Page 26 – iteration, making, building, and coding in education

Processing, Pong, and Kinetic Theory

I’ve been playing around with using Processing as a way to quickly get my Calculus students doing some programming. One of my experiments was in using what I’ve learned over the past couple months about object oriented programming to make the game have multiple balls in play at once.

Once I saw how well this worked, it turned rapidly into an attempt to max out my processor. The balls have random initial locations, and ‘speeds’ distributed uniformly between -2 and 2 pixels/frame.

The pong program keeps track of the bounces off of the left and right walls, and uses this as a basic way to calculate a score. When I saw this, it looked just like a kinetic theory simulation for ideal gases, though the particles are only bouncing off of the walls, not each other. That bounce variable keeps track of the collisions with the walls – can anything cool that can be calculated just from the picture alone and the number of collisions?

Processing sketch can be found here.

Electric Circuits – starting at the end.

We only have a couple weeks of class left, and there’s not enough time to do the traditional Physics B sequence that I’ve used for electricity with my seniors that asked for a non-AP physics course at the beginning of the year. Normally I do electrostatics for a couple of weeks, talk about electric fields and potential, and then use these concepts to motivate a treatment of electric circuits. I could have stretched that out, but given my freedom in pace and curriculum, I decided to switch everything around.

This year, I started at the end of my sequence to address a pretty big issue I’ve always seen with my students. As much as they talk about charging (mobile devices, laptops) and basic energy conservation such as turning lights off, they have a pretty fuzzy understanding of electricity and the origins of the energy they use everyday. Some of the last topics in my traditional sequence involve real voltage sources, batteries and internal resistance – the “real” electronics that you need to know if you want to actually build a circuit. You know, the actually interesting part.

There’s nothing interesting in looking at a circuit and calculating what current is going through an arbitrary resistor in a given circuit. It took me a while to come to this realization because I still have some brain cells clinging to the “theory first, application second” philosophy, the same brain cells I’ve been working to silence this year. These are the sorts of things I want my students to learn to do:

Build a charger for an iPod using a solar panel and some circuit components. What is involved in charging a battery in a way that the battery will actually charge up without blowing Nickel and Cadmium all over the classroom?
Create a circuit that lights up an LED with the right current so it can outlast an incandescent bulb.
Look at an AC adapter that isn’t made for a given device, and modify it so that it does work. The fact that it only costs $5 to buy a new one is irrelevant when you compare it to the feeling you get when you realize this is not hard to do. (Thanks Dad!)
Generate electricity. Figure out how hard you have to physically work to run your laptop.

This is what we did on day one:

I gave them a solar panel, some small DC motors and LEGO motors, a stripped down version of our FIRST Tech Challenge robot, some lemons, clip leads, and different kinds of wire, and said I wanted them to use these tools to generate the highest voltage they could. There was also a bag of green LEDs on the table there for them to play with. There was a flurry of activity among my five students as they remembered something vaguely from chemistry about sticking different metals into a lemon, and needing to connect one to another in a certain way. They did so and saw that there was a bit of a voltage from the lemons they had connected together, but that there wasn’t much there.

I then showed them one of the LEGO motors and had them see what happened on a connected voltmeter when the axle was rotated. They were amazed that this also generated an electrical potential. This turned immediately into a contest of rotating the motor as quickly as possible and seeing the result on the voltmeter. One grabbed an LED and hooked it up and saw that it lit up.

They then turned to the robot and its big beefy motors. They found I had a set of LED lights in my parts box and asked to use it. Positive results:

The solar panel was also a big hit as it resulted in us going outside. They were impressed with how “much” electricity was generated after seeing the voltmeter display over 15 volts – they were surprised then to see that it worked to turn on the LED display, but not any of the motors they tried.

At this point it was the end of the class block, so we put everything away and went on with our day.

Some of the reasons I finished the day with a smile:

There was never a moment when I had to tell any of the students to pay attention and get involved in the activity. The variety of objects on the table and the challenge were enough to get them playing and interacting with each other.
While I did show them how to play with one of the tools (i.e. DC motor acting as generator) , they quickly figured out how they might transfer this idea to the other items I made available.
They made bits of progress toward the understanding that voltage alone was not what made things work. This is a big one.

The next day’s class used the PHet circuit construction kit to explore these ideas further in the context of building and exploring circuits. We had some fantastic conversations about voltage of batteries, conventional vs. electron current, and eventually connected the idea of Ohm’s law (which was floating around in their heads from middle school science) to the observations they made.

I was struggling for a while about how to approach electricity because I have always followed the traditional sequence. In the end, I realized that I really didn’t want to go through electrostatics – I wasn’t excited to teach it this time around. I also realized that I didn’t need to do so, either in order to teach my students what I really wanted them to learn about electricity.

I think this approach will help them realize that electricity is not magic. They can learn to control it. I admit that doing so can be dangerous and expensive if one doesn’t know what he or she is doing. That said, a little basic knowledge goes a long way, even in today’s world of nanometer sized transistors.

Tomorrow we attempt the LED lighting assignment – feel free to share your comments or suggestions!

Results of a unit long experiment in SBG and flipping.

