## Reflections on EARCOS Teachers Conference 2012 - Friday

I decided to use a few digital tools to record my thoughts at the EARCOS conference. At other workshops, I tend to take notes on paper, leave them in a folder, and possibly go back to them when inspiration hits, if I remember I have them. Since I am on my computer so much of the time (and NOT digging around in a filing cabinet to see what is in there) I think this will keep the ideas from this conference fresh and nearby.

I attended a few fantastic workshops Friday and tweeted extensively about each one as important ideas came up. The #earcos12 archive and search function will be really useful for going back and reminding myself of the ideas that came to mind during those workshops.

### Workshop 1 - The Geometer's Sketchpad Workshop: Beyond Geometry with Nicholas Jackiw

It was really a treat hearing the person that defined dynamic geometry talk about the philosophy of his software that implements the model. Having learned mathematics using GSP back in 9th grade, I've always seen the dynamic geometry as a natural lens through which geometric concepts can be viewed. Nick mentioned that mathematicians initially had a problem with the concept because two triangles with vertices A,B, and C that aren't congruent are not the same. Since dynamic geometry defines triangles in terms of the relationships of vertices, two triangles with the same vertices connected in the same way represent the same geometric object. This means that any triangle ABC can be turned into any other triangle ABC just by dragging vertices around the screen.

We went through the basics of plotting points, lines, and measuring slope using the tools of Geometer's Sketchpad. I hadn't used it for a while, but it still remains a great program. Nick is a genuine guy with a love for mathematics and what his software can do for students learning concepts. He has a solid grasp and had some great activities that could be used for students to actively learn concepts through exploration rather than listening to a teacher go through a list of boring definitions.

I had the pleasure of sharing with Nick that I used Geometer's Sketchpad to use geometry in ninth grade and that I still had print outs of the assignments I did using the software. Back then we printed out computer assignments and turned them in, much different from today when turning things in electronically is quite easy. I was a little star-struck talking to him, but as with most good teachers I meet, he was really friendly and appreciative of my comments.

### Workshop 2 - The Harkness Method: The Best Class You Never Taught - Alexis Wiggins

One of the things I want help doing is improving the quality of classroom discussions. The shelf life of the discussions we have isn't much longer than the class period itself. I have been able to extend that a bit having students create wiki pages, interact onor create videos describing their understanding of problems.

I think this workshop provided a real possibility for restructuring my class to do this far more effectively.

Alexis shared how the Harkness method (originated at Exeter Academy) has transformed her classroom and itneraction with students. Students spend class time discussing, arguing, and critiquing arguments. In the process, they learn extensively how to be good community members, be constructive in their criticism, and communicate their ideas. She shrewdly hooked us math/science teachers at the beginning (why are we always the cynics?) by sharing that Exeter does this in their math department. Alexis also shared that she does need to do direct instruction once in a while - her ratio is around 60% discussion, 40% other methods. She also does not do this for the entire class period, particularly for the younger (9th grade) students. Modeling the process and explicitly teaching students skills that make this successful in her room is a key part of her process. She made clear that it takes time to get them to be good at it.

Alexis posted her materials at https://alexiswigginsharknessmethod.pbworks.com.

### Workshop 3 - Rules of Engagement - Using Technologies to Motivate Rather than Distract - Doug Johnson

We are constantly having discussions at our school (which is 1:1 Macbooks) about how to maximize student time on task during class - I think this is something almost everyone in schools is currently battling. The presence of technology has so many potential positive applications for learning. It is easy, however, to fixate on the negative aspects almost entirely and stall the process of making these potential benefits available to students in the classroom.

Along with having one of the most useful handouts I've ever received at a workshop, Doug Johnson made a number of fantastically relevant points about how school communities can think about the issue. The question he posed at the beginning was "How do teachers compete w/ tablets, smart phones, netbooks, mp3 players, portable games, etc?" What I found most interesting throughout was that he showed how it didn't need to be a competition. Instead teachers can capitalize on the opportunity

His emphasis on the distinction between entertainment and engagement really resonated with me, as I always wonder if the activities I do with my students are actually helping them learn or not. We then worked together to identify ways to make technology an active part of classroom activities, including a lot of modeling using gosoapbox.com and references to other similar sites such as socrative.com.

Doug's presentations can all be found at https://dougjohnson.wikispaces.com/engage

### Workshop 4 - Digital Citizenship: The Forgotten Fundamental Kim Cofino

This workshop from the excellent Kim Cofino was a perfect pairing with Doug's workshop and a good ending point for the day. She clearly described her process at the Yokohama International School of rolling out (all at once, which she said was the best idea ever) her 1:1 laptop program with students.

The most important takeaway was how much deliberate planning and community collaboration went into not only creating the acceptable use policy but actively sharing that philosophy with the students, teachers, and parents. The school year started with two days of 1:1 boot camp activities - students discussing and debating different aspects of the policy. She also mentioned that the students will soon repeat some elements of this training and discussion now that the community has been through several months of living out the policy.

An important element of this is that students are explicitly taught and engaged in activities that teach them digital citizenship. She made clear that this does not happen by accident, or by hoping that students will know how to act when they are suddenly given the power afforded them by technology. This is one of the key things I will be taking back with me to Hangzhou.

