## Exploring Functions (and Non-Functions) Interactively

Heeding Dan's encouragement to step things up in his NCTM talk, I revisited an introduction to functions activity that I put together three years ago. The idea is to get students to make observations about inputs and outputs and use the 'notice and wonder' parlance from the Math Forum to prompt conversations about these ideas.

I rewrote the activity with some deliberate changes and webified it to make it easy to access and share - you can find it here:
http://emwdx.github.io/functions-exploration/index.html

The activity has a few elements that I want to highlight with the hope that you might consider (a) trying the activity with your students or (b) downloading the code for the activity, tweaking it, and then re-sharing it with your enhancements.

### Students go through the modeling cycle multiple times.

The activity begs students to take a playful approach. Change the input value and watch the output. Predict what's going to happen and see if your mental model is correct. Then do the next one, and the next.

### Arithmetic isn't necessarily a prerequisite.

Some students were actually more puzzled by the functions that took text inputs. They experimented nevertheless to figure out what was happening, and some noticed that the pattern worked for numbers too.

### Controversy is built in.

Students working on Functions 5 and 6 saw nothing weird happening when they worked alone. When they then went to share their answers with classmates, the latter function started some really interesting interactions between students trying to figure out who was wrong.

### Students of different levels all succeeded and all struggled at some point.

One student zipped through the arithmetic exercises and then got stuck figuring out Function 3 or 7. Some of the weaker students jumped around and got Functions 1 and 4 and 8, which is enough to get in the game of finding patterns and drawing conclusions. A higher level student experimented with Function 7 to find that there was a well defined range for the outputs - random, but with limitations.

### The need for definitions came out of the activity, not the other way around.

Students felt the need to clearly define the behavior of Functions 6 and 7 as being different than the others in a fundamental way. Definitions for relations and functions weren't huge cognitive jumps for students since there was a recently established context. It's also important to notice that the definition for relations that aren't functions has to be more than just the lack of a pattern. Function 6 helps with this.

### Many of the CCSS standards for mathematical practice are embedded within.

...as are some of the high school standards for functions.

If you try this with students, let me know how it goes.

#### Technical Details:

If you want to try this yourself, you can download the code from Github here:
https://github.com/emwdx/functions-exploration/tree/gh-pages

I did this also as an attempt to whip together something using the React JS library which I've been learning recently. It makes for a really nice interface for building this type of interactivity into a webpage. There will be more, so stay tuned.

The React components for the eight functions are in lines 86-102 of the index.html file. The function definitions used by each component are defined toward the bottom of the code in that file. You could change these around using Javascript to make these functions fit with your vision of this activity for students. The file is self contained, so you share just the HTML file you change with students, the page will function correctly.

Happy coding!

1 Comment

Filed under computational-thinking, reflection, teaching stories

## The Incredible Growing Bricks

I put together this three-act activity two years ago, and decided to include it in the playlist for this year's Math 9 course. The students got right to work in figuring out the total mass of the three bricks together.

This time, I circulated the actual bricks among the students as they worked. I opted not to do this two years ago because I wanted to force them to use the dimensions in the image above to find their answers. The result was that some students chose to make the measurements themselves rather than use the image. This yielded some great interactions between students asking if the bricks were proportional to each other, and those assuming they were proportional. There were some excellent examples of strong explanations involving proportional reasoning among the student work, as well as typical examples of misconceptions, such as the mass being proportional to the scale factor between the sides.

I also did something new with this modeling task and asked students to predict their uncertainty. Often times, students see that they were close to the actual answer revealed in the third act (but not exactly equal), and subsequently classify their answers as wrong. The uncertainties allow more flexibility in this regard. It also revealed some misunderstanding of the relationship between uncertainty and reporting answers that wasn't unexpected: one student gave 16.895 grams, with an uncertainty of plus or minus 0.1 grams. This is a frequent issue in science classes, but not something I've addressed with mathematics students in the past.

