Category Archives: Connections to Prior Knowledge

The I in Instruction can be the same I in IEP and IDEA.

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).

2018-07-26 17.13.05.jpg

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.

2018-07-26 17.41.15


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More Than: Concept and Symbol

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.

alligator eats bigger number

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.

3 dots 5

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.

3 less than 5.

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.

concept of more chicken nuggets just plates

I think use the term “more than” by pointing to the plate with more and explain “this plate has more than this other plate.” I go on to use the quantities.

concept of more chicken nuggets more than words

Finally, I introduce the symbol to represent this situation.

concept of more chicken nuggets more than symbol

Below is the example my 3rd grade son used to explain less than to a classmate with autism. This method worked for the classmate!

Lucas less than example

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Sausage Fractions – Real Life Example

I have 3 kids and was cooking sausage for them.

2019-01-31 07.46.25.jpg

There were 5 sausage links available (below). How do I give each the same amount? Fractions!


Each child gets a full sausage link.


I then cut  the remaining 2 sausage links into 3 parts, 1 for each child. 1/3 of a link.



Each child gets 1/3 and another 1/3 or 2/3. So they get 1 full link and 2/3 of a link or 1 2/3. This is an entry point into mixed numbers (whole number and a fraction).




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Common Denominator – Why?

We explain steps in great detail to students but often omit the underlying concept. The topic of adding or subtracting fractions with unlike denominators is an example of this.

subtracting unlike denominators     adding unlike denominators

The example above right is a short cut for what is shown above left. These short cuts, which math teachers love to use, add to the student’s confusion because these rules require the student to use rote memorization which does is not readily retained in the brain.

I suggest using what I call a meaning making approach. I present the student 2 slices of pizza (images courtesy of Pizza Fractions Game) and explain the following setting. “You and I both paid for pizza and this (below) is what we have left. You can have the pizza slice on the left and I will have the pizza slice on the right. Is that OK?” The student intuitively understands that it is not because the slices are different sizes. I then explain that when we add fractions we are adding pizza slices so the slices need to be the same.

fourth pizza slice         half pizza

I then cut the half slice into fourths and explain that all the slices are the same size so we can now add them. Then the multiplying the top and bottom by 2 makes more sense.

fourth pizza slice          2 fourths pizza slices



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Concepts vs Skills – Need Both

In general math is taught by focusing on the steps. Conduct a Google search for solving equations and you will see the steps presented (below). You need a video to help your student understand solving and you typically get a presenter standing at the board talking through the examples. (I’ve posted on my approach to solving equations.)

When the math is taught through the skill approach the student may be able to follow the steps but often does not understand why the steps work (below). The brain wants information to be meaningful in order to process and store it effectively.

calvin hobbs toast

To help flesh this situation out consider the definitions of concept and skills (below). Concept: An idea of what something is or how it works – WHY. Skill: Ability” to execute or perform “tasks” – DOING.

definition conceptdefinition skill

Here is how the concept first approach can play out. One consultation I provided involved an intelligent 10th grader who was perpetually stuck in the basic skills cycle of math (the notion that a student can’t move on without a foundation of basic skills). He was working on worksheet after worksheet on order of operations. I explained down and monthly payments then posed a situation shown at the top of the photo below. I prompted him to figure out the answer on his own. He originally forgot to pay the down-payment but then self-corrected. Then I showed him the “mathy” way of doing the problem. This allowed him to connect the steps in solving with the steps he understood intuitively, e.g. pay the $1,000 down payment first which is why the 1000 is subtracted first. Based on my evaluation the team immediately changed the focus of this math services to support algebra as they realized he was indeed capable of doing higher level math.

solving equation with conceptual understanding first

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Mailbag Jan 26, 2019

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.

Image result for for in the road

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).

mapping traditional

comic book mapping

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.

Inclusion is not proximity.


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Retaining Information

Below is a model for information processing (retention and retrieval). Here are a couple key points I want to highlight:

  • A lot of information is filtered out so what gets through? Information that is interesting or relevant.
  • Information that is connected to prior knowledge, is relevant or that is organized has a better change of being stored effectively for retrieval.
  • Working memory has a limited capacity. Consider what happens to your computer when you have a lot of apps open. Your computer may start to buffer which is basically what happens to our kiddos if instruction involves opening too many apps in their brains.
  • Long term memory is basically retrieval of information. Think a student’s book bag with a ton of papers crammed in it. How well can he or she find homework? Compare this to a well maintained file cabinet that has a folder labeled homework with the homework assignment in question stored in this folder. That paper is much easier to retrieve. This is analogous to long-term memory. If the information is relevant or meaningful it will be stored in the file cabinet folder and more easily retrieved. In contrast, rote memorization like the rules teachers present students are papers crammed into an overflowing bookbag.

Information Processing.jpg

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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.

intro to solving equations

2019-01-18 22.16.51-2

First I develop an understanding of a balanced equation vis-a-vis an unbalance equation using the seesaw representation.intro to equations balanced vs unbalanced

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

intro to solving equations balance and unbalanced

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.

intro to solving equations adding

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

solving 1 step equations add scaffolded

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.intro to solving equations multiplying

The students are then provided a scaffolded handout followed by a regular worksheet.solving 1 step equations multiply scaffolded

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Trick for subtracting integers

I have algebra students, in high school and college, who struggle with evaluating expressions like 2 – 5.  This is a ubiquitous problem.

2-5 problem

I have tried several strategies and the one that is easily the most effective is shown below. When a student is stuck on 2 – 5 the following routine plays out like this consistently.

  • Me: “What is 5 – 2?”
  • Student pauses for a moment, “3”
  • Me: “So what is 2 – 5?”
  • Student pauses, “-3?”
  • Me: Yes!


I implemented this approach because it ties into their prior knowledge of 5 – 2. It also prompts them to analyze the situation – do some thinking.

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Example of Using Support Class to Support Current Math Content

I have posted on how to effectively provide support for current math topics. Here is an example (below) of how support can focus on both the current topic and prerequisite skills.

For example, on the 22nd in this calendar the current topic is solving equations. The steps for solving will include simplifying expressions and may involve integers. The support class can address the concept of equations, simplifying and integers which are all prerequisite skills from prior work in math.

examples of support class prerequisites

This approach allows for alignment between support and the current curriculum and avoids a situation in which the support class presents as an entirely different math class. For example, I recently encountered a situation in which the support class covered fractions but the work in the general ed classroom involved equations. Yes, equations can have fractions but often they do not and the concepts and skills associated with the steps for solving do not inherently involve fractions.

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