Her presentation and resources can be found at http://dctff.wikispaces.com/overview

This has been a really fantastic experience being at the conference this year - I am learning so much at the workshops and through meeting the incredible collection of teachers here. I appreciate that everyone has been so positive and open in sharing their work and ideas with me. I admit it - I'm addicted to this conference atmosphere. Thankfully, I'll be able to keep in touch with the people I have met here, and continue learning from them well after I have left Bangkok.

## Party games & geometry definitions

Today's geometry class started with a new random arrangement of student seats. It never fails to amaze me how the dynamics of the whole room change with a shuffle of student locations.

The lesson today was the first of our quadrilateral unit. Normally after tests, I don't tend to have homework assignments, but I decided to make an exception with a simple assignment:

Create a single Geogebra file in which you construct and label all of the quadrilaterals given in the textbook: parallelogram, rhombus, square, kite, rectangle, trapezoid, and isosceles trapezoid.

This appealed to me because I really dislike lessons in which we go through definitions slowly as a group. I also knew that giving the students some independence in reviewing or learning the definitions of these quadrilaterals was a good thing. Sometimes they are a bit to reliant on me to give them all the information they need. For this assignment, students would need to understand the definitions of quadrilaterals in order to construct them, and that was a good enough for walking into class today.

The warm-up activity involved looking at unlabeled diagrams of quadrilaterals, naming them, and writing any characteristics they noticed about them from the diagrams:

Some had trouble with the term 'characteristics', but a peek down at the chart just below on the paper helped them figure it out:

Based on what they knew from the definitions before class, I had them complete this chart while talking to their new partner. There was lots of good conversation and careful use of language for each listed characteristic.

This led to the next thing that often serves as an important (though often boring) exercise: new vocabulary. I used one of my favorite activities that gets students focused on little details - each student received one of the following four charts. The chart is originally from p. 380 of the AMSCO Geometry textbook, and was digitally ruined using GIMP.

The students had a good time filling in the missing information and conferring with each other to make sure they had it all. We then came up with some examples of consecutive vertices, angles, diagonals, and opposite sides.

From their work with the chart and using the new vocabulary whenever possible, we then did the following:

What information would you need in order to prove that a quadrilateral is... (use as much of the new vocabulary as possible!)

• a square?

• a rhombus?

• a parallelogram?

• a rectangle?

• a trapezoid? (an isosceles trapezoid?)

• a kite?

I was really pleased with how they did with this exercise - they really seemed to be interacting with the definitions and vocabulary well.

Finally, we arrived at the part that was the most fun. You know that annoying ice-breaker you sometimes are forced to do at professional development sessions where you wear something on your head and have to get the other attendees to tell you who you are?

I hate that activity. That usually means it's perfect for my students:

The students were all smiles during the ten minutes or so we spent going through it - yes, I had one too! They were using the vocabulary we had developed during the day and were pretty creative in getting each other to guess the dog names as well.

In the end, I feel pretty good about how today's set of activities went. The engagement level was pretty high and everyone did a good job of interacting with the definitions in a way that will hopefully lead to understanding as we start proving their properties in coming classes.

## Building meaning for momentum from discussions, definitions, and data.

Today we started our next unit in physics with a 'next time question' from Paul Hewitt:

My reason for giving this was specifically because of the fact that we haven't learned anything about it. I wanted the students to speak purely from their intuition. I asked them the following:

We aren't quite ready to answer this by calculation, but I do want you to make a guess.

Will they move together faster than, slower than, or with the same speed as the ball?

Student responses included:

• We need to know if he bends backwards when he catches it, because that will affect it.
• No matter how he does catch it, he will move slower. The larger mass will result in a smaller acceleration.
• The clown has a non-conservative force, so the kinetic energy will decrease.

Interesting responses. We talked a bit about collisions and throws and catches of objects and what they 'felt' doing this with different objects. I introduced the idea that it might be nice to have a physics quantity that contains the direction and rate information of velocity, as well as the mass.  I told them that physicists did, in fact, have such a quantity called momentum. They responded with a few non-physics related ways they had heard the term and described what it meant.

To figure things out about how momentum relates to collisions, I then had them analyze the three air track collision videos from the Doane Physics video library using Tracker. Their tasks were as follows:

• Find the momentum of each cart before and after the collision for the video you are assigned. Calibration information is contained in the first frame of each video.
• Find the total momentum of the system before and after the collision.
• Find the total kinetic energy of the system before and after the collision.
• What is thechange of the momentum of the system during the collision?
• What is the change of the kinetic energy of the system during the collision?

It was pretty cool to see them jump in with Tracker and know how to analyze things without too much trouble. Fairly soon afterwards, we had some initial velocities and final velocities, and changes in momentum to compare.

I was, of course, leading them toward something with the change calculations.
We calculated the changes in momentum, which were non-zero. Were the magnitudes significant? A student suggested looking at the percent change compared to the initial momentum. For the first two videos, the loss was less than 1%, though for the third it was around 20%.
A student proposed the possibility that the change should be zero if no momentum is lost during the transfer. Comments were made about how that made sense in the context of our previous unit on energy - things feeling right when all of a quantity can be accounted for.
I then did a little pushing (since we were almost out of time) about what this might mean about total initial momentum and total final momentum.  I also gave them definitions for elastic and inelastic collisions. I then assigned them a couple simple questions that I wanted them to figure out if we can say that the change in total momentum before and after is zero:
Then it was time for Calculus.
______________________________________
I don't usually like giving students information. I don't like giving it away without some sense of where it comes from. I also like when students can discover quantities without equation definitions. Sometimes though, the simplicity of an idea like momentum and its power can come from taking the calculation itself as a tool that can be used to analyze a situation.
In previous classes, I have given the definition, shown situations in which momentum is conserved, and then asked students to use this idea of momentum conservation with their math skills to find unknown quantities. I really liked this alternate approach today of using momentum itself to analyze a situation and then have the idea of conservation come out of discussion. I think its potential for 'stickiness' in the minds of students is much greater this way.

