calculus – Page 2

The TacoCopter? – a gimmick for integration review

I received an email sending me to this site yesterday about the TacoCopter, which of course was spot on given my interest in all things robotic. I also had PID control on the brain thanks to my course on driving a robot car from Udacity. Bits of python code were in my head already, and I had a strong need to put it all together. Given that it was also Sunday (a workday for most teachers) I had to plan for classes tomorrow, specifically Calculus and Physics.

All of this was in the context of the beautiful afternoon I spent on the balcony of the apartment looking out at the warmest, bluest Hangzhou skies of the year so far. It put me in the mood to do something a bit different for tomorrow’s Calculus class. The AP students will be reviewing related rates and implicit differentiation, but the regular students…they get to have a bit more fun.

This is the activity we will be looking at tomorrow in class: CW – TacoCopter Project

The full wiki page that students will be following is located here: http://wiki.hischina.org/groups/gealgerobophysiculus/wiki/42712/Calculus_Unit_8__The_TacoCopter.html

Some python code for simulating the TacoCopter rising to altitude, which can be found here at github.

Then Geogebra for plotting the data, which shows the lovely simulated accelerometer data with noise:

I don’t really know how it will go. At least students will have an excuse to grin as they review.

Volumes of Revolution & 3D Modeling

I had a conversation with a colleague a few years ago about volumes of revolution in Calculus. We were both a few years removed from our own Calculus courses in high school and college, and we were talking about how we thought about the concept visually.

For those that need a refresher, here is the idea behind a volume of revolution. Imagine you have a solid object that can be lined up with the x-axis so that its cross section looks like the image below. The object would have a pointy end at the origin (0,0) and a circular face located at x = 1. The closest real world object that fits this description is a Hershey’s Kiss.

The object is axially symmetric about the x-axis. If you were to cut the object with a knife so that the cut passes through the pointy end and the center of the flat face, the image at left would always be the cross section.

A volume of revolution is usually defined by an even simpler idea. Take a region of a graph and rotate it in a circle around some axis. The region at left is defined by rotating the area under the graph of y = x ^{2 around the x-axis.}

My colleague’s way of visualizing this idea started with the solid itself. Cut it into a series of discs, each of width dx , and then analyze a single differential disc to come up with an integral expression for the entire volume. This requires being able to visualize the entire solid first, and then see how it can be cut into discs.

I didn’t see it this way. I could visualize the solid usually, but to then mentally cut the solid into discs, and then construct a differential volume seemed to have one too many steps to make it simple. I focused on the step that made conceptual sense to me: start with a defined region and rotate it around an axis to create a solid. The differential strip of area we had been making underneath the graph since the first introduction of the definite integral was what I always visualized during integration. I could visualize taking that strip and rotating it around to form a disc, and using that concept for the differential volume. Then add up these discs through an integral to find the volume.

When I taught volume of revolution for the first time, I wanted to introduce it in a way that would emphasize how I had come to understand the concept. Granted, this assumes my way will work for the students, but so far it seems to be doing so pretty well.

Three dimensional computer modeling programs (Blender, Pro-E, Autodesk Inventor, etc) all have a function called ‘Revolve’ which is, by definition, how volumes of revolution are created. The idea is that you define a region, pick an axis, and then the software will create a 3D solid and display it. Having a copy of Pro-E from our FIRST Tech Challenge team, I was able to introduce the process with a series of demonstrations live with the software. Some examples:

The students immediately saw what was going on, and didn’t think much of the process. I could quickly make a sketch, revolve it, and then rotate the object around for students to see what it would look like if actually in front of them. We then proceeded to revolve strips under and between graphs to generate discs and washers. Writing the integrals was then a fairly simple process.

I think the difficulty that might come up with this type of problem is the visualizing step. Students must visualize the 3D shape in order to solve problems related to its volume. I think having this sort of tool available has made a big difference in my students seeing what it means to create a volume of revolution, which then leads to an easier time conceptualizing how to then find its volume using Calculus.

Turning random facts into logistics curves – ODE per day series continued.

I previously wrote about making sure that every class during our unit on differential equations starts with some differential equation they can see or feel in a concrete way.

During the last class, we investigated a draining tank using the video posted by Dan Meyer at his blog.

Today we did something different. I told them that I was doing an experiment with a simple task. They all needed to find the answers to some questions as quickly as possible:

When they found the answers, I wanted them to quickly throw a hand in the air to let me know. I told them to be honest – they didn’t know what I was doing with the information yet, so there really wasn’t a chance to skew it.

