Education ∪ Math ∪ Technology

Month: June 2012 (page 1 of 2)

Technology influences cognition

Technocriticism Wordle

That technology can influence cognition should be painfully obvious when you examine our primary technology of communication – language. Someone who knows a language cannot choose to ignore that language when confronted with it. If you are a literate person, letters arranged together do not appear randomly placed, they form words. When someone talks, you cannot hear it as babble, you are forced by how this technology has influenced your thinking to hear words.

When one looks at language, it is nonsensical to ask if one has a choice whether or not to use this technology. Once proficient in a language, barring a severe brain trauma, one remains proficient in that language and has it forever more alter their thinking. You can’t choose not to use a language once you have it and are exposed to it anymore than you can choose your parents.

Different languages and the accumulation of culture (another technology) that goes along with them result in different ways of thinking. One of the reasons why translation is so difficult between cultures is because quite often cultures have concepts which are unique only to their culture, which tells us that differences in the technology of culture results in different types of thinking. In the same way, people who are proficient in digital cultures have their own thinking altered by their participation in those cultures.

It is possible that there are technologies which do not influence our cognition. It may be, for example, that we are not influenced by our cell phones (which incidentally, whatever their value may be, a device which allows almost anyone in the world to interrupt you no matter what you are doing) to the degree that they influence our thinking.

I do not think this is true though. When you look at cell phone use in particular, you will no doubt recognize that possession of a cell phone and knowledge in its use means that if you are planning to meet someone else, who also posses a cell phone and knows how to use it, you are less likely to carefully plan exactly when and where you will meet them. So just possessing and knowing how to use a cell phone changes your behaviour, and changes how you plan your life. If one possesses a smart phone, and are at all proficient in its use, one generally stops planning exactly how one will travel somewhere in advance.

My strong suspicion, although I cannot yet prove this, is that all technologies include different types of thinking which are a necessary part of using the technology, and that while the influences of technology are not deterministic — we have some free will in our use of technology, I do not think that someone who is not cognicent of the limitations of their technology will see those limitations.

The benefits of technology use are generally easily apparent. What are usually less apparent are the drawbacks. So instead of blindly using technology without regard to the potential drawbacks, we need to be considerate of its use, and be critical of how it has changed us. We need to be technocritical as users, and those of us who are experts in technology use must especially be experts in critical reasoning around its use.

Glass is half-full

A CNN report on a survey done by the digital security company McAfee (which reads more like an ad than a report – what happened to investigative reporting?) has some startling statistics. According to the CNN sanitization of the survey:

  1. Clear browser history (53%)
  2. Close/minimize browser when parent walked in (46%)
  3. Hide or delete IMs or videos (34%)
  4. Lie or omit details about online activities (23%)
  5. Use a computer your parents don’t check (23%)
  6. Use an Internet-enabled mobile device (21%)
  7. Use privacy settings to make certain content viewable only by friends (20%)
  8. Use private browsing modes (20%)
  9. Create private e-mail address unknown to parents (15%)
  10. Create duplicate/fake social network profiles (9%)


On the flip side, this survey says something else as well:

  1. 47% of teenagers don’t clear their browsing history – either out of ignorance or because they trust their parents.
  2. 54% of teenagers feel no need to close their browser just because their parent walked in.
  3. 66% of teenagers either do not delete their IMs, do not delete videos, or both.
  4. 77% of teenagers tell the truth about online activities.
  5. 77% of teenagers use a computer their parents may check.
  6. 79% of teenagers do not have a smart phone.
  7. 80% of teenagers do not apply friends-only privacy settings (but I wonder how many of these teenagers are only posting innocuous content online).
  8. 80% of teenagers do not use private browsing.
  9. 85% of teenagers share all of their email addresses with parents.
  10. 91% of teenagers do not create fake or duplicate social network profiles.


While the Internet certainly has the potential to amplify poor behaviour, the fact that so many teens are using it in an open and public way, and that we have such little media coverage of their poor behaviour online (which it seems to me would be highly publicized if it existed), suggests that maybe teenagers today are okay. The Internet is a communication tool, and historically, teenagers have struggled to use communication tools in a highly appropriate way at all time – which is not surprising given that they are in a state of developing identity and understanding the nuances of society.

Kindergarteners programming

Here are two sample programs from a pair of kindergarten classes today (I took screen-shots of their program, and cropped them to fit in this blog).


