## From projectile motion to orbits using Geogebra

I was inspired last night while watching the launch of the Mars Science Laboratory that instead of doing banked curve problems (which are cool, but take a considerable investment of algebra to get into) we would move on to investigating gravity.

The thing that took me a long time to wrap my head around when I first studied physics in high school was how a projectile really could end up orbiting the Earth. The famous Newton drawing of the cannon with successively higher launch velocities made sense. I just couldn't picture what the transition looked like. Parabolas and circles (and ellipses for that matter) are fundamentally different shapes, and at the time the fact that they were all conic sections was too abstract of a concept for me. Eventually I just accepted that if you shoot a projectile fast enough tangentially to the surface of the Earth, it would never land, but I wanted to see it.

Fast forward to this afternoon and my old friend Geogebra. There had to be a way to give my physics students a chance to play with this and perhaps discover the concept of orbits without my telling them about it first.

The images below are the sorts of things I am hoping my students will figure out tomorrow. From projectile motion:

...to the idea that it is still projectile motion when viewed along with the curvature of the planet:

Continuing to adjust the values yields interesting results that suggest the possibility of how an object might orbit the Earth.

If you open the file, you can look at the spreadsheet view to see how this was put together. This uses Newton's Law of Gravitation and Euler's method to calculate the trajectory.You can also change values of the variable deltat to predict movement of the projectile over longer time intervals. There is no meaning to the values of m, v0, or height - thankfully the laws of nature don't care about units.

As is always the case, feel free to use and adjust this, as well as make it better. My only request - let me know what you do with it!

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.

I want to record a few things about the last couple of days of class here - cool stuff, some successes, some not as good, but all useful in terms of moving forward.

## Geometry:

I have been working incredibly hard to get this class talking about their work. I have stood on chairs. I've given pep talks, and gotten merely nods of agreement from students, but there is this amazing resistance to sharing their work or answering questions when it is a teacher-centric moment. There are a couple students that are very willing to present, but I almost think that their willingness overshadows many others who need to get feedback from peers but don't know how to go about it. What do I do?

We turn it into a workshop. If a student is done, great. I grab the notebook and throw it under the document camera, and we talk about it. (In my opinion, the number one reason to have a document camera in the classroom, aside from demonstrating lab procedures in science, is to make it easy and quick for students get feedback from many people at once. Want to make this even better and less confrontational? Throw up student work and use Today's Meet to collect comments from everyone.

The most crucial thing that seems to loosen everyone up for this conversation is that we start out with a compliment. Not "you got the right answer". Usually I tolerate a couple "the handwriting is really neat" and "I like that you can draw a straight line" comments before I say let's have some comments that focus on the mathematics here. I also give effusive and public thanks to the person whose work is up there (often not fully with their permission, but this is because I am trying to break them of the habit of only wanting to share work that is perfect.) This praise often includes how Student X (who may be not on task but is refocused by being called out) is appreciative that he/she is seeing how a peer was thinking, whether it was incorrect or not. I also noticed that after starting to do this, all students are now doing a better job of writing out their work rather than saying "I'll do it right on the test, right now I just want to get a quick answer."

## Algebra 2

We had a few students absent yesterday (which, based on our class size, knocks out a significant portion of the group) so I decided to bite the bullet and do some Python programming with them. We used the Introduction to Python activity made by Google. We are a 1:1 Mac school, and I had everyone install the Python 3 package for OS 10.6 and above. This worked well in the activities up through exercise 8. After this, students were then supposed to write programs using a new window in IDLE. I did not do my research well enough, unfortunately, as I read shortly afterward that IDLE is a bit unstable on Macs due to issues with the GUI module. At this point, however, we were at the end of the period, so it wasn't the end of the world. I will be able to do more with them now that they have at least seen it.

How would I gauge the student response? Much less resistance than I thought. They seemed to really enjoy figuring out what they were doing, especially with the % operator. That took a long time. Then one student asked if the word was 'remainder' in English, and the rest slapped their heads as they simultaneously figured it out. Everyone enjoyed the change of pace.

For homework, in addition to doing some review problems for the unit exam this week, I had them look at the programs here at the class wiki page.

## Physics

I had great success giving students immediate feedback on the physics test they took last week by giving them the solutions to look at before handing it in. I had them write feedback for themselves in colored pencils to distinguish their feedback from their original writing. In most cases, students caught their own mistakes and saw the errors in their reasoning right away. I liked many of the notes that students left for themselves.

