Why SBG is blowing my mind right now.

I am buzzing right now about my decision to move to Standards Based Grading for this year. The first unit of Calculus was spent doing a quick review of linear functions and characteristics of other functions, and then explored the ideas of limits, instantaneous rate of change, and the area under curves – some of the big ideas in Calculus. One of my standards reads “I can find the limit of a function in indeterminate form at a point using graphical or numerical methods.”

A student had been marked proficient on BlueHarvest on four out of the five, but the limit one held her back. After some conversations in class and a couple assessments on the idea, she still hadn’t really shown that she understood the process of figuring out a limit this way. She had shown that she understood that the function was undefined on the quiz, but wasn’t sure how to go about finding the value.

We have since moved on in class to evaluating limits algebraically using limit rules, and something must have clicked. This is what she sent me this morning:
[wpvideo 5FSp5JDn]

Getting things like this that have a clear explanation of ideas (on top of production value) is amazing – it’s the students choosing a way to demonstrate that they understand something! I love it – I have given students opportunities to show me that they understand things in the past through quiz retakes and one-on-one interviews about concepts, but it never quite took off until this year when their grade is actually assessed through standards, not Quiz 1, Exam 1.

I also asked a student about their proficiency on this standard:

I can determine the perimeter and area of complex figures made up of rectangles/ triangles/ circles/ and sections of circles.

I received this:
…followed by an explanation of how to find the area of the figure. Where did she get this problem? She made it up.

I am in the process right now of grading unit exams that students took earlier in the week, and found that the philosophy of these exams under SBG has changed substantially. I no longer have to worry about putting on a problem that is difficult and penalizing students for not making progress on it – as long as the problem assesses the standards in some way, any other work or insight I get into their understanding in what they try is a bonus. I don’t have to worry about partial credit – I can give students feedback in words and comments, not points.

One last anecdote – a student had pretty much shown me she was proficient on all of the Algebra 2 standards, and we had a pretty extensive conversation through BlueHarvest discussing the details and her demonstrating her algebraic skills. I was waiting until the exam to mark her proficient since I wanted to see how student performance on the exam was different from performance beforehand. I called time on the exam, and she started tearing up.

I told her this exam wasn’t worth the tears – she wanted to do well, and was worried that she hadn’t shown what she was capable of doing. I told her this was just another opportunity to show me that she was proficient – a longer opportunity than others – but another one nonetheless. If she messed up a concept on the test from stress, she could demonstrate it again later. She calmed down and left with a smile on her face.

Oh, and I should add that her test is looking fantastic.

I still have students that are struggling. I still have students that haven’t gone above and beyond to demonstrate proficiency, and that I have to bug in order to figure out what they know. The fact that SBG has allowed some students to really shine and use their talents, relaxed others in the face of assessment anxiety, and has kept other things constant, convinces me that this is a really good thing, well worth the investment of time. I know I’m just preaching to the SBG crowd as I say this, but it feels good to see the payback coming so quickly after the beginning of the year.

Winning the battle over Python programming

Two stories to share after this week’s activities with students about programming. I have posted previously about my interest in making Python a fundamental part of my classes this year, and so I am finding ways to include it when it makes sense to do so.

I have a couple of students that are bridging the gap between Algebra 2 and Precalculus with an independent study that I get to design. The tentative title of the course for their transcript is ‘Fundamentals of Mathematical Thinking’ and the overall goal is to get these students a chance to develop their fundamental skills to be successful in later classes. I see it as an opportunity to really dig in to some cool mathematical ideas and get them to, well, dig into the fundamentals of mathematical thinking. I don’t plan too much emphasis on the algorithms (though we will spend some time working on skills in algebra, polynomial manipulation, functions, and other crucial topics where they are weak). Looking at a situation, exploring the way different variables might be used to model that situation, and then really digging in to abstract the variables into a model.

