## 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 https://emwdx.github.io/groupData/index.html 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.

## 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.

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.

## 2015-2016 Year in Review: IB Mathematics SL/HL

This was my second year working in the IB program for mathematics. For those that don't know, this is a two year program, culminating in an exam at the end of year two. The content of the standard level (SL) and higher level (HL) courses cross algebra, functions, trigonometry, vectors, calculus, statistics, and probability. The HL course goes more into depth in all of these topics, and includes an option that is assessed on a third, one-hour exam paper after the first two parts of the exam.

An individualized mathematics exploration serves as an internally assessed component of the final grade. This began with two blocks at the end of year one so that students could work on it over the summer. Students then had four class blocks spread out over the first month of school of year two two work and ask questions related to the exploration during class.

I taught year one again, as well as my first attempt at year two. As I have written about previously, this was run as a combined block of both SL and HL students together, with two out of every five blocks as HL focused classes.

## What worked:

• I was able to streamline the year 1 course to better meet the needs of the students. Most of my ability in doing this came from knowing the scope of the entire course. Certain topics didn't need to be emphasized as I had emphasized in my first attempt last year. It also helped that the students were much better aware of the demands of higher-level vs. standard level from day one.
• I did a lot more work using IB questions both in class and on assessments. I've become more experienced with the style and expectations of the questions and was better able to speak to questions about those from students.
• The two blocks on HL in this combined class was really useful from the beginning of year one, and continued to be an important tool for year two. I don't know how I would have done this otherwise.
• I spent more time in HL on induction than last year, both on sums and series and on divisibility rules, and the extra practice seemed to stick better than it did last year in year one.
• For students that were self starters, my internal assessment (IA) schedule worked well. The official draft submitted for feedback was turned in before a break so that I had time to go through them. Seeing student's writing was quite instructive in knowing what they did and did not understand.
• I made time for open ended, "what-if" situations that mathematics could be used to analyze and predict. I usually have a lot of this in my courses anyway, but I did a number of activities in year one specifically to hint at the exploration and what it was all about. I'm confident that students finished the year having seen me model this process, and having gone through mini explorations themselves.
• After student feedback in the HL course, I gave many more HL level questions for practice throughout the year. There was a major disconnect between the textbook level questions and what students saw on the HL assessments, which were usually composed of past exam questions. Students were more comfortable floundering for a bit before mapping a path to a solution to each problem.
• For year two, the exam review was nothing more than extended class time for students to work past papers. I did some curation of question collections around specific topics as students requested, but nearly every student had different needs. The best way to address this was to float between students as needed rather than do a review of individual topics from start to finish.
• The SL students in year two learned modeling and regression over the Chinese new year break. This worked really well.
• Students that had marginally more experience doing probability and statistics in previous courses (AP stats in particular) rocked the conditional probability, normal distribution, and distribution characteristics. This applied even to students who were exposed to that material, but did poorly on it in those courses. This is definitely a nod to the idea that earlier exposure (not mastery) of some concepts is useful later on.
• Furthermore, regarding distributions, my handwaving to students about finding area under the curve using the calculator didn't seem to hurt the approach later on when we did integration by hand.
• This is no surprise, but being self sufficient and persevering through difficult mathematics needs to be a requirement for being in HL mathematics. Students that are sharp, but refuse to put in the effort, will be stuck in the 1-3 score range throughout. A level of algebraic and conceptual fluency is assumed for this course, and struggling with those aspects in year one is a sign of bigger issues later on. Many of the students I advised this way in year one were happier and more successful throughout the second year.
• I successfully had students smiling at the Section B questions on the IB exam in the slick way that the parts are all connected to each other.