Program 1:

Program 1


Program 2:

Program 2


I started the kindergarteners off the same way I started off third graders last week – they were to program me, and then program their partner. It worked fairly well, as most of the kindies could figure out how to get me to move in a square fairly easily, but an L turned out to be a stumper for a while, and a T was super hard. One could easily have done this entire activity with some adults (or older kids) willing to stand in as computers and be moved around by the kindies.

The idea of this activity is to get students thinking geometrically and systematically – if I want the computer to draw this shape, what do I need to do to get it to work. The key here is that the kindergarteners have to do the thinking, and what they showed me is that they are capable of some fairly advanced logical reasoning when pushed into it a bit. Most of the kindergarteners were able to get the computer to draw a square, or run way off the screen, and nearly all of their programs involved using the repeat function. I really found the students had to think to be able to do this activity, and to trouble-shoot when their programs didn’t work.

I would not advocate this activity replace moving around time, or other drawing time, but if you are stuck at the end of a year with nothing but worksheets to do, this could be an excellent replacement.

Many ways of learning how to ride a bike

When I learned how to ride a bicycle, I practiced with training wheels first because my parents thought that it would be too difficult for me to learn how to balance myself, steer, and pedal all at the same time. I eventually learned how to ride a bike without training wheels but it was challenging for me. 

With my son, we started him on a like-a-bike which would let him practice balancing on a bicycle (which some parents argued with us was the most difficult part of learning to ride a bike) before having to learn how to pedal. We then later gave him a bicycle with training wheels so he could practice pedalling separately from balancing himself but honestly, we didn’t find it helped much. When he finally had a standard bicycle, he still needed to learn how to pedal while keeping himself balanced.

I’m not sure either method is best. It probably depends on the kid which technique we should use. Maybe we should have just started our son on a regular bicycle.


So now look at mathematics education. Our goal is to have students think like mathematicians, and to know enough mathematics to be able to use it in their thinking. Is it entirely necessary that all of them learn it the same way? Can we find different ways to engage different students in the act of learning how to be a mathematician?

Logo(ish) programming in the browser

Blockly example


I found out through Reddit about a new visual programming language that runs in the browser called Blockly. The system looked pretty good, but wasn’t quite right for my students. Fortunately the Blockly code was fairly easy to figure out, and so I hacked around a bit this weekend, and put together a simplified version of the system for use with my kindergarten students tomorrow. This version allows students to use code to create simple animations. Unfortunately, at this stage the animations cannot be saved.

You can check it out here.

  • It does not run in Internet Explorer. It may not run in a few other browsers as well. I can confirm that it should work in Google Chrome and Firefox, and it probably works in Safari.
  • It’s not done yet. I plan on adding more of the advanced functionality, which exists already in the Blockly language, and should be easily implemented.
  • I plan on beefing up the forward and turn commands so they are similar in power to what can be done in Logo. One of the chief advantages of Logo is that it is both an easy to understand language, and powerful.
  • I plan on implementing save and share features if this system looks reasonably useful.
  • I think the graphics for the turn and forward code blocks could be better. For example, they should show an arrow either turning, or going straight.


Do you have any feedback about how I could improve this programming environment? 


Eduglogger survey

David Warlick shared a survey (due today) for education bloggers. Given the difficulty in gathering research information of this kind, I’ve decided to fill it out as well. Questions and answers below.


Blog URL:


What do you blog about?

I generally blog about education, often focusing on issues such as improving mathematics education, sharing my practice, educational technology, and systems in education.


Are you paid to blog?



What do you do professionally (other than blog)?

I’m a learning specialist for technology and math at an independent school in Vancouver, British Columbia. I’m also the author of a textbook, and I have been a consultant on programming projects.


How long have you been blogging at this site?

I started blogging in 2005, as a way of sharing what was going on in our lives (we were living abroad) with my friends and family back home. I started this particular education blog in November of 2008, when I was planning on looking for work as a portfolio of my thoughts and ideas.


Do you write in other platforms? (e.g. in a print magazine?)

I have 5 or 6 articles printed in a few different print magazines. I also write a monthly article for my schools Imprint magazine. I have several thousands tweets to my name. I have also guest blogged on a few other sites.


Can you remember why you started blogging?

My original purpose for blogging was to create a digital portfolio of some of my best ideas; for the purposes of hopefully showing it to employers later, and getting a job. At some point, I stopped recording what I was doing in my classes so much as my general thoughts on education.


What keeps you blogging?

I use my blog now primarily as a place to reflect and to start discussions on things in education that I see need improvement. Many of my posts are geared toward mathematics education specifically, since this is my main interest, but I have also blogged about school bell schedules, and a host of other issues related to the structure of schools. What keeps me blogging is the desire to get my ideas recorded, and out there in the world, to share my thoughts.


