## The effort to provide intervention to fill in gaps is challenging for different reasons. One reason is the effort to balance support for current content while filling in gaps. This post shows an example of how to fill in gaps while working through the current topic.

### Overview

Various rubrics used to assess teacher instruction includes an effort to build on or connect to prior knowledge. If the student has gaps with prior knowledge, the lesson becomes less accessible for students with the gaps. Previously, I addressed how to support both current content and fill in gaps. The idea is to systematically fill in gaps by addressing prerequisite skills as they arise in new lessons.

### Example

The handout out below shows an example of how this can play out. The first page is used as a do now for the content presented on page 2. If you are teaching a student how to solve 1 step equations and are moving into integers, page 1 is a a means of supporting the new content while filling in possible gaps. The first image shows the student will need to evaluate -13 – 3 as part of the solving in the lesson. This can be addressed in the do now, as shown in the 2nd image, page on the right. (Notice all the problems on page 1 are steps to solve on page 2 problems.) This is useful for students with special needs and for differentiation.

## When I am asked to consult or evaluate a student, often the student is years behind in math. As a result, I am often asked to determine the grade level of the student’s achievement. Regressing the math achievement to a single number is not viable. This post provides an explanation.

### Common Scenario

Here is a common scenario. A school official reported out the grade level in math for a student. The 7th grade student tested at a 4th grade level. As a result, the student spent much of her 7th grade year working on 4th grade math. When I started working with her, I discovered that she was very capable of higher level math. Six months later, she was taking algebra 1.

### The Math Spider Web

Unlike reading, math is not nearly as linear. It is more like a spider web of categories (called domains). For example, Geometry is not a prerequisite for Ratios and Proportions and Fractions is not a prerequisite for Expressions and Equations. Geometry and fractions may be included in problems associated with other domains but they are not foundational building blocks.

On the other hand, in reading, comprehension and decoding are essential in all grade levels. Unresolved trouble with decoding in 3rd grade causes major problems in 4th grade and beyond.

A student tests at a 3.2 in reading. This provides a clear picture of where the student is in the progression of reading ability. There are books written at that grade level.

If a student is reported to to test at a 3rd grade level in math, the student may have scored higher than 3rd grade in Geometry, at 3rd grade in measurement and data, and lower than 3rd grade in the other domains. True, in reading we have students who may decode at a high level and comprehend at a low level. That is more specific that sorting through 6 domains in math. Then consider that the comprehensive number of domains addressed by middle school increases to 11.

### The Domains

The image below shows a breakdown of the Common Core of State Standards math domains. In a video, I use this graphic to unpack why it is more challenging to determine a single level of ability for math.

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.

## Enrichment Resources for Math

When I work with advanced or talented and gifted students, I recommend that enrichment be used to “go wide, not deep.” The point is not to simply move the students vertically, but challenge them to go deeper with current content or use critical thinking activities.

Here is an online resource of problems from the Paul ErdÅ‘s International Math Challenge that will indeed challenge almost any student (check the problem below for 3rd and 4th graders).

Here is another online resource of problems from the Art of Problem Solving (AoPS). The link is to a search posts that address practice problems related to the American Mathematics Competition for 8, 10, 12. Below is an AMC 8 practice problem.

## Inclusion vs Proximity

Some educators and parents of students with special needs are unclear about what is meant by the term inclusion. Some think it is having the student with a disability in the same location as “nondisabled peers.” Some think it involves doing the same exact tasks or academic work.

Sesame Street figured this question out years ago. The girl in the red shirt in the video below (video set to start with her) was experiencing inclusion, not because she was next to the other kids. She was not jumping rope but was most certainly included and appeared to love it! (Note: “inclusion” is not defined in IDEA, so formally this issue would be one of least restrictive environment.)

Below is a genius representation of inclusion (not my idea).

It appears that inclusion is sometimes viewed as a dichotomous choice. For example, I observed the student in a school who was the most severely impacted by a disability sitting in a grade level history class during a lesson communism. This was an effort to provide inclusion but was he was experiencing proximity.

