IXL.com is a site that provides online practice for math (and other topics). It has a hidden feature that allows for very effective differentiation. This can be highly useful in a general ed math class and in settings for special education services. This includes special ed settings with students working on a wide ranges of math topics, for algebra students who missed a lot of class or enter the course with major gaps, and for the general algebra population to meet the range of needs. IXL can be used before the lesson or after, for intervention.

By way of example, assume you have a student or students working on graphing a linear function using an XY table (image below). Using a task analysis approach, this topic can be broken up into smaller parts: completing an XY table, plotting points and drawing the line, interpreting what all of this means. I will focus on the first two in this post.

IXL has math content for preschool up to precalculus. For the topic of graphing (shown above) many of the steps are covered in earlier grades. For example, plotting points is covered in 3rd grade (level E), 4th grade (level F), and 6th grade (Level H). To prepare students for the graphing linear functions, they can be provided the plotting points assignments below to review or fill in gaps.

The tables used to graph are covered starting in 2nd grade (level D) and up through 6th grade (level H). These can also be assigned to review and fill in gaps.

When it is time to teach the lesson on graphing a linear function, IXL scaffolds all of the steps. For example, the image below in the top left keeps the rule simple. The top right image below shows that the students now have an equation in lieu of a “rule.” The bottom image below shows no table. All 3 focus on only positive values for x and y before getting into negatives.

The default setting on IXL is to show the actual grade level for each problem. I did not want my high school students know they were working on 3rd grade math so I made use of a feature on IXL to hide the grade levels (below), which is why you see Level D as opposed to Grade 2.

In education, math especially, there exist a learning situation I call the patting head and rubbing belly phenomena. In this phenomena students are presented a math problem that consists of several steps they know how to do and then maybe one or two additional steps that are new. Adding the additional step is like adding the task of patting your head while you rub you belly. The additional math step seems so simple, but attempting it simultaneously with an additional task can make the entire effort exceedingly challenging. A related scenario is generalization to different settings, but that is different. This is true for all types of math, whether it is the general curriculum or life skills/consumer math.

This phenomena plays out in life skills math or consumer math in a stealthy manner because the steps or tasks seem so simple. For example, many of us have worked with a child or student who was learning to count money. When learning about a nickel or a quarter, the coin name and value are easily identified. Once both are introduced, many students confuse the two and may even freeze while attempting the work with the coins.

There is an ABA based process for addressing this using a task analysis and chaining in which steps are worked on in isolation before connecting (chaining) the steps together (and not all of them at once until the end). One related strategy to help implement this approach is through scaffolded handouts in which the steps are enumerated and the structure of the handout isolates the tasks. I have used this approach for 1 to 1 correspondence up to AP Statistics (see below).

When working out a draft of an IEP, I suggest having the task analysis and chaining explicitly identified in the accommodations page and ask for an example of what this looks like (using an example math topic).

There are numerous hidden tasks that we undertake while at the grocery store. We process them so quickly or subconsciously that we are not aware of these steps.

As a result, we may overlook these steps while educating students on life skills such as grocery shopping. Subsequently, these steps may not be part of the programming or teaching at school and therefore generalization is left for another day. Yet, the purpose of IDEA is, in essence, preparing students for life, including “independent living.”

To address this, we can take a task analysis approach in which we break down the act of shopping at a grocery store into a sequence of discrete steps or tasks (see excerpt of the task analysis document below).

Step 1 is to administer a baseline pretest during which we start with no prompting to determine if the student performs each task and how well each is performed. As necessary, prompting is provided and respective documentation is entered into the table (to indicate prompting as opposed to independent completion). For example, I worked with a client who understood the meaning of the shopping list but started off for the first item without a basket or cart. I engaged him with a discussion about how he would carry the items. At one point I had him hold 7 grapefruits and it became apparent to him that he needed a cart. (I documented this in the document.)

Other issues that arose were parking the cart in the middle of the aisle, finding the appropriate section of the store but struggling to navigate the section for the item (e.g. at one point I prompted him to read the signs over the freezer doors), and mishandling the money when prompted to pay by the cashier announcing the total amount to pay.

Step 2 is to identify a task or sequence of tasks to practice in isolation based on the results of the pretest. For example, this could involve walking to a section of the store and prompting the student to find an item. Data collection would involve several trials of simply finding the item without addressing any other steps of the task analysis.

Step 3 would be to chain multiple steps together, but not the entire task analysis yet. For example, having the student find the appropriate section and then finding the item in the section.

Eventually, a post-test can be administered to assess the entire sequence to identify progress and areas needing more attention.

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.

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.

Problems like the addition problem below are often viewed by adults as straight forward. This perception can make it difficult for adults, including teachers and even special education teachers to help students who struggle with it.

I find that the math teacher candidates and special education teacher candidates struggle with breaking down math topics, especially “easy” ones like the one below, into simple steps. To help students who struggle with math breaking down the math topic is imperative. The analogy I use is to break the topic down into bite-sized pieces like we cut up a hot dog for a baby in a high chair.

For new teachers I use a formal task analysis approach to teach candidates how to cut up the math into bite-sized pieces. A task analysis for the problem above was an assignment given to a group of graduate level special ed candidates. As is common, they overlooked many simple little steps hidden in the problem. These steps are hidden because they are so simple or so automatic in our brains that we don’t think about them. See below for how I break this topic into several pieces or steps. For example, before even starting the addition the person doing the problem has to identify that 43 is a 2-digit number with 4 in the TENS place and 3 in the ONES place. Understanding that the problem is addition which entails pulling the numbers together to get a total (sum) is an essential and overlooked step. If a student struggles with a step the step can be addressed in isolation, as I show in another blog post.

A reader asked about an algebra 2 problem and shared (below) his effort to cut up the math into bite-sized pieces. I greatly appreciate his effort because he is trying to meet student needs. While this post is very “mathy” I want to make a couple of points to the readers. First, I wrote out a detailed response (2nd photo below). Second, in both of our efforts we attempted unpack as much as possible. This is what our students need. Also, the reader is developing his ability to do this unpacking and if he continues he will become increasingly more adept at this skill (growth mindset). That means his future students will benefit!

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.

The purpose for having this website is to share my approach to teaching math. The approach is the use of special ed principles brought to bear on math. Specifically, I use a task analysis approach to break down a math topic into “bite-sized” pieces and to use a variety of instructional strategies and reinforcement to move the student through the individual tasks towards mastery of the math topic (including conceptual understanding).

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.