Education ∪ Math ∪ Technology

Month: October 2013 (page 1 of 2)

How do you define variable?

I recently read the Common Core standards for Math for grade 6, which is where the concept of variable appears to be introduced. The standard in question reads:

Write, read, and evaluate expressions in which letters stand for numbers.

I tweeted this out, and one person responded with this observation:

Clearly this definition of variable is limited, and possibly misleading.


How do you define variable with your students? How do you introduce the topic?



Getting started with programming JavaScript

JavaScript is one of the primary web programming languages. It is the language used in Google Apps Scripting, and in HTML5 interactive documents. Chances are pretty good that most of the interactive features that you see in websites were coded in JavaScript.

Much of the power of JavaScript comes from its power to manipulate and modify HTML, which is the markup language that websites are written in. Therefore, if you want to learn JavaScript, you will sooner or later need to learn HTML (and CSS) as well.

You will want these things to get started:

  1. At least one web browser. This will be where you run your code. I recommend using Firefox with the extension Firebug installed. Firebug is incredibly useful for finding bugs in your code, and for debugging. Google Chrome is also useful for some simple debugging as well. Note: JavaScript does not always run the same in every browser, so one reason to have multiple browsers is to test that the code you write works in all of them.
  2. A good text editor. I recommend Programmer’s Notepad for a PC, and TextWrangler for Mac, both of which are free.
  3. A language reference. I personally like Mozilla’s reference here
  4. Some understanding of how web technologies work. This reference guide, again by Mozilla, is useful.
  5. A folder on your computer dedicated to your projects. It is a good idea to be organized from the beginning.
  6. Blocks of time in your schedule to spend working on programming. Programming is something that will take concentration and focus, and is hard to learn in 10 minute intervals.
  7. I also recommend buying a reference/tutorial book. I found JavaScript for Dummies helpful as a starting place, for example.
  8. A JavaScript library that professionals use (I recommend jQuery). This will allow you to create more interesting projects, while at the same time helping you develop an understanding of JavaScript ideas.
  9. An understanding of the design cycle is useful.

Once you have the things above, my recommendation is try to duplicate someone else’s project. This will help you ensure that you have your development process set up correctly on your computer. Your JavaScript files are loaded by the HTML page you create, and so end up linked in it. Here is a sample template you can use for your JavaScript development. There are a lot of ideas that will want to unpack if you use this template.

Simple HTML template

My workflow for programming in JavaScript is to have the text files I am editing open in one program, and the output open in a browser. When I make changes to my code, and want to see the output, I simply save the changes, and then refresh the page that has my JavaScript in it.

In a future post, I will share a screen-cast of starting my own JavaScript project, and talk through the creation of this project.

So you want to learn to program…

When I was 8, my father gave me my first computer. There were two things I could do on this computer; program or write. I found programming to be much more interesting! So with a little bit of support from my father, and a copy of the Wang BASIC reference manual, I began the long process of teaching myself how to program.

Here is basically what you need to learn how to program:

  1. A goal. Decide on what you want to make. You may find that you can’t make it, but there will probably be some small part of it you can get started on.
  2. You need resources to help with syntax and structure. This could be a good book, or a website, a class, or any combination of the above. I tried programming classes a couple of times and found them pretty boring, but this likely does not apply to every programming class. There are lots of interesting books on learning how to program in a variety of languages.
  3. The patience to struggle, especially in the beginning. When you are first getting started, programming is like non-stop problem solving. It is challenging! 
  4. Time. Learning to program, and to do it well, is not something you pick up in an afternoon.

If this is something you are interested in, you can join a new section of the Edtech community on Google+. This is basically intended to be a peer support group, along with (hopefully) some mentors to help people get started, and to help people when they get stuck. While I cannot promise that joining this group will guarantee that you will learn how to program, I think it will help. Just be willing to lurk, and if needed, ask questions in the “Learn to Program” discussion forum of the group.

P.S. If you know anything about programming and are willing to help out, please join the group and introduce yourself in the section called “Learn to Program.” At the very least, feel free to offer suggestions for starting places for teachers.

We are homeschooling our son

My wife and I decided a couple of weeks ago to withdraw our son from our local community school and homeschool him. We realized that the constraints on the school, and the choices made at the school were going to prevent him from getting the exercise, play, and intellectual stimulation he needs to remain healthy in body and mind.

We were required to send a letter to the superintendent of the school district as the first step of our legal requirements in NY state in order to homeschool our son. Here is a copy of that letter (I have edited out the portions that identified which school and which teacher he had).

