## Let ’em Talk

We started the topic of Venn diagrams in Math 9 this week. In a class of international school students (and perhaps any group of students) the range of knowledge on a given topic is all over the place given their different backgrounds and school histories.

The teacher-me of ten years ago would have done an overview of the concept of a Venn diagram. I would have started by asking questions about different parts of what was there in a Socratic fashion. It would have been full of questions that I had written down in my lesson plan designed to get students to think deeply about the content. Based on asking questions of a sample individual students, I would have gotten an idea of what the class knew. The students who knew the material already would either raise their hands and try to answer every question, or stay silent and answer every question on the worksheet in a matter of minutes. The students that didn’t know the concepts, but wanted to, would likely stay quiet until either I approached them or until they could ask a friend for help. The students that were used to being defeated by math class would pass the time by doodling, pretending to be involved, or by distracting their friends.

This isn’t the teacher I am today. I’ve written about the power of social capital in the room before, so this is nothing new, but I don’t tend to do the ‘topic overview’ style lesson anymore. The one or two students that nod while we go through material aren’t representative of the class. The strength of my experience in the classroom is being able to observe students working and know what to do next. I can’t do this while standing at the front of the room and speaking.

My approach now is, whenever possible, to make an item of the topic a conversation starter. I gave them this image of a Venn Diagram, which appears in a collection of questions from old New York State Regents exams at http://www.jmap.org:

I gave them a series of questions that required them to figure out what they remembered, knew, or didn’t know about the topic. Students made arguments for the definitions. Their disagreement drove the need for clearer definitions of what the intersections of the sets meant, for example. I was free to circulate and figure out who knew the concepts and who did not. Many of the issues that arose were resolved within the groups. Those that still had lasting confusion were my targets for conversations later on.

As I’ve added years to my experience, I’ve become more comfortable relying on this system to drive what happens in my classroom. Every time I get the urge to just go over a topic, I remind myself that there’s a better way that involves students doing the heavy lifting first. There’s a reason students are in a room together for the purpose of learning, and that reason is not (all) about efficiency. Humans are social creatures, and learning is one of those processes that is driven by that reality. There are moments when direct instruction is the way to go, but those moments are not as frequent or necessary as we might think at first.

## Teaching from Anywhere

I use my phone as a document camera, which is nothing new. AirDrop is an option since my school computer is now running OS X Yosemite. I was using my own Python web application to upload these to the computer last year, but that was limited to one file at a time. Now I can send a whole stack of photos of student work at once, which makes it the obvious choice.

The laptop is parked to be plugged into the projector in a spot that doesn’t sacrifice student real estate, but is accessible if I need to get to it:

The thing that has always bugged me is having to be in one place in the room to do, well, anything. I like sitting with students. I have interesting and useful conversations with students when I’m among them, not while standing at the front of the room. My solution in the past has been to bring the laptop around the classroom with me and sit down next to students. Two things bother me about this:

• When move to join a table next to students, I always take up more room than any other person. This is because I’m there with a laptop, Wacom tablet, and some notes if I need them for the lesson. My students are too polite to actually object when I move in and they always consolidate their things to make room. I know the whole time, however, that they are wishing I wouldn’t. This whole process repeats if I want to move during the lesson, which I always do.
• I have an Apple TV that I’ve used in the past to wirelessly display my screen in this situation, but the lag between my movement and the display is enough to be uncomfortable for me, and render my handwriting into the illegible range if I’m not extremely careful. I can stream student work to the Apple TV from my phone directly, but without the ability to zoom in on what’s actually important or annotate, the capability limits more than it offers.

I have had the wireless kit for my Wacom tablet since last year, so that doesn’t need to be connected to the projector laptop anymore. To switch applications (which I do frequently), write more than a couple words on the screen (which is more efficiently done through typing), or upload student work, I’ve always needed to go back to the laptop. This additional step during class is a moment of dead time – a moment during which students have no choice but to wait and do nothing, or do worse. This moment of dead time has been an unavoidable consequence of my classroom design and configuration.

The arrangement that has minimized (if not eliminated) all of these issues for this new year is this set of devices:

I already mentioned the wireless Wacom tablet for handwritten work. The wireless keyboard (picked up during RadioShack’s sale of excess inventory this summer) lets me type from anywhere in the room. The Magic Trackpad lets me do the rest.

