The Reflective Educator

Education ∪ Math ∪ Technology

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Month: February 2013 (page 2 of 2)

Paper folding activities

I’ve been playing with paper folding recently, and exploring the mathematics involved. I’m simply amazed by the number of mathematical ideas that can be represented by paper folding, so I thought I would share a few of my discoveries here.


Folded in half

Paper folded in quarters

Paper folded into eighths

Paper folded into sixteenths

As you can see above, you can generate the sequence of numbers 1, 2, 4, 8, 16, 32 and so on, just by folding the paper in half again each time. This means that there is an exponential relationship between the number of folds you have made and the number of areas created on the paper.


Paper folded into thirds

Paper folded into ninths

Paper folded into 27ths

Notice that if I instead fold the paper into thirds each time, the sequence changes into 1, 3, 9, 27, etc… which suggests that folding a piece of paper is a little bit like multiplication.



three quarters

First, form the fraction 3/4 by folding the paper into quarters and shading three of them in.

two thirds time three quarters

Now fold the paper in the other direction into thirds, and shade 2/3, ideally in the other direction. Where your two shadings have overlapped is the product of your two fractions, in this case 6/12.



Paper folded into circular sixths

Paper folded into sixths, with cut-outs

Here is an example of folding the paper around the centre to produce rotational symmetry. I worked with a student to produce snowflakes with  9 points, 12 points, and other points, after watching this interesting video by Vi Hart



Tessellation folded up

Tessellation unfolded

If you fold a paper in half a bunch of times, you can create a tesselation by cutting portions of the paper out. The number of folds and the size of the repeated portion of the tessellation have an interesting relationship.


Circle geometry

Circle cut out

Circle cut out

Circle folded in half once

Circle folded in half in any other direction

If you very carefully cut a circle out of a piece of paper (which will finally give you a use for all of those CDs you have laying around you aren’t using anymore), you can prove quite a large number of the theorems from circle geometry by folding the paper in certain ways.

For example, if you fold the paper in half twice in two different directions, the intersection of the folds has a useful property.


For further resources on paper folding and mathematics, see this TED talk by Robert Lang, this book on the mathematics of paper-folding, and this useful PDF describing some geometry theorems that can be demonstrated through paper folding. See also this very interesting article on fraction flags (via @DwyerTeacher).

Two competing visions of the future of education

Which vision of computers would you prefer for your children?
(Image credit: Left – Multnomah County Library, Right: Sam Howzit)


If you ask people who attempt to predict the future of education, you will find out quickly that there are two very different, competing perspectives.

One camp believes that the future of education is in moving away from complete standardization of curriculum and focusing on nurturing students to become learners, so that when they need to learn something new, they are capable of doing so independently. They are less concerned with the media that students use to learn, and more concerned about ensuring that students have at least some say in what they learn, and how they learn it. They believe that computers are powerful devices for exploration, and that the full potential of computers in education has not yet been realized.

This first camp believes that learning is something best done within social contexts, while simultaneously believing that cultivating the ability to think independently of others is of critical importance in our life. They believe in students spending some time learning independently through self-exploration, and some time collaborating deeply with others. They believe in teaching kids how to think, not what to think. They believe the role of teachers is primarily to mentor students and to model being a learner with them.

The other camp believes that the future of education is in mechanical learning. They believe that if we can just find the right mixture of content, media, and machine-graded assessment, we can greatly reduce the costs of education, and deliver a personalized education experience to every child. They believe that a teacher’s job is to deliver content and assess the understanding of students, and they believe that these can both be done efficiently and effectively with a computer. They believe that if children just have the perfect explanation, they will learn.

This second camp believes that the future of learning is with children carefully isolated, sitting in cubicles, watching videos, and then answering questions prompted on the screen. They believe that social interaction with other children is at best a supplement to what happens on the computer, and at worst it is a distraction. This camp is usually more concerned with the cost of education than the quality of learning.

Both of my descriptions of these two camps are somewhat reductionist. Obviously there are shades of gray between these two camps. However, if you had to choose between these two visions, which would you choose? More importantly, what are you doing to make it a reality?


I’ve been working on improving ActivePrompt, and I decided to split it off to it’s own domain. This script was originally created by Riley Lark, and I’ve been working on my own fork of his project. The new site needs some serious work on the appearance (interested in helping? Let me know), but the functionality seems pretty solid.

