & Trigonometric Functions

    I wrote nearly a year ago about my adjustment to what I had done previously to develop the topic. The idea was based on what my own pre-Calculus teacher did in high school, a series of activities related to a 'wrapping function' moving around the unit circle. This lesson is for a group of Algebra 2 level students that will likely move into the IB program for next year. Mastery of trigonometric functions isn't necessary, but I do want students to feel comfortable converting between radians and degrees, locating angles on the unit circle, and evaluating trigonometric functions.

    In the last class, we talked about 30-60-90 and 45-45-90 triangles and the fact that we can evaluate trigonometric functions exactly using our knowledge of ratios and the Pythagorean theorem. We also did a series of exercises having students locate angles on the unit circle during the last class.

    Today's warm-up was a continuation of these ideas through these sets of questions:

    Screen Shot 2015-04-22 at 10.44.18 AM Screen Shot 2015-04-22 at 10.44.24 AM


    Normally at this stage, I show a development using similar triangles of finding what these coordinates are. Though I bring up this goal in a number of different ways, whether students are doing this at their seats, or I'm doing it for them, I can never the sense of understanding that I want. This development is also not what I want them to do when they are evaluating trigonometric functions either - I want them to figure out where they are on the unit circle, and then evaluate based on the x and y-coordinates of the point.

    Today I made a subtle change to my sequence. I directly told students that the coordinates of these points were some combination of a set of five lengths. Two of these lengths we found in a previous lesson, but I never made a connection to it here. I asked them to put the numbers in order from least to greatest

    Screen Shot 2015-04-22 at 10.20.16 AM

    Then I asked them to complete the coordinates in this blank unit circle. Here's a student's work, corrected by a classmate when it was shared:
    Screen Shot 2015-04-22 at 10.21.32 AM

    All of the conversations about sign and value that I had to force previously happened naturally this time. The handout was folded so that as students finished, I could then nudge them into the next step of finding angles that match to particular coordinates, an exercise on the other side.


    For most of the students, this wasn't a problem. Some even looked like they were enjoying it.

    It was only in the last few minutes of the class that I introduced the sine, cosine, and tangent as a shorthand way of asking the question of finding the x-coordinate, y-coordinate, or the ratio of the two. My students are pretty trusting, but they have also become used to asking why [Statement A] is true once they have the basic idea of what [Statement A] means. This lesson was just a continuation of this process. Almost every student was able to evaluate a cosine function of a different angle during the exit activity.

    I felt a little bad about giving the coordinates and putting off the understanding to later. This short bit of mathematical fact, however, was followed immediately by a task that required them to reason about what they mean. It builds the need to show why those coordinates are what they are, and this process of looking at 45-45-90 and 30-60-90 triangles on the unit circle will make much more sense in the context of the student experiences here.

    One student summed up my motivation for doing this beautifully as she was packing up - I love that I'm not making this quote up:

    It's good that you don't have to memorize it because you can just see the picture in your head and know what the answer is.



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    Filed under algebra 2, teaching stories

    My Application for ADE 2015

    I decided to apply for the Apple Distinguished Educator program this year. The primary reason is that the various ways I work toward my classroom goals tend to involve my use of their products. Their design aesthetic has had a strong influence on my own design tendencies as I create materials for the classroom, digital or not.


    I was not selected for this year's group. In hindsight, it's possible that my use of technology is platform independent enough that I don't really need Apple to do what I do. Oh well, maybe next time!

    The process of reflection is always valuable. If nothing else, my application stands as a pretty straightforward summary of my ed-tech philosophy these days.

    Here is my application video, and my answers to the questions:

    How have you as an educator transformed your learning environment?

    My major realization about technology in the classroom is that single-purpose devices are quickly losing their value. An iPhone in my pocket is simultaneously a document camera, graphing calculator, and assessment tool. My MacBook is a content recording studio, interactive whiteboard, and software development center. Student MacBooks combine authoring tools, answer manuals, problem generators, and nodes of an instant communication network in my classroom. All of us have access to the same tools; there is no way that I as a teacher am doing any sleight of hand. My students can learn to do what I do, make what I make, and then make completely new things on their own.

    In contrast, when I first started teaching, I had a number of useful (but single purpose) technological tools at my disposal: an interactive whiteboard, graphing calculators that networked together, and document cameras. My approach to integrating these tools into my lessons was to ask myself how I could use them to enhance my presentation of content to students.

    When my wife and I decided to move overseas to teach, it was to my current school which had a 1:1 MacBook program for the students I would be teaching. It felt awkward standing at the front of a classroom in front of desks of students behind screens. I was asking students in a whole class setting what they observed while I clicked through a program on an interactive whiteboard. The students had their own laptops in front of them - they should be the ones to be clicking, tapping, and sliding mathematical objects on screen. They could be making observations, drawing conclusions, and building intuition for what we were learning based on their experiences. No matter how good my direct instruction might be, students would be better served by spending more time actively working together.

