## A tale of two classrooms – which is yours?

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

Situation one:

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

Situation two:

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

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

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

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

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

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

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

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

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

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

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

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

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

What do you think?

## How good is your model? Angry Birds edition

With Algebra 2 this week, I decided it was time to get on the Angry Birds wagon. I didn’t even mention exactly what we were going to do with it – the day before, the students found the above image in the class directory on the school server, and were immediately intrigued. This was short lived when they learned they weren’t going to find out what it would be used for until the day after.

To maximize the time spent actually mathematical modeling, I used the video Frank Noschese posted on his blog for all students. They could pick any of the three birds and do the following:

Part A:
Birds are launched at 6, 13, and 22 seconds in the video. Let’s call each one Bird A, Bird B, and Bird C.
• Take a screenshot of any of the complete paths of birds A, B, or C.
• Import the picture into Geogebra. Create the most accurate model you can for the bird you selected. What is the equation that models the path? Does it match that of your neighbors?

Part B:
• Go back to the video and the part in the video for the bird that you picked. Move forward to a frame shortly after the bird is launched, take a screenshot, and put it again into Geogebra. Can you create a model that hits the landing point you found before using only the white dots that show only the beginning of the path?

If not, find the earliest possible time at which you can do this. Post a screenshot of your model and the equations for the models you came up with for both Part A and Part B.

My hope is not to just use the excitement of using Angry Birds in class to motivate knowing how to model using quadratic functions. That seems a bit too much like a gimmick. The most interesting and realistic use (and ultimately the most powerful capability of any model) of this source of data is to come up with as accurate of a prediction of the behavior of the trajectory as is possible using minimal information. It’s easy to come up with a quadratic model that matches the entire path after the fact. Could they do this only twenty frames after launch? Ten?

The students quickly started seeing how wildly the parabola changes shape when the points being used to model the parabola are all close together. This made obvious the importance of collecting data over a range of values in creating a model – the students caught on pretty quickly to this fact.

I think Angry Birds served as a cool “something different” for the class and has a lot of potential in a math class, as it does in physics. I am hoping to use this as a springboard to have students understand the power of models and ultimately choose something to model that allows them to predict a phenomenon that is of some importance to their own adolescent worlds. I don’t exactly know what this might be, and I have some suggestions for students to make if they are unable to come up with anything, but this tends to be one of those ideas that eventually results in a few students doing some very original work. Given my interest in ultimately getting students to participate in the Google Science Fair, I think this is just the thing to push them in the right direction of making their own investigation.

## Testing expected values using Geogebra

I was intrigued last night looking at Dan Meyer’s blog post about the power of video to clearly define a problem in a way that a static image does not. I loved the simple idea that his video provoked in me – when does one switch from betting on blue vs. purple? This gets at the idea of expected value in a really nice and elegant way. When the discussion turned to interactivity, Geogebra was the clear choice.

I created this simple sketch (downloadable here)as a demonstration that this could easily be turned into an interactive task with some cool opportunities for collecting data from classes. I found myself explaining the task in a slightly different way to the first couple students I showed this to, so I decided to just show Dan’s video to everyone and take my own variable out of the experiment. After doing this with the Algebra 2 (10th grade) group, I did it again later with Geometry (9th) and a Calculus student that happened to be around before lunch.

The results were staggering.

Each colored point represents a single student’s choice for when they would no longer choose blue. Why they chose these was initially beyond me. The general ability level of these groups is pretty strong. After a while of thinking and chatting with students, I realized the following:

• Since the math level of the groups were fairly strong, there had to be something about the way the question was posed that was throwing them off. I got it, but something was off for them.
• The questions the students were asking were all about winning or losing. For example, if they chose purple, but the spinner landed on blue, what would happen? The assumption they had in their heads was that they would either get \$200 or nothing. Of course they would choose to wait until there was a better than 50:50 chance before switching to purple. The part about maximizing the winnings wasn’t what they understood from the task.
• When I modified the language in the sketch to say when do you ‘choose’ purple instead of ‘bet’ on the \$200  between the Algebra 2 group and the Geometry group, there wasn’t a significant change in the results. They still tended to choose percentages that were close to the 50:50 range.

I made an updated sketch that allowed students to do just that, available here in my Geogebra repository. It lets the user choose the moment for switching, simulates 500 spins, and shows the amount earned if the person stuck to either color. I tried it out on an unsuspecting student that stayed after school for some help, one of the ones that had done the task earlier.

