Monthly Archives: October 2011

Dare to be silent.

I made a promise to myself today - I was going to force the physics class to speak. It isn't that they don't answer questions and participate, it's that usually they seem to do that to please me. Sometimes they will explain ideas to each other and compare answers, but it never works as beautifully as I want it to.

So today I told them I wasn't going to talk about a problem I gave them. They were. And then I sat on an empty table and waited. It was really difficult for me. Eventually someone asked someone else for an answer. I stayed quiet. Then another person nodded and agreed and then said nothing. I stayed quiet. Then someone disagreed.

Full disclosure - at this point I gestured wildly, but still stayed quiet.

After about five minutes of awkward silence punctuated with half explanations that trailed off, something happened - I don't know what the trigger was because if I did I would bottle it and sell it at educational conferences - a full discussion was suddenly underway. I was so amazed that I almost didn't think to capture it - thankfully I did get the following part:

Especially cool to see this knowing that English is not the first language of the students speaking.

I'm going to try to do this more often, though I again must point out that it was incredibly difficult working through the silence. The students in the end decided they had something to say, so they shared their thoughts with each other. I did nothing but wait for it to happen.

Build the robot the way you want...No, you're doing it wrong!

I teach an exploratory class for middle school students in robotics. The students rotate between robotics and some other electives during each quarter of the year, and there are no grades - just an opportunity to learn something interesting while doing. I like the no grades part, particularly because assessing progress in robotics is quite hard to do. My usual model is saying you get x points for doing the bare minimum (a D) and then incremental increases in the grade for doing progressively more challenging tasks.

It works, but I really like getting the opportunity to not have to do it. There are so many things you could measure to assess the students in their building and programming skills, but in my experience, the students don't tend to explore or tinker as much in that situation.

Today while working on the day's challenge on using sensors and loops, a student was fixated on building the following:

There weren't any instructions to do this - he just started putting things together, liked what he had created, and continued building it today.

I had to stop myself for a moment because teacher Evan started to come out and remind him to stay on task and contribute toward his team's solution to the challenge. Thankfully robotics Evan intervened and let it happen.

This is an exploratory class. It's supposed to expose the students to new situations that might interest them later on. Why in the world would I stop the exact sort of thinking and exploring the class was designed to provoke? It also turned out that this student, along with the other two in the group, were all taking turns in the programming and building so that each would have a chance to play in this way. In spite of their taking time to free build, this group actually solved the three challenges before the rest in the class.

It connects to a great TED talk on doodling by Sunni Brown. One idea from her talk was that doodling contributes to "creative problem solving and deep information processing." I think that these students (and all of us that 'play' with building toys like LEGO) are engaging in a similar process by free building. The connections students are making in figuring out how the tools work through play are not easy to measure. This is a horrible reason not to provide them time to do so. I do think there is an interesting connection between the tendency of these students to play and their ability to figure out the subtler parts of the class challenges.

After all, as David Wees pointed out, giving explicit step-by-step instructions on how to use creative tools like LEGO takes the creativity (and much of the fun) out of it. There has to be time to experiment and learn by doing how the tools work together, and that's exactly what this student was doing.

My final reaction during the class today was that I told them all that I wanted to take a picture of every off-topic LEGO design they create. Document it all. If it's cool enough to engage you for the time it takes to create it, I want a gallery of those designs to celebrate them.

The sad part? This made some of them stop. I can't win!

Are there too many people on this thing? (#anyqs)

Two images I want to share for the purposes of #anyqs. For those unaware, this hashtag means I want you to look at these images and let me know what mathematical questions jump out at you right away.

Once you've taken a moment to think about these, please send me a tweet (@emwdx) with the hashtag #anyqs letting me know what you think this is all about. That would be great if you could take a moment.

What would be even cooler, however, is if you could also take pictures of your own of these signs in elevators (or other places you might find them) as well as a little information about where it is being taken. For example, the left picture is taken in the elevator in my apartment building while the right is from the gondola that the 9th graders and I took to get down from Mount Tai on the rainy second-to-last day of the China trip.

I am especially curious to see how these signs vary between buildings, countries, and even between elevator banks within a building. I'd like to share these with students tomorrow, so please snap a quick picture and send it (or a link) my way. It hopefully goes without saying that I will give proper credit for photos that make it into the materials I use with students, and will share what comes of it with you all.