I’ve been a believer in the concept of standards based instruction for a while. The idea made a lot of sense when I first learned about the idea when Grant Wiggins visited my school in the Bronx a few years ago to present on Understanding by Design. Dan Meyer explored the idea quite a bit using his term of the concept checklist. Shawn Cornally talks on his blog about really pushing the idea to give students the freedom to demonstrate their learning in a way they choose, though he ultimately retains judgment power on whether they have or not. Countless others have been really generous in sharing their standards and their ideas for making standards work for their students. Take a look at my blogroll for more people to read about. For those unaware, here’s the basic idea: Look at the entire unit and identify the specific skills or you want your students to have. Plan your unit to help them develop those skills. Assess and give students feedback on those skills as often as possible until they get it. In standards based grading (SBG), reporting a grade (as most of us are required to do) as a fraction of standards completed or acquired becomes a direct reflection of how much students have learned. Compare this to the more traditional version of grading that consists of an average of various ‘snapshots’ on assignments, on which grades might be as much a reflection of effort or completion as of actual learning. If learning is to be the focus of what we do in the classroom, then SBG is a natural way of connecting that learning to the grades and feedback we give to students. My model for several years now has been, well, SBG lite. Quizzes are 15% of the total grade and test only a couple skills at a time. Students can retake quizzes as many times as they want to show that they have the skills in isolation. On tests, (60% of the total grade) students can show that they can correctly apply the set of all of their acquired skills on exercises (questions they have seen before) as well as problems (new questions that test conceptual understanding). As much as I tell students they can all have a grade of 100% for quizzes and remind those that don’t to retake, it doesn’t happen. I’ll get a retake here or there. I am still reporting quiz grades as an average of a pool of “points” though, and this might leave enough haziness in the meaning of the grade for a student to be OK with a 60%. For this unit in Geometry and Algebra 2, I have specifically made the quiz grade a set of standards to be met. The point total is roughly the same as in previous units. It is a binary system – students either have the standard (3/3) or they don’t (0/3), and they need to assess each standard at least twice to convince me they have it. I really like Blue Harvest, but my students didn’t respond so well to having twowhole websites to use to check progress. While a truly scientific study would have changed only one variable at a time, I also found that structuring the skill standards this way required me to change the way class itself was structured. This became an experiment not only in reporting grades, but in giving my students the power to work on things in their own way. This also freed me up to spend my time in class assessing, giving feedback, and assessing again. More on this ahead. The details:

Geometry

I started the unit by defining the seven skills I wanted the students to have by the end on this page. The unit was on transformational geometry, so a lot of the skills were pretty straight forward applications of different types of transformations to points, line segments, and polygons. I had digital copies of all of the materials I put together last year for this unit, so I was able to post all of that material on the wiki for students to work through on their own. I adjusted these materials as we moved through the unit and as I saw there were holes in their understanding. I was also able to make some videos using Jing and Geogebra to explain some concepts related to using vocabulary and symmetry, and these seemed to help some students that needed a bit of direct instruction in addition to what I provided to them one on one. I also tried another experiment – programming assignments related to applying transformations to various points. I said completing these assignments and chatting with me about them would qualify them for proficiency on a given standard. Assigning homework was simple: Choose a standard or two, and do some of the suggested problems related to those standards. Be prepared to show me your evidence of study when you come into class. Students that said ‘I read my notes’ or ‘I looked it over’ were heckled privately – the emphasis was on actively working to understand concepts. Some students did flail a bit with the new freedom, so I made suggestions for which standards students should spend a particular day working on, and this helped these students to focus. I threw together some concept quizzes for the standards covered by the previous classes, and students could choose to work on those question types they felt they had mastered. Some handed the quiz right back knowing they weren’t ready. I was really pleased with the level of awareness they quickly developed around what they did and didn’t understand. I quickly ran into the logistical nightmare of managing the paperwork and recording assessment results. Powerschool Blue Harvest, whatever – this was the most challenging aspect of doing things this way. I often found myself bogged down during the class period recording these things, which got in the way of spending quality face time with students around their understanding. Part of this was that I was recording progress for each standard, whether good or bad, in the comment field for each student. “Understands basic idea of translation, but is confusing the image and pre-image” is the sort of comment I started writing in the beginning. While this was nice, and I think could have led to students reading the comments and getting ideas for what they needed to work on, it was a bit redundant since I was having actual conversations with students about these facts. Here is where Blue Harvest shines – I can easily send students a quick message explaining (and showing) what they need to work on. Even more powerful would be recording the conversation when I actually talk to the student, but that would be more practical with an iPad/cell phone app to avoid lugging my computer from desk to desk. Still, I wanted the feedback to be immediate and be recorded, so I knew I had to change my approach. The compromise was to only record positive progress. If a student’s quiz showed no progress, it didn’t get a comment in Powerschool. If they showed progress, but needed to fix a small detail in their understanding, they might get a comment. If they clearly got it, they got a comment saying that they aced it. Two or more positive comments (and my independent review) led to a 3/3 for each standard. The other promise I made was that if they clearly demonstrated proficiency on the exam (which had non-standard questions and some things they needed to explain) I would give them credit for the standard. The other difficult issue was creating a bank of reassessment questions. My system of making a quiz on the spot and handing it out to individual students was too time consuming. I created an app(using my new Udacity knowledge) to try to do this, the centerpiece being a randomized set of questions that emphasized knowing how to figure out the answers rather than students potentially sharing all the answers. They quickly found all the bugs in my system, and showed that it is far from ready for being an actual useful tool for this purpose. I appreciated their humor and patience in being guinea pigs for an idea. As you might notice from the image above, there is a pretty strong relationship between the standards mastered and the exam scores. Most student exam scores were either the same or better following this system in comparison to previous exams. The most important metric is the fact that most students weren’t hurt by going to this more student-centered model. Some student took more notes while working to understand the material than they have all year. Other students spoke more to their classmates and both gave and received more help in comparison to when I was at the front of the room asking questions and doing mini-lessons. While there was a lot of staring at screens during this unit, there was also a lot of really great discussion. I would have focused conversations with every single student three to four times a class, and they were directly connected to the level of understanding they had developed. Some needed direct application questions. Others could handle deeper synthesis and ‘why is this true’ questions about more abstract concepts. It felt really great doing things this way. I have always insisted on crafting one good solid presentation to give the class – the perfect lesson – with good questions posed to the class and discussions inevitably resulting from them. I have to admit that having several smaller, unplanned, but ‘messier’ conversations to guide student learning have nurtured this group to be more independent and self driven than I expected before we started.