Below is an example of inclusion for a student with autism in an algebra 1 class. Below left is a typical math problem. To the right is one I created for the student with autism. It was designed to help him understand the concept of matching inputs and outputs without using a lot of the math terminology. In his case, the focus in math was on concepts.

## IXL.com – Excellent Tool for Differentiation

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.

## Simplifying not so Simple Equation Solving

Several special ed teachers identified solving multi-step equations as the most challenging math topic to teach in middle school math. Here is my approach to teaching multi-step equations like 3m + 4m + 1 = 15. .

First, I use a task analysis approach to break down the math topic like we cut up a hotdog for a baby in a high chair. MOST of the steps involved are prior knowledge or prerequisites skills. I present these in a Do Now (warm up, bell ringer, initiation) – see image below. This allows me to fill in the gaps and to lay the foundation for the lesson. The prerequisite skills include simplifying expressions and solving 2 step equations. I also present meaning for the equation with a relevant real life problem that is modeled by this equation. By attempting the walkathon problem without the “mathy” approach, the students will more likely understand the equation and why they add 3m and 4m.

After reviewing the Do Now I use Graspable Math, which is a free online application that allows users to enter their own expressions and equations. These can be manually simplified and solved by moving parts around. Here is a tutorial on how to do this. This allows them to manually work with the simplifying and the equation before working on the handout, in a concrete-representational-abstract approach.

This is followed by a scaffolded handout with the use of color coding. It is developed using a task analysis approach. I have student work on the first step in isolation as that is the new step (the other steps are prior knowledge and were addressed in the Do Now). This avoids all the work on the other steps that can result in sensory overload and allows me to address mistakes in the new content immediately.

This handout can have the equations removed and be used as a blank template to follow. In turn this would be followed with regular solving worksheets.

## Wicked Cool Simulation Site

I used this site, Explorelearning, with a 7th grader with Aspergers who tested at a 1st grade math and reading level. We used the Photo Synthesis Lab (screen shot below) to gather experimental data on the hypothesis “what helps flowers grow?” He won at the school level and went on to district competition.

(As of April 2020 you can get free 60 day unlimited access.)

## Randy on Dana Jonson’s Podcast: Lots of Information about Math

I was a guest on special ed attorney Dana Jonson‘s podcast. Below are links to items I cited in the discussion. There is a contact information form in this post.

Strategy to individualize instruction

Tandem of Online Resources: IXL.com and Google Classroom

Online Worksheets I recommend

Concepts vs Skills

Here is what we are doing at home during school closure time. I created a Google Classroom (anyone with Google account can create a class on the Google Classroom app) and posted links to IXL topics (photo at bottom). (NOTE: this is one method I use to differentiate at school.) If the school is providing online work, you can enter this into the classroom as well.

## Worksheet Websites I Use and Recommend

The following are screen shots of online math worksheet websites I use. The variety and the options in the criteria you select for your worksheets for some of these sites allows for differentiation in the classroom.

I will start with my favorite site, Math-Aids.com. This site allows for dynamic selection of criteria for each handout (see 2nd photo below) such as choosing the types of coins in problems for counting out the total value. The coin images are outstanding! It also offers content up to Calculus.

Super Teacher Worksheets is often used elementary schools. It offers content in science and language arts as well. It requires a \$25 annual subscription which I easily find to be worthwhile.

Common Core Sheets is very useful site if you want to find handouts for specific standards by grade level (see 2nd photo below). It offers multiple versions of each handout.

Dads Worksheets provides a large bank of worksheets – multiple versions of each worksheet.

Math Worksheets 4 Kids offers multiple versions of each worksheet and content in science and language arts. There are many worksheets that provide unique support in how the work is presented, e.g. the Ratio Slope worksheet shown in the 2nd photo below.

Visual Fractions – title speaks for itself.

Worksheet Works is my 2nd favorite. It offers options in the criteria you choose, e.g. difficulty level (2nd photo below). They also offer unique types of handouts such as a maze with math problems to solve to find the path (2nd photo).