Category Archives: Uncategorized

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.

dmv-reader

 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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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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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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Rate of Change in Real Life

61 cents per ounce is a rate of change. Graph the line modeled by this (y intercept is 0) and it becomes slope of the line. In referring to algebra we often hear, “when will I ever need this?” My response is “all the time!” Our job as teachers is to make this connection for students.

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Strategy to Individualize Instruction

It is difficult to individualize instruction in a whole class or small group setting. I created and taught the curriculum for a Consumer Math course at the high school where I teach. For a class of 10-12 students, all with an IEP, I developed an approach that allowed me to individualize the instruction for each students.

In the photo below is an example of a folder set up I used with the students in Consumer Math. Each student would have a dedicated folder, kept in the room and updated daily. The smaller paper shows the individualized agenda. The other paper shows an example of how the folder can be used as a resource. Student computer login information, accommodations like a multiplication table or notes can be secured inside the folder. The agenda would be changed out each day. (In case you are wondering about the label in the agenda, “Math Group 4.” This particular folder was used in a special education training session for teacher candidates.)

individualized folder

 

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Videos Making Algebra Accessible

Below is a screen shot of a video in a series of videos that provide instruction on algebra topics. The videos are designed to make algebra more accessible for almost all students.

screen shot relations video

The presentations include the following instructional strategies

  • A focus on conceptual understanding (not just teaching steps)
  • Connection to prior knowledge
  • Breaking the math topics down into “bite-sized” pieces (chunking)
  • Color coding
  • Making the math relevant

The videos can be used in the following ways

  • Differentiation for students who need an alternative presentation
  • Initial instruction for students who missed instruction
  • Initial instruction as part of a class, e.g. flipping a classroom
  • For use when a substitute is covering a class
  • Intervention based instruction
  • Part of math support services (especially for special ed teachers who are not well versed in algebra topics)
  • Homework support

The videos include a link to the handouts used in the presentation. Additional practice worksheets will be included as well.

NOTE: this is only a sample, with more samples to follow. Please share feedback or ask questions.

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Introduction to Imaginary Numbers

The topic of imaginary numbers is one of the most abstract and therefore difficult math topics to teach in algebra. Here is how I introduce it to students (emphasis that this is only an introduction).

0110190730_2 (1)

I write 1, 2, 3… on the board (see photo above) and explain to the student “at some point in life you learned to count on your fingers, 1, 2, 3…” These are called the Natural numbers.

toddler counting

Then I explain, “later you were told that no cookies means ZERO cookies. Zero is a new type of number. We call 0, 1, 2, 3… the Whole numbers. You learned a new type of number.”

This continues, “A little later on you were told you could have half a cookie and so you learned about a new type of numbers called fractions.”

This continues with negatives. Then I explain that all these number types can be found on the number line. We call the set of all of these numbers Real Numbers.

I conclude with “Now we have a new type of numbers that are not found on the number line. These are called imaginary numbers. Just like before you had number types you had before and now you have a new one to learn.”

The point of this approach is to help the students understand that a new number set simply builds on previous number sets. Also, the students have encountered this situation before.

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Summer Math Intervention Sessions

Overview

A graduate level math intervention course at USJ has a practicum component in which graduate students work 1 on 1 with students with special needs over a 5 week period. An experienced math interventionist oversees the work. The focus of the sessions is on an identified math objective selected by the family or teacher of the student with special needs in math (students with IEP or 504, who are receiving extra services like SRBI, who are below grade level or who have a history of struggling with math)

Each graduate student, in collaboration with the supervising instructor, conduct initial assessment, implement a variety of instructional strategies and conduct ongoing progress monitoring and provide lots of objective data!

Timeline

The sessions run 5-6PM every Thursday from July 12 to August 9 at the USJ campus.

Benefit

  • Parents are provided objective data from curriculum based assessments that can be provided to the home school educators.
  • Parents are provided a portfolio of an array of instructional strategies. This could provide insight for the home school educators on what works for the student with special needs.
  • Parents observe the sessions to gain first hand knowledge of how their student with special needs engages in math and knowledge of what effective math services look like.
  • At the end, the graduate student will meet 1 on 1 with the parents to explain the assessment data and instructional strategies and will provide a comprehensive report.
  • At the end, the instructor will meet with parents to explain how to leverage the information provided by the graduate students with the home school educators, especially PPT or 504 teams.

Next Step

If you are interested, please contact Randy Ewart (email address below). Note: these sessions are designed to benefit both the student with special needs and the graduate students. Hence, families who participate are expected to attend all 5 sessions.

Instructor

The instructor is Randy Ewart, a veteran math teacher and math interventionist who received his master’s in special education at USJ. He has provided consultation and services to multiple districts and over 100 students with special needs to make math accessible. You can visit his blog for examples of his work: www.ctspedmathdude.com. Email him at ctspedmathdude@gmail.com

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RTI – Response to Intervention

RTI Process

The photo above shows a model of the RTI (called SRBI in Connecticut) process. RTI is a systematic approach to addressing student academic needs. Here is a link to a video explaining the process and below is an outline of the process:

  1. Students are served in a classroom that provides high quality initial instruction. This includes the use of UDL, differentiation, formative assessment, instructional strategies to make content meaningful and concrete and to meet student needs in general. The general classroom is Tier I.
  2. Assessment is used to evaluate student progress AND the effectiveness of the instruction. If students are not understanding a math topic or unit (as demonstrated by data not observation) the student can be moved into Tier II which involves intensified focus of instruction and in a small group.
  3. Assessment is used again. If the student is not making sufficient progress despite changes in instruction the student can be moved into Tier III which involves maybe 1 on 1 or 1 teacher and 2 students. The level of intensity is ramped up further.

Here are a couple of key components:

  • The initial classroom includes an effort to meet individual needs.
  • Data is the key to decision-making. Assessment is objective.
  • The programming is evaluated using the data.
  • RTI is included in IDEA 2004.
  • Student placement at the different tiers is fluid. Students are moved into and out of tiers based on data.

 

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