The Case of the Missing Math Skills
And what it tells us about learning in the New Information Age
I have been marking first year physics exams for the last ten days or so, and I have come across a number of examples where the students are not getting correct answers because they don’t understand the physics, but because their lack of background mathematical knowledge is well below what I would consider adequate for a first year taking a numerate degree (mathematics, statistics, sciences, engineering and a number of social sciences where statistics are important). This is not down to any failings of the individual student, but is symptomatic of the poor preparation of increasing number of students coming to university fro the high schools. You will find lots of employers complaining about the lack of skills shown by recent university graduates. I can assure you that these complaints are echoed by large numbers of professors who teach introductory courses to students coming from high school. There is a definite drop in the skillset available to the recent high school graduate. This has absolutely nothing to do with the intelligence or work ethics of the students, but is the failure of the school system to equip the university-bound student with the basic tools of knowledge to enable them to settle into the university system. In this rant, I am concentrating on the mathematical skills necessary to make the jump from school to university. There are many other skills which also need working on, such as writing skills, the ability to manage time and work schedules effectively, and deliver good quality work at or before the appointed time.
The particular mathematical failure I want to address is the overworked symbol e. In mathematics it is known as the Euler Number and is the base of the natural logarithm. It is also used in the solution of the mathematical solution to a particular type of decay, known as exponential decay. This appears in many science and math contexts because it describes things like radioactive decay, the discharge of a capacitor in an electric circuit and the gradual decrease in motion of a pendulum under the influence of air resistance or other frictional forces. (For those interested, it is the solution of an equation where the rate of change of a certain quantity is proportional to the quantity itself. This is a first order differential equation. I am deliberately not putting in any equations, because they are just a tool to express these concepts symbolically)
In these cases, e takes the value of 2.7182818284590452353602874713527… However, in physics we also use e to mean something quite different, the value of the smallest possible free charge that can exist on its own. This has a value of +e for the charge of the protons in the atomic nucleus and –e for the charge on the electron, the particles responsible for carrying current around in an electrical circuit. In fact electrical current is defined as the rate of flow of charge. Now in this context, the value of e is approximately 0.00000000000000000016. As you can imagine, if you use the wrong value in your equation, you end up with wildly different answers.
So why do the students find this difficult? The answer is “Context”. If you meet e in mathematics, it will almost certainly be the Euler constant. If you meet e in physics, then it could mean the charge on an elementary particle, or it could mean the Euler constant, depending on how it is being used in the particular situation. This is not always immediately obvious; e often crops up in electrical circuit problems, as a solution to a time evolving equation, where it is the Euler application of e that is used, not the electronic charge value. In some other electrical problems, it is definitely the charge value which is used. These days we take it for granted that we can look things up on the internet very rapidly. So we rely on information retrieval to plug any gaps in our knowledge. The snag with this is, “Context”. You must interpret your answer to find out if it is indeed the solution to your original query, or whether it is entirely erroneous in your particular context. This is not as simple as it sounds, as anyone swamped with hits on any internet search engine will be able to testify. To use the power of internet search engines effectively you have to have some knowledge of the subject in the first place. You need a framework of knowledge in which to place the retrieved information.
Construction of the knowledge framework is what education is really all about. The New Age of Information has moved so rapidly that the present education system really has not got to grips with it yet. Academia still relies largely on the same education techniques used for the last hundred years or so — the lecture and seminar. What has shifted enormously is the number of students now attending university, their level of preparedness for higher education, and the way in which we construct the knowledge frameworks which enable successful and effective use of all of the information at our fingertips. It seems to me that we are not yet doing a good enough job in enabling our young people to construct the knowledge frameworks they need in order to be effective citizens in this new Age of Information. Do I have ideas on how to achieve this? Yes, but they will require more resources than the present Higher Education system is willing to devote to them. I will discuss possibilities in a subsequent article.