Page 23 – iteration, making, building, and coding in education

Telling students not to procrastinate solves the wrong problem.

In seeing my students working to prepare for semester exams over the last week, I have spent some time thinking about the advice I give students about how to manage the stress associated with this time of year. The reality for them (and for me, for that matter) is that there is a lot going on right now. A quick rundown of my obligations: exams have to be written, assessments marked, comments graded, recommendations written, assignments double-checked for accuracy in the grade-book…this doesn’t even mention the non-school related tasks on my plate. Some tasks I spread out over a few days usually in order to avoid the non-linear way that unpleasantness increases as a deadline approaches. Some tasks have to be done last minute, and there’s no way around them.

When I see students cramming and working feverishly to get things done, part of me wants to channel the oft repeated (and nonsense) advice that ‘if you had started earlier, you wouldn’t have this problem.’ And then I stop. Grand scheme of things, this is not really helpful. You don’t tell someone that just cut off his finger that doing so was a dumb idea. The important part is managing the situation in a way that balances all of the relevant costs and benefits to maximize the overall outcome. The biggest problems my students have is not (only) that they put things off. It’s that they think they can effectively manage the stress that comes with it by following some common, but misdirected principles. Here are my categories of guiding principles:

Ways students foolhardily trick themselves into doing what they do:

Principle of Work-Equivalence: As long as I am working on something I need to be working on, I am using my time effectively. After all, it all needs to get done, so why not just pick something and work on it?
Principle of Longevity: I’ve been doing this school thing for long enough – I know this has worked for me in the past, so I’m going to keep doing it. This comes from a major trend that I see with my students at the moment. Even more frightening is that the older they are, the better they think they are at managing things during stressful times. The way I see it, the opposite is true.
Principle of Education through Suffering: If I am not suffering as I get things done, I am not working hard enough. Carrying around stacks of papers, losing sleep, having unproductive (but fun) study parties seems to be par for the course. It certainly isn’t something that disappears after high school graduation.
Principle of Poor Prioritization: I know what I really should be spending my time doing, but this other mindless task seems like a much better use of my time. This is not about online distractions, though that is a big factor for all of us. This is when a student decides to white out all of the mistakes in his/her notebook from throughout the semester because he or she thinks this will make studying easier. Rewriting notes can be a useful exercise if it involves some sort of processing/summarizing/grouping of ideas. Simply copying them over is a passive activity that feels like it should help, but probably is less productive than other tasks.
Principle of Confidence: I’m going to work on the things I am already good at doing to boost my confidence. This will better make me able to tackle the things I don’t understand. I’ve had conversations with students that do know what they need to work on, but avoid those things like the plague because learning new things is difficult. Revisiting strengths is occasionally a good idea, but again, it is not truly productive.

Figuring out how to shake students of following these guidelines is really what we need to work on. We need to not just just lecture them about getting organized, planning out stressful times, taking effective breaks, and being deliberate about all of these processes, but model how to do these things. My question is one of practicality though – what are the best ways to do this? Is the best way integrated as part of existing courses? (My gut says yes.) Is it about going back to pencil and paper planners? Is it about using technology to help with reminders, calendars, etc?

The thing that I find most difficult about discussing this is that it always turns into a conversation about avoiding procrastination. I agree that this would help…if our students weren’t already told this hundreds of times per year. The design problem that needs to be solved is: given that our students are stressed, how do we help them work through it? Furthermore, how do we make the most of our own experience as adults working through stress, but deliver that experience in a way that doesn’t start by telling students what they believe is wrong?

A tale of two gradebooks – my SBG journey continues

I realized this morning that I could look back at the assignments from my PowerSchool gradebook from a year ago and see the distribution of assignments I had by the end of the semester:

My grades were category based – 5% class-work, 10% homework completion, 10% portfolio, 60% unit tests, and 15% quizzes. This comprised 80% of the semester grade, and was the grade that students saw for the majority of the semester. A semester exam at the end made up the remaining 20%.