Dear [Name Withheld],

We are sending this letter of intent as required of Section 100.10 of the Regulations of the New York State Commissioner of Education. We intend to homeschool our son, who is in grade two, for the remainder of the 2013-2014 school year beginning immediately. We would like explain to you why we are making this decision.

We moved here recently from Vancouver, British Columbia, where our son attended a public Montessori school. In this school, our son learned self-regulation, and as such had tremendous control over what he learned, and when he learned it. The school day started with Tai Chi, and our son had a morning recess, a lunch recess, and physical education every day. It was never difficult to get him to school, as he loved going to school.

This is not the situation this year. We are finding it much more difficult to motivate our son to wake up and get ready for school. Instead of telling us how much fun he had and how much he learned at school, we have to struggle to pull out of him what happened each day at school.

This is not because of our son’s current teacher. She is a caring, hard-working, and thoughtful person. We have no complaint with her character or her ability to teach.

We are concerned about the prescribed teaching method our teacher has been asked to use, specifically the excessive test preparation. We could easily offer this type of education ourselves after a trip to the local education supply shop. What then is the benefit of sending him to school? What the children need is to interact with each other and have at least some time to make their own discoveries which requires independent time and exploratory activities within the classroom.

Our son’s school does not offer a morning recess. At lunch time, unless they are participating in one of the supervised events occasionally run by staff, the children at my son’s school cannot run around. My son only has physical education once a week during which he has so far learned how to walk around a gym. Fortunately my son does have one dance class and one art class each week, but overall, he does not receive sufficient physical activity through school to keep him healthy and to help him focus.

We believe that children need these essential elements in order to become healthy adults; creativity, play, intellectual stimulation, exercise, and opportunities to collaborate with and learn from their peers. None of these elements is present in at my son’s current school in a sufficient degree. We can see that the school is fighting a losing battle to maintain some physical activity and art, and from the research we have done, these are often lacking in many of the public schools in New York City.

My son recently told us that his current school is the “No fun school” but that he is “learning to adjust to it”. Our fear is that in an effort to make school more academically rigorous, many of the things that make school worth attending are being removed.

The definition of rigorous, according to the Merriam-Webster dictionary, is:

  • very strict and demanding

  • done carefully and with a lot of attention to detail

  • difficult to endure because of extreme conditions

With the exception of the attention to detail, we do not believe that these things should be applied to school, particularly not school for children ages 5 to 11. Our son’s current school is teaching children that learning is a chore to be done, and not something to enjoy and to love.

We understand that relatively recent legislation in New York State, where teachers and principals are judged based largely on the test scores of their students, is to blame for this situation. This legislation seems like a gigantic experiment that lacks sufficient evidence to justify such a punitive policy and we see no reason to experiment with our son’s education in this way.

Instead of allowing our son’s school to drain the love of learning from our son, we are removing him from your school’s care.


David Wees
Vasilia Wees

Using technology to facilitate noticing and wondering

Today I observed a teacher using this tool built by Jennifer Silver to engage her students in mathematical reasoning. It was a powerful reminder to me of the intersection between effective uses of technology to provoke thinking in students, and the pedagogy used to support that student reasoning.

First, the teacher brought up the interactive diagram up on her Smartboard, and then she asked a student to come up to change the slider values. She repeatedly asked students to say what they noticed each time the slider was changed. She took the time to have multiple students clarify what they said, to have their peers restate and respond to each other’s reasoning, and to have students take the time to make mathematical observations. She engaged students in collaborative mathematical thinking for 30 minutes. At the end of the class, at least 10 students came up en masse to play with the interactive diagram themselves and continued to ask her questions and make observations. She had to promise them she would email them the link to the diagram so that they could continue to play with it themselves.

The point here is that the technology made the conversation easier. Instead of creating 20 different examples of graphs and seeing what happens as each variable is changed, students were able to visualize the changes, both in the graph representation, and in the formula representation. When asked if they noticed anything after the “Point on the line” slider was changed, one student said they noticed the Intercept-slope form of the equation did not change. Another student responded to him with “that form of the line doesn’t depend on which points you use.”

It was fantastic.

What does your online assessment actually measure?

What exactly does our assessment measure? I watched my 7 year old son complete an online assessment of his fluency with addition facts last week, and I noticed a few things the assessment measured unintentionally, at least to some degree.

  • It measured his ability to decode the symbols presented (eg. 2 + 10 = __).
  • It measured his ability to find the keys needed to answer the question on the keyboard.
  • It measured his ability to combine the pairs of numbers together to come up with an answer.

