Category Archives: reflection

June 13, 2013 · 5:22 pm

Three Acts – Counting with dots and first graders

I had an amazing time this afternoon visiting my wife’s first grade class. I’ve been talking forever about how great it is to take a step out of the usual routines in class and look at a new problem, and my wife invited me in to try it with her students.

Here’s the run-down.

Act 1

Student questions (and the number of students that also found the questions interesting):

Why do the dots come together? (8)
Why are the dots making pictures and not telling us what they mean? (8)
Why are some dots going together into big dots, and others staying small? (13)
Why do some of the dots form blue lines before coming together?

My questions (and the number of students that humored me):

How many dots are there at the end? (8)
What is the final pattern of dots after the video ends? (11)

Guesses for the number of dots ranged from a low of 20 to a high of 90.

Act 2

What information did they want to know?

They wanted to see the video again.
Seven students asked about the numbers of tens or ones in each group. (I jumped on the use of that vocabulary right away – it seemed they are comfortable using this vocabulary based on my conversations with them.)

I showed them the video and gave them this handout since I didn’t have video players for all of the students:
grouping dots

What happened then was a series of amazing conversations with some really energetic and enthusiastic kids. They got right to work organizing and figuring out the patterns.

Act 3

We watched the video and discussed the results and how they got their answers. Lots of great examples of student-created systems for keeping track of their counting. We then watched the Act 3 video:

While nobody had the total number correct, I was quite impressed with their pride in being close. More interesting was how little they cared that they didn’t get the exact answer. I asked who was between 70 and 80, and a few kids raised their hands, and then the same with 50 – 70. One student was one off. Most were within ten or so of the correct answer. The relationship between the guesses and their answers after analysis was something we touched upon, but didn’t discuss outside of some one-on-one conversations.

The absolute highlight of the lesson was when I asked why they thought nobody had the exact answer. One student walked up to the projector screen with out hesitation and pointed here:

She said “this is what made it tough” and then sat back down.

We had a little more time, so we watched a sequel video:

I asked what they saw that was different aside from the colors. One student said right away that he figured it out, the same student that first shouted out ‘tens!’ in Act 1. We lacked the time to go and figure it out, so we left it there as a challenge to figure out for the next class.

Footnotes:

Any high school or middle school math teacher that wants to see how excited students can be when they are learning math needs to go take a group of elementary students through a three act. I wish I had done this during the dark February months when things drag for me. My wife asked me to do this to see how it works, but I think I got a lot more enjoyment out of the whole experience.
I made a conscious decision not to include any symbolic numbers in this exercise. It adds an extra layer of abstraction that takes away from the students figuring out what is going on. I almost put it back in when I wasn’t sure whether it was obvious enough. I am really glad I left it out so the students could prove that they didn’t need that crutch.
This is written in Javascript using Raphael. You can see a fully editable version of the code in this JSFiddle.
All files are posted at 101 Questions in case you want to get the whole package.

Assessing assessment over time – similar triangles & modeling

I’ve kept a question on my similar triangles unit exam over the past three years. While the spirit has generally been the same, I’ve tweaked it to address what seems most important about this kind of task:

My students are generally pretty solid when it comes to seeing a proportion in a triangle and solving for an unknown side. A picture of a tree with a shadow and a triangle already drawn on it is not a modeling task – it is a similar triangles task. The following two elements of the similar triangles modeling concept seem most important to me in the long run:

Certain conditions make it possible to use similar triangles to make measurements. These conditions are the same conditions that make two triangles similar. I want my students to be able to use their knowledge of similarity theorems and postulates to complete the statement: “These triangles in the diagram I drew are similar because…”
Seeing similar triangles in a situation is a learned skill. Dan Meyer presented on this a year ago, and emphasized that a traditional approach rushes the abstraction of this concept without building a need for it. The heavy lifting for students is seeing the triangles, not solving the proportions.

If I can train students to see triangles around them (difficult), wonder if they are similar (more difficult), and then have confidence in knowing they can/can’t use them to find unknown measurements, I’ve done what I set out to do here. What still seems to be missing in this year’s version is the question of whether or not they actually are similar, or under what conditions are they similar. I assessed this elsewhere on the test, but it is so important to the concept of mathematical modeling as a lifestyle that I wish I had included it here.

