## 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.

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.

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.

## Simplifying Expressions (Combine Like Terms)

Simplifying expressions (see photo below) is one of the most challenging algebra tasks for many students receiving special education services. A major problem is that it is typically presented as symbol manipulation…addressed in very symbolic form.

My approach is to make math relevant and more concrete. Below is a scaffolded handout I use to help unpack the concept. In the handout I start with items the student intuitively understands, tacos and burritos or tacos and dollar bills. In the top left of this handout the student is asked how many tacos he or she has. 3 tacos eventually is written as 3T. See next photo to see how the handout is completed as NOTES for the students.

As I work with the problems below I remind the student that the “T” stands for taco so “3T” stands for 3 tacos. This takes the student back to a more concrete understanding of what the symbols mean.

To address negatives I use photos of eating a taco or burrito. “-2T” is eating 2 tacos.

So “3T – 2T” means I have 3 tacos and ate 2. I have 1 taco left… 1T. For students who may need an even more concrete representation, use actual tacos or other edible items.

## Introduction to Solving Equations

I introduce solving equations by building off of the visual presentation used to introduce equations. The two photos below show an example of handouts I use. Below these two photos I offer an explanation of how I use these handouts.

First I develop an understanding of a balanced equation vis-a-vis an unbalance equation using the seesaw representation.

I then explain that the same number of guys must be removed from both sides to keep the seesaw balanced.

I then apply the subtraction shown above to show how the box (containing an unknown number of guys) is isolated. I explain that the isolated box represents a solution and that getting the box by itself is called solving.

I use a scaffolded handout to flesh out the “mathy” steps. This would be followed by a regular worksheet.

I extend the solving method using division when there are multiple boxes. I introduce the division by explaining how dividing a Snickers bar results in 2 equal parts. When the boxes are divided I explain both boxes have the same number of guys.

The students are then provided a scaffolded handout followed by a regular worksheet.

## Graphing Linear Functions

Graphing linear functions and the underlying concept are challenging for many students. The video below shows a scaffolded approach to teaching how to graph. This approach also addresses the concept of the graph as a visual representation of all possible solutions (see photo above). Students often do not realize that the line is actually comprised of an infinite set of points which represent all the solutions. Here is a link to the document used in the video.

## 1 to 1 Correspondence Scaffolded

I have encountered several students who struggle with 1 to 1 correspondence with the educators struggling to figure out how to teach this to these students who continue to struggle. This post reveals an approach I used with a student.

I broke down the task using a formal task analysis approach. This approach involves identifying the different individual steps and to address these steps in isolation. Here is the sequence I use and suggest.

1. Conduct a pretest using a task analysis pretest data sheet I created for this topic. I do not use any scaffolding and prompt the student to count out the objects (in this case decks of cards) and to do so independently. I prompt the student after they show they cannot complete a step which allows the student to attempt the next step. (Think of teaching a student to get dressed and he cannot put his socks on. You help him with the socks then ask him to put on his shoes.)
2. I then focus on the movement of the objects. I provide scaffolding for start and stop piles (see mats with track photos above). The student is asked to move the cards one at a time without counting.
3. The student must learn the “rules of the game” which includes how to place the items in the stop pile. Students may be confused about placement, e.g. one student ran out of room while placing the decks in a straight line and I had to demonstrate that it was OK to place them on different spots on the mat. Once the student demonstrates mastery of moving the items we move on to the next step.
4. We then focus on counting in isolation. The card decks are labeled with numbers (photo below) and the student does not move anything but simply reads the numbers. (More on these numbers in a later step.) More numbers can be added as necessary.
5. The next step (photo below) is to have the student read the number on each card. I have a stack of decks of cars on the start pile with the numbers facing down. I show the student the number of the deck that I am moving to the stop pile and the student reads off the number. I place the used deck face down to hide the number. This activity forces the students to focus on each item as he reads the number. One student kept counting ahead to the next number and I prompted him to return his focus to the current number. This is the crucial step as it focuses on the 1 item 1 number aspect of counting.
6. The next step is to have the student move the decks from the start pile to the stop pile and to read each number while doing so. I turn each deck face up as a prompt for the student to move and read.
7. The student then is prompted to select the cards on his own and read (the cards can be in a pile in order by number).
8. Eventually 1 then 2 then 3 decks have the number missing which adds an extra task demand for the student – identify the next number as he is moving the item.
9. Finally the items do not have any numbers and the student counts, with the mats eventually be faded.

Note: this is especially effective for students with ADHD because it helps to focus and organize their task demand for the activity of counting.

## Kahoot Game

A Kahoot is an online and app quiz game that allows students to answer questions using a personal device (e.g. simulated phone in photo above). The teacher can create the questions (e.g. example question I created in photo above).

My approach is to use a Kahoot to scaffold learning. In this post I use plotting points as an example.

• I start with simple questions, e.g. identify the letter and number coordinates for the dog and chick below. Notice in the top photo below that I provide the actual coordinates in question 1 (“for the dog C4”) as a scaffold to show the students what to do.
• Then I show numeric coordinates for a point, but only with positive numbers.
• Eventually I present problems that address all 4 quadrants and ask questions about the parts of the coordinate plane (photo bottom one, below).
• Notice that the questions have times (in seconds). This indicates the time allotted to answer each question (teacher sets this). For students with special needs I print a hard copy to allow them more time to read the question. If necessary, they can respond by circling the answer on the handout.

## Elapsed Time Scaffolded

The photo above shows a scaffolded handout to break down elapsed time for a student. The problem is divided into 3 parts: time from 10:50 to 11:00, time from 3:15, time from 11:00 to 3:00 (see photo below). This is based on how we may compute elapsed time by focusing on minutes then on hours. Notice the 3 clocks (in photo above) with no hands which allows the student to engage the clocks by having to determine and show how many minutes passed, e.g. 10:50 to 11:00.

The final answer would be 4 hours and 25 minutes.

## One More or One Less Scaffolding

The photo above shows a screen for a hundreds table was shared by one of my students in a Math for Children graduate course. She found it on Pinterest for use a class presentation. I love this idea and came up with some revisions I think can make it more effective. It seems to me that this screen may be too busy with 4 different numbers showing. Additionally, the view of the other numbers outside the screen could be distracting.

Below are a couple revisions I would suggest.

## Making Slope Less Complicated

Slope is the rate of change associated with a line. This is a challenging topic especially when presented in the context of a real life application like the one shown in the photo. The graphed function has different sections each with a respective slope.

One aspect of slope problems that is challenging is the different contexts of the numbers:

• The yellow numbers represent time
• The orange numbers represent altitude
• The pink numbers represent the slopes of the lines (the one on the far right is missing a negative)

Before having students find or compute slope I present the problem as shown in the photo above and discuss the meaning of the different numbers. What I find is that students get the different numbers confused and teachers often overlook this challenge. This approach is part of a task analysis approach in which the math topic is broken into smaller, manageable parts for the student to consume. Once the different types of numbers are established for the students we can focus on actually computing and interpreting the slope.

This instructional strategy is useful for all grade levels and all math topics.