Building a need for algebraic reasoning - how can computers help?

I hear this all the time, and it drives me up a wall.

I haven't solved for x in years, and I'm doing just fine.

Few people realize that while they aren't using algebraic properties in their daily lives, they use the analog concept of finding missing values all the time. You won't win this argument with most people though. It just doesn't seem like algebra.

As math teachers, we also get annoyed when students are able to do this with nothing in between:

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Certainly in a Calculus class, this should not surprise us - at that level we would expect an ability to eyeball the solution. At the other end of the post elementary math progression, however, when we are teaching two step equations for the first time, our response might be this: "Yes, you got that one, but I could give you one that has negative numbers or (GASP!) decimals or fractions in it. Then what would you do? This is why it's important that you pay attention to this lesson. You have to do it this other way in order to get credit."

I've had this conversation, and it has always made me feel ridiculous. It's an arbitrary and crappy argument. It might be a valid one if standardized (or your own) tests of algebraic concepts are involved, but using tests as a motivation for doing anything makes the whole enterprise feel cheap, even when doing so needs to happen.

The bigger issue is that it perpetuates the reputation of math teachers and mathematicians as protectors of a sacred bag of secrets that nobody outside of a math classroom will need. It also presents a problem of artificiality. If I can suddenly make something harder by adding fractions or decimals, does doing so make it any easier for me to assess whether my students know what they are doing in solving an equation? I think we haven't done a great job of building in the need for algebra, especially in light of what computers can do. I've never had a student sarcastic and comfortable enough with me to do this, but bear with me. The theoretical argument in the back of my mind to what I said in response to the student I described earlier is this:

Really teach? With that college degree of yours, you could make up a question that I can't use my knowledge of arithmetic to solve? Impressive. I guess that even though I did everything my previous teachers told me to do - memorize multiplication tables, learn to add fractions with like/unlike denominators, draw lots of pie charts demonstrating equivalent fractions, AND draw lots of connect-the-dot dinosaurs as reviews of plotting in the coordinate plane, I still need you. Glad to be here. Oh, your tie is crooked. At least I can still help you out with that.

Furthermore, I wonder about the challenge of motivating algebra given that Wolfram Alpha, CAS, and even the lowly TI-83 solver can solve equations without breaking a sweat.

I'm not teaching introductory algebra right now, but the thinking I've done on how computers put the thinking back into process has me wondering how motivating the need for Algebra could be different, and better given how easy it is to compute these days. The most basic way that people interact with numbers is through tables and graphs - is it possible to motivate algebra through this familiar idea? Can we use the computer to compute a bunch of stuff, and see what it tells us?

Some food for thought:
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This is precisely the sort of thing we are looking for when we are solving an equation, but it's rare that we think about it this way. It's also something that most people outside of a classroom will do with a table of values in a newspaper or a website, for example. It is typically for more practical reasons (predicting value of a stock, figuring out when a bus will arrive at our location from a schedule that doesn't have every stop, etc) than simply finding 'x' as we ask students to do in the classroom. Is this algebra? Staring at a table of values is tedious, but I know people that would rather do this than solve an equation or do anything that smells like school math.

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Again, in our adult lives, we make estimations from given information from a table or graph from time to time, but few adults actually call this algebra. Is it obvious to an adult that changing the interval in the right way would allow the exact answer to be found? Is it obvious to a student? It's a subtle point here, but I think it's the sort of reasoning we want our students to be capable of doing. Is that type of understanding something inherently important in algebraic reasoning? How's that going for us now?

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We know there are algorithmic ways to solve this one, but I've already said here and in previous posts that I want to get away from mathematical thinking as a bag of algorithms. How good of an answer to this can we get from a table? I don't know about you, but I have yet to feel like I've taught well the idea of an irrational number in a good, intuitive way that doesn't result in students memorizing tricks. I think this hints at this concept in ways that is inaccessible without using computers. Even on a calculator, it's difficult to focus in on solutions as smoothly as I think can be done with a table of rapidly computed values.

I'm not suggesting that we shouldn't teach properties of numbers and inverse operations in the context of solving equations algebraically. I think we need to do a better job of selling the idea of algebra as being an enhancement of what we already have built in to our brains. We estimate what time we need to cross the street to not get hit by a truck but also to minimize our time waiting. We know that if the high is 68 degrees at 3 PM, that it will probably be a nice temperature outside at one-o-clock. This way of feeling our way to a solution through intuition, however, is not the optimal way to solve problems, especially when our intuition is wrong. There needs to be a better way.

