Education ∪ Math ∪ Technology

Month: September 2011 (page 1 of 2)

Scripted creation isn’t creation, it’s assembly

Lego - instructions for building a tree
(Image credit: toomuchdew on Flickr)


When I was a kid, I had a lot of Lego, most of which was given to me as birthday gifts, and came in nice neat boxes with instructions on how to build whatever was on the picture on the outside of the box. I would often follow the instructions carefully once, to make sure I could create the picture, and then that package of Lego joined the big bucket of Lego, and I never assembled that particular design again. I did spend a lot of time creating my own models, and learning different ways of putting the pieces together myself. I remember better the things I created than the things I assembled. 25 years later, I still remember when I used all of my Lego to create a sprawling metropolis on my floor.

My son and I play Lego together, and just like when I was a child, he wants to follow the directions. We’ve built a couple of things together that way, but we’ve also spent hours upon hours making our own designs. Recently, we’ve been constructing sling-shots with our Lego along with some rubber bands, and then using these to fire smaller pieces of Lego at targets. My son calls this "playing Angry birds." In this case, we’ve had a design (the Angry Birds slingshot) and we’ve reverse engineered a way to create this slingshot with Lego.

We need to be careful that we give students time for creation, which I see as a much different process than assembly. Creation helps kids develop entrepreneurial skills, use their imagination, and allows them to be inventors. Assembly allows them to learn how to follow instructions, and work toward a well established goal. There is also a middle ground between the two, which I call reverse engineering where you give students the final outcome, and they have to work to figure out how it was built.

When you assign a project and give students a highly structured way of completing the project, you are having them assemble their project, rather than create it. They may not be assembling pieces of Lego, but it is a form of assembly nonetheless. These kinds of projects can be valuable for your students, and can help focus students on the final product of the project but the more information you give students on what their project should look like, or how they should go about doing it, the less freedom they have to actually be creative.

There is research which shows that when kids are shown how to do something (as what happens when you give them a set of instructions to follow), they are less likely to "engage in spontaneous exploration and discovery" (Thanks to @jybuell and @andymikula for sharing those two links). We must then be very careful about our purpose in giving kids a set of instructions to follow, given that we may be shutting down some of their creative capacity in this area.

There are types of tasks for which instructions are pretty important. You can’t write a high quality academic paper without reading examples of other academic papers, and without some pretty careful instructions on formating. The language of academic papers is highly specialized and often cryptic.

However, I don’t want my son writing academic papers just yet. He’s young. I want him to explore the world, and see what is possible, and use his imagination as much as he can at this young age (thanks @allanalach for this link). He has plenty of time to learn about the dydactic academic world later. For now, I want him to play.

A subject like science, for example, can very much be taught either as an act of creation, or as an act of assembly. Give students all of the instructions on how to complete a lab, and they are assembling a lab. Give them a goal (figure out why this phenomena works) and they are reverse engineering. Give students time to play (safely) with the tools in a laboratory and come up with their own experiments, and they are creating.

Scientific inquiry is about asking questions and exploring the answers to those questions through experimentation. If we want kids to think like scientists, we need to give them the ability and option to experiment. As this comic suggests, the core of science is that ideas are tested by experiment, and that everything else is just bookkeeping.


Let’s make sure that our need for bookkeeping doesn’t disrupt our kids need for exploration. Let’s make sure we give kids lots of time to be creators, and not have them just assemble stuff.

Math in the real world: Balloons

This is part of a series of posts I’m doing on math in the real world.

Balloons in an office


The first question I thought of when I saw these balloons in my colleagues office was, how many of those would I need to be able to float? Clearly, this is a math problem, and one students can actually test themselves (I would recommend using inert ballast to test student guesses, rather than actual students). Students would first have find out the amount of weight one balloon can lift, and then use division to determine how ballons would be required to lift their weight.

If you want to make this problem much more complicated (and more of a calculus problem), you would point ouf that the density of air decreases as the balloon lifts, lowering its buoyancy, and putting a limit on how far the balloons will actually lift the student.

The shape of the balloons in this picture is also mathematically interesting, as is the shape of other balloons. Why do balloons form the shape that they do? How do the manufacturers of balloons know in advance what shape the balloons will have before they fill them up with helium?

Paypal and password security

This afternoon, I had to change a Paypal password. I went to Paypal, got to the screen to change my password, and after an attempt to choose a new password, I was confronted with this screen.


Paypal and password security screenshot


I definitely had at least eight characters in my password. I didn’t use my name or my email address. I used a mixture of upper and lowercase letters and numbers and symbols. Paypal just refused to change my password. I decided to test a longer password, specifically, InfinityIsCool4321! (I’m not actually using this password, so it’s safe to share it here) which according to this script would take 12.13 trillion, trillion centuries to break. Paypal still refused to accept my password, presumably because it contained some common words.

I’ve written about passwords before. It’s annoying that Paypal would rather that people created passwords they will forget (unless they write them down, kind of negating some of the security of a password) than to use some simple tips to create a secure password.

This is part of the reason people get frustrated with technology. When developers build forms which are broken like this, it makes the casual user feel like technology is something magical and incomprehensible.

New Math equals trouble, education expert says

The CBC just ran an article on the problems in our current math system which was terribly one-sided and an example of the worst kind of fear-mongering journalism. They are quoting an article by Michael Zwaagstra, an "educational expert" writing on behalf of the Frontier Centre for Public Policy.

First, let’s examine the article written by Zwaagstra.

A solid understanding of mathematics, also known as numeracy, is an important component of a well-rounded education. The ability to perform basic mathematical computations is a requirement of many entry-level jobs. In addition, careers in fields such as engineering, medicine, finance and all of the sciences require a solid background in higher-level university mathematics, including calculus, statistics and linear algebra.

The first thing to point out here is that the basic mathematical computations … for entry level jobs are much different than the higher-level university level mathematics needed for engineering, medicine, finance, and the sciences.

I have to agree with Zwaagstra that a solid understanding of mathematics is an important component of a well-rounded education, but his assertion that mathematics equals numeracy is definitely false, as I have had pointed out to me on a regular basis. There are many mathematicians, engineers, doctors, economics, and scientistis who struggle with basic computational math, but are fully capable of doing higher level mathematics, and this has been true for a long time; far longer than the new math has been used in schools.

Because math is such an important skill, schools have an obligation to ensure that students learn key math concepts. Unfortunately, schools are largely failing in this regard. First-year post-secondary students are increasingly unprepared for university-level mathematics, and this has led to a proliferation of remedial math courses at universities across Canada. Many parents choose to enroll their children in special tutoring sessions with organizations such as Kumon and the Sylvan Learning Centre to fill in the gaps left by the public school system. Unfortunately, many cannot afford extra tutoring, and this creates a two-tiered system that unfairly penalizes children whose parents cannot pay for extra math lessons.

Now Zwaagstra points out that remedial math courses are on the rise in universities, but he doesn’t mention a couple of key facts. First, under the old system of mathematics instruction, around 50% of students failed first year math courses, which were often included in programs as a tool with which to weed people out of university. Could it be that this issue has always been around, and universities are simply now doing something about the problem? What about the increase in students seeking a university education? Could these two issues be connected? Zwaagstra has assumed a correlation between the number of remedial math courses, and the effectiveness of k-12 math education, without actually finding research which supports his conclusion.

Further, he talks about parents enrolling their kids in after school tutoring programs without discussing the reasons why parents are doing this? Are parents increasingly enrolling their kids for extra tutoring because they are dissatisfied with their kids current educational attainment? Or do they have other reasons for paying for these tutoring services? We don’t know, and Zwaagstra doesn’t provide us with any evidence for the reasons for parents to choose tutoring, he just cherry-picks this fact because it seems to support his argument.

Although there is solid evidence supporting the traditional approaches to teaching math that involve mastering standard algorithms, practising skills to mastery and introducing concepts in incremental steps, most provincial math curricula and textbooks employ a different approach. Constructivism, which encourages students to come up with their own understanding of the subject at hand, is the basis for this new approach to teaching math. As a result, there is very little direct instruction of important mathematics algorithms or rigorous practising and memorization of basic math facts.

There is also solid evidence showing that the longer that people are out of school, the less likely they are to use the algorithms they use in school, but the more successful they are at solving mathematical problems they encounter, as Keith Devlin points out in his book, The Math Instinct. In other words, traditional school math seems to be a hindrance to people being able to actually solve real world mathematical problems. It’s worth pointing out that Devlin’s research is reasonably old, and most of the participants in the research learned mathematics in the traditional method. Is it even worth pointing out that Zwaagstra doesn’t actually include any of the solid evidence in his paper, and the footnote here (see the original article) leads to a definition of the word algorithm?

Our students deserve better. Pupils who are not taught math properly are being unfairly denied the opportunity to enter careers in many desirable fields. The public school system has an obligation to ensure that every child has the opportunity to learn the mathematics required for university-level mathematics courses.

It’s pretty important to note that the new math is not being taught evenly, and that when teachers are given proper training in how to use the new math materials, their students’ understanding improves. To say that the problems in our math education system are entirely due to the introduction of the new math curriculum, is pretty irresponsible, given that any number of other factors could be contributing to the problem. Further, many schools use the International Baccalaureate program, which itself relies on the "new math" with a focus on students understanding mathematics and being able to communicate their understanding and these students are highly sought after by universities. If the new math was so destructive, wouldn’t we see these students being turned away by universities in the sciences?