I can take all three of these anywhere in the classroom if I need to, though often one at a time will suffice. I can switch applications, write on the wall, and type from pretty much anywhere. For sharing, viewing, and cropping student work, I can use the trackpad to manage the stream of photos that I (or my students) send to the computer through Airdrop.

This freedom to run my class untethered from the computer and centered wherever student thinking is happening is worth every ounce of aluminum, glass, and plastic. This freedom makes a difference.

## 2014-2015 Year-In-Review: Standards Based Grading

This was my third year using standards based grading with my classes. I wrote last year and the year before about my implementation.

## What did I do differently?

• I had my WeinbergCloud implementation working from the beginning of the year, so it was part of the expectations I introduced on day one.
• I also adjusted this system a bit to make it easier to link the reassessments and the content of the standards. There seemed to be too much uncertainty about what each standard represented, which translated into more confusion when signing up for reassessments than I wanted. Creating a list of standards and resources associated with each standard shrank this gap.
• I did not limit the number of reassessments per day explicitly. I expected that students would not sign up for a ridiculous number given the limitations on their credits, which students earned by doing homework or coming to tutoring.
• I included time within at least one class a week per student during which students could do reassessments without having to come in outside of class time.
• Unit exams continued to be assessed purely on course standards, not points. Semester final exams were percentage based.
• I scaled all of my standards levels from 1 – 5 to be from 6 – 10 to make it easier to communicate the levels to parents and be consistent with our school grading policy of not giving numerical grades below 50%. No student actually received lower grades due to my system of adding a base grade to each standard, but the process of explaining to students and parents that a 1 was really a 60% (5 for the base grade + 1 for the standard level) was clearly more complex than it needed to be.
• For my combined IB HL/SL class, the HL students had standards that only they were responsible for learning, while also being responsible for the SL standards. More on this later.

## What worked:

• Students seemed to have a better understanding from the beginning of the year of what standards based grading and assessment was all about. I did a bit more deliberate instruction on the ideas behind it at the beginning of the year. I also had smaller classes than before, so I was better able to have individual conversations about signing up for reassessments and talking about the process.
• A small proportion of students were fully sold on the idea of reassessment as a learning tool. Some students reassessed at least twice a week throughout the semester, and these students had strong performances on the cumulative final exams.
• By the second unit exam, students were generally not leaving questions blank on assessments. They were trying their best to do some amount of work on each question.
• As with last year, I gave more challenging questions to assess the range of student ability. Most of these involved either multiple standards combined in one, more open ended responses, or questions requiring explanation. Assessing at the higher levels of mastery became strongly subjective, and students accepted this, though they occasionally advocated for themselves as to why they deserved to be marked higher. They generally felt that it was fair when arithmetic errors kept them in the 8/10 range.
• Having students report their mastery level when signing up for a reassessment made it much easier for me to know what problem type or category to give them. Furthermore, this made it easier to justify changing the mastery level higher after a successful reassessment, but not making it the highest level on the scale. A student that was a 6 and answered a couple of questions correctly might move to an 8, whereas a student that was previously an 8 would be given more challenging questions and some conversation explaining their understanding in order to move to a 10.
• It was my priority to get assessments back within the same period, and I estimate that I was able to do this more than 95% of the time. Simple, short, and carefully designed assessments can reveal quite a bit about what students do/don’t understand.

## What needs work:

• Similar to previous semesters, I had high participation of a small group of students, with far too many students choosing not to reassess until the very end of each semester. Some students did not initiate their own reassessments at all.
• Students again hoarded their credits to the end of the semester. I flirted with the idea of adding an expiration date to credits to discourage holding on to credits for long periods of time, but time constraints kept me from implementing this.
• As a consequence of credit-hoarding, students near the end of the semester signed up for absurd numbers of reassessments in a day – I believe the largest quantity was nine. I shared with students that a good rule of thumb for planning purposes is 10 minutes per reassessment, so doing five reassessments before school isn’t practical, but that didn’t come across well. Students that couldn’t do all of their reassessments in the morning simply pushed them to later in the day. This was a problem for me because I never knew if students were going to show up according to their scheduled time, or just do everything after school. Canceling after no-shows at the end fixed this problem pretty efficiently, however.
• When a student would answer all questions correctly on an unannounced standards quiz, I generally assigned this a mastery level of 8 on a 6 – 10 scale. Students that had less than an 8 in this case usually had trouble with the same questions on a unit assessment or reassessment on the same standard later on. In other words, the students that had trouble initially learning a concept did not necessarily get the help they needed to make progress before the unit exam. This progress often happened after the exam, but this led to a lot of students falling behind pretty early on. I need to introduce interventions much earlier.