Filed under geometry, reflection

## Trigonometric Graphs and the Box Method

Graphing trigonometric functions is a crucial skill, but there is a lot of reasoning involved in the process. I learned the method that follows from a colleague in my first years of teaching in the Bronx, and used it later on when I actually taught Algebra 2 and PreCalculus.

Here's an example for f(x) = 2 cos(2x) + 3:

Students first (lightly) draw a box that is one period wide, twice the amplitude tall, and centered vertically on the midline of the sinusoid. The process of doing this isolates the reasoning about transformations of the function from the actual drawing of the graph, which also takes some skill. I usually ask students to state the period, amplitude, and average value anyway, but this method implicitly requires them to find these quantities anyway. We use the language of the amplitude, period, and average value to describe this box and the transformations  of the parent function.

Once the box is in the right place, then we can focus on the graphing details. Is it a cosine or sine? How does the graph of each fit into the box we have drawn? Where does the curve cross the midline? This conversation is separate from the location and dimension conversations, and this is a good thing. The shape of the curve merits a separate line of reasoning, and encouraging that separation through this method reduces the cognitive demand along the way. I have also seen that keeping this shape conversation separate has reduced the quantity of the pointy sawtooth graphs that students inevitably produce.

I have considered doing this for other parent functions, but haven't been convinced of the potential payoff yet compared with the perfect fit thus far of the trigonometric family of functions. Thoughts?

Filed under Array

## Hosting Meteor JS Applications - My Process on Webfaction

There has been some interest expressed here and on Twitter for a description of the process of moving an application from the free meteor.com to my own server. I run my web applications (such as this blog) on Webfaction, which gives a lot of flexibility for setting up a number of different hosted services. I won't say each step below is easy and as straightforward as it sounds, so feel free to add your difficulties to the comments below, and I'll elaborate. Having the steps in one place will hopefully be as useful for you as it likely will be for me.

Note: Steps 1 - 3 in the list below are just for downloading data you may have collected in an online application onto your computer. If you don't care about the online data, you can skip down to step 4.

