Human senses are bombarded by external stimuli: smells, images, sounds, textures and flavors.
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.
Our working memory works to make sense of the stimuli and to package it for storage. It is like a computer, if there is too much going on, working memory will buffer.
The information will be stored in long term memory.
Either it will be dropped off in some random location and our brain will forget the location (like losing our keys)
Or it will be stored in a file cabinet in a drawer with other information just like it. This information is easier to find.
Analogy to Classroom Learning
Here is an analogy to what happens during school instruction. 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.
Other present stimuli may be filtered 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.
Impact on Students
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. One 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. Another with visual processing issues may struggle with picking out the turn arrows. Their brain may start to buffer, like a computer.
A major obstacle in math for many students with special needs is carrying in addition problems. Below is a task analysis approach.
First, I target the step of identifying the ONES and TENS place in the 2 digit sum in the ONES column (below it is 12). In a scaffolded handout I create a box to for the sum with the ONES and TENS separated. At first I give the sum and simply have the student carry the one.
Then I have the student find the sum and write it in the box (14 below). Once mastered I have the student write the sum and carry the 1.
They would have mastered adding single digit numbers before this lesson. I revert back to single digit numbers to allow them to get comfortable with writing the sum off to the side without the scaffolding. (In the example below I modeled this by writing 13.)
The last step is to add the TENS digits with the carried 1. I use Base 10 manipulatives to work through all the steps (larger space on the right is for the manipulatives) and have the student write out each step as it is completed with the manipulatives.
Finally, the student attempts to add without the scaffolding. I continue with color but then fade it.
When my son was in preschool I asked him who was in his class. He replied, ” Natalie, she’s the yellow heart.” Children learn color before they learn words because it is easier to process.
This is found in children’s toys with color used to guide use of toys.
The obvious use of color in real life in traffic lights. The colors represent different concepts with red being used universally in the U.S. as representing stop. Color is used to partition an object into sections, as often seen in maps of areas with different sections. Think of how many highlighters are sold to college students to help them highlight key passages in textbooks.
The use of color help convey information, especially sections of a whole is an effective and easy to use instructional or support strategy.
The top two images below show my earliest attempts to use color. The student for whom this was used was a 7th grade student with asperger’s who tested in math and reading at a 1st grade level.
In lieu of referring to the “horizontal line” I can refer to the “yellow line” as in “find the yellow 3” for plotting the point (3, -2). Color, as in the aforementioned yellow heart, is much more intuitive for students, especially those with a disability.
Color was used for the same student to represent positive and negative numbers, first with concrete tokens then with colored numbers on paper.
More examples are shown below. Color helps a student focus on the different parts of an equation or different parts of a ruler.
Color can also help organize a room into different parts. Each color represented different courses I taught, e.g. green was used for algebra 2. The room is more organized because of the sections outlined in color. Consider how this can help a student with ADHD, autism or an executive functioning disorder.
Are you a parent of a student with special needs who is struggling with a math topic? Are you a teacher figuring out how to differentiate for a particular student on a math topic? Pose your question and I will offer suggestions. Share your question via email or in a comment below. I will respond to as many as I can in future mailbag posts.
Here is a topic multiple educators and parents ask about:
I don’t want my child to be stuck in a room. He needs to be around other students.
Often we view situations in a dichotomous perspective. Inclusion in special education is much more nuanced.
In math if a student cannot access the general curriculum or if learning in the general ed math classroom is overly challenging then the student likely will not experience full inclusion (below) but integration (proximity).
For example, I had an algebra 1 part 1 class that included a student with autism. He was capable of higher level algebra skills but he would sit in the classroom away from the other students with a para assisting him. Below is a math problem the students were tasked with completing. Below that is a revised version of the problem that I, as the math teacher created, extemporaneously for this student because the original types of math problems were not accessible to him (he would not attend to them).
I certainly believe in providing students access to “non-disabled peers” but for students who are more severely impacted I believe this must be implemented strategically and thoughtfully. Math class does not lend itself to social interaction as well as other classes. If the goal is to provide social interaction perhaps the student is provided math in a pull-out setting and provided push-in services in other classes, e.g. music or art.
Here are the details of example of a push-in model I witnessed that had mixed effectiveness. A 1st grader with autism needed opportunities for social interaction as her social skills were a major issue. She was brought into the general ed classroom during math time and sat with a peer model to play a math game with a para providing support. The game format, as is true with most games, involved turn-taking and social interaction. The idea is excellent but the para over prompted which took away the student initiative. After the game the general ed teacher reviewed the day’s math lesson with a 5-8 minute verbal discussion. The student with autism was clearly not engaged as she stared off at something else.
Found this (above) cool example of corresponding angles (see photo below for explanation). This window photo could be a nice introduction to this type of problem by printing it out on paper and having students match angles as the teacher shows the photo on the Smart Board or screen.
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.
Slope is one of the the most important topics in algebra and is often understood by students at a superficial level. I suggest introducing slope first by drawing upon prior knowledge and making the concept relevant (see photo above). This includes presenting the topic using multiple representations: the original real life situation, rates (see photo above) and tables, visuals, and hands on cutouts (see photos below).
A key aspect of slope is that it represents a relationship between 2 variables. Color coding (red for hours, green for pay) can be used to highlight the 2 variables and how they interact – see photo above and below.
The photo below can be used either in initial instruction, especially for co-taught classes, or as an intervention for students who needs a more concrete representation of a rate (CRA). The clocks (representing hours) and bills can be left in the table for or cut out.
Students can hit a road block at the steps that appear to be very simple. For example, in the problem below the students are prompted to find the highest point on the graph. Many think the graph refers to the entire coordinate plane and they pick 5 as the high point. It is the highest point on the y-axis but not the graph. I introduce the problem by highlighting the actual graph in pink and explain that this highlighted line is what is meant by the graph.
The use of color also helps students distinguish between the x and y axes and what the variables x and y represent in the context of the problem (# minutes and # kilometers in this problem) – see photo above. This problem also involves plugging in a # for x (blue) IN the function (red). In the photo below you see how I use color to help emphasize this.
This is an example of color coding (highlighting) to help make a calculus problem accessible. You don’t have to know calculus to see that the yellow sections (left and right of the 0) are going up while the green section is going down. Color coding breaks a whole into parts that are easier to see and understand – works in preschool all through calculus!