Monthly Archives: February 2013

The post where I remind myself that written instructions for computer tasks stink.

It's not so much that I can't follow written instructions. I'm human and I miss steps occasionally, but with everything written down, it's easy to retrace steps and figure out where I went wrong if I did miss something. The big issue is that written instructions are not the best way to show someone how to do something. Text is good for some specific things, but defining steps for completing a task on a computer is not one of them.

Today I showed my students the following video at the start of class.
GEO-U6D2.1-Constructing Parallelogram in Geogebra

I also gave them this image on the handout, which I wrote last year, but students only marginally followed:
Screen Shot 2013-02-27 at 5.53.31 PM

It was remarkable how this simple change to delivery made the whole class really fun to manage today.

  • Students saw exactly what I wanted them to produce, and how to produce it.
  • The arrows in the video identified one of the vocabulary words from previous lessons as it appeared on screen.
  • My ESOL students were keeping up (if not outpacing) the rest of the class.
  • The black boxes introduced both the ideas of what I wanted them to investigate using Geogebra, and simultaneously teased them to make their own guesses about what was hidden. They had theories immediately, and they knew that I wanted them to figure out what was hidden through the activity described in the video. Compare this to the awkwardness of doing so through text, where they have to guess both what I am looking for, and what it might look like. You could easily argue this is on the wrong side of abstraction.
  • I spent the class going around monitoring progress and having conversations. Not a word of whole-class direct instruction for the fifty minutes of class that followed showing the video. Some students I directed to algebraic exercises to apply their observations. Others I encouraged to start proofs of their theorems. Easy differentiation for the different levels of students in the room.

Considering how long I sometimes spend writing unambiguous instructions for an exploration, and then the heartbreak involved when I inevitably leave out a crucial element, I could easily be convinced not to try anymore.

One student on a survey last year critiqued my use of Geogebra explorations saying that it wasn't always clear what the goal was, even when I wrote it on the paper. These exploratory tasks are different enough and more demanding than sitting and watching example problems, and require a bit more selling for students to buy into them being productive and useful. These tasks need to quickly define themselves, and as Dan Meyer suggests, get out of the way so that discovery and learning happens as soon as possible.

Today was a perfect example of how much I have repeatedly shot myself in the foot during previous lessons trying to establish a valid context for these tasks through written instructions. The gimmick of hiding information from students is not the point - yes there was some novelty factor here that may have led to them getting straight to work as they did today. This was all about clear communication of objectives and process, and that was the real power of what transpired today.

A computational approach to modeling projectile motion, continued.

Here is the activity I am looking at for tomorrow in Physics. The focus is on applying the ideas of projectile motion (constant velocity model in x, constant acceleration model in y) to a numerical model, and using that model to answer a question. In my last post, I detailed how I showed my students how to use a Geogebra model to solve projectile motion.

Let me know what I'm missing, or if something offends you.

A student is at one end of a basketball court. He wants to throw a basketball into the hoop at the opposite end.

  • What information do you need to model this situation using the Geogebra model? Write down [______] = on your paper for any values you need to know to solve it using the model, and Mr. Weinberg will give you any information he has.
  • Find a possible model in Geogebra that works for solving this problem.
  • At what minimum speed he could throw the ball in order to get the ball into the hoop?

We are going to start the process today of constructing our model for projectile motion in the absence of air resistance. We discussed the following in the last class:

  • Velocity is constant in the horizontal direction. (Constant velocity model)
  • x(t) = x_{0} + v t

  • Acceleration is constant in the vertical direction (Constant acceleration model)
  • v(t) = v_{0} + a t
    x(t)=x_{0}+v t +frac{1}{2}a t^2

  • The magnitude of the acceleration is the acceleration due to gravity. The direction is downwards.

Consider the following situation of a ball rolling off of a 10.0 meter high platform. We are neglecting air resistance in order for our models to work.
Screen Shot 2013-02-25 at 6.15.15 PM

Some questions:

  • At what point will the ball's movement follow the models we described above?
  • Let's set x=0 and y = 0 at the point at the bottom of the platform. What will be the y coordinate of the ball when the ball hits the ground? What are the components of velocity at the moment the ball becomes a projectile?
  • How long do you think it will take for the ball to hit the ground? Make a guess that is too high, and a guess that is too low. Use units in your answer.
  • How far do you think the ball will travel horizontally before it hits the ground? Again, make high and low guesses.