So, for the sake of an interesting idea that I think will start some cool discussions with students - skip the stairs today. Snap a picture in the elevator, and then reward yourself for your generosity (and for decision to postpone exercise for unselfish reasons) by eating a doughnut or other craved food of your choice.

Take Time to Tech - Perspectives after a Flip


Yesterday my calculus students reaped some of the benefits of a flipped class situation - I made some videos on differentiation rules and asked that they watch the videos sometime between our last class and when we met yesterday. We spent nearly the entire period working with derivatives rules for the first time. The fact that the students were getting their first extended period of deliberate practice with peers and me around (rather than alone while doing homework later on) will hopefully result in the students developing a strong foundation what is really an important skill for the rest of calculus.

They were using Wolfram Alpha to check their work, something that I paid lip-service to doing last year but did not introduce explicitly on the first day of learning these rules last year. There was plenty of mistake-catching going on and good conversations about simplifying and equivalent answers. I needed to do very little in this process - good in that the students were teaching themselves and each other and being active in their learning.

It was also interesting doing this so soon after discussing the role of technology in helping students learn on the #mathchat Twitter discussion. There were many great points made regarding the content of technology's effective use across grades. It made me think quite a bit about my evolution regarding technology in the classroom. Many comments were made about calculator use, teaching pencil and paper algorithms, and the role of spreadsheets and programming in developing mathematical thinking. I found a lot of connections to my own thoughts and teaching experiences and it has me buzzing now to try to explain and define my thinking in these areas. Here goes:

Developing computational and algorithmic fluency has its place.

In the context of my students learning to apply the derivative rules, I know what is coming up the road. If students can quickly use these rules to develop a derivative function, than the more interesting applications that use the derivative will involve less brain power and time in the actual mechanics of differentiation. More student energy can then be focused in figuring out how to use the derivative as a tool to describe the behavior of other functions, write equations for tangent and normal lines, and do optimization and minimization.

There was a lot of discussion during the chat about the use of calculators in place of or in addition to students knowing their arithmetic. I do think that good arithmetic ability can make a difference in how easily students can learn to solve new types of mathematical problems - in much the same way that skill in differentiation makes understanding and solving application problems easier. Giving the students the mental tools needed to do arithmetic with pencil and paper algorithms empowers them to do arithmetic in cases when a calculator is not available.

Technology allows students to explore mathematical thinking, often in spite of having skill deficiencies.

One of the initiatives my colleagues took (and I signed on since it made a lot of sense) when I first started teaching was using calculators as part of instruction in teaching students to solve single variable linear equations. There was a lot of discussion and protest regarding how the students should be able to manage arithmetic of integers in their head. It wasn't that I disagreed with this statement - of course the students should have ideally developed these skills in middle school. The first part of the class involving evaluating algebraic expressions and doing operations on signed numbers were done without calculators in the same way it had been done before.

The truth, however, was that the incoming students were severely deficient in number sense and arithmetic ability. Spending a semester or two of remediation before moving forward to meet the benchmarks of high school did not seem to make sense, especially in the context of the fact that students could use a calculator on the state test. So we went forward and used calculators to handle the arithmetic while students needed to reason their way through solving equations of various forms. They did learn how to use the technology to check the solutions they obtained through solving the equations step-by-step using properties. There were certainly downsides to doing things this way. Students did not necessarily know if the answers the calculators gave them made sense. They would figure it out in the end when checking, but it was certainly a handicap that existed. The fact that these students were able to make progress as high school math students meant a lot to them and often gave them the confidence to push forward in their classes and, over time, develop their weaknesses in various ways.

I have seen the same thing at the higher levels of mathematics and science. I used Geogebra last year in both pre-Calculus and Calculus with students that had rather weak algebra skills to explore concepts that I was taught from an algebra standpoint when I learned them. Giving them tools that allow the computer to do what it does well (calculate) and leave student minds free to make observations, identify patterns, and test theories that describe what is happening made class visibly different for many of these students. If a computer is able to generate an infinite number of graphs for a calculus student to identify what it means for a function graph to have a zero derivative, then using that technology is worth the time and effort spent setting up those opportunities for students.

