Page 18 – iteration, making, building, and coding in education

Moving in circles, broom ball, and Newton’s cannonball

In physics today, we began our work in circular motion. I started by asking the class three questions:

When do you feel ‘heaviest’ on an elevator? When do you feel ‘lightest’?
When do you feel ‘heaviest’ on an airplane? When do you feel ‘lightest?’
When do you feel ‘heaviest’ on a swing? Lightest?

We discussed and shared ideas for a bit. I tried my hardest not to nudge anyone toward thinking they were right or wrong, as this was merely a test for intuition and experience. We then played a few rounds of circular ‘book’-ball, a variation of the standard modeling curriculum activity of broom ball from the modeling curriculum in which students use a textbook to push a ball in a circular path on a table. The students could not touch the ball with anything other than a single book at one time. A couple of students quickly established themselves as the masters:

I then had students draw the ball in three configurations as well as the force and velocity vectors for the ball at those locations. Students figured the right configuration much more quickly than in previous years:
I think some of our work emphasizing the perpendicular nature of the normal force on surfaces in previous units may have helped on this one.

We then took a look at some vertical circles and analyzed them using what we knew from the last unit on accelerated motion together with our new intuition about circular motion.

We finished the class playing with my most recent web-app, Newton’s Cannonball. We haven’t discussed orbits at all, but I wanted them to get an intuitive sense of the concept of how a projectile could theoretically go into orbit. This was the latest generation of my parabolas to orbits exploration concept.

A new year in the cold: how can we play with this?

My wife and I spent our winter break exploring two places: Tibet and Harbin, China. Harbin is the location of the Harbin Ice and Snow Festival, where they do some impressive work building with ice and snow to show off for the rest of the world.

We spent our first day in Harbin’s sub zero temperatures wandering around on the frozen river. The locals have confronted the fact that their local river freezes over with a simple question: How do we play with this?

They have a number of simple answers.

From bumper cars, to ice slides, to team-driven ice sleighs…

…to ice bicycles, complete with a hilariously ineffective brake pedal:

The locals in Harbin have taken the brutal reality of their sub-zero temperatures (in both Fahrenheit and Celsius) and created some creative channels through which to enjoy that cold environment in ways that are enjoyable, cooperative, and unique.

The question “how do we play with this?” has become the organizing question for lessons in my classroom. I want to give students chances to explore, have fun, and work together in the process. Though I don’t always do so, I think the pursuit is a worthy one.

This blog has been silent for a while, not for any negative reasons, but because the realities of organizing for my classroom and school have compelled me to put my energy in places other than structured reflection. I hope to do more sharing, more reflection, and give more appreciation toward the members of my personal learning network in the latter half of this year. I wish everyone a productive, enjoyable, and satisfying 2014.

Just shut up and work with us, Weinberg

I have an issue with talking too much in class. I think many of us do.

I’ve already done some focused work identifying what my students need me to show them for a given topic, and it’s a lot less than I initially think. After a conversation with some smart educators, I decided to commit this week to not do whole class instruction unless it was absolutely necessary.

Sometimes I confuse necessity with convenience. The problem is that it’s always convenient to do whole class instruction. You look out and see eyes staring at you, and it seems at the moment to be maximally efficient to communicate to the entire group at once. The quality of that attention is never what it seems.

In my biggest class, I’ve been continuing to put direct instruction into videos. As I’ve written previously, these are videos (three minutes or so) that have the information distilled down to small chunks. In doing this, I get around to every student and make sure they are somehow engaging with that video through writing down important information, trying the problem being demonstrated, or completing the challenge I usually put at the end. It’s impossible for me to be instructing at the front of the class (or anywhere for that matter) and be aware of what every student is doing. With the video at every student’s seat, I can be there. I can ask them questions one-on-one to see what they understand. I can make notes of the students that are struggling. I can assess every student at some point while I walk around, leave alone those that are doing just fine without my dictating their attention, and focus on those that need more guidance.