Algebra 2

The unit focused on the students’ first exposure to logarithmic and exponential functions. The situation in Algebra 2 was very similar to Geometry, with one key difference. The main difference of this class compared to Geometry is that almost all of the direct instruction was outsourced to video. I decided to follow the Udacity approach of several small videos (<3 min), because that meant there was opportunity (and the expectation) that only two minutes would go by before students would be expected to do something. I like this much better because it fit my own preferences in learning material with the Udacity courses. I had 2 minutes to watch a video about hash functions in Python while brushing my teeth – my students should have that ability too. I wasn’t going for the traditional flipped class model here. My motivation was less about requiring students to watch videos for homework, and more about students choosing how they wanted to go through the material. Some students wanted me to do a standard lesson, so I did a quick demonstration of problems for these students. Others were perfectly content (and successful) watching the video in class and then working on problems. Some really great consequences of doing things this way:

Students who said they watched all my videos and ‘got it’ after three, two minute videos, had plenty of time in the period to prove it to me. Usually they didn’t.. This led to some great conversations about active learning. Can you predict the next step in the video when you try solving the problem on your own? What? You didn’t try solving it on your own? <SMIRK> The other nice thing about this is that it’s a reinvestment of two minutes suggesting that they try again with the video, rather than a ten or fifteen minute lesson from Khan Academy.
I’ve never heard such spirited conversation between students about logarithms before. The process of learning each skill became a social event – they each watched the video together, rewound or paused as needed, and then got into arguments while trying to solve similar problems from the day’s handout. Often this would get in the way during teacher-centered lessons, and might be classified incorrectly as ‘disruption’ rather than the productive refining and conveyance of ideas that should be expected as part of real learning.
Having clear standards for what the students needed to be able to do, and making clear what tools were available to help them learn those specific standards, led to a flurry of students demanding to show me that they were proficient. That was pretty cool, and is what I was trying to do with my quiz system for years, but failed because there was just too much in the way.
Class time became split between working on the day’s standards, and then stopping at an arbitrary time to then look at other cool math concepts. We played around with some Python simulations in the beginning of the unit, looked at exponential models, and had other time to just play with some cool problems and ideas so that the students might someday see that thinking mathematically is not just followinga list of procedures, it’s a way of seeing the world.

I initially did things this way because a student needed to go back to the US to take care of visa issues, and I wanted to make sure the student didn’t fall behind. I also hate saying ‘work on these sections of the textbook’ because textbooks are heavy, and usually blow it pretty big. I’m pretty glad I took this opportunity to give it a try. I haven’t finished grading their unit exams (mostly because they took it today) but I will update with how they do if it is surprising.

