Category Archives: instructional strategies

Math is a Language

The Gutenberg printing press was revolutionary because it provided a faster way to share words. In turn, these words and how they were structured were representations of ideas used to make sense of the world around us.

Math is a language with words and other symbols that also makes sense of the world around us. We consume and know more math than we realize or allow ourselves credit for.

When buying the latest iteration of an iPhone, we may call forth algebra. How much will you pay if you buy an iPhone for $1000 and pay $80 a month for service? Well, that depends on how many months you will use this iteration before moving on to the next iPhone. The number of months is unknown so algebra gives us a symbol to represent this unknown number of months, x (or n or whichever letter you want).

Just as there is formal and informal English (or other language), we can engage algebra formally or informally. You don’t need to write an equation such as y = 1000 + 80x to figure out how much you will pay. You can do this informally, compute 80 times 10 months + 1000 on the calculator. Then try 80 times 12 months etc.

Math provides us a means of organizing and communicating ideas that involve quantities like the total cost for buying an iPhone.

The difficulty in learning math is that it is often taught out of context, like a secret code. In contrast, a major emphasis in reading is comprehension through meaning, such as activating prior knowledge (see below).

In fact, math absolutely can and, in my view, should be taught by activating prior knowledge. My approach is to work from where the student is and move towards the “mathy” way of doing a problem.

Without meaning, students are mindlessly following steps, not closer to making sense of the aspects of the world that involve numbers.

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Counting Out Value of Coins

Counting out the total value for a set of coins can be very challenging for students who struggle with money.

Here is a video showing how to use a WORD document table a former colleague and I created to support students in a life skills class. The video shows how this handout is used but does so with a virtual version the can be completed on the computer (see image below). Here is a link to the virtual document.

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Online Floating Rulers to Measure Length

To help students learn how to measure with a ruler, I focus on minimizing the number of tic marks on the ruler at first. The image below shows an excerpt from a WORD document with a halves ruler that I use and an instructional strategy. It also contains a quarters and an eighths ruler that students can slide around the WORD document as shown above and in a video explaining this artifact and how I created it.

This is useful for distance learning as well as in class. Here is a link to the WORD document with the rulers shown in the video.

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Base Ten Using Popsicle Sticks

I use the following method as a entry point for double digit numbers.

The photo below shows 2 packs of Popsicle sticks counted as 10 each, followed by single sticks counted as 1 each. The student counts on from 20, with the use of the scaffolded handout (photo at bottom). The handout focuses only on counting on from 20 and shows a photo of 2 of the bundles of sticks. Similar handouts involve counting on from 10 or from 30 etc.

By engaging in the actual counting, the student learns the 10s by doing. This would be followed by counting on from each 10 without the handout.

The use of Popsicle sticks is useful for 2 reasons. First, a bundle of items like shown below is more concrete than the rods for Base 10 blocks. Second, pulling packs of sticks apart of bundling 10 sticks together is an act that is concrete for students and ties into their prior knowledge regarding the grouping of objects (e,g. pack of gum).

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Addition: Concept and Mechanics

The graphic organizer below is used to show the student the steps for addition. It also addresses the concept of addition (which I have addressed previously) as an act of pulling “together” two parts to form a whole.

The student is prompted to move the first part (set of coins in this case) to the number chart. This can be completed with 1 to 1 correspondence or with subitizing (identifying the number of items without counting). Then, the student is prompted to move the second part while counting on, e.g. 7, 8 etc. (as opposed to starting from the left and counting from the first coin: 1, 2 etc.). The chart scaffolds the counting on and allows the student to see the total as a magnitude.

It is important to first teach the students the “rules of the game”, i.e. how to use the graphic organizer. To do this have the student simply move the first part to the number chart then the second part. The student can also be prompted to state the addition problem (written at the top). When the student is fluent with these steps the counting on can be implemented.

The next step would be to replace the coins with the symbolic representation, numbers.

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Information Processing Analogy – Big Picture

Effective instruction is effective because it addresses the key elements of how the brain processes information. I want to share an analogy to help adults (parents and educators) fully appreciate this.

Below is a model of information processing first introduced to me in a master’s course at UCONN.

Here is a summary of what is shown in the model.

  1. Our senses are bombarded by external stimuli: smells, images, sounds, textures and flavors.
  2. We have a filter that allows only some of these stimuli in. We focus on the ones that are most interesting or relevant to us.
  3. Our working memory works to make sense of the stimuli and to package it for storage. Our working memory is like a computer, if there is too much going on, working memory will buffer.
  4. The information will be stored in long term memory.
    • Some will be dropped off in some random location and our brain will forget the location (like losing our keys)
    • Some will be stored in a file cabinet in a drawer with other information just like it. This information is easier to find.

Here is the analogy. You are driving down the street, like the one shown below.

There is a lot of visual stimuli. The priority is for you to pay attention to the arrows for the lanes, the red light and the cars in front of you. You have to process your intended direction and choose the lane.

