Making Sense of Testing

Testing (results shown on the Present Levels of Performance page shown below) is often confusing for parents, especially in regards to math. The results are often reported in broad terms, e.g. computation or IQ.

Standardized testing

Here is an analogy for the testing (in terms usefulness for determining instruction, performance and achievement). We go to the DMV and have to take an eye test. That test is used to determine if we have the physical ability to drive or what we need to ensure we have the physical ability to drive. If our vision is diminished maybe we need glasses in order to drive.


 Passing the vision test does not mean we are ready to drive. It means we have the potential to drive. In order to determine if we can actually drive we take a driver’s test.

learning to drive\

Similarly, in order to determine what we can actually do in math we need to take a math test (quiz, checkpoint or some type of curriculum based assessment).

Below is a problem aligned with the Common Core of State Standards for Math. I used it as part of a curriculum based assessment to determine the student’s current ability or present level of performance. She had all types of  standardized testing results on record but I needed to know if she could pass the actual driver’s test.

CCSS assessment



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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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Geometry Application

Obtuse angle on the left (see upside down “T” figure) and perpendicular lines (right angle) on the right.

how to walk on ice

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Data Collection for IEP Objectives

Here is an example of what data collection can look like. (The IEP objective should have been indicated on here as well.) It shows the data, any prompting from the teacher (P with a circle around it), notes and at the bottom is 3/9 for 33% correct.

Also note that I was working on finding the value of a set of nickels and pennies only before moving onto other combinations of coins and more coins.

data collecton for counting nps

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Mailbag Jan 29 2018

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!

dougs question about axis of symmetryaxis of symmetry problem broken down


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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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Submit Questions for the Daily Mailbag

If you have questions about math support, services or strategies share them using comment bar below or email me. I will answer as many questions as I can get to.

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Cutting Up the Math Into Bite-sized Pieces

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.

bite sized pieces

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.

Image result for task analysis

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.

soft cream icecream cone task analysis

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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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