This increased time away from blabbing at the front of the room means more assessment time. The class starts with a quick quiz (1-2 questions) that I can get back to students during the period. I can give every student some bit of feedback, and it ensures that I have a conversation with every single student during the class. That is awesome. It means I can ask higher level questions of the stronger students and push them forward. It means I can see what students are writing down within seconds of doing so.

Though I occasionally think to myself that the reason this works is because my students are well behaved and will stay on task when I am not directly focused on them, I don’t think this is why it has been successful. I’m in the middle of my students (rather than in one location) the whole time. I can see what they are all doing. If they do get off task, they know that I know if because chances are I’ll be there in a minute or so. The class is noticeably less structured, and I don’t feel as productive as I think I would if I was marching through a lesson plan. This is more a reflection of how I now have a more realistic awareness of how my students are doing with the material, rather than in ten minute chunks of independent work between lecture.

The students benefit most from interacting with each other. They do occasionally need help from me one-on-one, but the nature of that help varies greatly between students. I can give that help when I’m not spending so much time talking. The inverse is more powerful there – I can’t give that help if I’m talking too much.

I decided to give students a quick exit survey on whether they liked the new format, whether they wanted to go back, or whether they wanted something different from both classroom structures. Here’s what they said:

I’ve gotten this sort of strong message before, but I unfortunately go back to the old ways, for the old reasons. It’s easier to talk. It’s easier to do a developmental lesson. It’s easier to ask a question and conclude from a one or two student non-random sample that the class gets it. It just isn’t necessarily what works best for students. I need to keep that in mind.

Proofs in Geometry – The Modification Continues…

Two statements of interest to me:

I get more consistent daily hits on my blog for teaching geometry proofs than anything else. Shiver.
Dan Meyer’s recent post on proofs in Geometry gets to the heart of what bothers me about teaching proofs at all. Double shiver.

These statements have made me think about my approach in doing proofs with students in my 9th grade course, which has previously been a geometry course, but is morphing into something slightly different in anticipation of our move to the IB program. I like the concept of teaching proofs because I force students to confront the idea that there’s a difference between things they know must be true, might be true, and will never be true. I started the unit asking the class the following questions:

Will the sun rise tomorrow?
Will student A always be older than her younger sister?
Will the boys volleyball team win the tournament this weekend>

The clear difference between these questions was also clear to my students. The word ‘obviously’ came up at least once, as expected.

The idea of proving something that is obvious is certainly an exercise of questionable purpose, mostly because it confines student thinking in the mould of classroom mathematics. As geometry teachers, we do this as a scaffold to help students learn to write proofs of concepts that are not so obvious. The downside is the inherent lack of perplexity in this process, as Dan points out in his post. The rules of math that students routinely apply to solve textbook or routine problems already fit in this ‘obvious’ category either from tradition (‘I’ve done this since, like, forever’) or from obedience (‘My teacher/textbook says this is true, and that’s good enough for me.’)

I usually go to Geogebra to have students discover certain properties to be true, or give a quick numerical example showing why two angles supplementary to the same angle are congruent. They get this, but have a sense of detachment when I then ask them to prove it using the properties we reviewed in previous lessons. It seems to be very much related to what Kate Nowak pointed out in her comment to Dan’s post. Geometry software or numerical examples show something to be so obvious that proof isn’t necessary, so why circle back to then use the rules of mathematics to prove it to be true?

I had an idea this afternoon that I plan to try tomorrow to close this gap.
I wrote earlier about using spreadsheets with students to take some of the abstraction out of translating algebraic expressions. Making calculations with variables in the way a spreadsheet does shows very clearly the concept of variables, and also doing arithmetic with them. My idea here is to use a spreadsheet this way:

My students know that they should be able to change what is in the black cells, and enter formulas in the red cells so that they change based on what is in the black cells only. In doing this, they will be using their algebraic rules and geometric definitions to complete a formula. This hits the concrete examples I mentioned above – a 25 degree angle complementary to an angle will always be congruent to a 25 degree angle complementary to that same angle. It also uses the properties (definition of a complementary angle, subtraction property of equality, definition of congruence) to suggest the relationship between those angles using the language and structure of proof, which comes next in class.

