In general math is taught by focusing on the steps. Conduct a Google search for solving equations and you will see the steps presented (below). You need a video to help your student understand solving and you typically get a presenter standing at the board talking through the examples. (I’ve posted on my approach to solving equations.)
When the math is taught through the skill approach the student may be able to follow the steps but often does not understand why the steps work (below). The brain wants information to be meaningful in order to process and store it effectively.
To help flesh this situation out consider the definitions of concept and skills (below). Concept: An idea of what something is or how it works – WHY. Skill: “Ability” to execute or perform “tasks” – DOING.
Here is how the concept first approach can play out. One consultation I provided involved an intelligent 10th grader who was perpetually stuck in the basic skills cycle of math (the notion that a student can’t move on without a foundation of basic skills). He was working on worksheet after worksheet on order of operations. I explained down and monthly payments then posed a situation shown at the top of the photo below. I prompted him to figure out the answer on his own. He originally forgot to pay the down-payment but then self-corrected. Then I showed him the “mathy” way of doing the problem. This allowed him to connect the steps in solving with the steps he understood intuitively, e.g. pay the $1,000 down payment first which is why the 1000 is subtracted first. Based on my evaluation the team immediately changed the focus of this math services to support algebra as they realized he was indeed capable of doing higher level math.
A teacher friend of mine in the ASD community shared an anecdote for me to share. Her son had trouble figuring out perimeter. He was counting the squares and didn’t want to count the corner squares twice. The solution? The squares were pieces of sod to fill in the inside while the outer sides of the squares were fence pieces. This shows how kids can get caught up on the smallest details that teachers overlook.
This can be addressed using task analysis which is a process of breaking a skill into small steps. This is common in special education but not as much in math. While math teachers can be effective in the process, there are often little steps that are overlooked or not addressed as much as necessary. Effective task analysis also allows for more effective scaffolding.
The photos above are examples of task analysis. The top photo taken at Burger King shows a sequence of photos showing how to pour a soft cream ice cream. The bottom is a task analysis break down of all the steps in solving a linear equation – ALOT of steps!