teaching philosophy

Snacking on Statistics and Variability

One of my goals this year in Algebra 2 has been to include more discrete math, statistics, and probability when I can. I’ve been convinced by all sorts of smart people that as traditional as it may be to have Calculus as the ultimate goal for math students, statistics and probability are the math that people are more likely to need to use. It compels me to include it in my courses as more than a separate unit.

As if I didn’t need another reason, we are also in a spell of reviewing properties of radicals, and it’s refreshing to get my students thinking differently after a period of simplifying, multiplying, and rationalizing.

I gave them the following scenario:

Imagine yourself in twenty years – you are, of course, rich and famous. You are hiring someone to fly your personal jet. your last pilot fell asleep on the job, though he was luckily parked at the gate when it happened.

Two pilots have applied for the position, both equally qualified as pilots. In order to help you make your decision (and avoid the previous situation), you have asked them to keep track of how many hours of sleep they get over a two week period before the interview.

Two weeks later, they return to you with the following data:

What differences do you notice about the two pilots?

What calculations would you make to describe any quantitative differences between them?

Which one would you hire? Why?

Note: This data is completely made up. My new semi-obsession is in using normal distributions to mess up clean functions and force my students (in physics and math) to deal with messy data.

The students almost immediately started calculating means – exactly what I would have expected them to do given what they have been taught to do when faced with a table of data like this. Some did so manually, others used the Geogebra file that generated the data to make their calculation.

The results were fairly consistent – everyone chose the second pilot. When asked why, they said the pilot gets more sleep on average, and so would be the better choice.

When I asked who was more consistent in their sleep, they were easily able to identify the first pilot. When asked why, many had explanations that correctly danced around how most of the data was closer to the average. No students really brought up this fact before I asked though, which leads me to believe they observed one of two things:

The importance of the consistency doesn’t really matter given the difference in the means for how much sleep the pilots got.
They didn’t think to look at consistency at all.

Some other interesting tidbits:

None of the students thought to construct a histogram to look at the data. When asked, about half of the class said they knew how to construct a histogram. I didn’t dig any deeper to flesh this out. I was going to throw one together in Geogebra, but decided that might be something we should look at with more time available.
Half of the class that is taking AP Psychology didn’t think about finding standard deviation. Again, I didn’t dig any deeper to find if this was because they didn’t know that it might apply here, or because they thought the values of the means were more important.

There is plenty here to generate discussion, but the one thing I wonder about is if variation about a mean is a concept that comes naturally to students to consider when given a set of 1-D data. One of my professors mentioned offhand in an experimental design class that any measurement you take is a distribution, a point which I have never forgotten. Up to that moment, I had never really thought much about it either.

Sure, I had collected data in my biology, chemistry, and physics classes before and knew I had to take multiple data points. All I knew then was that doing so made my data “better”. More data makes things better. Get it? My understanding in high school science was also that you never measure the same quantity at the exact same value ten times in a row because someone in your lab group is always messing it up or doing it wrong. Averaging things together smooths that out. I don’t recall ever discussing in either math or science class that the true beauty of statistics comes from managing, communicating, and understanding variability in data that will never really go away. I have always shuddered when students write lab report conclusions that discuss how “the data are/is wrong because” rather than focusing on what the data reveals about an experiment. We definitely want to work to minimize experimental error, but sometimes the variation in the data is an important characteristic of what is being measured.

Maybe this is something that needs to be explicitly taught in the way we present statistics to our students. It seems like something that needs to be drawn out over time, rather than in one big statistics unit of a course that focuses on other things. I think using technology to handle the mechanics of calculating statistical quantities allows students to focus more on what the statistics say and develop their intuition about it. We risk letting the important ideas of variation and statistics collect dust and stagnate as another box of content for students to throw in the closet of their busy, distracted brains.

Experiencing an ODE per day

I don’t like how applications of math are presented as a “special topic” once the theoretical has been understood. There are, admittedly, some aspects of concepts that are more thoroughly understood with background knowledge. I subscribe to an approach that bounces between applied and theoretical whenever possible.

This especially applies to differential equations. I tell my calculus students the story of my time at college when I took a course in differential equations. I spent a lot of time trying to understand how the processes of solving differential equations actually worked. It was the same way I studied multi-variable calculus the semester before, lectures for which I found fairly intuitive. The lectures for differential equations, on the other hand, were extremely technical and involved processes that were not clearly motivated by, well, anything from my professor.