Here is the spreadsheet file I’ve put together:
02 – SPR – Congruent Angles

I plan to have them complete the empty cells in this spreadsheet and then move on to filling in some reasons for steps of more formal proofs of these theorems afterwards, as I have done previously. I’d like to think that doing this will make it a little more clear how the observations students have relate to the properties they then use to prove the theorems.

I’d love you to hack away at my idea with feedback in the comments.

Reassessment Web-App Update

I wrote last May about the difficulties I was having with doing reassessments efficiently. The short story – collecting reassessment requests, printing them, and distributing them to students was not scaling well. I had shared my progress on making a web application using Python to do this and was really excited to continue working on it.

After a summer of work and putting it into action this fall, Phases 1 and 2 are pretty much complete. I’m really happy with how it has been working for me. I host it on my personal laptop in the classroom and share its IP address with students so they can access their quizzes.

You can check out a mildly sandboxed version here:
http://quiz.evanweinberg.org/main/

UPDATE Mar. 2016: I’ve taken down the application to save memory on my hosting server. Write me if you are interested in learning more.

and the code is posted at Github:
https://github.com/emwdx/reassess

I took out a number of my questions (since students do occasionally read my blog) and made it so images can’t be uploaded. I hear that might be a security risk.

Some highlights:

Questions (with or without images) can be added, edited, and browsed all through a web interface.
Students can be assigned quizzes individually or through a page for the class they are in. They can also all be given different questions, which helps in my class that has students fairly close together.
Students each have their own url that can be bookmarked for easy access later.
The teacher view of the entire class has a link for each student that shows the quiz questions (and answers, if they are in the database) for easy grading.

What hasn’t been done:

Authentication for students and the admin side. Right now it’s all open, which bothers me a little, but my access log doesn’t show that this is being abused.
A way to capture their work digitally for each retake. I still have a pile of half-size A4 papers on my desk, and have to grade them while also having the answer page open. That isn’t the end of the world, but after my recent obsession with collecting as much student work as I can through a web interface, it’s something I’d like to have as an option. Students tend to lose these papers, and these are the formative assessment moments I’d love for them to include in their portfolios. Digital is clearly the way to go.
Randomization (phase 3 of my master plan), but in two different ways. I’m still manually choosing questions for students. I kind of want to keep it that way, since some students I do want to give different questions. But then I sometimes don’t – I’d rather it just choose questions from specific standards and students get the luck of the draw. I need an option that lets me waffle on this.
Question history – i.e. knowing which questions a student has been assigned, and integrating this into the program smoothly. This function is built into the database already, and won’t require a lot of work to make it happen, but I haven’t done it. Sorry.

There are a number of bugs features that still need to be worked out, but I’m aware of them all and know how to work through them when I have a bunch of students taking quizzes.

The most powerful aspect of having this working is that I can easily assess the whole class at the whole time on different questions if I want them to be different. I’ve been doing this at the beginning of the class this semester, and it increases the amount of time I spend talking to each student about their work regularly. Since student initiated reassessment still isn’t as widespread as I want it to be, I’ve started having students request which quiz they want to have in class the night before. They know it’s coming, and can get help or prepare in advance, rather than using their valuable lunch or after school time. More on that later.

Let me know if you’re interested in using this with your own class – it’s pretty portable and can be adapted without too much of a headache to different situations.

Computation & CAPM – From Models to Understanding

I wrote last spring about beginning my projectile motion unit with computational models for projectiles. Students focused on using the computer model alone to solve problems, which led into a discussion of a more efficient approach with less trial and error. The success of this approach made me wonder about introducing the much more simpler particle model for constant acceleration (abbreviated CAPM) using a computational model first, and then extending the patterns we observed to more general situations

We started the unit playing around with the Javascript model located here and the Geogebra data visualizer here.