1. I installed Mongo on my local machine from http://www.mongo.org
2. In a terminal on my local machine, I entered `meteor mongo --url my-app.meteor.com` to get the username, password, url, port, and database name from the *.meteor.com website. The result is a URL that contains the information you will need for the next step: (username:password@url:port/databasename)
3. save the data onto my local computer using `mongodump --host hostname:port --db databasename -u username -p password -o ~/Desktop ` where the information from the last step replaces the username, password, etc.
4. To create a packaged version of the app that runs on my computer, go to the Meteor project directory, and type ` meteor build --architecture os.linux.x86_64 ./ ` . This assumes that the server that will be running your project is a Linux machine...but this assumption is probably a good one for most hosting services.
5. Create a free account on mLab (http://www.mlab.com). The free account gives you 500 MB of space for databases. For comparison, the total size of my most-used application is only 2.7 MB. I think you'll have enough space. The important thing to get, once you have a database set up, is called the URI. If you create a database, there is a box at the top of the page that gives you information that looks like this: ` mongodb://:@dsXXXXXXX.mlab.com:PORT/DATABASENAME` which you will need in the next step.
6. I learned the Webfaction specific steps from reading this page, though you won't need to do the Mongo steps if you are using mLab as I am. On the Webfaction server, you must install node. You can do this by following the steps here. You must also create a custom WebSockets application. This will give you a port number that you will need in the step below where local variables are set.
7. Again, this is a Webfaction specific instruction, but you need to attach the application to a domain/URL where your application will be accessed. Save this URL
8. Once you have created the WebSockets app, upload the .tar file (from step 4) to the /webapps/web-sockets-application directory through ftp and unpack with `tar -xzf my-app.tar.gz`
9. Using the mLab information from step 5, the port number from step 6, and the location of the node application from step 6, you'll need to fill in the appropriate parts of the commands below:

``` export MONGO_URL="mongodb://:@dsXXXXXXX.mlab.com:PORT/DATABASENAME?autoReconnect=true" # 2 export MAIL_URL='email-address-for-sending-mail-from-your-app' export PORT="PORT-NUMBER-FROM-STEP-6" export ROOT_URL="DOMAIN-FOR-APPLICATION-FROM-STEP-7" export PATH=~/webapps/node/bin/:\$PATH ```

10. Almost there, folks. Inside the unpacked .tar directory (which will be called /bundle), enter the /bundle/programs/server/packages directory and run `npm install`
11. Go back to the /bundle directory and run `nohup node main.js `. This terminal command will run a command continuously on the shell, even if you close the terminal.
12. If you've made it this far, you might just have your Meteor app running on your own server.

I hope this helps you, but if you run into trouble, throw a comment below and I'll see if I can help.

Filed under computational-thinking, programming

## Taking Time Learning Math: A Student's Perspective

Yesterday was our school's student led conference day. I've written previously on how proud these days make me as an educator. Whens students do genuine reflection on their learning and share the ups and downs of their school days, it's hard not to see the value of this as an exercise.

During one conference, a student shared a fascinating perspective on her learning in math. This is not the usual level of specificity that we get from our students, so I am eager to share her thinking. Here's the student's comment during the conference:

“It isn’t that I don’t like math. Learning takes time in math, and I don’t always get the time it takes to really understand it.”

I asked her for further clarification, and this was her response:

...Math is such an interesting subject that can be “explored” in so many different ways, however, in school here I don’t really get to learn it to a point where I say yeah this is what I know, I fully understand it. We move on from topic to topic so quickly that the process of me creating links is interrupted and I practice only for the test in order to get high grades.

It's certainly striking to get this sort of feedback from a student who is doing all the things we ask her to do. The activities this student is doing in class are not day-after-day repetitions of "I do, we do, you do" - we do a range of class activities that involve exploring, questioning, and interacting with other students.

This student's comment is about limitations of time. She isn't saying that we aren't doing enough of X, Y, or Z - quite the contrary, she just is asking for time to let it sink in. She doesn't answer the question of what that time looks like, but that's not her job, it's ours.

I know I always feel compelled to nudge a class forward in some way. This doesn't mean I moving through material more quickly, but I do push for increased depth, intuition, or quality conversation about the content in every class period. Her comment makes me realize that something still stands to be improved. Great food for thought for the weekend.

Filed under reflection, teaching philosophy

## My Journey with Meteor as a Teacher-Coder

Many of you may know about my love for Meteor, the Javascript framework that I've used for a number of projects in and around the classroom. I received an email this morning alerting me (and the many other users) that the free hosting service they have generously offered since inception would be shutting down over the next month.

To be honest, I'm cool with this decision. I think it's important to explain why and express my appreciation for having access to the tool for as long as I have.

