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Visual Representations

Visual Representations

In preschool kids are assigned a color and shape as identification. Their names are too abstract at that age. This strategy is effective for teaching in general.

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Representations of Concepts

Representations of Concepts

Three ways to represent perimeter: I taught a lesson on perimeter to a 5th grade class. First I had them create a rectangular pen for their animals and they counted the number of fence pieces. Then we drew a rectangle to represent the pen. Finally we looked at the formula. This allows a deeper conceptual understanding of the concept. This is known as Concrete-Representation-Abstract – representing the concept at all three levels.

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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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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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Velocity, Acceleration, Speeding Up and Slowing Down

DISCLAIMER: This is a very mathy, math geek post but it also has value in demonstrating instructional strategies and multiple representations.

We all understand speed intuitively. Velocity is speed with a direction. Negative in this case does not indicate a lower value but simply which way an object is traveling. Both cars below are traveling at equivalent speeds.

velocity vs speed

The velocity can be graphed (the red curve below). Where the graph is above the x-axis (positive) the car is traveling to the right. Below is negative which indicates the car is traveling to the left. The 2 points on the x-axis indicate 0 velocity meaning the car stops (no speed). (I will address the blue line at the end of this post as to not clutter the essence of what is shown here for the lay people who are not math geeks.)velocity graph with tangents

Below is an example of using instructional strategies to help make sense of the graph and of velocity, acceleration, speeding up and slowing down.

As stated previously, the points on the x-axis indicate 0 velocity – think STOP sign. As the car moves towards a stop sign it will slow down. When a car moves away from a stop sign it speeds up.

The concept and the graph analysis are challenging for many if not most students taking higher level math. This example shows how instructional strategies are not simply for students who struggle with math. Good instruction works for ALL students!

speeding up and slowing down

It is counterintuitive that when acceleration is negative the car can be speeding up. The rule of thumb is when acceleration and velocity share the same sign (+ or -) the object is speeding up. When the signs are different the object is slowing down. This rule is shown in the graph but the stop sign makes this more intuitive.

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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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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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Function Notation for Algebra

Below is a video of a lesson I recorded on function notation using the Explain Everything app. The lesson starts by addressing the concept of function notation by connecting it to the use of the notation “Dr.” as in Dr. Nick of Simpson’s fame. The lesson builds on prior knowledge throughout with a focus on color coding and multiple representations.

This videos shows an instructional approach to teaching function notation and concepts in general and video lessons can be used for students who miss class or who need differentiation.

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AWESOME Online Graphing Calculator

Desmos graph pizza function

This can be a game changer for students with special needs who struggle with math. The Desmos graphic calculator allows students to interact with math equations through multiple representations. It is far superior to graphing calculators in terms of quality and ease of use and is free. The app for Smartphones is outstanding.

Here are features that make this calculator user-friendly and an outstanding instructional strategy.

  • Students can click on dots and the ordered pair will appear (see top photo below).
  • Students can change features of the equation and immediately see how the graph changes.
  • Students can collect data and create a graph and convert the data into “mathy” representations like equations (see top photo below).

Desmos regression price vs mileage of used cars


Desmos parabola



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Fractions! Meaning Making for Comparing Fractions


Fractions is one of the most challenging math topics. Many high school and college students struggle to some degree with fractions.  The Common Core of State Standards (CCSS), despite all the criticism, includes components to address the conceptual understanding of fractions. Below is a photo showing a 4th grade Common Core standard regarding fractions along with an objective for a class lesson I taught at an elementary school in my district. I subsequently presented on this at the national CEC conference in 2014. Notice the bold font at the bottom, ¨justify…using a visual fraction model.¨ The photo above shows an example of a model I used in class.

Screenshot 2017-03-16 at 9.40.19 AM

The photo below shows a handout I used in the lesson. The first activity involved having students create a Lego representation of given fractions. These would eventually lead to the photo at the top with students comparing fractions using Legos. The students were to create the Lego model, draw a picture version of the model then show my co-teacher or I so we could sign off to indicate the student had created the Lego model.

Screenshot 2017-03-16 at 9.44.45 AM

The Lego model is the concrete representation in CRA. In this lesson I subsequently had students use fractions trips (on a handout) and then number lines – see photos below.

Screenshot 2017-03-16 at 9.47.32 AMScreenshot 2017-03-16 at 9.51.37 AM

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