Tag Archives: exploration

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.

What my dad taught me about learning.

The first time I saw the word 'Calculus', I was staring at the spines of several textbooks that sat on the bookshelf at home. I didn't think much of them; I knew they were my parents', and that they were from their college days, but had no other awareness of what the topic actually was. I did assume that the reason there were so many of them was because my parents must have liked them so much. After further investigation, I learned that they were mostly my dad's books. His secret was out: he must have loved Calculus. I believed this for a while.

When my older brother took Calculus, these books came off the shelf occasionally as a resource, though I don't know if this was his decision or my dad's. From what I knew, my brother breezed through Calculus. I know he worked hard, but it also seemed to come fairly naturally to him. I remember conversations that my parents had about not knowing where my brother got this talent from. They admitted at this point that it couldn't have been from either of them. My dad had taken Calculus multiple times and the collection of textbooks was the evidence that hung around for no particularly good reason.

This astounded my young brain for a couple of reasons. It was mind-boggling to me that my parents ever had trouble doing anything. They always seemed to know just what to do in different situations - how could they not do well in a class designed to teach them something? It was also the first time I ever remember learning that my dad was not successful in everything he tried to do. This conflicted deeply with what I understood his capabilities to be.

As I understood it, he just knew everything.

When I was nine and my parents had bought me a keyboard to learn to play piano for the first time, there was no AC adapter in the box I had unwrapped only moments before. My dad scrounged around among his junk boxes and drawers and found one with the correct tip, but the polarity was wrong. I knew I wasn't going to be able to start jamming that night - it was late and a trip to the store wasn't an option. He wasn't going to submit to that as a possibility - he took the adapter downstairs to the basement and had me follow him. There was soldering involved, and electrical tape. I had no idea what he was doing. Moments later, however, he appeared with the same adapter and a white label that said 'modified'. We plugged it in to the keyboard and it lit up, ready for me to play and drive my parents crazy with my rendition of . I now understand that he switched the wires around to change the polarity - I did it myself with some students recently in robotics. At the time though, it seemed like magic. I just knew I had the smartest dad in the world.

His mantra has always been that if it can be fixed, it should be fixed, no matter the hilarity of the process. I watched him countless times take in the cast-off computers of other people who asked him if he knew how to fix them. Thinking back, I don't know that he ever specifically answered that question. His usual response was (and still is) "I'll take a look." So he would work long hours with a vacuum, various metal tools, and a gray multimeter (that I think he still has) laid out like a surgeon investigating a patient. I rarely had the patience to sit and watch. I would see the results of his work: sheets of yellow legal pad paper filled with notes and diagrams scrawled along the way. In the end, he would inevitably find a solution, though often at this point the person who had asked him to fix the item had gone and bought a new one. I don't recall ever believing my dad thought it was a waste.

We also worked on things together to try to get closer in my early teens. We both took tests to get amateur radio licenses. I came to really enjoy learning Morse code and got the preparation books to climb the license ladder. He commented repeatedly as I zipped through the books about memorizing the books and not understanding the underlying theory of resonant circuits and antenna diagrams. That was true – at the time I just wanted to pass the tests. I didn't understand that the process of learning was the valuable part, not the end point. I didn't see that. I just continued to believe that the tests were a means to an end, just as I viewed through my thirteen year old brain that his herculean efforts to fix things was a means to getting things fixed., and nothing more.

My dad is one of the smartest people I know. As I've grown older, however, I have come to understand that it wasn't that about knowing everything. He instead had been continuously demonstrating what real learning is supposed to be. It was never about knowing the answer; it was about finding it. It wasn't about fixing a computer, it was about enjoying figuring out how it can be fixed, however much frustration was involved. It wasn't just about saving money or avoiding a trip to the store to buy an electric adapter. It was about seeing that we can understand the tools we use on a regular basis well enough to make them work for us.

I have seen time and time again how he mentors people to make them better at what they do. I have seen it in the way he mentors FIRST robotics teams as a robot inspector at the Great Lakes regional competition in Cleveland. I have seen it in the way he has spent his time since selling the company he founded with partners years ago. He chooses to do work that matters and makes sure that others are right there to learn beside him. There were times growing up when, admittedly, I just wanted him to fix things that needed to be fixed. To his credit, he insisted on involving me in the process, even when I protested or became impatient.. I didn't see it when I was younger. Knowing how to go about solving problems is among the most important skills that everyone needs. I was getting free lessons from someone that not only was really good at it, but cared enough about me to want me to learn the joy of figuring things out.

