Education ∪ Math ∪ Technology

Month: October 2011 (page 1 of 3)

Math in the real world: Marshmallow constructions

This is another post in my series of posts on math in the real world.

Building materials

My wife, son, and I  went to a kids science event at SFU today, and at one table they had some marshmallow diagrams set up to demonstrate molecules. They let the kids play with the marshmallows and toothpicks, so my son made a giraffe. When we got home, he helped himself to some marshmallows and toothpicks and continued to make things with them.

Simple diagram


My son noticed that the most stable form included triangles (with some help from mommy), so he started to construct everything with triangles. When he moved into three dimensions, he noticed that the tetrahedron was the most stable of the forms he could build and so his construction soon began to look very mathematical in shape.

More complicated diagram


Now in his most complex form, he has started to build a three dimension tesselation. If he hadn’t been called away to dinner, or if we hadn’t been running low on toothpicks, I’m sure he would have continued the pattern.

Very complicated diagram


This activity involves both 2d and 3d geometry, tesselations, sequences and other patterns. Can you think of other mathematics that can be found in this activity?

BC’s personalized education plan

At 1:53 of this video, the BC Ministry of Education shares a need to give more flexibility into the current system for how, when, and where they learn, but they’ve forgotten an important option: what they learn.

In a world where total knowledge is increasing at an exponential rate, it is impossible to determine what the best subset of knowledge should be that everyone learns. Further, true personalization of learning comes when one has choice over what one learns; when one is the director and producer of one’s life, and not just an actor in it.

A couple of ideas I like for addressing this need are the Learning in Depth project, and something akin to Google’s 20% time. Students could use the time to learn more about their own culture, get involved in mentorship in a workplace, improve their free-style skateboarding, or anything other learning that they would be willing to document for their school and community.

Nowhere Was Home: The Street Kids and Their Truth

In 1993, my father interviewed street kids to find out what living on the streets was really like, and how kids could end up on the street. He self-published his work, but unfortunately, I don’t feel like his book got the audience it deserved (or needed).

Two and a half years ago, my father passed away. He left a lot of stuff unfinished in his life, so I’m trying to complete some of the tasks that he was working on. One of these tasks is sharing the stories he collected from street kids.

A few days ago, I found the text of his book in a file on his website and decided to publish it through the Amazon Kindle store, so that it can be shared more widely. The stories in his book are sometimes hard to read, but they are an honest account of what it was like to live on the streets in the 1990s. They an indictment of the realities that street kids still face in our society. These stories need to be read and shared.

Here’s an excerpt from his book:

Even things that I didn’t know if they were right or not, I really wanted to experience them so that I could make up my own mind. By the time I was fourteen, I was beginning to assume that everything my parents told me was lies, so many things had been. So I just wanted to go out and experience what most people would call the seedier side of life. But I had to be in by six o’clock every night.

Nowadays you might say, "Well, compared with living on the streets I would have put up with the rules." But I didn’t expect to be kicked out of the house and never allowed back. But once it was done, it was done, and I really relished the freedom.

I have always felt that even a bad experience has some value. It was very exciting being on the streets. It was terrifying and it was dangerous. And it was all new experience.

I used to sit long hours and talk to all the winos. They’d tell me their life stories.

Life on the street gave me a new perspective on people I had been like just a few short months before; people like my family and the kids and families I had grown up with … just how unwilling they are to become involved and lend a hand in anything that they are afraid will happen to them. I would sit on the street and watch all the people going by who were exactly like my family, and like I used to be, up until a couple of years before. I would think on that and how hard they were trying to keep their blinders on. What an effort it took for them. I never realized that before. How much work went into maintaining their illusion that nothing was wrong. It was a very enlightening experience. I can’t do it again. I could never ever live on the streets again because I know a lot more now, and it’s too dangerous. Back then .. some nights I wouldn’t sleep all night because I was so terrified.

But other nights because of youth and the feeling of immortality that comes with it, it would just be exciting and interesting. But now there is no way I could spend even one night on the streets because I know what could happen, and that it’s very likely to happen.

I kind of miss that stupid courage. It was ridiculous. But I’d like to have that feeling. I was really tough back then because I realized I could be tough. I was accosted day and night by slimy men. They would come on to me "Oh, let me buy you a drink." I had to be very very tough to protect myself from that. I miss that toughness. It was a real shield that I had. I felt very strong when I was on the streets. I miss that.