The first activity was to take some position data for an object and model it using the CAPM model. I explained that the computational model was a mathematical tool that generated position and velocity data for a particle that traveled with constant acceleration. This was a tedious process of trial and error by design.

The purpose here was to show that if position data for a moving object could be described using a CAPM model, then the object was moving with constant acceleration. The tedium drove home the fact that we needed a better way. We explored some different data sets for moving objects given as tables and graphs and discussed the concepts of acceleration and using a linear model for velocity. We recalled how we can use a velocity vs. time graph to find displacement. That linear model for velocity, at this point, was the only algebraic concept in the unit.

In previous versions of my physics course, this was where I would nudge students through a derivation of the constant acceleration equations using what we already understood. Algebra heavy, with some reinforcement from the graphs.

This time around, my last few lessons have all started using the same basic structure:

Here’s some graphical or numerical data for position versus time or a description of a moving object. Model it using the CAPM data generator.
Does the CAPM model apply? Have a reason for your answer.
If it does, tell me what you know about its movement. How far does it go? What is its acceleration? Initial velocity? Tell me everything that the data tells you.

For our lesson discussing free fall, we started using the modeling question of asking what we would measure to see if CAPM applies to a falling object. We then used a spark timer (which I had never used before, but found hidden in a cabinet in the lab) to measure the position of a falling object.

They took the position data, modeled it, and got something similar to 9.8 m/s² downward. They were then prepared to say that the acceleration was constant and downwards while it was moving down, but different when it was moving up. They quickly figured out that they should verify this, so they made a video and used Logger Pro to analyze it and see that indeed the acceleration was constant.

The part that ended up being different was the way we looked at 1-D kinematics problems. I still insisted that students use the computer program to model the problem and use the results to answer the questions. After some coaching, the students were able to do this, but found it unsatisfying. When I assigned a few of these for students to do on their own, they came back really grumpy. It took a long time to get everything in the model to work just right – never on the first try did they come up with an answer. Some figured out that they could directly calculate some quantities like acceleration, which reduced the iteration a bit, but it didn’t feel right to them. There had to be a better way.

This was one of the problems I gave them. It took a lot of adjustment to get the model to match what the problem described, but eventually they got it:

Once the values into the CAPM program and it gave us this data, we looked at it together to answer the question. Students started noticing things:

The maximum height is half of the acceleration.
The maximum height happens halfway through the flight.
The velocity goes to zero halfway through the flight.

Without any prompting, students saw from the data and the graph that we could model the ball’s velocity algebraically and find a precise time when the ball was at maximum height. This then led to students realizing that the area of the triangle gave the displacement of the ball between being thrown and reaching maximum height.

This is exactly the sort of reasoning that students struggle to do when the entire treatment is algebraic. It’s exactly the sort of reasoning we want students to be doing to solve these problems. The computer model doesn’t do the work for students – it shows them what the model predicts, and leaves the analysis to them.

The need for more accuracy (which comes only from an algebraic treatment) then comes from students being uncomfortable with an answer that is between two values. The computation builds a need for the algebraic treatment and then provides some of the insight for a more generalized approach.

Let me also be clear about something – the students are not thrilled about this. I had a near mutiny during yesterday’s class when I gave them a standards quiz on the constant acceleration model. They weren’t confident during the quiz, most of them wearing gigantic frowns. They don’t like the uncertainty in their answers, they don’t like lacking a clear roadmap to a solution, they don’t like being without a single formula they can plug into to find an answer. They said these things even after I graded the quizzes and they learned that the results weren’t bad.

I’m fine with that. I’d rather that students are figuring out pathways to solutions through good reasoning than blindly plugging into a formula. I’d rather that all of the students have a way in to solving a problem, including those that lack strong algebraic skills. Matching a model to a problem or situation is not a complete crap shoot. They find patterns, figure out ways to estimate initial velocity or calculate acceleration and solidify one parameter to the model before adjusting another.