While I did enter some information about the homework assignments, my grade was just a reflection of how they completed it relative to the effort I expected them to make while working on it. No penalty for being wrong on problems, but a cumulative penalty developed over time for students tending not to turn it in. This, however, was essentially a behavior grade, and not an indication of what they were actually learning. The homework was the most frequent way for students to get feedback, and it did help students improve in what they were learning, but the completion grade was definitely not a measure of what they were learning at all. There were six quizzes that fit into my reassessment system. Not important enough to matter, I realize now with 20/20 hindsight.

The entire Standards-based-grading community shoots me a look saying ‘we told you so’, but only momentarily and without even a hint of snark. They know I am on their side now.

Here is a screen shot of the assignments in my grade-book as of this morning:

There is a clear indication of what my students have been working on here. With the exception of the portfolio, a student can look at this (and the descriptions I’ve included for each standard) and have a pretty good idea of what they did and didn’t understand over the course of the semester. They know what they should be working on before the semester exam next week. The parents can get a pretty good idea of what they are looking at as well. I knew making the change to standards based grading (SBG) made sense, but there have been so many additional reasons I am happy to have made the change that I really don’t want to go back to the old system.

I’ll do more of a post-game analysis of my SBG implementation in PowerSchool soon. I will be making changes and enhancing parts that I like about what I have done so far. I have to first make it through the busy time ahead of marking exams, submitting comments, and getting my life ready for the extended winter break that is peeking its beautiful head over the piles of reassessments on my desk. It is really satisfying to see that my students have weathered the transition to SBG beautifully. Their grades really do emphasize the positive aspects of learning that a pure assignments & points system blurs without thinking twice.

Crowdsourcing a learning-to-teach framework

After a good conversation with a friend that is getting started with teaching, I was thinking a bit about the process of learning to teach. Things that I obsessed about as a first year teacher come much more naturally now, but if you asked me what I needed to learn in the beginning, I would have babbled on like an idiot. Knowing what to focus on when everything is so new, not to mention feeling you aren’t good at any of it, you understand why it is so easy for students to shut down when we ask them to ‘be responsible’ without helping them understand what we mean. Our job as teachers is to provide students with a framework that will help them be successful in learning what we teach them.

You would hope that guidance in this would be an essential component of teacher preparation programs, but it often doesn’t, particularly in cases where observation is a box to be checked, not a pathway to improvement. There are many frameworks for observation, but I haven’t seen one that specifically gives guidance (or even a curriculum?) for what new teachers should be looking for when in a mentor teacher’s classroom. Most of the observation forms I’ve seen are in evaluating teachers for teacher quality. When I go to watch a colleague, I’m thinking about how I’m going to use what I see to improve what I do, not how to make them a better teacher. I know what I am looking for because I’ve had the keys to my classroom for a little while.

C’mon internet, let’s work together to create this and help our newbies. We were all new to this once, and there’s a lot that we may not realize we are thinking about after pulling out our hair and having teaching nightmares for so long. (Do they ever stop?)

To be clear, the goal is to start conversations between new teachers and their mentors, not put new teachers in a position to evaluate those who are being observed. We want to make the most of this time that is probably the most valuable teacher preparation tool outside of standing in front of a class yourself.

I’ve put a document designed to compile these ideas here:

So you’re a new teacher. What should you focus on this week?

Please add to the list and snarky-up the title. There may even be a better way to organize this so that it isn’t a big list that again serves only to intimidate. Maybe along the lines of Emergency Compliments?

Cell phone tracking, Processing, and computational thinking

I gave a survey to my students recently. My lowest score on any of the questions was ‘What I learn in this class will help me in real life.’ I’ve given this question before, and am used to getting less than optimal responses. I even think I probably had a higher score on this question than I have received previously, but it still bothers me that we are having this discussion. Despite my efforts to include more problem solving, modeling, and focusing on conceptual understanding related tasks over boring algorithmic lessons, the fact that I am still getting lower scores on this question compared to others convinces me that I have a long way to go.

I came up with this activity in response. It combines some of the ideas I learned in my Udacity course on robotic cars with the fact that nearly all my students carry cell phones. While I know many cell phones have GPS, it is my understanding that phones have used cell towers for a while to help with the process of locating phones. It always amazes me, for example, how my cell service immediately switches to roaming immediately when driving across the US-Canada border, even when I had a non-GPS capable phone.

