“Do you want a doctor who just knows what they have been told by others? I want a problem solver, someone who can make observations and inferences.”
The above quote is what inspired this blog and, really, haven’t we heard similar things to this so many times before? I’ve lost track of how many times someone has dismissed memorisation of facts & instead prioritised problem solving.
I should say that I have no issue with an end goal of increasing student’s fluency at problem solving, I think it’s an admirable goal.
The trouble is, though, that often the first step in complex problem solving is to make sure the students have very solid declarative (factual) knowledge. Problem solving often relies on the application of multiple combinations of declarative, conditional & procedural knowledge. If you’re missing one of these components, then you won’t know how solve the problem. Even worse, you might not even know how to look up how to solve the problem.
To demonstrate this point, I’m going to take an example physics question from a past PAT paper. These papers are challenging and involve a lot of….complex problem solving.
Apologies, as the description will be physics heavy…but I’ll highlight the necessary bits of knowledge needed to answer it successfully. The question is:
An electron gun in a cathode ray tube accelerates an electron with mass m and charge −e across a potential difference of 50 V and directs it horizontally towards a fluorescent screen 0.4 m away. How far does the electron fall during its journey to the screen? Take m ≈ 10−30 kg and e ≈ 1.6 × 10−19 C.
Declarative knowledge 1: To know how far the electron falls, we need to know how long its journey is (time before it hits the screen).
Declarative knowledge 2: If we know the time of the electron’s journey, we can then use an equation of motion to solve for how far the electron falls. We do not yet have enough variables to calculate this, we need more.
Declarative knowledge 3: If you accelerate an electron across a potential difference of 50 V, it will end up with a kinetic energy of 50 eV.
Declarative knowledge 4: The equation for kinetic energy is Ek=0.5mv2.
Declarative knowledge 5: We need to convert the kinetic energy of the electron to Joules. To do this, we need to multiply by 1.6 × 10−19.
Conditional/procedural knowledge 1: To get the horizontal velocity of the electron, we need to rearrange the kinetic energy equation.
Declarative knowledge 6: The equation for speed is v = s/t.
Conditional/procedural knowledge 2: We can rearrange this equation to give t=s/v. We now have the time of the electron’s journey.
Declarative knowledge 7: Acceleration due to gravity is 9.81 m/s2.
Declarative knowledge 8: Initial vertical velocity of the electron is 0 m/s.
Conditional/procedural knowledge 3: Now that we have these variables, we can solve the equation of motion for how far the electron falls in that time.
Now, I make that at least eight separate facts that a student would have to have committed to memory before they can successfully answer this question. Probably more.
Some of these facts you could easily look up, for example the equation for kinetic energy. But the trouble is that, if there were too many gaps in knowledge, the student would have absolutely no idea in how to start this problem. If they knew *none* of the declarative information, how would they look this up? They wouldn’t know that this would involve multiple equations; including kinetic energy, speed & an equation of motion.
Google wouldn’t be able to tell them how to answer this (unless they literally googled the question and found a model solution…but then that wouldn’t really aid understanding…).
So my point is, yes…problem solving is an admirable aim. But don’t neglect the role of declarative factual recall in this. In fact, it’s probably the most important aspect to problem solving.
(If you’re interested…my model solution is below)