Computational models form one of the only ways I’ve found that successfully allows students of different skill levels to go from concrete to abstract reasoning in the context of problem solving in physics. Here’s the way the progression goes up the ladder of abstraction for the example I showed above:

The maximum height of the ball occurred at that time. Student points to the graph.
The maximum height of the ball happened when the velocity of the ball went to zero in this situation. I’ll need to adjust my model to find this time for different problems.
The maximum height of the ball always occurs when the velocity of the ball goes to zero. We can get this approximate time from the graph.
I can model the velocity algebraically and figure out when the ball velocity goes to zero exactly. Then we can use the area to find the maximum height.
I can use the algebraic model for velocity to find the time when the ball has zero velocity. I can then create an algebraic model for position to get the position of the ball at this time.

My old students had to launch themselves up to step five of that progression from the beginning with an algebraic treatment. They had to figure out how the algebraic models related to the problems I gave them. They eventually figured it out, but it was a rough slog through the process. This was my approach for the AP physics students, but I used a mathematical approach for the regular students as well because I thought they could handle it. They did handle it, but as a math problem first. At the end, they returned to physics land and figured out what their answers meant.

There’s a lot going on here that I need to process, and it could be that I’m too tired to see the major flaws in this approach. I’m constantly asking myself ‘why’ algebraic derivations are important. I still do them in some way, which means I still see some value, but the question remains. Abstracting concepts to general cases in physics is important because it is what physicists do. It’s the same reason we should be modeling the scientific method and the modeling process with students in both science and math classes – it’s how professionals work within the field.

Is it, however, how we should be exposing students to content?

Programming and Making Use of Structure in Math

A tweet from James Tanton caught my eye last night:

Quickie: What is the sum of all products in a 10-by-10 multiplication table? What is sum of all products axbxc in 10-by-10-by-10 table?

— James Tanton (@jamestanton) October 13, 2013

Frequent readers likely know about my obsession with playing around the borders of computational thinking and mathematical reasoning. This question from James has some richness that I think brings out the strengths of considering both approaches quite nicely. For one of the few times I can remember since starting my teaching career, I went to a computational solution before analyzing it analytically.

A computational approach is pretty simple. In Python:

sum = 0 for i in range(1,11): for j in range(1,11): sum += i*j print(sum)

…and in Javascript:
sum = 0 for(i=1;i

The basic idea is the same in both languages. We iterate over each number in the first row and column of the multiplication table and add them up. From a first look, one could call this a brute force way to a solution, and therefore not elegant from a mathematical standpoint.

Taking this approach does, however, reveal some of the underlying mathematical structure that is needed to resolve this using other techniques. The sequence below is exactly how I analyzed the problem once I had written the program to solve it:

For a single row of the table, we are adding together the elements of that row. Instead of adding the individual elements together one by one, we could instead think about finding the sum of the elements of a single row, and then add together all of the rows. For example: \(1 + 2 + 3 + ... + 10 = 55\). This is a simple arithmetic series.
Each row is the same as the row before it, aside from each element being multiplied by the first element in the row. Every row's sum therefore is being multiplied by the numbers in the first column of the table. \(1(1+2+3...+10)+2(1+2+3...+10)+3(1+2+3+...+10)+...+10(1+2+3+...+10)\).
Taking this one step further, this is equivalent to the sum of that first row multiplying the sum of the first column: \((1 + 2 + 3 + ... + 10)(1 + 2 + 3 + ... + 10)\). In other words, the answer to our problem is really the square of the sum of that first row (or column), or 55*55.

I bring up this problem because I think it suggests a useful connection between a practical method of solving a problem, and what we often expect in the world of classroom mathematics. This is clearly a great application of concepts behind a traditional presentation of arithmetic series, and a teacher might give this as part of such a unit to see if students are able to see the structure of the arithmetic series formulas within it.