  • The site now requires logins for all pages except the prompts themselves.
  • When you create a prompt, it is added to your list of prompts.
  • It is also added to the gallery.
  • You can now edit and delete prompts.
  • The gallery should only show prompts that include unique pictures, rather than the gigantic number of prompts from before.

If you created prompts, they still exist, but they are not currently attributed to you. Please create an account on ActivePrompt, and email me and I’ll try and link you to your prompts manually. Include in this email:

  • Your user name on,
  • The direct link to the prompt you created (you should be able to find it in the gallery or perhaps you had the original link bookmarked).



Sam Wineberg on the need to change how research is shared

If you are involved in educational research or you are interested in learning more from educational research, I strongly recommend watching this presentation by Dr. Sam Weinburg (via Dan Meyer).

Dan does a good job of highlighting the strengths of this video, however I have this to add: most academic writing might as well be written in Ancient Greek and buried at the bottom of the sea for all the good the research does society. If you write in language which is incomprensible to most people and only available to a very select few, you are doing very little to actually change your chosen field.

We should really prepare students for university

I don’t think we are doing a good enough job of preparing high school students for the university experience. We need to do more!

  • We should increase class sizes in high school up to 500, so that students get the experience of being in a large lecture hall. We may want to ease students into this experience, so we should gradually increase up to 500 students per class, perhaps at 20 students a year. Working backwards, this would mean we should start with kindergarten class sizes of 260 students.
  • We should make high schools larger. No more measly 2,000-student high schools. They should have 20,000 to 40,000 students at least!
  • We should hire mostly teachers who have little formal training in teaching and are mostly interested in pursuing their own research. The lowest level classes in the school should be taught by graduate students with little to no teaching experience.
  • We should reduce summative assessment in our schools to two exams per semester and use little to no formative assessment. If the students do not understand, they need to study more.
  • If our students are struggling, we should just keep putting them into remedial courses until they drop out. Why would we offer them any support? They will be on their own in university!
  • We should charge students ever increasing amounts for tuition and force students to take out gigantic student loans in order to complete high school if they cannot afford to pay.
  • We should drop all of the ‘soft’ courses from our schools. Students do not need to take home economics, planning 10, or shop class. We should also make physical education optional. After all, our job is to prepare students minds for academia, not prepare them for life.
  • We need to teach students how to navigate depersonalized bureacracy. Therefore we should make high school as depersonalized and bureacratic as possible.
  • We should encourage our high school students to drink, so that we can replicate the drinking cultures prevalent on many university campuses.

(Or maybe we should stop backwards designing from university and instead focus on building effective practice, whether or not it "prepares students for university"?)

Math in the Real World presentation

I am presenting in Hope, British Columbia today, on the topic of Math in the Real World. Here are my presentation slides.

You will probably notice that sections 6 and 7 of my presentation are not completely focused on the topic of “math in the real world” but I feel like they are such important concepts for mathematics teachers to understand that I needed to include them in my presentation.

Presentations for tomorrow

I’m presenting at the ISABC professional development conference tomorrow, twice. Once on using Twitter as a professional development tool, and the other time on the use of technology in math class. I’ve embedded the slides for these presentations below. Both presentations are intended to be run as workshops so that participants will be expected to do just that, participate.

Six word story

In the spirit of a story Ernest Hemmingway probably never wrote, I was going to offer this as my six word story for #etmooc.

"For sale: Master’s degree, never used."

Unfortunately, it seems that this particular short story has been thought of before. It’s also worth noting that this particular piece of fiction does not accurately describe my life. So I went back to the drawing board and came up with this:

"Student died in math. Nobody noticed."

This is obviously fictional, but it better describes what has become my life’s work to avoid happening to students.

Education as vaccination

"We seem to think that education is a thing—like a vaccine—that can be designed from afar and simply injected into our children." ~ Governor Edmund G. Brown Jr

This is a quote from the State of the State address by the Governor of California, and it is one of my new favourite quotes on education. Thanks to Scott Mcleod for spotting it.

It’s a pretty interesting analogy. I wonder how many parents, students, teachers, administrators, and education policy makers believe that education is like a vaccination against ignorance. I see education as something developed within a community, by a community, for their community.

The challenge I see here is when a community begins to trample on the rights of its community members, neglects its duty to ensure everyone has equal access to happiness and opportunity, and therefore creates an education system which does not serve everyone in their community. In this case, I think the state should be able to intercede, but otherwise (unless they are coordinating projects between communities where scale matters) the state should stay out of the way.