## Perplexed in Monument Valley

I just finished playing Monument Valley. It's a beautiful game that uses M.C. Escher style impossible geometry to create a mesmerizing series of challenges moving a character around the screen. It's no wonder that Apple awarded it one of its coveted design awards this year.

This game evoked for me many of the ideas that Dan Meyer shared in his latest blog post and talk about video games and math class. Its tremendously easy to get started: tap on a spot, and the character goes there if doing so is possible. There are many ways to find a solution - an open middle, in his words.

One other attribute is that there is no sliding backwards, so there is little to no penalty for making mistakes. Once you've solved part of the puzzle, you never die and have to start over (unless you quit the game.) This means there's no practice involved (which is where the math class analogue of independent practice might be applied with care) which is a good thing. This game is consequently pure problem solving, replete with many moments of realization and discovery of how to make progress. I was in the position below for a while before realizing that a single rotation would allow me to reach the striped target and complete the level.

This game got me thinking about designing lessons with similar levels of constructive confusion and forcing new ways of thinking. If you are looking for an enjoyable way to feel the type of perplexity that we strive to offer our students, give the game a try. It's a few hours time very well spent.

If you haven't seen Dan Meyer's talk on using the structures of video games to make math class resemble things students like, you need to do so now. You could wait until after Christmas, I guess, but not too much longer.

There's an interesting mix of comments on that blog post. The thread that interests me most is that on the relationship between the story telling aspect of video games, and the equivalent story telling that happens in good math problems. I'm not convinced that there needs to be a good backstory for a game to be compelling, just as a real world context doesn't tend to be sufficient to get most students enthusiastic about a particular problem.

One comment from Kevin Hall, however, tapped into an idea I've also been mulling over since finding one of my own Choose-Your-Own-Adventure books during a trip back home in November. Here's Kevin writing in a comment on Dan's post:

I’ve thought about embedding videos in a Google Form so students can choose their own adventure and see the consequence of their choices. For example, if each pizza is \$6 and delivery is \$1.50, you could ask how much it would cost to get 2 pizzas delivered. If a student selected \$13.50, you’d take them to a video of a single delivery guy bringing 2 pizzas. If the student said \$15, you’d show a guy bringing 1 pizza, driving back to the pizza place, and bringing the other pizza separately. But it’s a lot of work and, I think, a critical aspect of making math more like video games.

The work of putting together such a task is not to be ignored. I do think though that getting students thinking about their thinking in a way that doesn't require whole class discussion is worth investigating. Some carefully crafted questions, ones that we might ask the entire class based on student responses, might also have some power for individual students to go through before sharing thoughts with others.

I also recently learned about an online tool called Twine that takes away some of the difficulty of putting these together. You can edit your adventure in the browser, link pages together without too much hassle, and add links to pictures or videos online using standard HTML. If you know Javascript, you can use it to add even more interactivity to the story. The tool allows you to piece together a truly individualized path.

I'm interested in piecing together some activities using Twine as a starting point for some explorations next semester. I've done things like this on paper before, but the limitations of paper are such that it's impossible to progressively reveal questions based on student responses. The way that Twine reduces the friction for doing this seems just enough to make this an option to explore. I'm writing this out now as a way to get some of you to push me to actually do it.

I'd love to see what happens when the math-twitter-blog-o-sphere gives Choose Your Own Adventure a try. Give it a go, and let me know what you create. I'll be here.

## Analyzing IB Physics Exam Language Programmatically

I just gave my IB physics students an exam consisting entirely of IB questions. I've styled my questions after IB questions on other exams and on homework. I've also looked at (and assigned) plenty of example questions from IB textbooks.

Just before the exam, students came to me with some questions on vocabulary that had never come up before. It could be that they hadn't looked at the problems as closely as they had before this exam. What struck me was that their questions were not on physics words. They were on regular English words that, used in a physics context, can have a very different meaning than otherwise. For these students that often use online translators to help in decoding problems, I suddenly saw this to be a bigger problem than I had previously imagined. An example: a student asked what it meant for an object to be 'stationary'. This was easily explained, but the student shook her head and smiled because she had understood its other meaning. On the exam, I saw this same student making mistakes because she did not understand the word 'negligible', though we had talked about it before in the context of multiple ways to say that energy was conserved. Clearly, I need to do more, but I need more information about vocabulary.

It got me wondering - what non-content related vocabulary does occur frequently on IB exams to warrant exposing students to it in some form?