Do you have any idea of the size or character if your audience? How?

According to Google Analytics, I’ve had 127, 822 page views of this blog during 2011 (about 350 views per day), which only counts people that come to the blog to visit it, and not people subscribed to the blog, so the actual number of views of my blog entries is probably higher. Given the nature of what I posted, I suspect that most of these views are from educators, and this is born out from most of the comments on my blog entries.


What’s your attitude to/ relationship with people who comment on your blog?

I’m grateful for everyone who comments on my work. It’s nice to see people reading what I’ve written, and then responding to it. I’ve received a small number of very negative comments on my posts, and given the over-whelming majority of positive and/or thought provoking comments, it’s pretty easy to shrug these off.


Do you feel as if you fit into any particular community, network or genre of blogging? (e.g. schools, science, education, museums, technology)

I meander a bit, between technology, mathematics, and other general education topics. I’d say this is really a blog about learning.


If so, what does that community give you?

The primary benefit of the community that my blog fits into is feedback. If I write something that doesn’t make sense, or is wrong, people call me out on it. When I write stuff that people agree with, they often will comment to extend my ideas.


What do you think are the advantages of blogging? What are its disadvantages/ limitations?

The primary advantage of blogging, as I pointed out in the previous question, is feedback from your peers. I think of it as a less formal way of publishing peer reviewed content, much like the formal feedback process in an academic journal. It has also been a place where I can store ideas that I’ve had, and so it in part acts as a memory of the various educational ideas I’ve been experimenting with. A further advantage is that if I share something useful, I may change someone else’s practice, and of course when I read other people’s blogs, it changes my teaching practice.

An inherent disadvantage of blogging, particularly in a society that is heavily focused on avoiding failure, is that when you post a half-baked idea, people can jump all over it, instead of seeing it as an opportunity for you to grow. Also, another problem for beginners who do not end up with a decent audience quickly (I started with an audience of 1 person, my mom!) they get frustrated and many of them quit, which means that some of the reflection they were doing on their teaching practice ends. It’s difficult at times to post your ideas out as they may be controversial or against the standard voices in education. 


Do you tell people you know offline that you’re a blogger? (e.g. your grandmother, your boss)

Yeah, my boss knows I blog here. I doubt he would agree with everything I’ve written, but this space is for my voice, not his. He probably agrees with much of it though as he is a fairly progressive educator. My family knows I blog. Every once in a while one of them will read a post of mine. I even have a comment from my wife on one of my posts! My first comment ever on one of my blog posts was from my mom.

I don’t call myself a blogger though. I think of myself as a philosopher in education who uses writing as his vehicle to record and share his ideas, and my blog is where I store my writing.


Is there anything else you want to tell me about I haven’t asked?

I actually am participating in a few blogs. For example, I’ve written for the Cooperative Catalyst, Edutopia, Dialogue Online, Science with my son, Questions about technology, Educational comics, and I have both a Tumblr and Twitter account as well. 

Programming with 3rd graders

Group 1


I did two things I’ve never done before – I taught 3rd graders, and I introduced elementary school kids to programming.

Yesterday, I started off by talking to the kids about how programs work on a computer, using the analogy that a programming language is like talking to a computer. If you can speak the language of the computer, you can make it do what you want it to do. I pointed out that all of them were much smarter than a computer since they can speak and understand English, and no computers can do this – at least they can’t carry on a conversation that passes the Turing test (yes, I mentioned the Turing test with a group of 3rd graders).

The first activity we did I called "Program your partner." It’s a pretty easy activity to set up. I told the students that computers only understand simple instructions, like step forward and turn left. So I partnered up the students, and I had one act as the programmer, and the other act as the computer. The programmer was to tell the computer what to do but they could only use the commands ‘step’ (which meant take one step forward) and ‘turn’ (which meant turn one quarter turn left).

The first challenge was to get their computer to trace out the path of an L on the floor with their movement. They switched roles for the next challenge, which was to trace out a T. The next challenge was to trace out a square, and then I introduced a new commend which I called repeat. I asked the students, "How could you use the repeat command to make tracing out a square easier?" They had a bunch of ways of describing the command, which all boiled down to exactly what one student said which was "Repeat 4 times – Step then Turn." The final challenge was to trace out 4 squares connected to each other, and for this activity I asked them to double check their program by tracing it out using pencil and paper. It was interesting to me that none of the groups seemed to have any difficulty abstracting from taking steps on the floor, to writing those steps on paper. In fact, every group came up with a slightly different way of writing down the output of their program (many of them included arrows so that they could keep track of the path actually traced).

