Developing mathematical prowess

The book above by Imre Lakatos is an important read for mathematics educators. In fact, a single sentence gets me thinking about what we are actually trying to achieve when we attempt to teach adults and children mathematics.

Speaking of the argument he is about to make in his book, Lakatos states: 

[The book's] ... modest aim is to elaborate the point that informal, quasi-empirical, mathematics does not grow through a monotonous increase of the number indubitably established theorems but through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations. [Emphasis mine]

In other words, mathematical prowess does not primarily improve by adding the knowledge of algorithms and formula to our strategic repertoires.  Thus, if you decided as an adult to learn mathematics (and I hope you do!) you would not really benefit from attempting to memorize all the methods taught in your maths books.  

Ah, okay. What then should you do?

You would improve your use of mathematics substantially by developing your ability to make guesses (speculating) and then criticizing the guess.  That means improving the quality of HOW you make guesses, and improving the quality of your criticisms.  This requires argument, proof and refutation - the building blocks of mathematics.

This has been referred to as zig-zagging toward the truth.  Much how a sailing boat will tack toward a point. Tacking consists of making the first guess, the following tack is proving why the guess was wrong. Tack more accurately this time, and then prove it wrong. With each successive attempt you move toward accuracy. This is how knowledge is developed and by extension, ought to be a model of learning mathematics. Contrast it with traditional models in which any guess, or tack, is considered a mistake, not a natural part of a process, and usually corrected by someone else.  

It is certainly very different to how things are reflected in the NZQA numeracy standards and how content is taught in the adult sector.  We could argue about the differences between numeracy and mathematics, in particular that numeracy ought to be concerned only with giving people the skills to complete a job.  However, this argument generally hits the rocks when we realise that 'mathematical understanding' and 'agency' are increasingly pivotal given that mathematical demands are increasing rapidly. The ability to tack and gybe enables adults to learn and adapt. Showing them the way to a single destination (the answer) is no longer sufficient in a dynamic, not static, mathematical environment.   

Anyway, if you want to improve your own, your learners', or your children's math skills then think about how you might improve their ability to guess and then to criticize their guess.  Because it is these two skills that help the learner to 'tack' in a zig zag way toward correct answers.