Teaching mathematicians shouldn't be like programming a computer

Traditional methods of teaching maths have more in common with how we programme a computer than what we might do if we wanted to engage our students in mathematical thinking. We shouldn’t be overly surprised then when our students consider mathematics to be all about learning a set of rules that they need to apply in the right order so as to output the correct response. But is there a better way?

The product of most computational thinking, an algorithm is in essence just a step by step list of instructions that can be followed by a human or machine. An algorithm is defined by Google as "a process or set of rules to be followed in calculations or other problem-solving operations, especially by a computer". This result was returned as a result of the very special and complex algorithm that is deployed by Google to make sense of my search and deliver results that are likely to meet my needs. Google also provides me with a nifty graph revealing the frequency at which the word ‘algorithm’ is used. It shows a rapid increase in the words use beginning around 1960; the sort of growth curve that corporations dream of. The rapid rise in our use of the word algorithm reflects that it is increasingly normal for students to learn to code and in doing so create a list of instructions that are followed by a machine/computer.

Most algorithms are much simpler than the one that powers Google and the algorithms we deal with in most school mathematics lessons are many orders of magnitude more simple. Indeed, it may be because of this lack of complexity that we do not recognise that so much of the mathematics taught in a traditional maths lesson could easily be translated into an algorithm to be followed by a machine. We may not even notice that the methods we use to teach maths to our children are painfully similar to how we would programme a computer.

Typically, the lesson begins with the teacher presenting the required method to the students. The teacher begins with step one being demonstrated on the board. Once step one is complete, the teacher demonstrates step two, and then step three and sometimes steps four and five. With triumphant zeal the teacher indicates the correct answer with a flourish of whiteboard marker and perhaps a double underline for effect. In phase two the students copy the process they have been shown with the teacher looking on to ensure the steps have been followed accurately. Naturally there are some bugs and errors that require correction. By the end of the lesson most students are able to accurately follow the steps and arrive at a desirable answer even if some of the numbers are changed.

Compare this to how a computer is programmed. The ‘coder' determines the steps to be completed and enters them into the machine ensuring accuracy; this equates closely to phase one of our lesson although with our students the coding occurs visually and aurally rather than via keyboard. The coder then runs the code on the computer and looks for bugs in the code which may cause unwelcome results; this is phase two of our lesson. Finally, having checked the code and feeling confident that it is bug free and fit for purpose the coder releases their programme into the world where it runs on a range of subtly different systems and with a mix of inputs; a very near comparison to phase three of our lesson.

If you have some awareness of the limitations of a computer that result from the strict manner in which it follows rules, you are not surprised when a piece of software fails completely when asked to perform a task it was never intended to perform. Even the supposedly ‘intelligent' software that outsmarted Chess master Gary Kasparov would not have been much use to the staff of Pixar Films as they animated Toy Story and for the most part we are not surprised by this. What does surprise is when our students are unable to apply their mathematical knowledge to new situations even though they have been ‘coded’ in the same way that a computer is, one logical step after the other.

What our students lack as a result of their mathematical programming is a true understanding of mathematics. The manner in which they have been taught instils a belief that mathematics is a discipline of rules and procedures to be followed accurately so as to produce the correct answer. Errors occur as a result of missed or inaccurately followed steps. Without an algorithm to guide them they are lost in a sea of numbers and a forest of symbols with no rules to show them the way.

This approach to teaching mathematics as a set of rules to be memorised is reflected in the thinking of mathematical educators such as Ed Southall who writes:

Mathematics gradually became a mysterious entity, whose rules and steps I was expected to unquestioningly memorise – which I dutifully did. However, the process of storing numerous algorithms and their quirky properties became increasingly tedious, and I fell out of love with the subject that once intrigued and excited me. (Southall, 2017 p1)

Fortunately, Ed persisted and has searched for the understanding that his mathematics lessons failed to provide. Many students do not persist and turn away from mathematics as soon as they have the opportunity. It is sad that many of the teachers who have presented mathematics as this dry, sterile subject of rules and procedures believe that they are doing their students a service by sharing methods that make maths easy.

What is needed is a fresh approach, one that begins with an exploration of essential ideas or concepts. Students need opportunities to play with numbers, create visualisations of what might be going on, search for patterns and ask questions. When we approach mathematics as a discipline full of creativity and inquiry we also provide our students with opportunities for true mathematical thinking. We build number sense and flexible fluency where students understand that numbers can be manipulated, decomposed and recomposed. We build with our students an understanding of the effect that mathematical operations have and why a particular process produces a given result. In doing so we teach to the intent of the curriculum. For those in New South Wales this means that we teach our students to ‘wok mathematically' as follows:

Students develop understanding and fluency in mathematics through inquiry, exploring and connecting mathematical concepts, choosing and applying problem-solving skills and mathematical techniques, communication and reasoning . . . As an essential part of the learning process, Working Mathematically provides students with the opportunity to engage in genuine mathematical activity and develop the skills to become flexible and creative users of mathematics. (NESA)

This delightful description of mathematical learning may not apply where you teach but it is not an uncommon declaration of what curriculum planners see as a most vital element of instruction. In the Common Core State Standards, mathematical understanding is valued and the "NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections” are included as the first of mathematical processes and proficiencies.

When the moment is right, we may still teach students a particular process, after all there is a great deal of mathematical knowledge on which we can build. What we change is the place that learning procedures has in our curriculum. Rather than being the starting point for our mathematical instruction we teach the processes at the point of need. In a mathematical exploration where students arrive at the point where a procedure is required we teach the method. We unpack it, pull it apart, visualise what is going on, play with moving its parts around and in the end our students have a new tool with which to think and an understanding of its utility.

By Nigel Coutts

Southall, Ed. Yes, but why? Teaching for understanding in mathematics (p. 1). SAGE Publications.