(Students) thinking like computer scientists

It generally isn’t too difficult to program a computer to do exactly what you want it to do. This requires, however, that you know exactly what you want it to do. In the course of doing this, you make certain assumptions because you think you know beforehand what you want.

You set the thermostat to be 68º because you think that will be warm enough. Then when you realize that it isn’t, you continue to turn it up, then down, and eventually settle on a temperature. This process requires you as a human to constantly sense your environment, evaluate the conditions, and change an input such as the heat turning on or off to improve them. This is a continuous process that requires constant input. While the computer can maintain room temperature pretty effectively, deciding whether the temperature is a good one or not is something that cannot be done without human input.

The difficulty is figuring out exactly what you want. I can’t necessarily say what temperature I want the house to be. I can easily say ‘I’m too warm’ or ‘I’m too cold’ at any given time. A really smart house would be able to take those simple inputs and figure out what temperature I want.

I had an idea for a project for exploring this a couple of years ago. I could try to tell the computer using levels of red, green, and blue exactly what I thought would define something that looks ‘green’ to me. In reality, that’s completely backwards. The way I recognize something as being green never has anything to do with RGB, or hue or saturation – I look at it and say ‘yes’ or ‘no’. Given enough data points of what is and is not green, the computer should be able to find the pattern itself.

With the things I’ve learned recently programming in Python, I was finally able to make this happen last night: a page with a randomly selected color presented on each load:

Sharing the website on Twitter, Facebook, and email last night, I was able to get friends, family, and students hammering the website with their own perceptions of what green does and does not look like. When I woke up this morning, there were 1,500 responses. By the time I left for school, there were more then 3,000, and tonight when my home router finally went offline (as it tends to do frequently here) there were more than 5,000. That’s plenty of data points to use.

I decided this was a perfect opportunity to get students finding their own patterns and rules for a classification problem like this. There was a clearly defined problem that was easy to communicate, and I had lots of real data data to use to check a theoretical rule against. I wrote a Python program that would take an arbitrary rule, apply it to the entire set of 3,000+ responses from the website, and compare its classifications of green/not green to that of the actual data set. A perfect rule for the data set would correctly predict the human data 100% of the time.

I was really impressed with how quickly the students got into it. I first had them go to the website and classify a string of colors as green or not green – some of them were instantly entranced b the unexpected therapeutic effect of clicking the buttons in response to the colors. I soon convinced them to move forward to the more active role of trying to figure out their own patterns. I pushed them to the http://www.colorpicker.com website to choose several colors that clearly were green, and others that were not, and try to identify a rule that described the RGB values for the green ones.

When they were ready, they started categorizing their examples and being explicit in the patterns they wanted to try. As they came up with their rules (e.g. green has the greatest level) we talked about writing that mathematically and symbolically – suddenly the students were quite naturally thinking about inequalities and how to write them correctly. (How often does that happen?) I showed them where I typed it into my Python script, and soon they were telling me what to type.

In the end, they figured out that the difference of the green compared to each of the other colors was the important element, something that I hadn’t tried when I was playing with it on my own earlier in the day. They really got into it. We had a spirited discussion about whether G+40>B or G>B+40 is correct for comparing the levels of green and blue.

In the end, their rule agreed with 93.1% of the human responses from the website, which beat my personal best of 92.66%. They clearly got a kick out of knowing that they had not only improved upon my answer, but that their logical thinking and mathematically defined rules did a good job of describing the thinking of thousands of people’s responses on this question. This was an abstract task, but they handled it beautifully, both a tribute to the simplicity of the task and to their own willingness to persist and figure it out. That’s perplexity as it is supposed to be.