Using skill level as a prerequisite for doing interesting or applied problems in mathematics is the wrong approach.

Saying you can't drive a car until you can demonstrate each of the involved skills separately makes no sense. Saying that students won't appreciate proportional reasoning until they have cross-multiplied until their pencils turn blue makes no sense. Saying that learning skills through some medium makes all the other projects and applications that some of us choose to explore in class possible does not make sense. It makes mathematics elitist, which it certainly should not be.

Yes, having limited math skills is a limit on the range of problem solving techniques that are available to students. A student that can't solve an equation using algebra is destined to solve it by guess and check. Never underestimate the power that a good problem has to entice kids to want to know more about the mathematics involved. Sometimes (and I am not saying all the time) we need to work on the demand side in education, on the why, on the context of how learning to think in different ways applies to the lives of our students.

Emphasizing algorithms without providing students opportunity to develop context or some level of intuitive understanding (or both) has significant negative consequences.

I don't mean to suggest that teaching algorithms on their own can't result in students performing better on a type of problem. The human brain handles repetition extremely so well that learning to do one skill through repetition is not necessarily a bad way to learn to do that one thing.

One problem I see with this has to do with transferring this skill to something new, especially when the depth of available skills is not great. Toss a weak student ten one-step equations of the form x + 3 = -8, and then give them something like 0.2 x = 25, and chances are that student won't solve it correctly without some level of intuition about the subtle differences between the two. Getting this right takes practice and feedback really good opportunity for students to be reflective of their process.

It is also far too easy when applying an algorithm to stop thinking critically about intermediate steps. I spoke to a colleague this week about his students learning long division and we both questioned the idea that the algorithm itself teaches place value. We looked at a student's paper that was sitting on the desk and instantly found an example of how the algorithm was incorrectly applied but through a second error resulted in a correct answer. If we teach algorithms too much without giving activities that allow students to show some sort of understanding of some aspect of how the algorithm fits into their existing mathematical knowledge, it's undercutting a real opportunity to get students to think rather than compute. I like the concepts pushed by the Computer-Based Math movement in using computers to compute as they do best, and leave the thinking (currently the strength of the human brain) to those possessing one.

As often as we can, it is important to get students to interact with the numbers they are manipulating. Teaching the algorithms for multiplying and adding large numbers does provide students with useful tools and does reinforce basic one digit arithmetic. I get worried sometimes when I hear about students going home and doing hundreds of these problems on their own for various reasons. If they enjoy doing it, that's great, though I think we could introduce them to some other activities that they might see as equally if not more stimulating.

I do believe to some extent that full understanding is not necessary to move forward in mathematics, or any subject for that matter. I took a differential equations course in college trying to really understand things, and my first exam score was in the seventies, not what I wanted. I ended up memorizing a lot after that point and did very well for the rest of the course. It wasn't until a systems design course I took the following year that I actually grasped many of the concepts that eluded me during the first exposure. This same thing worked for me in high school when I took my first honors track math class after being behind for a couple years. My teacher told me at one point to "memorize it if I didn't understand it" which worked that year as I was developing my skills. Over time, I did figure out how to make it make sense for myself, but that took work on my part.

Uses of technology to apply/show/explore mathematical reasoning comprise the best public relations tool that mathematics has and desperately needs.

I really enjoyed reading Gary Rubenstein's recent post about the difference between "math" and mathematics. I read it and agreed and have been thinking a lot along the lines of his entry since then.

Too many people say "I'm not good at math." What they likely mean is that they aren't good at computing. Or algorithms. Or they aren't good at ________ where __________ is a set of steps that someone tried to teach them in school to solve a certain type of problem.

On the other end of the perceived "math" ability spectrum, parents are proud that their children come home and do hundreds of math problems during their free time. These students take the biggest numbers they can find and add them together or multiply them and then show their parents who are impressed that their normally distracted kids are able to focus on these tasks long enough to do them correctly.

It makes sense that most people, when asked to describe their experiences in math, describe pencil and paper algorithms and repetitive homework sets because that's what their teachers spent their time doing. This, unfortunately, is the repetitive skills development process that is part of mathematical learning, but should not be the main course of any class. We show what we value by how we spend our time - if we spend our time on algorithmic thinking, then this is what students will think that we as teachers and as thinkers value as being important in mathematics.