Let's model this information in a spreadsheet. The table of values is nothing more than repeated calculations of the algebraic models from the previous page. You will construct this yourself in a bit. NBD.
Screen Shot 2013-02-25 at 6.39.23 PM

  • Estimate the time when the ball hits the ground. What information from the table did you use?
  • Find the maximum horizontal distance the ball travels before hitting the ground.

Here are the four sets of position/velocity graphs for the above situation. I'll let you figure out which is which. Confirm your answer from above using the graphs. Let me know if any of your numbers change after looking at the graphs.

Screen Shot 2013-02-25 at 6.42.35 PM

Now I want you to recreate my template. Work to follow the guidelines for description and labels as I have in mine. All the tables should use the information in the top rows of the table to make all calculations.

Once your table is generating the values above, use your table to find the maximum height, the total time in the air, and the distance in the x-direction for a soccer ball kicked from the ground at 30° above the horizontal.

I'll be circulating to help you get there, but I'm not giving you my spreadsheet. You can piece this together using what you know.

Next steps (not for this lesson):

  • The table of values really isn't necessary - it's more for us to get our bearings. A single cell can hold the algebraic model and calculate position/velocity from a single value for time. Goal seek is our friend for getting better solutions here.
  • With goal seek, we are really solving an equation. We can see how the equation comes from the model itself when we ask for one value under different conditions. The usefulness of the equation is that we CAN get a more exact solution and perhaps have a more general solution, but this last part is a hazy one. So far, our computer solution works for many cases.

My point is motivating the algebra as a more efficient way to solve certain kinds of problems, but not all of them. I think there needs to be more on the 'demand' side of choosing an algebraic approach. Tradition is not a satisfying reason to choose one, though there are many - providing a need for algebra, and then feeding that need seems more natural than starting from algebra for a more arbitrary reason.

Struggling (and succeeding) with models in physics

Today we moved into exploring projectile motion in my non-AP physics class. Exhibit A:
Screen Shot 2013-02-22 at 4.29.27 PM

I launched a single marble and asked them to tell me what angle for a given setting of the launched would lead to a maximum distance. They came up with a few possibilities, and we tried them all. The maximum ended up around 35 degrees. (Those that know the actual answer from theory with no air resistance might find this curious. I certainly did.)

I had the students load the latest version of Tracker on their computers. While this was going on, I showed them how to use the program to step frame-by-frame through one of the included videos of a ball being thrown in front of a black background:
Screen Shot 2013-02-22 at 4.34.45 PM
Students called out that the x-position vs. t graph was a straight line with constant slope - perfect for the constant velocity model. When we looked at the y-position vs t, they again recognized this as a possible constant acceleration situation. Not much of a stretch here at all. I demonstrated (quickly) how the dynamic particle model in Tracker lets you simulate a particle on top of the video based on the mass and forces acting on it. I asked them to tell me how to match the particle - they shouted out different values for position and velocity components until eventually they matched. We then stepped through the frames of the video to watch the actual ball and the simulated ball move in sync with each other.

I did one more demo and added an air resistance force to the dynamic model and asked how it would change the simulated ball. They were right on describing it, even giving me an 'ooh!' when the model changed on screen as they expected.
Screen Shot 2013-02-22 at 5.04.31 PM

I then gave them my Projectile Motion Simulator in Geogebra. I told them that it had the characteristics they described from the graphs - constant velocity in x, constant acceleration of gravity in y. Their task was to answer the following question by adjusting the model:

A soccer ball is kicked from the ground at 25 degrees from the horizontal. How far and how high does the ball travel? How long is it in the air?

They quickly figured out how it works and identified that information was missing. Once I gave them the speed of the ball, they answered the three questions and checked with each other on the answers.