My students know how to find distance using the distance formula and sets of coordinates, but they were intrigued by the idea of going backwards – if you know your distance from known locations, can you figure out your own location? The idea of figuring this out isn’t complicated. It can most easily be done by identifying intersections of circles as shown below:

One of my students recalled this method of solving the problem from what he saw in the movie Taken 2 , and was quickly able to solve the problem this way graphically in Geogebra. Most students didn’t follow this method though – the general trend was to take a guess and adjust the guess to reduce the overall error until the distances were as close to the given distances as possible.

I got them to also look at other situations – if only two measurements to known locations are known, where could the cell phone be located? They played around to find that there were two locations in this case. I again pointed out that they were following an algorithm that could easily be taught to a computer.

I then showed them a Processing sketch that went through this process. It is not a true particle filter that goes through resampling to improve the guessed location over time, but it does use the idea of making a number of guesses and highlighting the ones with the lowest error. The idea of making 300,000 random guesses and choosing the ones that are closest to the set of distances is something that computers are clearly better at than humans are. There are analytical ways of solving this problem, but this is a good way of using the computational power of the computer to make a brute force calculation to get an approximate answer to the question.

You can look at the activity we did in class here:
Using Cell Phones to Track Location

Who’s gone overboard modeling w/ Python? Part II – Gravitation

I was working on orbits and gravitation with my AP Physics B students, and as has always been the case (including with me in high school), they were having trouble visualizing exactly what it meant for something to be in orbit. They did well calculating orbital speeds and periods as I asked them to do for solving problems, but they weren’t able to understand exactly what it meant for something to be in orbit. What happens when it speeds up from the speed they calculated? Slowed down? How would it actually get into orbit in the first place?

Last year I made a Geogebra simulation that used Euler’s method to generate the trajectory of a projectile using Newton’s Law of Gravitation. While they were working on these problems, I was having trouble opening the simulation, and I realized it would be a simple task to write the simulation again using the Python knowledge I had developed since. I also used this to-scale diagram of the Earth-Moon system in Geogebra to help visualize the trajectory.

I quickly showed them what the trajectory looked like close to the surface of the Earth and then increased the launch velocity to show what would happen. I also showed them the line in the program that represented Newton’s 2nd law – no big deal from their reaction, though my use of the directional cosines did take a bit of explanation as to why they needed to be there.

I offered to let students show their proficiency on my orbital characteristics standard by using the program to generate an orbit with a period or altitude of my choice. I insist that they derive the formulae for orbital velocity or period from Newton’s 2nd law every time, but I really like how adding the simulation as an option turns this into an exercise requiring a much higher level of understanding. That said, no students gave it a shot until this afternoon. A student had correctly calculated the orbital speed for a circular orbit, but was having trouble configuring the initial components of velocity and position to make this happen. The student realized that the speed he calculated through Newton’s 2nd had to be vertical if the initial position was to the right of Earth, or horizontal if it was above it. Otherwise, the projectile would go in a straight line, reach a maximum position, and then crash right back into Earth.

The other part of why this numerical model served an interesting purpose in my class was as inspired by Shawn Cornally’s post about misconceptions surrounding gravitational potential and our friend mgh. I had also just watched an NBC Time Capsule episode about the moon landing and was wondering about the specifics of launching a rocket to the moon. I asked students how they thought it was done, and they really had no idea. They were working on another assignment during class, but while floating around looking at their work, I was also adjusting the initial conditions of my program to try to get an object that starts close to Earth to arrive in a lunar orbit.

Thinking about Shawn’s post, I knew that getting an object out of Earth’s orbit would require the object reaching escape velocity, and that this would certainly be too fast to work for a circular orbit around the moon. Getting the students to see this theoretically was not going to happen, particularly since we hadn’t discussed gravitational potential energy among the regular physics students, not to mention they had no intuition about things moving in orbit anyway.

I showed them the closest I could get without crashing:

One student immediately noticed that this did seem to be a case of moving too quickly. So we reduced the initial velocity in the x-direction by a bit. This resulted in this:

We talked about what this showed – the object was now moving too slowly and was falling back to Earth. After getting the object to dance just between the point of making it all the way to the moon (and then falling right past it) and slowing down before it ever got there, a student asked a key question:

Could you get it really close to the moon and then slow it down?

