Category Archives: instructional strategies

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 conceptdefinition 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

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Cutting Up the Math Into Bite-sized Pieces

When I train new math and special education teachers I explain that teaching math should be like feeding a hot dog to a baby in a high chair. Cut up the hot dog into bite-sized pieces. The baby will still consumer the entire hot dog. Same with math. Our students can consume the entire math topic being presented but in smaller chunks.

bite sized pieces

My approach to doing this is through a task analysis. This is very similar to chunking. It is a method to cut up the math into bite-sized pieces just as we would break up a common task for students with special needs.

Image result for task analysis

While waiting for my coffee order at a Burger King I saw on the wall a different version of a task analysis. It was a step by step set of directions using photos on how to pour a soft cream ice-cream cone. I thought it was amazing that Burger King can do such a good job training its employees by breaking the task down yet in education we often fall short in terms of breaking a math topic down.

soft cream icecream cone task analysis

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Meeting Student Needs

One of my beliefs about the education is that teaching is built on a delivery based model. If teachers take certain steps the learning will happen – an educator’s version of Field of Dreams. Often the result is a focus on having students assimilate into the teacher’s class environment. 

assimilate

I subscribe to the exact opposite approach. Teacher’s should accommodate student needs as the focus of the classroom environment.

accommodate

Below is a quote from a parent whose child benefited from my effort to be hyper responsive to her daughter’s instructional needs. The child had veto power over any activity or strategy I attempted. If what I used didn’t work for her I would try something else.

“Working with Randy has been life changing for my daughter. 

Math was her biggest source of frustration and no matter how hard she worked it never made sense. Teachers would tell me she was ‘doing awesome’ but she was really just following steps without understanding any of it. I thought she was going to go through life unable to even buy a candy bar without being taken advantage of.

Randy changed all that. He is able to break math down in a way that makes sense. He is able to identify what is confusing her and find different ways to explain it. He makes it meaningful for her.”

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Retaining Information

Below is a model for information processing (retention and retrieval). Here are a couple key points I want to highlight:

  • A lot of information is filtered out so what gets through? Information that is interesting or relevant.
  • Information that is connected to prior knowledge, is relevant or that is organized has a better change of being stored effectively for retrieval.
  • Working memory has a limited capacity. Consider what happens to your computer when you have a lot of apps open. Your computer may start to buffer which is basically what happens to our kiddos if instruction involves opening too many apps in their brains.
  • Long term memory is basically retrieval of information. Think a student’s book bag with a ton of papers crammed in it. How well can he or she find homework? Compare this to a well maintained file cabinet that has a folder labeled homework with the homework assignment in question stored in this folder. That paper is much easier to retrieve. This is analogous to long-term memory. If the information is relevant or meaningful it will be stored in the file cabinet folder and more easily retrieved. In contrast, rote memorization like the rules teachers present students are papers crammed into an overflowing bookbag.

Information Processing.jpg

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Strategy to Individualize Instruction

It is difficult to individualize instruction in a whole class or small group setting. I created and taught the curriculum for a Consumer Math course at the high school where I teach. For a class of 10-12 students, all with an IEP, I developed an approach that allowed me to individualize the instruction for each students.

In the photo below is an example of a folder set up I used with the students in Consumer Math. Each student would have a dedicated folder, kept in the room and updated daily. The smaller paper shows the individualized agenda. The other paper shows an example of how the folder can be used as a resource. Student computer login information, accommodations like a multiplication table or notes can be secured inside the folder. The agenda would be changed out each day. (In case you are wondering about the label in the agenda, “Math Group 4.” This particular folder was used in a special education training session for teacher candidates.)

individualized folder

 

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Introduction to Equations – (Meaning Making)

This is a meaning making approach to introducing equations. I will walk through the parts shown in the photo in the space below this photo. (A revised edition of this handout will be used in a video on this topic.)

intro to equations

First I explain the difference between an expression (no =) and an equation (has =). An equation is two expressions set equal to each other (21 is an expression).

intro to equations definition equation

I then develop the idea of a balanced equation and will refer to both sides of the see saw as a prelude to both sides of the equation. I also focus on the same number of people on both sides as necessary for balance.intro to equations balanced vs unbalanced

At this point I am ready to talk about an unknown. Here is the explanation I use with the photo shown below.

  • I start with the seesaw at the top. The box has some guys in it but we don’t know how many.
  • We do notice the seesaw is balanced so both sides are equal.
  • This means there must be 2 guys in the box.
  • I follow by prompting the students to figure out how many guys are in the box(es) in the bottom two seesaws.
  • Finally, I explain that the number of guys in the box is the solution because it makes the seesaw balanced.

intro to equations definition solution

There are multiple instructional strategies in play.

  • Connection to student prior knowledge – they intuitively understand a seesaw. This lays the foundation for the parts of an equation and the concept of equality.
  • Visual representation that can be recalled while discussing the symbolic representation, e.g. x + 1 = 3
  • Meaning making which allows for more effective storage and recall of information.
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Visuals Aid Memory

This research has major implications for math for students with special needs…but some of us already knew this!

brain memory

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