I then asked them to use the Geogebra model to simulate the launcher and the marble from the beginning of the class. I asked them to match the computer model to what the launcher actually did. My favorite part of the lesson was that they started asking for measuring devices themselves. One asked for a stopwatch, but ended up not needing it. They worked together to figure out unknown information, and then got the model to do a pretty good job of predicting the landing location. I then changed the angle of the launcher and asked them to predict where the marble would land. Here is the result:

Nothing in this lesson is particularly noteworthy. I probably talked a bit too much, and could have had them go through the steps of creating the model in Tracker. That's something I will do in future classes. When I do things on the computer with students, the issues of getting programs installed always takes longer than I want it to, and it gets away from the fundamental process that I wanted them to see and have a part of - experiencing the creation of a computer model, and then actually matching that model to something in the real world.

My assertions:

  • Matching a model (mathematical, physical, numerical, graphical, algebraic) to observations is a challenge that is understood with minimal explanation. Make a look like b using tool c.
  • The hand waving involved in getting students to experiment with a computer model is minimized when that model is being made to match actual observations or data. While I can make a computer model do all sorts of unrealistic things, a model that is unrealistic wont match anything that students actually see or measure.
  • Students in this activity realized what values and measurements they need, and then went and made them. This is the real power of having these computer tools available.
  • While the focus in the final modeling activity was not an algebraic analysis of how projectile motion works mathematically, it did require them to recognize which factors are at play. It required them to look at their computed answer and see how it compared with observations. These two steps (identifying given information, checking answer) are the ones I have always had the most difficulty getting students to be explicit about. Using the computer model focuses the problem on these two tasks in a way that hand calculations have never really pushed students to do. That's certainly my failure, but it's hard to deny how engaged and naturally this evolved during today's lesson.

The homework assignment after the class was to solve a number of projectile motion problems using the Geogebra model to focus them on the last bullet point. If they know the answers based on a model they have applied in a few different situations, it will hopefully make more intuitive sense later on when we do apply more abstract algebraic models.

Algebra is very much not dead. It just doesn't make sense anymore to treat algebraic methods as the most rigorous way to solve a problem, or as a simple way to introduce a topic. It has to start somewhere real and concrete. Computers have a lot of potential for developing the intuition for how a concept works without the high bar for entry (and uphill battle for engagement) that algebra often carries as baggage.

What do I have wrong here? Computational thinking obsession continues

Another installment of my Hong Kong presentation titled 'Why Computational Thinking matters.' This is where my head is these days in figuring out how computers relate to what we do in class. My view is that activities like the one I describe in the video is more active than the way we (and I include myself in this group) usually attack word problems as part of our sequence.

Help me flesh this out. I think there's a lot here.

When things just work - starting with computers

Today's lesson on objects in orbit went fantastically well, and I want to note down exactly what I did.

Scare the students:

Screen Shot 2013-02-05 at 3.23.59 PM

Push to (my) question - how close is that?

Connect to previous work:

The homework for today was to use a spreadsheet to calculate some things about an orbit. Based on what they did, I started with a blank sheet toward the beginning of class and filled in what they told me should be there.
orbit calculationsScreen Shot 2013-02-05 at 3.30.08 PM
Some students needed some gentle nudging at this stage, but nothing that felt forced. I hate when I make it feel forced.

Play with the results

Pose the question about the altitude needed to have a satellite orbit once every twenty four hours. Teach about the Goal Seek function in the spreadsheet to automatically find this. Ask what use such a satellite would serve, and grin when students look out the window, see a satellite dish, and make the connection.

Introduce the term 'geosynchronous'. Show asteroid picture again. Wait for reaction.

Screen Shot 2013-02-05 at 3.23.59 PM

See what happens when the mass of the satellite changes. Notice that the calculations for orbital speed don't change. Wonder why.

See what happens with the algebra.

Screen Shot 2013-02-05 at 3.36.51 PM

See that this confirms what we found. Feel good about ourselves.

Wonder if student looked at the lesson plan in advance because the question asked immediately after is curiously perfect.

Student asks how the size of that orbit looks next to the Earth. I point out that I've created a Python simulation to help simulate the path of an object moving only under the influence of gravity. We can then put the position data generated from the simulation into a Geogebra visualization to see what it looks like.

