problem-solving skills

The following post pertains to most scientific word problems, i.e. physics, chemistry & math problems. It doesn't necessarily apply to humanities or other subjects. The information here is based on my own views and experiences. Science here refers to a generalized curriculum that also includes mathematics.

There is a thing that isn't taught frequently in schools: problem-solving skills. More often than not, people often understand the basic concepts that were taught during lectures/classes, but are unable to apply them to homework or test problems. Worse still, some students who couldn't understand a concept would be forced to "memorize" concepts, which doesn't work most of the time. Personally, I usually assume the responsibility of teaching how to understand concepts and work through problem to the instructor, but this is hardly true, mostly because of the lack of time and attention devoted to scientific curriculums. Unfortunately, this leads to more of a vicious cycle, because the lack of a solid scientific foundation in elementary school, middle school, and high school could only cause further problems upstream. And this is problem one reason why many students dislike science: to them, it's an exotic thing that they could never understand.


From the perspective of someone who knows science, I find in impossible to believe why people have such a difficult time with science. Neither do many of the science teachers. What they often do (and to my disappointment) is that they would try to teach things at a level that is beyond what the students are capable of. They often attempt to explain pure concepts and fail to establish links between what is being explained and what has already been explained. The most crucial link (as far as I believe) to understanding a concept is to link it do something that is already known. From a psychological perspective, this quite logical: a concept that is isolated - has not been linked to any other concepts - will be very difficult to remember. This is one reason why I'm not an advocate of using mnemonics. Many use mnemonics purely for fun and often do grasp the reason behind why things are. Of course, there are some things that do need blatant memorization, but in general, scientific topics often do not need them, because science is based on logic and empirical knowledge (however, there are things that are purely "conventional" [e.g. "7" colors of a rainbow, direction of a current, right-handedness of the 3D-Cartesian system, etc.] and students must know so.


In order for students to understand science, it would become necessary to show how problems should be tackled. One approach - which some instructors use - is to provide an "exhaustive" list of the ways in which a problem, based on some concept, could manifest itself in various different ways, and to teacher the students how those problems should be solved. An alternate approach is to provide a few examples of how problems are solved, and hope the students can extrapolate those skills. I personally find neither satisfying. The first approach is by no means exhaustive: there is an infinite variation in which problems could come, figuratively speaking, and this often requires the student to memorize unnecessary knowledge. The second approach expects the student to know how problem-solving skills can be extrapolated. But the problem is that some students may not even know how to solve a problem.


My suggestion is to actually teach the problem-solving skills in the most general way possible. And this is what this post will be about - the most general way (that I can think of) in which scientific problems can be solved. This makes it redundant to teach specific steps for solving each & every type of problem that may or may not be encountered, since a student can simply learn just the basic rules for problem-solving.


Clarify
The first step to any word problem is to strip the problem of useless information. In some sense, a word problem is really the convoluted version of something simple. Some types of information will be useful to solve one particular problem, while the same information may be useless in some other problem. So there's no general rule to determine which things to remove and which shouldn't be removed.

Other than removing redundant information, it is necessary to highlight important information, and - especially important - understand what the question is asking for.

Let's consider the following word problem:

A baseball falls from the rooftop in 5 seconds. Determine the height of the rooftop.

It's quite obvious that the falling object itself is not going to be useful here. So let's make that word problem a bit simpler:

object falls from rooftop;
time taken (t) = 5 s;
height (H) of rooftop = ?

I've removed some adjectives and verbs to make things especially clear (even though this problem itself might be pretty easy to understand for someone familiar with basic physics). Often, in science problems, numbers are quite important, although this is not necessarily always true, since some problem-writers tend to put in redundant numbers which aren't going to be used at all.

Sketch
When a problem has been simplified by removing redundant information, it becomes easier to sketch the problem. The usual way to do it is to start with a blank page, and start relating the information that you have read from the problem with knowledge and concepts learned during classes. Basically, try to construct a diagram of some sort (and don't be concerned about neatness) that describes the situation described in the problem. For example, in the problem discuss above, you might consider drawing an object falling from height = H to height = 0. This step is not absolutely necessary, but it will help you if you get stuck.

Divide & Conquer
When a problem is too big and contains too much information, it becomes necessary to convert it into smaller problems. This is how several equations can be chained together to find a single result using multiple variables. For example

A photon strikes an electron of a gaseous helium atom, causing the electron to be ejected. Given E_I(He) = 1520 kJ mol^-1, what is the wavelength of the photon?