Zwaagstra then goes on to bash the results of the PISA examinations, citing an article (claiming it is research) written that suggests that Finnish students are not as good at math as the PISA results would claim, and that by extension, neither are Canadian students.

There is a strong consensus [emphasis mine] among math professors that the math skills of these students are much weaker than they were two or three decades ago.

Zwaagstra links to two articles (neither of which is a research study) that state that some professors have found a drop in numeracy skills (again, these are associated with mathematical ability, but are not equivalent), and the other of which makes no mention of math skills at all. In this case, Zwaagstra is completely misrepresenting the articles themselves. He then points to two professors who have done research on the computational abilities of graduates and noticed a decline, but he does not clarify whether or not this is correlated with a decline in their ability to do university level mathematics.

Zwaagstra continues by bemoaning the lack of standards and emphasis on accurate calculations by the National Council of Mathematics Teachers (NCTM) and the Western and Northern Canadian Protocol (WNCP). Clearly the research these two organizations have done for decades is not sufficient for Zwaagstra, especially considering Zwaagstra’s credentials (Hint: He’s never been a math teacher, nor has he any credentialed expertise in mathematics education Update: Apparently, Zwaagstra spent 7 years as a middle school math teacher, so I’m retracting at least this part of my response).

However, there is a big difference between demonstrating a conceptual understanding of mathematics and actually being able to solve equations accurately and efficiently. Just as most people would be very uncomfortable giving a driver’s licence to someone who merely demonstrates a conceptual understanding of how to drive a car, we should be concerned about a math curriculum that fails to emphasize the importance of mastering basic math skills.

To extend Zwaagstra’s analogy, we should similarly be afraid of giving the keys to someone who has no real world experience driving. If someone has spent all of their time in a flight simulator, but never actually driven a car, should they be allowed to do so? Does an emphasis on the mechanics of driving a car (or the mechanics of mathematics) turn someone into who is capable of driving a car (or able to use mathematics)?

Zwaagstra’s solution to improving math education is to move "back to basics" which is as unoriginal an idea as I’ve heard, and it is arrogant of Zwaagstra to assume that this approach hasn’t been tried before. Perhaps Zwaagstra could instead address the issue of elementary school teachers often lacking support and training in how to teach math? Zwaagstra points out (correctly) that having mastered one computation, students are then better able to learn another computation, but this leaves students learning a series of computations, and not spending any time actually using them.

JUMP math is mentioned in Zwaagstra’s article as an antidote to the problem, but he doesn’t talk about the issue of the associated training, or the lack of diverse assessment used in the JUMP math system. I think that the training manuals which go along with the JUMP math curriculum, for example, actually address the misconceptions of the people teaching the math (mostly elementary school teachers) rather than itself being a significantly better system. As one educator has told me, JUMP math is pretty useless without the training materials for teachers.

Just as someone who does not practise the piano will never learn to play well, someone who does not practise basic math skills will never become fluent in math.

Similarly, someone who has not had time to play with a piano, to improvise, and to perform music for others will never develop an appreciation for the instrument. Zwaagstra is suggesting that we should discard the extra parts of math education, like problem solving, and focus on computations, which is the musical equivalent of only learning scales, and never getting to perform music.

No one would stand for that in music education, so why should we accept it in math education?


Update: Here’s another good rebuttal to Zwaagstra’s article.

What is Edcamp?

Edcamp is a (relatively) new form of professional development which is highly flexible, and based on the needs of the participants. Here is a presentation on Edcamp I’ve created to share one form of Edcamp which was used for Edcamp Vancouver last year.


Photos of Edcamp Vancouver, taken by Darren Yung.


Kristen Swanson presented on Edcamp at TEDxPhiladelphiaEd last June. Her TED talk is embedded below, and she goes into much more detail about what Edcamp can look like.

Moebius Noodles

A couple of weeks after I posted some resources for parents looking to teach their young kids about math, Maria Droujkova has introduced the Moebius Noodles project which is intended to build a book and a support site for parents who would like some support teaching math to their children.

In her own words, the reason she started this project is:

  1. There are very few materials and no community support for smart math for babies and toddlers. Just try to find anything that is not about counting or simple shapes! Mathy parents create opportunities for their own kids, of course. But without support and resources, it’s very hard even for the rocket scientist mothers and fathers. We want to change that!
  2. Peer-to-peer learning, research and development groups in mathematics education need a process for crowd-funding their projects. We are the trailblazers for other fabulous communities that want to make open and free math materials with the support of their members, such as the group developing materials for learning mathematics through music, the play math network, and the math circle problem-solving depository project.
  3. We are creating OERs – Open Educational Materials. It means people can access, use, modify and share the materials for free [emphasis mine]. Imagine the project you support translated into any language in the world, and used freely to support young kids everywhere!
  4. The activities are sustainable in many senses. You can use everyday household items and recycle materials for Moebius Noodles games.
  5. If you are a parent or teacher who loves arts and crafts but is afraid of math, the book will help you teach your kids mathematics through your talents. If you are a math or science geek who envies other families always doing neat art projects, the arts-math bridge in the book goes both ways!