I decided to use a computational solution because I didn't have time to go through multiple exams and circle words I thought students might not get. I wanted to know what words were most common across a number of recent exams.

Here's what I did:

• I opened both paper 1 and paper 2 from May 2014, 2013, 2012 (two time zones for each) as well as both papers from November 2013. I cut and pasted the entire text from each test into a text file - over 25,000 words.
• I wrote a Python script using the pandas library to do the heavy lifting. It was my first time using it, so no haters please. You can check out the code here. The basic idea is that the pandas DataFrame object lets you count up the number of occurrences of each element in the list.
• Part of this process was stripping out words that wouldn't be useful data. I took out the 100 most common words in English from Wikipedia. I also removed some other exam specific words like instructions, names, and artifacts from cutting and pasting from a PDF file. Finally, I took out the command terms like 'define','analyze','state', and the like. This would leave the words I was looking for.
• You can see the resulting data in this spreadsheet, the top 300 words sorted by frequency. On a quick run through, I marked the third column if a word was likely to appear in development of a topic. This list can then be sorted to identify words that might be worth including in my problem sets so that students have seen them before.

There are a number of words here that are mathematics terms. Luckily, I have most of these physics students for mathematics as well, so I'll be able to make sure those aren't surprises. The physics related words (such as energy, which appeared 177 times) will be practiced through doing homework problems. Students tend to learn the content-specific vocabulary without too much trouble, as they learn those words in context. I also encourage students to create glossaries in their notebooks to help them remember these terms.

The bigger question is what to do with those words that aren't as common - a much more difficult one. My preliminary ideas:

• Make sure that I use this vocabulary repeatedly in my own practice problems. Insist that students write out the equivalent word in their own language, once they understand the context that it is used in physics.
• Introduce and use vocabulary in the prerequisite courses as well, and share these words with colleagues, whether they are teaching the IB courses or not.
• Share these words with the ESOL teachers as a list of general words students need to know. These (I think) cut across at least math and science courses, but I'm pretty sure many of them apply to language and social studies as well.

I wish I had thought to do this earlier in the year, but I wouldn't have had time to do this then, nor would I have thought it would be useful. As the semester draws to a close and I reflect, I'm finding that the free time I'll have coming up to be really valuable moving forward.

I'm curious what you all think in the comments, folks. Help me out if you can.

## You Don't Know Your Impact Until You Do.

There comes a time, often at the end of the semester, when you look around your classroom once the students have left, and let out a big sigh.

Am I doing the right things?

Am I helping students grow in ways that are best for them?

Then you get an email from a former student that says things like this:

I got selected to be a part of a research group in the department of PHYSICS! Can you believe it? The one subject I did not like at all is the first research opportunity for me!
...
All these great opportunities wouldn’t have happened to me if you didn’t have patience to make me understand physics. I now understand why you wanted me to figure out how to approach a problem all by myself instead of telling me what to do step by step.

I never realized how important it is to be able to do more than calculations until recently because I have been helping out a friend with her chemistry homework. However, I feel like that is all I do - help her finish her homework instead of helping her understand how to analyze a problem before jumping to equations.

I don’t want her to jump to equations because, at the end of the day, chemistry is a science, not math. We use math to help us, but a calculated answer means nothing by itself. It is the ability to analyze and interpret numbers than differentiates us from computers. Going to back to my friend and her chemistry homework, I noticed a lot of things that she says that reminded me of myself and physics.

For example, she would say “I don’t get it, it seems so easy, but I just don’t know which equation to use.” Then when I try to guide her to figure out which equations to use, she just interrupts me with “Just tell me which equation to use, and I can do the math.”

Doesn’t that sound like me in physics class? It frustrates me how she takes such a mathematical approach to a scientific problem. I mean it’s great that she can do math, but so can the computer.

I am telling you about my experience because I want to first let you know how much I appreciate your patience with me, and second, I want to apologize for that things I said about physics. It must not have been very pleasant to hear someone talk about something you are obviously interested in in such an aggressive tone.

I am sorry for complaining about physics the way I did last year, and if you students in the future complain about a subject feel free to relate my experience with physics to them. Also, 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.

Moreover, do continue to do explorations with your students because they are so helpful when it comes to critical thinking....

...I know you always take the opinions of your students seriously, and I know that you have stepped away from doing explorations because our class had such a negative attitude towards them; however, knowing how to use a different program can help student develop their problem solving skills, which makes them a more competitive student.