## Students #flipping class presentations through making videos

Those of you that know the way I usually teach probably also know that projects are not in my comfort zone. I always feel they need to be well defined in such a way to make it so that the mathematical content is the focus, and NOT necessarily about how good it looks, the "flashy factor", or whether it is appropriately stapled. As a result, I often avoid them like the plague. The activities we do in class are usually student centered and involve  a lot of student interaction, and occasionally (much to my dismay) are open ended problems to be solved.

Done well, a good project (and rubric) also involves a good amount of focused interaction between students about the mathematical content. I don't like asking students to make presentations either - what often results is a Powerpoint and students awkwardly gesturing at projected images of text that they then read to the group in front of them. In class, I openly mock adults who do this to my students - I keep the promise that I will never ask them to read to me and their peers standing at the front of the room. Presentation skills are important, don't get me wrong, but I don't see educational gold in the process, or get all tingly about 'real-world skill development' from assigning in-class presentations. They instill fear in the hearts of many students (especially those that are students of ESOL) and require  tolerance from the rest of the class and involved adults to sit through watching them, and require class time in order to 'make' students watch them.

I'm also not convinced they actually learn content by creating them. Take a bunch of information found on Wikipedia or from Google, put it on a number of slides, and read it slowly until your time is up. Where is the synthesis? Where is the real world application of an idea that the student did? What new information is the student generating? If there's very little substantive answer to those questions, it's not worth it. It's no wonder why they go the Powerpoint slide route either - it's generally what they see adults doing when they present something.

In short, I don't like asking students to do something that even adults don't typically do well, and even then without the self-esteem and image issues that teenagers have.

All of that said, I really liked seeing a presentation (a good one, mind you) from Kelly Grogan (@KellyEd121) at the Learning 2.011 conference in Shanghai this past September. She has her students combine written work, digital media, audio, and video into digital documents that can be easily shared with each other and with her as their teacher. The additional dimension of hearing the student talking about his/her work and understanding is a really powerful one. It is but one distilled aspect of what we want students to get out of the projects we assign.

The fact that it isn't live also takes away a lot of the pressure to get it all right in one take. It also takes advantage of the asynchronous capability that technology affords us - I can watch a student's product at home or on my iPad at night, as can the other students. I like how it uses the idea of the flipped classroom to change the idea of student presentations. Students present their understanding or work through video that can be watched at home,  and then the content can be discussed or used in class the next day.

It was with all of this in mind that I decided to assign the project described here:

http://wiki.hischina.org/groups/gealgerobophysiculus/wiki/57f0c/Unit_5__Living_Proof_Video_Project.html

The proofs were listed on a handout given in class, and students in groups of two chose which proof they wanted to do. Most students submitted their videos today. I'm pretty pleased with how they ran with the idea and made it their own. Some quick notes:

• The mathematical content is the focus, and the students understood that from the beginning. While the math isn't perfect in every video, the enthusiasm the students had for putting these together was pretty awesome to watch. There's no denying that enthusiasm as a tool for helping students learn - this is a major plus for project based assignments.
• Some students that rarely volunteer to speak in class have their personalities and voices all over these. I love this.

My plan to hold students accountable for watching these is to have variations of them on the unit test in a couple weeks. I don't have to force the students to watch them though - they had almost all shared them before they were due.

Yes, you heard that right. They had almost all shared their work with each other and talked about it before getting to class. I sometimes have to force this to happen during class, but this assignment encouraged them to do it on their own. Now that's cool.

I have ideas for tweaking it for next time, but I really liked what came out of this. I've been hurt(stung?)  by projects before - giving grades that meet the rubric for the project, but don't actually result in a grade that indicates student learning.

I can see how this concept could really change things though. There's no denying that the work these students produced is authentic to them, and requires engagement with the content. Isn't that what we ultimately want students to know how to do when they leave our classroom?

## A tale of two classrooms - which is yours?

Consider the two scenarios below in the context of your own classroom, or if you are an administrator, in the context of how you might react to the following situations occurring in your teacher's classrooms. Assume the class skill level is normally distributed from weak to strong.

Situation one:

You are teaching a lesson in a mathematics class on a skills-heavy topic - perhaps solving a quadratic equation with rational roots. You have a lesson completely planned, a great intuitive hook problem at the beginning, and plenty of pivotal questions to shape student understanding around the process. Perhaps you have a carefully crafted exploration that guides students to figure out for themselves exactly how the procedure works. You have students work in groups to create a set of procedures to follow, and then students individually solve practice problems and compare to each other to check their work and help each other.