I’ve put the price of the book at the minimum that the Kindle bookstore will allow. For just $1 US, you can download his book to a Kindle device or to any Kindle application. I plan on giving what proceeds of this book I do earn to my sister, who is currently struggling to make ends meet on a disability pension, so this is also an opportunity to help out someone in need.

You can find Nowhere Was Home: The Street Kids and Their Truth here.


First review:

I bought this book yesterday and tore through it in a couple hours. It is absolutely amazing.

The stories of these individuals range from the empowering to the heart-wrenching. They will stay with you for as long as you let them.

I would love to find a way to share this book with as many people as I can. I don’t normally write reviews for the books I read (who cares what I think?) but I had to write one for this one. It’s too powerful. People need to know that this is something worth reading.



Grading is a compression algorithm

The objective of traditional grading is to compress information teachers have gathered about a student down into a single score to make understanding the information easier. One of the original reasons for this compression was the limitation on how much information could be shared on a single piece of paper. One of the purposes of comments is to uncompress the grade a bit, so that parents and students have some ideas on how to improve their grades.

This process is used to change the size and quality of pictures as well. Compare the two pictures below, and ask yourself, which one conveys more information?

high quality Low quality


Is there a way we can share information parents and students can understand, while not reducing the information too much?

I flipped my math classroom

I’ve normally started my classes with a description of what math we will be learning, and a class discussion about what the math means. 

When I first started teaching, I would lecture for 30 minutes, and students would work for 60 minutes (I started in with a double block of math) during double block math classes, and in a 45 minute lesson, I would still lecture for 30 minutes, and students would get 15 minutes to practice and do other activities.

I discovered early on in my teaching that the less time I talked, the more time students had to work on activities and exercises, and this led to improved understanding. I read research suggesting that adolescents could actively pay attention for about 10 – 15 minutes, so I focused on getting the lecture portion of my lesson down to this length, and on embedding more questions and subsequent discussion into my lecture.

Today I tried something new. I found questions (with an emphasis on real world application) related to exponential functions that students had never seen before, and started class by handing them out as a package, and asking students to work on these problems in groups. I then spent class circulating around the room, answering the occasional student question (but being very careful what types of questions I answered) and pushing students to try finding multiple solutions to the problems. When students were completely stuck, I offered support, but by asking them questions, rather than just giving them the solution.

Now, I’ve definitely had classes where I haven’t taught an idea to the entire class before, but this is the first time I’ve introduced a completely new topic without either presenting a lecture on the topic ahead of time or using some sort of guided instructional aid for the students (like a video prepared in advance of the lesson).

Here are some observations I had while I was circulating around the classroom.

  • Not one student asked me "is this solution right?"
  • Students were actively engaged in the problem solving process.
  • The questions I overheard from students (to each other) were often about the nuances in the problems, rather than "how did you do this?"
  • Every group of students found the most efficient standard solution to the problem, as well as 2 other ways of solving the problem.
  • No one attempted to Google for the solutions, or even open their textbook to see what information it had.
  • My students were thinking.

At the end of class, I asked students to continue working in groups and come up with notes to explain the topic. As the students will be taking an exam in about a year and half on all of the material they are writing, I recommended that they write the notes for their future self that might not remember having worked on these problems. Next class, I plan on having students form new groups, and collaborate to construct meaningful notes for the future, and then work on some more related problems.

I’ve flipped the classroom. Instead of me presenting the ideas, my students look for solutions, and I help them. Instead of me giving notes to students, they make their own notes. Instead of the classroom being about the content, it’s about the process.

There were no videos, no notes in advance, no computer assessed exercises; just a focus on changing who was doing the thinking.


It’s time to redesign the report card

The typical report card looks like this (click to embiggen):

image credit: rutlo

image credit: Richard Giles

image credit: clintjcl


A problem with these reports is they do not share with parents information that can be used to help their children improve their learning. What they share is information that is helpful to rank their children with respect to the other children in their classes. They are essentially an autotopsy of learning, rather than a document which can be used to help students improve.

