There are several layers to solving equations that can be unpacked using a task analysis approach. This includes written and mental steps (such as what we teachers mean when we tell a student to do the same thing to “both sides of the equation”). Here is how I develop the concept of equations and solving.
To unpack the layers for students, I have had a lot of success with the scaffolded handouts below (see last photo of example of what to write.) Here is a link to a Dropbox folder with all 4 handouts, in WORD format. Feel free to use and revise as desired.
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.
All too often math topics are introduced first with the skills and steps. This is backwards. The photo above shows how I introduced solving equations a high school student with autism using the concept as an entry point.
We discussed what was involved in buying a car, including payments (no interest) then I posed the problem seen at the top. I asked him to figure out the monthly payment. He worked out the problem, overlooking the down payment. With a minimal prompt he self corrected. I followed this by “showing him the mathy way of doing the problem.” (Seen in the bottom half of the photo). He conceptually understood why the -1,000 was the first step and x had meaning.
Dragon Box (link to website here) presents solving equations, proportions (and fractions and expressions) using an alternative representation and in a highly engaging game format (different platforms!).
In the photo above, the goal is to get the treasure chest by itself. To do this, the fly looking thing and the snake have to be eliminated. First eliminate the snake with the black background (a night card) by placing the other snake card on top of it (a day card). The day card snake must be placed on the right side as well. The day and night card (representing positive and negative numbers) become a circulating hurricane looking card (which represents a zero).
I used it with my 5 year old son (video here) and he could solve 2 step equations (on the game) independently within a couple hours. This is a game changer in teaching kids algebra.