Bingo. I didn’t get to adjust the model during the class period to do this, but by the next class, I had implemented a simple orbital insertion burn opposite to the object’s velocity. You can see and try the code here at Github. The result? My first Earth – lunar orbit design. My mom was so proud.

The real power here is how quickly students developed intuition for some orbital mechanics concepts by seeing me play with this. Even better, they could play with the simulation themselves. They also saw that I was experimenting myself with this model and enjoying what I was figuring out along the way.

I think the idea that a program I design myself could result in surprising or unexpected output is a bit of a foreign concept to those that do not program. I think this helps establish for students that computation is a tool for modeling. It is a means to reaching a better understanding of our observations or ideas. It still requires a great amount of thought to interpret the results and to construct the model, and does not eliminate the need for theoretical work. I could guess and check my way to a circular orbit around Earth. With some insight on how gravity and circular motion function though, I can get the orbit right on the first try. Computation does not take away the opportunity for deep thinking. It is not about doing all the work for you. It instead broadens the possibilities for what we can do and explore in the comfort of our homes and classrooms.

Who’s gone overboard modeling in Physics? This guy, part I.

I’ve been sticking to my plan this year to follow the Modeling Instruction curriculum for my regular physics class. In addition to making use of the fantastic resources made available through the AMTA, I’ve found lots of ways to use Python to help drive the plow through what is new territory for me. I’ve always taught things in a fairly equation driven manner in Physics, but I have really seen the power so far of investing time instead into getting down and dirty with data in tables, graphs, and equations when doing so is necessary. Leaving equations out completely isn’t really what I’m going for, but I am trying to provide opportunities for students to choose the tools that work best for them.

So far, some have embraced graphs. Some like working with a table of data alone or equations. The general observation though is that most are comfortable using one to inform the other, which is the best possible outcome.

Here’s how I started. I gave them the Python code here and asked them to look at the lines that configure the program. I demonstrated how to run the program and how to paste the results of the output file into Geogebra, which created a nice visualization through this applet. Their goal through the activity was to figure out how to adjust the simulation to generate a set of graphs of position and velocity vs. time like this one:

Some used the graph directly and what they remembered from the constant velocity model (yeah, retention!) to figure out velocity and initial position. Others used the table for a start and did a bit of trial and error to make it fit. While I have always thought that trial and error is not an effective way to solve these types of problems, the intuition the students developed through doing came quite naturally, and was nice to see develop.

After working on this, I had them work on using the Python model to match the position data generated by my Geogebra Particle Dynamics Simulator. I had previously asked them to create sets of data where the object was clearly accelerating, so they had some to use for this task. This gave them the chance to not only see how to determine the initial velocity using just the position data, as well as use a spreadsheet intelligently to create a set of velocity vs. time data. I put together this video to show how to do this:

[wpvideo bQyM2woe].

It was really gratifying to see the students quickly become comfortable managing a table of data and knowing how to use computational tools to do repeated calculations – this was one of my goals.

The final step was setting them free to solve some standard Constant-Acceleration kinematics problems using the Python model. These are problems that I’ve used for a few years now as practice after introducing the full set of constant acceleration equations, and I’ve admittedly grown a bit bored of them.Seeing how the students were attacking them using the model as a guide was a way for me to see them in a whole new light – amazingly focused questions and questions about the relationship between the linear equation for velocity (the only equation we directly discussed after Day 1), the table of velocity data, and what was happening in position vs. time.

One student kept saying she had an answer for problem c based on equations, but that she couldn’t match the Python model to the problem. In previous classes where I had given that problem, getting the answer was the end of the story, but to see her struggling to match her answer to what was happening in her model was beautiful. I initially couldn’t do it myself either until I really thought about what was happening, and she almost scooped me on figuring it out. This was awesome.

They worked on these problems for homework and during the beginning of the next class. Again, some really great comments and questions came from students that were previously quiet during class discussions. Today we had a learning standard quiz on constant acceleration model questions, and then decided last night during planning was to go on to just extending the constant acceleration model to objects in free fall.