Our students (and many adults) often don't know how to create tools to help them solve the problems they face. They choose to do things that are tedious because they don't know a better way, and the math skills they have developed previously are disconnected and seem irrelevant as a result. We do understand the idea of computation, but we often aren't good at doing it ourselves. If nothing else, it's pushing people to become more confident that they know what they are looking at when we see a bunch of numbers together.

Playing with robots - a weekend well spent

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This weekend marked the culmination of a few months of work from my robotics students. We traveled to Shanghai to compete in the FIRST Tech Challenge tournament with 47 other teams. I got involved in FIRST nine years ago when teachers at my school in the Bronx tracked me down after hearing of my engineering background. They had just won the Rookie of the Year award the season before, and were excited to have an engineer around to help. Given that it was my first year teaching, I wasn't able to be nearly as involved as I wanted to be. It was enough of a hook to get me to see how powerful programs like FIRST really are for working on the 'demand' side of the educational system, the problem-solving-hands-on-building stuff that makes students see what the end game of education can be. Playing with robots on a competition field is no more 'real world' than estimating the number of pennies in a pyramid, but the learning opportunities in both are rich and demanding. Nine years later, I am still as convinced as ever that these are the types of activities our students need to understand the context of the skills we teach them in our classrooms.

This weekend, we met stiff competition from our Chinese competitors. They built cascaded elevator systems, scissor lifts, and sensor systems that helped to play this year's game, a tic-tac-toe variant played using colored rings on a set of horizontal pegs. More impressive for me was seeing the mentors noticeably bored and checking their phones while the students were the ones focused on tweaking their robots and fixing programming snafus.

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This, however, was not our main challenge. The biggest issues that we faced were of our own creation - how to achieve consistency in our lifting mechanism using a web of zip-ties, or discovering just how unstable our own lifting mechanism was. The students were constantly sawing different parts of the robots off to make room for the solution to the last problem they created while trying to solve another. Clearing the complex residue of multiple good ideas to leave a simple, capable solution is the ultimate goal of a good design. The overall process of doing this is difficult, even with experience. They are early enough on the curve to know that there is much that they do not know though, and their positive and cheerful manner throughout was inspiring. Even after multiple technical issues and defeats on the field, they left the competition today feeling accomplished and full of ideas.

I was most inspired by my students' reactions to seeing the clever designs of their Chinese counterparts. I have witnessed students wandering the pits at FIRST events and greeting unique and capable designs with accusation as the immediate reaction. "They could do that because they have so much more money" or "the mentors did all the work - it isn't fair because we do everything on our team." I understand the sentiment, but have always passed it off as being overly pessimistic. Some skilled teams make it look easy without always making obvious the associated level of effort required to execute such designs.

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What made me particularly proud of my students this weekend was seeing them look at other designs and go through two stages of processing them. First, they would remark how cool it was that the team was able to solve the problem in such a unique way. Second, with some thinking about just how, they would say something along the lines of we could have done that.

While our ranking was closer to the bottom than I (or they) will reluctantly reveal, I don't care much at this point. The team is young and will hopefully have more opportunities to learn and build together over the next couple of years. Their satisfaction was evident in watching the final matches with a clear sense of accomplishment, even while not being part of them. Their sense of togetherness is stronger than ever.

Our bus lost its headlights on the way back, forcing us to spend an hour and a half at a repair place while the driver and nine other people figured it out while the usual pattern of loud Mandarin was punctuated with hacking and drags off cigarettes. The team, meanwhile, procured a healthy supply of snacks and seemed content to sing along to music played off their school laptops. This is a close group that has only grown closer. Easily the highlight of the whole weekend right there.

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

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

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

I also gave them this image on the handout, which I wrote last year, but students only marginally followed:
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It was remarkable how this simple change to delivery made the whole class really fun to manage today.

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

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

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

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

When things just work - starting with computers

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

Scare the students:

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

Connect to previous work:

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

Play with the results

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

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

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

See what happens with the algebra.

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

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

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

Simulate & Visualize

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

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


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

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

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

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

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

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

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

Angry Birds Project - Results and Post-Mortem


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