My question is what a teacher does if he or she presents this problem and the students don't make that connection. Is the next step a whole class discussion about how to proceed? Is it a leading question asking how arithmetic series applies here? This, by the way, zaps the whole point of the activity if the goal was to see if students see that underlying structure based on what they already know. Once this happens, it becomes yet another 'example' presented to the class.

I wonder what happens if a computer/spreadsheet solution is consistently recognized throughout the class as a viable tool to investigate problems like this. A computer solution is really nothing more than an abstraction of the process of adding the numbers together one by one. If a student did actually do this by hand, we'd groan and ask if they thought there was a better way, and the response inevitably is 'yes, but I don't know a better way'. In the way I found myself thinking about this problem last night, I started from the computational method, discovered the structure from those computations, and then found a path toward a more elegant solution using algebraic techniques.

In other words, I made use of the structure of my program to identify an analytical approach. Contrast this with a more traditional approach where we start with an abstract definition of an arithmetic series (by hand), do practice problems (by hand) and once we understand how it works, use computational shortcuts.

The consistent power that I see in approaching and developing ideas with students from a computational standpoint first is not that it often makes it easier to find an answer, though that can be a good thing when the goal is to find an answer. Computational methods can make it easy to change things around and generalize a problem - what Polya termed generalization. It's easy to change the Javascript program to this and ask what multiplication table it models:

sum = 0 for(i=5;i

Computation makes the process of finding a more elegant way seems much more natural - in the best situations, it builds intellectual need for an easier way. It is arbitrary to say that a student should be able to do a problem without a calculator. Computational tools demand that we find a more compelling reason to solve problems by hand if computers are able to do them rapidly once they are set up to solve them through programming. It is a realistic motivation to show that an easier way speeds up finding a solution to a problem by a factor of 10. It means less waiting for a web page to load or an image to post.

The language of mathematics is difficult enough to throw in the additional complications of computer language syntax. I fully acknowledge that this is a hurdle. I also think, however, that this syntax is more closely related to the concepts that we are trying to teach our students (3*x is three times x) than we sometimes think. The power of computer programming to be a bridge between the hand calculations that our students do and the abstractions of the mathematical content we teach is too great to ignore.

Intermediate Value Theorem & Elevators

I’ve used the elevator analogy with the intermediate value theorem before, but only after talking students through the intermediate value theorem first. This time, I took them through the following thought experiment first:

Step 1:

You enter the elevator on floor 2. You close your eyes and keep them closed until you arrive at floor 12, twenty seconds later.

Questions for discussion:

At approximately what time was the elevator located at floor 7? How do you know? What assumptions are you making?
Was there a time when the elevator was at floor 3? Floor 8? How do you know?
Were you ever at floor 13? How do you know? Are you really sure?

Step 2:

Another day, you again enter the elevator on floor 2. You again keep your eyes closed, but another person gets on from some floor other than floor 2. You keep your eyes closed. The other person leaves the elevator at some point. After 60 seconds, you are on floor 12, and you open your eyes.

Questions:

Was there a time at which the elevator was at floor 7? How do you know?
Was there a time at which the elevator was at floor 13? How do you know?
What was the highest floor at which you can guarantee the elevator was located during the minute long trip? The lowest floor?

Step 3

On yet another day, you are once again entering the elevator at floor 2 to go to floor 12. You close your eyes, same story as before. Another person gets on the elevator and leaves. This time, however, you open your eyes just long enough to see that the person leaves the elevator at floor 15. As before, the entire trip takes 60 seconds.

Questions:

Was there a time at which the elevator was at floor 7?
Was there a time at which the elevator was at floor 13? How do you know?
Make a list of all of the floors that you can guarantee that the elevator could have stopped at during the 60 second trip.
Can you guarantee that the elevator was never located at floor 17?

We then visited the driving principle to why we can do this thought experiment: why can we come to these conclusions without opening our eyes in the elevator? What is it about our experiences in elevators that makes this possible?