Situation two:

You are again teaching students to solve a quadratic equation with rational roots. You give them the set of practice problems at the beginning of the class and briefly review what it means to solve an equation - what should your final answer look like on your paper? You then give them textbooks, laptops with internet access, Geogebra, graphing calculators, whiteboards - all sorts of materials and tell the students your expectation is that they learn using whatever method works for them how to solve the equation. Some look on Youtube for hints. Some students might already know how to solve the equation - those students quickly tell their friends how to do so. Some decide to graph the quadratic function, get the solutions to the equation first, and then try to get those answers algebraically. You find that some students are struggling, so you are able to give additional help to those students, and they do seem to understand the general procedure after getting some help from online videos and their peers in the class. By the end, everyone has solved at least a couple of these types of problems on their own.

Suppose also that the next day you give the students a quiz with two of these problems, the second with an additional layer of difficulty. The strongest students get both questions correct, and the rest get at least the simplest question correct, with some fundamental flaw in reasoning or procedure for the second. In other words, I want the measured outcome of both situations to be roughly the same.

Before I go on, let me be clear about my own background here. When I was first trained to teach in New York City public schools, I was expected to teach lessons fitting the mould of the first scenario. The "I-do, we-do, you-do" model or the developmental lesson were the names often given to this type of classroom. The principal expected teachers to stick to a well defined structure for each lesson, and he was in and out of classrooms frequently to ensure that this was the case. The idea was that the structure helped with classroom management, made learning objectives clear to students, and made it easy for students to take notes and keep track of what they learned. Another part of doing things this way was that there was some level of control over how students were guided to an answer. If the activities or examples are shrewdly selected, a lesson doesn't devolve into situations in which it is necessary to say "Yes, [generic shortcut that students will find if it exist] works in this case, but it won't always do so."

Since leaving that school, I've taught in environments in which I've been able to experiment a bit more and try new instructional methods. In my current school, I am supported to use whichever methods I choose to help my students learn. I find, however, that since my mind is not really made up, I go back and forth. I am more likely to use the first situation in Calculus and geometry, and the second in physics and algebra two, but there are exceptions.

Which of these classrooms is yours? What are the advantages and disadvantages of each? Since I'm the one writing, I get to share first.

Situation one has always been my go-to model for helping students that are weak in arithmetic, algebraic skills, or overall organization. These students benefit from seeing clear examples of what to do, and then from getting opportunities to practice either with guidance through whole class, pair, or independent work. In many cases these students are not sure how best they learn, so they follow steps they are given and trust that the path their teacher has selected will be one that will eventually lead them to success. In addition, my presentation and activities can be carefully chosen to make it so that students are not just memorizing a procedure, but are required to go through thinking to understand the mathematical thinking involved.

In the larger context of teaching mathematical thinking, however, this method can lead to students expecting or relying on the teacher to provide the frame work for learning. It can (though does not necessarily, depending on the group) lead to a mindset on the part of students that it's the teacher's job to explain everything and make it easy to learn. I do believe in my responsibility to know how to explain or present material in many different ways to help students, but there are some concepts that just aren't easy. They may take work, practice, and interaction with me and the other students to understand and apply.

Situation two offers a bit more in terms of empowering the students to take control of their learning. It lets the students choose how they learn a concept best, whether by direct instruction, watching a video, reading example problems, or working with peers. If students learn the material on their own, have seen it before, or grasp the concept quickly, this offers many opportunities for using that knowledge to help other students or challenge them with more difficult questions. It does not require that material be presented in a linear fashion, from simple to complex, because it offers opportunities to jump back and forth, working backwards and from different representations to eventually come to an understanding.

In many cases, this offers the opportunity for the teacher to show what it looks like when figuring something out or learning something for the first time. I have read many people that refer to this position as the 'learner-in-chief', a concept I really like because I think students need to see that learning is non-linear, filled with mistakes and the testing of theories. Getting it right the first time, while nice when it happens, is not the norm. Sharing this fact with students can be a valuable learning experience. While it is nice to see a concept presented perfectly, it contrasts with the real learning process that is a lot more messy.

I have seen a couple negative factors that need to be considered in implementation, the first concerning the weak students. These are often the students that perhaps lack the background knowledge to figure out a mathematical procedure, or the self control to sit and figure something out on their own. What is nice in the second situation, assuming the other students know they must complete the assigned lesson and work for the day, is that the group of these students is a smaller one than the entire class. It is an example of differentiation in action - the students that need direct instruction to learn, get it. Those that do not, are able to reinforce and apply their learning habits by learning on their own. This situation also presumes the students are motivated to learn the concepts, though being able to do so in their own way and being held accountable for their learning may improve how some students react to the prospect of coming to your classroom each day.

Another downside that I've seen in practice is also a downside of students teaching each other mathematical processes. Students will often teach 'just the steps' and none of the understanding. While this is not the end of the world, it is something that teachers must reinforce with their students. The idea that mathematics is not just a list of problems, but a way of thinking, is strengthened by the arrangement in situation two. If the arrangement of resources available to students is sufficiently broad, the students will be able to piece together the overall concepts as a group. This entire process needs to be modeled, however, early on in the year to teach students both how to do it and what the expectations are.

For administrators, I imagine that walking into a classroom like this can result in an initial feeling of chaos or disorder, and might therefore lead to the feeling that this is less ideal than that presented in situation one. To be clear - it is possible to run a classroom poorly in both situations, and classroom management is essential to maximize the student learning occurring in both. Ultimately, a classroom filled with students that are all learning in their own way to reach a given set of learning standards, is the holy grail. It is important to be given the opportunity, training, time to interact with colleagues, and the necessary resources to make this feasible in every classroom. The important part, chaos or no chaos, is to determine whether (or not) learning is happening in the classroom. My main point is that there is fundamental difference in the philosophy of learning between the two classrooms.