    This has since become the new ideal for my classroom. I do not start with the technology, and then decide what I could do with it to make my teaching better. I start by asking myself what I want my classroom environment to be, how I want students to interact, and what students should do there in order to learn. Technology then serves to help me build that classroom. My planning time consists of making or searching for tools that let students construct knowledge themselves. When direct instruction seems necessary to help students learn, I work to reduce it to its essential elements. I have recorded videos of content that students watch during class. This frees me to circulate amongst the students and listen to the conversations students have with each other.

    Technology helps maximize the quality of social interaction between students and me in the classroom. It helps minimize the time spent collecting student answers and responses in one place, which then maximizes the time we can all spend discussing and analyzing that work. It provides structure to keep me and my students organized, which maximizes the brain space available to manage abstract thinking in mathematics and physics. It reduces the clerical work associated with selecting questions for a quiz or making copies, and instead moves students and me quickly to the point where we can have crucial conversations about learning.

    Illustrate how Apple technologies have helped in this transformation.

    The simplest shift came from unplugging my MacBook from the projector screen. I can sit anywhere in the classroom and project notes, problems, and student ideas wirelessly through an AppleTV using AirPlay. I use a USB tablet and stylus to make handwritten notes during class. I use the same set up to record short instructional videos and share them with students for use during class, or when they are on their own.

    There are many applications and online tools that exist to make it easy to collate responses in a classroom, make collaborative documents, and share images. The reality of accessing these tools through Chinese internet filters makes use of these applications is unreliable and difficult, if not impossible. The features of these tools, however, would be valuable for helping create the learning environment I want for my students. I have learned to use Python and JavaScript to build tools with some of these features for my classroom. I host these applications on my MacBook and students access them through Safari over the school network.

    I created a web based application that allows me to take a picture of student work with my iPhone, and then upload the file directly to a folder on my computer. We can then flip through different responses using Preview and discuss the content as a class. Students can also share images of their work using their phones or computers, anonymously or not.

    I have implemented standards based grading for almost all of my courses so that students have multiple opportunities to demonstrate mastery of what they have learned. I wrote another application that sends individualized quizzes on specific learning standards to students through a web page, also hosted on my laptop. Students can access their individual quiz site through whichever device they have availiable. I experimented with the Meteor JavaScript framework and built a site that lets students sign up for these quizzes at any time, from anywhere.

    I let the technology handle the collecting, organizing, displaying, and calculating, as these are what computers do best. As a result, the valuable but limited time that I have with my students can be spent learning to do the thinking and develop the skills that are uniquely human, and that will be necessary long after students leave my classroom. The versatility of the tools that Apple provides makes that process possible.

    What successes have you seen with your learners?

    I survey my students frequently on what is or is not working well in the classroom. Listening to me talk and go through problems, though it is easiest for me in terms of planning, is consistently at the bottom of student preferences. The more student-centered methods are, by far, the most effective and preferred methods for students to learn in my classes. My presence in the classroom is most valuable when spent moving from student to student, listening to conversations, and asking questions based on my assessment of their comprehension level. In the lessons that involve my recorded videos, the ELL students appreciate being able to pause the videos and switch their focus between the concepts being taught and the language. The more advanced students often start with the assigned problems, and then work backwards with the video content when they need to get unstuck in solving a problem. I can monitor how students are engaging with these videos through written notes and solving problems, and can provide assistance on an individual basis.

    Many of the students in my classes are used to rote instruction, as this is what they experience in schools in their home countries. My use of technology as a tool for investigation, and emphasis on sharing student ideas to develop understanding, helps reduce the belief that memorization and obtaining answers are the primary goals in mathematics and science. My students understand that there are many tools available to help them arrive at an answer. They use one tool to verify the results of another.

    I have had excellent results with students in my AP Calculus and AP Physics courses over the past five years. I attribute much of this success to the positive learning habits that students have developed through my classes. Students know how to get unstuck. They know how to use each other's presence in the classroom to build on their understanding.

    The best feedback on my teaching often comes from students that are no longer in my classroom. One student from last year's physics class was often frustrated that I would not generally not lecture on how to solve every type of problem. Here is an excerpt from an email I received from this student earlier this year:

    "...I am very happy that you made me struggle with physics last year because now when I don’t see how to solve a problem immediately, I know how to use the tools available to me to experiment to find the right answer. "

    I often wonder if I am doing what is best for my students. Comments like this one lead me to believe that I am moving in the right direction.

    How do you share these successes to influence the broader education community?