Over the course of working with the sketch, the thing he started looking for was not when the best point to switch was, but when the switch point resulted in no difference in the amount of money earned in the long run by spinning 500 times. This, after all, was why when both winning amounts were \$100, there was no difference in choosing blue or purple. This is the idea of expected value – when are the two expected values equal? When posed this way, the student was quickly able to make a fairly good guess, even when I changed the amount of the winnings for each color using the sketch.

I’m thinking of doing this again as a quick quiz with colleagues tomorrow to see what the difference is between adults and the students given the same choice. The thing is, probably because I am a math teacher, I knew exactly what Dan was getting at when I watched the video myself – this is why I was so jazzed by the problem. I saw this as an expected value problem though.

The students had no such biases – in fact, they had more realistic ones that reflect their life experiences. This is the challenge we all face designing learning activities for the classroom. We can try our best to come up with engaging, interesting activities (and engagement was not the issue – they were into the idea) but we never know exactly how they will respond. That’s part of the excitement of the job, no?

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.

## 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.

## Graphical Systems in Geogebra and crashing LEGO robots in Algebra 2

In the Algebra 2 class, we started our unit on solving systems of equations. From a teaching perspective, this provides all sorts of opportunities for students to conceptualize what solutions to systems mean from a graphical, algebraic, and numerical perspective. Some students seem to like the topic because it tends to be fairly straight forward, is algorithmic, and has many ways to check and confirm whether it has been done correctly.

I used this as my warm-up activity today:

a) Estimate the solution of the system.

b) Write an equation for each line in standard form.

c) In Geogebra, select CAS view and type the following using your two equations: Solve[{7x+3y=6,3x-4y=12},{x,y}]

d) Use your calculator and convert these values to decimals. How close are these to your estimate?

We had some great discussions about the positives and negatives of graphical solutions to equations. Weaker students got some much needed practice writing equations for lines. For all students, this led to some good conversations about choosing two points that the lines clearly pass through for writing equations (if possible) rather than guessing at the y-intercept. The students also got the idea of how Geogebra can solve a system of equations exactly as a quick check for their algebra, an improvement over substituting (which is at times more trouble than it’s worth for students with poor arithmetic) and slightly faster than solving for y on a graphing calculator and finding the intersection.

I also like the unit, though I don’t tend to like the word problems. It’s hard to convince students about the large scale importance of coin problems (especially in an international school with everyone used to different currency) or finding how many tickets were sold at the door or advance since anyone with a brain would just ask the person tallying the tickets.

I also found myself thinking about Dan Meyer’s post over the summer about how many word problems are made up for the purposes of math, rather than using mathematics to analyze cool situations and create problems out of the situations. Getting students to figure out how to use the math to do this is ultimately what we want them to learn to do anyway. Figuring out when trains pass each other is not exciting to students, but I realized this morning while brushing my teeth that doing this problem with real robots either crashing into each other or racing adds a neat dimension to this problem. The question of figuring out both when they will crash or catch up to each other, and also where they will do so is a clear motivation for finding a solution to a system of equations describing their positions as functions of time.

So I gave the students the two robots (videos of them posted at http://bit.ly/vIs0lu and http://bit.ly/u9jSPB) . I told them I was going to set them apart a certain distance that was tentatively 80 centimeters, but said I wanted the ability to change that at any time. I wanted them to predict when and where they would collide.

The rules:

No, you can’t just run the experiment and see where they crash. That not only defeats the purpose of this exercise, but we will be doing this sort of activity in a couple different ways during the unit, so being able to do this analytically is important. You also can’t run both robots at the same time – that’s for those of you that are going to try to be lawyers and break that first rule.

You can measure anything you want using any units that you want using either robot individually.

At some point, you should be able to show me how you are modeling the position of each robot as a function of time.

And I set them off to figure things out. Despite the fact there were only two robots, the 12 kids naturally divided themselves up into a couple teams to characterize each robot, and there was some good sharing of data amidst some whining about how annoying it was to actually measure things. In the end, most students at least had some idea of how they were going to put together their models, and some had actually written out what they were. As one would hope for these types of activities, there were plenty of examples of students helping others to understand what they were doing. The engagement was clearly there, as confirmed by students visibly excited to run the robot and time how long it took for it to move around.

It was a fun exercise that I plan to return to in a few ways during this unit – perhaps some interrobo-species interaction (my iCreate robot is charging up as we speak). Fun times.

UPDATE: This is the video of the next day’s class when students solved their functions. I set the robots apart from each other and the students did the rest.

## 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.

## “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.

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.