Which is better? I'm not sure. I go back and forth between the two, depending on the concepts we are exploring on a particular day, or he problems we are looking at. Some of the most fulfilling lessons I have taught have involved giving the students a challenging problem and letting them figure it out in their own way. Yesterday in Calculus we did a number of activities that led to the Fundamental Theorem, but I was guiding the way. I think keeping it balanced is the way to go, but that's partly because I haven't structured my courses to be taught completely one way or the other. Maybe, in moving to Standards Based Grading, it might make it more natural to move toward more of situation two.

What do you think?

## Relating modeling & abstraction on two wheels.

Over the course of my vacation in New Zealand, I found myself rethinking many things about the subjects I teach. This wasn't really because I was actively seeing the course concepts in my interactions on a daily basis, but rather because the sensory overload of the new environment just seemed to shock me into doing so.

One of these ideas is the balance between abstraction and concrete ideas. Being able to physically interact with the world is probably the best way to learn. I've seen it myself over and over again in my own classes and in my own experience. There are many situations in which the easiest way to figure something out is to just go out and do it. I tried to do this the first time I wanted to learn to ride a bicycle - I knew there was one in the garage, so I decided one afternoon to go and try it out. I didn't need the theory first to ride a bicycle - the best way is just to go out and try it.

Of course, my method of trying it was pretty far off - as I understood the problem , riding a bicycle first required that you get the balancing down. So I sat for nearly an hour rocking from side to side trying to balance.

My dad sneaked into the garage to see what I was up to, and pretty quickly figured it out and started laughing. He applauded my initiative in wanting to learn how to do it, but told me there is a better way to learn. In other words, having just initiative is not enough - a reliable source of feedback is also necessary for solving a problem by brute force. That said, with both of these in hand, this method will often beat out a more theoretical approach.

This also came to mind when I read a comment from a Calculus student's portfolio. I adjusted how I presented the applications of derivatives a bit this year to account for this issue, but it clearly wasn't good enough. This is what the student said:

Something I didn't like was optimisation. This might be because I wasn't there for most of
the chapter that dealt with it, so I didn't really understand optimisation. I realise that optimisation applies most to real life, but some of the examples made me think that, in real life, I would have just made the box big enough to fit whatever needed to fit inside or by the time I'd be done calculating where I had to swim to and where to walk to I could already be halfway there.

Why sing the praises of a mathematical idea when, in the real world, no logical person would choose to use it to solve a problem?

This idea appeared again when reading The Mathematical Experience by Philip J. Davis and Reuben Hersh during the vacation. On page 302, they make the distinction between analytical mathematics and analog mathematics. Analog math is what my Calculus student is talking about, using none of "the abstract symbolic structures of 'school' mathematics." The shortest distance between two points is a straight line - there is no need to prove this, it is obvious! Any mathematical rules you apply to this make the overall concept more complex. On the other hand, analytic mathematics is "hard to do...time consuming...fatiguing...[and] performed only by very few people" but often provides insight and efficiency in some cases where there is no intuition or easy answer by brute force. The tension between these two approaches is what I'm always battling in my mind as a swing wildly from exploration to direct instruction to peer instruction to completely constructivist activities in my classroom.

Before I get too theoretical and edu-babbly, let's return to the big idea that inspired this post.

I went mountain biking for the first time. My wife and I love biking on the road, and we wanted to give it a shot, figuring that the unparalleled landscapes and natural beauty would be a great place to learn. It did result in some nasty scars (on me, not her, and mostly on account of the devilish effects of combining gravity, overconfidence, and a whole lot of jagged New Zealand mountainside) but it was an incredible experience. As our instructors told us, the best way to figure out how to ride a mountain bike down rocky trails is to try it, trust intuition, and to listen to advice whenever we could. There wasn't any way to really explain a lot of the details - we just had to feel it and figure it out.

As I was riding, I could feel the wind flowing past me and could almost visualize the energy I carried by virtue of my movement. I could look down and see the depth of the trail sinking below me, and could intuitively feel how the potential energy stored by the distance between me and the center of the Earth was decreasing as I descended. I had the upcoming unit on work and energy in physics in the back of my mind, and I knew there had to be some way to bring together what I was feeling on the trail to the topic we would be studying when we returned.

When I sat down to plan exactly how to do this, I turned to the great sources of modeling material for which I have incredible appreciation of being able to access , namely from Kelly O'Shea and the Modeling center at Arizona State University. In looking at this material I have found ways this year to adapt what I have done in the past to make the most of the power of thinking and students learning with models. I admittedly don't have it right, but I have really enjoyed thinking about how to go through this process with my students. I sat and stared at everything in front of me, however - there was conflict with the way that I previously used the abstract mathematical models of work, kinetic energy, and potential energy in my lessons and the way I wanted students to intuitively feel and discover what the interaction of these ideas meant. How much of the sense of the energy changes I felt as I was riding was because of the mathematical model I have absorbed over the years of being exposed to it?

The primary issue that I struggle with at times is the relationship between the idea of the conceptual model as being distinctly different from mathematics itself, especially given the fact that one of the most fundamental ideas I teach in math is how it can be used to model the world. The philosophy of avoiding equations because they are abstractions of the real physics going on presumes that there is no physics in formulating or applying the equations. Mathematics is just one type of abstraction.

