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.
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 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.
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).
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.
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.
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.
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.
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.
In the effort to assess student ability performance factors are likely present. It is incumbent upon the educators to mitigate the performance issues to assess true ability.
For example, I conducted an evaluation on a student in middle school who has ADHD. All of her testing records indicated that she would lose focus during the assessment and that the focus was problematic for testing. Before we met I surveyed her on her favorite snack (didn’t know Sour Skittles is a thing), brought this reinforcer along with a bottle of water. She sat through an entire 1 1/2 hour KeyMath Assessment without incident.
Link to Drop Box folder for webinar on Task Analysis
Link to Drop Box folder for webinar on Making Math Meaningful (note: the folder is not populated with handouts with excerpts shown on the video. These documents will be available in the folder by Oct 16.
Teaching students to add appears to be a very linear, skill driven endeavor. Hidden in this is the concept of what it means to add and how to assess this conceptual understanding. Here is an approach to address and assess the concept of adding.
In the photo above a student is prompted to pull both groups into 1 pile (see photo below). The word, add, is not addressed. The symbol is absolutely not introduced yet.
Once the student has demonstrated a consistent performance of pulling the groups into 1 pile (addition) two other tasks are introduced, taking away and sorting. The student is presented each of these individually (field of 1).
After showing consistent performance in demonstration of these skills, the skills are then presented using a generalized mat (see below).
Then two skills as pairs. First “pulling together” and “taking away” are randomly prompted individually, e.g. “pull into a pile” using the generalized mat above. Then combine “pull together” and “sort” then “sort” and “take away.” Finally all 3 are randomly chosen (field of 3).
A key to intervention for math is to drill down into a topic to see which step is causing a student problems. This is a big reason why ongoing progress monitoring is vital to intervention.
In this case a student in a previous session had occasionally added the percent to the dollar amounts – the step that was problematic. He conceptually wasn’t thinking about the meaning of the values but just added or subtracted numbers he saw. In response the next session focused on helping the student discern between the percent rate and the monetary values.
In the photo above is the work of the student as review of the previous session. This was followed by highlighting the dollar amounts in green and the percent amount in yellow. It was emphasized that the yellow was not used in the calculation in the bottom row.
This was followed by the task seen in the photo below. The focus is strictly on the one step that was problematic. This was followed by work on IXL.com (2nd photo below) with the student writing in values on the handout shown on the bottom photo to help the student focus on this tep
List all the steps for the objective. Use this table (above) as a pretest to identify gaps.
Provide instruction on the gaps. In the photo below I used color coding to show what to multiply and scaffolding to align the digits in ONES and TENS place. NOTE: I provide the problems with some steps already completed to focus on the steps for which gaps were identified.
After providing instruction on the steps with gaps data is collected on mastery of these isolated steps. NOTE: The problems are identical in nature to the gaps and the problems used in instruction. (Link to the data sheets – WORD so you can revise.)
The photo shows a pre-posttest for a student in a consumer math class. In the course I taught we would conduct a pretest at the start of the class to determine which of the related skills a student lacked mastery. The course focus for this student was on the identified skills – highly individualized. The assessment also provided present level of performance information, allowed us to monitor progress and to evaluate instruction.