Warning: some philosophizing ahead. Don’t say I didn’t warn you. I like experimenting with the way my classroom is structured. I especially like the standards based philosophy because it is the closest I’ve been able to get to recreating my Montessori classroom growing up in a more traditional school. I was given guidelines for what I was supposed to learn, plenty of materials to use, and a supportive guide on the side to help me when I got stuck. I have seen a lot of this process happening with my own students – getting stuck on concepts, and then getting unstuck through conversation with classmates and with me. The best part for me has been seeing my students realize that they can do this on their own, that they don’t always need me to tell them exactly what to do at all times. If they don’t understand an idea, they are learning where to look, and it’s not always at me. I get to push them to be better at what they already know how to do rather than being the source of what they know. It’s the state I’ve been striving to reach as a teacher all along, and though I am not there yet, I am closer than I’ve ever been before. It’s a cliche in the teaching world that a teacher has done his or her job when the students don’t need you to help them learn anymore. This is a start, but it also is a closed-minded view of teaching as mere conveyance of knowledge. I am still just teaching students to learn different procedures and concepts. The next step is to not only show students they can learn mathematical concepts, but that they can also make the big picture connections and observe patterns for themselves. I think both sides are important. If students see my classroom as a lab in which to explore and learn interesting ideas, and my presence and experience as a guide to the tools they need to explore those ideas, then my classroom is working as designed. The first step for me was believing the students ultimately ~~want~~need to know how to learn on their own. Getting frustrated that students won’t answer a question posed to the entire class, but then will gladly help each other and have genuine conversations when that question comes naturally from the material. All the content I teach is out there on the internet, ready to be found/read/watched as needed. There’s a lot of stuff out there, but students need to learn how to make sense of what they find. This comes from being forced to confront the messiness head on, to admit that there is a non-linear path to knowledge and understanding. School teaches students that there is a prescribed order to this content, and that learning needs to happen within its walls to be ‘qualified’ learning. The social aspect of learning is the truly unique part of the structure of school as it currently exists. It is the part that we need to really work to maintain as content becomes digital and schools get more wired and connected. We need to give students a chance to learn things on their own in an environment where they feel safe to iterate until they understand. That requires us as teachers to try new things and experiment. It won’t go well the first time. I’ve admitted this to my students repeatedly throughout the past weeks of trying these things with my classes, and they (being teenagers) are generous with honest criticism about whether something is working or not. They get why I made these changes. By showing that iteration, reflection, and hard work are part of our own process of being successful, they just might believe us when we tell them it should be part of theirs.

Why I’m thinking today about the Tufts class of 2012.

I had my first group of ninth grade students during my second year teaching in the Bronx. It’s a unique experience being an adult mentor to a group of students fresh out of middle school. I’ve always gotten a kick out of seeing them first test the rules in their new high school environment, and this group being my first, it was new to me as well.

It has been a while since I’ve heard from many of them. I’m proud to say that a number of students from this group will be donning caps and gowns over the next couple of weeks to celebrate their earning undergraduate degrees. There’s a whole list of superlatives that describe the magnitude of pride I feel for this group and their accomplishments. As a digital pack-rat, I’ve held on to the spreadsheets I used to keep track of grades. I took a look at them just before writing this, which prompted a slideshow of smiling faces as I went through the list, name by name. I think I could vaguely place them in their seats in the classroom, but in all likelihood, this was just as likely my brain coddling me in my hope that I could remember such minimal details.

One student in particular in this group is pushing me to assert my bragging rights.

As I’ve mentioned other times on my blog, I am a proud graduate of Tufts University, majoring in Mechanical Engineering as a member of the class of 2003. It was through my work as a resident tutor in math and physics that I discovered that I had an interest in teaching, and this prompted me to apply to alternative certification programs that would help me do this. I could have applied my engineering credentials to be one engineer in the working world. Another option was to teach students to become engineers too, in effect, multiplying my own influence on the field. Through the New York City Teaching Fellows , I joined the faculty at Herbert H. Lehman high school in the Bronx during the fall of 2003 to teach math.

The first year was a blur. It was the fastest I’ve ever needed to learn a multidimensional set of skills and the most agonizing; I knew when I wasn’t getting across to my students and had few tricks to use in managing a class. The one thing I figured out very quickly though was that the students in my classes were sharp. They were good at picking up on things presented in the right way. Their skills were not necessarily where they needed to be, but that is a work in progress that can be managed through classroom work. I saw there was tremendous opportunity to help those students that were interested to become engineers.

I’ve had a number of students follow this route through my courses in math, engineering, and AP Physics. They chime in from time to time to let me know what they are doing, and I am always really impressed with their work. I’ve also had the occasional graduate write me to ask if it’s alright with me to not study physics or engineering as they originally planned to do. I am, of course, fine with this! I am always telling my students to go where their passions are, and am always a bit amused when they are afraid they are letting me down with such an admission.

The special case that I am writing about today is a young man that not only followed the engineering path, but decided to go to Tufts himself after leaving Lehman. He was a member of my first group of ninth graders, and though he was quickly switched into another section that year, he joined me for physics during his senior year. He also frequently contributed to the robotics team, never shying away from tackling the big challenges of robot design or from the small tasks of sweeping the shop floor at the end of the day. He also honored me during his senior year by speaking at a ceremony at which I received an award for my work, and his very kind words have stuck with me ever since.

It isn’t a miracle that he will cross the stage to receive his Tufts diploma today. Far from it – he did the hard work to get where he is, and I can’t take credit for the great things he learned both in my presence and away from it. And his story is far from over – I hope he (like many other students I’ve told this) keeps me in mind if I ever need a job. His story, and those of the rest of his class earning degrees this month, make me incredibly proud to be a teacher.

That said, there is something special about our story. The unique way that Tufts now connects us is unlike any I’ve ever had with others, even with my own Tufts classmates in the class of 2003. I hope that he can look back fondly to his times on campus as I do from time to time. For whatever small part I served in getting him there, I am glad to have helped him out.

I have nothing but excitement and pride for the adventures that lay ahead of him and his classmates.

Congratulations, Class of 2012!

What my mom taught me about patience.