This was also a time when my tolerance for pure mathematics was fairly low. As an engineering student, I needed to have an application nearby in order to push through the theoretical, otherwise my internal ‘what’s the point’ light would start flashing and I would tune out. There was very little of this in the lectures, but I pushed for understanding in my completion of problem sets and studying for the first exam.

My grade on the first exam was a 73. I was shocked. I also decided to give in to the suggestions of the sophomore students that had taken it before, who said just to memorize it all. I didn’t like it, but I did it, and my grade subsequently shot up. It was not until I took courses in system design, heat transfer, dynamics, and control systems, when I saw how differential equations actually worked and could work to understand much of the theory behind them.

Footnote: This entry is not in any way going to be an indictment of my university mathematics education (which was on the whole fantastic), a commentary on the perils of testing (which I do like discussing), or on how pure mathematics is not a rich course of study (which I further believe is NOT the case after teaching for several years and actually doing recreational mathematics on my own and with students). I don’t care to be boxed into any of those categories – the point is coming, I promise.

Last year was my first time teaching Calculus. Knowing how powerful differential equations are, I had prepared a full day where we spent looking at various differential equations and how they are used to model real phenomena from a bunch of different fields. What I found, however, was that my Calculus students reacted in much the same way as my other math students would when they sensed a day of word problems was ahead. There had to be a better way.

Here’s the plan:

Every day, I will show some physical situation with changes that can be modeled using a differential equation. No simulation allowed unless I can also show an actual apparatus that the students can visibly see, feel, hear, or otherwise sense changing over time, distance, etc. There’s something really powerful about generating data real-time – especially when it is related to something students can sense themselves.

My progress thus far:

Day 1: Newton’s Law of Cooling

I started class by asking a student to bring a mug of hot water from the water dispenser around the corner. When he returned, I tossed in a temperature sensor that I had connected to a National Instruments myDAQ board, and without much other commentary, started some review of antiderivatives. Close to the end, I stopped the LabVIEW program from running, and showed the resulting lovely graph of Temperature vs. time.

This resulted in all sorts of questions and discussions- when was the temperature changing the fastest with respect to time? What would happen to the derivative of temperature with respect to time as time went on? What is the physical meaning of this?

One of the students noted that this happened because the temperature of the cup was higher than the temperature of the room. This started a mini-discussion about situations where the temperature of the cup would rise. This all motivated the idea behind Newton’s law of cooling beautifully.

Day 2: Newton’s 2nd Law

The physics students weren’t impressed by this one. Part of the homework assignment from the previous day was to research and post information on the class wiki about a differential equation that (genuinely) held some meaning or interest for them. A couple of the students independently put on Newton’s 2nd, and I accepted it since they did it in slightly different ways. I then showed the students this apparatus (again, not a surprise to the physics students).

This time though, the focus was on the dynamics of an object on a spring. Giving the mass a nudge downward starting it oscillating nicely.

This led us to figure out what the forces acting on it were, namely gravity, the spring force, and possibly friction. This led to the differential equation form of Newton’s 2nd law. I did make available a Processing sketch I put together that contained the differential equation so they could see that this really was what governed the motion of the object.

We didn’t talk too much about the specifics of the program, as lines of computer code thrown at students tend to result in glassy eyes fairly quickly without proper preparation. We will look at programming again later on in the year though, so I’m not too upset that we didn’t talk about the details.

Future topics?

My hope is to include some lights attached to capacitors and resistors to show an RC circuit, a draining tank of water, deflection of a cantilevered beam, maybe even monitoring an oxygen sensor with a candle in a closed container. Part of me also wants to do a bar heated at one end, maybe a bit trickier since it is a partial differential equation, but I think it might also serve to get students thinking about how temperature might vary as functions of time and position. I don’t know what else, but I’m excited about the possibilities.

What are your favorite demonstrations of differential equations in action?

Students #flipping class presentations through making videos

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

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

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

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

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

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

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

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

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

The mathematical content is the focus, and the students understood that from the beginning. While the math isn’t perfect in every video, the enthusiasm the students had for putting these together was pretty awesome to watch. There’s no denying that enthusiasm as a tool for helping students learn – this is a major plus for project based assignments.

Some students that rarely volunteer to speak in class have their personalities and voices all over these. I love this.

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

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

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

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

A tale of two classrooms – which is yours?

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

Situation one:

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

Situation two:

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

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

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

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

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

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

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

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

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

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

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

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

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

What do you think?

Relating modeling & abstraction on two wheels.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[youtube http://www.youtube.com/watch?v=RR74vJo-okI&w=420&h=315]

Getting started with LEGO Robotics – the book and the real thing.