If you know me at all, you know that this hits many of the questions I have about my own teaching. One perspective is certainly not every perspective. I'm certainly not going to stop questioning. That said, this message made me grin with pride. It means a lot to hear that something you do in the classroom enables students to make opportunities for themselves.

With the student's permission, I was eager to share the email as a way to help others remember why we do this job. You might never know the impact you have as a teacher until you do.

Keep this in mind as you approach the last teaching days of the year, everyone.

## Releasing my IB Physics & IB Mathematics Standards

Our school is in its first year of official IB DP accreditation. This happened after a year of intense preparation and a school visit last March. In preparation for this, all of us planning to teach IB courses the next year had to create a full course outline with details of how we would work through the full curriculum over the two years prior to students taking IB exams.

One of the difficulties I had in piecing together my official course outline for my IB mathematics and IB physics courses was a lack of examples. There are outlines out there, but they were either for the old version of the course (pre-2012) or from before the new style of IB visitation. The IB course documents do have a good amount of detail on what will be assessed, but not the extent to which it will be assessed. The math outline has example problems in the outline which are helpful, but this does not exist for every course objective. The physics outline also has some helpful details, but it is incomplete.

The only way I've found to fill in the missing elements is to communicate directly with other teachers with more experience and understanding of IB assessment items. While some of this has been through official channels (i.e. the OCC forums), most has been through my email and Twitter contacts. Their help has been incredible, and I appreciate it immensely.

At the end of the first semester for Mathematics SL, Mathematics HL (one combined class for both), and Physics SL/HL (currently only SL topics for the first semester), I now have the full set of standards that I've used for these courses in my standards based grading (SBG) implementation. I hope these get shared and accessed as a starting point for other teachers that might find them useful.

For my combined Mathematics SL/HL class:
Topics 1 - 2, IB Mathematics SL/HL

For my combined Physics SL/HL class:
Topics 1 - 2, IB Physics SL/HL

The third column in these spreadsheets has the heading 'IB XXXX Learning Objective' - these indicate the connection between the unit standard (e.g. Standard 3.1 is standard 1 of unit 3) to the IB Curriculum Standard (e.g. 2.3 is Topic 2, content item #3). Some of these have sub-indices that correspond with the item in the list of understandings in the IB document. IB Mathematics SL objective 1.3.2 refers to IB Topic 1, content item #3, sub-topic item #2.

If you need more guidance there, please let me know.

## If you are a new IB Mathematics/Physics teacher accessing these...

...please understand that this is my first year doing the IB curriculum. There will be mistakes here. In some cases, I also know that I'll be doing things differently in the future. If these are helpful, great. If not, check the OCC forums or teacher provided resources for more materials that might be helpful.

## If you are an experienced IB Mathematics/Physics teacher accessing these...

...I'd love to get your feedback given your experience. What am I missing? What do I emphasize that I shouldn't? What are the unspoken elements of the curriculum that I might not be aware of as a first year? Let me know. I'd love it if you could give me the information you wish you had (or may have had) to be maximally successful.

I've benefited quite a bit from sharing my materials and getting feedback from people around the world. I've also gotten some great help from other teachers that have shared their resources. Consider this instance of sharing to be another attempt to pay that assistance forward.

## Direct Instruction Videos - What's your Workflow?

I've written before about my experience recording my direct instruction into short, Udacity style videos and having students watch them during class. This enables me to circulate and have a lot more conversations with students as they are learning than when I'm talking at the front of the room. It also puts me in a position to see how my students are engaging with this material since I'm walking around and see what they are writing down, where they are stopping the videos, and can listen to their conversations. The quality of my interactions (and the student-to-student interactions) is so much higher with this approach.

The main obstacle to my doing this more, however, is the investment of time in creating the videos. With a consultant meeting with us this week and asking us to examine our technology practices, I'm wondering whether others have cracked the code and found ways to be efficient.

Most of my time is spent editing. I do one video at a time for each piece of what I want my students to watch before they try something on their own. I also want my videos to be short (ideally less than 3 minutes each), so I find I'm editing out spoken flubs, unclear descriptions, extra pauses, and time spent writing by hand to reach that ideal. Camtasia is my tool of choice. I know there are videos out there that I could assign rather than recording my own, but I'm convinced I can still work on my efficiency with some good advice.

