We’ve all been there. Someone shows us a method to solve a particular kind of problem and helps us as we try to go through the steps ourselves. Eventually, we feel confident we know how the method works. Then our instructor says, “Here is a similar problem. Try this one completely on your own.”
And we have no idea how to begin!
The instructor told us it is similar to the one we just saw. But we can’t quite see how it is similar.
Clearly, I’m not talking about elementary-grade worksheets designed to provide repetitive practice at one specific mathematical operation, such as addition or multiplication. Though even there, beginners can experience the same feeling of not knowing how to proceed, even if just one or two specific numbers change.
More significantly, however, instructors are almost certainly familiar with the situation where students in, say, a physics class are unable to solve a problem the requires the very linear algebra techniques they just applied to pass a test in the math class. Simply being in a different class can make a big difference.
The problem is even more pronounced when it comes to using mathematics techniques to solve real-world problems. Even when we are told explicitly that the same technique just used successfully to solve one problem can be used to solve the new one, fewer than 10% of us can actually do that.
This is not a sign of intellectual weakness. It’s a built-in feature of the human brain that cognitive scientists have known about and studied for many years. The good news is, it can be overcome. Those same cognitive science studies show us what we need to do to overcome the difficulty.
The starting point is the recognition that the evolution by natural selection of the human brain equipped it to learn naturally from experiences, to better ensure its survival, or to perform in a more self-advantageous fashion, when next faced with a similar experience. Learning through experience in this fashion is automatic, and tends to be robust, but is heavily dependent on the particular circumstances in which that learning occurred. It does not take much variation in the circumstances to render what was learned ineffective.
Let me give you an example. Imagine you are a doctor faced with a patient who has a malignant tumor in their stomach. It is impossible to operate on the patient, but unless the tumor is destroyed the patient will die. There is a kind of ray that can be used to destroy the tumor. If the rays reach the tumor all at once at a sufficiently high intensity, the tumor will be destroyed. Unfortunately, at this intensity the healthy tissue that the rays pass through on the way to the tumor will also be destroyed. At lower intensities the rays are harmless to healthy tissue, but they will not affect the tumor either. What type of procedure might be used to destroy the tumor with the rays, and at the same time avoid destroying the healthy tissue?
You might want to think about this for a while before reading on. Fewer than 10% of subjects are able to solve it at first encounter.
Have you solved it?
Are you one of that ten percent?
Okay, let’s move on.
The solution is to direct multiple low-intensity rays toward the tumor simultaneously from different directions, so that the healthy tissue will be left unharmed by the low-intensity rays passing through it, but when all the low-intensity rays converge at the tumor they will combine to destroy it.
Easy, no? At least, it’s easy once you are shown how to solve it!
Here is a second problem. A General wishes to capture a fortress located in the center of a country. There are many paths radiating outward from the fortress. All have been mined so that while small groups of men can pass over the paths safely, any large force will detonate the mines. A full-scale direct attack is therefore impossible. What does the General decide to do?
Again, you might want to think about this before proceeding.
This too is solved by at most 10% of subjects when they first meet it.
How did you do with this one?
The General’s solution is to divide his army into small groups, send each group along a different path, and have the groups converge simultaneously on the fortress.
This is, of course, logically the same problem as the cancer treatment, and the same “hub and converging spokes” solution works in both cases. Except that there is no “of course” about it. A substantial proportion of people fail to recognize the similarity, even when presented with one problem right after the other, as I just did here.
Specifically, a number of studies have shown that, whereas only 10% of people solve the radiation problem at first encounter, if they are first shown the General’s problem and its solution and then presented with the radiation problem, about 30% are able to solve the radiation problem. They see the similarity. That’s up from the 10% of subjects who can solve the radiation problem in the absence of any priming, but the do not see a connection between the two problems.
Notice that, in this two-problem scenario, where the subjects are not told to look for a similarity, fewer than a third of them are able to spontaneously notice it. Moreover, this disparity arises despite the fact that, knowing they are subjects in a psychology experiment, one might expect that all subjects would consider how the first part might be related to the second. But they do not make that connection.
On the other hand, if a group is shown the two problems one after the other, and told to look for a similarity, then around 75% of them can solve the second problem. (But fully one quarter still cannot!)
In both study cases with the two problems, a substantial number of subjects see one as a problem about warfare and the other as a problem about medical treatment. They do not see an underlying logical structure that is common to both. That’s a significant finding, from which we can learn a lot.
Cognitive scientists use the term “inflexible knowledge” to refer to knowledge that is closely tied to the surface structure of the scenario in which it was acquired. It takes time, effort, and exposure to two or more different representations of what is in actuality the same problem in order to recognize the underlying structure that is the key to making that knowledge flexible—something to be applied to any novel scenario having the same underlying structure.
In the case of the radiation problem, 90% of people see it purely in terms of radiation—essentially, a physics problem, unconnected to the General’s problem (a problem about military strategies).
Yet those two instances of inflexible knowledge become mere applications of a more general strategy of a hub-and-spokes wheel, where activity that follows the spokes from the circumference inwards converges into a single combined action at the hub—once you have acquired that wheel concept. When you have, it’s easy to recognize new instances where it can be applied. Being on an abstract level, your knowledge is flexible. But getting to that more abstract, flexible-knowledge structure is not automatic. It requires work. In the case of the hub-and-spokes solution, it takes exposure to a third problem scenario before most people “get” the trick. (See Note 1 at the end.)
Here’s why this is relevant to mathematics. In order to solve mathematical problems (or, more generally, to solve problems using mathematical thinking), you generally need to identify the underlying logical structure, so you can apply (possibly with some adaptation) a mathematical solution you or someone else found for a structurally similar problem. That means digging down beneath the surface features of the problem to uncover the logical structure.
To be sure, once you have mastered—by which I mean fully understood—a particular mathematical concept, structure, or method, it becomes a lot easier to see it as the underlying structure (if indeed it is). For example, if you understand linear algebra, you will be able to identify many problems where it can be applied, in math, in physics, in economics, or whatever. Quite simply, your (flexible) knowledge of the (abstract) method makes it possible for you to recognize when it may be of use.
Here’s the educational rub. Actionable, flexible knowledge cannot be taught, as a set of rules. It has to be acquired, through a process of struggling with a number of variants until the crucial underlying structure becomes apparent. There is no fast shortcut.
How do you recognize an underlying abstract structure or achieve understanding of an abstract method in the first place? It seems important to make those connections between the abstract-and-general, to the concrete-and-particular. We learn best from concrete examples we experience, and the mind’s natural inclination is to store knowledge in a fashion closely bound up with the scenarios in which it was first acquired. Overcoming that constraint to learning to recognize abstract structural or logical patterns requires effort, and often the study of more than just a couple of examples. You need several examples and you need to go deep into them.
To sum up, in order to develop the flexible thinking required to tackle novel problems using mathematics—regardless of where those problems come from, what specific mathematical concepts they may embed, and what specific techniques their solution may involve—what is required is experience working in depth on a small number of topics, each of which can be represented and approached from at least two, and ideally more, different perspectives.
NOTE 1: For a more expansive discussion of these issues, with more examples, see my December 1 Devlin’s Angle post for the Mathematical Association of America.
NOTE 2: For anyone curious as to why the brain works the way it does, and why it finds some things particularly difficult to master, particularly the recognition of abstract structure, check out my book The Math Gene, published in 2001, where I draw on a wide range of results from several scientific disciplines in an attempt to shed light on just how the human brain does mathematics, how it acquired the capacity to do so, and why it finds abstraction so challenging.