Then I realized I was falling back into old patterns just telling them that all objects in free fall near Earth’s surface accelerate downward at roughly 9.81 m/s^2. Why not give them another model to play with and figure this out? Here’s what I put together in Python.

The big plus to doing it this way was that students could decide whether air resistance was a factor or not. The first graph I showed them was the one at right – I asked whether they thought it could represent the position versus time graph for an object with constant acceleration. There was some inconsistency in their thinking, but they quickly decided as a group after discussing the graph that it wasn’t. I gave them marble launchers, one with a ping-pong ball, and another with a marble, and asked them to model the launch of their projectiles with the simulation. They decided what they wanted to measure and got right to it. I’m also having them solve some free fall problems using the gravity simulation first without directly telling them that acceleration is constant and equal to g. They already decided that they would probably turn off air resistance for these problems – this instead of telling them that we always do, even though air resistance is such a real phenomenon to manage in the real world.

A bit of justification here – why am I being so reliant on the computer and simulation rather than hands on lab work? Why not have them get out with stopwatches, rulers, Tracker, ultrasonic detectors, air tracks, etc?

The main reason is that I have yet to figure out how to get data that is reliable enough that the students can see what they have learned to look for in position and velocity data. I spent an hour working to get a cart on an inclined air track to generate reasonable data for students to use in the incline lab in the modeling materials from AMTA on constant acceleration, and gave up after realizing that the students would lose track of the overall goal while struggling to get the mere 1 – 2 seconds of data that my 1.5 meter long air track can provide. The lab in which one student runs and other students stand in a line stopping their stopwatches when the runner passes doesn’t work when you have a small class as I do. The discussions that ensue in these situations can be good, but I have always wished that we had more data to have a richer investigation into what the numbers really represent. The best part of lab work is not taking data. It’s not making repetitive calculations. Instead, it’s focusing on learning what the data tells you about the situation being measured or modeled. This is the point of spending so much time staring and playing with sets of data in physics.

I also find that continuing to show students that I can create a virtual laboratory using several simple lines of code demonstrates the power of models. I could very easily (and plan to) also introduce some random error so the data isn’t quite so smooth, but that’s something to do when we’ve already understood some of the fundamental issues. We dealt with this during the constant velocity model unit, but when things are a bit messier (and with straight lines not telling the whole picture) when acceleration comes into play, I’m perfectly comfortable with smooth data to start. Until I can generate data as masterfully as Kelly does here using my own equipment, I’m comfortable with the computer creating it, especially since they can do so at home when they think nobody is looking.

Most of all, I find I am excited myself to put together these models and play with the data to model what I see. Having answered the same kinematics questions many times myself, being able to look at them in a new way is awesome. Finding opportunities for students to figure out instead of parrot responses after learning lists of correct answers is the best part of teaching, and if simulations are the way to do this, I’m all for it. In the future, my hope is to have them do the programming, but for now I’m happy with how this experiment has unfolded thus far.

Simulations, Models, and the 2012 US Election

After the elections last night, I found I was looking back at Nate Silver’s blog at the New York Times, Five Thirty Eight.

Here was his predicted electoral college map:

…and here was what ended up happening (from CNN.com):

I’ve spent some time reading through Nate Silver’s methodology throughout the election season. It’s detailed enough to get a good idea of how far he and his team have gone to construct a good model for simulating the election results. There is plenty of description of how he has used available information to construct the models used to predict election results, and last night was an incredible validation of his model. His popular vote percentage for Romney was predicted to be 48.4%, with the actual at 48.3 %. Considering all of the variables associated with human emotion, the complex factors involved in individuals making their decisions on how to vote, the fact that the Five Thirty Eight model worked so well is a testament to what a really good model can do with large amounts of data.

My fear is that the post-election analysis of such a tool over emphasizes the hand-waving and black box nature of what simulation can do. I see this as a real opportunity for us to pick up real world analyses like these, share them with students, and use it as an opportunity to get them involved in understanding what goes into a good model. How is it constructed? How does it accommodate new information? There is a lot of really smart thinking that went into this, but it isn’t necessarily beyond our students to at a minimum understand aspects of it. At its best, this is a chance to model something that is truly complex and see how good such a model can be.