This week I got a special early holiday present in my mailbox from my friend Mark Gura. Mark had invited me a couple years ago to be part of a book for helping teachers new to the field of LEGO robotics get started with their students. We had a great conversation one evening after school over the phone during which we discussed the educational goldmine that building with LEGO is for students.

Mark did this with a number of people with a range of LEGO robotics experiences, wrote up our conversations, and then combined them with a set of resources that could be immediately useful to novices in the book.

The book, Getting Started with LEGO Robotics, is published by the International Society for Technology Education. If you, or anyone you know, is just getting started with this exciting field, you will find some great stuff in here to help you work with students and get organized.

It was a particularly perfect time for the book to arrive because we have a new group of students at my school getting started themselves with building and programming using the NXT. My colleague Doug Brunner teaches fifth graders across the hall. He volunteered (or more realistically, his students volunteered him) to take on coaching a group of students in the FIRST LEGO League program for the first time. After building the field for this year’s challenge, today the fifth graders actually got their hands on the robots and programming software. I had the robots built from the middle school exploratory class available so the students could immediately start with some programming tasks.

We started with my fall-back activity for the first time using the software – program the robot to drive across the length of the table and stop before falling over the edge. The students were into it from the start. I stepped back to take pictures and Doug took over directing the rest of the two hours. He is a natural – he came up with a number of really great challenges of increasing difficulty and wasn’t afraid to sit down with students to figure out how the software worked. By the end of the session, the students were programming their robots to grab, push, and navigate around obstacles by dead reckoning. It was probably the most impressively productive single session I’ve ever seen.

It’s always interesting to see how different people manage groups of students and LEGO. Some want to structure things within tight guidelines and teach step by step how to do everything. Others do a mini lesson on how to do one piece of the challenge, and then send the students off to figure out the rest. Some show by example that it’s perfectly fine to get something wrong in the process of solving a challenge by working alongside students. It was impressive to see Doug think on his feet and create opportunities for his students to work at different paces but all feel accomplished by the end of the day. It is also really unique to have to tell a bunch of students at 5:15 PM on a Friday to go home from school, and yet this has now been the norm in the classroom for a couple straight weeks.

Having robotics in my teaching load means that I am thinking about these ideas interspersed among planning activities for my regular content classes. There’s no reason why the philosophies between them can’t be the same, aside from the very substantial fact that you don’t have to tell students how to play with LEGO but you do often have to tell them how to play with mathematics or physics concepts. It’s easy for these robotics students to fail at a challenge twenty times and keep trying because they are having fun figuring it out. The holy grail of education is how to pose other content and challenge problems in the right way so it is equally compelling and motivating.

Note that I am not saying making content relevant to the real world. One of my favorite education bloggers, Jason Buell, has already made this point about why teaching for “preparation for the real world” as a reason to learn in the classroom is a flawed concept to many students that have a better idea than we do about their reality. I never tell people asking about the benefits of robotics that students are learning to make a robot push a LEGO brick across a table now because later on they will build bigger robots that push a real brick across the floor. Instead I cite the fact that seeing how engaged students are solving these problems is the strongest level engagement I have ever seen. The skills they develop in the process are applicable to any subject. The self esteem (and humility) they develop by comparing their solutions to others is incredible.

This sort of learning needs to be going on in every classroom. I used to believe that students need to learn the simple stuff before they are even exposed to the complex. I used to think that the skills come first, then learning the applications. Then I realized how incongruous this was with my robotics experiences and with the success stories I’ve had working with students.

Since this realization, I’ve been working to figure out how to bridge the gap. I am really appreciative that in my current teaching home, I am supported in my efforts to experiment by my my administrators. My students thankfully indulge my attempts to do things differently. I appreciate that while they don’t always enthusiastically endorse my methods, they are willing to try.

Rubrics & skill standards – a rollercoaster case study.

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

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

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

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

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

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

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

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

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

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

Him: I can’t figure it out.

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

Why this seems slippery and slopey:

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

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

Since the quizzes are formative assessments anyway, it’s a chance to see where he needs help. This quiz question gave me that information and I know what sort of thing we need to go over. He doesn’t need help with FBDs. He needs help knowing what happens in situations where an object is on the verge of leaving uniform circular motion. This is not a summative assessment, and there is still time for him to learn how to do problems like this on his own.

This is a perfect example of how a student can learn from his/her mistakes. It’s also a perfect example of how targeted feedback helps a student improve.

