My tutor's name is Geogebra CAS.

When I first started teaching, I learned that the best thing to have students do after factoring a trinomial was to have the students check by multiplying out the binomials. At the time, it naively made total sense - students don't even need me to be there to practice! They can do this on their own while sitting on the subway or waiting for the bus - whatever dead time they have. The students that need to practice factoring can do as much of this as they need until they can factor with some degree of automaticity.

Some (not all) students took my advice. Of those that did, I often saw stuff like this:

x² - 4 = (x - 2)(x - 2)
= x² - 2x + x(2) + 4 = x² - 4

This was a worse situation than how we started - not only were they factoring incorrectly, but their inability to multiply binomials was giving them the false idea that they were doing a good job of factoring! This frustrated me to no end - even if I did give students time during class to practice and develop these skills, what could I tell them to do to improve outside of class? One colleague considered stopping giving homework because he saw it repeatedly reinforcing student errors. I didn't go that far, but I did start grading homework to try to find mistakes.

The missing piece for these students is the lack of useful and correct feedback. Most of them learned the procedures, but made arithmetic or careless errors such as leaving out terms when simplifying. Without any correct data to make decisions on, these students were just going through a procedure and generating incorrect results, and using the incorrect results to validate an incorrect procedure. If they had a way to generate correct feedback, this experience would stop being worthless and instead become a useful method for developing student skills!

This is where CAS systems come in - Wolfram Alpha is nice, but Geogebra CAS is even better because of speed. I worked with a student that needed practice both in simplifying polynomial expressions and factoring polynomials completely. This is what I had him do while he sat with me:

  • Make up a pair of binomials of the form (x - 5)(4x - 5), multiply them, and then find the quadratic and linear coefficients. When you are ready, use the Simplify[] command to check your answer.
  • Make up a product of polynomials of the form 4x^2(x+5)(2x-5) . Multiply it out all the way on paper, and then check your result using the Simplify[] command.

After this step, we talked about how he could do this on his own and check his work. While we were sitting there, he made mistakes, but was able to catch them himself. He was the source of the problems, and was able to check and see if his final answers were correct. We then moved on to factoring practice:

  • Write out 15 products of binomials (3x-1)(x+5). For some of them, add a monomial factor. Include a couple sum and difference polynomials as well. Multiply any three of them out manually and check using Simplify[].
  • Use Geogebra to multiply any ten of the the rest of them and write down the resulting polynomials on a separate sheet of paper.
  • Eat dinner, watch TV, or something that has nothing to do with factoring.
  • Return to the paper and factor the ten polynomials you wrote down completely. Use Factor[] to check and make sure your final answers match what Geogebra produces - if there are differences, check to see if you have actually factored completely or not. Make a note of any repeat mistakes.

There is a whole lot of extra busy work involved in this process, but part of that is because it's easy to factor a polynomial that you just generated moments before if you still remember the factors. For some students, this won't matter, but it helps ensure that the exercises generates are actually useful. This student was on fire during class today, even though we were looking at a different topic entirely. I should have asked him directly whether this is the case, but perhaps the boost of confidence going through this process gave him is part of the reason. I also really like that this method allows the student to simultaneously work on multiplying polynomials and factoring them. My method beforehand would have been to stick to multiplying, then factoring, and then mix them up - there's no reason to do this.

Computer algebra has been around for a while. The reason I think it's now to the point where it can be transformational is that it's easily accessible, easy to use, and almost instant. This idea of using technology (and particularly Geogebra) to help students develop their pencil and paper skills is one that really excites me. I'm excited to see if it works with the students that came in a bit behind but are willing to put in the time to catch up. I don't want my class time to be spent learning algorithms - that defeats my strong belief that we should focus on teaching mathematical thinking, modeling, and problem formulation instead of algorithms. That said, students do need to be able to develop their skills, and this offers a personalized way to help them do so on their own.


Impressing the parents with Wolfram's for your own good, kids.

I received a few emails from parents recently wondering how to help their children get better in math. Parents often apologize for not being strong at math themselves, and the students, in my case all teenagers, have trouble communicating with parents about, well, a lot of things, let alone math. Creating a genuine way for children and parents to communicate with each other about math has always been difficult. Thankfully, tools like Wolfram Alpha can come to the rescue.

Here is the advice I gave one parent this week whose child is learning to factor quadratic trinomials:

Think of four numbers, keep them between 1 and 8. For example, 2, 1, 5, 7.

You can then write them like this: (2x+1)(5x+7) or make some negative: (2x - 1)(5x+7).