I see this as another piece of evidence that computational thinking is a necessary skill for students to learn today. Seeing how to create a computational model of something in the real world, or minimally seeing it as an comprehensible process, gives them the power to understand how to ask and answer their own questions about the world. This is really interesting mathematics, and is just about the least contrived real world problem out there. It screams out to us to use it to get our students excited about what is possible with the tools we give them.

Automating conference scheduling using Python

I’ve always been interested in the process of matching large sets of data to a set of constraints – apparently the Nobel committee agreed this past week in awarding the economics prize. The person in charge of programming at my school in the Bronx managed to create an algorithm that generated a potential schedule for over four thousand students given student requests and needs. There was always some tweaking that needed to be done at the end to make it work, but the fact that the computer was able to start the process always amazed me. How do you teach a computer to do this sort of matching in an efficient way?

This has application within my classroom as well – generating groups based on ability, conflicting personalities, location – all complex situations that required time and attention to do correctly. In the end though, this is the same problem as arranging the schedules. It’s easy to start with a random arrangement and then make adjustments based on experience. There has to be a way to do this in an automated way that teaches the computer which placements work or do not. Andy Rundquist does this using genetic algorithms – I must know more about how he does it, as this is another approach to this type of problem.

This became a more tangible challenge for me to attempt to solve last year when I saw that the head of school was doing the two days of parent-teacher conference scheduling by hand. This is a complex process given the following constraints he was working to fulfill:

Parent preferences for morning/afternoon conference times.
Consecutive conference times for parents that had siblings so that the amount of time parents had to wait around was minimized.
Balanced schedules between the two days for all teachers.
Teachers with children had breaks in their schedule to attend conferences of their children.

This was apparently a process of 4 – 5 hours that sometimes required starting over because he discovered that the schedule he had started putting together was over constrained and could not meet all requirements. During this process, however, he had figured out an algorithm for what was most likely to work. Schedule the families with the largest number of children first, and work down the list in order of decreasing size. Based on the distribution of younger vs. older children in the school, start by scheduling the youngest children in a family first, and move to the older ones. Save all families with single children for last.

Hearing him talk about this process was interesting and heartbreaking at the same time – he works incredibly hard on all aspects of his job, and I wanted to provide some way to reduce the requirements of at least this task on his schedule. I was also looking for a reason to really learn Python, so this challenge became my personal exercise in problem based learning.

It took a while to figure out all of the details, but I broke it down into stages. How do you input the family data based on how it is already stored by the front office? (I didn’t want to ask the hard-working office staff to reformat the data to make it easier for me – this was supposed to make things easier on everyone.) How do you create a structure for storing this data in Python? How do you implement the algorithm that the head of school used and balance it with the idea of fairness and balance to all families and teachers?

Over the following few months, I was able to piece it together. It was, needless to say, a really interesting exercise. I learned how to ask the right questions that focused on the big picture needs of the administration, so that I could wrestle with the details of how to make it happen. The students learned that I was doing this (“Mr. Weinberg is using his robots to schedule conferences!”) and a few wanted to know how it worked. I have posted the code here as a gist.

I put in more than the 4-5 hours required to do this by hand. It was a learning experience for me. It also paid serious dividends when we needed to schedule conferences again for this year. We wanted to change the schedule slightly to be one full day rather than two half days, and it was a simple task adjusting the program to do this. We wanted to change the times of conferences so that the lower and upper schools had different amounts of time for each, rather than being a uniform twenty minutes each. (This I was not able to figure out before we needed conferences to go out, but I see a simple way to do it now.)

The big question that administration was about the upper school conferences. Last year we had seven different rooms for simultaneous conferences, and the question was whether we could reduce the number to five. I ran the program with five rooms a number of different times, and it was unable to find a working schedule, even with different arrangements and constraints. It was able to find one that worked with six rooms though, which frees administrators from needing to be in individual conference rooms so that they can address issues that come up during the day. Answering that question would not have been possible if scheduling was done by hand.