## What needs work:

### For year one:

• I lean far too hard on computer based solutions (Geogebra, Desmos) than on the graphing calculator during class. The ease of doing it these ways leads to students being unsure of how to use the graphing calculator to do the same tasks (finding intersections and solutions numerically) during an assessment. I definitely need to emphasize the calculator as a diagnostic tool before really digging into a problem to know whether an integer or algebraic solution is possible.
• Understanding the IB rounding rules needs to be something we discuss throughout. I did more of this in year one on my second attempt, but it still didn't seem to be enough.
• ### For year two:

• Writing about mathematics needs to be part of the courses leading up to IB. Students liked the mini explorations (mentioned above) but really hated the writing part. I'm sure some of this is because students haven't caught the writing bug. Writing is one of those things that improves by doing more of it with feedback though, so I need to do much more of this in the future.
• I hate to say it, but the engagement grade of the IA isn't big enough to compel me to encourage students to do work that mattered to them. This element of the exploration was what made many students struggle to find a topic within their interests. I think engagement needs to be broadened in my presentation of the IA to something bigger: find something that compels you to puzzle (and then un-puzzle) yourself. A topic that has a low floor, high ceiling serves much more effectively than picking an area of interest, and then finding the math within it. Sounds a lot like the arguments against real world math, no?
• I taught the Calculus option topics of the HL course interspersed with the core material, and this may have been a mistake. Part of my reason for doing this was that the topic seemed to most easily fit in the context of a combined SL/HL situation. Some of the option topics like continuity and differentiability I taught alongside the definition of the derivative, which is in the core content for both SL and HL. The reason I regret this decision is that the HL students didn't know which topics were part of the option, which appear only on a third exam section, Paper 3. Studying was consequently difficult.
• If for no other reason, the reason not to do a combined SL/HL course is that neither HL or SL students get the time they deserve. There is much more potential for great explorations and inquiry in SL, and much more depth that is required for success in HL. There is too much in that course to be able to do both courses justice and meet the needs of the students. That said, I would have gone to three HL classes per two week rotation for the second semester, rather than the two that I used throughout year one.
• The HL students in year two were assigned series convergence tests. The option book we used (Haese and Harris) had some great development of these topics, and full worked solutions in the back. This ended up being a miserable failure due to the difficulty of the content and the challenge of pushing second semester seniors to work independently during a vacation. We made up some of this through a weekend session, but I don't like to depend on out-of-school instruction time to get through material.

Overall, I think the SL course is a very reasonable exercise in developing mathematical thinking over two years. The HL course is an exercise in speed and fluency. Even highly motivated students of mathematics might be more satisfied with the SL course if they are not driven to meet the demands of HL. I also think that HL students must enjoy being puzzled and should be prepared to use tricks from their preceding years of mathematics education outside of being taught to do so by teachers.

## QuestionBuilder: Create and Share Randomized Questions

I've written previously about my desire to write randomized questions for the purpose of assessment. The goal was never to make a worksheet generator - those exist on the web already. Instead, I wanted to make it easy to create assessment questions that are similar in form, but different enough from each other that the answers or procedures to solve them are not necessarily identical.

Since January, I've been working on a project called QuestionBuilder. It's a web application that does the following:

• Allows the creation of assessment questions that contain randomized elements, values, and structures.
• Uses regular Javascript, HTML, and the KaTEX math rendering library to create and display the questions
• Makes it easy to share questions you create with community members and build upon the work of others to make questions that work for you.

Here's a video in which I convert a question from the June 2016 New York State Regents exam for Algebra 2 Common Core into a randomized question. Without all of my talking, this is a quick process.

I've put a number of questions on the site already to demonstrate what I've been using this to do. These range from simple algebra to physics questions. Some other folks I appreciate and respect have also added questions in their spare time.

For now, you'll need to create an account and log in to see these questions in action. Go to http://question-builder.evanweinberg.org, make an account, and check out the project as it exists at this point.

My hope is to use some time this summer to continue working on it to make it more useful for the fall. I'll also be making some other videos to show how to use the features I've added thus far. Feel free to contact me here, through Twitter (@emwdx), or by email (evan at evanweinberg.com) if you have questions or suggestions.

## Endings and Beginnings

Today, I bid farewell to my home away from home for the past six years.

When I first moved away from New York, I had shed all doubts that the teaching career was for me. I knew that learning and exploring were important elements of a meaningful existence on this planet, both for me and my students. I knew that few things were more satisfying than spending time with good people around plates of food. I knew that not knowing the local language or the location of the nearest supermarket was a cause for excitement, not fear. Purposely putting one's self into situations with unknown outcomes is not a reckless act. It is precisely these challenges that define and refine who we are so that we are better prepared for those events that we do not expect.

