Maths makes a lot of sense when it is taught in a way that makes sense. If you teach something that makes sense in a way that doesn’t, at the end of the day it doesn’t make sense.

In other words, if we teachers don’t get our conceptual thinking right, then we can’t really expect students to, can we ?

The most important dimension of this conceptual thinking depends on the age group you are teaching to. However, there is definitely one dimension of conceptual thinking you cannot overlook from secondary school on: **the logical progression**.

The logical progression is the order in which you choose to expose and develop the successive aspects of a new Maths topic or chapter. Rather than being theoretical about it, let’s look at a practical example from HIgher GCSE and A-Level Maths: **quadratic equations.**

Here is a progression (taken from a widely circulated textbook):

- Factorising quadratic expressions
- Factorising a quadratic with a unit coefficient of x
^{2} - Difference of two squares
- Solving quadratic equations by factorisation

- Factorising a quadratic with a unit coefficient of x
- Solving quadratic equations
- Solving quadratic equations by factorisation
- Solving the general quadratic by the quadratic formula

- Completing the square
- Solving a quadratic equation by completing the square
- Quadratic equations with no solution (discriminant)

- Quadratic graphs
- Drawing quadratic graphs
- Reading values from quadratic graphs
- Using graphs to solve quadratic equations

What are the issues with this progression ?

- The first issue is that students are asked to work on quadratics without knowing what a quadratic looks like and what the major difference is between a quadratic and a linear graph. We must remember that, at the initial stage of the progression, students only know about linear graphs and expressions. To understand what a quadratic is about, the first thing to do is to draw one and realize that, unlike linear graphs, a quadratic graph is a
**curve**. In other words, a logical progression that increments harmoniously from student knowledge would be to start with quadratic graphs. - The second issue is that students are asked to factorise quadratics without understanding why. This is aggravated by the (late) discovery that not all quadratics can be factorised. There again, if the progression starts with quadratic graphs, it becomes visually obvious that some quadratics cross the x-axis twice, some once, while some don’t. To use ordinary English, it simply depends ‘how high’ the quadratic is.
- The third issue is that the quadratic formula is introduced before completing the square. Whereas it is precisely completing the square that explains both the discriminant and the quadratic formula.
- The fourth issue is the the discriminant is identified and explained separately from the quadratic formula, whereas it is by far the most important and useful part of the formula.
- The fifth issue is that quadractic factorisation methods are introduced in disorder, whereas there is a logical order and the one thing to remember for students is that using the quadratic formula is always a last resort (unless explicitly asked on the exam paper).

Then what would be a good logical progression for quadratics ? **The first quality of a logical progression is that it should be powered by necessity.**

Considering the five issues below, a sound logical progression for quadratics could go like this:

- Quadratic graphs
- Drawing quadratic graphs
- Reading values from quadratic graphs
- Understanding the relationship between the shape and position of the quadratic graph and its coefficients

- Making the link between quadratic graphs, factorisation and solving the equation “quadratic = 0”
- Visually understanding that the equation “quadratic = 0” has 2, 1 or 0 solution.
- 2-way link: expanding / factorising from solutions

- Method to factorise quadratic expressions (and solve quadratic equations)
- Classics: squaring brackets, difference of 2 squares
- When the factorisation is easy to find (looking at a possible pattern linking x coefficient and constant)
- If not so easy: completing the square
- Generalisation of ‘completing the square’: the discriminant (or how to know whether a quadratic can be factorised or not)
- Last resort: using the quadratic formula
- Application to all quadratic equations

Are we saying that, for any given Maths topic, there is only one acceptable logical progression. No, of course not. But there is certainly not a great number of those…

Why is a rigorous logical progression important ? Because it brings:

- A clear structure
- Pace: kids understand, learn and progress faster when your presentation is immacutely logical
- Compression: logic is the building block of knowledge compression. Compression is one of the essential brain processses that makes Maths a non-entropic activity. Maths as a non-entropic activity is the gateway to its healing dimension. Sorry, that was a bit quick, but I’ll write more about that later…

So, ever wondered why it’s actually better to have specialists teach a specialist subject ? Well, this is an example. Among many others…