Other notes:

One of the most powerful applications of computers in the classroom is getting students hands on real data – gobs of it. There is a visible level of satisfaction when students can talk about what they have done with thousands of data points that have meaning that they understand.
I happened upon the perceptron learning algorithm on Wikipedia and was even more excited to find that the article included Python code for the algorithm. I tweaked it to work with my data and had it train using just the first 20 responses to the website. Applying this rule to the checking script I used with the students, it correctly predicted 88% of the human responses. That impresses me to no end.
A relative suggested that I should have included a field on the front page for gender. While I think it may have cut down on the volume of responses, I am hitting myself for not thinking to do that sort of thing, just for analysis.
A student also indicated that there were many interesting bits of data that could be collected this way that interested her. First on the list was color-blindness. What does someone that is color blind see? Is it possible to use this concept to collect data that might help answer this question? This was something that was genuinely interesting to this student, and I’m intrigued and excited by the level of interest she expressed in this.
I plan to take a deeper look at this data soon enough – there are a lot of different aspects of it that interests me. Any suggestions?
Anyone that can help me apply other learning algorithms to this data gets a beer on me when we can meet in person.

5 Comments

Filed under computational-thinking, reflection, teaching stories

April 10, 2013 · 4:08 pm

Building a need for math – similar polygons & mobile devices

The focus of some of my out-of-classroom obsessions right now is on building the need for mathematical tools. I’m digging into the fact that many people do well on a daily basis without doing what they think is mathematical thinking. That’s not even my claim – it’s a fact. It’s why people also claim the irrelevance of math because what they see as math (school math) almost never enters the scene in one’s day-to-day interactions with the world.

The human brain is pretty darn good at estimating size or shape or eyeballing when it is safe to cross the street – there’s no arithmetic computation there, so one could argue that there’s no math either. The group of people feeling this way includes many adults, and a good number of my own students.

What interests me these days is spending time with them hovering around the boundary of the capabilities of the brain to do this sort of reasoning. What if the gut can’t do a good enough job of answering a question? This is when measurement, arithmetic, and other skills usually deemed mathematical come into play.

We spend a lot of time looking at our electronic devices. I posed this question to my Geometry and Algebra 2 classes on Monday:

The votes were five for A, 5 for B, and 14 for C. There was some pretty solid debate about why they felt one way or another. They made sure to note that the corners of the phone were not portrayed accurately, but aside from that, they immediately saw that additional information was needed.

Some students took the image and made measurements in Geogebra. Some measured an actual 4S. Others used the engineering drawing I posted on the class blog. I had them post a quick explanation of their answers on their personal math blogs as part of the homework. The results revealed their reasoning which was often right on. It also showed some examples of flawed reasoning that I didn’t expect – something I now know I need to address in a future class.

At the end of class today when I had the Geometry class vote again, the results were a bit more consistent:

The students know these devices. Even those that don’t have them know what they look like. It required them to make measurements and some calculations to know which was correct. The need for the mathematics was built in to the activity. It was so simple to get them to make a guess in the beginning based on their intuition, and then figure out what they needed to do, measure, or calculate to confirm their intuition through the idea of similarity. As another chance at understanding this sort of task, I ended today’s class with a similar challenge:

My students spend much of their time staring at a Macbook screen that is dimensioned slightly off from standard television screen. (8:5 vs. 4:3). They do see the Smartboard in the classroom that has this shape, and I know they have seen it before. I am curious to see what happens.

2 Comments

Filed under geometry, reflection, Uncategorized

Tagged as devices, intuition, ratio, similarity

March 12, 2013 · 9:40 pm

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:

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:

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.

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?

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.

3 Comments

Filed under computational-thinking, reflection

March 2, 2013 · 10:41 pm

Playing with robots – a weekend well spent

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.

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.

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.

JavaScript required to play 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:

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, but 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.

4 Comments

Filed under geogebra, geometry, reflection

Tagged as clear instructions, exploration, parallelograms, video, written instructions

February 5, 2013 · 8:32 am

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:

Screen Shot 2013-02-05 at 3.23.59 PM http://neo.jpl.nasa.gov/news/news177.html

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.
orbit calculations
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.

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.

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.

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.

