Education ∪ Math ∪ Technology

Year: 2011 (page 1 of 28)

Calculator vs Slide Rule vs Hand calculations

In an excellent post Michael Doyle says the following:

Electronic calculators are abstract, abacuses are not.
Slide rules fall in-between.
Our sense of quantities has become abstract.

Here is my comment back:

It’s not so much that numbers are abstract to students, but our representations of those numbers (the numerals that make up 122) are abstract.

Base 10 is a much more complicated concept than we give credit to in our teaching of it.

How many people are fooled by the numbers thrown out on the news (example: 12 millions dollars over-budget, on a budget of 1.3 trillion)?

The scale on a slide rule is a logarithmic one, and so it breaks the intuitive linear sense of numbers that we are born with. Hence, I don’t think it’s necessarily the case that the scale helps us understand the operations any better. A slide rule is a calculator, it just uses slides instead of buttons, and the operations are just as mystifying to most people.

Similarly the operations on a calculator (or the standard algorithms we teach kids to use on pencil and paper) are just as mystifying. Regardless of which algorithm one uses, one should have an understanding first of what the expected outcome of the algorithm should be, which unfortunately, we spend almost no time teaching in our mathematics curriculums.

It has become more important to do operations quickly and accurately to demonstrate "computational fluency" and error checking, a MUCH more important skill, has fallen by the wayside. Being able to estimate or predict the outcome of an algorithm is not only a far more useful life-skill than accurately calculating it (when one wants accuracy, one should use the correct tool) and further, it requires actual understanding of the process.

I think Michael is right that our sense of quantities is abstract, but I think that this has always been true for most people, regardless of the form of technology we’ve used to do our calculations. It is not so much the calculator that is the problem, but the embedded abstraction in numerals themselves. Unfortunately, we spend much more time teaching kids how to do operations on numbers, and I don’t think we spend enough time teaching about what the numbers themselves mean (or at least on reminding students what they learned in previous years on numbers). For example, in the BC curriculum, the discussion on the relative size of numbers seems to stop in grade 6.

It is too easy to blame kids lack of fluency as adults with numbers on the technology used to do calculations with those numbers. Instead we should ask ourselves, where have we taught them fluency with numbers? How have we helped them learn about the abstraction numbers represent?

Actual engagement

Online Schools is now following you!


A Twitter user named "Distance Education" with the user name ‘onlinecourse’ has followed me 25 times. This suggests that they have also unfollowed me at least 24 times. This kind of behaviour I’ve heard called ‘follower-churn.’

I may have accidentally unfollowed someone, and then followed them back. I certainly haven’t been using this to gain followers. I follow people because they are interesting to me, and I manually follow each person I find, after reading through some of their tweets. It’s a bit onerous, but it does tend to ensure that I get a better selection of information from the people I’m following.

Distance Education is currently at 6565 followers. I have roughly the same number.  Who do you think is interested in actual engagement, and who is just looking to pad their numbers? Hint: It’s called social media for a reason.

We’ve had email for 40 years – and people still don’t know how to use it

The topic of email has come up recently at my school, and while I appreciate that my staff knows how to use email fairly effectively, it is surprising to me that 40 years after its invention, many people still don’t know how to use email.

  • People still occasionally hit reply all instead of reply
  • People send emails to huge lists of their friends using CC instead of BCC.
  • Many, many people have hundreds of unread emails in their inbox.
  • People respond to scams sent via email.
  • People download attachments or click on links in email from people they don’t know, and infect their computer with viruses.
  • Many people do not how to either search through their email, or organize it into folders, leading them to email requests like "send that email to me again."
  • People use email to forward chain letters.
  • People use email to have conversations that would be faster and more efficient in person.

To be fair, I’m sure that other forms of communication over the years have been inproperly used. It just seems strange to me that something which could have such an obvious value; asynchronous communication with anyone, anywhere in the world, for free, isn’t more firmly established in the educational community.

On another note, just as most educators are using it, email may be dying. Spam, an overload of subscriptions, and the abuses of email I listed above, are all combining to make this tool, which could be useful, less productive than more restrictive communication environments, and so the younger generations aren’t using it. One of the chief advantages of Facebook and Twitter for asynchronous communication, is the ability to disable specific people from sending you messages, and to turn off messages from people you don’t know.

