Education ∪ Math ∪ Technology

Month: August 2011 (page 3 of 4)

Geogebra simulations in math

I love using Geogebra! Take a look at the diagram below (use the slider to change the value of n) and then think about how difficult this one simple interactive diagram would be to recreate without the technology.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

For more Geogebra resources, either check out the official Geogebra website ( or the awesome resources shared below to get a feel for what you can do with Geogebra. by @stefras by @MrHonner by @dak721


Explain Constructivism in 140 characters

This video of Michelle Bachmann complaining about Mathematics education is just wrong. 


My take-away from this video is that Michelle Bachmann has a fundamental misunderstanding of what constructivist teaching is. She doesn’t get it. To be honest though, when you look at educational materials which are shared with parents, they very rarely, if ever, define the purpose behind the pedagogy in language that is accessible to people outside of the field of teaching.

My thought is that we need to define some of the terms we use in language which makes sense for the average person. For example, the definition of Constructivism on Wikipedia is:

Constructivism is a theory of knowledge (epistemology) that argues that humans generate knowledge and meaning from an interaction between their experiences and their ideas.

This definition is problematic, not because it is inaccurate, but because it is incomprehensible to a non-academic. It comes across as knowledge is arbitrary.

A few weeks ago, I asked people to attempt and define constructivism themselves (in 140 characters or less) via Twitter. Here are some of the definitions.

A belief that knowledge exists in and is constructed by the mind of the learner, not transmitted from outside. @MathEdNet and @Wedaman

Making meaning and building understanding starting from what you already know. @josieholdford

Learning is the result of our individual experiences & the social support given around those experiences. @delta_dc

Making meaning and building understanding starting from what you already know. @pwacmdh

Learning is the result of our individual experiences & the social support given around those experiences. @mssandersths and @bonzimmer

While these meanings may not be enough to explain constructivism to parents in enough detail, they are more comprehensible to the average person, and a much easier place to begin a conversation about what the impact the Constructivist learning theory has on how we should approach education.

Schools are…

I tried a little experiment today with Google. I started by typing "learning is " into Google and waiting to see what the auto-suggest feature would come up with.

Learning is fun


Next, I typed "Teaching is " into the auto-suggest.

Teaching is fun


Finally, I typed "Schools are " into

Schools are prisons


As I understand it, Google pulls the auto-suggest phrases from the most commonly typed search phrases people use. In other words, the auto-suggest phrases represent the opinions of people using the Google search engine.

Now while I think lots of schools are fun, and it would be wrong to characterize most schools as prisons, at the very least we have a marketing problem. One would assume that if people think learning is fun, and teachers think teaching is fun, then the institutions where both of these things happen should be fun, right?

Why then do the search results above come up? Is everyone searching for the song "Schools are prisons" by the Sex Pistols? Or do we have a larger problem?


Update: @sjhughes shared this one. Google "school makes me" and you’ll see some more opinions of students about school.

School makes me

A really free market approach to schools

I read a great post this morning from Mary Beth Hertz ( @mbteach ) where she shares her insight on her problem with the KIPP and Mastery charter schools. In her article she says:

OK, I get it. KIPP works, Mastery works. But are they really offering the choices they claim they offer to students and families in Philadelphia? If they’re so similar, what’s the choice there? ~ Mary Beth Hertz

The problem is, when you only use one way to judge the success of a school (external test scores), you prevent real innovation from happening. Every school starts to look more similar rather than having freedom to try out different solutions to the "education problem", because each school has to turn out the same product. There are only so many ways you can produce kids when you have single measure of the quality of their education.

In a true market approach to schools, you would let the market decide what accountability measures the consumers want. Since this seems so obvious to me, it is clear that the reformers pushing standardized tests as the only effective measures of schools do not actually want a free market approach to schooling. In fact, the very notion that we should think of kids as being products produced by a factory-like system is nauseating to me.

This problem is exasperated by the fact that in almost all school districts in Canada (and the US, UK, and Australia), the curriculum the kids are expected to cover looks exactly the same. Again, if you want schools to have the freedom to experiment with different models of education, standardizing the curriculum means they have much less choice on what they offer.

At my school we are fortunate. For our 11th and 12th grades, our students are essentially exempt from covering the British Columbia curriculum. We’ve gotten this waiver because we use the International Baccalaureate curriculum, and the BC Ministry of Education presumably considers this curriculum rigorous enough. We do have a couple of requirements from the BC Ministry for our students, but these are easily covered through our program.

In British Columbia overall, we actually have a tremendous amount of choice for our students. When we were looking around for schools for my son, we found Montessori schools, democratic schools, unschools,  fine arts schools, and lots of other types of schools, which are all publicly funded.

When organizations like the Fraser Institute started ranking schools by their standardized test scores, and then further suggest that teachers should be judged by these same scores, and then in the same breath recommend school choice, you have to start wondering what their real aim is.

Standardized curriculum, and standardized testing are the antidote to school choice, not the solution. 

Peace One Day – What can we do?