This was after reading about Frank Noschese's experience doing this with his students after a quiz. I realize that this is something powerful that should be done during the learning cycle rather than with a summative assessment - but it also satisfied a lot of their needs to know when they left how they did. Even getting a test back a couple days later, the sense of urgency is lost. I had them walking out of the room talking about the physics rather than talking about how great it was not to be taking a test anymore.

Today we started figuring out circular motion. We played broom ball in the hallway with a simple task - get good at making the medicine ball go around in a circle using only the broom as the source of force.

We then came in and tried to figure out what was going on. I took pictures of all of their diagrams showing velocity and the applied force to the ball.

It was really interesting to see how they talked to each other about their diagrams. I think they were pretty close to reality too, particularly since the 4 kilogram medicine ball really didn't have enough momentum to make it very far (even on a smooth marble floor) without needing a bit of a tangential force to keep its speed constant. They were pretty much agreed on the fact that velocity was tangent and net force was at least pointed into the circle. To what extent it was pointed in, there wasn't a consensus. So Weinberg thinks he's all smart, and throws up the Geogebra sketch he put together for this very purpose:

All I did was put together the same diagram that is generally in textbooks for deriving the characteristics of centripetal acceleration. We weren't going to go through the steps - I just wanted them to see a quick little demo of how as point C was brought closer to B, that the change in velocity approached the radial direction. Just to see it. Suddenly the students were all messed up. Direction of change of velocity? Why is there a direction for change in velocity? We eventually settled on doing some vector diagrams to show why this is, but it certainly took me down a notch. If these students had trouble with this diagram, what were the students who I showed this diagram and did the full derivation in previous years thinking?

Patience and trust - I appreciate that they didn't jump out the windows to escape the madness.

_______________________________________________________

All in all, some good things happening in the math tower. Definitely enjoying the experimentation and movement AWAY from lecturing and using the I do, we do, you do model, but there are going to be days when you try something and it bombs. Pick up the pieces, remind the students you appreciate their patience, and be ready to try again the next day.

## Testing physics models using videos & Tracker

I've gotten really jealous reading about how some really great teachers have stepped up and used programming as learning tools in their classes. John Burk's work on using vPython to do computational modeling with his students is a great way to put together a virtual lab for students to test their theories and understand the balanced force model. I also like Shawn Cornally's progression of tasks using programming in Calculus to ultimately enable his students to really understand concepts and algorithms once they get the basic mechanics.

I've been looking for ways to integrate simple programming tasks into my Algebra 2 class, and I think I'm sold on Python. Many of my students run Chrome on their laptops, and the Python Shell app is easily installed on their computers through the app store. It would be easy enough to ask them to enter code I post on the wiki and then modify it as a challenge at the end of beginning of class.. It's not a formal programming course at all, but the only way I really got interested in programming was when I was using it to do something with a clear application. I'm just learning Python now myself, so I'm going to need a bit more work on my own before I'll feel comfortable troubleshooting student programs. I want to do it, but I also need some more time to figure out exactly how I want to do it.

In short, I am not ready to make programming more than just a snack in my classes so far. I have, however, been a Tracker fan for a really long time since I first saw it being used in a lab at the NASA Glenn Research Center ten years ago. Back then, it was a simple program that allowed you to import a video, click frame by frame on the location of objects, and export a table of the position values together with numerically differentiated velocity and acceleration. The built-in features have grown considerably since then, but numerical differentiation being what it is, it's really hard to get excellent velocity or acceleration data from position data. I had my students create their own investigations a month ago and was quite pleased with how the students ran with it and made it their own. They came to this same conclusion though - noisy data does not a happy physics student make.

I wanted to take the virtual laboratory concept of John's vPython work (such as the activities described here) for my students, but not have to invest the time in developing my students' Python ability because, as I mentioned, I barely qualify myself as a Python novice. My students spent a fair amount of time with Tracker on the previous assignment and were comfortable with the interface. It was at this point that I really decided to look into one of the most powerful capabilities of the current version of Tracker: the dynamic particle model.

My students have been working with Newton's laws for the past month. After discovering the power of the dynamic model in Tracker, I thought about whether it could be something that would make sense to introduce earlier in the development of forces, but I now don't think it makes sense to do so. It does nothing for the notion of balanced forces. Additionally, some level of intuition about how a net force affects an object is important for adjusting a model to fit observations. I'm not saying you couldn't design an inquiry lab that would develop these ideas, but I think hands-on and actual "let me feel the physics happening in front of me" style investigation is important in developing the models - this is the whole point of modeling instruction. Once students have developed their own model for how unbalanced forces work, then handing them this powerful tool to apply their understanding might be more meaningful.