I started writing programs to use in my classroom in earnest in 2012. Most of these tended to be pretty hacky - a simple group generator and a program to randomly generate practice questions on geometric transformations were among these early ones. The real power I saw for these was the ability to collect, store, and filter information that would be useful for teaching so that I could focus my time on using that information to decide on the next steps for my students. I took a Udacity course on programming self-driving cars and on web applications and loved what I learned. I learned to use some Python to take some of the programs I had written early on and run them within web pages. I built some nifty online activities inspired by the style of Dan Meyer and put them out for others across the world to try out. (Links for these Half-Full and Shapes tasks are below.) It was astounding how powerful I felt being able to take something I created and get it out into the internet wilderness for others to see.

It was also astounding how much time it took. I learned Javascript to manage the interactivity in the web page, and then once that was working, I switched to Python on the server to manage the data coming from users. For those that have never done this sort of switching, it involves a lot of misplaced semicolons, tabs, and error messages. I accepted that this was the way the web worked - Javascript in front, and Python (or PHP, Rails, Perl, etc.) on the back end. That extra work was what kept someone like me from starting a project on a whim and putting it together. That cost, in the midst of continuing to do my actual job of teaching and assessing students five days a week, was too great.

This was right around the summer of 2013 when a programmer named Dave Major introduced me to Meteor. I did not know the lingo of reactivity or isomorphic Javascript - I just saw the demonstration video on YouTube and thought it was cool. It made the connection between the web page and the server seamless, eliminating the headaches I mentioned earlier. Dave planned to put together some videos and tutorials to help teachers code tools for the classroom using Meteor, and I was obviously on board. Unfortunately, things got in the way, and the video series didn't end up happening. Still, with Dave's help, I learned a bit about Meteor and was able to see how easy it was to go from an idea to a working application. I was also incredibly impressed that Meteor made it easy to get an application online with one line: `meteor deploy (application-name here) `. No FTP, no hostname settings - one line of code in the terminal, and I could share with anybody.

With that server configuration friction eliminated, I had the time to focus on learning to build truly useful tools for myself. I created my standard based grading system called WeinbergCloud that lets students sign up for reassessments, earn credit for the homework and practice they did outside of class, and see the different learning objectives for my course. I created a system for my colleagues to use to award house points for the great things that students did during the school day. I made a registration and timing system for our school's annual charity 5K run that reduced the paperwork and time required of our all volunteer staff to manage the hundreds of registrants. I spoke at a Meteor DevShop about this a year and a half ago and have continued to learn more since then.

Most importantly to me, it gave me knowledge to share with a class of web programming students, who have learned to create their own apps. One student from last year's class learned about our library media specialist's plan to hold a read-a-thon, and asked if he could create an interactive website to show the progress of each class using, you guessed it, Meteor. Here's a screenshot of the site he created in his spare time:

And yes, all of these apps have been hosted on the free deploy server at *.meteor.com, and yes, I will have to do the work of moving these sites to a new place. The public stance from Meteor has been that the free site should not really be used for production apps, something I've clearly been doing for over two years now. I re-read that line on the documentation website back in January and asked myself what I would do if I no longer had access to that site. The result: I did what I am paid to do as a master learner, and learned to host a site on my personal server. That learning was not easy. The process definitely had me scratching my head. But it also meant that I had a better understanding of the value that the free site had given me over my time using it.

The reality is that Meteor has clearly and publicly shifted away from being just being that framework that has a free one line deployment. The framework has so much going for it, and the ability to create interesting apps is not going away. The shift toward doing what one does best requires hard choices, and the free site clearly was something that did not serve that purpose. It means that those of us that value the free deploy as a teaching tool can seek other options for making it as easy to get others in the game as it was for us.

Meteor has helped me be better at my job, and I appreciate their work.

As promised, here are those learning task sites I mentioned before:

Filed under programming, teaching philosophy, Uncategorized

## Choosing the Next Question

If a student can solve $3x - 1 = 5$ for x, how convinced are we of that student's ability to solve two step equations?

If that same student can also solve $14 = 3x + 2$ , how does our assessment of their ability change, if at all?

What about $-2-3x= 5$ ?

Ideally, our class activities push students toward ever increasing levels of generalization and robustness. If a student's method for solving a problem is so algorithmic that it fails when a slight change is made to the original problem, that method is clearly not robust enough. We need sufficiently different problems for assessing students so that we know their method works in all cases we might through their way.

In solving $3x-1 = 5$ , for example, we might suggest to a student to first add the constant to both sides, and then divide both sides by the coefficient. If the student is not sure what 'constant' or 'coefficient' mean, he or she might conclude that the constant is the number to the right of the x, and the coefficient is the number to the left. This student might do fine with $10 =2x-4$ , but would run into trouble solving $-2-3x = 5$ . Each additional question gives more information.