I wonder if one of the following would work better:

• Record all of the writing with no narration first. Add voiceover second to match the text.
• Record all of the direct instruction for an entire class. Edit out flubs, writing, then split into multiple videos for a lesson.
• Write out all of the written parts before recording. Cut and paste them in the video frame one by one as I speak on top of the video. Gesture and highlight as needed.

I've sacrificed perfection for getting my ratio of recording time to video time down to about four to one. That's still a sizable investment of time, and it certainly benefits my students, but as is, I'm leaving the classroom after 5 PM pretty regularly.

Any experienced flipped classroom folks care to weigh in on this?

## Coding for the Classroom with Meteor Series: ImageShare

When I visited Meteor headquarters for their monthly DevShop, I gave a lightning talk on my use of the Meteor framework for developing tools that helped me do my job as a teacher. Both during and after my talk, I was asked how I thought I could help other teachers learn to do what I had done. I pledged at the time that it was my goal to make some videos and tutorials about how to use it for classroom specific applications. Since then, I've had some ideas for what I might do.

When I asked idea-man Dan Meyer what he thought the first project should be, the response came back surprisingly quick:

As usual, Dan's expectations were high. I was waiting for Meteor to release its 1.0 version before getting started, so when that happened this week, I hit the books interwebs hard to figure out how to make the response viewer a reality with Meteor.

Thankfully, it actually came together quite quickly. This is an amazing testament to the power that the Meteor framework has for minimizing the idea-to-app lifecycle, and making it easy to get these tools in the hands of teachers.

You can check out my 26 minute tutorial video below. I made it almost real-time (minus some edited video flubs) to show how quick it is to get started.

I have also included the files that I made in the tutorial on Github here:
https://github.com/emwdx/image-share

Take a look and let me know what you think. I would like to do others if there are requests for teaching-related apps out there. Keep me posted on what you would like to see.

## Computational Thinking in Teaching and Learning (Re-post)

A modified version of this post appeared on the Techsmith Blog here and in their quarterly newsletter, the Learning lounge. I appreciate their interest in my perspective. I hope to continue this important discussion here with my readers.

The idea of computational thinking has radically changed my approach to teaching over the past few years. This term, first coined by Jeanette Wing, a professor of computer science at Carnegie Mellon University, refers to several key ideas of thinking that are essential to computer science. The paper clearly identifies the reality that there are some tasks that computers do extremely well, and others that are better suited to the human brain. Traditionally, computer scientists have worked to outsource the calculating, organizing, searching, and processing work for task X to a computer so that they can focus on the more complex, challenging, and engaging aspects of the same task. According to Wing, one of the most essential skills we should develop in students is sorting tasks into these two groups.

My classroom, at its best, is a place where maximum time is spent with students wrestling with an engaging task. They should be working together to develop both intuition and understanding for required content. I can read the smiles or frowns and know whether I should step in. I can use my skills to nudge students in the right direction when I think they need it. Knowing precisely when they need it can't easily be determined by an algorithm. For some students, this moment comes early on after encountering a new concept. Others require just one more minute of struggle before the idea clicks and it's in their brains for good. Knowing the difference comes from the very human experience of time in classrooms with learners.

This is the human side of teaching. It is easy to imitate and approximate using technology, but difficult to produce authentically. Ideally, we want to maximize these personal opportunities for learning, and minimize the obstacles. For me, the computer has been essential to doing both, specifically, identifying the characteristics of tasks that a computer does better. If a computer can perform a task better than me or my students alone, I'm willing to explore that potential.

The most consistent application of this principle has been in the reduction of what I call 'dead time'. I used to define this as time spent on tasks required for learning to be possible, but not actually a learning task itself. Displaying information on the board, collecting student answers, figuring out maximum and minimum guesses for an estimation problem - these take time. These sorts of tasks - displaying, collecting, processing - also happen to be the sort at which computers excel. I wrote a small web application that runs from my classroom computer that allows students to snap a picture of their work and upload it to my computer, anonymously if they choose. We can then browse student answers as a class and have discussions about what we see. The end result is equivalent to the idea of students writing their work on the board. The increased efficiency of sharing this work, archiving it, and freeing up class time to build richer activities on top of it makes it that much more valuable to let the computer step in.