The idea behind using the dynamic particle model in Tracker is this: any object being analyzed in video can be reduced to analyzing the movement of a particle in response to forces. The free body diagram is the fundamental tool used to analyze these forces and relate them to Newton's laws. The dynamic particle model is just a mathematical way to combine the forces acting on the particle with Newton's second law. Numerical integration of acceleration then produces velocity and positions of the particle as functions of time. Tracker superimposes these calculated positions of the particle onto the video frames so the model and reality can be compared.

This is such a powerful way for students to see if their understanding of the physics of a situation is correct. Instead of asking students to check order of magnitude or ask about the vague question "is it reasonable", you instead ask them whether the model stops in the same point in the video as the object being modeled. Today, I actually didn't even need to ask this question - the students knew not only that they had to change something, but they figured out which aspect of the model (initial velocity or force magnitude) they needed to change.

It's actually a pretty interesting  progression of things to do and discuss with students.

• Draw a system schema for the objects shown in the video.
• Identify the object(s) that you want to model from the video. Draw a free body diagram.
• Decide which forces from the diagram you CAN model. Forces you know are constant (even if you don't know the magnitude) are easy to model. If there are other forces, you don't have to say "ignore them" arbitrarily as the teacher because you know they aren't important. Instead, you encourage students start with a simple model and adjust the parameters to match the video.
• If the model cannot be made to match the video, no matter what the parameter values, then they understand why the model might need to be adjusted.  If the simple model is a close enough match, the discussion is over. This way we can stop having our students say "my data is wrong because..." and instead have them really think about whether the fault is with the data collection or with the model they have constructed!
• Repeat this process of comparing and adjusting the model to match the observations until the two agree within a reasonable amount.

Isn't the habit of comparing our mental models to reality the sort of thing we want our students to develop and possess long after they have left our gradebook?

It's so exciting to be able to hand students this new tool, give them a quick demo on how to make it work, and then set them off to model what they observe. The feedback is immediate. There's some frustration, but it's the kind of frustration that builds intuition for other situations. I was glad to be there to witness so we could troubleshoot together rather than over-plan and structure the activity too much.

Here is the lab I gave my students: Tracker Lab - Construction of Numerical models If you are interested in an editable version, let me know. I have also posted the other files at the wiki page. Feel free to use anything if you want to use it with your students.

I am curious about the falling tissue video and what students find - I purposely did not do that part myself. Took a lot of will-power to not even try. How often do we ask students to answer questions we don't know the answer to? Aren't those the most interesting ones?

I promise I won't break down and analyze it myself. I've got some Python to learn.

## Presenting the MVT In Calculus w/ Geogebra...tech as a game changer.

During our warm-up activity today, we looked at a function and identified critical points, relative, and absolute extrema for this function:

It was kind of neat talking about this and the extreme value theorem from last time. Since the domain is not defined over a closed interval, the EVT doesn't guarantee the existence of an absolute maximum or minimum value. The students seemed to really get the idea this year that this function specifically has no absolute maximum over the domain because it is an open interval - last year there was a lot of confused faces on this idea. There were a couple really insightful comments about whether there would be an open interval domain over which the function did have an absolute maximum, even though the hypothesis wasn't satisfied. The theorem just tells you whether or not you are guaranteed to find one, not that there isn't one at all. Really good stuff, and I'm proud of the way everyone was chiming in to talk about what they understood.

The most important thing was that this led perfectly into introducing the idea of an existence theorem. This idea is different from other theorems (especially in comparison to geometry) that students have learned because the information it gives you is not as specific as "alternate interior angles are congruent" or "the remainder of polynomial P(x) upon division by (x - c) is P(c)". All it does is tells you whether you can find what the theorem says is there. I didn't plan on having this discussion today, but it was perfect for then introducing the mean value theorem, and I will definitely repeat it in the future.

I then gave my students this geogebra applet to play with today.

The students understood pretty quickly what they had to do, and didn't seem to have a hard time. It was kind of interesting to watch them rediscover the concept of forming a tangent line using two points, as that concept has been a bit overshadowed by other things as we looked at derivative rules before the test they took last week. Some students moved P and Q so that they were tangent, and then adjusted the domain using C and D to find a domain over which the tangent line and line AB were parallel.

From this, I showed them what the slope of line AB represented (average rate of change over the interval) and came up with the right side of the MVT. We then talked about what the slope of the tangent line they identified represented - a couple immediately referenced the derivative of the function. What is the relationship between parallel lines? What would make it so that you couldn't find this value? Ideas of continuity and differentiability jumped out. There it was: the entire mean value theorem.

