## Covertization- what is it?

Covertization sounds like an incredibly fancy concept that seems complicated. Although it comes in many forms, it is not difficult to explain what it is.

I first came across the word “Covertization” in Naveen’s blog about using it with the difference of two squares which can be found by using the following hyperlink: http://conceptionofthegood.co.uk/?p=777 . This concept really interested me and I wanted to know more so I decided to buy Engelmann’s Magnus Opus which is quite a large book if you did not already know!

Engelmann defines Covertization as “instruction that involves prompt shifts progressively from highly-prompted examples to unprompted examples”. The examples are given in the most simplest context to start with, and then the context changes, but the concept remains constant.

Naveen has taken this to mean a sequence of examples with explicit features transitioning to implicit features. This allows a process where students take small steps to go from each example that is presented. This means that the first example pupils encounter is structured within the simplest context.

I tried Covertization with simultaneous equations and getting students to realise when the two equations are in the form ready to be added or subtracted. As you can see from the picture below I have moved from a standard example where x can be eliminated to one where y is the subject of each equation. Going from one set of equations to another involves a small step that students can follow with ease. These examples allow students to see that x can be eliminated, y can be eliminated, the x and y terms may not always be lined up properly and they allow for deeper thinking into whether adding or subtracting is always the easiest way of eliminating a variable.

I do not use this type of covertization often because I believe that it is incredibly specific to certain topics. Naveen has used this process with the difference of two squares. I strongly advise you check her blog out! My most common use of covertization is “Dropping steps”. I call this “fading”. Many examples on this website are based on “fading”.

## DBRS Process for Changing the subject of a formula

I promised I would post this! I have now presented about the process of Decide, Break, Repair and Simplify at Bristol and Sheffield maths conferences for changing the subject of a formula. This has been a pleasure and has been received incredibly well by a lot of maths teachers across the country.

The main idea of these workshops were to show how Kris Boulton’s method of solving linear equations can be adjusted to changing the subject of a formula.

The key difference for changing the subject of a formula when compared to linear equations is when we have to factorise… this can be a little daunting for students to do during a question as they normally get it wrong! However, during my presentation I mentioned how I have got around this problem with slides like below: