Convergent and divergent thinking
Suppose we are asked this question: "What is the square root of 5?" How would we go about solving it?
We would start by narrowing the scope of our search. Obviously 2-squared is 4 and 3-squared is 9, and so the answer should lie between 2 and 3. In fact, 2.5-squared is 6.25, so the answer should be between 2 and 2.5. Also, 2.2-squared is 4.84 and so the answer should lie between 2.2 and 2.5.
This way we "converge" to an answer that is close enough to the required solution.
This kind of thinking is characteristically called convergent thinking. In such thinking processes, we are looking for "a" solution. Our process constitutes taking into account several factors, combining them together, eliminating what is not necessary, fine tuning what we already have, until we arrive at the desired solution.
Now consider the following kind of problem. Take an object that you can see -- say a pen. Now come up with as many possible uses of the pen as you can think of.
This would of course, include the "standard" use of a pen -- to write. But then, we could use a pen for a variety of other purposes. Maybe we can use it as a measurement device to measure the length of a table in pen units. Maybe we can use it as a lock to hold an open window from closing. We can perhaps sell the pen and get some money. We can use a pen to hold rubber bands in place. And so on..
When I ask such questions, one typical response is "anything goes" as an answer. But that is not true. In fact, if you were to state that "we can eat the pen" as an answer, it is wrong (for a typical pen). So, it is not that "anything goes" is an answer. There are correct answers and incorrect answers. But the thing is, there are several correct answers!
This is an example of what is called divergent thinking. In this style of thinking, we start from a given problem state (the pen, in the example above) and try to fit it consistently in different worlds.
Divergent thinking is a necessary ingredient for any kind of design activity. Suppose you have to design a car. All cars have more or less the same components -- an engine, transmission, wheels, steering, seats, etc. But then, not all cars look and feel the same. A Brio is different from an Alto, which is different from an Indica, which is different from a Verito, and so on.
The way these different components can be put together, so that the end result is an efficient, ergonomic, sturdy and safe automobile, are several.
For some reason, we have forgotten to inculcate, teach and nurture divergent thinking in our curricula. I find the dearth of divergent thinking especially acute among techies.
Sometimes I have heard this argument that the emphasis on precision, especially in mathematically inclined fields like science and engineering, is detrimental towards divergent thinking. But it is not very convincing.
Divergent thinking also requires a lot of precision. Especially, when there are physical constraints involved. For instance, consider again the task of putting together different automobile components to design a car. We can think of putting them together in several ways. But, each such combination has precise bearing on the overall performance of the car. One combination could be more efficient than the other in terms of fuel usage, but more expensive in terms of overall cost. Another combination may make the car safer to drive, but may make it inefficient. And so on.
So, divergent thinking does not imply lack of precision.
There is also another mental block I've often seen -- that which equates divergent thinking to shallowness, and convergent thinking to depth.
A convergent thinking process may go deep into a subject, but that does not make divergent thinking shallow. Besides, there is nothing macho about "depth;" it has its own follies. Let me explain with an example.
Recently, we were conducting interviews for a graduate program, where the incumbents all had an engineering degree. As part of the interview, I posed two puzzles to each candidate -- one which required a convergent thinking process and another, which required a divergent thinking process.
There was this candidate, who swore by depth. He had been interested in some specific subjects in the curriculum and had done vast amounts of reading and exploration on his own. So much so that his performance in other subjects suffered and he had failed in a couple of them. It was however, evident that his interest in the subjects that he liked, were genuine. He spoke at length about signals and signal processing and the projects he had worked on his own. He clearly had depth in his understanding. His main peeve was that our education system does not encourage people who are "truly passionate" about learning, as his couple of failures in subjects that he didn't care about, had proven to be a hurdle at every step.
I could definitely empathize at some level, with such sentiment. But then, when I posed the puzzles, I was in for a surprise.
His performance in the divergent thinking puzzle, was understandably bad. I had given 30 seconds for him to come up with as may answers as possible. But he remained silent for 30 seconds thinking (deeply?), and came up with 3 answers, all of which revolved around electrical engineering solutions.
Not much surprise there. The real surprise was with the puzzle requiring convergent thinking. His performance was equally bad there too! The reason was not hard to see. This puzzle required him to think in terms of permutations and combinations to arrive at an answer. But, given any problem, he just tended to think in terms of signals and signal processing!
If all you have is a hammer, everything looks like a nail.
That is the problem with focusing only on depth and treating it as a macho trait. It just conditions our mind to think within a single paradigm.
