Archives (page 47 of 97)

October 12, 2011 / 4 Comments

Constructivist teaching is not “unassisted discovery”

I’ve been challenged recently to provide research which supports "unassisted discovery" over more traditional techniques for teaching math. This is not possible, as there are no teachers actually using "unassisted discovery" in their classrooms.

First, it is not possible to engage in the act of "unassisted discovery" as a student. Just knowing the language to describe what you are working on is a clear sign that at the very least you have the support of your language and culture in whatever you attempt.

Second, if a teacher has chosen the activity for you, or designed the learning objects you will be using, then they have given you an enormous amount of help by choosing the space in which you will be learning. Even Seymour Papert’s work with Logo was assisted discovery, after all, Logo is itself going to direct the inquiry toward what is possible to do with the language.

I can’t give examples of research which supports unassisted discovery, but I can give research which supports discovery learning in general. Without searching too hard, I found the following supportive research:

Bonawitza, Shaftob, Gweonc, Goodmand, Spelkee, Schulzc (2011) discovered that if you tell children how a toy works, they are less likely to discover additional capabilities of the toy than if you just give it to them, suggesting that direct instruction is efficient but comes at a cost: "children are less likely to perform potentially irrelevant actions but also less likely to discover novel information."

Chung (2004) discovered "no statistically signicant differences" between students who learned with a discovery based approach based on Constructivist learning principles as compared to a more traditionalist approach.

Cobb, Wood, Yackel, Nicholls, Wheatley, Trigatti, and Perlwitz (1991) discovered that students who learned mathematics through a project based approach for an entire year had similar computational fluency compared to a more traditional approach, but "students had higher levels of conceptual understanding in mathematics; held stronger beliefs about the importance of understanding and collaborating; and attributed less importance to conforming to the solution methods of others, competitiveness, and task-extrinsic reasons for success."

Downing, Ning, and Shin (2011) similarly found that a problem based learning approach to learning was more effective than traditional methods.

Wirkala and Kuhn (2011) very recently discovered that students who learned via problem based learning "showed superior mastery…relative to the lecture condition."

In a meta-study of nearly 200 other studies on student use of calculators in the classroom the NCTM concluded that "found that the body of research consistently shows that the use of calculators in the teaching and learning of mathematics does not contribute to any negative outcomes for skill development or procedural proficiency, but instead enhances the understanding of mathematics concepts and student orientation toward mathematics." (I’ve included this piece of research since many traditionalists oppose the use of calculators in mathematics education.)

Keith Devlin, in his book The Math Instinct, cited research by Jean Lave which found that people had highly accurate algorithms for doing supermarket math which were not at all related to the school math which they learned. In fact, people were able to solve supermarket math problems in the market itself with a 93% success rate, but when face with the exact same mathematics in a more traditional test format only answered 44% of the questions correctly. Later in the same chapter of his book, Devlin revealed more research suggesting that the longer people were out of school, the more successful they were at solving supermarket math questions.

It should also be noted that this discussion on what should be done to improve mathematics education shouldn’t be restricted to either traditional mathematics education, or discovery based methods, but that we should look at all of our possible options.

October 8, 2011 / 5 Comments

8 alternatives to traditional mathematics education

Guided discovery

Learning math instead of computations

WCYDWT and Anyqys

Problem solving

Khan Academy

Math without words

Learning math through games

Real world mathematics

To suggest implicitly that there are two opposing views of how math should be taught is to create a false dilemma. There are many different perspectives on how mathematics should be taught, and some of them have not even been tried on any significant scale yet.

October 6, 2011 / 1 Comment

The Foucault and Chomsky debates

In 1971, Michael Foucault and Noam Chomsky had a debate on Dutch Television, and a recording of a portion of that debate is available on Youtube (see below). I found out about these debates through the r/education section on Reddit, which I highly recommend following.

Foucault seems to have a pessimistic perspective on change in our society, suggesting that our very notions upon which we might use as levers for change are themselves dependent on the flawed structures in our society. He suggests that since our notion of education and justice are based on what these look like in a classed society, that they are themselves flawed notions. The corollary of this is that actual social change is likely impossible, since one cannot separate the levers for change from their origins. We could consider that if knowledge is relative to the society in which it exists, then change is either extremely difficult, or potentially impossible.

On the other side of the debate, Chomsky believes that there is an absolutely definition of truth and from it a related notion of justice that is fundamental to the human condition. His approach is definitely more optimistic than Foucault’s, as it leaves a path forward for change. Chomsky might agree that much of the definitions of the terms we use, are grounded in the society from which they came, but that this still leaves open the door for alternative definitions, from outside of the society, that can be worked toward.

Both men believed that our society is unjust, and it would be pretty foolish to disagree with this assertion, even 40 years after this debate. That our society has advanced at all has been through the tireless work of people working under the assumption of an absolute form of justice, and another group of people making sure that our definition of what is just is continually examined, as our society changed. Change in our society requires the optimism of the absolutists, and the scrutiny and pessimism of the relatists.

Our structure of education, if it is to be reformed, therefore requires both people working toward what they feel is a concrete target, and people helping push the target.

October 4, 2011 / 6 Comments

Math of the game Portal

A few weeks ago, I spotted one of my students playing an interesting looking game on his computer, so I asked him about it. Turns out the game he was playing is called Portal (created by a company called Valve), and it’s still a fairly popular game today.