A child who does "poorly" is rarely given sufficient advice to help them improve via their report card. Most comments from teachers are of the "what did David do wrong" variety, rather than "David should do x to help improve learning." A child who does well on their report card is given a free pass, and rarely pushed to extend themselves. The comments you put on your report cards should be ones that help students improve. Canned report card comments are a waste of time! If teachers do not have time to give appropriate comments for each of their students, that points to a systemic problem with classroom size (and workload) and that can’t be rectified by adding useless comments to report cards that teachers can just select.

I don’t think that online grading systems are the answer either. These lead to situations where teachers are forced into an unhealthy practice (grading everything students do) just so parents can always keep on top of the "progress" of their students. It is counter to the purpose of formative assessment to include it in an overall summative grade, and it is counter to the purpose of a summative assessment to include everything students do. Students also need a bit of freedom from their parents in order to experience this learning process themselves, and having their helicopter parents whirring around all the time checking in on them is counter-productive to them developing their own sense of independence.

We’ve used student led conferences at our school, which are an opportunity for students to show to parents directly, the results of their learning, based on portfolios our students have constructed. We are hoping to eventually have these portfolios be online, and I’d like to see the specific assessments students have done linked from their report cards. These experiences are far more valuable, both for "strong" and "weaker" students. They do have issues; not every parent takes the time to come and see their student’s work, and not every student is able to adequately explain how well they did.

I wonder what a report card that was sent home that just listed student’s (apparent) strengths and weaknesses would look like? Could you send home information that helps students improve, rather than information which helps them be numbered?

Could we design an electronic report card that gave far more information than our current ones do? What would a report card with each assessment criteria for the year, and how well our students did on each look like? Would that be too over-whelming? What if we sent home a link to an eportfolio for each student, with suggestions and comments for how to improve attached to each assessment the student does? What would happen if we gave students more ownership (perhaps with some oversight to start) over how they reported their learning to their parents?

I don’t have the answer to what the report card of the future should look like, but I do know that our current report cards need improvement.

Lessons Learned While NOT Teaching

At the keynote talk on Friday at the BCAMT conference, Dr. Peter Liljedahl shared three very interesting pieces of his research from the past ten years as an observer of classroom practice, in his lecture entitled "Lessons Learned while NOT Teaching."

  1. Given a choice between a sitting at a desk working on paper, sitting at a desk working on a whiteboard, standing up working on a flipchart, or standing up working on a whiteboard, how would you expect students to learn best?

    According to Dr. Liljedahl, students learn best while working on a non-permanent surface, while standing. He suggested that this because the non-permanence of the surface makes mistake-making easier to cope with, and that by standing up, they are on-the-stage, and so can’t hide their thinking.

    How many of our classrooms are set-up to encourage this type of learning environment?

  2. According to Dr. Liljedahl, note-taking is an activity which takes very little mental energy (from most students). Hence, if our objective is to get students to think more, we should rethink how we share notes. He recommends sharing the notes for your class after the class, and not having expectations that students will take notes from your class. Interestingly enough, this is a practice that many university professors are now moving toward as well, and it is standard practice at meetings to have minutes to share afterward. In other words, note-taking is a less useful skill today, than it might have been 20 years ago, and so it is less important that students learn the skill of taking notes. Note that I think that making notes on a topic is a different activity, and is still useful.

    Is taking notes still a useful activity?

  3. Similarly, the types of questions we answer for students has an impact on their thinking as well. From Dr. Liljedahl’s experience, students tend to ask one of three types of questions: proximity questions (which are questions asked because you are near them), stop-thinking questions (such as: "Is this right?"), and start-thinking questions (such as: "What would happen if…?"). His recommendation is to stop answering the first two types of questions. The result is that students have to spend a bit more time thinking, and that peer to peer interactions become more important.

    Does this change how we ask questions?

Alan Lishness – Indigenous Innovation: How Small Places can Change the World

I’ve read about this information about the Finnish schools before, but it is nice to have it all collected into one talk. My thanks to Adam Burk for sharing it.

How does Finland do so much better than the US in education? They do everything differently.

Almost no standardized testing, well prepared teachers, huge respect for the teaching profession, professional autonomy for teachers, competitive entry system for teachers, a system controlled from the bottom up, and a much reduced emphasis on competitiveness in their education system.