Like an embroidery, mathematics has a front and a back end.
The completed sum is the front of the embroidery. This is the showing off part of the embroidery. This is the end teachers think they can explain to learners. If these explanations were successful, we would have had an oversupply of mathematicians in this world. In fact, learners would love maths and would not cover their sum with their hand when the teacher comes closer. Current teaching and assessment systems only focus on the in the front end and they bear the fruits thereof.
The secret of sums, as with embroideries, is at the back end where the yarns are sewn and knotted.
First, there are Thinking Tools to enable the learner (not the teacher) to unravel the sum. Most sums cannot be done by learners because they do not understand the question (sum). There are Thinking Tools which stood the test of time which enable learners to unpack/unravel/analyse sums themselves.
When unravelling a sum, Thinking Tools enable learners get all the facts on the table, gain insight into relations, for example, if 70 apples go into 2 boxes, how many apples go into …. This synthesis of the sum is the learner’s AHA or insight. When this is achieved, it becomes clear to the learner that the sum is not difficult, it is doable.
This enables the learner to embark on the front end of the embroidery and demonstrate how the sum is done.
The same applies to "y = mx + c" type of sums.
The surface Thinking Tools enable the learner first to determine the unravel the back end of the embroidery, i.e. what is the gradient, in which quadrant does the line fit and the coordinates. Learners who are Thinking Tools skilled know the drill, they bite the bullet and start with the Thinking Tools which guide them to deeper thinking, while keeping the fault-finding Thinking Tool nearby.
The result is that learners clarify sums before doing the sums. Once this is in place the front end of the embroidery is a walk in the park.