In contrast to the above, was another candidate, who was nothing spectacular in terms of her grades. As expected, she did not fare too well with the convergent thinking puzzle. But when given the divergent thinking puzzle, she immediately rattled off with some 15 solutions in 30 seconds -- 2 seconds per solution on average! At the end of it she says, "That was fun!!! If you gave me 30 more seconds, I'll come up with several more answers!"
And her answers came from a wide variety of scenarios -- from technological solutions to business solutions to household use to emergency care and so on. Many of which caught me completely by surprise and yeah, it was fun!
Divergent thinking requires us to keep hopping between mental models. As I had argued in a previous post, changing mental models frequently is an emotionally draining process. When mental models change suddenly, it takes a while for the brain to understand and compute repercussions; and our initial reaction is an emotional one. A large variety of humour is based on suddenly changing mental models with trivial repercussions -- our initial reaction would be emotional, in terms of laughter.
There is another theory I have heard that divergent thinking constitutes the "female" style of thinking while convergent thinking constitutes the "male" style of thinking.
Apparently, men think of the world in terms of neatly separated boxes, each of which are distinct and separate from one another. And for women, everything is connected with everything else and they can keep hopping from one subject to another with much more ease than men.
Well, then, it must be a male who made up such a theory -- that categorizes thinking styles into neatly separated boxes called male style and female style. And according to this theory then, females should not accept this theory of neat separation of thinking styles into male style and female style. If they do, then they are male. :-)
Needless to say, I am not very convinced by such "neat separation" theories. I think everybody has both styles of thinking -- convergent and divergent -- perhaps expressed to different levels. Someone may be pre-disposed towards convergent thinking, but that does not mean that they cannot think in a divergent fashion at all. And vice versa.
But more fundamentally, I think, convergent theory is a special case -- a focused kind of divergent thinking. If you are familiar with game theory, this is somewhat like pure strategies and mixed strategies. In pure strategies, you choose one action among a given set of actions. In mixed strategies, you choose all actions with some probability. There is no dichotomy between pure and mixed strategies. The space of mixed strategies subsumes pure strategies. Pure strategies are just a special boundary case of mixed strategies.
Similarly, thinking is about making connections. So compartmentalized thinking is not different from connectionist thinking. It is just connectionist thinking within a single mental model.
If we keep the mental model intact and keep connecting things together within the model, we eventually achieve depth. If instead, we allow for connections to span across mental models, we achieve divergent thinking.
We would start by narrowing the scope of our search. Obviously 2-squared is 4 and 3-squared is 9, and so the answer should lie between 2 and 3. In fact, 2.5-squared is 6.25, so the answer should be between 2 and 2.5. Also, 2.2-squared is 4.84 and so the answer should lie between 2.2 and 2.5.
This way we "converge" to an answer that is close enough to the required solution.
This kind of thinking is characteristically called convergent thinking. In such thinking processes, we are looking for "a" solution. Our process constitutes taking into account several factors, combining them together, eliminating what is not necessary, fine tuning what we already have, until we arrive at the desired solution.
Now consider the following kind of problem. Take an object that you can see -- say a pen. Now come up with as many possible uses of the pen as you can think of.
This would of course, include the "standard" use of a pen -- to write. But then, we could use a pen for a variety of other purposes. Maybe we can use it as a measurement device to measure the length of a table in pen units. Maybe we can use it as a lock to hold an open window from closing. We can perhaps sell the pen and get some money. We can use a pen to hold rubber bands in place. And so on..
When I ask such questions, one typical response is "anything goes" as an answer. But that is not true. In fact, if you were to state that "we can eat the pen" as an answer, it is wrong (for a typical pen). So, it is not that "anything goes" is an answer. There are correct answers and incorrect answers. But the thing is, there are several correct answers!
This is an example of what is called divergent thinking. In this style of thinking, we start from a given problem state (the pen, in the example above) and try to fit it consistently in different worlds.
Divergent thinking is a necessary ingredient for any kind of design activity. Suppose you have to design a car. All cars have more or less the same components -- an engine, transmission, wheels, steering, seats, etc. But then, not all cars look and feel the same. A Brio is different from an Alto, which is different from an Indica, which is different from a Verito, and so on.
The way these different components can be put together, so that the end result is an efficient, ergonomic, sturdy and safe automobile, are several.
For some reason, we have forgotten to inculcate, teach and nurture divergent thinking in our curricula. I find the dearth of divergent thinking especially acute among techies.
Sometimes I have heard this argument that the emphasis on precision, especially in mathematically inclined fields like science and engineering, is detrimental towards divergent thinking. But it is not very convincing.