Last year I presented the students with the MVT, and then we drew graphs to represent what it was saying. They kind of got it, but it wasn't a sticky idea. I was doing all the developing. This approach today started with something visual that they were doing, that they could understand intuitively, and then that intuition was applied to develop an abstract concept out of that understanding.

I continued doing what I had done last year - answering some multiple choice questions about the MVT (See here for today's handout) analytically, and I immediately lost a couple students. So I showed them how to throw the new function into Geogebra and adjust the domain to match the problem. They could then solve the problems graphically - they immediately located the points to be able to answer the questions.

The group is a mix of AP and non-AP exam bound students. I will introduce them all to the analytic ways of identifying these points, and we did some of it today. It was really nice that the moment things got a bit too abstract, I could push students to identify how the question being asked was the same as the idea of the MVT, and they were then able to solve it.

Without the technology, these students would have been done for the rest of the period. Those that could handle the algebra, would. Those that couldn't would spend the rest of the period feeling like they were in over their heads. Introducing how to use the technology to really understand what was being said by the abstract theorem enabled many more students to get in on the game. That made me feel all warm and fuzzy inside. The rest of the class focused on definitions of increasing functions using the derivative, something that was made incredibly easy by referring back to the activity at the beginning of the period.

We'll see how well they remember the ideas moving forward, but it felt great knowing that, at least for today's lesson, everyone in the room had a way into the game.

## 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.

## The museum in your classroom - exploration, discovery, play, and authentic learning

I was visiting the Great Lakes Science center with family during high school. I was busy reading information printed on a horizontal rotating triangular prism - the three sides had facts about the phenomenon being demonstrated at the exhibit, though I don't remember what the exhibit was about. While I was reading, a younger student came along and stood in front of me to view the exhibit. He barely paid any attention to me; his attention was piqued when he placed his hand flat on the prism and it rotated slightly under the weight. He then proceeded to flick the prism as hard as he could to see how fast he could make it turn. That was how he spent a couple of minutes while I looked on, flabbergasted by both his lack of interest in the flashing exhibit in front of him and the fact that all the exhibit inspired him to do was to challenge himself to a spinning competition. Once he was satisfied that he had peaked in the spinning task, he went off to a new exhibit.

I was pretty annoyed as a sixteen year old science enthusiast. How dare he not take the time to read what was printed at the exhibit? Not even try? There was so much good knowledge there to be learned - why bother coming to the museum if you weren't going to try to learn something new? The best part for me was coming and playing with the exhibits and then seeing what science principle was being demonstrated. This was (as I understood it at the time) what science was all about.

For me, this was not something that was limited to museum visits. I had a pretty good arrangement for doing investigation at home as well. I was always able and encouraged to go outside and explore in the woods, burn stuff with magnifying glasses (not of course while doing the previous activity), and do experiments mixing things in the kitchen. I am incredibly grateful that my mom allowed me to do these experiments in spite of my frequent habit of rushing away afterwards without cleaning up. I didn't realize at the time how unique it was that she let me do some of the things I did, and probably would have cleaned up myself more often  if I had. (I did do so marginally more often after a particularly stern chat about the difficulties of removing hardened candle wax from the good silverware. She clearly explained that the experiments would stop if I didn't do this sort of cleaning myself.) The other major time I realized I should be grateful was when I accidentally removed the gold coating on a fork during an electroplating experiment. (Sorry mom.)

One that sticks in my mind was after I first learned about objects from space reentering the atmosphere and burning up.  I didn't understand how metal objects could burn - I had seen metal melt before on TV, but could it really burn? I took a penny and some cooking tongs to the gas stove and held the penny in the flame for a long time. I was able to see the penny get hot and ultimately glow. I had a bowl of cold water there to drop the penny into afterwards. The colored patterns on its surface reminded me of a picture in a magazine that showed the oxidation patterns on a sample of material that had survived reentry. I also tried wires and aluminum foil in the flame, and the way both materials twisted around themselves and changed both in appearance and material properties gave me some insight into what it meant for metal to burn.

Was there a goal? Not really. I didn't write up a lab report or keep a notebook recording my observations. These were just experiences in which I explored what I could do with the stuff in front of me. I did get the sense that this sort of thing was distinctly different from what I was doing in school because there was no assessment. I don't know if she ever talked to others about her son "playing on the stove" as she called it. At the time I objected to her calling it that because I thought it made it sound like I was being reckless. I had a purpose to my experiments. I was creating meaning on my own as I had done throughout my Montessori elementary education. And I was careful when carrying out these investigations.