My students were primed to bring up continuity given that they worked through the concepts during the previous class. That said, there were quite a few lights that went on when I asked what it would be like to ride in a discontinuous elevator. Skipping floors, feeling the elevator move upwards and then arriving at a floor lower than where we started, or arriving at different floors just from closing or opening the doors.

Once we were comfortable with this, I threw the standard vocabulary of the intermediate value theorem:

Suppose f(x) has a maximum value M and a minimum value L over an interval [a,b]. There exists a value c in [a,b] such that L≤f(c)≤M as long as…

…and I left it there, hanging in the air until a student filled the silence with the condition of continuity over [a,b]. This was also a great time to introduce the idea of an existence theorem – it tells you that a mathematical object exists, and might give you some information on where to find it, but won’t definitively tell you exactly where it is located. Fun stuff.

We then talked about other examples of functions that are or are not continuous. Students brought up crashing into a wall after moving at a non-zero velocity. I also have this group of students the following period for physics, so I brought up what the velocity versus time graph actually looks at when you zoom in to the time of impact. (I like that this wasn’t a cognitive stretch for them given their experience zooming in on data on their calculators and graphs from Logger Pro.) The student that brought this up quickly argued himself back from saying that this was truly discontinuous.

This was a fun activity, and I’m glad I went through it. The concept of IVT is fairly intuitive, but we often present it in a way that doesn’t emphasize why it is special. In previous years, I started with the graph of a polynomial function bouncing up and down, asked students for the max/minimum value, and then asked them to identify whether they could do this for any value in the range between the maximum and minimum. They could, but never really saw the point of why that was special. Forcing them to imagine closing their eyes, limiting the information available to them, and then seeing how far they could take that limited knowledge made a difference in how this felt on the teaching end. I’ve seen some pretty good responses on my assessments of this concept as well, so it seems to have done some good for the students as well. (Phew!)

Math Caching and Immediately Useful Teaching Data

Last July, I posted a video in which I showed how to create a local, customized version of the Math Caching activity that can be found here.

I was inspired to revisit the idea last weekend reading Dan Meyer’s post about teacher dashboards. The part that got me thinking, and that stoked the fire that has been going in my head for a while, is identifying the information that is most useful to teachers. There are common errors that an experienced teacher knows to expect, but a new teacher may not recognize is common until it is too late. Getting a measure of wrong answers, and more importantly, the origin of those wrong answers, is where we ideally should be making the most of the technology in our (and the students’) hands. Anything that streamlines the process of getting a teacher to see the details of what students are doing incorrectly (and not just that they are getting something wrong) is valuable. The only way I get this information is by looking at student work. I need to get my hands on student responses as quickly as I can to make sense of what they are thinking.

As we were closing in on the end of an algebra review unit with the ninth graders this week, I realized that the math cache concept was good and fun and at a minimum was a remastering of the review sheet for a one-to-one laptop classroom. I came up with a number of questions and loaded it into the Python program. When one of my Calculus students stopped in to chat, and I showed her what I had put together, I told her that I was thinking of adding a step where students had to upload a screenshot of their written work in addition to entering their answer into the location box. She stared at me and said blankly: ‘You absolutely have to do that. They’ll cheat otherwise.’

While I was a bit more optimistic, I’m glad that I took the extra time to add an upload button on the page. I configured the program so that each image that was uploaded was also labeled with the answer that the student entered into the box. This way, given that I knew what the correct answers were, I knew which images I might want to look at to know what students were getting wrong.

This was pure gold.

Material like this was quickly filling up the image directory, and I watched it happening. I immediately knew which students I needed to have a conversation with. The answers ranged from ‘no solution’ to ‘identity’ to ‘x = 0’ and I instantly had material to start a conversation with the class. Furthermore, I didn’t need to throw out the tragically predictable ‘who wants to share their work’ to a class of students that don’t tend to want to share for all sorts of valid reasons. I didn’t have to cold call a student to reluctantly show what he or she did for the problem. I had their work and could hand pick what I wanted to share with the class while maintaining their anonymity. We could quickly look at multiple students’ work and talk about the positive aspects of each one, while highlighting ways to make it even better.