For a student stressed about assessments anyway (as many tend to be) this is an example of how we might work to change that view. Assessments can be additional sources of feedback if they are carefully and deliberately designed. If we are to ever change attitudes about getting points, showing students how assessments are designed to help them learn instead of being a one-shot deal is a really important part of this process.

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

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

Giving badges that matter.

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

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

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

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

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

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

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

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

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

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

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

Having conversations about and through homework

I’ve been collecting homework and checking individual problems this year. I grade it on completion, though if students tell me directly that they had trouble with a question before class (and it is obvious it isn’t a case of not being able to do ANY of it because they waited until the last minute to try) I don’t mind if they leave some things blank. I did this in the beginning since I had heard there were students that tried to skip out on doing homework if it wasn’t checked. We do occasionally go over assigned problems during class, but I tend not to unless students are really perplexed by something.

I have lots of opinions on homework and its value. Some can use the extra practice and review of ideas developed in class. Some need to use homework time to make the material their own. In some cases, it gives students a chance to develop a skill, but in those cases I insist that students have a reliable resource nearby that they know how to use (textbook, Wolfram Alpha, Geogebra) to check their work. I don’t think it is necessary to assign it just to “build character” or discipline. I read Alfie Kohn’s The Homework Myth, and while I did find myself disagreeing with some aspects of his arguments, it did make me think about why I assign it and what it is really good for. I do not assign busy work, nor do I assign 1 – 89 – each problem I assign is deliberately chosen.
Among the many ways I try to assess my students, I admit that homework doesn’t actually tell me that much about the skill level of a student. Why do I do it then?

My reason for assessing homework is for one selfish reason, and I make no secret of it with my students:

The more work I see from students relating to a concept, the better I get at developing that concept with students.

I would love to say that I know every mistake students are going to make. I know many of them. If I can proactively create activities that catch these misconceptions before they even start (and even better, get students talking about them) then the richness of our work together increases astronomically. You might ask why I can’t get this during conversation or circulation with students during the class period. I always do get some insight this way. The difference is that I can have a conversation with the student at that point about their thinking because he or she is in the room with me. I can push them in the right direction in that situation if the understanding is off. The key is that most of my students are alone when they do their work, or at least, have only online contact with their classmates. In that situation, I can really see what students do when they are faced with a written challenge. The more I see this work, the better I get.

I am not worried about students copying – if they do it, it always sticks out like a sore thumb. Maybe they just aren’t good at copying. Either way, I don’t have any cases of students that say ‘I could do it in the homework, but can’t do it when it comes to quizzes or tests.’ Since I can see clearly when the students can/can’t do it in the homework, I can immediately address the issue during the next class.

The other thing I have started doing is changing the type of feedback I give students on homework. I still fall into the habit of marking things that are wrong with an ‘x’ when I am not careful. I now try to make all feedback a question or statement, as if I am starting a conversation with a student about their work through my comments, whether positive or negative:

Great explanation using definition here.
Does x = 7 check in the original equation? (This rather than marking an x when a solution is clearly wrong.)
(pointing out two correct steps and then third with an error) – mistake is in here somewhere.
You can call “angle CPK” “angle P” here.
Good use of quotient rule – can you use power rule and get the same answer?

The students that get papers back with ink on them don’t necessarily have wrong answers – they just have more I can chat with them about on paper. The more I can get the students to understand that the homework is NOT about being right or wrong, but about the quality of their mathematical thinking, I think we are all better off.

This does take time, but it is so valuable to me, and I think the students not only benefit from the feedback, but appreciate the effort on my part. I don’t check every problem, just key ones that I know might cause trouble. If a student has everything right on the questions I am checking, it’s a chance to give feedback on one of the others. If there’s nothing to say because the paper is perfect (which is rare), I can praise the student for both their clear written solutions, hard work, and attention to detail.

I decided at the beginning of this year to look at more student work, and checking homework in this way is letting me do this. I am lucky to have prep time in the morning, and I have committed to using morning time for looking at student work almost exclusively. I have had to force myself to do this on many mornings because it’s so easy to use the time for other things. Some of my best ideas and modifications to lessons come after seeing ten students make the same mistake – it feels good to custom fit my lessons to the group of students I have in front of me.

In the end, it’s just one more way the students benefit from having a real teacher working with them instead of a computer. Every mark I make on the paper is another chance to connect with my students and conversation that can help make them better thinkers and learners. I don’t think I really need to justify my presence in the classroom, but it feels good to say that this is one of the reasons it’s good I’m there.

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