Category Archives: special education in general

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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Karen Linder (advocate and parent of a son with a disability) and I are collaborating on a series of Parent User Guide materials for parents. The goal is to scaffold parent efforts to use the IEP. We are presenting documents in smaller chunks to make it easier for parents to understand the steps they can take without being overwhelmed with an already challenging endeavor of advocating for your child’s needs.

Our first installment is part 1 in understanding IEP objectives (see screen shot of front page above). There is a document with templates to use and an accompanying video to provide additional support in how to use the templates. Please share feedback so we can improve our efforts to help parents.

Please feel free to reach out to us if you think we can be of service or if you have questions.

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Real Life Math VS “Mathy” Math

In working with students, parents and IEP teams, I find that there is an assumption that math at some point, possibly beyond arithmetic, is simply a science fiction movie that is minimally related to real life. I hear from students as well as adults, statements like, “algebra, when are we ever going to use it?” My response is, ALL THE TIME!

The math we often present in school is a “mathy” version of the math we encounter in life. For example, the top photo below shows a pizza menu and a situation that is realistic. The calculator screen shot below the menu shows how we likely would solve the problem using a calculator on our phone.

Below is the same type of problem, but solved using “mathy” math. How many of us (besides me) are doing this at the pizzeria?

The point is, we engage in algebra but maybe do not use all the symbols and vocabulary of algebra, e.g. when we typed in 2.25 repeatedly in our calculator, we were working with the math term “slope.”

This has implications for secondary students whose post-secondary plans do not include college. If the math class is teaching “mathy” math but you want your student to learn math as it is used in real life, then an alternative math course is needed. This could be addressed through the IEP.

 

 

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Problems with Determining Math Grade Level for a Student

A common scenario involves a school official reporting out the grade level in math for a student. For example, a 7th grade student I was helping had tested at a 4th grade level. As a result, the student spent much of her 7th grade year working on 4th grade math.

There are a couple problems in establishing a grade level in math. First, unlike reading, math is not nearly as linear. The image below shows a breakdown of the Common Core of State Standards math categories, called domains. In a video, I use this graphic to unpack why it is more challenging to determine a single level of ability for math. In short, the reason is the student could be doing well in some categories and doing poorly in others. Second, the testing used to establish ability level can be problematic for the student. For example, the student may not have the stamina or attention span to endure a longer assessment.

If you are presented with a single grade level as an indicator of math ability, I recommend that you ask for a breakdown by category and how your student will be provided differentiation to address gaps. This is more appropriate than plowing through all of the math at a lower grade level.

 

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Long Range Planning in Regards to Math for Students Receiving Special Ed Services

Below is a photo of a hyper-doc that I use to map out a long range plan for math services and academics for students receiving special education services. Here is a link to a video explaining how the document is organized and how it “works.” (Note, the image of the document on the video is not crisp, so I suggest you look at the handout while watching the video.)

The document contains several links to resources such as videos, websites and blog posts that provide additional information. Feel free to reach out to me using the Contact Form on this page if you have questions or would like input. I am happy to help.

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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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Long Term Effects of Disabilities

Often we view disabilities in the context of the individual as a student, or a child or adolescent. The long term effects may be had to understand or extrapolate based on what we see at the younger ages.

There was teacher candidate whom I trained who had ADHD and struggled in the program in which we worked. He shared his struggles to keep up with the programming, organization, and in general, keeping up with the demands placed upon him.

I askedĀ  him to write a statement explaining his challenges that I could share with others. The statement is shared below. I hope this can help parent and educators make a more refined connection between the setting at an earlier age with the settings and outcomes the individual will face later in life. I explain to sped teacher candidates whom I train that we have an awesome responsibility and opportunity in how we can impact young lives…when they are no longer young.

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Shopping at the Grocery Store

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.

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Asking for Examples of Mastery for IEP Objectives

To ensure the IEP team is on the same page as to what mastery of an objective looks like, the person writing the objective can take two steps:

  1. provide an example problem that would be used to assess mastery (and the example problem would have the same language as used in the objective)
  2. provide an example of a response to the example problem cited above that would be considered mastery level work

The graph below is not data. A graph is a representation of summary statistics. This summarizes the data.

The chart below does not show the actual prompts, e.g. what number was shown to Kate, but it does show the individual trials. This is data, with a summary statistics at the end of each row. Here is a link to more discussion about data, with an example of a data sheet I use.

 

The data shown below addresses the student’s effort to solve an equation. Problem 21 is checked as correct and the error in problem 22 is identified. I can use this data to identify where the student is struggling and how to help. NOTE: the math objective would use the same verb as the problem: solve the linear equation.

 

The excerpt of a data sheet, shown below shows trials in a student’s effort to compare numbers.

 

Data below shows a student’s effort to evaluate integer expressions.

 

This applies to all areas beyond math. The chart above or the data sheet I linked above show data sheets that indicate the prompt and the results, with notes. For example, if I am asking my son to put on his shoes, each row of the data sheet is a trial with the outcome and notes.

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