Expanding Limits

What is half of 8? Most people would immediately think that the right and only answer is the number 4. Why? Because we all learn mathematics in school, and it teaches us that half of 8 is 4. But what if we look at the number visually? We can cut or fold the number 8 into two equal parts. If we cut or fold 8 horizontally, we would get the number 0 or the letter o, depending on whether the number or the alphabet come to mind first, as the half of 8. And if we cut or fold 8 vertically, we would get 3 or , depending on which way we look at it. And if we fold randomly (not horizontally or vertically) from the middle of 8, we could get even more answers.

Now look at this for a moment: O. What is it? Again, most people would think about the number 0, the letter O, or a circle. How many of you would come up with answers like an egg, a top view of a cup, the earth from outer space, a coin, a snake eating away at its own tail? Not many maybe, and if you are one of the few who saw possibilities beyond a number, letter or…a cirlce, consider yourself lucky as you have retained the ability to think beyond the obvious.

If you are one of the majority, don’t feel bad either, as most of us have been conformed either by what we have learnt in school and how we think people (our boss, our teacher from school, the society) expect us to answer the question. Humans are social beings who follow external expectations to various extents. The tighter our relationship is with other people and the tighter the society we live in, the more pressure we seem to get to fulfill others’ expectations.

Here’s a story that will highlight both how most of us have become conformist thinkers, and how we can break out of that mold to become creative thinkers.

In 1905 Niels Bohr sat a physics exam at the University of Copenhagen, in which he was required to answer a question on how to use a barometer to measure the height of a building. According to Arthur J. Cropley in “Creativity in Education and Learning” (2001), the expected answer was “to use the barometer to measure the air pressure at ground level and at the top of the building, calculating the height of the building from the difference between the two.”

Bohr’s response to the question was to attach the barometer to a long string, and lower the barometer until it reached the ground. By measuring the length of the string and the height of the barometer, one could obviously get the height of the building. Not a wrong answer, but not the answer expected by the teacher. Bohr failed the exam. Upon Bohr’s appeal at the outcome, it was determined that although Bohr’s answer was not incorrect, it “showed no knowledge of physics”. Bohr was given a few minutes to demonstrate such knowledge.

Apart from giving the right answer expected by the teacher, Bohr gave other alternatives that demonstrate his knowledge of physics. One of the alternatives was to “throw the barometer off the roof and count the seconds until it hits the ground. Calculate the height of the building with the formula s=1/2at².” Another alternative was to “measure the length of the barometer and of its shadow and calculate the ratio of the two. Measure the length of the shadow of the building and multiply it by the same ratio.”

He even mentioned several other alternatives unrelated to physics. They include “climbing up the fire escape and marking off the height of the building in barometer lengths (using the barometer as yardstick)”, and “offering the janitor the barometer as a bribe to reveal the height of the building (using barometer as an object with monetary value)”! (Cropley, 2001.)

Other than being amusing, this story draws our attention to three issues. The first issue we would like to highlight is the type of education most of us went through: we were always expected to come up with one right answer determined by the teacher. None of Bohr’s responses were wrong per se, but only one met his teacher’s expectation of a predetermined answer. This issue shows how most of us become accustomed to coming up with the ‘right’ solution, thus directing our thinking to what is expected by others – be they our teachers and our parents (when we were young), our boss, or the society at large.

The second issue we would like to emphasis from Bohr’s example is that there are actually many right answers to a problem (even in a field as exact as physics!), but most of us are used to stopping at one solution. We limit ourselves by coming up with only one solution – we consider the solution we have thought of is the one right solution for the problem and stop exploring what perhaps could be more exciting possibilities.

The third issue we would like to underline is the fact that although Bohr knew how he was expected to respond to the problem, he challenged and entertained himself in finding what other possible solutions there were to solve the problem. In other words, he knowingly challenged the system of ‘one problem, one solution’ conventionally used in many schools.

Bohr’s inquisitive mind had enabled him to make fundamental contributions to human understanding of the atomic structure and quantum mechanics that won him the Nobel Prize for physics in 1922.

So we have seen how when it comes to thinking, external expectations, or rather, what we perceive as external expectations, can be very limiting. James L. Adams in “Conceptual Blockbusting” (2001) calls this as the “tendency to delimit the problem area poorly”, defined as our inclination to impose too many constraints upon solving the problems we are facing (p.25).

Going back to the examples at the beginning of this article, the most common answer to “what is half of 8?” is still 4, even when we are no longer in math class. The most common answer to “what is: O?” is still the letter O, the number 0, or the shape circle – answers that appeal again to what we know as the right answer back when we were in schools. The problem is, we are no longer in school.

In real life situations, no one actually gives us problems just for the sake of testing our ability. As such, no one has predetermined solutions for real life problems that have been handed out for us to solve. Yet, most of us are still confined by looking for ‘the one right answer’. So how do we challenge ourselves to find other possible solutions to a problem? How do we break away from conformist thinking?

The answer is to be aware of the limits that we are imposing on ourselves when finding solutions to a problem. Are they actually there? Outside of the school context, no one expects us to limit the answer to “what is: O?” to resemble a known letter, number, or shape. Once we realize this, we can move away from the expected answers and come up with answers like the steering wheel of a car, an eyeball, an unsharpened end of a pencil, etc.

So far, so good; but our responses are limited to the visual realm. Once we realize that we do not have to stop at one answer, we can think of more exciting possibilities, such as O as a sound. It’s the sound one makes when one finally realizes something, as in “Oh, I see.” It’s the sound people use in yoga (Om). It’s the sound of disbelief, as in “Oh no! What have you done!. Like Bohr, it takes courage to give unexpected answers, are you ready to break away?

In James L. Adam’s words, “limits are negotiable”. Expand your limits!