Years later, I have a different understanding the role of play in learning. I really like this TED talk by Stuart Brown that talks about some of the reasons why play is important. Much of what I have learned about building with LEGO is in the context of playing in an unguided way. Another major influence on my philosophy on play was K.C. Cole's book Something Incredibly Wonderful Happens, which I heard about during the summer before teaching biology for the first time to ninth graders. The book describes physicist Frank Oppenheimer (brother of Robert Oppenheimer) and the full story of his life as a father, rancher, teacher, and ultimately creator of the world class Exploratorium.

I have visited the Exploratorium twice. There is no other museum in the world that has influenced me in such a visceral way as that museum. as I can still picture numerous things at that museum and what I learned from doing the exhibits. Reading about Frank and his process of seeing the museum as vital was really important to defining something that I think I hadn't officially acknowledged in the preceding six years I had been teaching.

First, some Frank Oppenheimer quotes from a speech he gave upon receiving an award from the American Association of Museums, along with my thoughts:

Many people who talk about the discovery method of teaching are really talking about arranging a lesson or an experiment so that students discover what they are supposed to discover. That is not an exploration. The whole tradition of exploration is being lost for entire generations.

There is a role for discovery in our classrooms. This is not, as is often thought, the expectation that students will spontaneously figure out Newton's laws or the quadratic formula. These are instead carefully designed activities through which students arrive at an idea. Our world needs more interactivity. People, not just students, are spending less time constructing their own understanding, and more time (since we are all inherently busy) hoping that others can explain things to us since it will invariably be faster this way.

If one of the things, however, we want to teach students is how to construct their own understanding, this is not going to come from giving them information and then telling them how to use it. Any way we can engage students to interact with the material actively instead of merely receiving content moves us closer to that goal.

It is, therefore, more important than ever that museums assume the responsibility for providing the opportunities for exploration that are lacking for both city and suburban dwellers. It would be fine, indeed, if they would, but it will take a bit of doing to do so properly. If museums are too unstructured, too unmanageable, people get lost and simply want to get back to home base. On the other hand, if they are too rigid, too structured or too channeled, there are no possibilities for individual choice or discovery.

It may be useful to note that these quotes are from 1982. Certainly these issues are no different nearly thirty years later. If lesson activities for students are too unstructured, they may have no idea of the learning goal, what they are supposed to figure out, or how to get from one point to another. They get lost. They get cranky. They would rather just be told information. This, however, is the opposite extreme. While some students demand the structure, there are serious limitations to the quality of learning experiences under a classroom model that is too rigid.

Exploring, like doing basic research, is often fruitless. Nothing comes of it. But also like basic research, as distinct from applied or directed research, exploring enables one to divert attention from preconceived paths to pursue some intriguing lead: a fragrance, a sight or smell, an interesting street or cave, an open meadow encountered suddenly in the woods or a patch of flowers that leads one off the trail, or even a hole in the ground! Often it is precisely as a result of aimless exploration that one does become intensely directed and preoccupied.

A museum that allows exploration does not have to be disorganized either physically or conceptually. It does, however, mean that the museum must contain a lot of which people can readily miss, so that discovery becomes something of a surprise, a triumph, not so much of personal achievement as of personal satisfaction. It is the kind of satisfaction that invariably leads me to tell someone about the experience.

When people in a museum find something that engages them, that moment of engagement is what justifies the museum's existence. That may be what the visitor remembers about their museum experience. It may also be the sort of experience that causes the visitor to come back, and ideally, bring a friend or child. A well designed exhibit involves its visitors in its operation, tries to engage them, and along the way provides interesting information in the off chance a visitor is interested. A good museum has many of these experiences.

Here is the key idea that changed the way I decorate my classroom, organize my lessons, and structure my time with students:

You cannot entirely control what your students will get out of their time with you.

I have spent lots of time designing what I thought were perfect lessons only to have students remember the fact that I used colors in my handout, even months later, because that was what stuck with them.

You can tell them what you want them to get out of an activity. You can assess that they got out of the activity what you wanted them to get out of it. You can also try to tell them why something should be interesting to them. (Not recommended) None of these work well, at least authentically well when it comes to evaluating our use of an activity to reach specific learning goals.

What you can do is provide a range of activities, approaches, and experiences for your students. Providing students a chance to play in your classroom is one of the most powerful tools in our tool chest. You can't play incorrectly. You can't get playing wrong. Play is one of the few times when the only judgments being made belong to the individual that is playing. In the world of math education where students still see math as a class where there is always a right answer, and that right answers are inherently worth more than wrong answers, we need play more than ever.