Go to Wolfram Alpha, and in the main input bar, type what you wrote, as shown below:

A page will load with a part that looks like this, a bit of the way down the page:

Give the top one (10x^2+9x-7) to him, and say to factor it. A groan at this point is natural. And then he will remember how to do it. The final answer should be the same as what you entered into the website. You can come up with new numbers and do this as much as you want - it will only make him stronger. If he has trouble, make the 1st and 3rd numbers you pick be 1, and it will simplify it a bit.

Yes, it will result in at least some expression of teenage 'come-on-mom/dad-ery'. But that's probably going to happen anyway, right?

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.

Math is everywhere! - fractals on the Franz Josef glacier

One of the stops on our New Zealand adventure was at the Franz Josef glacier on the West coast. We went on the full day hike which gave us plenty of time to explore the various ice formations on the glacier under the careful eye of our guide. Along the way up the glacier, I took the following series of pictures:

All of these were taken on the way up the glacier. Can you tell in what order I took them? If you're like my students (and a few others I have shown these to), you will likely be incorrect.

I realized as I was walking that this might be because of the idea of self-similarity, a characteristic of fractals in which small parts are similar to the whole. When I showed this set of pictures to my geometry class, I then showed them a great video video zooming in on the Mandelbrot fractal to show them what this meant.

The formations in the ice and the sizes of the rocks broken off my the glacier contributed to the overall effect. Here is another shot looking down the face of the glacier in which you can see four different groups of people for a size comparison:


The cooler thing than seeing this in the first place was discovering that it's a real phenomenon! There are some papers out there discussing the fact that the grain size distribution of glacial till (the soil, sand, and rocks broken off by the glacier) is consistent throughout a striking range of magnitudes. The following chart is from Principles of Glacier Mechanics by Roger Leb. Hooke:








In case you are interested in exploring these pictures more, here are the full size ones in the same A-B-C-D order from above:

Oh, and in case you are wondering, the correct order is B-C-A-D.

Relating modeling & abstraction on two wheels.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Bugs on your windshield - An introduction to definite integrals

Considering how tired I was this morning on the first day back to school, I could only imagine how the students might be feeling. Today was the first day of our definite integrals unit in Calculus, and I decided to start off class today nice and easy with the following question:

Suppose I pay you to clean the classroom according to the following plan. I'll give you $400 for the
first hour, $200 for the second, $100 for the third, and so on. If it takes you 6 hours to clean the room,
how much do you make?

They joked about the silliness of the plan and what they would do given this opportunity. Then they got down figuring out the solution. They were a bit rusty and many assumed there was something complicated going on, so some started recalling geometric series and writing functions involving 2^x. These students quickly gave in to peer pressure and just calculated the total directly. It's always interesting to see how more experienced students decide not to take the simplest route (though in high school, I think I was one of them.)

The other warm-up question I gave for the day was the following:

The images below represent the windshield of the bus during one of Mr. Weinberg's trips in New Zealand.
The windshield initially had no bugs on it.

The students were a bit annoyed at having to do this, but they got a much needed review of approximating derivatives. Most students used a central difference, with only a couple using just a forward or backward difference. The fact that they did both was really useful during discussions later on about using left, right, and midpoint calculations for integrals.

As tends to occur with my students, especially at this point in the year when they know most of my ideas don't come out of nowhere, they demanded to see some of the pictures I took. I was, of course, happy to oblige:

I was able to show them a few more actual scenic pictures, which kept things light as they needed to be before diving into the tedium of calculating areas under curves manually.

The rest of the lesson went great and was essentially unchanged from last year, with the exception of using the following data table instead of a table of velocity vs. time:

Originally I was going to start the lesson with this, but added the second warm-up activity when I thought it might seem a bit too contrived to just throw a table like this at them without any feasible way of actually generating it. I also gave the warning that though the values in this table was made up (though some thought that it seemed completely in character for me to actually take the pictures every hour for the purposes of Calculus), it would be possible to generate such a table using the procedure they followed in the warm-up question.

We talked about how we might estimate the total number of bugs during each two hour interval if we knew the rate and assumed that rate was constant. The left hand and right hand sums came straight out of this. A couple students immediately thought about averaging together the two rates to do midpoint, and later on that led very nicely to a visual discovery of the trapezoidal rule. When we looked at what this process then meant graphically, most students seemed to find the overall concept pretty simple.

The mechanics of doing a left/right/midpoint sum with a function initially appeared more complicated, but having them set up the calculation using a table to organize the values (as with the smash rate table) made a big difference.

Overall, I think the students last year got along fine with the more traditional introduction finding displacement from a table of velocity vs. time data. They got the concept fine, as did the students this year when I showed them how it was really the same as what we did. I think it made a difference to be able to introduce the topic in a more quirky way that grabbed their attention slightly more than something that was just plain easy to understand.

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

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

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

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

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

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

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