This fact is one of the main reasons I started thinking how to change my class structure. My students were talking about not being good at a certain type of problem ("I don't get this problem...I can't do problems that need you to...") rather than having difficulties with concepts ("I don't get why linear functions have constant slope...I don't get why x^2 + 9 is not factorable while x^2 - 9 is).

If we as teachers want students to value mathematics as more than learning a set of problems to be solved on a test, then we have to invest time into those activities that allow students to experience other types of mathematical thinking. This is where technology shines. The videos of Vi Hart, Wolfram Alpha, the antics of Dan Meyer, the Wolfram Demonstrations Project, the amazing capabilities of Geogebra - all of these offer different dimensions of what mathematical thinking really is all about.

We can share these with students and say "check these out tonight" at the end of a lesson and hope that students do so. Sometimes that works for a couple students. That isn't enough.

I think we need to invest in technology with our students with our time. We need to deliberately use valuable class time to take them through how to use it and why it makes us excited to use it with them. It's really the only way students will believe us. Show that it's important, don't just tell your students it is. That's right - that valuable class time that we often plan out too carefully and structure so that they reach the well-defined goals we have for them - that time. Plan to use a specific amount of class time, and enough time, to let students play around with a mathematical idea using any of the amazing technology tools out there. Show them how you play with the tools yourself, but don't make this the focus of this time - do so afterwards, perhaps.

To be clear - I am not saying do this all the time. Students need to learn algorithms, as I have already stated. Students also need to be looking at interesting problems. We should not wait to show them these problems until after students have demonstrated automaticity because it gives students the impression that the algorithms came before the thinking that went into them.

I am saying that balance is key.

The only way we are going to change the perception of what mathematical thinking really looks like is by living it and sharing it with our students.

Physics #wcydwt - Indirect Measurement

While cleaning up after robotics class today, I noticed a statics problem involving an object hanging from a couple wires that was poking out from under one of my many piles of papers. We had looked at this question earlier during the week in class. A couple students were out for a volleyball tournament in Beijing, so I wanted to do something hands on and multimedia-esque that the missing students wouldn't feel too upset about missing, but could somehow still be involved and connected with the class work from today.

I realized that we hadn't yet used the spring scales during our discussion of forces. My obsession with #wcydwt lately has been on using the novelty of a minimum amount of information to get students to see a problem jump off the page/screen. I also wanted the students in class to get the joy of holding back information from their classmates to see if they could figure out the missing info. Lastly, I wanted there to be a simple physics problem that would serve to assess whether all of the students understood how to solve a 2D equilibrium problem.

So I grabbed the spring scales, some string, slotted weights, and told the students to put together a few pictures using these materials. We briefly discussed what information could be given, and what they wanted to leave out for the athletes to figure out on their own. I admit - I pushed them along, and given more time I would have given them more choice, but I don't think my selfishness and excitement in doing this was too much. The other factor - the vice principal had given us an extra pizza to share - they were also really pushing for efficiency. It wasn't all me.

And thus the spring scale picture project was born, thanks to one student's iPhone and Geogebra:

The complete link of the assignment is at http://wiki.hischina.org/groups/gealgerobophysiculus/wiki/e495b/Unit_2__Spring_Scale_Challenge.html.

I'm sure I am not the first to do this, but it was so simple to execute that I had to give it a shot, and I am sharing it because I'm trying to share everything I can these days. We will see what happens when the results come in next week.

Scheming with Schema...

When teaching physics before, I found the process of building free body diagrams with students to be a fairly smooth process. It took a lot of feedback to get there, but they way I introduced the topic was along the lines of the chart below:

This chart was based on one I had from my own physics notes taken during class with Mr. Bob Shurtz who influenced me both as a student (helping me explore the love of physics and engineering I didn't know I had beforehand) and then as a colleague while designing my own AP Physics course in the Bronx.

