Algebra 1: Concepts and Skills

I had an interesting discussion through a Facebook post recently regarding concepts vs skills. I want to share some information I have gathered regarding this topic. I do so, because there were a substantial number of teachers advocating for skill based learning. I hope to initiate some meaningful discussion.

Below left is a photo of an information processing model presented in a graduate level course on learning I took at UCONN. A key element I want to highlight is that information is more effectively processed if the information is meaningful. A theory behind this is Gestalt Theory in which the brain want to make information meaningful or organize it, e.g., the closure model in which our brains complete the triangle in the middle of the circle portions.

The meaning underlying math skills originates in the concepts. Below are the definitions for both, with the concepts being the “how or why” underlying the skills which are the “what to do” part.

I am not arguing that skills are unimportant or that rote practice is wrong. My position is that the concepts should drive the process. Here is a cartoon I think highlights the challenges with students having only skill based knowledge for topics that have important underlying concepts. I witnessed this first hand as a college adjunct instructor and found that a substantial number of students only understood slope by its formula. I also see a substantial number of students receiving special ed services who are taught at a skill level only to allow for progress. Often this is challenging for them when they have working memory or processing issues.

I will summarize in my own words an interpretation an article I read on the definition of Math, which stated there is no singular definition. The following was a theme that appeared to emerge. Math is a set of quantitative related ideas that can help explain the phenomena and the world. The mathematical symbols are used to represent these ideas. There are different ways to represent these ideas, e.g., we represent functions with tables, graphs, and equations. Formal proofs in Western Civilization are not the same a those in the East. Computer based proofs are not fully accepted by many math experts.

Technology has provided amazing ways to represent mathematical ideas. The most genius approach I have encountered is Dragonbox. The image below shows their initial representation of an equation through their algebra app. It develops the concept and the skills simultaneously.

Below is a list of some algebra 1 topics and some of the associated concepts. These are largely derived from math sources but include some massaging by me. I am happy to hear the working definitions of others.

Randy on Dana Jonson’s Podcast: Lots of Information about Math

I was a guest on special ed attorney Dana Jonson‘s podcast. Below are links to items I cited in the discussion. There is a contact information form in this post.

Strategy to individualize instruction

Tandem of Online Resources: IXL.com and Google Classroom

Online Worksheets I recommend

Concepts vs Skills

Concepts vs Skills – Need Both

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.

calvin hobbs toast

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.

definition concept
definition skill

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.

solving equation with conceptual understanding first

Address Skills at Performance Points

performance-point

In the photo above you see a contrast between how children learn and how educators often teach necessary skills. Children learn to ride a bike by actually performing the target skills. This is a performance point – the setting in which the child actually performs. In school students are often taught necessary skills in isolation, away from the performance points. Imagine teaching a child to ride a bike by having him sit at a desk while the parent points out all the steps for riding a bike.

Often accommodations and supports are provided in isolation or out of context. Students with autism have lunch buddies in a contrived setting with an educator leading conversation. Students with ADHD have a weekly time to organize their notebooks. Students who have trouble functioning in a general ed classroom may be pulled out as a result.

Below are a couple of examples of how support can be provided at the points of performance. The photo below shows a checklist I used for a students with autism in my algebra class. They would follow the checklist and self-evaluate by checking off each step as it was completed. They were learning how to perform necessary skills at the point of performance.

checklist-in-class

Another overlooked point of performance is in organizing a notebook. Students should organize a notebook while IN CLASS and on a DAILY basis. I use the rubric below to help support students with this task.

checklist-for-notebook

Dr. Russell Barkley, an expert on ADHD, talks about performance points for students with ADHD in his book and in his ADHD Report. This focus at the “points of performance” can and should apply to any student with a disability (and students in general).

Task Analysis

ice cream cone task analysis

task analysis two step equation

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!