Comment on “How Deeply Are Our Students Learning?”
I enjoyed the essay on student learning, (“How Deeply Are Our Students Learning?” MIT Faculty Newsletter, Vol. XXX No. 1) and generally agree with its findings. And I can also confirm that the concept of free-body diagrams is one of the most difficult for students to master, despite it being “simple” and “obvious.” But it is also “subtle.”
What do you mean?
There exists a common difficulty that may complicate life to students during an exam: It is the way in which a question is actually phrased. If the wording leaves room for interpretation, then chances are good that Murphy’s dictum will come into full force: if something can be misunderstood, then it will be! This is especially true when the question uses everyday concepts and language that leave room for interpretation. As they say, if you correctly understand a question, then you already have solved half of that problem. So when I have given exams at MIT – for a good many years I taught a graduate course in Structural Dynamics within the School of Engineering – I made sure that the questions were clear by testing these on the TA first. Professors very often make the mistake of believing that what is clear to them will be clear to the students, especially after having taught a subject for a while.
For example, consider the concepts of velocity and speed of a particle in motion on a semi-circular path as used in the first example in the FNL article. Now, velocity is a vector that has both magnitude and direction. So is speed the same as velocity? Or is speed= abs(velocity)? If yes, then that ought to have been explained explicitly in the question. And then, of course, is the added complication that “speed” has no sign, but tangential velocity does, even if it shares the magnitude with the speed. So even if in that first example the speed had been specified as being constant – but in what sense? – the velocity would not have been so, because of the change in direction.
But here comes also into play the everyday life experience: say you drive your car on a curved highway using cruise control, and that you set the speed at 55 mph. Is that a “constant” velocity? Not in the sense described previously, but certainly acceptable in the context of driving on the road, especially if at some point you are stopped by the police stating that you were going too fast, that you exceeded the local maximum velocity. In this context, the change in direction is irrelevant and velocity is the same as speed. From the perspective of the driver, he is certainly not accelerating, the centripetal acceleration notwithstanding! But in the example given, and even if the students were fully aware that velocity is a vector, then what is meant by “average velocity”? (the question specifically asked “what is the average velocity” and not what is the average speed). If the speed had been constant instead of rising slowly, does that mean that the average velocity equals the constant speed? Or is the horizontal component zero, since the particle fully reversed direction from the upper entrance point to the exit point vertically below, so there is no net lateral motion during the travel time? And what about the average vertical velocity? My sense is that it is these ambiguities that cause most of the troubles observed.
Another example comes from the natural sciences: Suppose you were asked in an entrance exam (or in the SAT) about figs and tomatoes, and you had to decide if these were fruits or vegetables. In everyday life, a fig is a fruit and a tomato is a vegetable. But in botany, the fig is not a fruit and the tomato is not a vegetable.
Instead, a fig is an enclosed inflorescence (or syconium) and a tomato is a berry, i.e., a fruit. So which is the correct answer? I’d say that the everyday meaning came first, and that botanists’ definition came later. So how does the student decide in a test what the examiner actually meant?
If it is obvious to me it should be obvious to you too, or shouldn’t it?
Consider also the elastic steel marble dropped onto a table. Yes, the kinetic energy in the ball is mgh, and the elastic energy stored in the table when the ball comes to a full stop is ½ F*u, so F=mg(2h/u), and since (2h/u)>>1, it is now clear that F>>mg. But is this really trivial or obvious, especially so if you haven’t solved problems like this before, i.e., have no training? This brings me to the second observation: There exist many problems that may be quite obvious to an experienced person with deep knowledge, but it isn’t so for an undergraduate student who must drink water from a firehose while applying the principle of selective neglect. That is, chances are good that the instructor overestimates the “obviousness” of most questions. As C.E. Inglis (FRS, James Forrest Lecture, 1944) once stated:
In problems relating to vibrations, nature has provided us with a range of mysteries which for their elucidation require the exercise of a certain amount of mathematical dexterity. In many directions of engineering practice, that vague commodity known as common sense will carry one a long way, but no ordinary mortal is endowed with an inborn instinct for vibrations; mechanical vibrations in general are too rapid for the utilization of our sense of sight, and common sense applied to these phenomena is too common to be other than a source of danger.
Then there are strong differences between undergraduate and graduate students. The former are there to get an education and a degree to move on in life, while the latter wish to specialize and acquire depth in some area. Thus, they have different motivations. Most undergraduate students warm up for exams, enough to master – or even excel in – the quizzes, but by the time that they enroll in next semester’s continuation, they may have completely forgotten a good part of what they had learned before – and I know this for a fact! This is because to them the first priority is to learn enough to do well and pass the exams and then in due time graduate. But deep learning comes only with repetition and training, not to mention motivation.
Manipulation without understanding
Many students become quite proficient in the use of mathematical tools, but that by itself does not in any way make them experts in physical model building. Mathematics is a tool that is very useful to evaluate physical phenomena, but mastery of math it is not per se an enabler of model building. These are two wholly separate “intelligences.” For example, without having seen an example of application of the convolution integral, the vast majority of students will simply fail to see the connection between some phenomenon that could be described by a convolution and the mathematical operation they learned in calculus or signal processing – or at least they would do so in a quiz. Why should they see the connection? Model building is not an intuitive, natural process, but an art that is learned in part from mentors, instructors, and experience.
My own sense is that most undergraduate students learn about many different technical disciplines, and in so doing develop mental muscle – not unlike those who lift weights. They also learn how to think and acquire tools so that they can later continue learning on their own. After leaving school, many (or most) of them will have largely forgotten what they learned, including how to integrate or differentiate. But they will keep the mental mass that will allow them to be successful engineers and scientists.