We are starting with what I think is the most fundamental application of this: sequences and series. Even simpler, the first task I gave the students was to look at the number of bricks in the rows of a triangular tower and use Python to add up the bricks in each row. This started as a couple of exercises getting to know Python’s syntax. They are then taking programs I wrote to model this problem and adjusting them to find other sums, including the sum of even and odd numbers. One student that completed this task was intrigued that the sum of the latter consisted of perfect squares, but we didn’t explore it any further at this point.

I then gave this student a bunch of sequences. His task was simple: model each one in Python and generate the given terms. This is a standard exercise for Algebra & Precalc students by hand, but I figured that if he could do this with Python, clearly he was able to figure out the pattern. I showed him how to write fractions using string concatenation (e.g. 1/3 = 1 + “/” + 3) which enabled him to develop the harmonic series. Today he figured out Fibonacci and a couple other new ones. It was really fascinating to see him mess around think deeply about the patterns associated with each one. I did tap him slightly in the right direction with Fibonacci, but I have otherwise been hands off. I am also having him write about his work to give him opportunities to work on his writing too. When he feels comfortable sharing it (and I have already warned him that this is the plan), I will post links to his work here.

The other new thing was in Calculus. I have shortened my review of Pre-Calculus concepts substantially, and have made the first unit a survey of limits, rate of change, and definite integrals. Most of this has required technology to explore local linearity and difference quotients. On Thursday, I introduced using rectangular sums to find area – they were otherwise stuck on counting boxes, and I could tell they felt it was like baby math. They really didn’t know any other way.

In showing them rectangular sums, we had some pretty good discussions about overestimating and underestimating. The students had conversations about how rough the approximation with only 3 – 5 rectangles gave for area under a parabola. A couple of them figured out how to use more rectangles. I told them I was going to write a program to do this while they were sitting and working. I created this program and talked them through how it works. They thought it was too complicated to be worth the time, but I think they did understand the basic idea. I then changed the value of N and asked them what they thought that meant. They got it right the first time. I then pushed the value to higher and higher values of N and they immediately saw that it was approaching a limit. Game, set, match.

Today I had the AP students together working on another definite integral activity that focused on the trapezoidal rule. I showed them the code again and gave them the line that calculates area. It wasn’t too much of a stretch for them to work their way to adjusting the program to work for the Trapezoidal rule. We ran out of time to discuss comparisons between the two programs, but they stayed late after class and into their lunch getting it working on their own computers and playing a bit. Here is what we came up with.

The big battle I see is two-fold.

  • Help students not be intimidated by the idea of writing a program to do repetitive calculations.
  • Give students opportunities to see it as necessary and productive to use a computer to solve a problem.

Sometimes these battles are the same, other times they are different. By using the built-in version of Python on their Macs, I have already started seeing them run commands and use text editors to create scripts without too much trouble. That’s the first battle. My plan is to give lots of examples supporting the second one in the beginning, and slowly push the burden of writing these programs on to the students as time goes by and they become more comfortable with the idea. So far I am feeling pretty good about it – stay tuned.

The perils of playing cards and probability. What do you assume your students know?

One of the topics taught in the first semester of my first semester teaching was probability. Flipping coins and rolling dice both serve to bring the kids to understand how it is used in games, but the first thing a couple teachers told me to do once they got the basics was to go to playing cards. This seemed like a natural fit to get the students excited – I figured they had seen people playing cards on the street as I had seen countless times wandering around the city. There are also so many opportunities to talk about intersection and union of sets. How many cards are hearts or face cards? How many are hearts and face cards? Sounded like a good idea to me.

When I actually did this with my class the first time, there were a couple really big issues that came up. Being a new teacher, I wasn’t as strong in terms of preventing students from calling out answers. When I did write up some fairly simple questions on the board (such as find P(red card) if a single card is selected) the enthusiasm for three or four students in answering these led me to believe that this small sample was representative of the class. If these four knew it (or so assumed my naive first year teacher brain), the rest probably knew it, but just didn’t feel comfortable answering. This was a ridiculously inaccurate assumption. In fact, I think it’s a painfully clear example of self-selection bias that all teachers should consider when asking any question of an entire class. Who is going to raise their hand for the purposes of establishing that he or she does not know what I am talking about?