One of my students this year was really into electronic circuits and microcontrollers. He soldered 120 LEDs into a display and wanted to use an Arduino to program it to scroll text across it. The student's program wasn't working and he didn't know why. I had only been tangentially paying attention to the issues he was having, and when he was visibly frustrated, I pulled up a chair and sat next to him, and then said 'let's take a look.” We went through lines of code and found some missing semicolons and incorrectly indexed arrays, and I asked him to tell me what each line did. I was only a couple steps ahead of him in identifying the problem, but we laughed and tried making changes while speaking out loud what we thought the results would be. At one point, he said to me “Mr. Weinberg, you're so smart. You just know what to do to fix the program.”

I immediately corrected him. I didn't know what was wrong. We were able to make progress by talking to each other and experimenting. It wasn't about knowing just what to do. It was about figuring out what to try next and having strategies to analyze what was and was not working. I learned this from a master.

On this Father's day (that also happens to be the day before my dad's birthday), I celebrate this truth: much of what I do as a teacher comes from trying to channel my dad's habits while confronting big challenges. I don't want my students to memorize steps to pass tests; I want them to understand well enough to be able to solve any challenge set before them. I don't want to fix my students' problems – I want to help them learn to fix problems themselves. I don't want my students to be afraid to fail; I want them to understand through example that failure leads to finding a better way.

I am grateful for all that I have learned from him., and I try to teach my students what he has taught me about learning at every opportunity. It would be fine by me if I ever need to do Calculus for him - I'd still be in the red.

Electric Circuits - starting at the end.

We only have a couple weeks of class left, and there's not enough time to do the traditional Physics B sequence that I've used for electricity with my seniors that asked for a non-AP physics course at the beginning of the year. Normally I do electrostatics for a couple of weeks, talk about electric fields and potential, and then use these concepts to motivate a treatment of electric circuits. I could have stretched that out, but given my freedom in pace and curriculum, I decided to switch everything around.

This year, I started at the end of my sequence to address a pretty big issue I've always seen with my students. As much as they talk about charging (mobile devices, laptops) and basic energy conservation such as turning lights off, they have a pretty fuzzy understanding of electricity and the origins of the energy they use everyday. Some of the last topics in my traditional sequence involve real voltage sources, batteries and internal resistance - the "real" electronics that you need to know if you want to actually build a circuit. You know, the actually interesting part.

There's nothing interesting in looking at a circuit and calculating what current is going through an arbitrary resistor in a given circuit.  It took me a while to come to this realization because I still have some brain cells clinging to the "theory first, application second" philosophy, the same brain cells I've been working to silence this year. These are the sorts of things I want my students to learn to do:

  • Build a charger for an iPod using a solar panel and some circuit components. What is involved in charging a battery in a way that the battery will actually charge up without blowing Nickel and Cadmium all over the classroom?
  • Create a circuit that lights up an LED with the right current so it can outlast an incandescent bulb.
  • Look at an AC adapter that isn't made for a given device, and modify it so that it does work. The fact that it only costs $5 to buy a new one is irrelevant when you compare it to the feeling you get when you realize this is not hard to do. (Thanks Dad!)
  • Generate electricity. Figure out how hard you have to physically work to run your laptop.

This is what we did on day one:

I gave them a solar panel, some small DC motors and LEGO motors, a stripped down version of our FIRST Tech Challenge robot, some lemons, clip leads, and different kinds of wire, and said I wanted them to use these tools to generate the highest voltage they could. There was also a bag of green LEDs on the table there for them to play with. There was a flurry of activity among my five students as they remembered something vaguely from chemistry about sticking different metals into a lemon, and needing to connect one to another in a certain way. They did so and saw that there was a bit of a voltage from the lemons they had connected together, but that there wasn't much there.

I then showed them one of the LEGO motors and had them see what happened on a connected voltmeter when the axle was rotated. They were amazed that this also generated an electrical potential. This turned immediately into a contest of rotating the motor as quickly as possible and seeing the result on the voltmeter. One grabbed an LED and hooked it up and saw that it lit up.

They then turned to the robot and its big beefy motors. They found I had a set of LED lights in my parts box and asked to use it. Positive results:

The solar panel was also a big hit as it resulted in us going outside. They were impressed with how "much" electricity was generated after seeing the voltmeter display over 15 volts - they were surprised then to see that it worked to turn on the LED display, but not any of the motors they tried.

At this point it was the end of the class block, so we put everything away and went on with our day.

Some of the reasons I finished the day with a smile:

  • There was never a moment when I had to tell any of the students to pay attention and get involved in the activity.  The variety of objects on the table and the challenge were enough to get them playing and interacting with each other.
  • While I did show them how to play with one of the tools (i.e. DC motor acting as generator) , they quickly figured out how they might transfer this idea to the other items I made available.
  • They made bits of progress toward the understanding that voltage alone was not what made things work. This is a big one.

