## Ratios & Proportions - Day One Antics

Yesterday was our first day into a unit on similarity with the ninth grade students.

The issue that comes up every year is that students like to cross multiply, but are incredibly mechanical in their understanding of why they do so. They don't like fractions that aren't simplified, and can usually simplify them well. They bring up the fact that multiplication of numerator and denominator by the same number is equivalent to multiplying by one. They seem to have very little understanding of how this relates to units and unit conversions as well.

I changed my approach this year to be much less review of how to solve proportions. I wanted to get at the aspects of measurement that are inherent to math problems involving similarity. I wanted to get them to ask themselves a bit more about why they took the steps they took in solving proportions in the process.

I started with a couple simple problems in the warm-up. Here was one:

I took pictures of two students' work, put them side by side, and asked the class which one they thought was a better answer to the question:

The resulting vote and conversation was especially spirited, particularly for a class that normally rejects whole class discussion. We talked about the ideas of approximate and exact answers, a couple of students pointing out that substitution of the approximate answers would result in a false statement in the equation.

After this, I showed them another picture and asked if the LEGO pieces in this picture would go together:

Every hand went up.

I then showed them the bricks, which I had made on our school's new 3D printer:

Pause for groans. Some key things were said in response to my 'playing dumb' question of why the two bricks won't fit together. One student even directly said that they looked similar to each other, but that they weren't the same size. I wanted them to have in the back of the heads that I was going to be pushing them to always think about figure with the same shape, different size.

We then made it to the second task of the warm-up activity. I asked them to estimate (and subsequently measure) the ratio of one of my heads in this image to the next:

I developed the following points:

• When communicating ratios to another person, begin explicit and clear about order is extremely important.
• Despite the different units, these ratios are all communicating the same relationship from one head to the next. This relationship is even more obvious when we write the ratio as a fraction instead of using the colon notation.
• The approximate values of this fraction are all roughly the same. We don't need to convert units either for this to happen - the units divide themselves out in the fraction.

I went on to define a proportion and reviewed the idea of cross products. They were a bit surprised when I showed them that cross products were equal for equivalent fractions. Part of this was because they saw me equate 2/5 and 4/10 and immediately said they were equal because one simplified into the other. I gave them 2/5 and 354453764/886359410 and they were a bit more willing to see that cross products can be a slicker way to check equality.

One more point that I made was that a proportion with a variable in it was really a question. If we are saying two fractions are equal to each other, and one (or more) of the fractions has a variable, what does that mean about the value of the variable? It led to a bit more conversation about the reasons for cross multiplication as a method of solving proportions, and I was satisfying then leaving students to work through some more review problems on their own.

They were able to come up with some, but struggled to make ratios that were more than simple multiples. This was surprising, as their mental calculation skills are generally quite strong. As shown in the example, I gave them one way to see how to come up with an arbitrary set of lengths that fit the requirement.

I then showed them this question:

Some of the students realized (and explained eloquently) that they could divide the length by 7 and find the length of a single 'unit', and then multiple that unit by 3 or 4 to get the length. Explanations for why this worked didn't really materialize. I introduced the algebraic approach, and students saw it as an explanation, but seemed to be fine with just remembering it as a method rather than as a rationale.

The more that I teach proportions and similarity, the more I feel compelled to have students ground the concepts in measurements. Making measurements, especially by hand, is not something they typically do on a day-to-day basis, so there's a bit of a novelty factor there. These conversations about measurements, units, and fractions were authentic - there was a need to talk about these ideas in the context I established, and the students did a great job of feeling and then filling that need during the class. Nothing we did was a particularly real world task though. What made this real was my attempts to first frame the skills that we needed to review in the context of a need for those skills. I try to do this often, and I'd like to mark this as a success story.

Filed under geometry, teaching stories

## Magnetic Fields, Data Collection, and You(r dog)

Assignment 1: Read the following abstract for a scientific paper.

Dogs are sensitive to small variations of the Earth’s magnetic field

Not too bad, right? Now read this more palatable explanation of what this really means:
Dogs align their bodies along a North-South axis....

Finally, here's a composite image I've put together:

For all of these, the top of the phone was facing the same direction as my dog's nose during the act. These were all in different locations in the same yard, so it was clear to me that he wasn't just finding the same spot every time. It took copious treats and showing my dog the photos to convince him that I was not taking a picture of him while he did his business.

It's possible to get over the social peculiarity of remembering to pull out your phone, start the compass application, and take a screenshot whenever your dog pops a squat. To me, this seems like a ripe opportunity for a student project in statistics and data analysis. Furthermore, the potential for doing this now (compared to just a few years ago) is better than ever. Why?