Looking back over the students I’ve taught over the past nine years, I can say that I’ve worked with some phenomenal youngsters. Many of the proudest moments have been those that have required a great deal of patience in moving them forward and helping them develop. There are many times when I’ve felt I owe it to the world to be patient because, well, I know others were patient with me. When a toddler sits behind me and plays the ‘kick-the-seat’ game on a flight, I just sit and take it. I played that game. Actually, I did worse – I perfected an imitation of the call-button ping so that flight attendants would hear the sound, and then look around frantically for the light indicating which row needed attention. I would giggle hysterically; my mom (I assume) hid her face and shook her head.

ejmd — Me, Josie, and my parents in a Shanghai garden, during my parents’ visit to China last fall.

My mom’s patience has always been boundless. When I would make messes in the kitchen with my experiments, she would kindly ask that I clean up after myself. In the many cases that I didn’t, she would remind me, often while I stirred my chocolate milk, loudly. Then I would slurp it, spoon by spoon, each successive clink of the spoon on the glass louder until she would snap, screaming my name sharply to tell me to just drink it. One more clink, then compliance.

I wasn’t the only one that pushed the limits of her sanity. As the middle child of three brothers, we were the worst/best when we worked toward the common goal of mayhem in her midst. Shopping trips at the grocery store were opportunities to get extra things into the cart. In spite of her vigilance, we often succeeded in getting giant rubberband balls, quart containers of honey, and boxes of sugar cereals she subsequently kept from us.

In spite of all of the ways we tested her, she still went out of her way to give us the enriching experiences that shaped who my brothers and I have become. She signed me up for magic lessons at the library. She not only tolerated my interests in collecting insects and animals and getting unbelievably muddy during the process, but scrounged up things like mason jars and film canisters and all the books, field trips, and camps she could find to learn to do these things well. She has always kept me honest. She would look up the facts I claimed were true to see if I was full of it, as I had repeatedly proven I could be. She was the one that broke the news to me that the reason my hamster couldn’t walk that morning because it had a tumor. After tolerating my tears and anger in the midst of the devastating tragedy this was for me at the time, she followed with a completely straight-faced phone conversation with a veterinarian about how one might go about putting down a hamster.

One of the reasons I can maintain a positive outlook on things is that I know that good people are looking out for me. I do my best when people demand the best I have to offer, but understand that there will always be setbacks and failures along the way. My mom was doing this long before I ever realized or appreciated it. Striking the balance between being strict and direct with rules and directions and granting the freedom to try and explore and learn from one’s mistakes is the hardest part of being a teacher. But I get to go home and try again the next day with my students. She put up with my stomping around and singing for no pay in the same house, and had only a crossword puzzle to hide behind.

She managed this balance like a pro, despite the working conditions. I still push her buttons and put my smelly feet on the kitchen table. She shoots the same look she gave me when I was nine. This sort of consistency is rare. It is also what makes me smile knowingly when my students start playing the button-pushing game with me. I just smile and nod to defuse the situation, and that works well enough for me.

The thing I can never get right in the moment, however, the secret that I think my mom had figured out from the beginning is this: she always let me think I had won. I could go on to torture one of my brothers; she could get back to taking care of the important stuff, and being entertained by seeing us battling with each other. I can’t say for sure that this was her tactic. She knew a lot more than she let on when I was younger, but has always been modest enough to just say that I knew how to drive her crazy. I think that is true. I have this sneaking suspicion though that she has always had the upper hand.

I wish her a wonderful Mother’s day. I am committed to trying to be as patient with my students as she was with me, as well as to leaving my dirty socks on her computer in the near future. For the record though: I maintain that Ben was involved in the sandwich incident that resulted in my head getting cracked open.

Geometric Optics – hitting complexity first

I started what may end up being the last unit in physics with the idea that I would do things differently compared to my usual approach. I taught optics as part of Physics B for a few years, and as many things end to be in that rushed curriculum, it was fairly traditional. Plane mirrors, ray diagrams, equations. Snell’s law, lenses, ray tracing, equations. This was followed by a summary lesson shamefully titled “Mirrors and lenses are both similar and different” , a tribute to the unfortunate starter sentence for many students’ answers to compare and contrast questions that always got my blood boiling.

This time, given the absence of any time pressure, there has been plenty more space to play. We played with the question of how big a plane mirror must be to see one’s whole body with diagrams and debate. We messed with a quick reflection diagram of a circular mirror I threw together in Geogebra to show that light seems to be brought to a point under certain conditions. Granted, I did make suggestions on the three rays that could be used in a ray diagram to locate an image – that was a bit of direct instruction – but today when the warm up involved just drawing some diagrams, they had an entry point to start from.

After drawing diagrams for some convex and concave mirrors, I put a set of mirrors in front of them and asked them to set up the situation described by their diagrams. They made the connection to the terms convex and concave by the labels printed on the flimsy paper envelopes they were shipped in – no big introduction of the vocabulary first was needed, and it would have broken the natural flow of their work. They observed images getting magnified and minefied, and forming inverted or upright. They gasped when I told them to hold a blank sheet of paper above a concave mirror pointed at one of the overhead lights and saw the clear edges of the fluorescent tubes projected on the paper surface. They poked and stared, mystified, while moving their faces forward and backward at the focal point to find the exact location where their face shifted upside down.