What does play look like? Not like the majority of lessons I do, admittedly, but I'm working on that. The real reason this doesn't happen as often as I want it to is that it doesn't necessarily feel productive. I force myself to push through this because I've had the idea of clear learning goals and measurable objectives drilled into my head from the moment I started teaching. The problem is that real learning doesn't look like this. When we figure things out, it isn't with the end goal in mind. Unstructured time to just be in the presence of an idea that motivates itself is enough to get students to think as they do during play.

The biggest tool that we have at our disposal though is the use of technology. I've seen students discover by accident that when you hit the equal sign on some calculators, it repeats the previous operation with the answer. What do most students do when they discover this after multiplying? They hit it a bunch of times until the calculator overflows. Sometimes they will do the same thing with multiplying a decimal, and the number of zeroes to the right of the decimal point increases. Is there a lesson on place value or exponential functions there? Sure. The moment you tell them that though, it suddenly ceases to be exploration and starts becoming Math Class.

It's also easy to create a Geogebra sketch of a quadrilateral with measured angles and tell students to "play" with it for five minutes. The goal is not to have them discover the sum of the interior angles is 360 degrees, though they might observe that. The goal is instead to give them a chance to interact with a mathematical object and have an experience that is all their own. Then start the lesson. See what happens. This is exactly what Noah Podolefsky from the PhET physics simulation project at the University of Colorado recommends students be allowed to do for 5 - 10 minutes before telling students what you want them to use the simulation to do.

The other aspect of this is in decorating my classroom. I don't want so much on the walls that students will be continually distracted. I do want things that create interesting learning experiences without much effort. I hung a spring between two corners of the room as an example of a catenary curve - students don't care about that. They do walk by it all the time and make it bounce up and down. Sometimes they see how long it takes for the vibrations to die down. Sometimes they hang things on it to see how it changes the droop in the overall spring. I have a bicycle wheel that normally is used as a demonstration of conservation of angular momentum. Students have instead spun it and observed that it stays upright like a top on the table. I have a checker board with checkers, the game Set, little metal puzzles, and a bunch of other things that don't require a whole lot of explanation to be interesting. It's amazing to see how the students use their down time to interact with these objects and with each other - it makes my classroom the same safe learning environment of a museum. The dream is to create this environment during every single lesson.

Looking back, the kid that stepped in front of my at the Great Lakes Science Center wasn't learning what the exhibit designers intended him to learn. He was, however, constructing his own knowledge when he spun the prism as fast he could. He might have gotten some notion of what feels different about a force and a torque. He might have seen that the rotation only increased in speed while his hand was in contact with it - an intuitive concept related to Newton's 2nd law.

Or not. It was pretty hypocritical of me to judge and potentially hamper his learning process when so many others (including my mom, who had many good, flammable reasons to do so) did not. He wasn't using the museum wrong - I was. He was just doing what came naturally.

Entirety of Frank Oppenheimer's speech to the American Association of Museums, 1982 can be found here.
I also find myself going back to this article written by Oppenheimer about teaching as a quick reminder of all sorts of important ideas.

## Exploring Point Slope Form through Geogebra

In geometry we are studying parallel and perpendicular lines and the theorems that can be proven about them. In thinking about how to present the connection between algebra and geometry for this unit, I wanted to include an exploration of what exactly makes lines (and linear functions) so special: constant rate of change, or slope.

We did not get a chance to do this entire exploration in class, but I am expecting students to look at it at some point over the weekend. I know they have seen slope-intercept form, but point-slope form is convenient in many ways, especially in the way it applies the concept of slope between points located on a line.

## Lessons from the CME project - Verbal Systems

In contrast to what I wrote in a previous post about disliking word problems relating to solving systems, I found myself returning to the topic with a new approach that I really liked. I've read through the presentations of the Center for Mathematics Education (CME) project, and have gotten an idea of what they do through the examples they present. I've been very interested in getting actual copies of their textbooks, but haven't gotten around to it both because of my location (no shipping to China as far as I know) and because, well, other things have occupied my time.

I really like the general theme of how mathematical thinking is closely aligned with the sort of logical thinking we already do. The concept of 'guess-check-generalize' makes sense especially in the context of what I always find my students doing anyway when I present them with a word problem. See this post on the CME blog to get an idea of what it's all about. In the past I have tended to use verbal problems, especially in the context of systems of equations, as a way of reinforcing solution methods of these systems. I have also found many students will naturally use a brute-force guess-and-check method of trying to solve them. I was consistently impressed when kids with low levels of number sense and arithmetic ability would fall upon a solution to a system of equations after a period of deliberate and focused trial and error. Why were these students so willing to spend ten minutes trying a bunch of solutions while being unable to sit and listen for five minutes on how to solve it methodically? Was what I was presenting so abstract and disconnected that the obvious method that made sense to them but was a lot more work was clearly the better choice? Clearly so.

