Category Archives: instructional strategies

Trig Fingers

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Place Value Representation

Here is an easy way to create and implement strategy to unpack place value for students (created by one of my former graduate students). I suggest using this after manipulatives and visual representations (drawing on paper) in a CRA sequence. It is hands on but it includes the symbolic representation (numbers). Hence is another step before jumping into the mathy stuff.

pace value representation just numbers.jpg

The focus can shift to money as well.

2017-06-07 18.06.49

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The I in Instruction can be the same I in IEP and IDEA.

I am consistently surprised by the reliance on canned items for students who struggle. There are different reasons students struggle but we know that there are secondary characteristics and factors that inhibit effective information processing that can be addressed with some Individualization.

In a math intervention graduate course I teach at the University of Saint Joseph, my graduate students are matched up with a K-12 student with special needs. The graduate student implements instructional strategies learned from our course work. Below is the work of one of my grad students. From class work and our collaboration we developed the idea of using the fish and a pond as base 10 blocks for the student my grad student was helping. He likes fish and fish will get his attention. The grad student explained that if he has 10 fish the 10 fish go into a pond. In the photo below the student modeled 16 with a TEN (pond) and 6 ONES (fish).

2018-07-26 17.13.05.jpg

Similarly, another student likes Starbursts and that student’s respective grad student created Starburst packs to represent TENS and ONES (there are actually 12 pieces in a pack so we fudged a little).  The point is that it was intuitive and relevant for the student. The student understood opening a pack to get a Starburst piece.

2018-07-26 17.41.15

 

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More Than: Concept and Symbol

The alligator eats the bigger number is the common approach for student to use inequality symbols (<  > <  > ). I find that students remember the sentence but many do not retain the concept or use the symbols correctly, even in high school. The reason, I believe, is that we introduce additional extraneous information: the act of eating, the mouth which is supposed to translate into a symbol, the alligator itself. For a student with processing or working memory challenges this additional information can be counter productive.

alligator eats bigger number

I use the dot method. By way of example here is the dot method. I show the symbols and highlight the end points to show one side has 2 dots and the other, 1.

Compare-with-Dots

Then I show 2 numbers such as 3 and 5 and ask “which is bigger?” In most cases the student indicates 5. I explain that because 5 is bigger it gets the 2 dots and then the 3 gets the 1 dot.

3 dots 5

I then draw the lines to reveal the symbol. This method explicitly highlights the features of the symbol so the symbol can be more effectively interpreted.

3 less than 5.

That is the presentation of the symbol. To address the concept of more, especially for students more severely impacted by a disability, I use the following approach. I ask the parent for a favorite food item of the student, e.g. chicken nuggets. I then show two choices (pretend the nuggets look exactly the same) and prompt the student to make a selection. This brings in their intuitive understanding of more.

concept of more chicken nuggets just plates

I think use the term “more than” by pointing to the plate with more and explain “this plate has more than this other plate.” I go on to use the quantities.

concept of more chicken nuggets more than words

Finally, I introduce the symbol to represent this situation.

concept of more chicken nuggets more than symbol

Below is the example my 3rd grade son used to explain less than to a classmate with autism. This method worked for the classmate!

Lucas less than example

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Meeting Needs Part 2

In the past year I have helped two 7th grade students who are categorized as twice exceptional (2e). Both had more severe math anxiety that impacted their performance and masked their ability. When we started both were working on elementary school level math. Within a couple of months both were working on algebra. (Both had gaps but I was testing their ability by test running higher level math with them.)

As I shared in a previous post my approach is to focus on meeting needs. I want to elaborate on this. My secret is I listen to the student… In other words, the student drives the instruction.

Here’s an analogy. You go to a frozen yogurt or ice cream store and they offer you a sample. You try a couple then go with the one you like. That’s what I do. I try out different types of instruction (samples of the ice cream) and the student tells me (verbally or by the response to the instruction) which one they want. That is the I in IDEA and in IEP.

icecream samples

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Common Denominator – Why?

We explain steps in great detail to students but often omit the underlying concept. The topic of adding or subtracting fractions with unlike denominators is an example of this.

subtracting unlike denominators     adding unlike denominators

The example above right is a short cut for what is shown above left. These short cuts, which math teachers love to use, add to the student’s confusion because these rules require the student to use rote memorization which does is not readily retained in the brain.

I suggest using what I call a meaning making approach. I present the student 2 slices of pizza (images courtesy of Pizza Fractions Game) and explain the following setting. “You and I both paid for pizza and this (below) is what we have left. You can have the pizza slice on the left and I will have the pizza slice on the right. Is that OK?” The student intuitively understands that it is not because the slices are different sizes. I then explain that when we add fractions we are adding pizza slices so the slices need to be the same.

fourth pizza slice         half pizza

I then cut the half slice into fourths and explain that all the slices are the same size so we can now add them. Then the multiplying the top and bottom by 2 makes more sense.

fourth pizza slice          2 fourths pizza slices

 

 

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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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