I held students to the requirement in the beginning that every time they constructed a FBD they must make one of these charts because my feeling was it would help both in identifying the important forces acting on a single object and in discussions of Newton's 3rd law. The students grumbled as they tend to do when we expect them to use organization scaffolds like this that they feel they don't need. As time went on and FBDs were drawn correctly, I would loosen that requirement to the point that students were drawing diagrams and, minimally, felt guilty if they weren't at least thinking to make sure all of those forces could be identified. Those charts were admittedly annoying, but I felt they at least got students in the right mindset for drawing free body diagrams, so it was a good thing to require.

When my fans on carts exploration with the students went long last week, I decided to push the introduction of FBDs to this past Monday. We did have time last week to talk out different types of forces (normal, gravity) so they at least had some ideas of what different forces could be included in the chart. This extra time gave me the weekend to take a closer look at Modeling Instruction, and more specifically, at the concept of drawing system schema. I had never heard this term, but it appeared all over the modeling literature, so I decided to take a closer look at the Arizona State University site on modeling where I found an excellent paper that details using them as part of the FBD development process. It seemed harmless enough. Worst case, it would be a scaffold like the chart I mentioned earlier, used in the beginning and then taken away over time.

It was especially lucky that shortly after reading this, Kelly O'Shea had posted an excellent guide on how she introduces the Balanced Force Particle model to her class. It seemed like such a natural way to analyze problems, so I introduced it to the class as part of drawing free body diagrams for the first time on Monday.

Some really interesting things happened during that class and during Wednesday's class that deserve to be shared here. First, I was impressed how naturally students took to the idea of drawing the schemata. Not a complaint in the room.
They shared with each other, pointed things out, and quickly came to an agreement of what they should look like.
It was incredibly natural for them to then draw a dotted circle around the object they were analyzing and see the free body diagram nearly jump out at them. The discussions about directions and what should be in the diagram were matter of fact and clear with virtually no input from me. Score one for the schema.

The second thing that came up during class on Wednesday was in discussing a homework problem about a bicycle moving down a hill at constant speed due to a drag force of magnitude cv. The schema that one student had put together looked something like this:

The students were wondering how they would combine the friction from the ground and the air drag force into one to use the given information.

I was floored - after giving this problem for four years in a row, this was the first time the students even thought to think of anything about the friction on the ground. They decided to neglect this force after we thought about whether drag force had anything to do with the ground, but the fact that we even had this discussion was amazing and really shows the power of the schema to get students to think about what they are doing.

The final thing the class pointed out was an inconsistency that had again never even occurred to me. On Wednesday, we were looking at the following sketch as part of a kinetic friction problem:

The block was moving at constant velocity across a surface with coefficient of friction of 0.7. I asked the students to draw a schema, FBD, and figure out what the magnitude of the force F must be. They started working on their schemata, but then had these uncomfortable looks on their faces shortly afterwards.

Looking at the diagram, they had no problem identifying the effects of the entire earth and the ground, and they were fairly sure based on the situation that drag was not an important part of it. The thing they really didn't know how to handle was that disembodied force F.

What object was causing it? Where was it coming from? How in the world could they include it in the diagram if they didn't know what interaction was governing its presence?

At this point in previous years, students didn't generally mind that random forces were being applied to blocks, spheres, or other random shapes - they just knew that they had to do a sum of the net force in x and y and solve for unknowns in the problem. In the context of the schema, however, the students were clearly thinking about the situation in exactly the way I had taught them to do and were genuinely concerned that there was no clear source of this force. This goes back to the fact that they were seeing the system schema as a representation of real objects, which is really what we want students to be doing! I had never thought about this before, but it was so amazing to know that they were having these thoughts on the second day of meeting the free body diagram.

We agreed on the spot, given my omnipresent power as a physics teacher, that any time a force appeared in a problem diagram that had no clear source, that it had to be because of an interaction with me, and they could include me in the schema to indicate that interaction. For the purposes of satisfying their newly found need for a source for every force (a possible catch phrase for schemata?) they now have permission to do this in their schemata.

I admit that my students in the past have gotten away with abstracting the process of equilibrium problems into barely more than a math problem. That capability has still gotten them to analyze some interesting situations and pushed them to explain phenomena that they observe in their own lives. Still, the way using the schema changed our conversations over the past couple days is an impressive piece of evidence in favor of using them.

In short? I'm sold. I'll take twenty.

What do you do when they don't need you?

