## 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.

## Add vs Subtract Concepts – Kinesthetic Approach

In working with students who have fallen significantly behind in math, a common challenge is adding and subtracting. This can manifest in word problems or simple add and subtract problems. When this is the case, the first thing I check is whether the student understands conceptually what addition and subtraction are. Here is a Google Slides file that shows the approach I use. (You can make a copy and then edit.) I copy and paste slides for subsequent days so the Google Slides file services as a repository of the trials – data collection.

Each slide as the same format.

• The operation in bold font.
• The primary pile with the objects to work with.
• The secondary pile with objects used for addition prompts.
• The garbage can for objects discarded in subtraction prompts.

I like to use pennies as objects but will use Google Images of topics the student likes if I need something extra to keep their interest and attention. The student is tasked with performing an action with the objects and to distinguish between tasks to show discernment of what action to perform, relative to the prompt. There are 3 operations types: addition, subtraction, and sorting. The sorting is used simply as a distractor. Each image shows the original slide and a slide of the final product. The slides can be copied to for additional prompts. This first round focuses on common language that speaks to addition and subtraction.

In the next round, the sorting is removed and the language is focused on the terms add and subtract as a step in shaping understanding of the eventual symbols.

Finally, the actual symbols are used. If the student gets confused the previous language can be used as a prompt.

## Multi-step Word Problems – Create Groups of Items

Here is a Google Slides file as a follow up to the Multiplication Word Problems Matching and Creating Groups post.

Each slide has a multi-step word problem (multiplication and either addition or subtraction) that continues the use of the grouping approach. The boxes (for groups) and dots (for items) and dynamic and can be copied as needed. I suggest having an example that can be a We Do to guide the students through the use of this application. For subtraction, groups of items can be created and the dots taken away and maybe changed to red.

## Progress, but in What Direction!?

In the video, Charles Barkley has made great progress getting to Annapolis for the NCAA Basketball Tournament. Problem is, the tournament was in Indianapolis.

For obvious reasons, in special education we frequently discuss and recognize progress. As in the commercial, there can be a lot of progress, but in what direction? Is it moving the student towards postsecondary goals the family has established and will the student be prepared for postsecondary life?

When I work with IEP teams or with families, I look to establish the postsecondary goals and evaluate progress accordingly. Often, I am called in to help when a student is in middle or high school and is years behind in math. With only a few years remaining with support through IDEA, it is crucial that progress be evaluated in terms of a long range plan that gets the student ready for postsecondary life and the student is ready BEFORE leaving high school (or a transition program). In other words, the student arrives in Indianapolis and is there before the game starts.

## College Preparation and Retests

In a recent Facebook group for high school math teachers, an interesting discussion arose about retests and how to respond to students who have a low score on a given test. A common perspective that I shared for a time is that we should eschew retests as we are preparing students for college. In this post I offer counter points to this argument, and attempt to unpack the college preparation issue. The focus is on math.

It appears that colleges universally have placement tests. Below are excerpts from universities in Florida, Texas, California, Washington, Nebraska, and Connecticut, along with a sprinkling of 2 year schools from some of these states. If a student makes an A in precalculus in high school, that does not qualify the student to take calculus or even to retake precalculus! A test score is required, whether it is the SAT or the college’s placement test. Indirectly, this appears to show that the colleges are not taking high school results at face value. Whether the student took a retest or not, or didn’t need one he or she still must prove content mastery to some degree.

Then consider what is happening at colleges. Dr. S.A. Miller of Hamilton College wrote an essay about grades in college (excerpt below left). In it, she cites Dr. Stuart Rojstaczer‘s article in the Christian Science Monitor about grade inflation in college. The focus here is not on how the grades are inflated, e.g., with a retest, but the evidence shows grades skewed towards the higher end and has increasingly become more pronounced over the past few decades. The rigor and accountability cited in the argument against retests is not what it appears.

Given the need to demonstrate content mastery through placement tests, and college academic expectations that are not quite what they seem, it makes sense to focus on mastery of content. This is illustrated by what is happening in Connecticut, a state that ranked 3rd in US News and World Report state K-12 education rankings and 2nd in Wallet Hub’s rankings of state school systems.

In a Connecticut State Board of Education report, over 40% of students entering a state community college or state university (not including UCONN) needed to take a remedial course.

If this is happening in Connecticut, it may shed light on what is happening in other states. It certainly appears that there is something not working in terms of content mastery in math. I am not attempting to place blame and certainly do not suggest that the problem is a lack of retests. Another issue is self-help study skills and how well students are playing their part of the learning process. A report of a survey conducted by Manchester Community College in Connecticut included the most common reason people struggle in their classes. The 2nd most cited response, by 60% of faculty and 61% of students, was that the students do not know how to study effectively.

It appears the issue of retests is more an issue of why is there a need for retests to be considered and the implications of not filling in the content knowledge gaps. I will conclude with a bit of irony. UCONN is ranked as the 23rd best public university. Students are allowed 3 attempts on the math placement test.

## In working with students with special needs on math programming and services, a common and important issue is that the student is behind and there is a tension between math intervention to fill gaps and addressing ongoing grade level content.

### Unpacking the situation

There is no single grade level for math, as is the case for reading. Math progression is more like a web, not a line. For example, if a student can do 5th grade geometry but only 3rd grade level fractions, do we average out the grade level math to be 4th grade? (No.) Do we identify the student as working at a 3rd grade level? (No.) 5th grade level? (No.)

Like a suitcase, there is a capacity to the daily time a student has for school services. I often encounter situations in which the services recommended involve the student working on grade level content and catching up on the gaps during support time. If the student has only been learning 75% of the math content each year, he or she needs that support time to help learn the new content to get closer to 100%. There is too much being stuffed into the suitcase. Something has to give.

The focus of the services and programming often shifts away from post-secondary plans, with a focus on the short term. Like the situation facing the man in the image below, there are long term implications.

### Recommendations

There are two recommendations I make in regards to addressing the gaps, without overstuffng the suitcase.

• Use triage to shift focus to the priority topics. For example, the parents of a student in 7th grade but working on math from lower grade levels wanted to pursue a math track that would allow the student to go to community college. I mapped out a long range plan (image below) that focuses on algebra as that is the type of math most likely encountered in a math requirement. Here is another plan which was to prepare a student to possibly work in a field related to cars.