My response early on in my teaching career was then to give systems of equations that they would NOT be able to solve by brute force. Systems with solutions that were decimals and fractions were much less likely to be figured out. Doing this though felt so arbitrary. If I have to modify the questions I was asking in a contrived way in order for my algebraic method to finally become the better solution to this group of students, there was something wrong with MY presentation and application of the mathematics, not with the students' method.

This is part of the reason I would get frustrated teaching verbal systems of equations as part of solving systems of equations. The situations that came up (in most textbooks that I read) were made in a way that they fit the solution methods that were simple to solve using elimination or substitution. A person could know almost no English and still figure out a system of equations that most likely solved the given problem.

The thing that seems different about the guess-check-generalize framework though is that it encourages the type of self-aware mathematical thinking that we want students to do. This was the first time I really presented a problem this way, but it seemed to work well, particularly in the case of some of the students that have demonstrated both weaker math skills and/or a limited English proficiency. I gave them a problem of this type:

A store has a sale on sneakers and shirts. Tyrone buys three shirts and two pairs of shoes for \$225. Maria buys two pairs of sneakers and five shirts and pays \$325. What are the prices for a pair of sneakers and a shirt?

When I asked students to guess a solution to the problem, one student immediately 'guessed' that the answer was "2x + 3y = 225". It was a great moment telling the student that if he went into a store and asked a salesperson how much a shirt and pair of shoes was, and the salesperson started spouting off an equation, that salesperson would most likely be smacked in the head and fired for being unhelpful. It makes no sense to respond to a verbal question with an equation, but that is what students (including mine, unfortunately) have been conditioned into doing. With that expected response out of the way, we could move on with the guess-check-generalize model.

I decided to call the answer a "model answer" instead of a guess - I have an ongoing battle with students about how much I hate the word "guesstimate" because people tend to use it to make a true guess sound more authoritative by connecting it to the very different word estimate. I asked a student what a possible answer could be to the question.

What was pretty interesting was that the rest of the creation of the mathematical system came naturally from this guess. What were the variables? Since what the question was asking us to find was the unknowns, the quantities found in the model answer were what we would probably model in a system of equations. There was no argument this time about how X was not equal to "SHIRTS" and Y equal to "SHOES" - instead it was plainly obvious from the model answer that we were guessing a price of a shirt and a price of a pair of shoes. Here was how the legend appeared:

The system of equations came just as easily. No teaching the formulaic way I once did that "for a cost type equation, you multiply the x-cost by the x variable, add it to the y-cost multiplied by the y-variable, and set it equal to the total cost." Instead, we just found what the cost would be using our model answer:

If the model answer had been correct, then the cost would have been \$70. I targeted this question toward one of the students that I was more concerned might not understand the whole process, and it seemed to come naturally. Clearly the \$70 was wrong, but the students were actually thinking about this fact rather than blindly putting together an equation. The calculation using the model answer not only did this for them, it screamed out to us what the actual equation had to be. Smooth as silk.

It was exhilarating seeing this work with my group. Granted, they are generally a pretty strong group, but verbal problems like these (especially given the international make up of this class) tend to make them all visibly uncomfortable. This worked much more smoothly than any of my previous lessons. I certainly have tried to get students to think this way before, but never explicitly used a guess to generate the rest of the equation. For those ESOL students, it seems like a non-threatening first step to come up with an example of what an answer to the question might look like. This idea could help all students that have a tendency not to read questions all the way through and guess what they are being asked to do.

It is very possible that I'm just late to the guess-check-generalize party and teaching using this method is obvious. If that is the case, I apologize to my students for getting it wrong for so long. I see a lot of the similarities between this and modeling, which I've really enjoyed using with my students through exploration of Newton's laws. Maybe the parity between them is why I'm suddenly so excited about the overall concept.

In the end, I do have some more tricks up my sleeve for how I want to use some actual, interesting, realistic, and authentic problems with this group. The robot crash went really well and the students enjoyed that activity. I go back and forth as to the benefit of giving them word problems like the one we worked on. They exist in math world, the world of math textbooks, but not so much in the reality of us as math teachers trying to teach what authentic mathematical thinking looks like.