## Under consideration for next year:

These are the ideas I am mulling over implementing before school gets started in a month, and I’d love to hear what you think.

• Make credit expiration happen. This has been an issue for the year and a half of WeinbergCloud’s existence. I threatened implementing this in speaking with students, and they were immediately asking me not to because it would prevent them from putting off reassessments as they preferred to do. This includes students that were doing the practice problems between classes anyway, so this wasn’t just about losing the credits. Adding a “why not just give a reassessment a try” argument worked in face-to-face conversation with students that were hoarding credits, so forcing the process might be worth the effort. I understand that learning takes time, but many of the students putting off reassessment weren’t actively reviewing the standards over time any way. I’d rather force the feedback cycle through more iterations since that is when students seem to learn the most.
• Introduce submitting work into the process of reassessment. This could be electronic (“To complete your sign up, submit a scan/photo of the work you have done to prepare”) or could just be shown before I give them a reassessment. This would reduce some of the sign-ups that happen only based on the mastery score rather than reviewing the concepts that come with it. Students earn credits by doing practice problems or coming to tutoring, and these let them sign up for reassessments – this won’t change. To actually go the final step and take the reassessment, I need to see what students have done to prepare. In some cases (students that see me the day before, for example) I may waive this requirement.
• Require X number of reassessments per two week cycle of the block schedule. This might be in lieu of the previous change, but I’m afraid this might encourage (rather than prevent) a rush of reassessments at the end of a two week period. On the other hand, if the goal is to increase opportunities for feedback, this might be more effective.
• Make it possible for students to sign-up for an appointment to go over (but not be assessed) material on a given standard. Reassessments are great opportunities for feedback, but sometimes students want to come in to go over material. I get emails from students asking this, but it might be easier to just include this within WeinbergCloud.
• Introduce skills/definition standards for each unit. This would be a standard for each unit that covers basic recall of information. I’ll discuss why I want these (particularly in physics) in more detail within a later post. The short story is that I want to specifically assess certain concepts that are fundamental to all of the standards of a unit with a single binary standard.
• Classify standards mastery levels in terms of ‘likelihood of success’. This is a lower priority, and when I tried to explain this to a colleague, she wasn’t convinced it would be worth the effort. If you have a 10, it means you have a 95% or higher likelihood of answering anything I give you correctly. The probabilities might not scale linearly – a 9 might mean between 90-95%, an 8 between 75% and 90, etc. I don’t know. The reason I want to do this is to justify giving a 10 to students that have demonstrated solid proficiency without requiring perfection, and have a better reason for only raising a student from a 6 to an 8 after answering a couple questions on a single reassessment.

Right now the difference between an 8, 9, and 10 are defined (in order) by answering questions correctly on a single standard quiz, a comprehensive unit exam, and correctly answering stretch questions correctly. A student that gets an 8 on a standards quiz before an exam might then answers related questions incorrectly on the multi-standards exam and remains an 8. If this student then takes a quiz on a single standard and answers that question correctly, does it make sense to then raise their mastery level above 8? This is what I often do. I can also control for this by giving a more challenging question, but I’m not sure I need to.

In short, something is fishy here, and I need to think it out more in order to properly communicate it to students. In my head, I understand what I want to communicate: “yes, you answered these questions correctly, but I’m still not convinced that you understand well enough to apply the concepts correctly next time.” This is not the highest priority out of the ones I’ve mentioned here.

## Social Interactions and Time

Social work is important but social work will require, by its nature, more wait time than automated work.
–p. 131, Functionary: Learning To Communicate Mathematically In Online Environments by Dan Meyer

This quote from Dan’s dissertation gets to a theme of my lesson design this year. The time requirements of social interactions in the classroom are critical to honestly working them in to classroom routines. Dan is referring to the time required waiting for another students to refactor and resubmit a verbal description online. My takeaway from this point gets at a reality of making student socialization a tool for learning in the classroom.