I've tried an experiment over the last two days - my advanced algebra students and geometry students each had some challenging tasks that I sort of left to them to figure out. Last year, I taught them very explicitly how to do the tasks at hand, modeled some examples along side their own work, and then gave them time in class to practice. For homework, I gave them more problems that were similar to those we did in class, giving them more chances to practice what I had assigned them.

This year I turned it around. In geometry, we are starting proofs. I gave them a couple relatively simple ones, and asked them in groups of two to construct some sort of logical reasoning to go from a starting point to proving the statement I had given them. There was a lot of struggling, difficulty stating using facts why one logical statement led to another. Over time, they did start communicating with each other and sharing what they were thinking. I did occasionally poke one group in a certain direction, but didn't lead the whole group in that way. Eventually they were all thinking along the lines that I envisioned at the beginning. I could have modeled for them what I did last year, but I saw a lot of really good conversations along the way. By the end, they were much closer to making their own proofs than they had in the beginning. By the end, they were clearly seeing the connections between thoughts. This was only the second class period during which we had talked about proofs. While I don't think any of them would wager large sums of money over constructing geometric proofs, I think they at least see how the system can be used to make logical statements that are irrefutable.

I did something similar with the advanced algebra group which was to figure out graphing absolute value functions during our lesson last Friday. I gave them an exploration that was, in hindsight, confusing and didn't do much aside from frustrate them with Geogebra commands. I told them that I wanted them to use Geogebra, the textbook, Wolfram Alpha, and any other resources available to learn how to graph any arbitrary absolute value function by hand. At the end of the class I broke down and apologized for giving a poorly designed exploration. I told them I would put together a video on graphing functions, and I did - posted it on the wiki.

When we had class on Tuesday, I found out that none of the students had actually taken advantage of the video. They had looked in the textbook. They had graphed functions over and over. When I asked students to share what they had figured out, one student used a table of values and a piecewise function based on the sign of the argument of the absolute value function. Another student had graphed both the argument of the absolute value function and its opposite since that was what this student had observed, and then erased either the top half or bottom half of the graph. Another student broke down and did what the book said to do. By the end, all of them were graphing absolute value functions using their own method. I wasn't sure about understanding, but in the end, I admit that I didn't quite mind. They all had their own models for what was going on, and they were confident that they could use the technology to confirm whether what they were doing was right or not.

I have always wondered about what I would do in the situation where it was clear a group of students didn't need me to be there. In a way, this is part of my fear of tools like Khan academy - if there are others out there that are more engaging, better at explaining ideas, or better at coming up with really interesting questions that got students thinking about what they were doing, and these people happened to make videos: what would I do in my classroom if students got hold of these videos? Would I be mad? Or would I figure out a way to take advantage of the fact that students had figured something out on their own and use it to do something even more interesting or impressive?

I think I do a good job of engaging students - today we were talking about differentiability and we were joking like crazy about whether functions would be differentiable at a given point or not. Really? Were we really joking about this? It seemed like everyone was minimally entertained, but based on the questions I was bouncing around from person to person, it seemed like they also understood the concept. I know teachers that are more effective though at making kids understand and be entertained and do problem after problem until concepts are so painfully clear that they become automatic. These teachers could easily make a career in stand-up or television based on their comedic brilliance and presence - what if they decide to make videos?

What then? What do I do? If I get to that point, is it the end of my usefulness as a teacher?

Or is that just the beginning? Maybe that's the point where I can assume my students have a certain level of basic knowledge and I can then build off of that level to do even cooler stuff. Maybe that's where I can assume, for once, that my students have a base level of skills, and can then rise above to analyze bridges or patterns in nature or create a mathematical model that inspires a student to choose to be a doctor or engineer where he or she never would have if I hadn't done the right project or group activity or lesson. Maybe the fact that I entrusted my students to try to figure out something on their own is enough that they feel empowered to try things even if failing again and again is a possibility. Make mistakes and come back asking questions about why their theories were incorrect. I remember worrying at one point in my career what horrible things would happen if I somehow introduced students to a concept in a way that it caused them get a question wrong and cause them just one more failure in a line of failures. If I could teach in a way that makes students feel OK coming in the next day saying "I didn't get it, but this is what I tried" I'd feel pretty good about myself. That is, ultimately, the sort of resilience that a person needs to survive in this world.