I've also dabbled in making videos of direct instruction, but I have students watch and interact with them while they are in the classroom. During whole class instruction, I can't really keep track of what each student is and isn't writing down because I am typically in a static location in the classroom. With videos simultaneously going throughout the classroom, I can see what students write down, or what they might be breezing through too quickly. I get a much better sense of what students are not understanding because I can read their faces. I can ask individualized questions of students to assess comprehension. The computer distributes and displays what I've put together or curated for my students – one of its strengths. My own processing power and observation skills are free to scan the room and figure out what the next step should be.

Letting the computer manage calculation (another of its strengths) enables students to focus on the significance of calculations, not the details of the calculations themselves. This means that students can truly explore and gain intuition on a concept through use of software such as Geogebra or a spreadsheet before they are required to manage the calculations themselves. For students that struggle with arithmetic operations, this enables them to still make observations of mathematical objects, and observe how one quantity affects another. This involvement has the potential to inspire these same students to then make the connections that underlie their skill deficiencies.

Full disclosure though: I don't have a 100% success rate in doing this correctly. I've invested time in programming applications that required much more effort than an analog solution. For instance, I spent a week writing all of my class handouts in HTML because the web browser seemed like a solution that was more platform independent than a PDF. That ended when I realized the technology was getting in the way of my students making notes on paper, a process I respect for its role in helping students make their own learning tools. There are some tasks that work much more smoothly (or are just more fun) using paper and a marker.

I value my student’s time. I value their thoughts. I want to spend as much class time as is possible building a community that values them as well. Where technology gets in the way of this, or adds too much friction to the process, I set it aside. I sit with students and tell stories. I push them to see how unique it is to be in a room for no other reason but to learn from each other. When I can write a program to randomize groups or roll a pair of dice a thousand times to prove a point about probability, I do so.

Knowing which choice is better is the one I wish I could write an algorithm to solve. That would take a lot of the fun out of figuring it out for myself.

## Sensors First - Progress Report

I wrote previously about my plans to change how I teach programming to my LEGO robotics students. By including sensor use as a starting point, my hope is to equip students with the experience to know when sensors can do a better job than simply aiming the robot toward the target and hoping for the best.

Yesterday was my first open ended challenge after beginning this approach. Students needed to build and program their robots to retrieve the loops located at the ends of the black line paths. The time available for them to do so was kept short. As one more way to advantage sensors over a trial and error approach, I told them that I might tell them to start their robot anywhere along the line, and that they could only pick up their robot once while retrieving the two loops.

I really didn't need that final requirement. Students quickly figured out how to adapt the line following tricks I taught them to this task. In a forty minute period, all of the teams made progress and were able to make contact with the loop using a collection mechanism.

The most satisfying result? Not a single group spent significant time aiming their robot. They clearly didn't feel the need, which is a step in the right direction.

## Standards Based Grading(SBG) and The SUMPRODUCT Command

I could be very late to the party finding this out. If so, excuse my excitement.

I gave a multiple choice test for my IB Physics course last week. Since I am using standards based grading (SBG), I wanted a quick way to see how students did on each standard. I made a manually coded spreadsheet eight years or so ago to do this. It involved multiple columns comparing answers, multiple logical expressions, and then a final column that could be tallied for one standard. Multiply that by the total number of standards...you get the drill.

I was about to start piecing together an updated one together using that same exhausting methodology when I asked myself that same question that always gets me going: is there a better way?

Of course there is. There pretty much always is, folks.

For those of you that don't know, the SUMPRODUCT command in Excel does exactly what I was looking for here. It allows you to add together quantities in one range that match a set of criteria in another. Check out the example below:

The column labeled 'Response Code' contains the formula '=1*(B6=E6)', which tests to see if the answer is correct. I wanted to add together the cells in F6 to F25 that were correct (Response Code = 1) and had the same standard as the cell in H6. The command in cell I6 is '=SUMPRODUCT((F6:F25)*(E6:E25=H6))'. This command is equivalent to the sum F6*(E6=H6) + F7*(E7=H6)+F8*(E8=H6)+...and so on.

If I had known about this before, I would've been doing this in some way for all of my classes in some way since moving to standards based grading. I've evaluated students for SBG after unit exams in the past by looking at a student's paper, and then one-by-one looking at questions related to each standard and evaluating them. The problem has been in communicating my rationale to students.

This doesn't solve the problem for the really great problems that are combinations of different standards, but for students that need a bit more to go on, I think this is a nice tool that (now) doesn't require much clerical work on my part. I gave a print out of this page (with column F hidden) to each student today.

Here is a sample spreadsheet with the formulas all built in so you can see how it works. Let me know what you think.
Exam Results Calculator