The question of using computers to automate processes that are repetitive has been in my head all this year. I’ve come to recognize when I am doing something along these lines, and try to immediately switch into creating a tool in Python to do this automatically. On the plane during a class trip last week, we needed to arrange students into hotel rooms, so I wrote a program to do this. I used it this week to also arrange my Algebra 2 students in groups for class. Generating practice questions for students to use as reassessment? I always find myself scrambling to make questions and write them out by hand, but my quiz generator has been working really well for doing this. Yesterday I had my first day of generating quizzes based on individual student needs.

The students get a kick out of hearing me say that I wrote a Python program to do XXX or YYY, and their reactions certainly are worth the effort. I think it just makes sense to use programming solutions when they allow me to focus on more important things. I have even had some success with getting individual students to want to learn to do this themselves, but I’ll write more about that later.

First day of Geometry proofs – Refining my process

Last year, I figured out about a week or two after the first introduction to proofs in Geometry last year that I should have started with a more clear connection to the ideas we had been working on in the classes before. We did a progression of logical statements, conditional statements, working on biconditionals as definitions, and then the laws of detachment and syllogism. I realized then that I never made strong references to these concepts and how they all fit together – I just hoped that the students would see how the proofs were built out of these ideas without formally telling them as such.

This year I was much more explicit in how the ideas fit together, particularly by showing a paragraph proof as a series of conditional statements with true hypotheses. I was really happy with the results. I created two videos to use as part of the instruction:
[wpvideo 4LLQ8TEa]

Students watched the video and then worked on identifying the properties of equality and congruence being applied in a few different situations, and completing statements given that a particular property is being applied. This led to some great conversations about subtle differences between the transitive property of equality and the substitution property of equality. (‘If a = 5 and b = 5, then a = b’ is the latter, not the former. This assumes only one property is being applied at a time, of course.)

Once I was satisfied with their progress, I sent them on to watch this video:
[wpvideo EY4lUecB]

Some students immediately took the equation, solved it, and said they were done. This led to more good discussions about the purpose of this lesson. We already knew how to solve equations – what was new this time was justifying each step using a property. It was an opportunity to push these students to then focus on what was happening in each step and not so much on the algebra that they did quickly. Once I was convinced that they did understand what was going on in each step of the video, I had them move to either some problems with the steps written out, missing only the justifications (for weaker students), and for others, full algebraic problems that they do from start to finish.

The thing I did differently here (and which was made easier by the magic of video) is emphasizing that the different steps in a proof are really all either conditional statements, statements of fact (the given information), and possibly steps of arithmetic simplification. Each line should be connected to the previous one in the form of a conditional statement. I have said it in the past, but never explicitly written it out each time so that the students think of it this way.

The whole point of doing things this way is so that students are not introduced to two concepts simultaneously: writing steps in a proof, and proving a statement deductively from scratch. Having a good sense of algebra, this lesson focused on introducing students to the process first. Next time we will move on to actually finding and proving theorems about line segments, with the idea that they already have a basic sense for how different thoughts can be linked together logically in a proof. I am hoping that being this deliberate will pay off – this was definitely the smoothest this lesson has ever gone for me.

Differentiation Rules – Making it Interactive

I always struggle during the days spent going over differentiation rules. The mathematician in me says the students need to see where the rules come from so that they aren’t just a recipe. On the other hand, I see students glazing over a bit with notation and getting lost in the midst of the overall goal: how do we find shortcuts for finding derivative functions outside of using the limit definition every time?

I have also tried going through the derivations in class and having them just watch and see the progression on their own, without copying things down. Some compulsively copied despite my repeated requests not to do so – I think it was a situation of seeing copying notes down as an alternative to really digging in to what was actually going on. It’s mindless to copy down notes, a great alternative to actually going through the steps of understanding.

Last year I made videos of the derivations and asked students to watch them outside of class in a one-off attempt at flipping. That didn’t work – students said they watched but ‘didn’t get it’, so my attempt to quiz them when they arrived in class was a bust.

This is my compromise this year: for finding the derivative of a constant, a constant times a function, and the power rule, students will be guided through what has essentially my lesson plan for previous lessons. Sums of functions, products, and quotients will be given first as applications of the limit rules, but the details of getting from the start to the finish will be kept as an exercise for later.

See my handout for today here:
03 – CW – Differentiation Rules

Thank you to Patrick Honner and Dan Anderson for their comments pushing me on this.

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