The three equations look different. The operation that is done as a first step to solving all three is the same, though the position of the constant is different in all three. Students that are able to solve all three are obviously proficient. What does it mean that a student can solve the first and last equations, but not the middle one? Or just the first two? If a student answers a given question correctly, what does that reveal about the student's skills related to that question?

It's the norm to consider these issues in choosing questions for an assessment. The more interesting question to me theses days is that if we've seen what a student does on one question, what should the next question be? Adaptive learning software tries to do this based on having a large data set that maps student abilities to right/wrong answers. I'm not sure that it succeeds yet. I still think the human mind has the advantage in this task.

Often this next step involves scanning a textbook or thinking up a new question on the spot. We often know the next question we want when we see it. The key then is having those questions readily available or easy to generate so we can get them in front of students.

Filed under teaching philosophy, Uncategorized

## Standards Based Grading & Streamlining Assessments

I give quizzes at the beginning of most of my classes. These quizzes are usually on a single standard for the course, and are predictably on whatever we worked on two classes before. I also give unit exams as ways to assess student mastery of the standards all together. Giving grades after exams usually consists of me looking at a single student's exam, going standard by standard through the entire paper, and then adjusting their standards grades accordingly. There's nothing groundbreaking happening here.

The two downsides to this process are that it is (a) tedious and (b) is subject to my discretion at a given time. I'm not confident that I'm consistent between students. While I do go back and check myself when I'm not sure, I decided to try a better way. If you're a frequent reader of my blog, you know that either a spreadsheet or programming is involved. This time, it's the former.

One sheet contains what I'm calling a standards map, and you can see this above. This relates a given question to the different standards on an exam. You can see above that question 1 is on only standard 1, while question 4 spans both standards 2 and 3.

The other sheet contains test results, and looks a lot like what I used to do when I was grading on percentages, with one key difference. You can see this below:

Rather than writing in the number of points for each question, I simply rate a student's performance on that question as a 1, 2, or 3. The columns S1 through S5 then tally up those performance levels according to the standards that are associated with each question, and then scale those values to be a value from zero to one.

This information was really useful when going through the last exam with my ninth graders. The spreadsheet does the association between questions and standards through the standards map, so I can focus my time going through each exam and deciding how well a student completed a given question rather than remembering which standard I'm considering. I also found it much easier to make decisions on what to do with a student's standard level. Student 2 is an 8 on standard 1 before the exam, so it was easy to justify raising her to a 10 after the exam. Student 12 was a 7 on standard 4, and I left him right where he was.

I realize that there's a subtlety here that needs to be mentioned - some questions that are based on two or three standards might not communicate effectively a student's level with a single 1, 2, or 3. If a question is on solving systems graphically, a student might graph the lines correctly, but completely forget to identify the intersection. This situation is easy to address though - questions like this can be broken down into multiple entries on the standards map. I could give a student a 3 on the entry for this question on the standard for graphing lines, and a 1 for the entry related to solving systems. Not a big deal.

I spend a lot of time thinking about what information I need in order to justify raising a student's mastery level. Having the sort of information that is generated in this spreadsheet makes it much clearer what my next steps might be.

You can check out the live spreadsheet here:

Standards Assessment - Unit 5 Exam

## Boat Race, Revisited

A couple of years ago, I was impressed with Dan Meyer and Dave Major's creation of Boat Race, an activity that involved navigating around buoys with some knowledge of bearings. I hoped to use his creation for my ninth graders two years ago, but Boat Race in its original form was zapped from the interwebs. At the time,  I did an analog version, which you can find here in PDF form:

07 - CW - Boat Race

This year, when looking at my materials in the revamped Math 9 course, I felt compelled to take a crack at my own digitization of this activity.  Here's the result:

You can also visit the live site here and try it out yourself.

Boat Race

The moving circle moves painfully slow by design. Students will (hopefully) be compelled to do a good job of calculating distances and angles accurately. I plan to give them the analog version on paper for planning purposes. Shortest time by the end of the class wins fame and glory.

Filed under computational-thinking, geometry

## An Easy Transformation: Right Triangle Trigonometry

From Haese and Harris MYP 9:

I was looking for problems to give my students as applications of the right angle trigonometry from our previous class. The problem is essentially equivalent to the basic questions requiring them to find an unknown side or angle - the work is all done for them. One of my standards is all about parsing a word problem for the information needed to answer it, and this question does not require students to do any parsing.

I removed all of the measurements, and this problem became remarkably more demanding:

This will certainly prompt more conversation than in its original form.

It's embarrassing how easy it was to make this change - I anticipate a nice payoff in class.