Simulate & Visualize

Introduce how to use the simulation
Use the output of the spreadsheet to provide input data for the program. Have them figure out how to relate the speed and altitude information to what the simulation expects so that the output is a visualization of the orbit of the geosynchronous satellite.

Screen Shot 2013-02-05 at 3.55.37 PM

Not everybody got all the way to this point, but most were at least at this final step at the end.

I've previously done this entire sequence starting first with the algebra. I always would show something related to the International Space Station and ask them 'how fast do you think it is going?' but they had no connection or investment in it, often because their thinking was still likely fixed on the fact that there is a space station orbiting the earth right now . Then we'd get to the stage of saying 'well, I guess we should probably draw a free body diagram, and then apply Newton's 2nd law, and derive a formula.'

I've had students tell me that I overuse the computer. That sometimes what we do seems too free form, and that it would be better to just get all of the notes on the board for the theory, do example problems, and then have practice for homework.

What is challenging me right now, professionally, is the idea that we must do algebra first. The general notion that the 'see what the algebra tells us' step should come first after a hook activity to get them interested since algebraic manipulation is the ultimate goal in solving problems.

There is something to be said for the power of the computer here to keep the calculations organized and drive the need for the algebra though. I look at the calculations in the spreadsheet, and it's obvious to me why mass of the satellite shouldn't matter. There's also something powerful to be said for a situation like this where students put together a calculator from scratch, use it to play around and get a sense for the numbers, and then see that this model they created themselves for speed of an object in orbit does not depend on satellite mass. This was a social activity - students were talking to each other, comparing the results of their calculations, and figuring out what was wrong, if anything. The computer made it possible for them to successfully figure out an answer to my original question in a way that felt great as a teacher. Exploring the answer algebraically (read: having students follow me in a lecture) would not have felt nearly as good, during or afterwards.

I don't believe algebra is dead. Students needed a bit of algebra in order to generate some of the calculations of cells in the table. Understanding the concept of a variable and having intuitive understanding of what it can be used to do is very important.

I'm just spending a lot of time these days wondering what happens to the math or science classroom if students building models on the computer is the common starting point to instruction, rather than what they should do just at the end of a problem to check their algebra. I know that for centuries mathematicians have stared at a blank paper when they begin their work. We, as math teachers, might start with a cool problem, but ultimately start the 'real' work with students on paper, a chalkboard, or some other vertical writing surface.

Our students don't spend their time staring at sheets of paper anywhere but at school, and when they are doing work for school. The rest of the time, they look at screens. This is where they play, it's where they communicate. Maybe we should be starting our work there. I am not recommending in any way that this means instruction should be on the computer - I've already commented plenty on previous posts on why I do not believe that. I am just curious what happens when the computer as a tool to organize, calculate, and iterate becomes as regular in the classroom as graphing calculators are right now.

Computational Thinking & Spreadsheets

I feel sorry for the way spreadsheets are used most of the time in school. They are usually used as nothing more than a waypoint on the way to a chart or graph, inevitably with one of its data sets labeled 'Series 1'. The most powerful uses of spreadsheets come from how they provide ways to organize and calculate easily.

I've observed a couple things about the problem solving process among students in both math and science.

  • Physics students see the step of writing out all of the information as an arbitrary requirement of physics teachers, not necessarily as part of the solution process. As a result, it is often one of the first steps to disappear.
  • In math, students solving non-routine problems like Three Act problems often have calculations scrawled all over the place. Even they are written in an organized way, in the event that a calculation is made incorrectly, any sets of calculations that are made must be made again. This can be infuriating to students that might be marginally interested in finding an answer in the first place.
  • Showing calculations in a hand written document is easy - doing so in a document that is to be shared electronically is more difficult. There are also different times when you want to see how the calculation was made, and other times that you want to see the results. These are often presented in different parts of a report (body vs. appendix) but in a digital document, this isn't entirely necessary.