Exit tickets at the end of the class are quick ways to assess specific skills presented during a class period, but they are essentially one way channels since they can’t be acted upon until next class. Time in class for lightly structured conversation around a lesson reveals understanding (or a lack thereof) is not just interactive for students, but allows me to hear a range of responses and parse them for what my students have learned. This conversation can be limited to small chunks of one or two minutes, so the payoff to investment ratio is big if those conversations are carefully designed and motivated.

Identifying what is and is not useful in those conversations is essential to working in an environment with peers. This is a valuable skill for students to develop. It’s difficult impossible to plan for every possible response students will have to everything that is said, and there will always be unexpected or off topic elements. This ‘noise’ can be managed but shouldn’t be eliminated. Doing so denies the ebb and flow of real conversations that students have outside our classrooms all the time. If we are to leverage socialization in our classrooms for learning, we have to acknowledge that the efficiency will never be perfect. This is especially the case as Dan’s research suggests that students best learn to communicate mathematically through revision and feedback.

I could go much faster through material if all I used was direct instruction. My students would be forced to be compliant to such a structure, and probably wouldn’t enjoy my class as much, which I’ve decided is important to me. It is satisfying as a teacher to see students working through their understandings without my help, and this can only happen if I provide time for it during class. Scheduling time for it is a way to show students that I value what comes out of these conversations.

## On Grant Wiggins

Like many others in the world of education, I was saddened by the loss of Grant Wiggins on May 26th. Before I begin my summer period of writing on what I’ve learned this year, it seems appropriate to share just how much Grant and his ideas helped shape my classroom into the place of learning it has become.

I was lucky to have met Grant when he came to my school in the Bronx in my fourth year teaching. My assistant principal at the time worked to bring him and was understandably excited to share the news of his approaching visit. I had not read Understanding by Design from start to finish in my education courses, but the principles described were frequently referenced. I was embarrassed to learn that I knew of Grant’s ideas but not his name. My wife pulled out her copy of UbD when I told her who was coming to visit us and pointed to Grant’s name on the cover, and I realized this wasn’t going to be just another disconnected day of PD staring at a PowerPoint presentation.

The time he spent with us began a transformative period of refining my planning process, possibly the most significant I’ve had over my twelve year career.

His beliefs around assessing content skills independently pushed me to experiment with standards based grading. His famous analogy identifying the distinction between practicing soccer skills and playing in a game revealed clearly the mismatch between the different types of assessments I was using and the mixed levels of success my students had on them. I experimented more with open ended problems to give my students the experience of playing the game of mathematics. I came to shed my fear of exposing students to problems that they hadn’t seen before, and instead embraced them as opportunities to expand student intuition around the associated skills. This shift away from the ‘skills first, application later’ philosophy became central to my teaching. It would take a bit longer for me to successfully integrate essential questions into my unit planning routine. I changed my lesson planning routine to be end goal oriented rather than being decided by sections in a textbook or pacing guide. It took longer to feel comfortable using essential questions to plan lessons, but I knew when I first learned about their power that I wanted to develop my ability to do so.

I also learned a great deal about the power of sharing ideas from reading Grant’s blog. It was clear that he saw his work helping teachers as a process leading them to discover these truths for themselves, and not as a keeper of secret knowledge to be doled out by buying the next book. He was always describing his experiences with teachers as they were developing their craft. He wrote openly about the struggles he faced along the way. When I started blogging myself, I felt obligated to service my own teaching through a similar level of honesty in writing. I was honored that he also discussed and shared my ideas on a couple occasions.

A colleague of mine once said that much of the professional development we receive as teachers is little more than stating the obvious. The ideas that Grant shared were not new, but they also were not what I was told from the beginning of my training as a teacher. They should have been. Start from the end, give students opportunities to think big, and assess authentically what you want your students to be able to do. Keeping these ideas at the front of my teaching has not always resulted in the outcomes I expected, but I love how they have shaped my priorities when sitting down to plan what comes next.

It is often the small shifts in thinking that make the big differences in what we do daily. I am thankful to Grant starting this process for me. I know his work lives on in the many classrooms that have been touched by his ideas, and students are the ultimate benefactors of the changes he promoted in our classrooms.
Thank you, Grant, for sharing your life with us.

## 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?

## 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.

### Update:

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.

## 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.

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:

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:

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.

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 $latex \frac{7 \pi}{6}$ to the coordinate of $latex 2 \pi$ 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:

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

## Formula Sheet – A Toolbox or Takeout Menu?

During the IB Exams, students get a set of equations and constants to use. Part of the motivation behind them is to reduce the amount of memorization required. There’s no sense in students memorizing Planck’s constant or the Law of Cosines in a context that emphasizes application of these ideas.