2 Comments

Filed under computational-thinking, physics, reflection

January 30, 2013 · 1:22 am

Angry Birds Project – Results and Post-Mortem

In my post last week, I detailed what I was having students do to get some experience modeling quadratic functions using Angry Birds. I was at the 21CL conference in Hong Kong, so the students did this with a substitute teacher. The student teams each submitted their five predictions for the ratio of hit distance to the distance from the slingshot to the edge of the picture. I brought them into Geogebra and created a set of pictures like this one:

After learning some features of Camtasia I hadn’t yet used, I put together this summary video of the activity:

I played the video, and the students were engaged watching the videos, but there was a general sense of dread (not suspense) on their faces as the team with the best predictions was revealed. This, of course, made me really nervous. They did clap for the winners when they were revealed, and we had some good discussion about modeling, which videos were more difficult and why, but there was a general sense of discomfort all through this activity. Given that I wasn’t quite able to figure out exactly why they were being so awkward, I asked them what they thought of the activity on a scale of 1 – 10.

They hated it.

I should have guessed there might be something wrong when I received three separate emails from the three members one team with results that were completely different. Seeing three members of one team work independently (and inefficiently) is something I’m pretty tuned in to when I am in the room, but this was bigger. It didn’t sound like there was much utilization of the fact that they were in teams. I need to ask about this, but I think they were all working in parallel rather than dividing up the labor, talking about their results, and comparing to each other.

Some things I want to remember about this:

I need to be a lot more aware of the level of my own excitement around activity in comparison to that of the students. I showed one of the shortened videos at the end of the previous class and asked what questions they really wanted to know. They all said they wanted to know where the bird would land, but in all honesty, I think they were being charitable. They didn’t really care that much. In the game, you learn shortly after whether the bird you fling will hit where you want it to or not. Here, they had to go through a process of importing a picture, fitting a parabola, and finding a zero of a function using Geogebra, and then went a weekend without knowing.
While it is true that using a computer made this task possible, and was more enjoyable than being forced to do this by hand, the relativity of this scale should be suspect. “Oh good, you’re giving me pain meds after pulling my tooth. Let’s do this again!”
A note about pseudocontext – throwing Angry Birds in to a project does not by itself does not necessarily engage students. It is a way in. I think the way I did this was less contrived than other similar projects I’ve seen, but that didn’t make it a good one. Trying to make things ‘relevant’ by connecting math to something the students like can look desperate if done in the wrong way. I think this was the wrong way.
I would have gotten a lot more mileage out of the video if I had stopped it here:

That would have been relevant to them, and probably would have resulted in turning this activity back around. I am kicking myself for not doing that. Seriously. That moment WAS when the students were all watching and interested, and I missed it.

Next time. You try and fail and reflect – I’m still glad I did it.

We went on to have a lovely conversation about complex numbers and the equation $x^{2}+4 = 0$ . One student immediately said that $ \sqrt{-2} $ was just fine to substitute. Another stayed after class to explain why she thought it was a disturbing idea.

No harm done.

P.S. – Anyone who uses this post as a reason not to try these ideas out with their class and to instead slog on with standard lectures has missed the point. I didn’t do this completely right. That doesn’t mean it couldn’t be a home run in the right hands.

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Filed under algebra 2, computational-thinking, reflection, teaching stories

Tagged as angry birds, geogebra, reflection, video

January 26, 2013 · 11:58 pm

Why computational thinking matters – Part I

My presentation at 21CLHK yesterday was an attempt to summarize much of the exploration I’ve done over the past year in my classroom into the connection between learning mathematical concepts and programming. I see a lot of potential there, but the details about how to integrate it effectively and naturally still need to be fleshed out.

After the presentation, I felt there needed to be some way to keep the content active other than just posting the slides. I’ve decided to take some of the main pieces of the presentation and package them as videos describing my thinking. I’m seeing this as an iterative process – in all likelihood, these videos will change as I refine my understanding of what I understand about the situation. Here is the start of what will hopefully be a developing collection:

JavaScript required to play computational thinking – what is it?.

JavaScript required to play computational thinking – an example.

JavaScript required to play computational thinking – what it isn’t.

I want to express my appreciation to Dan Meyer for his time chatting with during the conference about my ideas on making computation a part of the classroom experience. He pushed back against some my assertions and was honest about which arguments made sense and which needed more definition. I think this is a big deal, but the message on the power of computational thinking has to be spot on so it isn’t misunderstood or misused.

With the help of the edu-blogging community, I think we can nail this thing down together. Let’s talk.