I knew these things already. And yet, I leave China today as a changed teacher. I met students from all around the world. I made connections not just with new people in the same building as me, but with teachers in many distributed time zones. People that I respected and admired for their ideas humbled me as they invited me to join in their conversations and explore ideas with me. I found opportunities to present at conferences and get to know others that had also fallen in love with the international teaching lifestyle. I started this blog, and surprisingly, had people read it with thoughts of their own to share.

I also learned to accept the reality that life continues in twenty four time zones. News from home made it seem more foreign and paradoxically more connected to my own experiences here. When opening my eyes and my various devices in the morning to see what had happened while I slept, I again never knew what to expect. I lost family members both suddenly and over stretches of time. Kids grew up. Our parents sold their houses and apartments. Friends put prestigious letters at the end of their names.

Our world changed as well. We added new countries to our passports and got lost in cities that refused to abide by a grid system. We fell in love with our dog and his aggressive sneezing at harmless bystanders. We tried to address the life and work balance through weeknight dinners and mini vacations. We repeatedly overcommitted to traveling during our summers off and time went too quickly. We became parents.

I write this not because anything I'm saying is especially new. The 'time marches on' canon is well established. That does not invalidate the reality that we're all experiencing life and its passage for the first time ourselves. This is the magic that we, as teachers, witness between the end of one year and the beginning of the next. We tweak our lessons from the previous year with the hope that they prompt more questions and productive confusion on the next iteration. Our students do experience some of the ideas we introduce for the first time in our classrooms, and it is unique that we get to design those experiences ourselves.

The best way to understand the rich range of emotions that our students experience while in our care is to live deeply and richly in our own lives. We need to learn to know and love others, explore and make mistakes, and be ready to move forward even when the future is uncertain. My time abroad thus far has given me numerous journeys through these human experiences. I would not give them up for the world, and luckily, I do not have to do so.

I'll write more about my next move in a future post.
Until then, I wish you all a summer full of good times with good people.

## Hacking The 100-Point Scale - Part 1

One highlight of teaching at an international school is the intersection of many different philosophies in one place. As you might expect, the most striking of these is that of students comparing their experiences. It's impressive how the experienced students that have moved around quickly learn the system of the school they are currently attending and adjust accordingly. What unites these particularly successful students is their awareness that they must understand the system they are in if they are to thrive there.

This is the case with teachers, as we share with each other just as much. We discuss different school systems and school structures, traditions, and assessment methods. Identifying the similarities and differences in general is an engaging exercise. In general, these conversations lead to a better understanding of why we do what we do in the classroom. Also, in general, these conversations end with specific ideas for what we might do differently on the next meeting with students.

There is one important exception. No single conversation topic has caused more argument, debate, and unresolved conflict at the end of a staff meeting than the use of the 100-point scale.

The reason it's so prevalent is  that it's easy to use. Multiply the total points earned by 100, and then divide by the total possible points. What could go wrong with this system that has been used for so long by so many?

There a number of conversation threads that have been particularly troublesome in our international context, and I'd like to share one here.

### "A 75 isn't a bad score."

For a course that is difficult, this might be true. Depending on the Advanced Placement course, you can earn the top score of 5 on the exam by earning anywhere between around 65% and 100% of the possible points. The International Baccalaureate exams work the same way. I took a modern physics exam during university on which I earned a 75 right on the nose. The professor said that considering the content, that was excellent, and that I would probably end up with an A in the course.

The difference between these courses and typical school report cards is that the International Baccalaureate Organization (IBO), College Board, and college professor all did some sort of scaling to map their raw percentages to what shows up on the report card. They have specific criteria for setting up the scaling that goes from a raw score to the 1 - 5 or 1 - 7 scores for AP or IB grades respectively.

What are these criteria? The IBO, to its credit, has a document that describes what each score indicates about a student with remarkable specificity. Here is their description of a student that receives score of 3 in mathematics:

Demonstrates some knowledge and understanding of the subject; a basic sense of structure that is not sustained throughout the answers; a basic use of terminology appropriate to the subject; some ability to establish links between facts or ideas; some ability to comprehend data or to solve problems.