Another issue appeared when I started walking around the room during independent work. I saw that the students were struggling both with the idea of probability AND with the details of the different types of cards. It was hard separating the two bodies of knowledge because I had framed the topic only in terms of these concepts. Students that didn’t understand what the various cards were couldn’t answer the questions because they couldn’t figure out which cards were desired outcomes. Students that didn’t get probability in general didn’t understand how the sample space and the desired outcomes fit together to calculate theoretical probability. Some didn’t understand either idea.

After the class, I talked to a few teachers about it. One said a phrase that makes my blood boil every time I hear it: “Come on – they really should know _______”. In this case, the phrase in the blank was “the types of playing cards.” The assertion that there is something wrong with a student because he or she doesn’t know an arbitrary fact is not an argument we should be making. The biggest reason it is a problem is this: if your lesson predicates itself on students knowing a fact, and you haven’t made any effort to establish as part of your lesson whether or not students actually know that fact, your lesson is going to backfire. Hard. It will be like pulling your own teeth while simultaneously telling your students “look, you can do it too!”

I understood more about this in talking to my mentor teacher. He pointed out that using playing cards is one of the worst ways teachers could teach probability because of the cultural bias inherent in assuming students have the required background knowledge. Reasons why:

  • Alright kids, we’re going to do some probability, but make sure you know what these words mean first, because I’m going to be using them all with the assumption you do: suit, hearts, diamonds, clubs, spades, face card, king, queen, jack, ace, joker. Don’t forget that there are red cards and black cards.
  • Wait, English isn’t your first language? OK, so spend your time learning these words in addition to the math content terms I really want you to learn: probability, sample space, and outcome. Uh…I guess that learning this esoteric set of words will be good for you because it will help you understand spoken English better. The more words you know, the better your English, right?
  • What’s that? How can you not understand that something can be a face card and a club? Face cards are jacks, queens, kings, and aces – get it? And there are four different suits, so there have to be four face cards that are also clubs – get it?. Well, yes, spades are also black, but clubs are black and have the little clover shape. Yes, the symbol tells you the suit. No, the card doesn’t actually say “spades” or “hearts”. But it’s easy because the heart is for hearts, the diamond is for diamond, and well, you might just have to remember the others. Oh wait, the spade is shaped like a shovel – did you know shovels are sometimes called spades? That will help you remember it. Get it? [By now, the student is nodding to get you out of his face.]
  • So now that we’ve covered all the vocabulary, what is the probability of randomly picking a card that has a value of 10 or greater? Oh, you don’t know about the value of cards? Sure, well that’s just fine. Obviously the jack is above the 10 because it has a guy on the front. It has the lowest value of the face cards, because the queen and king are higher. The king is of higher value than the queen because of the patriarchal culture that has dominated the globe for, well, forever. And then there’s the ace. Sometimes the ace is the highest card. Other times it has the lowest value. That’s life. Who has an answer?

How much math content has actually been explored during this entire (imagined) dialogue? Furthermore, if we assume that playing cards is an engaging and authentic application of probability, shouldn’t understanding the math content be made easier by the presence of all of this extra knowledge? Think about the reverse situation – should a student that knows her probability, but does not know the details of the card system, get a 50% on a quiz of this topic in a math class?

I don’t know about you, but I didn’t actually play cards that much as a kid. It’s a dangerous assumption to make that all kids have. If you don’t know if your students have this knowledge or not, and don’t want to guess from looking at them (which is always good policy not to do), and don’t want to spend class time reviewing, it probably isn’t a good idea.