The tool used appeared to make an incorrect assumption – that it was measuring fluency with an addition fact exclusively. In fact the feedback it offered to me the parent said exactly that.

However, there were some important things not measured by this online tool.

  • It did not measure his ability to explain his reasoning to others.
  • It did not ask him to show multiple solutions for finding his answer.
  • It did not present a meaningful context, and measure my son’s ability to apply his understanding to that context.
  • It did not check to see if my son had gained any transferable understanding.
  • It did not allow my son to talk to peers about his solution.

When we look at any assessment tool, we should ask ourselves, what is possible to measure with this tool, what may be unintentionally measured with this tool, and most importantly, what is not measured with this tool.

Virtually all digital assessment tools I have looked at are very good at the low-hanging fruit of automated responses to trivial questions, and almost none of them help answer more important questions about student understanding. 

Benny’s Conception of Rules and Answers in IPI Mathematics

My colleagues and I have formed a journal study group where we intend to share pieces of research which are interesting, and have some compelling story to tell about understanding research. I’ve chosen Benny’s Conception of Rules and Answers in IPI Mathematics by Stanley Erlwanger. In order to support our discussion of the research, I’ve created a few slides which I have shared below. I recommend opening up the speaker’s notes in order to understand the presentation better.

Ten rules for living

  1. Life isn’t fair. Do something about it.
  2. The world may not care about your self-esteem, but having a healthy self-image is more likely to lead to positive relationships with the people to whom you are close and having healthy relationships is the key to having a happy life.
  3. Don’t focus on the money. Focus on improving yourself and doing good work. Both of these are more valuable than money.
  4. Don’t work for a boss. Work for yourself. Be tough on yourself, but not so much that you crack. Life is hard work, and it’s short, and there’s no reason you should spend it following someone else’s orders. If you work for yourself, you don’t need to start at the bottom and work your way up.
  5. You will make mistakes. Don’t look to blame someone, learn from them.
  6. Always clean up after yourself. Fix your mistakes, own up to them, before you look to fix other people’s mistakes. Expect others to own up to what they’ve done wrong.
  7. In life you’ll find that those in power define winning and success so they are doing it, and you are not. Don’t let other people define for you what it means to be successful.
  8. Don’t waste time worrying about what other people think of you. Vanity can be the death of innovation because every moment spent worrying about what the world thinks is time spent not thinking for yourself.
  9. Don’t watch television. It will fill your mind with stereotypes, none of which will help you find your own path in life.
  10. "Be kind, for everyone you meet is fighting a hard battle." ~ Plato
  11. Bonus: Be insatiably curious about the world.

Hacking the webinar format

How screen space is allocated in webinars

I’ve recently been tinkering with the format of an online webinar in an effort to make what happens in a webinar more engaging and "minds-on" for teachers.

The problem, as I see it, for most webinars is that the presenter spends 99% of the time talking during the webinar, while the participants, if they are lucky, are in a chat window, occasionally discussing what the presenter is sharing. I feel fairly certain that almost everyone in a webinar will at some point switch the window to work on some other task, whether it is tweeting, reading their email, or something else not related to the presentation.

It is important to remember that a webinar is not worth having if no one learns anything from the experience, and no one even considers if their current practice needs to change. It is equally important to remember that a webinar is not worth having if the participants would have been better off reading a book on the topic instead. For the cost of some webinars, participants could potentially buy several books to read themselves at their leisure, rather than have someone they cannot see read it to them. A free webinar is not really free; it has the cost of the time participants spend watching it.

What I have discovered is making a webinar something people can participate in more easily is challenging for a variety of reasons.

  1. Most webinar software is designed to be something a presenter uses to share information. The bulk of webinar mechanics seem to be to ensure that participants are able to hear a presenter read their slides clearly.
  2. While many webinar packages allow for participants to share their screens, the software limits screen-sharing to one participant at a time. I have yet to see a single webinar software package that enables participants to form smaller groups and share what they are doing with each other.
  3. Screen space in a webinar is almost entirely devoted to what the presenter is sharing, rather than the discussion space where participants could potentially talk about the presentation. Imagine the physical space of your classroom was apportioned in the same way. Wouldn’t that seem a little bit ridiculous?
  4. Application sharing, if it is present in the webinar software, virtually always restricts access to the application being shared to the presenter. Even if I wanted to give a participant a chance to play with a particular piece of software in front of the whole group, it is not possible to do so.


Here are some things I have done to try and alleviate some of these issues.