After a while with this, I took out some lenses. Each got two to play with. They instantly started holding them up to their eyes and moving them away and noticing the connections to their observations with the mirrors. One immediately noticed that one lens flipped the room when held at arms length but didn’t when it was close, and that another always made everything smaller like the convex mirror did. I asked them to use the terms virtual and real, and they were right on. They were again amazed when the view outside was clearly projected through the convex lens was held in front of a student’s notebook.

I hope I never take for granted how great this small group of students is – I appreciate their willingness to explore and humor me when I am clearly not telling them everything that they need to know to analyze a situation. That said, there is really something to the backwards model of presenting complexity up front, and using that complexity to motivate students to want to understand the basics that will help them explain what they observe. Now that my students see that the lenses are somehow acting like mirrors, it is so much easier to call upon their curiosity to motivate exploring why that is. Now there is a reason for Snell’s law to be in our classroom.

Without planting a hint of why anyone aside from over excited physics teachers would give a flying fish about normals and indices of refraction, it becomes yet one more fact to remember. There’s no mystery. To demand that students go through the entire process of developing physics from basic principles betrays the reality that reverse engineering a finished product can be just as enlightening. I would wager that few people read an instruction manual anymore. Even the design of help in software has changed from a linear list of features in one menu after another to a web of wiki-style tidbits of information on how to do things. Our students are used to managing complexity to do things that are not school related, things that are a lot more real world to them. There is no reason school world has to be different from real world in how we explore and approach learning new things.

Planning for instruction: Not just for humans!

My wife and I welcomed a new member to our family a couple months ago. Meet Mileaux:

His name is a play on the more standard Milo, with the end spelled in the Cajun way as a tribute to Josie (my wife’s) roots. He’s now around six months old. We’re not exactly sure what he is – the current theory is a mix of a Pekinese and a Pomeranian, but there are hints of a whole bunch of other dogs in his behavior. His hobbies include chewing on towels and begging on command with his paws clenched together like an Italian soccer player trying to get out of a yellow card call. You have to see it to understand how spot on this description is.

Training him has been really interesting. As with every other part of my life since I started teaching, it serves as yet one more source of data on how learning occurs naturally. A disclaimer:

Yes, I know that my students are not dogs. I am saying, for the purposes of understanding the learning process, that outside of the supremely unnatural structure currently called ‘school’, that some aspects of learning are universal. As another comparison with my students, I can say for sure that Mileaux doesn’t like when I lecture him either.

Mileaux shows a lot of behavior that makes sense when thinking about how learning really should happen. He responds more strongly to positive reinforcement than negative, and the negative (when we do resort to it) has the consequence of sometimes leaving him confused rather than corrected. He sometimes gets tired of learning when he’s had enough. Sometimes he just needs to take a break in order to get it the next time.

One command we hadn’t tried until today was to lay down. We hadn’t really figured out the best way to do it. Yes, there are videos with suggestions on how to do it, but it’s fun to try to figure out how to communicate what we want him to do. I went for a quick 20-minute run to think of how I wanted to approach it. Here was my process:

I knew what he already knew how to do – specifically to sit. That seemed like a good entry point into getting him to lay down.
He just had his Lepto shot yesterday and was consequently a bit stiff and sore today. I didn’t want to use a leash or pressure to urge him into the down position. I wanted him to be able to figure out what we wanted him to do, and do it on his own.
There would, of course, be treats involved in the process when he did exactly what I wanted him to do.

Since he knew how to sit, I could put a treat within his reach laying down on the floor in my fingers. Any time he got up to move toward the treat, I would again give the sitting command. After around five minutes of doing this, he figured out that he needed to stay seated, and chose to stretch out into an awkward leaning position with his head arched down toward the ground. Then came strained reaching and pawing toward the treat on the floor. Soon after, he realized that laying down was a much more comfortable option for getting the treat, and started doing that every time. Copious petting, treats, and praise followed.

The connections to teaching content?

There is no paragraph in the textbook introducing the concept of laying down. Mileaux and I didn’t read it together and then do a share-out. I just needed to clearly define what I wanted him to learn, and this didn’t involve words.
While it is true that the skill of ‘sitting’ is one that he needed to have beforehand for my method to work, if he didn’t, I would have chosen another entry point to the activity. He lays down every day. He knows what it is. My goal for him was to make the connection between this skill of laying down with the verbal command. The knowledge he already had was really useful in helping him understand what he needed to do, but the background knowledge was not necessarily a prerequisite for the task we were doing.
I posed the problem in a way that had constraints that he figured out on his own. I couldn’t tell him not to move his hind legs. That limitation needed to be obvious to him as part of the activity. Managing this limitation as part of getting the delicious snack was what led him to learn the command as quickly as he did.
I had him go through this activity from a number of different starting points – standing up in the kitchen, sitting next to the couch, begging in the doorway – because I needed him to see that in these different contexts, the one skill I wanted him to learn was to lay down on command. He figured out that it was the common thread, and not any of the other simpler cues or tricks he could have used as a crutch or shortcut.
He didn’t do exactly what I wanted him to do, and felt alright about that. He knew it was just fine to get things wrong. The key to his getting it right in the end was clearly communicating when he did what he was supposed to do.