You can donate to her cause by clicking on the image below. At the time I posted this entry, Maria is about $4000 away from her goal.

Moebius Noodles Fundraiser Badge

Disguising flash cards as a game is deceptive

I’m reading The Connected Family by Seymour Papert, and ran into a quote which I found appropriate.

"…learning multiplication facts by putting flash cards on the screen is not a new way of learning math. It is a polished-up version of the old ways and promotes to greater heights their worst and most mechanical features. Moreover it is often done in a spirit which I see as dangerously dishonest: Disguising [emphasis mine] flash cards as a game introduces an element of deception that undermines two fundamental educational principles.

First, learning works best when the learner is a willing and conscious participant. Second, deception and dishonesty in the teaching process make a mockery of the idea that schools should develop moral values as well as knowledge of math or history." ~ Seymour Papert, p19, The Connected Family, 1996

It was timely, because just this morning, I saw this tweet from Jason Klein:

Most Education category apps (iOS, ChromeOS, Android) are low-level, fact-based. Just give me Internet & Creation apps.

Just searching now on my iPhone for math in the App Store, these apps showed up.

Top five math apps on the iPhone

All five of these applications are based on learning math facts and arithmetic, and two of them even have the word "flash" in their name. Of the top twenty five math applications that I saw, 23 of them are essentially flash cards disguised as games. Two of them aren’t, one is called "Equation Genius" (it solves algebra equations), the other is called "Motion Math" (which lets students learn the relationship between fraction as symbols and visual representations of those fractions).

Could we please get more educators programming these apps? (If someone would donate me a Mac to work on, I’ll happily do it myself.) 

Heading to the Computer Based Math conference in London, England

So I just got confirmation (and have paid for registration and my airfare AND found a place to stay – mostly) that I get to attend the Computer Based Math conference happening London, England on November 10th and 11th. I’m very excited about it!

I’m flying out of Vancouver on Tuesday, November 8th (after being in workshops all day with my colleagues), and arriving in London midday on the 9th. I’m at the conference on the 10th and 11th, and flying to Toronto on Sunday, November 13th, where I’ll be attending the Mindshare Learning Canadian Edtech conference on Monday, November 14th.

I have a place to stay arranged for my time in London (actually many offers of places to stay) but I could use a place to stay for the Sunday night I’ll be in Toronto. I’m trying to make this trip more economical for my school (since they are footing the bill) by staying with friends.

My hope is to find out more about how different people are using technology in math education specifically at the Computer Based Math conference, and to be part of the team trying to build a curriculum for math based on the assumption that students have computation devices with them whenever they need them. The big questions I have are, what does that kind of curriculum look like, and would it be effective for teaching math?

It will be strange to be at these conferences as it will be the first time in 2 years that I’ve attended a conference, and not presented. Maybe I’ll find a way to get to talk about some of what I do at one of these conferences anyway… even if I’m not officially on the schedule to present.

I’m posting this to let the people in my PLN know, and I’d love to connect with anyone else heading to these conferences that I’ve met via Twitter.

Am I failing at social media?

I can't unfollow you
(Image credit: docpopular)

When I first got started with Twitter, I set up a filter so that whenever I got a notification from Twitter that someone followed me, it was sent to a special folder in my Gmail inbox.

That folder now has 9581 emails in it. So 9581 times, I’ve gotten a notification that I was just followed by someone. There are a couple of people that follow and unfollow me repeatedly, trying to get me to follow them (while never engaging in me in dialog at all, I might add), but almost all of those notification emails are from new people. 

However, I don’t have 9500 followers, I actually have about 6600 followers.

Some quick subtraction shows that of the 9500 or so people that followed me at some stage during the past three years, nearly 3000 of them have unfollowed me. To put it another way, nearly a third of the people who follow me eventually unfollow me. Are these 3000 lost opportunities to connect with people? 

Why do these people unfollow me?

  • My friend told me that she unfollowed me because I was tweeting too much and overwhelming everyone else in her network.
  • Some of the "people" who unfollowed me were actually spam bots and eventually got blocked by Twitter.
  • Some of them unfollow me because the information I post doesn’t meet their needs.

The key thing is, most of the people who unfollowed me because of their needs, not because of what I’m doing or not doing. It’s easy to take being unfollowed or blocked personally, but I really try hard not to. After all, the whole point of a personal learning network is that it is personal, and that you need to meet your needs.