I just watched this amazing TED talk by Jeremy Gilley.


I thought to myself, what I can do on this day to support peace? I’ve decided that one thing I will do is blog about peace, and share Jeremy’s message with everyone I know. I’m going to share the idea with my students, and we will brainstorm ways we too can get involved. World peace isn’t the job of one person, it’s everyone’s job.

I’d like to invite all of you to make similar vows. Follow @PeaceOneDay on Twitter and offer them our support. Use the hashtag #PeaceOneDay to share your blog posts about this idea. Share this idea with your students and ask them, if just for one day, can they set aside their differences and be at peace with each other? Can we set aside our differences as adults and be at peace together, at least for one day? Let us all set aside September 21st as a day where we all work toward peace.

People are (not) born bad at maths

There is an article in Telegraph newspaper, shared with me by @bucharesttutor which suggests that people are born bad at mathematics. While this may be true, the research cited by the article cannot be used to make this claim.

From the article:

The research, led by Dr Melissa Libertus, focuses for the first time on children too young to have had lessons in maths.

Dr Libertus said: "Our study shows the link between ‘number sense’ and maths ability is already present before the beginning of formal math instruction.

"The relationship between ‘number sense’ and maths ability is important and intriguing.

"Maths ability has been thought to be highly dependent on culture and language and takes many years to learn.

"A link between the two is surprising and raises many important questions and issues."

During the study, 200 four-year-olds underwent several tests.

The problem with this article is that it makes the claim that this means that the ability to do math could be inborn. In fact, the article goes on to claim:

According to the research team, this means that being good at maths could be inborn.

There is a serious flaw in this research. By the time the kids are 4 years old, they may not have had any formal math instruction, but they have had lots of informal math instruction, from their parents and other adults in their lives. It’s possible that this is accounted for in the research, but it is not mentioned at all in the article. Articles like this make me upset because they are intended to be sensationalist, rather than really informative.

My son and I play numerical games. We play Go Fish, and roll dice as part of board games. We count everything. We count in 2s and 5s and 10s. We play with blocks and build intricate patterns. We talk about fractions, and split halves into halves to get quarters, add up halves to get wholes.

We play a game that @JohnTSpencer suggested which we call "How can we get ____?". I choose a number, and my son tries to figure out a bunch of different ways to get that number through addition. For example, I’ll ask my son, "How can we get 7?" He responds with, "Uh… (thinking) … 1 and 2 and 2 and 1 and 1 is 7!" I’ll ask him, "What are some other ways to get 7?" He’ll come back with, "Uh… (more thinking) … 1 and 1 and 1 and 1 and 1 and 1 and 1 makes 7. Also, 2 and 5 makes 7!" He used to use his fingers a lot when playing this game, but he’s switched to doing it in his head. He then gives me a number (usually much larger) and I model playing the game as well, talking aloud when I’m "figuring out" how to make the number he’s given me.

The point is, because I am mathematically numerate, I pass along this numeracy to my son through informal conversations and numeracy games. One cannot assume that simply because children have no formal mathematics instruction that they have no math learning. Our world is filled with mathematics, and the people who recognize that will share it with kids. By the time kids are 4 years old, they will likely have had literally thousands of interactions with numeracy.

We will never end bullying in our schools

We will never end bullying in our schools while we accept it in society.


Children learn from the examples set by adults, and we provide many examples in society of how we tacitly accept bullying. Question period in Canada is brutish and childish. Listen to the video above, and tell me that these men and women are setting a good example for our children.

Professional sports, particularly hockey, allow bullying of players and full-scale violence to occur. Listen  to the people cheering in the background of the video below! How can we possibly end school-yard violence when we embrace it in our entertainment?


Glenn Beck, a horrific talk show host who was finally removed from the Fox network, was allowed to spew his hatred for years before he was finally pulled from the air. Why was he allowed to bully people for so long on the air? Why did we allow Ann Coulter to bully gays and lesbians for so long on the radio? Why is it that (great) projects like the "It Gets Better" project take so long to happen in our society?


We must remember that children do as we do. This message really hits home if you watch the video below (shared by @jenmarten).


I’m okay with doing everything we can to end bullying in schools. I just don’t think that we should pretend it will make a damn bit of difference while we continue to accept it in our society.

Math in the real world: Leaning Plants

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


View all pictures


When plants lean over due to being pulled by gravity, they often form a similar shape. With some exploration, we can determine what shape this is (at least approximately). First, I opened up one of these pictures and embedded it in Geogebra. Next, I added some points to my diagram, following along the shapes of one of the plants.

Next, I exported these points over to MS Excel, so I could find a regression on the points. A quick glance at the shape the curve seemed to be representing suggested I should try fitting the points to a parabola.

Graph of points - parabola regression model

The shape does appear to be a parabola, however, I know from experience that not all parabolic shapes are what they appear. For example, a hanging line is actually a catenary.

What would you have to do to confirm that this shape is a parabola? Is it possible that it is only approximately a parabola?