A system schema is another abstraction of the real world. It also happens to be a really effective one for getting students to successfully analyze scenarios and predict what will subsequently happen to the objects. Students can see the objects interacting and can put together a schema to represent what they see in front of them. Energy, however, is an abstract concept. It's something you know is present when observing explosions, objects glowing due to high temperature, baseballs whizzing by, or a rock loaded in a slingshot. You can't, however, observe or measure energy in the same way you can measure a tension force. It's hard to really explain what it is. Can a strong reliance on mathematics to bring sense to this concept work well enough to give students an intuition for what it means?

I do find that the way I have always presented energy is pretty consistent with what is described in some of the information on the modeling website - namely thinking about energy storage in different ways. Kinetic energy is "stored" in the movement of an object, and can be measured by measuring its speed. Potential energy is "stored" by the interaction of objects through a conservative force. Work is a way for one to object transfer energy to another through a force interaction, and is something that can be indicated from a system schema. I haven't used energy pie diagrams or bar charts or energy flow diagrams, but have used things like them in my more traditional approach.

The main difference in how I have typically taught this, however, is that mathematics is the model that I (and physicists) often use to make sense of what is going on with this abstract concept of energy. I presented the equation definition of work at the beginning of the unit as a tool. As the unit progressed, we explored how that tool can be used to describe the various interactions of objects through different types of forces, the movement of the objects, and the transfer of energy stored in movement or these interactions. I have never made students memorize equations - the bulk of what we do is talk about how observations lead to concepts, concepts lead to mathematical models, and then models can then be tested against what is observed. Equations are mathematical models. They approximate the real world the same way a schema does. This is the opposite of the modeling instruction method, and admittedly takes away a lot of the potential for students to do the investigating and experimentation themselves. I have not given this opportunity to students in the past primarily because I didn't know about modeling instruction until this past summer.

I have really enjoyed reading the discussions between teachers about the best ways to transition to a modeling approach, particularly in the face of the knowledge (or misinformation) they might already have . I was especially struck by a comment I read in one of the listserv articles by Clark Vangilder (25 Mar 2004) on this topic of the relationship between mathematical models and physics:

It is our duty to expose the boundaries between meaning, model, concept and representation. The Modeling Method is certainly rich enough to afford this expense, but the road is long, difficult and magnificent. The three basic modeling questions of "what do you see...what can you measure...and what can you change?" do not address "what do you mean?" when you write this equation or that equation...The basic question to ask is "what do you mean by that?," whatever "that" is.

Our job as teachers is to get students to learn to construct mental models for the world around them, help them test their ideas, and help them understand how these models do or do not work. Pushing our students to actively participate in this process is often difficult (both for them and for us), but is inevitably more successful in getting them to create meaning for themselves on the content of what we teach. Whether we are talking about equations, schema, energy flow diagrams, or discussing video of objects interacting with each other, we must always be reinforcing the relationship between any abstractions we use and what they represent. The abstraction we choose should be simple enough to correctly describe what we observe, but not so simple as to lead to misconception. There should be a reason to choose this abstraction or model over a simpler one. This reason should be plainly evident, or thoroughly and rigorously explored until the reason is well understood by our students.

## Why my trip to New Zealand will make me a better teacher this week....

I just returned today from an amazing three week tour of New Zealand with my wife. My plan is to post photos and captions somewhere in cyberspace, though I haven't figured out exactly where, and given the start of the new semester this coming week, it may take some time before I am able to do so.

Given that it was the end of the semester before we left, there was no need to even think of bringing work along. Instead, I was able to spend my time focused on the most breathtaking 3,500 kilometers of driving I've ever done, giving mountain biking a try (with the scars to show for it), and staring down trails like this:

It amazes me how taking time to completely take my mind off of work and teaching somehow tends to result in doing some of my best brainstorming about work and teaching. Making time for genuine renewal is a real productivity booster. I read The Way We're Working Isn't Working by Tony Schwartz a couple years ago towards the end of the school year, an excellent book which explores this idea in depth. I found myself agreeing with all of the concepts then, even though I had done the complete opposite throughout the year. It is counter-intuitive to take a break in the midst of stress - you think about how many little tasks you can get done in the ten minutes you might spend taking a walk, or the thirty minutes you might spend running a few miles, and it becomes too easy to rationalize not  taking a break even though there is plenty of evidence to show that it does good things for you.  It's the same principle behind the Google twenty percent rule through which employees are given 20% of their work week to work on whatever projects they want to work on.

I made the decision to keep most of my tech toys at home on this trip. I checked email occasionally and looked at tweets, but was otherwise fully immersed in the various adventures we had scheduled for ourselves. It was the right decision, including from a teaching standpoint for this reason: I find myself starting the semester with a big list of ideas for activities and potential projects to engage and involve students through my classroom. I am excited to share my vacation with students on a basic level, but am more excited to show how bug splatters lead to finding definite integrals, or how hiking on a glacier made me think about self similarity. I will share those ideas as I put some structure to them and share them with students over the next week or so.

