Education ∪ Math ∪ Technology
I recommend this talk by Eric Mazur on why he switched his teaching from lecture based teaching to peer instruction based approach. It’s more than an hour long, but it really is worth it.
How does this change how we teach? How much of what students learn in our classes is actually learned? If a student can only apply the concepts they have learned to very familiar contexts, and are completely unable to apply them in different contexts, can we really say they have learned the concepts?
As an experiment, I started out the beginning of this year and tried flipping my classroom, but with a slight twist: I have extra instructional time, so students were to watch the instructional videos (from the Khan Academy and IBVodcasting.com) during classroom time. We spent about 1/3 of classtime using the Khan Academy videos and exercises, about 1/3 doing problem solving activities (like what is available on www.mathpickle.com and projecteuler.net), and the rest of the time attempting to put the knowledge we were learning into a useful context for the students. While students were involved in these activities, I spent my time circulating the classroom and providing individual and small group support and instruction.
After a month I ended my experiment and am currently in a state of transition while I explore other possible ways of running my classroom. Here are some of the reasons I ended it.
I’m hoping to implement the RME model and looking for resources that will help support the course curriculum I’m required to cover in the International Baccalaureate program. If I can’t find resources to support this, I’m switching back to my style where I spend some time with students doing experiments in math, some time working on practice problems, and some time with me explaining mathematical concepts. I’m definitely not using the Khan Academy videos again (but I will probably use the IBVodcasting.com videos as additional support for students).
See this Slideshare presentation for a description of what the RME model looks like.
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First some misconceptions about science:
What if we create really clear explanations to address these misconceptions?
What if we run experiments with students? What if they design their own experiments to test out their hypothesis? (Really recommend watching this until the very end.)
Are there some scientific facts which are useful to know? Definitely. We could teach most those facts in a single science course if that was the purpose of science education. Why then do we teach science for 13 years in school?
A recent article suggesting that labs are "a waste of time" in science assumes that the purpose of science education is to transfer information. Instead, I believe that kids should learn that science is about experimentation and testing ideas, and that the facts which comprise scientific knowledge have been discovered through experimentation. They should know that science is not a collection of permanent facts about the world, but that instead, what is considered true in science changes. Science is more of a way of thinking about the world than a collection of isolated facts. Science is a philosophical perspective on the world wherein we recognize that through observation, experiment, and analysis, we can learn about the world.
What would you prefer? Should students know a lot of science facts, but perhaps don’t understand how those facts were derived? Or would you prefer that students understand the scientific method deeply, but might not know as much existing scientific knowledge?
A common argument for continuing to use grades, awards, competitions, and point systems in schools:
Students will experience extrinsic motivation outside of school, so we help prepare them for this experience by using similar reward systems in schools.
Our monetary system is a reward system. You work, you get rewarded with money, which you can use to purchase goods and services to improve your life. Work harder, smarter, longer, you get more money. The recent attention on the income disparity in the world, and the fact that people will lie, cheat, steal, and murder people for this reward should tell you that there are perverse side effects of our monetary system simply because of its existence.
Every reward system I can think of has unintended negative side effects. While every system has people who play by the rules, all of them similarly have people who have focused too much on the reward and then engaged in immoral acts.
Instead of basing what we do in schools on what is not working in our society, why don’t we look at other alternatives for to motivate our students? If you can think of a system based on motivating participants with a reward that doesn’t have serious problems with internal corruption and cheating, please let me know…
I recommend listening to this interview of Douglas Rushkoff on CBC Spark by Norah Young.
Rushkoff’s recommendation is that children should learn a little bit of a taste of programming right after they learn long division. His reasoning is basically this; once students see an algorithm like long division, and they learn how to make a computer compute long division for them, they’ll see that computers are devices which compute algorithms, not places for them to visit.
I’d like to add that teaching a computer to program something like long division would be very empowering for children. Having been through this process of learning what is one of the most complicated sequences of steps they’ve likely been exposed to in their young lives, they can then conquer this algorithm and "teach" a computer how to do it. As a happy consequence of teaching the computer the algorithm, they’ll probably understand how it works better.
I started learning origami again this past weekend. So far I’ve built a swan, and a couple of paper airplanes that are more advanced than what I usually make but none of it has been particularly complicated to make. I’ve often thought that origami would be a fun hobby, but that I wouldn’t find much use for it in my teaching.
Today, I watched a TED talk (thanks to @BobbycSmith for sharing it with me today) that definitely changed my mind. Origami is way up there now on my list of things I need to learn.
"Unless you have used a computer to learn something yourself, you are not in a good position to think about how it can help children learn." ~ Seymour Papert (1996) The Connected Family, p85
This statement by Seymour Papert is true both of people who promote the use of computers in schools, and those who argue against their use in schools.
Further, I suspect this is almost certainly true of any instructional strategy. You cannot effectively evaluate an instructional technique from a distance, because when you are immersed in the activity, you have a much different perspective than when you attempt to evaluate the activity without experiencing it. It is far too easy to look at a collection of data and use this to evaluate an educational practice and miss critical benefits of the practice that are invisible in your data.
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.
Learning math instead of computations
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.
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.