## Math Portfolio - Sharing my own story.

In Calculus, I use the third edition of Finney, Demana, Waits, and Kennedy. I love the selection of activities and explorations that are used to get students where they need to be for the Calculus AB exam. A colleague recommended that I check out Dan Kennedy's website as a treasure trove of resources both mathematical and philosophical about teaching. One of the things I found there that I decided to bite the bullet and do this year is having students put together a math portfolio detailing their work over the year.

The reasons for doing this are many, some of them more selfish than others, but they include the following:

• By having a record of student work, I can easily look back and remind myself of some of the major mistakes and misconceptions that students have at a particular moment in time.
• I like reading and seeing how students respond to their own work. I often have students reflect on their work on short time scales ("I should have studied X or Y to do better on the unit test") but don't do as much over long periods of time ("I've become much better at graphing lines in comparison to when we first met linear functions in class.") Part of this is because my students don't tend to hold on to their papers for very long. I take partial responsibility for this, never holding them accountable for it, though I do occasionally remind them that the easiest way to study for a final exam is to look at old exams.
• I think students selecting what work represents their progress often means things that are very different than what teachers see as their best work. Sometimes students are afraid of sharing their failures, though we as teachers see those as being the most meaningful learning experiences. Whichever is right, having students actively evaluating their own work and thinking about their own learning process is valuable for being able to identify how they learn best.

My introduction to the concept of the portfolio took a lot from Dan Kennedy's document describing them, and I am incredibly thankful for his decision to publish his document online. My own document describing the content of the portfolio and how it is integrated into the grade is here.

At the beginning of the year I introduced the idea, and the response wasn't applause. It was, incidentally, very similar to the introduction this year of true student-led conferences. The students wanted to know why we were demanding they do much more work just for parents and teachers that see their work anyway on the report card. My responses, fully sincere, included the ones I gave above: portfolios are opportunities to highlight not the grade that was received, but the learning process that it describes. Conferences, however, went extremely well as reported by teachers, parents, and most impressively, the students. Since requiring students to also produce the portfolio, I have been equally impressed by some of the thoughts shared by students about what they do and do not understand, the mistakes they tend to make, and also some of the things that go through their minds when thinking about learning.

One of my requirements is that students write a reflection and scan in their skills quizzes any time they want to retake a quiz. This is my current implementation of standards-based-grading, though I am considering expanding it significantly soon. This raises the bar somewhat for what students have to do to retake, but I don't object to this requirement at all. Sometimes I have to tell them to do the reflections a second time - in this situation, they usually look something like "I didn't get it but now I studied and I get it" without any detail as to what it is, what "not getting it" means, or what "studied" actually looks like. Once I get them past this point to do some serious thinking about what they have difficulty understanding, I am very pleased with the responses.

I tried handling the start of the portfolio myself since I wanted to make sure they all looked similar in case these did become official school documents at some point. This was a lot of work keeping track of quiz retakes, reflections, scanning them in, etc - I finally turned over the files as they were last week and have given them to students to keep up to date. Some strong students, however, have nothing in their portfolios because they weren't retaking quizzes, and the only thing I had time to really check up on before the end of the first quarter was that the bios were in place.

What I decided to do to show ALL students what I was looking for in the math reflection portion (with the mathematics exploration to be added soon) is to share my own portfolio with some artifacts from high school that I still happen to have. I've always guarded my test, quiz, and project papers from high school as really authentic sources of material not only to use with my own classes, but also to show students that might not believe I ever had any difficulty in math.

Here is my own math portfolio, complete with biography and student (namely my own) work:  Weinberg portfolio example

I shared this today with students and had some really interesting responses:

• "This is really your work from high school? Why in the world did you save it?"
• "You had a 63 on a math test?"
• "That looks like really hard math"

I got to tell them (1) to read it all the way through to see my comments and (2) that I was proud to show them some of my work along the way to becoming the math student that I was when I left high school. If nothing else, I am hoping that they will read it first because of the inherent fascination students have with their teachers as actual people (I love when they say things like 'It's cool to know you are a real person) and second to get some inspiration for the sort of thinking and reflection I want them to put together.

I know it is difficult to expect reflection to be a perfect process when it is new - it takes time and effort and it doesn't immediately pay dividends. I want students to understand that reflection is not only a really beneficial process, but that over time becomes enjoyable. It shows that learning is a continual process, that you don't just suddenly "get it". This is the same process that I am enjoying about writing on this blog. It takes time, I have to make time to do it - in the end, I really enjoy looking back at my thoughts and holding myself to the commitments I make to my own practice and my students.

So I am leading by example. This group of students continues to really impress me when I expect great things out of them - here's just one more way I am hoping to help them grow.