Unfortunately, communication tools are only useful if many people are using them, and what we will lose if and when email stops being used, is the ability to send messages to people we don’t know very well. This is especially problematic given that the postal service is also seeing a huge decline in usage. Will we end up in a world where the technology exists to communicate with people who are different from us, and we just won’t be using it?


How can we evaluate our use of educational technology?

At Edcamp Fraser Valley, I facilitated a conversation on the use of educational technology, focusing on the issue of evaluating its use. The framework I shared (created by Bates and Poole, see below) was the SECTIONS framework, which I have found to be a reasonably comprehensive approach to examining educational technology, depending on how you unpack it. This framework reminds us that we need to think about students, ease of use, cost structure, teaching and learning, interactivity, organizational issues, novelty, and speed, when examining educational technology.

At the beginning of the session, I started by recommending starting with students, teaching, and learning when using these frameworks since a primary purpose in using technology should be to improve teaching and learning, with a focus on the effect on student learning. Here is a summary of some of the various points that were brought up during the session.

  • Students:

    Is the technology accessible for all of your students? Many (if not most) Flash based applications are nearly useless for your severely visually impaired students, for example. Videos without subtitles are extremely difficult for hearing impaired students to follow. Text based technology might be gibberish to students who are struggling with literacy issues.

  • Ease of use

    How easy is the technology to use? Will learning how to use the technology interfere with the ability to use the technology as a tool? Will students spend ages booting up computers (4 minutes a lesson is 12 hours of wasted instructional time over the course of a year!)?

  • Cost structure:

    Compare the cost of the technology to what it allows you to do. Some technology is cheap and can be used in a variety of different ways. Some technology is expensive (both in terms of time and money) and can only be used in one or two ways. The benefit to cost ratio of technology should be as high as possible. Further, do not forget to evaluate the environmental cost of the technology. 

  • Teaching and learning:

    This is what you should look at when evaluating a new technology right after establishing it’s accessibility. What pedagogy am I trying to support with this technology? Does this technology support new modes of pedagogy? Will this technology improve student learning (or at least provide affordances for student learning)? How will I know if this technology is effective? Has any (independent from a vendor) research been done to evaluate this technology?

  • Interactivity:

    Is this technology interactive? Does it provide opportunities for the learner to get feedback about what they are doing? Is the technology doing something that you can’t already do without it?

  • Organizational issues:

    Can your organization support this technology? How will you maintain it (if needed)? Does its use fit your organizations mission and values? Do you have sufficient staff to provide training and support for the technology? If not, what staffing ratios will it require? How will you train staff to use the technology (if necessary)? Will the technology even be used? One participant noted that, when cleaning out a closet, he found unopened packs of teacher resources. What a waste of money!

  • Novelty:

    Choosing a technology because the novelty of it will temporarily engage learners is a bad idea. If this is the primary reason you are using a technology, you’ll find yourself needing to switch to a new technology on a regular basis without sufficient time to evaluate each of the successive technologies. Further, it will make it challenging to evaluate the effectiveness of the technology for long-term use. Was the technology useful for students because of inherent qualities, or because they just hadn’t used it before?

  • Speed:

    How fast is the technology? Will significant time be spent waiting for the technology to do its job? Is there another equivalent technology that could be faster?

After more time considering the issue, I’d add one more item to the SECTIONS framework which I think is hardly ever considered.

  • Culture:

    How will the use of this technology affect the culture of your organization? How will this technology affect our greater society? As Neil Postman notes, no technology is culture-neutral. Will our society change as a result of this technology? Is this change desirable? What would we do if we used technology, and found that the changes in our society (or environment) were undesirable? Could we stop using the technology? There is enormous inertia behind the use of technology in society, and technologies are very rarely abandoned.

Technology which is unevaluated before its use could be ineffectively used, wasteful, or even harmful for students. It is irresponsible to use technology with students without some expectation of the outcomes of the technology. While I support experimentation in the use of technology (and maybe not having all of the answers about the effectiveness of the technology in advance), this does not mean that one should not think critically about the use of the technology before using it. 



Bates, A.W., and G. Poole. (2003). Effective teaching with technology in higher education. San Francisco: Jossey-Bas.

Postman, N. (1998). Five Things We Need to Know About Technological Change.



#Pencilchat – 10,000 Tweets and counting

When my friend John posted some tweets using pencils as an allegory for educational technology, he didn’t expect the #pencilchat hashtag he used to go viral. He’s suggested some reasons why the topic went viral, and he’s probably right. The one addition to his list that occured to me is that the idea of critiquing some of the arguments against educational technology is timely, given many teachers’ current struggles with the use of technology in their classrooms.

