Tag: mastery

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Viability of a Grade Level in Math

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.

Reading Rockets

Grade Level

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.

Addressing Grade Level Metric

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.

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Asking for Examples of Mastery for IEP Objectives

To ensure the IEP team is on the same page as to what mastery of an objective looks like, the person writing the objective can take two steps:

  1. provide an example problem that would be used to assess mastery (and the example problem would have the same language as used in the objective)
  2. provide an example of a response to the example problem cited above that would be considered mastery level work

The graph below is not data. A graph is a representation of summary statistics. This summarizes the data.

The chart below does not show the actual prompts, e.g. what number was shown to Kate, but it does show the individual trials. This is data, with a summary statistics at the end of each row. Here is a link to more discussion about data, with an example of a data sheet I use.

 

The data shown below addresses the student’s effort to solve an equation. Problem 21 is checked as correct and the error in problem 22 is identified. I can use this data to identify where the student is struggling and how to help. NOTE: the math objective would use the same verb as the problem: solve the linear equation.

 

The excerpt of a data sheet, shown below shows trials in a student’s effort to compare numbers.

 

Data below shows a student’s effort to evaluate integer expressions.

 

This applies to all areas beyond math. The chart above or the data sheet I linked above show data sheets that indicate the prompt and the results, with notes. For example, if I am asking my son to put on his shoes, each row of the data sheet is a trial with the outcome and notes.

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Levels of Learning

Learning is not a singular threshold to be met. There are different levels of learning – a continuum (see photo below taken from the book Teaching Mathematics Meaningfully).

continuum of learning

A student demonstrating proficiency (fluency) is far different from a student simply showing some level of understanding (acquisition). I remember learning to drive a car with a stick shift. During acquisition (initial understanding) I was looking down at the pedals and the stick shift as I thought through the steps. It is not surprising that many students who only show acquisition of a math topic soon forget it. Despite this, the acquisition stage is often were math in schools resides.

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This extends beyond math fact fluency to all math topics and the students should take the next step and demonstrate maintenance. To do this, I recommend that a curricular based assessment be given a couple of weeks after a student initially showed what is considered mastery – the student successfully performing problems aligned with a given math objective.

Below is a excerpt from the book with an explanation of the topics. I use this text in the math for special ed courses I teach at different universities.

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