Divergent thinking also requires a lot of precision. Especially, when there are physical constraints involved. For instance, consider again the task of putting together different automobile components to design a car. We can think of putting them together in several ways. But, each such combination has precise bearing on the overall performance of the car. One combination could be more efficient than the other in terms of fuel usage, but more expensive in terms of overall cost. Another combination may make the car safer to drive, but may make it inefficient. And so on.
So, divergent thinking does not imply lack of precision.
There is also another mental block I've often seen -- that which equates divergent thinking to shallowness, and convergent thinking to depth.
A convergent thinking process may go deep into a subject, but that does not make divergent thinking shallow. Besides, there is nothing macho about "depth;" it has its own follies. Let me explain with an example.
Recently, we were conducting interviews for a graduate program, where the incumbents all had an engineering degree. As part of the interview, I posed two puzzles to each candidate -- one which required a convergent thinking process and another, which required a divergent thinking process.
There was this candidate, who swore by depth. He had been interested in some specific subjects in the curriculum and had done vast amounts of reading and exploration on his own. So much so that his performance in other subjects suffered and he had failed in a couple of them. It was however, evident that his interest in the subjects that he liked, were genuine. He spoke at length about signals and signal processing and the projects he had worked on his own. He clearly had depth in his understanding. His main peeve was that our education system does not encourage people who are "truly passionate" about learning, as his couple of failures in subjects that he didn't care about, had proven to be a hurdle at every step.
I could definitely empathize at some level, with such sentiment. But then, when I posed the puzzles, I was in for a surprise.
His performance in the divergent thinking puzzle, was understandably bad. I had given 30 seconds for him to come up with as may answers as possible. But he remained silent for 30 seconds thinking (deeply?), and came up with 3 answers, all of which revolved around electrical engineering solutions.
Not much surprise there. The real surprise was with the puzzle requiring convergent thinking. His performance was equally bad there too! The reason was not hard to see. This puzzle required him to think in terms of permutations and combinations to arrive at an answer. But, given any problem, he just tended to think in terms of signals and signal processing!
If all you have is a hammer, everything looks like a nail.
That is the problem with focusing only on depth and treating it as a macho trait. It just conditions our mind to think within a single paradigm.
In contrast to the above, was another candidate, who was nothing spectacular in terms of her grades. As expected, she did not fare too well with the convergent thinking puzzle. But when given the divergent thinking puzzle, she immediately rattled off with some 15 solutions in 30 seconds -- 2 seconds per solution on average! At the end of it she says, "That was fun!!! If you gave me 30 more seconds, I'll come up with several more answers!"
And her answers came from a wide variety of scenarios -- from technological solutions to business solutions to household use to emergency care and so on. Many of which caught me completely by surprise and yeah, it was fun!
Divergent thinking requires us to keep hopping between mental models. As I had argued in a previous post, changing mental models frequently is an emotionally draining process. When mental models change suddenly, it takes a while for the brain to understand and compute repercussions; and our initial reaction is an emotional one. A large variety of humour is based on suddenly changing mental models with trivial repercussions -- our initial reaction would be emotional, in terms of laughter.
There is another theory I have heard that divergent thinking constitutes the "female" style of thinking while convergent thinking constitutes the "male" style of thinking.
Apparently, men think of the world in terms of neatly separated boxes, each of which are distinct and separate from one another. And for women, everything is connected with everything else and they can keep hopping from one subject to another with much more ease than men.
Well, then, it must be a male who made up such a theory -- that categorizes thinking styles into neatly separated boxes called male style and female style. And according to this theory then, females should not accept this theory of neat separation of thinking styles into male style and female style. If they do, then they are male. :-)
Needless to say, I am not very convinced by such "neat separation" theories. I think everybody has both styles of thinking -- convergent and divergent -- perhaps expressed to different levels. Someone may be pre-disposed towards convergent thinking, but that does not mean that they cannot think in a divergent fashion at all. And vice versa.
But more fundamentally, I think, convergent theory is a special case -- a focused kind of divergent thinking. If you are familiar with game theory, this is somewhat like pure strategies and mixed strategies. In pure strategies, you choose one action among a given set of actions. In mixed strategies, you choose all actions with some probability. There is no dichotomy between pure and mixed strategies. The space of mixed strategies subsumes pure strategies. Pure strategies are just a special boundary case of mixed strategies.
Similarly, thinking is about making connections. So compartmentalized thinking is not different from connectionist thinking. It is just connectionist thinking within a single mental model.
If we keep the mental model intact and keep connecting things together within the model, we eventually achieve depth. If instead, we allow for connections to span across mental models, we achieve divergent thinking.
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