Now this problem is actually quite involved. Let's start by clarifying the problem:

1 photon strikes 1 electron of 1 He_(g) atom;
electron ejected;
E_I(He) = 1520 kJ mol^-1;
photon wavelength (λ) = ?

So we are given ionization energy (E_I), and we are asked to find wavelength. Let's start by working backwards: how does one get the wavelength (as far as chemistry is concerned)? The equation that is usually used to find wavelength in chemistry problems is:
c = λ ν (c = speed of light, ν = frequency)
Which variables are known? c is known - no problem there. What is ν? What is the usual way of finding frequency (other than using the previous equation)? Which equation contains a frequency term? Here it is:
E = h ν (E = energy, h = Planck constant)
h is, again, a constant that can be found easily. But where does energy, E, come in? It seems that the problem has mentioned the word "energy" somewhere, but it's not in the right form: it's the ionization energy (of every mole of gaseous He). Did we ionize anything? Yes, we "ejected an electron" - that's ionization right there. How many moles of He did we ionize? One atom. How many moles is that? (These are a chain of questions that should be thought of while analyzing the problem, and don't have to be written down)

The relationship between atoms and moles is (to derive this, just look at the units of N_A, which is mol^-1, and compare that with the units of number of moles, which is mol; see how that cancels out so that the number of atoms has no units?):
(number of atoms) = (number of moles) × N_A
Since we have the number of atoms (just 1), and we know what N_A is, we can easily find the number of moles. With the number of moles at hand, we can use:
(ionization energy of n mol) = n mol × (ionization energy of each mol)
Again, this can simply be derived by looking at the units of ionization energy, which implies that we must multiply it by the number of moles to get rid of the mol^-1. Now, with the ionization energy of n mol (which in this case will be for 1 atom), we can equate this energy with the energy that we desperately needed for E = h ν. Now the problem is complete, we just need to plug in the numbers. This is a typical example of how a problem can lead to several smaller problems, and each individual problem must be dealt with separately

Algebra
Most of the time, one should not solve the problem without equations, especially for unfamiliar problems. Try to list every formula that might be related to the problem, and see which ones are applicable here (this is why when remembering formulas, do remember when they are applicable). For example, an object in free fall (the problem discussed above) will typically accelerate at a constant rate due to gravity (air resistance ignored). Therefore, the equations of motion for objects in free fall can be used here:
s = u t + 0.5 a t²
v = u + a t
v² = u² + 2 a s
In this problem, none of the variables are given directly. But some are given implicitly. For example, a = g = 9.81 m s^-2 (acceleration of gravity), while u = 0 m s^-1 (initial velocity) and t = 5 s (time taken). Hence, the first equation is most applicable because we have just one variable missing: s, which is the distance traveled by the falling object, i.e. the height of the rooftop. As you can see here, it is important to establish relationships between things that are already known (knowledge) and things that might vary from situation to situation (the problem). With sufficient practice, this will become very easy for known problems.

It is also important to realize that sometimes equations don't give the answer directly, or one may have to construct the equation using a set of known rules. For example:

The mean of 3 consecutive numbers is 5. What are these 3 numbers?

In this case, one should always start by constructing an equation in terms of unknown variables. Don't fear unknown variables. Without them, many complicated problems could go unsolved. So start by writing the equations in terms of them, and worry about finding them later.

In this problem, let's call the 3 consecutive numbers: a, b, and c. From the equation for mean:
(a + b + c) / 3 = 5
Because there are 3 different unknown variables, we need 3 non-equivalent equations (by that, I mean 3 equations that are not equivalent to each other). It's easy to construct a few more:
b = a + 1
c = b + 1
Now we have 3 equations. The next step would be to solve them, and this is why it is necessary to learn basic algebraic skills. (Be careful with equations, label the variables correctly: some variables may not be equal to another, so subscripts may be required to differentiate them.)

Skill
The process of solving problems is a skill that can only be acquired through practice. Therefore, it is necessary for a student to learn this skill from day one of science class. Over a sufficiently long period of time, a student should be capable of solving problems with ease. But this is by no means a short period of time. Thus, I would say that this skill is absolutely essential and should've been taught from the beginning. Once a student has grasped this skill even partially, practice and further practice will be able to allow the student to learn the skill even better (like a positive feedback loop). Somethings can only be acquired with hindsight - when a students learns a method of solving something from the instructor, hindsight and practice will be able to provide the foresight necessary to predict how a certain problem should be solved, without going through all these stages mentioned. At the optimal level, a students will be capable of solving problems without even thinking any of these steps. It will become a natural habit that always goes unnoticed.