The basic premise of the game is that you have a special kind of gun which can create two portals, and your character can use these portals to travel instantly between two locations in the level. Each level of the game is a 3D puzzle that you have to solve. I decided tonight to look to see if I could find this game online, and I found this interesting 2D version of Portal, built in Flash. To really understand how the game works, I recommend playing it.

Step 1

What I noticed, as I played the game, is that the puzzles are very much logic puzzles. You need to both strategize what a good move will be in the level, and also experiment a fair bit to figure things out. Every time your character dies in the game, you get to try over again to solve the puzzle, and so you get as many chances as you need to try to figure the puzzles out. Some of the puzzles involve a bit of reflexes, and some of them just involve some reasoning.

Step 2

I also noticed that the portals themselves introduce a little bit of topological reasoning to the game. Once you start playing, you quickly realize that two positions in the game are equivalent, if there is an easy way to generate a portal between them. You also learn some tricks like the "infinite loop", where you create two vertical portals which you can fall through endlessly. Sometimes, I felt a little like I was playing Towers of Hanoi (itself a fairly mathematical game) because I would have to plan my moves ahead and choose the order of my portals carefully.

Step 3

The perspective of the 2D version of the game is quite distorted (to allow for more surface area upon which to place one’s portals) and this got me thinking about perspective. "What’s wrong with the perspective in this game?" I thought. "Oh right, that wall and that floor are at the wrong angle with each other." The line of site of the portal gun also reminded me of intersecting lines, and I found myself visualizing the intersection of a wall and a line from my current position, and wondering if I was "going to hit that wall or not."

Step 4

Timing is fairly critical on some of the levels. I found myself occasionally timing how long it took me to do an action, or a series of actions, so that I could time myself to be "in good position" to avoid a deadly (to my character) obstacle in the game. This is a little bit like algebraic reasoning, wherein I work backwards from one time and attempt to calculate (at least roughly) a good time to begin the sequence of actions.

Step 5

I experimented a fair bit, and would systematically move one of my portals a little bit each time, so that I could see how this changed the outcome of my movement through the portals. This is similar to a strategy used to solve some problems (and is a little bit like the scientific method) in mathematics. Sometimes to find a pattern, you have to build up representations in carefully thought out sequences, and the same is true in this game.

Step 6

Portal is challenging like interesting problems in mathematics are. In the game, you can keep trying to work on the puzzle for as long as it takes. The only feedback you get from the game is the progress you have toward completing your goal, or (as often happened to me) your character’s death.

Level completed

You could formalize some of the mathematical ideas that are part of this game, much in the same way that formalization of the use of Angry Birds to teach physics has been done.

I’d be interested to hear of anyone has some others on how this game could be useful in a math classroom, so please share any ideas you have.

October 1, 2011 / 3 Comments

What I learned from making waffles

When my son woke up this morning, he asked me to make him waffles. Having never made waffles before, I was going to refuse, but then I decided to take a chance, and just learn how to make waffles. Most of my adventures in the kitchen in the past 6 years have happened with the help of my wife, but I really need to spend more time cooking by myself, like I did when I was a bachelor. Both my wife and I agree that more balance needs to happen between us in terms of who makes meals (although she’s pretty happy with me doing all of the dishes…).

I looked up a recipe for waffles online and decided to make sure I had all of the ingredients. With my son’s help, we looked through the kitchen and found all of the ingredients for the waffles, except we only had 1 egg, and the recipe called for two. We also didn’t have enough vegetable oil, so I had to do a couple of substitutions.

Together, my son and I measured out the ingredients for the waffles and put them into a bowl and mixed them all up. I then pulled out the waffle maker, and figured out how it worked, with my son’s help. It certainly makes waffles easy to make!

Making waffles

Unfortunately, I didn’t know how much of the wafflie mix to put into the waffle cooker. I decided to take a guess and glopped some mix into the cooker. As you can see, this didn’t work out so well.

Mess on the counter

The mixture overflowed from the waffle cooker, and onto the counter. Oops! I’d put too much in! After some experimentation, and more messes, I figured out how much was the right amount to cook.

The big moment came, when I actually got to try my waffles for the first time.

Yummy wafflies ,

My wife and son agree with me, my waffles were yummy! I was pretty pleased with myself, and although I realized afterward that making waffles is really not all that difficult, I still felt a sense of accomplishment.

As I ate my waffles, I thought about how this experience should translate to student learning.

I picked a project which was meaningful to me.
I created a plan to complete my project.
I followed through on my plan, which required me to trouble-shoot, revise my plan, and clean up after my mistakes.
I enjoyed and shared the fruits of my labours at the end of the project.
I learned a skill I can almost certainly use later.
I took a risk and met the challenge successfully, while overcoming some obstacles in my way.

While it’s clear to me that not every learning experience can be as successful, or as self-directed as my waffle-making experience, it’s also clear to me that too few experiences of children in schools mirror my experience at home. We spend a lot of time directing the lives of students, and I’d like to see more schools with structures in place that allowed students to be in charge of at least some of their learning.

September 27, 2011 / 2 Comments

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.

September 27, 2011 / 5 Comments

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?

September 26, 2011 / 0 Comments

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.

September 21, 2011 / 19 Comments

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.

September 19, 2011 / 2 Comments

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.

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The Reflective Educator