I then showed them the slide with the questions:

I also simultaneously started the following Python program. (UPDATE: Code is posted here.) This let me easily record any time a student raised his/her hand.

I then pasted the data directly into a Geogebra spreadsheet and graphed the data…

…and then fit a logistics curve to the data:

They had seen and heard the concept of learning/performance curves before, but it was really great to be able to develop one on the spot with the class. I was impressed with how good the data turned out. It was then neat to be able to show the differential equation that describes this type of phenomenon and solve it to get this type of function.

As is probably obvious, I only have ten students in this group. It would be really cool to try something like this with a bigger group and see if the data fits as nicely.

Experiencing an ODE per day

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

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

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

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

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

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

Here’s the plan:

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

My progress thus far:

Day 1: Newton’s Law of Cooling

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

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

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

Day 2: Newton’s 2nd Law

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

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

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

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

Future topics?

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

What are your favorite demonstrations of differential equations in action?

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.

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.

Download link here.

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.

Using #Geogebra to Predict and then Verify

Last year’s class introducing logarithmic and exponential differentiation was a bust. I tried to include it as an application of implicit differentiation, but I knew afterwards then, and still believe now that doing so was an incredibly horrible idea. There’s no way students are going to ‘see’ an application of an abstract concept like implicit differentiation better…by using it in another abstract concept. I’ve accepted that, and vowed this year to do a much better job.

I also had a shocking moment yesterday when a Calculus student came to me after school and asked me ‘what is the derivative?’ We had started the unit with a conceptual development of the derivative using limits and average rate of change, and had since moved to applying differentiation rules, so we were deep in that process – power rule, quotient rule, product rule, chain rule…really the primary ‘rules’ section of any Calculus course. I was taken aback by the comment – had I really stopped emphasizing the definition of the derivative in our class activities? In a way, yes. We had been writing equations for tangent lines and graphing them, but we hadn’t seen the limit definition (which I’ve been impressed by students remembering) in a little while. This proved that not only did I need to do a better job with logs and exponential functions, but that a little conceptual basis in that process would be useful.

I always like using Geogebra as a tool to pre-load information I am about to give students – what is about to happen? What should my result look like when I do this on pencil and paper? The graphing capabilities make it really easy to do this and set this up – I created this file and made it look the way I wanted in a few minutes.

You can direct download the file here.

These were the instructions I gave students:

Sketch what you would expect the derivative of y = 2^x to look like. Then click the ‘Show Derivative Function’ to graph the actual derivative. How close were you?

How would you expect your sketch to change for the derivative of y = 3^x?

Graph and make a prediction of the graph of the derivative of y = 2^-x. Check and see how close you were using the Geogebra tool.

Can you adjust the slider value for a so that the derivative is the same as the function itself? Use the arrow keys to adjust the slider more precisely.

Go through this same process to sketch the derivative of y = ln(x) in a new Geogebra window. Create this by going to the ‘File’ menu and selecting ‘New Window’.

It was really great seeing students predicting what the derivative would be, and then using the applet to confirm what they thought. There were lots of good conversations about scale factors and reflections, and some of them pretty much nailed what the general forms were going to be. This made the algebraic derivation a piece of cake – they knew where it was headed.

I also sprung this on them:

I’ve been really getting into the idea of standard based grading, and have been doing a form of it through my quizzes for a while, but it is still a small component of the overall grade calculation. While their grades aren’t being calculated any differently at the moment, I shared that this list would make a really good tool as we prepare for the unit exam on derivatives next week, and most started going through on their own and deciding what they needed to work on.

I’m still getting caught up after a couple very busy weeks, but I really like how this group in Calculus has been developing and maturing as math students in only a couple months. Their questions are more directed: ‘I don’t understand this application of the chain rule’ compared to ‘I don’t get it’. Their written work is detailed and clear, making it easy to locate errors. As a group, they get along really well, and class periods are filled with moments of furious productivity and camaraderie as well as humor and smiles throughout.

It was raining hard all day. I watched some students walk into class, look outside at the afternoon sky, and sink into their chairs, clearly feeling a bit down. I told them it was perfect Calculus weather – why not sit inside and do some differentiation?

Probably not what they had in mind. By the end of class, everyone left the classroom looking much more positive than when they walked in, and at least feeling good about the work they had in front of them.

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

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

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

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

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

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

Take Time to Tech – Perspectives after a Flip

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

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

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

Developing computational and algorithmic fluency has its place.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

I am saying that balance is key.

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

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