Granted, this may be strained. I accept that this may not be immediately be applicable to everyone’s classrooms. I do think it’s important to think about what we are asking our students to do, how we are communicating those objectives, and how we are helping them develop a healthy mindset toward learning along the way. We need to be thinking about knowledge in the context of figuring out problems. Solving them is an innate part of living in the world, whether as a snail, a dog, or as a human. The more we can create learning experiences that connect to this need to challenge and interact with our world, the more effective these experiences can be for our students.

Computational Thinking – Why do we need to do this?

I’ve been pushing programming tasks on my Algebra 2 class. Pushing is the operative word.

Specifically, I’ve given them tasks that require them to use programs written in Python to do something related to what we are learning. Previously I showed them a program I wrote for a homework assignment as part of my work for Programming a Robotic Car , which went just fine for what it was: a short activity. I’ve also given them a few tasks during previous classes asking them to adjust a program I had written. IN one case this was to calculate a solution to a linear system; in another to evaluate the quadratic formula.

What I admit I did a bad job of, however, of making it clear why we would even want to do so. I think I erred on the side of them seeing how great it was that the computer could do this, which is silly considering I’m also pushing applications like Wolfram Alpha that do much more than my programs, and in a much more eye-catching way. I imagine it’s sort of like showing my students a typewriter that can post on twitter and insisting they find it cool because such a thing exists. The power of programming is in making the computer do work that makes sense for a computer to do. The power of showing programming to students is to both demonstrate how a computer can do this type of work for them, and then empower the students to apply this knowledge on their own.

After reading a couple posts by David Wees on programming and mathematical thinking, I realized that I was doing things backwards. I needed to establish a reason for doing and teaching computational thinking. I was going to have some students out for athletics last week, and those tend to be the days when I experiment. I mentioned in the previous class with this group that I was planning to do a lesson with Python, and the reaction was instant and violently vocal. Those that were going to be out were thrilled. The others looked about as excited as if they learned that lunch for the following week would be nothing but spinach.

I designed the lesson knowing that if there was going to be any Python at all, it would need to be a task that justified itself quickly, clearly, and with as little trouble as possible.

The topic of the day was composition of functions. The warm up activity was this:

Suppose f(x) = x^2. What is f(f(2))?
This should also work for complex numbers. What is f(f(1 + i))?

What was neat was that a couple of the students were stuck on the first question playing around with it. I didn’t have to ask a student to perform the composition three, then four, then five times to its own value – they did it on their own to postpone having to deal with the complex numbers. Not a problem. They got the point that they could do this, but that it was tedious. That was phase 1.

This came next:

The program they used can be found at http://repl.it/CMz .

They quickly figured out how to use the program to run a whole bunch of times. Some found that they weren’t sure if an initial value escaped or not because the value was close to 1 or -1, so they figured out that they could change the number of iterations in the program to give them a better indication. Others realized they were looking at scientific notation and that they needed to review that that meant.

We talked about why the computer was the way to go for this, and then related it to the second warm up problem. Could it be possible to use the computer to do this for the more tedious task of managing complex numbers? Enter my second program at http://repl.it/CMz/3.

The students used the second program to determine whether these points diverged/converged (I occasionally slipped these words in with hand motions to link them to escaping or being trapped) and found the concept pretty straight forward, as it had been with the points on the number line. I continued to ask them how they could do this manually using pencil and paper, and was met with groans – the computer clearly was the more logical tool to use for this. (Yes!)

My final task for them was simple: plot the points that are trapped. I gave them this:

I said I wanted them to color in any of the squares with bottom left hand corners that represented values that converged. That’s a lot of grid squares, but that was kind of the point. There were nine in class that day, so they started dividing up the space. Some picked points randomly. Others were more methodical, with some starting to trace the border of the region, but generally there was only a scattering of filled points on the class copy where I had asked them to record their colored in grid squares.

Having predicted how this would likely end, I was ready with this modified version of a Processing sketch written by Daniel Shiffman that basically did the entire task on the computer. The students were staring at the clock after doing this for about ten minutes and only having filled in a minimal portion of the plane. The students understandably said that this process was stupid and that there most likely was a better way for them to be spending their time. That’s when I said they were right.

You know when you design activities to carefully manipulate your students’ emotions so as to realize a particular point, and it totally works? My students have a pretty distinctive facial expression that they each make when they realize I’ve done this, and right about then, it spread like wildfire through the class.

I showed them this:

This didn’t contradict what they had come up with, but it was significantly more complete. I asked if there was any recognition, but there was none. So I decreased the pixel size more:

Still no recognition, but there was more recognition that the shape of the region was an odd one. One more iteration made it pretty clear:

We had looked at a video zooming into the Mandelbrot set earlier in the year, so they had seen it before. I wanted to push that the computer made it possible to investigate this sort of mathematics. This sort of thing could not have been done by hand at this level. Having the computer available to do repetitive calculations and construct graphs according to simple rules made it possible to investigate the mathematics of these areas in ways that were never known before. They were impressed that the mathematics of fractals was not investigated until 1980, which is recent enough for them to perhaps see that math is actually a dynamic field, in contrast to the way it is usually presented.

I liked this lesson, and plan to continue to push my students to use computation when necessary. Our first unit problem for the exponential and logarithmic functions unit was a twist on the penny problem (get one penny the first day,two the second, four the third, and continue on for a month, or just take a lump sum of $50,000) and I insisted they use some computational tool to answer this, or just straight out find it extremely difficult.