    When I first moved abroad, I left a large department of teachers to be a member of a one person team at my current school. While this team has since grown to include amazing collaborators, I get a lot of my best ideas and encouragement from teachers that I have never met in person. They push back when I think I have everything figured out, and never let me stop tweaking a lesson to be its best. I am in communication with this network of teachers from around the world regularly through Twitter, blogs, and email. Many of these teachers are already in the ADE community, and their feedback was important in deciding to apply to the program myself.

    Any time I have an experience in the classroom, successful or not, I turn to my online community. It has been important to share the good ideas, but it is increasingly more beneficial to also share uncertainty. I blog whenever possible at my website about my experiences with students. When an activity has materials that can be shared in their raw form, I make these materials available on my website. Otherwise, I include enough details that teachers that want to imitate what I have done can do so with minimal effort. When computer code is involved, I share it through Github or other online repositories.

    I have presented at conferences in my region about my use of technology for teaching. This includes the EARCOS Teachers Conference in Bangkok, the 21st Century Learning conference in Hong Kong, and Learning 2.0. On my personal website, I post videos of these workshops and presentations so that anyone can benefit from what I have to share. I also have presented to my colleagues about mathematics, technology, and assessment.

    These experiences have led to invitations to join online communities for teacher education. I have collaborated with leaders in mathematics education to build online learning experiences for students around the world. I have spoken to online groups such as the Global Math Department, Global Physics Department, and a Google Hangout on computational thinking.

    In short, I am eager to share my ideas and learning with others. Doing so helps me develop as a teacher and stay active as a learner, which also lets me model life long learning for my students.


    Filed under teaching philosophy

    Dot Circle - An Introduction to Vectors

    After learning from Jessica Murk before our spring break about the idea of revising mathematical writing in class, I decided to try it as part of an introduction to the fourth topic in the IB Mathematics curriculum: vectors. The goal was to build a need for the information given by vectors and how they provide mathematical structure in a productive way.

    I started by adapting Dan Meyer's activity here with a new set of dots.

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    I asked all students to pick one dot, and then asked a student to give the class instructions on which one they picked. They did a pretty good job with it, but there was quite a bit of ambiguity in their verbal descriptions, as I wanted. This is when I sprung Dan's helpful second slide that made this process much easier:
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    Key Point #1: A common language or vocabulary makes it easy for us to communicate our ideas.

    I then moved on to the next task. Students individually had to write directions for moving from the red dot to the blue dot. I gave them this one to start as a verbal task, but nobody was willing to take the bait after the last activity:
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    Fair enough.

    I then gave one of the following images to each pairs of students, with nothing more than the same instruction to write directions from the red to the blue dot.

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    Screen Shot 2015-04-10 at 5.34.51 PM

    Screen Shot 2015-04-10 at 5.34.56 PM

    Screen Shot 2015-04-10 at 5.35.02 PM

    Here is a sampling:

    • Move across 5 dots on the outermost layer counter-clockwise, with the blue dot at the bottom of paper (closest to you)
    • Move 7 units to right, and move about (little less) 3 units up so that the blue dot is right on the vertical line
    • Fin the dot that is directly opposite to the red dot that is across the diagram. Once there, move down one dot along the outermost layer of dots.
    • Stay on the circle and move right for five units
    • Move from coordinate \frac{7 \pi}{6} to the coordinate of 2 \pi on the unit circle.

    After putting the written descriptions next to the matching image, students then rotated from image to image, and applied Jessica's framework for students giving written feedback for each description they saw.

    Here is some of the feedback they provided:
    Screen Shot 2015-04-10 at 6.05.13 PM

    Then, without any input from me, I had students sit down and each write a new description. Just as Jessica promised, the descriptions were improved after students saw the work of others and focused on what it means to give specific and unambiguous directions.

    This is where I hijacked the results for my own purposes. I asked how the background information I gave helped in this task? They responded with:

    • Grid/coordinate system in background of the dots
    • Circle connecting dots - use directions and circles to explain how to move
    • Connected all dots - move certain number of 'units'

    One student also provided a useful statement that the best description was one that could not be misinterpreted. I identified the blue dot as (3,0), and asked if anyone could give coordinates for the red dot. Nobody could. One student asked where (0,0) was. I pointed to some other points as examples, and eventually a student identified the red dot as (3,8). Another said it could also be (3,-5). I pointed out that if I had asked students to plot (3,-5) at the beginning of the class, the answer would have been totally different.

    This all got us to think about what information is important about coordinates, what they tell us, and that if we agree on common units and a starting point, the rest can be interpreted from there. This was a perfect place to introduce the concept of unit vectors.

    We certainly spent some time wandering in the weeds, but this ended up being a really fun way to approach the new unit.

    If you are interested, here is the PDF containing all of the slides:
    Point Circle

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    Filed under IB, teaching philosophy, teaching stories