  1. I share links to Google Docs and Spreadsheets in my webinars, so that participants can leave the webinar software and collaborate with each other to build something useful.
  2. In a recent presentation on Programming in Math Class, I set aside 20 minutes of the webinar time so that participants could actually use the software I was recommending, and see what they could build with it.
  3. I plan for questions from teachers to drive at least a portion of the presentation, and I modify my presentation as much as I can in order to suit the needs of the participants. This addresses the question; why I am teaching this, to this particular group of learners, in this particular way? If you don’t at least respond to, and modify your presentation based on questions from participants, you might as well have sent them a link to a video instead. At least they could pause and rewind the video as needed.
  4. There are some software packages out there that address some of these concerns, but all of them have some pretty serious limitations. Google Hangouts allow for live document editing and application sharing within the Hangout window, but are limited to 10 participants. Mikogo addresses some of these issues as well, but is limited to 25 participants.


Here is what I would like to be able to do with webinar software.

  1. I need to be able to form participants into groups, which are either randomly chosen, self-selected, or grouped by the presenter. In each group, participants should have shared access to software as designated by the presenter, or as each participant feels the need to share.
  2. I’d like to be able to switch the view of the presentation as necessary so that the slides can shrink when they are unimportant and the shared workspace or chat window could expand as necessary.
  3. I really want to be allow participants to collaborate on objects in any software during a presentation which they can take away with them later, and therefore exist outside of the presentation space.
  4. I want to be able to designate presentation pathways, based on decisions I make during the presentation. If a group wants to see an example in more detail, or has a particular set of needs, I’d like to be able to switch the direction of the presentation on the fly, and choose a different set of slides to use to frame our discussion as needed.
  5. This is a minor point, but I’d like to be able to do polls and get responses from participants within the software itself, in a way which makes sharing the results with all of the participants visible and intuitive to understand.


Imagine you are in charge of designing webinar software for the future. What do you want it to include?


Explaining subtracting negative numbers to my wife

Tonight my wife asked me why 4 – (-3) = 7. Apparently my son had "explained" it to her earlier tonight, but she hadn’t really understood his explanation. So I gave it a shot.

First I tried the same explanation that seemed to work for my son when he asked me what 4 – (-3) would be earlier today.

Me: "Imagine you had 4 blue circles, representing 4 positive 1s or 4 total. To this you add 1 positive circle and one negative circle. What would be the new total?"
My wife: "Still four. You added 0."
Me: "Okay, so let’s add 0 like this 3 times so the total will still be 4, right?"
My wife: "I get it. I don’t know why we are doing this, but okay."
Me: "So now, I take away the three negative circles, and therefore I am taking away a total of negative three from this picture and I’m left with 7. Therefore 4 – (-3) = 7."
My wife: "I don’t get it."


I tried a number line representation.

My wife: "I like looking at things on the number line."
Me: "Now imagine I’m on the number line at 4. If I subtracted 3, I’d end up at 1, so if I subtract -3, I must do the opposite, and so I end up at 7."
My wife: "I don’t get it."


Next, I tried operation consistency.

Me: "Okay, let’s try again. You agree that 4 + -3 = 1, right?"
My wife: "Yes. I understand that."
Me: "And 4 + 3 = 7 is obvious to you, as is 4 – 3 = 1. 4 – (-3) can’t be the same as 4 – 3, so it must be that 4 – (-3) = 7."
My wife: "That makes no sense."


I tried going back to the number line.

Me: "Okay, so what’s the distance between 5 and 2 on the number line?"
My wife: "3."
Me: "Right, since 5 – 2 is 3. Basically, one way to think of subtraction is that it gives you the distance between two points on the number line."
My wife: "Aaaaaah, I get it now."
Me: "So what’s the distance between 4 and -3 on the number line?"
My wife: "7. I get it, thank you so much. Who invented this rule anyway?"

(Aside: This is not quite true. -5 – -3 is a pretty good counter-example, but I’ll talk to my wife about that tomorrow.)

I decided to go back to mathematical consistency in a different way.

Me: "Okay, let’s look at the following pattern. 4 – 3 = 1, 4 – 2 = 2, 4 – 1 = 3, 4 – 0 = 4. What do you notice?"
My wife: "Well the thing you are subtracting is getting smaller, and the answer gets bigger."
Me: "What do you think would happen if I subtracted -1? What would make this pattern consistent?"
My wife: "Well 4 – (-1) would have to be 5 then."
Me: "Right. And so then 4 – (-2) = 6, and 4 – (-3) = 7."
My wife: "Okay. I’m going to ask our son to try and explain this to me tomorrow, pretending I don’t get it. I’ll let you know if his explanation makes more sense then."


Do you have any other models I can use, should this question come up again?