Here's my model for how a spreadsheet can address some of these issues:
Screen Shot 2013-02-01 at 7.47.59 PM

Why I like it:

  • The student puts all of the given information at the top. This information may be important or used for subsequent calculations, or not. It minimally has all of the information used to solve a problem in one place.
  • The coloring scheme makes clear what is given and what is being being calculated.
  • The units column is a constant reminder that numbers usually have units. In my template, this column is left justified so that the units appear immediately to the right of the numerical column.
  • Many students aren't comfortable exploring a concept algebraically. By making calculations that might be useful easy to make and well organized, this sets students up for a more playful approach to figuring things out.
  • Showing work is easy in a spreadsheet - look at the formulas. Depending on your own expectations, you might ask for more or less detail in the description column.

Some caveats:

    • A hand calculation should be done by someone to confirm the numbers generated by the spreadsheet are what they should be. This could be a set of test data provided by the teacher, or part of the initial exploration of a concept. Confirming that a calculation is being done correctly is an important step of trusting (but verifying, to quote Reagan for some reason) the computer to make the calculations so that attention can be focused on figuring out what the numbers mean.
    • It does take a bit of time to teach how to enter a formula into a spreadsheet. Don't turn it into a lecture about absolute or relative addressing, or about rows and columns and which is which - this will come with practice. Show how numbers in scientific notation look, and demonstrate how to get a value placed in another cell. Get straight into making calculations happen among your students and in a way that is immediately relevant to what you are trying to do. Then change a given value, and watch the students nod when all of the values in the sheet change immediately.
    • Building off of what I just said, don't jump to a spreadsheet for a situation just to do it. The structure and order should justify itself. Big numbers, nasty numbers, lots of calculations, or lots of given information to keep track of are the minimum for establishing this from the start as a tool to help do other things, not an end in and of itself.
    • Do not NOT


      hand your students a spreadsheet that calculates everything for them. If a student wants to make a spreadsheet for a particular type of calculation, that's great. That's the student recognizing that such a tool would be useful, and making the effort to do this. If you hand them a calculator for one specific application, it perpetuates the idea among students that they have to wait for someone else that knows better than them to give them the tool to use. Students should have the ability to make their own utilities, and this is one way to do it.

Example from class yesterday:

We are exploring the way Newton's Law of Gravitation is used. I asked students to calculate the force of gravity from different planets in the solar system pulling on a 65 kilogram person on Earth, with Wolfram Alpha as the source of data. Each of them used a scientific or graphing calculator to calculate their numbers, with the numbers they used written by hand (without units) on their papers with minimal consistency. They grumbled about the sizes of the numbers. When noticeable differences arose in magnitude between different students, they checked each other until they were satisfied.

I then showed them how to take the pieces of data they found and put them in the spreadsheet in the way I described above. In red, I highlighted the calculation for the magnitude of the force for an object on Earth, and then asked a student to give me her data. This was the value she calculated! I was quickly able to confirm the values that the other students also had made.

I then had them calculate the weight of an object on Earth's surface using Newton's law of gravitation. This sent them again on a search for data on Earth's vital statistics. They were surprised to see that this value was really close to the accepted value for g = 9.8;m/s^2. I then asked them in their spreadsheet how they might figure out the acceleration due to gravity based on what they already knew. Most were able to figure out without prompting that dividing by the 65 kilogram mass got them there. I then had them use that idea and Newton's Law of Gravitation to figure out how to obtain the acceleration due to gravity at a given distance from the mass center of a planet. I then had them use the spreadsheet model on their own to calculate the acceleration due to gravity on a couple of different planets, and it went really well.

The focus from that point on was on figuring out what those numbers meant relative to Earth. Often with these types of problems, students will calculate and be done with it. These left them a bit curious about each other's answers (gravity on Jupiter compared to the Moon) and opened up the possibilities for subsequent lessons. I'll write more about how I have grown to view spreadsheets as indispensable computing tools in the classroom in the future. A pure computational tool is the lowest level on the totem pole of applications of computers for learning mathematics or science, but it's a great entry point for students to see what can be done with it.


Spreadsheet Calculation Template

Centripetal Acceleration of the Moon - a comparison we used two days ago to suggest how a 1/r^2 relationship might exist for gravity and the moon.