Author: David Wees (page 42 of 97)

At Edcamp Delta, Brad shared the following parable with me.

In the parable, researchers supposedly put 5 monkeys together in a cage, and in the cage there were some steps leading to some bananas. Whenever any of the monkeys would start to go up the steps to get a banana however, the hypothetical researchers would spray all of the monkeys in the cage with cold water. Every time any one of the monkeys went to get a banana, the entire group would get sprayed with cold water. After time, the monkeys would self-regulate and would attack any monkey that tried to make a move for the bananas, and the researchers stopped spraying the monkeys with water.

In the next stage of the experiment, they removed one of the original monkeys, and replaced it with a new one. The new monkey, not knowing about the cold water, would eventually decide to make an attempt to get one of the bananas, at which point the other four monkeys would attack the new monkey, preventing it from getting a banana. Over time, the new monkey learned not to make an effort to get a banana, and even though it had not experienced the spraying of cold water, it learned about the aversion to going to get a banana.

Over time, the researchers replaced each of the original monkeys with a new monkey, and each time, the new monkey learned not to attempt to get a banana as the other monkeys would beat it up. In other words, the researchers learned that the monkeys had passed along a cultural adaptation (avoidance of getting bananas) without having the original reason for the adaptation still present.

There was an experiment done with monkeys in the 1960s which is likely from where this parable was derived. Stephenson was able to train monkeys to avoid touching an object with blasts of air, and discovered that these monkeys could transmit this aversion to a ‘naive’ colleague (Stephenson, 1967). So while the parable seems to have never actually been tested, the results from Stephenson’s experiment suggest the parable is believable.

The point of this parable is that we can often repeat patterns as a form of peer pressure without understanding the original reasons for the behaviour. While we have an advantage over monkeys, which is that we can clearly articulate to each successive generation the reasons for a particular action that we take, unfortunately, we often do not share our reasons with the next generation, and only share the rules that we have established.

We see this experimented repeated over and over again in society. For example, suppose that in a subway station a door was locked in the early morning, and another door was unlocked. The first people to arrive at the subway station, not seeing anyone around, will test both doors, and enter through the one that works. Eventually traffiic gets a bit higher, and the door that works is always open because of constant traffic through the door. The other door, which doesn’t work, is assumed to not work since no one seems to be using it. If a custodian came by and unlocked the door, it wouldn’t be used, until someone came and actually went against the grain and tested the other door. We could call this person who tests the other door, even though no one else seems to be using it, an innovator.

Cultural pressure to follow what we’ve always done before, without really understanding the reasons why we follow the actions we do, is quite strong. How many of our structures in schools are done in certain ways even though the restrictions on doing them in other ways no longer exist?

References:

Stephenson, G. R. (1967). Cultural acquisition of a specific learned response among rhesus monkeys. In: Starek, D., Schneider, R., and Kuhn, H. J. (eds.), Progress in Primatology, Stuttgart: Fischer, pp. 279-288.

In an excellent post Michael Doyle says the following:

Electronic calculators are abstract, abacuses are not.
Slide rules fall in-between.
Our sense of quantities has become abstract.

Here is my comment back:

It’s not so much that numbers are abstract to students, but our representations of those numbers (the numerals that make up 122) are abstract.

Base 10 is a much more complicated concept than we give credit to in our teaching of it.

How many people are fooled by the numbers thrown out on the news (example: 12 millions dollars over-budget, on a budget of 1.3 trillion)?

The scale on a slide rule is a logarithmic one, and so it breaks the intuitive linear sense of numbers that we are born with. Hence, I don’t think it’s necessarily the case that the scale helps us understand the operations any better. A slide rule is a calculator, it just uses slides instead of buttons, and the operations are just as mystifying to most people.

Similarly the operations on a calculator (or the standard algorithms we teach kids to use on pencil and paper) are just as mystifying. Regardless of which algorithm one uses, one should have an understanding first of what the expected outcome of the algorithm should be, which unfortunately, we spend almost no time teaching in our mathematics curriculums.

It has become more important to do operations quickly and accurately to demonstrate "computational fluency" and error checking, a MUCH more important skill, has fallen by the wayside. Being able to estimate or predict the outcome of an algorithm is not only a far more useful life-skill than accurately calculating it (when one wants accuracy, one should use the correct tool) and further, it requires actual understanding of the process.

I think Michael is right that our sense of quantities is abstract, but I think that this has always been true for most people, regardless of the form of technology we’ve used to do our calculations. It is not so much the calculator that is the problem, but the embedded abstraction in numerals themselves. Unfortunately, we spend much more time teaching kids how to do operations on numbers, and I don’t think we spend enough time teaching about what the numbers themselves mean (or at least on reminding students what they learned in previous years on numbers). For example, in the BC curriculum, the discussion on the relative size of numbers seems to stop in grade 6.

It is too easy to blame kids lack of fluency as adults with numbers on the technology used to do calculations with those numbers. Instead we should ask ourselves, where have we taught them fluency with numbers? How have we helped them learn about the abstraction numbers represent?

I just remembered this quote from a participant at the Computer Based Math summit, and it is one of my favourite from the day.

"Math is not done on paper or by a computer. Math is done by the brain."

I wish I knew the name of the gentleman who said it. I believe he introduced himself as a Professor of Mathematics. If anyone knows, please let me know.