In the meantime, here is just a taste of another #anyqs that is brewing at the moment:

Finally, a video look at this curious landmark from the North Island:

## Rubrics & skill standards - a rollercoaster case study.

• I gave a quiz not long ago with the following question adapted from the homework:

The value of 5 points for the problem came from the following rubric I had in my head while grading it:

• +1 point for a correct free body diagram
• +1 for writing the sum of forces in the y-direction and setting it equal to may
• +2 for recognizing that gravity was the only force acting at the minimum speed
• +1 for the correct final answer with units

Since learning to grade Regents exams back in New York, I have always needed to have some sort of rubric like this to grade anything. Taking off  random quantities of points without being able to consistently justify a reason for a 1 vs. 2 point deduction just doesn't seem fair or helpful in the long run for students trying to learn how to solve problems.

As I move ever more closely toward implementing a standards based grading system, using a clearly defined rubric in this way makes even more sense since, ideally, questions like this allow me to test student progress relative to standards. Each check-mark on this rubric is really a binary statement about a student relative to the following standards related questions:

• Does the student know how to properly draw a free body diagram for a given problem?
• Can a student properly apply Newton's 2nd law algebraically to solve for unknown quantities?
• Can a student recognize conditions for minimum or maximum speeds for an object traveling in a circle?
• Does a student provide answers to the question that are numerically consistent with the rest of the problem and including units?

It makes it easy to have the conversation with the student about what he/she does or does not understand about a problem. It becomes less of a conversation about 'not getting the problem' and more about not knowing how to draw a free body diagram in a particular situation.

The other thing I realize about doing things this way is that it changes the actual process of students taking quizzes when they are able to retake. Normally during a quiz, I answer no questions at all - it is supposed to be time for a student to answer a question completely on their own to give them a test-like situation. In the context of a formative assessment situation though, I can see how this philosophy can change. Today I had a student that had done the first two parts correctly but was stuck.

Him: I don't know how to find the normal force. There's not enough information.

Me: All the information you need is on the paper. [Clearly this was before I flip-flopped a bit.]

Him: I can't figure it out.

I decided, with this rubric in my head, that if I was really using this question to assess the student on these five things, that I could give the student what was missing, and still assess on the remaining 3 points. After telling the student about the normal force being zero, the student proceeded to finish the rest of the problem correctly. The student therefore received a score of 3/5 on this question. That seems to be a good representation about what the student knew in this particular case.

Why this seems slippery and slopey:

• In the long term, he doesn't get this sort of help. On a real test in college, he isn't getting this help. Am I hurting him in the long run by doing this now?
• Other students don't need this help. To what extent am I lowering my standards by giving him information that others don't need to ask for?
• I always talk about the real problem of students not truly seeing material on their own until the test. This is why there are so many students that say they get it during homework, but not during the test - in reality, these students usually have friends, the teacher, example problems, recently going over the concept in class on their side in the case of 'getting it' when they worked on homework.

Why this seems warm and fuzzy, and most importantly, a good idea in the battle to helping students learn:

• Since the quizzes are formative assessments anyway, it's a chance to see where he needs help. This quiz question gave me that information and I know what sort of thing we need to go over. He doesn't need help with FBDs. He needs help knowing what happens in situations where an object is on the verge of leaving uniform circular motion. This is not a summative assessment, and there is still time for him to learn how to do problems like this on his own.
• This is a perfect example of how a student can learn from his/her mistakes.  It's also a perfect example of how targeted feedback helps a student improve.
• For a student stressed about assessments anyway (as many tend to be) this is an example of how we might work to change that view. Assessments can be additional sources of feedback if they are carefully and deliberately designed. If we are to ever change attitudes about getting points, showing students how assessments are designed to help them learn instead of being a one-shot deal is a really important part of this process.

To be clear, my students are given one-shot tests at the end of units. It's how I test retention and the ability to apply the individual skills when everything is on the table, which I think is a distinctly different animal than the small scale skills quizzes I give and that students can retake. I think those are important because I want students to be able to both apply the skills I give them and decide which skills are necessary for solving a particular problem.

That said, it seems like a move in the right direction to have tried this today. It is yet one more way to start a conversation with students to help them understand rather than to get them points. The more I think about it, the more I feel that this is how learning feels when you are an adult. You try things, get feedback and refine your understanding of the problem, and then use that information to improve. There's no reason learning has to be different for our students.

## Testing expected values using Geogebra

I was intrigued last night looking at Dan Meyer's blog post about the power of video to clearly define a problem in a way that a static image does not. I loved the simple idea that his video provoked in me - when does one switch from betting on blue vs. purple? This gets at the idea of expected value in a really nice and elegant way. When the discussion turned to interactivity, Geogebra was the clear choice.

I created this simple sketch (downloadable here)as a demonstration that this could easily be turned into an interactive task with some cool opportunities for collecting data from classes. I found myself explaining the task in a slightly different way to the first couple students I showed this to, so I decided to just show Dan's video to everyone and take my own variable out of the experiment. After doing this with the Algebra 2 (10th grade) group, I did it again later with Geometry (9th) and a Calculus student that happened to be around before lunch.

The results were staggering.

Each colored point represents a single student's choice for when they would no longer choose blue. Why they chose these was initially beyond me. The general ability level of these groups is pretty strong. After a while of thinking and chatting with students, I realized the following:

• Since the math level of the groups were fairly strong, there had to be something about the way the question was posed that was throwing them off. I got it, but something was off for them.
• The questions the students were asking were all about winning or losing. For example, if they chose purple, but the spinner landed on blue, what would happen? The assumption they had in their heads was that they would either get \$200 or nothing. Of course they would choose to wait until there was a better than 50:50 chance before switching to purple. The part about maximizing the winnings wasn't what they understood from the task.
• When I modified the language in the sketch to say when do you 'choose' purple instead of 'bet' on the \$200  between the Algebra 2 group and the Geometry group, there wasn't a significant change in the results. They still tended to choose percentages that were close to the 50:50 range.