Compare this to their description of a score of 7:

Demonstrates conceptual awareness, insight, and knowledge and understanding which are evident in the skills of critical thinking; a high level of ability to provide answers which are fully developed, structured in a logical and coherent manner and illustrated with appropriate examples; a precise use of terminology which is specific to the subject; familiarity with the literature of the subject; the ability to analyse and evaluate evidence and to synthesize knowledge and concepts; awareness of alternative points of view and subjective and ideological biases, and the ability to come to reasonable, albeit tentative, conclusions; consistent evidence of critical reflective thinking; a high level of proficiency in analysing and evaluating data or problem solving.

I believe the IBO uses statistical and norm referenced methods to determine the cut scores between certain score bands. I'm also reasonably sure the College Board has a similar process. The point, however, is that these bands are determined so that a given score matches

The college professor used his professional judgement (or a bell curve, I don't actually know) to make his scaling. This connects the raw score to the 'A' on my report card that indicated I knew what I was doing in physics.

The reason this causes trouble in discussions of grades in our school, and I imagine in other schools as well, is the much more ill-defined definition of what percentage grades mean on the report card. Put quite simply, does a 90% on the report card mean the student has mastered 90% of the material? Completed 90% of the assignments? Behaved appropriately 90% of the time? If there are different weights assigned to categories of assignments in the grade book, what does an average of 90% mean?

This is obviously an important discussion for a school to have. Understanding the meaning of the individual percentage grades and what they indicate about student learning should be clear to administrators, teachers, parents, and most importantly, the students themselves. These is a tough conversation.

Who decided that 60% is the percentage of the knowledge I need to get credit? On a quiz on tool safety in the maker space, is 60% an appropriate cut score for someone to know enough? I say no. On the report card, I'd indicate that a student has a 50 as their grade until they demonstrate he or she can get 100% of the safety questions correct. Here, I've redefined the grade in the grade book as being different from the percentage of points earned, however. In other words, I've done the work of relating a performance measure to a grade indicator. These should not be assumed to be the same thing, but being explicit about this requires a conversation defining this to be the case, and communication of this definition to students and teachers sharing sections of the same course.

Most of this time, I don't think there is time for this conversation to happen, which is the first reason I believe this issue exists. The second is the fact that a percentage calculation is mathematically simple and understood as a concept by students, teachers, and parents alike. Grades have been done this way for so long that a grade on the 100-point scale is generally assumed to be this percentage mastered or completed concept.

This is too important to be left to assumption. I'll share more about the dangers of this assumption in a future post.

## My Journey with Meteor as a Teacher-Coder

Many of you may know about my love for Meteor, the Javascript framework that I've used for a number of projects in and around the classroom. I received an email this morning alerting me (and the many other users) that the free hosting service they have generously offered since inception would be shutting down over the next month.

To be honest, I'm cool with this decision. I think it's important to explain why and express my appreciation for having access to the tool for as long as I have.

I started writing programs to use in my classroom in earnest in 2012. Most of these tended to be pretty hacky - a simple group generator and a program to randomly generate practice questions on geometric transformations were among these early ones. The real power I saw for these was the ability to collect, store, and filter information that would be useful for teaching so that I could focus my time on using that information to decide on the next steps for my students. I took a Udacity course on programming self-driving cars and on web applications and loved what I learned. I learned to use some Python to take some of the programs I had written early on and run them within web pages. I built some nifty online activities inspired by the style of Dan Meyer and put them out for others across the world to try out. (Links for these Half-Full and Shapes tasks are below.) It was astounding how powerful I felt being able to take something I created and get it out into the internet wilderness for others to see.

It was also astounding how much time it took. I learned Javascript to manage the interactivity in the web page, and then once that was working, I switched to Python on the server to manage the data coming from users. For those that have never done this sort of switching, it involves a lot of misplaced semicolons, tabs, and error messages. I accepted that this was the way the web worked - Javascript in front, and Python (or PHP, Rails, Perl, etc.) on the back end. That extra work was what kept someone like me from starting a project on a whim and putting it together. That cost, in the midst of continuing to do my actual job of teaching and assessing students five days a week, was too great.