We then gathered together in front a projector, where I showed the students how to start the Turtle Art software, and showed them the Forward and Right functions. At this point, I got stuck! I had used Turtle Art in Ubuntu, which has a slightly different interface, and I didn’t know how to actually run my program! So we experimented together and played around with the buttons on the screen, making sure to make copious use of the help button, and eventually we discovered the magic wand together. Getting stuck here was a valuable lesson for all of us, because while I explored how to debug my problem, the students got to see my thinking process live.

Next, we had the students self-organize into pairs and get a computer out, and try out Turtle Art for themselves. The first few programs the students created were pretty simplistic, but fairly quickly the students learned how to create more and more complicated shapes. One of their favourite things to do was to use the Forever loop, and see the screen flickering as a particular square or other shape they had created was repeatedly drawn. I didn’t really have anything specific for the students to do with the program at this stage, as I thought some free exploration would be more valuable.

Group 2


Some of the students created complicated loops, and others used the commands to draw. This particular student, as you can see from their screen, had 4 different commands, which she learned how to alternate to produce the diagram above. So she hadn’t learned how to create a procedure to draw the figure above in one go, instead she had internalized what each of the commands did, and then used them as one would use a paint brush.

At the end of the sessions, the students gave it an enthusiastic thumbs up. I discussed the success of the two days with their teacher, and she was pretty impressed with the students ability to come up with representations of their own, and to create somewhat complicated figures within a few minutes of using Turtle Art. We are considering doing a longer unit next year related to the use of Turtle Art.

The popularity of the event was summed up by one student who told me "I give this activity an infinity out of ten!"


ISTE standards without references to technology

Read the ISTE NETS for students with references to technology to technology stripped where possible.


1. Creativity and Innovation

Students demonstrate creative thinking, construct knowledge, and develop innovative products and processes.

a. Apply existing knowledge to generate new ideas, products, or processes
b. Create original works as a means of personal or group expression
c. Use models and simulations to explore complex systems and issues
d. Identify trends and forecast possibilities

2. Communication and Collaboration

Students use appropraite media and environments to communicate and work collaboratively, including at a distance, to support individual learning and contribute to the learning of others.

a. Interact, collaborate, and publish with peers, experts, or others employing a variety of environments and media
b. Communicate information and ideas effectively to multiple audiences using a variety of media and formats
c. Develop cultural understanding and global awareness by engaging with learners of other cultures
d. Contribute to project teams to produce original works or solve problems

3. Research and Information Fluency

Students apply appropriate tools to gather, evaluate, and use information.

a. Plan strategies to guide inquiry
b. Locate, organize, analyze, evaluate, synthesize, and ethically use information from a variety of sources and media
c. Evaluate and select information sources and tools based on the appropriateness to specific tasks
d. Process data and report results

4. Critical Thinking, Problem Solving, and Decision Making

Students use critical thinking skills to plan and conduct research, manage projects, solve problems, and make informed decisions using appropriate tools and resources.

a. Identify and define authentic problems and significant questions for investigation
b. Plan and manage activities to develop a solution or complete a project
c. Collect and analyze data to identify solutions and/or make informed decisions
d. Use multiple processes and diverse perspectives to explore alternative solutions

5. Citizenship

Students understand human, cultural, and societal issues and practice legal and ethical behavior.

a. Advocate and practice safe, legal, and responsible use of information and tools.
b. Exhibit a positive attitude that supports collaboration, learning, and productivity
c. Demonstrate personal responsibility for lifelong learning
d. Exhibit leadership for citizenship

6. Technology Operations and Concepts

Students demonstrate a sound understanding of technology concepts, systems, and operations.

a. Understand and use technology systems
b. Select and use applications effectively and productively
c. Troubleshoot systems and applications
d. Transfer current knowledge to learning of new technologies


What you no doubt notice is that the only standard that requires the use of technology is the sixth standard, which in my opinion is the weakest reason to use technology. Technology is more than a tool, it shapes us, it changes us, it gives us affordances that are otherwise not possible without the technology. It is for these purposes that we should use technology, and our standards should reflect these purposes.

The first five standards are admirable goals, but none of them requires technology specifically, and when we attach the technology terminology with them, and label them digital literacies, we are sending the message that if "my school doesn’t use technology" that these standards are some how optional, and that they are only possible with technology.

This is the wrong message to send.