I made an updated sketch that allowed students to do just that, available here in my Geogebra repository. It lets the user choose the moment for switching, simulates 500 spins, and shows the amount earned if the person stuck to either color. I tried it out on an unsuspecting student that stayed after school for some help, one of the ones that had done the task earlier.

Over the course of working with the sketch, the thing he started looking for was not when the best point to switch was, but when the switch point resulted in no difference in the amount of money earned in the long run by spinning 500 times. This, after all, was why when both winning amounts were \$100, there was no difference in choosing blue or purple. This is the idea of expected value - when are the two expected values equal? When posed this way, the student was quickly able to make a fairly good guess, even when I changed the amount of the winnings for each color using the sketch.

I'm thinking of doing this again as a quick quiz with colleagues tomorrow to see what the difference is between adults and the students given the same choice. The thing is, probably because I am a math teacher, I knew exactly what Dan was getting at when I watched the video myself - this is why I was so jazzed by the problem. I saw this as an expected value problem though.

The students had no such biases - in fact, they had more realistic ones that reflect their life experiences. This is the challenge we all face designing learning activities for the classroom. We can try our best to come up with engaging, interesting activities (and engagement was not the issue - they were into the idea) but we never know exactly how they will respond. That's part of the excitement of the job, no?

The social aspect of being in a classroom is what makes it such a unique learning environment. It isn't just a place where students can practice and develop their skills, because they can do that outside of the classroom using a variety of resources. In the classroom, a student can struggle with a problem and then ask a neighbor. A student can get nudged in the right direction by a peer or an adult that cares about their progress and learning.

If students can learn everything we expect them to learn during class time by staring at a screen, then our expectations probably aren't what they should be. Our classrooms should be places in which ideas are generated, evaluated, compared, and applied. I'm not saying that this environment shouldn't be used to develop skills. I just mean that doing so all the time doesn't make the most of the fact that our students are social most of the time they are not in our classrooms. Denying the power of that tendency is missing an opportunity to engage students where they are.

I am always looking for ways to justify why my class is better than a screen. Based on a lot of discussion out there about the pros and cons of Khan academy, I tried an experiment today with my geometry class to call upon the social nature of my students for the purposes of improving the learning and conversations going on in class. As I have mentioned before, it can be a struggle sometimes to get my geometry students  to interact with each other as a group during class, so I am doing some new things with them and am evaluating what works and what doesn't.

The concept of badges as a meaningless token is often cited as a criticism of the Khan academy system. It may show progress in reaching a certain skill level, it might be meaningless. How might this concept be used in the context of a classroom filled with living, breathing students? Given that I want to place value on interactions between students that are focused on learning content, how might the concept be applied to a class?

I gave the students an assignment for homework at the end of the last class to choose five problems that tested a range of the ideas that we have explored during the unit. Most students (though not all) came to class with this assignment completed. Here was the idea:

• Share your five problems with another student. Have that student complete your five problems. If that student completes the problems correctly  and to your satisfaction, give them your personal 'badge' on their paper. This badge can be your initials, a symbol, anything that is unique to you.
• Collect as many people's badges as you can. Try to have a meaningful conversation with each person whose problems you complete that is focused on the math content.
• If someone gives a really good explanation for something you previously didn't understand, you can give them your badge this way too.

It was really interesting to see how they responded. The most obvious change was the sudden increase in conversations in the room. No, they were not all on topic, but most of them were about the math. There were a lot of audible 'aha' moments. Some of the more shy students reached out to other students more than they normally do. Some students put themselves in the position of teaching others how to solve problems.

In chatting with a couple of the students after class, they seemed in agreement that it was a good way to spend a review day. It certainly was a lot less work for me than they usually are. Some did admit that there were some instances of just having a conversation and doing problems quickly to get a badge, but again, the vast majority were not this way. At least in the context of trying to increase the social interactions between students, it was a success. For the purpose of helping students learn math from each other, it was at least better than having everyone work in parallel and hope that students would help each other when they needed it.

It is clear that if you want to use social interactions to help drive learning in the classroom, the room, the lesson, and the activities must be deliberately designed to encourage this learning. It can happen by accident, and we can force students to do it, but to truly have it happen organically, the activity must have a social component that is not contrived and makes sense being there.

The Khan academy videos may work for helping students that aren't learning content skills in the classroom. They may help dabblers that want to pick up a new skill or learn about a topic for the first time. Our students do have social time outside of class, and if learning from a screen is the way that a particular student can focus on learning content they are expected to learn, maybe that makes sense for learning that particular content. In a class of twenty to thirty other people, being social may be a more compelling choice to a student than learning to solve systems of equations is.

If we want to teach students to learn to work together, evaluate opinions and ideas, clearly communicate their thinking, then this needs to be how we spend our time in the classroom. There must be time given for students to apply and develop these skills. Using Khan Academy may raise test scores, but with social interaction not emphasized or integrated into its operation, it ultimately may result in student growth that is as valuable and fleeting as the test scores themselves. I think in the context of those that may call KA a revolution in education, we need to ask ourselves whether that resulting growth is worth the missed opportunity for real, meaningful learning.