There are reasonable arguments against the use of technology in schools, or at least in being critical about our use. Unfortunately, these are not the arguments people use to attack the use of technology in schools.

#Pencilchat highlights the issues educators face in their use of technology. If you want to see what these issues are for yourself, you can download this archive of 10,000 of the tweets (which unfortunately, include some spam from when the discussion was trending) from the viral discussion.

It could make an interesting research project to go through the tweets and find all of the various arguments people outline for and against the use of educational technology. While some of the tweets are quite funny, and there’s a fair bit of repetition, there are also some insights into issues facing schools today.

Experiments in assessment

Here a few experiments in assessment I’m considering for next year.

  1. Compare the results between an oral assessment (as in, find out what they can tell me they know verbally) and a traditional test. . Question: How much of a difference does the mode of assessment make?
  2. Compare the results between a 10 minute quiz and a full length test. Question: Do I find out significantly more with a longer assessment?

  3. Give my students an assessment where I only give them written feedback and no numbers or check marks. Compare this with an assessment where I only give check marks, and another where I only give numeric feedback. Question: Are the numbers and check marks necessary?

What other experiments would you suggest that I try?

Mathematics in the real world: World Statistics

This is another post in my series on mathematics in the real world.



Thanks to a colleague of mine, I rediscovered the Google Public Data explorer. Within 10 minutes, I had constructed the above graph, which shows adolescent fertility rate for 15 to 19 year olds, versus life expectancy, measured against (look at the colors) average income for all of the countries in the world. If you click play, you can see a happy trend; life expectancy is increasing across the world for almost all countries, and the fertility rate is also decreasing.

This type of graph also lends itself well to questions from your students. For example, they may ask why so many teenagers have babies in some countries. They may also why there is a relationship (and from the above graph, it looks like the relationship is reasonably strong), between births from teenage moms, and life expectancy. They may also ask about trend itself, and why that is happening. Further, they may ask, how strong is this relationship? They may also confuse correlation with causation, which in itself can lead to an interesting conversation.

A natural extension of an activity related to this graph would be to have students construct their own graphs, perhaps even collecting their own data. What kind of social data do you think would interest your students?

What is math?

This image is an attempt to capture the important stages of doing mathematics. As pointed by other people, mathematics is not a linear process, which I am attempting to share via this image. I see analytical reasoning, flashes of insight, and exploratory calculations as the glue that holds these stages of mathematical thinking together.


The stages to doing math


How do you see the process of "doing math"? Is it possible that what sets mathematics apart from other disciplines is the formalism, and the calculations involved? How does this process compare to other things that we do in life?

Why math instruction is unnecessary

This TED talk by John Bennett raises an important question; why do we teach middle school and high school math?


I don’t know if using "puzzles" is a scalable solution for the problems in mathematics instruction in middle schools and high schools. It would probably work for many math teachers, but wouldn’t necessarily work for all math teachers. Puzzles and games are good for teaching analytical skills, provided you have someone around who models the use of analytical skills during the game. I’ve noticed, over many, many years of playing games, that many of my friends do not use much deductive reasoning during games. What I would support is much more use of puzzles and games during mathematics class than what is currently considered acceptable practice.

John’s argument that middle school and high school mathematics is unnecessary should actually be restated: our current middle school and high school mathematics curriculum is unnecessary. John is essentially arguing for a different curriculum, rather than discarding the practice of developing mathematical reasoning in students.

I think we need a variety of approaches. What we are doing right now works when students have a strong mathematics instructor, but isn’t working for every student. Instead of assuming that there is one solution to the mathematics education "problem", we should recognize that there are a variety of solutions. What works for John Bennett may not work for every mathematics teacher. I’d like to see these different solutions compete more with each other, and be able to do more research on the effectiveness of each of these approaches. We definitely need more flexibility in mathematics instruction, especially with regard to the curriculum outcomes.

I think we should be focusing less on curriculum outcomes, and more on the holistic goals of a mathematics education. I don’t think it matters if every student learns about the quadratic formula (for example), but all students would benefit from learning deductive and inductive reasoning, pattern finding, modelling of data, and problem formulation. Curriculum should be a vehicle for these goals, rather than the goal itself.