Category Archives: Concrete-Representation-Abstract

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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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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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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WAR Card Game Modified for Height Comparision

Comparing numbers is very challenging for some students and likely speaks to a major gap in their number sense. It is also very challenging to address effectively with these students. The photo below shows an entry point into I have used with success in helping such a student.

Even for these struggling students, taller and shorter are likely prior knowledge that students understand intuitively. The people outline on the vertical number chart leverages this intuitive understanding to compare numbers.

  • I start by showing 2 different outlines and asking which is “taller” and stick use that term until the student gets the idea. This isolates the focus to comparing items.
  • Then I redo all the comparisons using the term “taller” and when the student makes a selection, 10 in this case, I reply “yes, 10 is MORE!” and have them repeat “more.”
  • Finally, I redo the comparisons asking which is more and for improper responses I ask which is taller then restate that the taller item is “more.”

For a change of pace, I created a revised version of the card game WAR by using these outlines (no face cards). The cards are 5″x7″ which I purchased on Amazon. Here is a link to the WORD document with the outlines I cut out and taped to the cards. After a student shows success with this revised game, I play regular WAR but use the people outlines for feedback to make a correction or as a prompt as necessary, with the intent to fade their use.

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Superhero Math

I was recruited to help a middle school student who is having a very rough time at this time in his life. It was shared with me that he likes Marvel superheroes and he is struggling with counting money and multiplication. Below are some ideas I presented for a test run and photos of the items I ordered for these suggested activities.

  • For multiplication

    • Put the heroes (or villains) in groups of 2 and have him count out 4 groups and compute. Use different groups and number per group. (IGNORE the numbers on the cards)

    • Get a group of 10 villain cards. Pretend heroes have to travel in groups of 2 and ask how many groups to get 10 heroes to fight the 10 villains. (IGNORE the numbers on the cards). Variations of this.

    • After gets the idea of groupings, focus on the number on the cards and show him two 5s and have him compute. Variations of this.

    • Play a game where he draws two cards and has to multiply the cards (start with very low numbers or maybe show him a 2 card and he has to pick another card to multiply by 2.

marvel playing cards

  • For Money

    • Tell him he earn money to buy these figures, one at a time – a monetary version of a token economy. Have him rank them by his favorite to least favorite and come up with a price for each with his favorite figures costing more. Start with the least favorite and make the price such that with a little practice he could count out the coins to pay for it. Maybe 17 cents with dimes and pennies. He has to count out the money correctly and independently to actually buy the item.

Avengers Figurines

  • Other options

    • If he needs work with addition you can play WAR in which 3 cards are played and each person adds to find the total. For subtraction do the same with 2 cards.

    • You can play subtraction in which one person has superheroes and the other has villains. In order for a villain to win a villain card has to be higher than a hero card by 3 or more.

You can write an 11, 12 and 13 on the J, Q, K cards respectively. All the games can be presented though Direct Instruction – I do, we do, you do. The You do can be used as daily progress monitoring. If he needs prompting this can be recorded. This can be used for your progress reports. Attached is a data sheet I use for activities.

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Symmetry Made Accessible

The work shown below posted on LinkedIn by Maria Priovolou. I think this is awesome.

The photo below shows a focus on just the vertical axis and the student has to reflect one object at a time. This is a nice task analysis approach. The stamp creates the objects which makes it hands on and a little different from just mathy work.

symmetry vertical axis only

This hands on work can be followed with work on this website. In the photo at the bottom you see an example problem. This can make reflection more concrete and eventually more intuitive for the student.

top marks symmetry

top marks symmetry problem.JPG

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Lego Fractions

The photo below is courtesy of Robert Yu, Head of Lego Education China, as shared by Jonathan Rochelle, Director for Project Management at Google.

The use of Legos shown here is a classic (and wicked clever) example of manipulatives.

Lego fraction model

Before writing the actual fractions students can use drawings as shown below. The sequence of manipulatives, drawings then the actual “mathy” stuff constitutes a Concrete-Representational-Abstract (CRA) model. Concrete = manipulative, Representational = picture, abstract = symbolic or the “mathy” stuff.

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Comparison Shopping – Real Life Example

Saw the following price tags, shown in the two photos, at an office supply store. $4 for 4 batteries or $9 for 8 batteries. To compare we can double the smaller pack to see that 2 packs would cost $8 for 8 batteries for a better deal.

Another method is to use unit rates. Rates are a measure of one quantity, with units, per 1 unit of another quantity, e.g. you make $10 per hour. To compute

$4/4 batteries = $1/1 battery  vs $9/8 batteries = $1.13/1 battery

2018-11-18 15.49.442018-11-18 15.49.40

Below is an example of instruction for unit rates to help a student conceptually understand (pretend that gas price shown on this pump is $2 per gallon). Say you pumped 3 gallons and it cost $6. Show the 3 1-gallon gas cans together and the 6 $1 bills together. Separate them to you have equal groups to get $ per 1 gallon. You can use actual gas cans (unused) or cutouts from Google Images.

gas price rate

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Sales Price Entry Point

A pseudo- concrete representation of a sales price problem is shown below. This is what I use as an entry point for teaching these problems.

  • The entire shape represents the total price of $80. This is 100%, which in student language is “the whole thing.”

  • The discount rate is 25%. Cut with scissors to lop off the 25% which also lops off $20, which is the actual discount. Explain to the student that this 25% is part of the “whole thing.”

  • What remains is 75% or $60. This is the “new price” which is called the sales price.

IMG_20180526_165232.jpg

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