The difference between instrumental and relational understanding

Stanley Park map


I recently found this article written by Richard Skemp that Gary Davis (@republicofmath) highlighted on his blog . I recommend reading the whole article. Skemp describes the difference between instrumental and relational understanding, and how the word understanding is used by different people to mean different types of understanding. He also makes the observation that what we call mathematics is in fact taught in two very distinct ways. Skemp uses an analogy to try and explain the difference between relational and instrumental knowledge which I would like to explore. 

Imagine you are navigating a park, and you learn from someone else some specific paths to follow in the park. You move back and forth along the paths, and learn how to get from point A to B in the park, and you may even be able to move quickly from point A to B. Eventually, you add more points to your list of locations to which you know how to navigate. Step off any of your known paths though, and you are quickly completely lost, and you might even develop a fear of accidentally losing your way. You never really develop an overall understanding of what the park looks like, and you may even not know about other connections between the points you know. This is instrumental understanding.

Imagine that instead of navigating the park by specific paths shown to you, you get to wander all over the park. For some parts of the park you may be guided, through other parts of the park, you wander aimlessly. In time, you develop an overall picture of the park. You might discover the shortest paths between two points, and you might not, but you would understand the overall structure of the park, and how each point in the park is related to each other point. If someone showed you a short-cut in the park, you’d probably understood why it worked, and why it was faster than your meandering path. You wouldn’t worry about stepping off the path though, since even if you get lost, you’d be able to use your overall understanding to come to a place you know. This is relational understanding.

Here’s Richard Skemp’s description of the analogy.

“The kind of learning which leads to instrumental mathematics consists of the learning of an increasing number of fixed plans, by which pupils can find their way from particular starting points (the data) to required finishing points (the answers to the questions). The plan tells them what to do at each choice point, as in the concrete example. And as in the concrete example, what has to be done next is determined purely by the local situation. (When you see the post office, turn left. When you have cleared brackets, collect like terms.) There is no awareness of the overall relationship between successive stages, and the final goal. And in both cases, the learner is dependent on outside guidance for learning each new ‘way to get there’.

In contrast, learning relational mathematics consists of building up a conceptual structure (schema) from which its possessor can (in principle) produce an unlimited number of plans for getting from any starting point within his schema to any finishing point. (I say ‘in principle’ because of course some of these paths will be much harder to construct than others.) This kind of learning is different in several ways from instrumental learning.” ~ Richard Skemp, Mathematics Teaching, 77, 20–26, (1976)

Instrumental understanding is really useful when you have to know how to do a specific task quickly, and aren’t too concerned about how this task fits into other similar tasks. Relational understanding is useful when you want to explore ideas further, are unconcerned about your destination, and are more concerned with the process.

Unfortunately, our system tends to favour instrumental understanding too much. While it is useful to be able to get from point A to point B quickly, if one is not aware of one’s surroundings, and doesn’t get to enjoy the scenery, it hardly makes the trip worthwhile.

Children are not railroad trains

"Timetables! We act as if children were railroad trains running on a schedule. The railroad man figures that if his train is going to get to Chicago at a certain time, then it must arrive on time at every stop along the route. If it is ten minutes late getting into a station, he begins to worry. In the same way, we say that if children are going to know so much when they go to college, then they have to know this much at the end of this grade, and that at the end of that grade. If a child doesn’t arrive at one of these intermediate stations when we think he should, we instantly assume that he is going to be late at the finish. But children are not railroad trains. They don’t learn at an even rate. They learn in spurts, and the more interested they are in what they are learning, the faster these spurts are likely to be." ~ John Holt, How Children Learn (1984), p155

John has certainly identified the problem, the question is, how would we build our system differently?

A lot of people have identified this problem, but I have seen less solutions to it than people expressing their outrage at it. It is certainly true, we do treat children like railroad trains, and expect far too much regularity in how they learn.

Further, our education system has become more like an accelerating railroad train in which each year children are expected to be able to do more sooner. Algebra in 8th grade. Reading in kindergarten. Essays in 5th grade. Why do we feel the need to keep up with the Joneses?

Designing a new system will be tremendously difficult. We have an enormous amount of cultural inertia in our current system. It is a difficult problem! How can we take a system wherein we fund students to attend school at a ratio of one teacher for every 20 children (on average) and find ways for each of these children to learn everything we feel is important in order for them to become adults?

Here are some suggestions, which are by no means exhaustive.

  1. Trim the list to that which is really important.
  2. Cultivate a desire to learn more, and the ability to learn for oneself.