One of the other teachers with whom I discussed this issue gave out a reference sheet with all of the vocabulary, pictures, and cards in order of value, and let them use it for quizzes and tests that included this topic. I think that’s fine. An even better solution though? Find a topic that doesn’t require so much background knowledge. Flip a coin and roll a 20 sided die. Put numbers on index cards, throw them in a bag, and ask for probabilities of drawing a card that is even or prime. At least in that case, students need to use mathematical knowledge to classify the outcomes. That’s what you want to assess anyway, right?

Making connections to background knowledge is one of the most powerful ways to help students learn. Making assumptions about what background knowledge students have is an easy way to make a lesson a dud. Assess, don’t assume.

On not getting in the way.

Today continued a run of some great class time working on electric circuits. Our whole class period (85 minutes) consisted of looking at the following six circuits:

Circuit 1

Describe what you would expect to happen when this circuit is connected.

A student blurted out this would be a short circuit, so things would get hot. Everyone immediately agreed (WHY DID YOU BLURT THAT OUT!). Still time to save it; I ask why it is a short circuit?

Answer: Because the path through the wire is shorter than traveling through the resistors.

I smirk.

Circuit 2

Groans from students…then a more refined answer about differences of resistance between the two branches of the wires.

Circuit 3

Tell me anything you can tell me about this circuit. If you see something to calculate, calculate it. Build the circuit on the PHET circuit constructor and show that your calculations confirm what happens in the simulator.

Insert student-centered-learning opportunities and fantastic conversations between students here. Students seeing a difference between their answers and what the simulation is telling them causes conflict that they help each other to resolve. Some students bring up the term ‘parallel’, which I’ve never said in class. Others don’t understand what that is, so there is some fighting. One student describes qualitatively what should happen and then shows he is correct in the simulation, no calculations. Furthermore, this student usually is one that gets anxious when there are limited formulas to cling to, which is the norm in my class.

Circuit 4

Repeat the same procedure. Calculate what you can calculate. Build the circuit in the simulator and verify. Explain away differences, or see if there is something you are missing.

Continued progress in recognizing this is a combination of series and parallel resistors, but I don’t make a big deal out of this. A couple students look up formulas and discover the idea of finding equivalent resistance (which I have never mentioned to them). This helps, but the simulator telling them what the correct answers are is key. They are compelled to get the simulator’s answers through calculation – it’s almost as if they feel the simulator is cheating by giving them the answers, so they must understand how to get it on their own. Eventually, they are convincing each other why they are right.

Circuit 5

Same as before, tell me anything you can tell me about this circuit. If you see something to calculate, calculate it. Build the circuit on the PHET circuit constructor as a last resort and to show that your reasoning has led to a correct analysis of the entire circuit.

This time the students hit a wall. Some continued finding equivalent resistance and the battery current, but weren’t sure how to find the current through the 10 ohm and 30 ohm resistors. One reasoned it would split proportionally, and confirmed the answer using the simulator. Another measured the voltage across one of the 10 ohm resistors using the simulated voltmeter, measured the current, and then calculated the voltage difference across it. Repeating for the other resistor, they figured out the voltage difference across the parallel resistors, which then led to a current calculation. Again, I only had to tap students in the right direction – the rest was them helping each other.

Circuit 6

Try to analyze this circuit completely using what you have learned today. Once you are convinced you have accounted for all voltage differences and all current, build it in the simulator to confirm your answers. Find a way to calculate the power used by the 20 ohm and the 10 ohm resistors separately – look it up if you want.

They did a fantastic job of figuring this out – some very quickly and quantitatively. One student that oftenstruggles with concepts figured out how the current between the different branches would compare, and reasoned which ones would have the greatest voltage difference across them.

Then I started lecturing about the equivalence of electrical power and mechanical power, and the magic disappeared. They stopped talking and returned to compliance mode. I saw that happening, so I stopped. Anything I could do at this point would only ruin what was quite possibly a perfect learning experience for them.