This was right around the summer of 2013 when a programmer named Dave Major introduced me to Meteor. I did not know the lingo of reactivity or isomorphic Javascript - I just saw the demonstration video on YouTube and thought it was cool. It made the connection between the web page and the server seamless, eliminating the headaches I mentioned earlier. Dave planned to put together some videos and tutorials to help teachers code tools for the classroom using Meteor, and I was obviously on board. Unfortunately, things got in the way, and the video series didn't end up happening. Still, with Dave's help, I learned a bit about Meteor and was able to see how easy it was to go from an idea to a working application. I was also incredibly impressed that Meteor made it easy to get an application online with one line: meteor deploy (application-name here) . No FTP, no hostname settings - one line of code in the terminal, and I could share with anybody.

With that server configuration friction eliminated, I had the time to focus on learning to build truly useful tools for myself. I created my standard based grading system called WeinbergCloud that lets students sign up for reassessments, earn credit for the homework and practice they did outside of class, and see the different learning objectives for my course. I created a system for my colleagues to use to award house points for the great things that students did during the school day. I made a registration and timing system for our school's annual charity 5K run that reduced the paperwork and time required of our all volunteer staff to manage the hundreds of registrants. I spoke at a Meteor DevShop about this a year and a half ago and have continued to learn more since then.

Most importantly to me, it gave me knowledge to share with a class of web programming students, who have learned to create their own apps. One student from last year's class learned about our library media specialist's plan to hold a read-a-thon, and asked if he could create an interactive website to show the progress of each class using, you guessed it, Meteor. Here's a screenshot of the site he created in his spare time:

And yes, all of these apps have been hosted on the free deploy server at *.meteor.com, and yes, I will have to do the work of moving these sites to a new place. The public stance from Meteor has been that the free site should not really be used for production apps, something I've clearly been doing for over two years now. I re-read that line on the documentation website back in January and asked myself what I would do if I no longer had access to that site. The result: I did what I am paid to do as a master learner, and learned to host a site on my personal server. That learning was not easy. The process definitely had me scratching my head. But it also meant that I had a better understanding of the value that the free site had given me over my time using it.

The reality is that Meteor has clearly and publicly shifted away from being just being that framework that has a free one line deployment. The framework has so much going for it, and the ability to create interesting apps is not going away. The shift toward doing what one does best requires hard choices, and the free site clearly was something that did not serve that purpose. It means that those of us that value the free deploy as a teaching tool can seek other options for making it as easy to get others in the game as it was for us.

Meteor has helped me be better at my job, and I appreciate their work.

As promised, here are those learning task sites I mentioned before:

## Choosing the Next Question

If a student can solve $3x - 1 = 5$ for x, how convinced are we of that student's ability to solve two step equations?

If that same student can also solve $14 = 3x + 2$ , how does our assessment of their ability change, if at all?

What about $-2-3x= 5$ ?

Ideally, our class activities push students toward ever increasing levels of generalization and robustness. If a student's method for solving a problem is so algorithmic that it fails when a slight change is made to the original problem, that method is clearly not robust enough. We need sufficiently different problems for assessing students so that we know their method works in all cases we might through their way.

In solving $3x-1 = 5$ , for example, we might suggest to a student to first add the constant to both sides, and then divide both sides by the coefficient. If the student is not sure what 'constant' or 'coefficient' mean, he or she might conclude that the constant is the number to the right of the x, and the coefficient is the number to the left. This student might do fine with $10 =2x-4$ , but would run into trouble solving $-2-3x = 5$ . Each additional question gives more information.

The three equations look different. The operation that is done as a first step to solving all three is the same, though the position of the constant is different in all three. Students that are able to solve all three are obviously proficient. What does it mean that a student can solve the first and last equations, but not the middle one? Or just the first two? If a student answers a given question correctly, what does that reveal about the student's skills related to that question?

It's the norm to consider these issues in choosing questions for an assessment. The more interesting question to me theses days is that if we've seen what a student does on one question, what should the next question be? Adaptive learning software tries to do this based on having a large data set that maps student abilities to right/wrong answers. I'm not sure that it succeeds yet. I still think the human mind has the advantage in this task.

Often this next step involves scanning a textbook or thinking up a new question on the spot. We often know the next question we want when we see it. The key then is having those questions readily available or easy to generate so we can get them in front of students.