Do we understand understanding?
I saw this on Twitter today
I strongly suspect it’s meant to be a joke. At least I hope so. The use of the word “research”, however, got me thinking.
Most of you, I feel, have undergone a process similar to mine over the last couple of years. Trying to wrap our heads around an unfolding health ‘crisis’ and the responses to it. At some point along the way we probably all had a sort of WTF moment, or maybe a collection of what the mini-fuck moments that eventually built into something more significant.
Whatever the process, we’ve all (I think) arrived at a point where our understanding is that the response to this newish virus was all, or mostly, wrong. There are many different explanations we might have for why those responses happened, but I feel confident that most of us would agree those responses were not good ones from the perspective of health outcomes.
When I was working at a university, I remember many discussions with colleagues about some of our students’ lack of understanding and what we could do about it. How could we help improve things? One colleague noted that a significant fraction of students suffered from what he called the “goats and sheep problem”.
You have a problem in terms of goats (the goat is running down the hill this fast and takes this amount of time etc) which you solve in class. You change the numbers a bit and the students are doing great and get it right. You give them the same problem for homework, but this time in terms of sheep, and a good number get stuck.
The actual example he used was a problem in which you’re looking at a rocket launch and, knowing how fast the rocket is going up, you need to calculate how fast the angle for the ‘line of sight’ is going up. He showed them how to do this in class. Made sure they could answer the problem on homeworks etc. Come the exam, he cast the problem in terms of a balloon, not a rocket - and stumped about a third of his class who then complained they “hadn’t seen this problem before”.
I don’t want to get into a discussion about the implicit coddling of the students you will have noted, at least not here. Nor do I wish to address the absurdity of students expecting to only solve problems they have already seen. I will say, however, that during my first semester of teaching after having just arrived, and after having spent a couple of decades doing research at an industrial lab, I set what I thought was a reasonable exam. It matched what I expected my students ought to be able to do and the topics I’d covered in class. It matched my own understanding of an appropriate ‘standard’ for that level and was easier than the exams on the same subjects I had taken at university. Things did not work out as I had planned or expected, and the Dean forced me to adjust the grade boundaries in order to achieve an “acceptable” GPA. It was my first exposure to the realities of the decline in university standards.
But what fascinates me here is the nature of ‘understanding’ and how to quantify that (if at all).
So many times I had students come to me and say things along the lines of “I did so much work, how did I get such a bad score?”. I accepted my share of the blame in that and tried to adjust my own approaches every semester so that I could help more of my students. But one thing really stuck out like the sorest of thumbs; many students had no understanding of what understanding is.
I’m pretty certain I don’t have a perfect understanding of what understanding is either, but I know that it’s not about simply reading stuff, or learning stuff, or just plugging in the numbers into some formula. I tried my best to tell them that a technical subject, like maths or physics, was more about doing than knowing, but many of them were having none of it.
At the risk (or perhaps I should say near-certainty) of tuning most of you out, I want to discuss what’s involved in solving a very simple physics problem.
It’s not, admittedly, a very exciting problem, but such is the nature of elementary physics. We have to go through the drudgery of getting the basics firmly in place before we can do the fun stuff.
The problem is this : you’re a police officer enjoying your donut and coffee in your cruiser parked at the side of the road. You notice a car coming up behind you very fast, travelling at a constant speed, and certainly over the speed limit. Just as the car overtakes you, you step on the gas, and give chase. If your acceleration is constant how long does it take you to catch up with the speeding car?
Like I said, not a very exciting problem, but it is fairly typical of the kind of question I would expect students to be able to answer after the first couple of weeks of a typical university PhysicsI course1. It is a very straightforward, and routine, problem in what is called kinematics - which is basically working out how things move when you have constant acceleration.
The first thing to notice here is that we actually do need to know some stuff beforehand. We do have to have learned (and hopefully understood) a few things before we can start to tackle problems like these.
The second thing to notice is that there are no numbers here. It’s not really an “open-ended” question, but it is starting to bring a bit of real-world thinking into the classroom. Problems we face in the real-world are often not beautifully laid out, or precisely specified, and we have to work to even frame the problems correctly, or usefully.
I could have used numbers. I could have said that the car whizzes past at 100km per hour, and that you accelerate at 0.25 metres per second squared. But what, really, do the numbers give us extra here? They add nothing to our understanding at this point except to concretize the problem for those not used to thinking in more abstract terms.