There is other stimuli that you filter out because it is not pertinent to your task: a car parked off to the right, the herbie curbies (trash bins), the little white arrows at the bottom of the photo. There is extraneous info you may allow to pass through your filter because it catches your eye: the ladder on the right or the cloud formation in the middle.

Maybe you are anxious because you are running late or had a bad experience that you are mulling over. This is using up band width in your working memory. Maybe you are a relatively new driver and simple driving tasks eat up the bandwidth as well.

For students with a disability that impacts processing or attention, the task demands described above are even more challenging. A student with ADHD has a filter that is less effective. A student with autism (a rule follower type) may not understand social settings such as a driver that will run a red light that just turned red. A student with visual processing issues may struggle with picking out the turn arrows.

Effective instruction would address these challenges proactively. Here is a video regarding learning disabilities (LD) that summarizes the need in general for teachers to be highly responsive to student needs. Here is a great video that helps makes sense of what autism in terms of how stimuli can be received by those with autism (look for the street scene). Here is a video of a researcher explaining how ADHD responds to sensory input (he gets to a scenario that effectively encapsulates ADHD).

To address these challenges:

Ironically, this is likely a lot of information for your brain to process…

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Counting Money as a Game

The Allowance Game is retail game tailored to students who are learning to count money.

 

I revised it to make it more authentic and more functional.

  • I changed some money amounts to necessitate the use of pennies (see below)
  • I use real dollars and cents to provide more opportunities to handle real money
  • I differentiate by creating different task demands. For example
    • I was using the modified version with 2 clients
    • One was learning to count out dimes and pennies only, but could manage ONES simultaneously
    • This student would also be provided a coin chart I use to teach students how to count with coins
    • The other was practicing with ONES, and all coins up to a quarter
    • I collected data on a data sheet – 1 per student

The students loved playing the game, it was engaging so they practiced the counting out money, I was able to collect data, and I was able to differentiate. When I co-taught a Consumer Math course, I would assign a para (instructional assistant) to facilitate the game with a couple students and to collect data.

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Token Sheet to Address Target Behaviors

Perhaps the vast majority of students with disabilities need support with math. Their challenges with math can be directly related to their disability or can be the result the effects of an ongoing struggle with math. The later results in what is termed secondary characteristics.

When I work with students with a disability, I first seek out background information about the student to identify what interests them, what reinforcers (rewards) can be used to enhance their performance, and what challenges and behaviors need to be addressed. Upon gather this information, I often decide to use a token sheet that is personalized for each student.

Below is an image of such a token sheet. At the start of our work together I felt the student in question needed immediate reinforcement for work completed to get him into a groove. I was also targeting a behavior in which he would draw dots on each digit he wrote, which slowed him down considerably. He would earn a Scooby (I would circle it) in the middle column for completing his work and an extra Scooby in the right column if he wrote digits appropriately (no dots). After 2 sessions, his dot writing dropped significantly to the point that I was able to remove the column on the right. As you can see at the bottom, 5 Scoobies resulted in iPad time.

This can be particularly effective for students who have more severe math anxiety, a fear of failure, or have ADHD. Such a token sheet can be included in the accommodations page of the IEP.

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Multiplying and Carrying a Tens Digit

Carrying the TENS digit in a multiplication problem is a sticking point for many students. To address this, I use a task analysis approach to zero in on the step of identifying the product for the ONES as a prelude to carrying.

In the example below, 5 and 4 are in the ONES place and the product is 20. The task analysis steps involved:

  • compute the product
  • identify the digits in the product
  • identify the digit in the ONES
  • identify the digit in the TENS
  • Understand that the TENS digit must be carried to the TENS column

By creating a place holder for the product and scaffolding it to differentiate between the TENS and the ONES, the student can visualize the product. This reduces the demand placed on working memory. Once mastery with the place holder is demonstrated, it can be faded (and used as necessary as part of corrective feedback).

NOTE: I started this mini-lesson for a student with ADHD by having him warm up with problems without carrying. Also, extra line below the 60 and 20 are used for multiplying by 2 digit numbers (next in the sequence).

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Making Sense of Fractions – Regrouping with Mixed Numbers

It is easy to get caught up in the steps and rote memorization when working with fractions. The brain processes information more effectively when the information is meaningful. ADHD makes paying attention to rote memorization of steps even more challenging.

Below is an excerpt of work I completed with a middle school student who has ADHD. This was completed extemporaneously as intervention (you see his initial attempt was incorrect) but can be used as Universal Design in whole class instruction.

Here is a break down of how I helped the student after seeing his mistake in his initial attempt. First, I modeled the first mixed number as pizza pies.

Then I presented the problem in pizza terms. “You have 3 pies and 1 slice and you are going to give me 1 pie and 2 slices. Do you have enough slices?” <wait for response> “You don’t, so what can we do?” <wait for response> “We cut up one of the pies.” I have the student cut the pie into fourths.

I then make the connection with the mixed number and guide the student to taking away 1 pie and writing 4/4. This provides more concrete meaning for writing 1 as 4/4.

In turn, this provides meaning for the new mixed number and meaning for the subtraction of the whole numbers (pies) and the fractions (slices).

 

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