This is an important skill, and I believe in it. To make it happen, I am committing to deliberately committing time for students to learn how to use computers to do the computation work so we don’t have to. I hope my days of seeing students solve the bee/train problem through tedious methods of manually adding terms together will soon be over.

Topic for #mathchat: Do we need students to reach automaticity?

I was honored when asked recently to offer a topic for discussion on #mathchat.

My suggested topic:

Is it necessary for students to develop automaticity in their pencil and paper mathematics skills? Why or why not?

First some definitions and examples to clarify the intent of the question.

By automaticity, I also mean procedural fluency. A student that has developed automaticity is familiar enough with the mechanics of a particular task to not have to devote substantial thought to how to do it. It also is connected to retention over time – how well do the details stick with a student as more information is learned over time?

In an Algebra class, for example, do the details of arithmetic need to be automatic so that the student can focus on applying algebra knowledge to solving an equation? In Calculus, should students be able to apply the product and quotient rules efficiently when working on optimization or related rates? Or is it reasonable for them to figure out the derivative using basic principles or use a computer algebra system to take care of this step when it comes up?

I also refer specifically to pencil and paper skills because, for what I would guess is a majority of us that teach math, we tend to assess students by pencil and paper at the end of the day. A student can use a graphing calculator, Geogebra, or other piece of technology to explore a concept and check her/his work. The thing I often wonder about is how the use of activities and technologies help students perform mathematical tasks when these technologies are not available.

Is it necessary to do these tasks when these tools are not available? I don’t know. I think that’s open to interpretation and individual opinion. There are some cases, however, when that choice is not up to us. Standardized tests are one example. Given that they do exist (and independent of whether or not we agree with their content/quality/use), standardized tests are not typically electronic and are timed. These are often posed as opportunities for students to choose an appropriate method of finding answers to questions and then find those answers with a limited set of resources available.

Let me be clear – I am wildly inconsistent on this, because I don’t have a good answer to the question. I emphasize understanding through the activities I do in my classes – very rarely will I directly tell students how to solve a problem, have them practice the skills with me, and then send them home to practice those skills in isolation from others. I really appreciate Conrad Wolfram’s point about using computers to handle the calculating, and leave the thinking to us and our students. I have decided on occasion not to assign #1-30 for students to practice differentiation because my feeling at that time is that if they can apply it correctly several times, they get the point, and are ready to apply that knowledge to more interesting contexts.

But when these same students that complete the short assignment, later struggle in finding anti-derivatives, I wonder if I should have drilled them more. My decision not to burden them with repetitive exercises because they are repetitive often has implications for the future of the students in class. Do I need to drill this to automaticity so that next year’s teacher doesn’t come complaining to me about how “your old students can’t XXXXXXXXXX” where XXXXXXXXXX = [arbitrary math skill that either (a) will mean the difference between getting into a top choice school during Senior year or (b)won’t matter at all ten years after leaving the classroom]?

So I call upon the collective brilliance of the #mathchat community to help find an answer.

For those unaware, #mathchat is a Twitter based chat held every Thursday night at 8PM in which all respondents use the hashtag #mathchat in their post so that everyone else following that hashtag is updated with the latest responses. If you aren’t up on using Twitter for professional development, you need to be. It completely changed my perception of how Twitter is useful and has put me in contact with some pretty amazing folks from around the world.

Socializing in Geometry – Similar Triangles

Another successful experiment getting my participation-challenged geometry class to interact with each other yesterday.

Each student received a cut-out triangle from the image at left. The challenge:

One (or possibly two) people in this room have triangles similar to yours. Your task is to find the person and do the following:

Find the similarity ratio between your triangle and your match in the order big:small.

Determine the ratio of the perimeters of each of your triangles.

Determine the ratio of the areas of each of your triangles.

I then cut them loose. Almost immediately they started scrambling around the classroom holding up triangles and calculating as quickly as possible. (I didn’t totally get why they were in a hurry, actually.) They clustered on tables and rapidly shifted partners until everyone found they were in the right place. The calculating began for perimeter – that was the easy part. Then the area question took center stage.

Some asked me how to find the heights of the triangles, and I shrugged my shoulders with the smirk of someone with ideas that isn’t sharing them. (I call this my ‘jerk’ mode that I love taking on during class for the sole reason that it gets them finding and figuring on their own.) Some recreated the triangle in Geogebra. Some superimposed it over graph paper and counted to get an estimate. One student cleverly found Heron’s formula. It was really entertaining watching them excitedly explain the formula without writing it down (something else I didn’t understand) and share how it quickly and easily allows the area to be calculated. The energy in the room was apparent as they ran from person to person trying to get everyone to complete the task. Eventually they found out themselves that the similarity ratio was a square relationship. I didn’t have to do a thing.

Part of my justification in doing this was to get them thinking about the important ideas necessary in solving another problem I threw their way during the previous class comparing the old iPad to the new one. The two different groups that had worked on it were generally on the right track, but there were some serious errors in their reasoning that I hinted at but didn’t explicitly point out to them. I think this activity closed the gap. There should be some interesting answers to discuss in class when we next meet.

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