When I taught AP Physics, we spent a day on series circuits and deriving resistance formulas, a day on parallel circuits and deriving equations for parallel resistors, and then another day on analyzing circuits that have both. Before today, I had never used the term ‘parallel’ with my students. This time they brought it up. They now have the ability to analyze the same level of circuits as my former AP students, but this group was able to figure much of it out on their own, with no mention of memorization of formulas and no extended periods spent listening to me blabber on about how ‘going through the theory helps you understand’.

There is lots I could say about this, but I think the points made are pretty clear. Let’s just say that I’m really proud of my students work today.

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:


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 wantneed 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!

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.

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.

Bringing robotic cars and Udacity to my classroom

I was really excited to learn about Udacity, a new online education system that premiered two courses on February 20th. That a course on programming a robotic car would appeal to me is probably not surprising to anyone that knows me. I also love having yet one more excuse to continue learning Python, especially one that gets me working with an expert in the field such as Professor Sebastian Thrun. I recall reading about him shortly before his team’s successful bid at the DARPA Grand Challenge, and have since seen his name repeated at many key moments along my development as a robotics enthusiast.

The course is structured really well, with short videos introducing concepts, quizzes and programming tasks (with solutions) along the way to check comprehension, and homework assignments. The students love that I have homework.

I am busy, but this was too cool to pass up.

I also have a pretty hard time hiding the things I’m enthusiastic about in my classroom, so the content of the class has been something I’ve mentioned and shared with students at the start or end of planned activities. The whole classroom gasped at this video from the 25th second onward:
[youtube http://www.youtube.com/watch?v=bdCnb0EFAzk?feature=player_embedded&w=640&h=360]

Based on that reaction, I really wanted to give them a sense for the things I was learning to do. The first week centered on learning about localization – a process that uses probability calculations to estimate the location of the car using sensor readings and a map of the surroundings. I did a quick overview of what this meant as a filler activity to break up work during class, but wanted to find a way to do much more.

Today’s Algebra 2 class was going to be missing a couple students that are attending a Model UN conference, so I figured it would be a good time to try something different.

We started with the following warm-up problems:

Mr. Weinberg tells you we are guaranteed to have a quiz one of the days between Monday and Friday. He tells you that the probabilities of the quiz happening Monday through Thursday are 0.1, 3/8, 1/16, and 36%. What is the probability that the quiz will be on Friday? On which day is the quiz most likely to occur?

This helped review the total probability principle which is key to understanding the localization algorithm. We also did a review of finding the probability of compound independent events, first with a tree diagram, and then using multiplication and the counting principle.

We then went through the following activity for the rest of the period:
Robot Localization activity

I adapted parts of the course material provided by Udacity, primarily simplifying language, cleaning up diagrams, and adjusting the activities for my students who do not have any programming ability. We did have a Python activity back in October, but installing and running Python was a hassle on the 1-1 Macbooks with OSX since I was trying to do it with Python 3 and IDLE. It was only shortly afterward that I learned that an earlier version of Python was automatically installed. Oops. For this activity, we used http://repl.it/ to do the programming. This worked fantastically well.

The students seemed to do really well with the introductory material and filling things in, and modifying the basic programming went smoothly. They ran into some trouble around problem 7, which I half expected – that was the first part of the activity when I told them to do something without any rationale behind it. Most were generally able to implement the procedure and get to problem 9, but at this point at the end of the day on a Friday afternoon, fatigue started to take over. This was after around 45 minutes of working on the activity.

I added a section on motion for possible use in another class, as I ultimately would like them to be able to throw my own homework solution code into a simulator provided by Udacity user Anna Chiara. I did not deal with any of the sensor probability or move probability. The intuition for understanding how those apply in the algorithm is a bit subtle for the background of my students, and would take more of an investment of time than I think my students have the patience for at this state. I think it would be easier to talk about how these issues exist, and then have them observe what they mean by looking at the output of the program.

All in all, it was a cool, low-key way to share my own learning with students after an exhausting week. I think we all needed a bit of a change.