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.
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.
I am consistently surprised by the reliance on canned items for students who struggle. There are different reasons students struggle but we know that there are secondary characteristics and factors that inhibit effective information processing that can be addressed with some Individualization.
In a math intervention graduate course I teach at the University of Saint Joseph, my graduate students are matched up with a K-12 student with special needs. The graduate student implements instructional strategies learned from our course work. Below is the work of one of my grad students. From class work and our collaboration we developed the idea of using the fish and a pond as base 10 blocks for the student my grad student was helping. He likes fish and fish will get his attention. The grad student explained that if he has 10 fish the 10 fish go into a pond. In the photo below the student modeled 16 with a TEN (pond) and 6 ONES (fish).
Similarly, another student likes Starbursts and that student’s respective grad student created Starburst packs to represent TENS and ONES (there are actually 12 pieces in a pack so we fudged a little). The point is that it was intuitive and relevant for the student. The student understood opening a pack to get a Starburst piece.
Some differences are directly related to ADHD. Others are the result of secondary characteristics. In special education these are characteristics of a student that result not from the disability but from how the disability plays out in an academic and other settings. For example, a student with a speech impediment may be very timid and anxious in situation in which he or she may need to speak.
In math a major secondary characteristic is math anxiety. This is a performance issue vs an ability issue and it must be addressed as a legitimate obstacle for the student. I work with graduate students who still suffer math anxiety years later.
The alligator eats the bigger number is the common approach for student to use inequality symbols (< > <> ). I find that students remember the sentence but many do not retain the concept or use the symbols correctly, even in high school. The reason, I believe, is that we introduce additional extraneous information: the act of eating, the mouth which is supposed to translate into a symbol, the alligator itself. For a student with processing or working memory challenges this additional information can be counter productive.
I use the dot method. By way of example here is the dot method. I show the symbols and highlight the end points to show one side has 2 dots and the other, 1.
Then I show 2 numbers such as 3 and 5 and ask “which is bigger?” In most cases the student indicates 5. I explain that because 5 is bigger it gets the 2 dots and then the 3 gets the 1 dot.
I then draw the lines to reveal the symbol. This method explicitly highlights the features of the symbol so the symbol can be more effectively interpreted.
That is the presentation of the symbol. To address the concept of more, especially for students more severely impacted by a disability, I use the following approach. I ask the parent for a favorite food item of the student, e.g. chicken nuggets. I then show two choices (pretend the nuggets look exactly the same) and prompt the student to make a selection. This brings in their intuitive understanding of more.
I think use the term “more than” by pointing to the plate with more and explain “this plate has morethan this other plate.” I go on to use the quantities.
Finally, I introduce the symbol to represent this situation.
Below is the example my 3rd grade son used to explain less than to a classmate with autism. This method worked for the classmate!
Very clever activity implemented by the teacher who runs the Life Skills program at our school. She created envelopes (below) for each teacher. The envelopes do not contain any content but are used for practice sorting mail for the students in the program. The students in the program sort and deliver them to our mailbox. We return them to this return bin for reuse.
Such experiences should be available to all of our students who are more severely impacted. Many will need YEARS of practice to develop skills which means a transition program from 18-21 years old may not be enough.
Learning is not a singular threshold to be met. There are different levels of learning – a continuum (see photo below taken from the book Teaching Mathematics Meaningfully).
A student demonstrating proficiency (fluency) is far different from a student simply showing some level of understanding (acquisition). I remember learning to drive a car with a stick shift. During acquisition (initial understanding) I was looking down at the pedals and the stick shift as I thought through the steps. It is not surprising that many students who only show acquisition of a math topic soon forget it. Despite this, the acquisition stage is often were math in schools resides.
This extends beyond math fact fluency to all math topics and the students should take the next step and demonstrate maintenance. To do this, I recommend that a curricular based assessment be given a couple of weeks after a student initially showed what is considered mastery – the student successfully performing problems aligned with a given math objective.
Below is a excerpt from the book with an explanation of the topics. I use this text in the math for special ed courses I teach at different universities.
In the past year I have helped two 7th grade students who are categorized as twice exceptional (2e). Both had more severe math anxiety that impacted their performance and masked their ability. When we started both were working on elementary school level math. Within a couple of months both were working on algebra. (Both had gaps but I was testing their ability by test running higher level math with them.)
As I shared in a previous post my approach is to focus on meeting needs. I want to elaborate on this. My secret is I listen to the student… In other words, the student drives the instruction.
Here’s an analogy. You go to a frozen yogurt or ice cream store and they offer you a sample. You try a couple then go with the one you like. That’s what I do. I try out different types of instruction (samples of the ice cream) and the student tells me (verbally or by the response to the instruction) which one they want. That is the I in IDEA and in IEP.