At this point I would tell my students to put their pens down and take a few seconds to make sure they’ve really understood what is being asked of them. They’re being asked to figure out a time. So whatever else they might need we already know that the problem is going to be one requiring the unfolding of time. We’re going to start at some time, a bit later on in time the police car will have set off but will still be behind the car it’s chasing. At some specific point in time later the police car will have caught up. We need to know what that specific time is.
What else do we know? We know the car we’re chasing is travelling with a constant speed, and that we’re applying constant acceleration. We also know that we start to accelerate as the car is passing us.
All of that before we’ve even started to write those squiggly hieroglyphics down on paper.
We now have, we hope, a decent understanding of the actual situation and what we’re being asked to figure out. A diagram often helps here too - so we might produce something like this and even begin to annotate it with a few symbols to represent time, distance, speed and acceleration - all parameters we think might be important in solving the problem.
I’ve used v here for velocity, because we frame things more often in terms of velocity than speed (but in this particular problem it makes no difference). Velocity includes a direction - speed just tells you how fast you’re going in that direction. In this case where everything’s in a straight line and in the same direction there’s no essential difference between the speed and the velocity.
We set the ‘start’ time at t = 0. We’re only interested in an elapsed time - so where we start our counting of time makes no difference. Setting t = 0 here is a matter of convenience. Same for the distance travelled (which we’re not, yet, sure we’ll need to solve the problem) which we set as x = 0 when t = 0.
Just the process of initializing the problem with setting t = 0 and x = 0 requires some understanding of why it’s OK to do that.
At this point we haven’t really even done any maths. We’re just setting up the problem in anticipation of the maths we’re going to do. We know our answer for the time will involve speed and acceleration, and it might even need the distance travelled. A large part of the difficulty in answering these kinds of questions is involved with setting things up and figuring out what’s important or not. This is the process of modelling, and it’s not easy and is a significant part of why a lot of people find physics problems more challenging than others.
In order to solve the problem, we have to have some model of how the ‘world’ works (insofar as that affects our particular circumstance). That’s true, I would suggest, of almost every problem we face - whether it’s about the best time to hang out the washing or figuring out when the police car is going to be able to issue me with that speeding ticket.
When we’ve done all that, we might, if we’re lucky, have the key insight that the ‘catch up’ happens when both cars have travelled the same distance (from the starting point) and that this happens at the same time (from the starting time).
At this point most students are screaming inside. And probably many of you are too. Just give us the formula to use. Enough with all this wordy nonsense. But without all the ‘wordy nonsense’ we can’t really claim to have ‘understood’ the problem, or what might be involved in solving the problem.
As you get more experience and actually do more of these kinds of problems, the process of ‘understanding’ becomes a lot easier because of the familiarity - and all the ‘wordy nonsense’ gets processed in your mind very quickly. This is why the doing and the practicing are so essential when it comes to solving technical problems.
Is it boring doing all that practice? Oh God, yes.
As a moderately decent piano player I still practice scales - and it’s about as interesting as the latest instalment of the Rings of Power.
So, we’re now in a position to actually solve the problem.
Here’s where we need to use some existing stuff we’ve already learned. We can use the kinematic equations that relate distance travelled to time and constant acceleration.
We call the catch-up time, T (we could have just left it as ‘catch-up time’, but isn’t it easier to just use a single letter instead of this whole phrase?) and we’ll label the catch-up distance as d.
For the speeding car (constant speed) : d = vT
For the police car (constant acceleration) : d = aT^2/2
(That’s T squared, or T times T).
There’s a few simple technical things I’ve left out at arriving at those equations - but I didn’t want to overdo things even further!
The equations are both true (assuming the kinematic equations themselves are true). The first equation is true for the speeding car. The second equation is true for the police car. So, this means that vT = aT^2/2 (or, more bluntly, d = d)
Now it’s just algebra to give us our final answer for this problem :
T = 2v/a
The final step is checking. Does it make sense? First thing we do is to check the units. If we have something in seconds on the LHS we’d better have something that gives us seconds on the RHS. Yup - that works out.
What about if the speeding car was going faster? That would mean it would take longer for the police car to catch up. So, if we increase v then T should increase (assuming the police car doesn’t change its acceleration). Yup - that checks out.
What about if the police car used more acceleration (make a bigger)? That would mean it would catch up with the speeding car quicker. So T would be smaller. Yup - that checks out too.