In this problem, we had a fantastic discussion about communicating both reasoning and process:

The next step that I’d like to make is to have this process of seeing all of the responses be even more transparent. I’d like to see student work popping up in a gallery that I can browse and choose certain responses to share with the class. Another option to pursue is to get students seeing the responses of their peers and offer advice.

Automatic grading certainly makes the job of answering the right/wrong question much easier. Sometimes a student does need to know whether an answer is correct or not. Given all the ways that a student could game the system (some students did discuss using Wolfram Alpha during the activity) the informative part on the teaching and assessment end is seeing the work itself. This is also an easy source of material for discussion with other teachers about student work (such as with Michael Pershan’s Math Mistakes).

I was blown away with how my crude hack to add this feature this morning made the class period a much richer opportunity to get students sharing and talking about their work. Now I’m excited to work on the next iteration of this idea.

Computational Thinking and Algebraic Expressions

I am still reviewing algebra concepts in my Math 9 course with students. The whole unit is all about algebraic operations, but my students have seen this material at least once, in some cases two years running.

Not long ago, I made the assertion that the most harmful part of introducing students to the world of real-world algebra looks like this:

Let x = the number of ________

Why is this so harmful?

For practiced mathematicians, math teachers, and students that have endured school math for long enough, there are a couple steps that actually happen internally before this step of defining variables. Establishing a context for the numbers and the operations that link them together are the most important part of producing a correct mathematical model for a given situation. A level of intuition and experience is necessary if one is going to successfully skip straight to this step, and most students don’t have this intuition or experience.

We have to start with the concrete because most people (including our students) start their thinking in concrete terms. This is the reason I have raved previously about the CME Project and the effectiveness of using their guess-check-generalize concept in introducing word problems to students. It forms an effective bridge between the numbers that students understand and the abstract concept of a variable. It encourages experimentation and analysis of whether a given answer matches the constraints of a problem.

This method, however, screams for computers to take care of the arithmetic so that students can focus on manipulating the variables involved. Almost all of the Common Core Standards for Mathematical Practice point toward this being an important focus for our work with students. I haven’t had a great point in my curriculum since I really started getting into computational thinking to try out my ideas from the beginning, but today gave me a chance to do just that.

Here’s how I introduced students to what I wanted them to do:

I then had them open up this spreadsheet and actually complete the missing elements of the spreadsheet on their own. Some students had learned to do similar tasks in a technology class, but some had not.
02 – SPR – Translating Algebraic Expressions

The students that needed to have conversations about tricky concepts like three less than a number had them with me and with other students when they came up. Students that didn’t quickly moved through the first set. I’d go and throw some different numbers for ‘a number’ and see that they were all changing as expected. Then we moved to a more abstract task:

It’s great to see that you know how to use different operations on the number in that cell. Now let’s generalize. Pick a variable you like – x, or N, or W – it doesn’t matter. What would each of these cells become then? Write those results together with the words in your notebook and show me when you’re done.

The ease with which students moved to this next task was much greater than it has ever been for me in past lessons. We also had some really great conversations about x*2 compared with 2x, and the fact that both are correct from an arithmetic standpoint, but one is more ‘traditional’ than the other.

Once students got to this point, I pushed them toward a slightly higher level task that still began concrete. This is the second sheet from the spreadsheet above:

Here we had multiple variables going at once, but this was not a stretch for most students. The key that I needed to emphasize here for some students was that the red text was all calculated. I wanted to put information in the black boxes with black text only, and have the spreadsheet calculate the red values. This required students to identify what the relationship between the variables needed to be to obtain the answer they knew in their head had to be true. This is CCSS MP2, almost verbatim.

This is all solidifying into a coherent framework of using spreadsheet and programming tools to reinforce algebra instruction from the start. There’s still plenty to figure out, but this is a start. I’ll share what I come up with along the way.

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