1. There are lots of dogs around, because dogs are awesome. They all have to unload at some point in the day. That makes for a large potential sample set of data to work with.
2. It's all about number two, which is (of course) hilarious and engaging to all of us. I chuckled first when I read the headline of this story, and then a second time when I realized that scientists observed 1,893 defecation events and then sorted them according to magnetic field activity. I propose that this paper might include the term 'defecation events' more frequently than any other academic paper, and for that reason alone, it is special.
3. Now more than ever, we have these great devices in our pockets that are not just capable of capturing such screenshots easily, but combine other useful information that might be important factors for students to consider. Time, date, geotagging information for location - all useful things students might choose to analyze in seeing if the results of this study are repeatable.
4. Crowdsourcing. I took eight of these pictures, my wife took a few more. Imagine the potential of getting a photo stream of defecation event data for students to analyze from dogs around the world. Wolfram Alpha pegs the number of dogs in the US at 78.2 million.

Here's what I propose. If you're into this, take some data on the next outing with your dog. Some suggestions to maintain data integrity:

• Stand behind the test subject and align the phone so that the top of your phone points in the direction your dog is looking while he/she concentrates on the task at hand.
• Make sure your compass is calibrated when you take data.
• Snap a screenshot of the compass screen. On iOS, hold the Home and Power buttons simultaneously, then release the power button. I found out from Lifehacker that in Android 4, you just hold down the power and volume-down buttons.
• Now's the time to take a big (data) dump - upload those screenshots to Flickr, Instagram, etc. with the hashtag #DogsPoopNorth .

I don't teach statistics, but I'd love to see a class take a chunk of data and show that there is signal in the noise. The original researchers clearly showed this, but it's a great experience to have students do their own analysis work and come to their own conclusion about whether dogs have this unique ability or not.

Get to work, interwebs. I'm really interested to see what comes out.

Filed under computational-thinking

## What do you mean by 'play with the equation'?

My professional obsessions lately have focused on using technology to turn traditional processes of learning into more inquiry based ones. A really big part of this is purely in the presentation.

• Calculate the force between the Earth (mass 5.97 x 10^24 kg) and an apple (100 g) at the surface of the Earth (which has a radius of 6.37 x 10^6meters.
• If the Earth somehow doubles in mass, but keeps the same size, what would happen to the force of gravity?
• Find at what distance from Earth's center the gravity force would be equal to 0.5 Newtons.

My observation in this type of sequence is that the weaker algebra students (or students that use the presence of mathematics as a way out of things) will turn off the moment you say calculate. The strong students will say 'this is easy', throw these in a calculator, and write down answers without units.

A modified presentation would involve showing students a spreadsheet that looks like this:

• Calculate the force between the Earth (mass 5.97 x 10^24 kg) and an apple (100 g) at the surface of the Earth (which has a radius of 6.37 x 10^6meters.
• If the Earth somehow doubles in mass, but keeps the same size, what would happen to the force of gravity?
• Find at what distance from Earth's center the gravity force would be equal to 0.5 Newtons.

I would argue that this is, for many students, just as much of a black box as the first. In this form, however, students are compelled to tinker and experiment. Look for patterns. Figure out what to change and what to keep the same. In the last question, students will likely guess and check, which may get tedious if the question changes. This tedium might motivate another approach. A mathematically strong student might double click on the force cell and look up the function, or look up Newton's Law of Gravitation on Wikipedia and try to recreate the spreadsheet on his or her own. A weaker student might be able to play with the numbers and observe how doubling the mass doubles the force, and feel like he or she has a way to answer these questions, albeit inefficiently. Both students have a path to wrestle with the question that forms the basis of physics: how do we model what we observe in our universe?

This approach makes obvious what it means to play with a mathematical object such as an equation. Playing with an equation is something that I've admittedly said to students before in a purely algebraic context. I know that I mean to rearrange the equation and solve for the variable that a given question is asking for. Students don't typically think this way in math or science, or any equation that we give them. If they do, it's because we've artificially trained them to think that this is what experimentation in math looks like. I think that this is primarily because the user interface of math, which has been paper and pencil for thousands of years, doesn't lend itself to this sort of experimentation easily. Sure, the computer is a different interface, and has its own input language that is sometimes quite different from mathematical language itself, but I think students are better at managing this gap than we might give them credit for.

Filed under computational-thinking, reflection

## 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.

Filed under physics, reflection

## 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.

Filed under teaching philosophy

## 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.

1 Comment

Filed under reflection

## 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.

1 Comment

Filed under computational-thinking, geometry

## 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/

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.

Filed under computational-thinking, programming

## 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:

1. 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.
2. Does the CAPM model apply? Have a reason for your answer.
3. 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:

1. The maximum height of the ball occurred at that time. Student points to the graph.
2. 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.
3. 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.
4. 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.
5. 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?

Filed under computational-thinking, physics

## Programming and Making Use of Structure in Math

A tweet from James Tanton caught my eye last night:

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<=10;i++) { for(j = 1;j<=10;j++) {

 sum+=i*j } } console.log(sum) 

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<=10;i++) { for(j = 5;j<=10;j++) {

 sum+=i*j } } console.log(sum) 

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.