At this point, if we’ve done all that, we have some confidence we’ve got the right answer and that we’ve done a decent job of ‘understanding’.
As a lecturer I might, at this point, want to check to see how much of the basics the students have really understood and so I might pose the following question:
Suppose that at some point the speeding car notices the police car coming up behind and applies the same (constant) acceleration. Without doing any further algebra, does the distance between the cars increase, or decrease, or stay the same? Explain your answer.
This is a tricky question to answer because it could be any of the above answers. However, by posing it in this way it ‘misleads’ the student into thinking there is one right answer. This then sets up a further questioning in the student who really has understood what’s going on. “Have I got this right? There seems to be one ‘right’ answer here”. A good student will challenge the wording of the question at this point - and will be forced to check and re-check their own understanding.
Another follow-on question might be:
Without doing any algebra explain why the police car’s speed must be greater than the speed of the other car when it catches up.
These kinds of follow-on questions really test the understanding of the student.
If you’ve stayed with me this far - my thanks and condolences. Although I’ve expressed it all in terms of a (very) boring physics problem, look at the amount of work and thought that goes into properly understanding even this simple problem.
Now pat yourselves on the back, because you’ve all done this in order to figure out that the covid responses were wrong. You’ve gone through a very similar process to work from what you know (or to learn it), to frame the problem, and make some logical deduction. You didn’t rely on what you were told. You didn’t blindly follow a ‘recipe’. You looked at the evidence, you thought about it, and came to the conclusion that the government had given you the wrong answer.
Might you (and me) still be wrong? Yes, of course, although that is looking somewhat unlikely at this stage (in what might be the understatement of the decade). But I’ll be willing to place a very large bet on the fact that you all understand the issues to a far greater depth than those who just went along with everything.
To be honest, I would really expect most students to be able to answer this kind of question before reaching university.
Riggery, you have put your finger on exactly the problem. Education as historically constituted was never meant to help people think, learn (in an appreciable sense of the word), OR problem-solve. It was meant to produce technically proficient, compliant "citizens" and workers. The gold standard would be someone who could solves complex equations and amenably work on teams making the "betters" who run society richer and more powerful. Universal, compulsory education nationwide (and its "core" standards) in the U.S. (I don't know about the U.K. and other places) was basically determined top-down from a handful of elites from a handful of universities (Stanford, Harvard, Columbia, etc.) and then basically imposed on a populace. This was awash with the zeal for "efficiency" (whatever that means) and internalization of orthodoxy. What you are proposing is the technical equivalent of meta-learning ("learning how you learn") and not learning what they want you to think and do. Your exercise is subversive precisely because it poses a simple physics problem (who could protest that!) as a tool to critical engagement with how the world works and the foundational PRACTICAL as well as theoretical principles that guide perception and lead to sound conclusions. All the tripe about "critical thinking" in colleges which became just another shallow exercise in syllogism and logic, only underscores what, by contrast, you pose as an alternative: REAL and APPLIED analysis that gets to the sense of things and what makes them tick. This requires operationalized independence of thought, perception, and relationship and a discovery/experimental mindset to see how these independent things coherently relate. Right now we have neither. Nothing is independent. It is "captured" (i.e. the Covid policy and pronouncements). Nothing is related. We have stacked statistics pulled out of context to prove the opposite of what the complete data set confirms (i.e. younger people, especially boys are FAR more likely to be injured by vaccines than helped, if they are helped at all, since there has never been a single study showing benefit for those basically under that age of 30). How can we educators compete with a monolith of social media, academia, legacy media, governmental agencies all operating as a megachurch? We do it by going to the basics of sanity and sense UNDERNEATH the hype. We do it by educating the whole person, not just the compliant citizen and worker. We do it by showing the cartoonish errors of captured orthodoxy, and we do it with our own choices to stand up and tell a plain truth without flinching.
In fairness to the students, depending on their education before college, they may barely have been taught to read. The decline in education has been precipitous. My own pre-university education even in the 60s and 70s was appalling (possibly due to my social and economic circumstances). It’s a wonder I made it through university; in retrospect I had absolutely no idea how to even think.
In more recent times, the education of children has been defined by its almost universal, grinding mediocrity except possibly for a few islands of light, and even they are being systematically destroyed.
State-run, unionized, education is going to turn out to have been a grave mistake. Wokeness will be the final mail in the coffin for much of what passes for civilization until the next Renaissance.