That said, I’ve heard variations on the following from different students just in the past three days:

• I thought I was right, then I looked at the formula sheet, and realized I was wrong. (She was right the first time.)
• I didn’t study it because I knew it was on the formula sheet.
• I don’t know what formula to use.

If you read my blog, you know that I don’t test formula memorization for all sorts of reasons. You get it. I get it. It has a place, but that place isn’t one I want to be spending my time.

You might also know that I’ve experimented with different versions of resources available to students during a test. I’ve done open note-card, open A4 sheet, open A5 sheet, open computer/closed network, open computer/open network, open notebook, and open people (i.e. a group test) formats.

I believe that the act of students creating their own formula sheets is more effective than handing one to them. The process of seeing how a formula is applied in different contexts and deciding what needs to be remembered is valuable on its own. Identifying that one problem is similar to another for reasons of physics shows understanding. I want to make opportunities for that to happen. Reducing the size of the resource requires students to prioritize. These are all high level skills.

The difficulty is that students see formulas directly as a pathway from problem to solution. Most problems worth solving don’t fit with that level of simplicity. Formula sheets give you the factual information, and rely on the user to know how to connect that information to a problem. The student thinks that the answer is staring at them in the face, and they just have to pick the right one. As teachers, we want students to identify information they need, then look at the reference to get it.

This is part of the reason I like standards based grading, as it justifies assessing students through conversation. A student asks me for a specific piece of information. If it’s how to calculate something, I’ll tell them if the related learning standard is about applying a concept, not calculating a quantity. If their request directly asks for the answer to the question, I don’t tell them. If they ask for a hint, I give them enough to get them moving, and adjust their proficiency level for the related standard according to the amount of help I give them.

In the long run, however, students need to know how to use the resources available to them. This is one of those big picture skills everyone talks about. Students need to know how to use Google to effectively find what they are looking for. They need to know that typing the text of a question into Yahoo Answers is not going to get them the answer they are looking for. I do know that if a student directly says “I can’t remember a formula for [ ]”, and I give them an equation sheet, they can usually find it. If they use the formula sheet as step one, they are not likely to complete the problem on their own. Having the sheet there in front of them makes it far too easy to start a problem that way. Would having students tally the number of times they looked at their sheet be enough of a feedback mechanism to keep this in check?

I don’t know what the answer is right now.

How do you help students treat a formula sheet more like a tool box, and less like a restaurant take-out menu?

## The (Un)changing Role of Teachers

I happened upon this article today from The Atlantic titled The Deconstruction of the K-12 Teacher. Here is a highlight from the article:

The relatively recent emergence of the Internet, and the ever-increasing ease of access to web, has unmistakably usurped the teacher from the former role as dictator of subject content. These days, teachers are expected to concentrate on the “facilitation” of factual knowledge that is suddenly widely accessible.

This line of reasoning inevitably comes up in my conversations with those that don’t teach, including those that have children currently in the system. What is the role of the teacher in today’s classroom?

My response usually pays lip-service to the idea that the role of teachers is certainly changing in response to the presence of technology. I think it’s obvious that is the case. I don’t believe that most of us are turning our classrooms into rows of students doing computerized lessons because of their effectiveness – that certainly isn’t he case either. My arguments for there being a place for teachers in the classroom surround the social situation that exists in having learners together in one place. In the best classrooms, historically, it has never really been about transferring knowledge from the front to the back.  It has instead always been about the community.

Here are my main ideas on this concept:

• Making the social network of the classroom into a learning resource requires careful planning and experience in managing the process.
• Students need to learn that it is normal to make mistakes along the road to understanding. This isn’t easy when done in isolation.
• Making big picture connections is done best in conversation with others having a diversity of experiences and understandings.
• Some skills are learned best in context with someone knowledgable in their use.
• Asking a question of a source you know and trust is easier than taking a shot in the dark on an online forum or through a chat window.

I’m not saying these processes can’t be completed online. Our students certainly have experience communicating through online channels. They need our guidance as teachers in using these networks for learning, however, and the classroom is a great place to give them that guidance. In light of the social, emotional, and finally academic needs of teenagers, I think we will be needed for a while yet before computers can fully take over the classroom for good.