The next day's class used the PHet circuit construction kit to explore these ideas further in the context of building and exploring circuits. We had some fantastic conversations about voltage of batteries, conventional vs. electron current, and eventually connected the idea of Ohm's law (which was floating around in their heads from middle school science) to the observations they made.

I was struggling for a while about how to approach electricity because I have always followed the traditional sequence. In the end, I realized that I really didn't want to go through electrostatics - I wasn't excited to teach it this time around.  I also realized that I didn't need to do so, either in order to teach my students what I really wanted them to learn about electricity.

I think this approach will help them realize that electricity is not magic. They can learn to control it. I admit that doing so can be dangerous and expensive if one doesn't know what he or she is doing. That said, a little basic knowledge goes a long way, even in today's world of nanometer sized transistors.

Tomorrow we attempt the LED lighting assignment - feel free to share your comments or suggestions!

Geogebra for Triangle Congruence Postulates

It has been busy-ville in gealgerobophysicsulus-town, so I have barely had time to catch my breath over the last few days of music performances, school events, and preparations for the end of the semester.

My efforts over the  past couple days in Geometry have focused on getting in a bit of understanding of congruent triangles. We have used some Geogebra sketches I designed to have them build a triangle with specific requirements. With some feedback from some Twitter folks (thanks a_mcsquared!) and students after doing the activities, I've got these the way I want them.

Constructing a 7-8-9 triangle: Download here. (For discovering SSS)

Constructing a 3-4-45 degree triangle: Download here. (For discovering SAS)

Looking for an ASA postulate. Download here. (Clearly for ASA explorations.) - This one I made a quick change before class to making it so that the initial coordinates of the base of the triangle are randomized when loading the sketch. This almost guarantees that every student will have a differently oriented triangle. This makes for GREAT conversations in class. Here are three of the ones students created this afternoon:

I'm doing a lot of thinking about making these sorts of activities clearly driven by simple, short instructions. This is particularly in light of a few of the students in my class with limited English proficiency. Creating these simple activities is also a lot more fun than just asking students to draw them by hand, guess, or just listen to me tell them the postulates and theorems. Having a room full of different examples of clearly congruent triangles calls upon the social aspect of the classroom. Today they completed the activity and showed each other their triangles and had good interactions about why they knew they had to be congruent.

Last year I had them construct the triangles themselves, but the power of the end message was weakened by the written steps I included in the activity. Giving them clear instructions made the final product, a slew of congruent (or at least approximately in the case of 7-8-9) triangles a nice "coincidence" to lead to generalizing the idea.

Exploring Point Slope Form through Geogebra

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

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

Download my Geogebra file here.

Comments welcome!

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.

Build the robot the way you want...No, you're doing it wrong!

I teach an exploratory class for middle school students in robotics. The students rotate between robotics and some other electives during each quarter of the year, and there are no grades - just an opportunity to learn something interesting while doing. I like the no grades part, particularly because assessing progress in robotics is quite hard to do. My usual model is saying you get x points for doing the bare minimum (a D) and then incremental increases in the grade for doing progressively more challenging tasks.

It works, but I really like getting the opportunity to not have to do it. There are so many things you could measure to assess the students in their building and programming skills, but in my experience, the students don't tend to explore or tinker as much in that situation.

Today while working on the day's challenge on using sensors and loops, a student was fixated on building the following:

There weren't any instructions to do this - he just started putting things together, liked what he had created, and continued building it today.

I had to stop myself for a moment because teacher Evan started to come out and remind him to stay on task and contribute toward his team's solution to the challenge. Thankfully robotics Evan intervened and let it happen.

This is an exploratory class. It's supposed to expose the students to new situations that might interest them later on. Why in the world would I stop the exact sort of thinking and exploring the class was designed to provoke? It also turned out that this student, along with the other two in the group, were all taking turns in the programming and building so that each would have a chance to play in this way. In spite of their taking time to free build, this group actually solved the three challenges before the rest in the class.

It connects to a great TED talk on doodling by Sunni Brown. One idea from her talk was that doodling contributes to "creative problem solving and deep information processing." I think that these students (and all of us that 'play' with building toys like LEGO) are engaging in a similar process by free building. The connections students are making in figuring out how the tools work through play are not easy to measure. This is a horrible reason not to provide them time to do so. I do think there is an interesting connection between the tendency of these students to play and their ability to figure out the subtler parts of the class challenges.

After all, as David Wees pointed out, giving explicit step-by-step instructions on how to use creative tools like LEGO takes the creativity (and much of the fun) out of it. There has to be time to experiment and learn by doing how the tools work together, and that's exactly what this student was doing.

My final reaction during the class today was that I told them all that I wanted to take a picture of every off-topic LEGO design they create. Document it all. If it's cool enough to engage you for the time it takes to create it, I want a gallery of those designs to celebrate them.

The sad part? This made some of them stop. I can't win!