The Learning Curve
Calculus and learning
Calculus is the mathematical analysis and description of things that are continuously changing. It allows us to answer important questions such as where is the change fastest, where are the highest and lowest points, how is quantity accumulating etc. Such questions relate directly to the theoretical study of curves and shapes, as well as real world problems involving motion, growth, heat, populations, economics and many other things. A particularly important aspect of this analysis is the ability to determine values and rates precisely at an instant, without having to resort to approximations or averages.

A simple model of learning

As an example, think about learning something new—first you know nothing, but then your knowledge gradually increases. As you become more comfortable and familiar with the subject matter you probably learn faster, until eventually you know most of what you need, and so your rate of learning necessarily slows.

We can plot this process if we make the horizontal axis indicate the passage of time (T), and the vertical axis indicate the knowledge acquired (a value of 1 means learning complete). The curve below is one possible representation of the process of smooth acquisition of knowledge. This is called a learning curve.

We can plot the slope (i.e. the learning rate) vs time as well. Click the Toggle slope curve button to see this curve. The tools of calculus enable us to calculate the slope curve—describing the rate of change, from the function curve—describing the value at each point in time.

Consider this curve, and think about the rate of learning it indicates:

  • When is it fastest?
  • When is it slowest?
  • Click the Toggle tangent button, and drag to see the tangent at each point in time.
  • Can you see the relationship between the slope of the tangent and the rate of learning?

A more complicated model of learning

Of course learning does not always proceed so uniformly. A more realistic model might include effects such as an initial period of rapid learning and naïve confidence, followed by a realisation of impending difficulty and some associated loss of confidence, before stabilisation and recovery leading to slower but more certain progress and eventual mastery.

This situation is shown in the following plot:

Consider this curve, and think about the rate of learning it indicates. Display the tangent to help if necessary.

  • Drag the green estimate crosses to construct what you believe the slope curve (describing the learning rate) might look like in this case.
  • When you are confident with your estimate, plot the slope curve and see how close you were.
  • Consider the points where the learning rate is maximum, zero, and negative. Do these points relate to particular features of the original function?

Practice constructing slope curves

The plot below will show various curves. Use the green estimate crosses to practice estimating the shape of the corresponding slope curve in each case. As before, you can display the tangent line to help, and check your answer by displaying the actual slope curve.

Click Next to bring up a new curve, and if Easier is selected, some of the harder examples will be excluded.

About the activity

Calculus comes in two flavours, but both share the fundamental component of dealing with infinitesimals—vanishingly small quantities. Differential Calculus is the calculation of instantaneous rates of change by considering the change in a function over an infinitely small region, and Integral Calculus which allows us to calculate how much of something there is by accumulating infinitely small pieces.

The study of continuous, smooth change has been part of mathematics all the way from Zeno of Elea, Eudoxus of Cnidus, and Archimedes of Syracuse in Ancient Greece, through Descartes (the inventor of co-ordinate geometry) and Fermat (famous for his Last Theorem), to Newton and Leibniz who independently invented calculus in the last third of the 17th century (and argued over priority). Yet more than 150 years would pass before calculus was made fully rigorous through the work on limits by Cauchy, Weierstrass and others in the mid 1800s.