If the students show us that they don't need us to show them how to solve a specific problem, then we as teachers should honestly accept this fact. Our goals, if they can do the problem in a way that is mathematically correct, should shift to applying that ability to doing something more profound and relevant, be it communicating that solution or applying the solution to a new situation that is different but connected in a subtle way. Our job puts us in contact with some really amazing minds that are eager to do what we say in some circumstances. In the cases that they demonstrate that they don't need us - that is when we must apply our professional judgment and teach them to expand their knowledge to something bigger than themselves.

Not sure why I'm waxing so philosophical today, but I've been really impressed with my students this week, and it's only Wednesday! After these few days, it feels like what I'm doing with this group is like using a super computer to do word processing. I only hope they are enjoying the process as much as I am.

Nice work when you can get it....

I graded my first set of physics tests today. With the group deciding not to take the class at the Advanced Placement level, we've been able to slow down and spend time experimenting and really engaging with kinematics and projectile motion. I assigned them problems and helped them learn, but I was more impressed with the experiments they worked on and their engagement level during those activities. I was concerned about what would happen when we returned to solving problems, but I was very pleasantly surprised.

I'm interested in sharing student work, so this is the first time I will be doing it. When I asked the student if it would be OK to share, the student agreed and was really excited that I would want to show the work to other teachers.

This student started out solving problems in a very scattered way: calculations here, sketches there, units nowhere to be found. When I showed the structure I wanted students to use to solve problems, it was initially a burden. The student didn't like doing it. Upon grading, I was very happy to see this:

The degrees vs. radians issue is one that I always battle, made especially difficult this year because students have me for physics (when I insist on degree mode almost exclusively) and then calculus (when I change my insistence to radian mode) right in a row. Yes, the student should have noticed that multiplying by the ratio should not have resulted in a negative sign for initial y-velocity. Yes, it should have again become obvious that something was up when he found he needed to 'add' a negative sign in the answer at the end to make the sign of the answer make sense according to his own sign convention.

The fact that the student can notice these things (and that I can see where the errors are) would not even be something I could discuss if it wasn't for the structure I put in place. By learning to use the structure to organize thoughts, this student became able to solve problems in a logical manner rather than with calculations all over the page. I don't like teaching procedures, but this is an example of where it pays off.

We like students exploring and experimenting and constructing their own knowledge. These are really good ways for them to spend time in our classroom. I include using correct mathematical notation, showing steps, labeling axes, learning terminology, and other things of that nature as part of my class expectations - at times a battle that seems unimportant in the context of what I really want students to know how to do in five or ten years. There is room for procedural knowledge, however, and this student's success is evidence of why we do it.

The main point (and the thing I've been working to change compared with how I did things for a while) is that these procedures should not be the meat of a lesson or the main focus of instruction all the time. These things CAN be taught by computers or videos and don't necessarily need a human in the room. It is important for students to have skills and have access to resources that help them develop those skills.

But getting answers is not the point - this is the tricky part that we have to do a better job of selling to students, at least I do. Clear communication of reasoning and sharing the logic of our ideas are some of those "21st century skills" that students should have when they leave our classrooms. If a student needs to learn a structure to help them with this process, it is worth the time needed to help them learn it.

"What can you do with this?" (WCYDWT) - Flood Gates Open

I've been making an effort to look for as much WCYDWT material as possible on a regular basis. This is not so much because I've had students asking 'when are we going to use this' though that is always brewing under the surface. Instead, I've been making an effort this year to spend less time in class plodding through curriculum, and more time getting students to get their hands dirty with real data, real numbers, and using their brains to actually figure things out. By recording screencasts, doing demos, and using Geogebrs, I've made some progress in getting the students to see the benefit of learning the routine skills-based stuff on their own for HW so we can use class time to do more interesting things. I've quizzed and am feeling pretty good about this thus far, but we'll see.

During my trip with the ninth graders to Shandong and my week off due to the national holiday when my parents visited, I've kept my eyes open on reasonable, non-contrived problems that might serve as applications of linear functions. I've wanted some problems with non-trivial answers along with some low-hanging fruit that might give all of the students in the class a way in.

