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.
Here is a link to a Dropbox folder with the documents I will address in my presentation. (Note: documents will not be uploaded until Jan 19, 2017.)
Imagine being asked to explain the climate in Spain and the photo above is the resource you had to use. If you didn’t speak Spanish this would be a challenging task for two reasons. First, you may not know the climate so this is new learning. Second, you don’t know the language used to explain the content – a double whammy!
The photo below shows a resource you are more likely to encounter. The language (if you are an English speaker) is natural for you which leaves you to focus on the content alone. Language is a barrier to learning math.
The photo below is an excerpt from “Why Do Some Children Have Difficulty Learning Mathematics? Looking at Language for Answers” by Joseph E. Morin and David J. Franks. It shows another element to the language barrier in learning math. In this example the term over is used to describe the location of the the white square (bottom frame) but the students understand over as a term used in 3-dimensional space (top frame). The misunderstanding of a single term can throw a student totally off in learning a new math topic.
Below is an excerpt from Malcom Gladwell’s book Outliers. He explains that the languages of Math and English do not get along very well. “Thirty” has to be translated into the concept of 3 TENS. Compare this to Chinese and Math. The problem in Chinese is read as “three-tens-seven” which is already in Math terms. This extra step of translating is often a problem for our students, especially those with special needs.
In teaching math the issue of the language of Math is an additional issue to address separately. I teach students to learn math in their own language (informal English, using manipulatives etc.) and after the concept is learned I show students the “mathy” way of talking about it. This follows the Concrete-Representation-Abstract (CRA) approach to presenting math.