I'm pretty happy with how things have ended up with the top three contenders. There are some other things in the works, but I'm hoping to keep those under wraps for the moment. Click on the links to read the details.

Climbing Mount Tai

This one I already started talking about in a previous post, but I spiced it up just a bit by putting images together and throwing the head image I've now used in a few places to be cute.

Ms. Josie and the 180 Days

I like this one especially since it has a good story behind it. My students know my wife, and I defer to her awesomeness quite a bit in class. Students certainly love it when their teacher is willing to knock him/herself down a few pegs, especially when it's for their entertainment and for comedic effect in class. I think this challenge is a good combination of mathematical reasoning and drama - I don't think I can lose!

Moving on up at the Intercontinental Hotel

I was looking for a third one that really jumped out as kinda cool and visually stunning since the others, though cool, weren't particularly impressive visually. On the last day my parents were in town, we went to the Intercontinental hotel in Hangzhou and the problem smacked me in the face.

The videos aren't all up yet - in addition to the two outside videos, the more enlightening videos (which I will post tomorrow before class) have a view of the elevator doors and the digital floor display as the elevator moves up and down. In addition, there is a nice reflection of the view out the glass wall of the elevator, beautiful in its own right, but perhaps a wee bit distracting from the really useful stuff in this problem. If I wanted to go the full-eye-candy route, I suppose I could have gotten a reflection of the elevator doors and floor display in the glass wall of the elevator. Maybe next time.


My plan is to let students choose which of the three projects they want to work on, and then give them tomorrow's class (and finishing up for HW) to put something together. I plan to grade according to this rubric:

I think it gives them enough detail on what I want them to do, without being overly difficult to grade. I am even thinking of giving them a chance to grade each other since they will all be posting their work (from groups) on the wiki page.

I've had these things in my mind for a little while - I admit, after how this particular class made an impressive effort I am really excited to see what happens next.

Modeling anyone? Fans on carts edition.

After reading a lot about the success that others have had with teaching physics using the modeling method, I'm giving it a shot as I start Newton's laws with my physics class. When I taught this with my AP physics previously, I did a traditional development of Newton's laws describing (I admit it - lecturing) about Newton's understanding of what caused acceleration. We talked about acceleration being proportional to net force and inversely proportional to mass, and then went from there exploring what it meant for net force to be zero through a series of problems involving net forces, components, etc.

What I did seemed to work in so far as students were able to solve the problems I gave them. The undying assumption of course is that what I did was efficient and made me feel that I had got across the material to students, but along the way I wasted an opportunity for students to SEE the principles in action and try to figure things out on their own. Since my students this year are not taking the course at the AP level, I see no reason not to try this and see how it compares in the long run to student understanding and enjoyment of the exploration of physics concepts. It is the sort of thing that I can see doing even in the Physics B curriculum, as dense as it is, given the fact that students really need a chance to play to connect the mathematics of the equations to the fact that physics describes the real world, not just idealized situations.

Here's where I'd love to get some input though - I am giving my students a test in the first half of the 85 minute period tomorrow, and then my plan is to let them spend the rest of the time watching some videos that I took this afternoon of toy fans attached to cars on an air track. The students will get to play with the actual air track, but I want to introduce to the way I want them to play by seeing these videos that I created.

I have posted the series of videos here at my wiki site. The general instructions for what I want them to do are there, but I might as well run through them here as well.

First, I want them just to watch all the videos. No physics, just observation. After they have done this, I've posted a number of questions I want them to use to classify, analyze, and predict based on constant velocity and non-uniform velocity cases. I plan to have them sketch what effect a single fan would have on the motion of the cart. My plan in the end is to have them construct a situation with the fans that results in a given scenario. For example: arrange the fans on the cart so that The cart has zero initial velocity and an acceleration to the left. Draw position, velocity, and acceleration graphs, and then use Tracker to confirm/refute what their models suggest will happen.

Let me know your thoughts either here or through Twitter (@emwdx) - I am excited to try this, and excited to give the students a chance to get some first hand experience testing their own ideas. I had a blast playing with it this afternoon, and while I do have a different standard for what is 'fun' at times, I don't think this is one of those times.

Wiki site: http://wiki.hischina.org/groups/gealgerobophysiculus/wiki/52698/Unit_2__Carts_with_Fans.html