Connections: Standards ~ Goals ~ Teaching
In the previous posts, we considered a math standard for Grade 6, and unwrapped it to identify key vocabulary, skills and concepts required for mastery of the standard.
6.NS.a.1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.
We determined that, in order to demonstrate mastery of this standard, a student would need to
- know and understand (as evidenced by correctly using) the terms fraction, dividend, divisor and quotient;
- demonstrate understanding of the concepts of fraction, quotients of fractions, word problems as a representation of division problems involving fractions, and the concept of division;
- interpret and compute quotients of fractions;
- solve word problems involving division of fractions by fractions.
Next, we considered what obstacles to learning students might have, which would interfere with successful mastery of the standard. To do this, we thought about the kinds of questions students asked when working on work based on this standard.
Now that we’ve analyzed the standard, and anticipating where students might struggle with foundational skills, concepts, and vocabulary, it’s time to consider a student with disabilities, and determine the specific areas of support that this student might have.
Case Study: Student K, Grade 7:
Student K is a 7th grade boy in a self-contained, alternative placement. His most recent standardized math testing shows him to be near grade-level, in all areas of math. He made great academic gains over the past school year, as he worked to gain control over his own behavior.
In order to use this standard as a way to identify specific IEP goals for Student K, let’s take a look at some of our observations of him over the past year:
It typically takes a great deal of time for Student K to transition between areas and begin work, especially first thing in the morning: most days, it is 30 minutes before he begins his morning work, during which time he swears at staff, walks out of the room, and sometimes shrieks in the stairwell. When he’s ready, he sits down to work, and can work independently on the task through to completion.
- Issue #1: difficulty transitioning between activities, especially when they are in different areas
- Issue #2: difficulty with task initiation, even when the task is within his capability
Neither of these issues are specific to this standard; in fact, they persist across all content areas and across time, making them important areas for IEP Goals.
Once Student K starts working, he works VERY slowly – on everything. His work output is very low, although he works steadily, requires little help, and produces high-quality material. His Woodcock-Johnson IV scores show a low-average IQ but extremely low processing speed. Scores on classroom assessments show him at risk due to lack of fluency (computational, math fact, reading), but the only area of fluency he actually shows deficits in is rate: his accuracy, prosody and understanding are grade-appropriate.
- Issue #3: minimal work output due to slow processing speed
Again, while this issue isn’t specific to this standard, it will impact his learning across content areas, and bears greater emphasis in his IEP.
When working, Student K appears to lack a repertoire of known mathematical procedures (e.g., long division algorithms, solving for an unknown, computing using order of operations). Once he has instruction, he learns the new procedure quickly, and can use it correctly. He tends to choose numerical models (i.e., equations) to solve math problems, and avoids visual models, even when they would be more efficient.
- Issue #4: lack of procedural knowledge and fluency
- Issue #5: difficulty using different modes of representing and solving the same problem
If we work under the hypothesis that Student K’s behavioral issues interfered in the past with his ability to be available for grade-level learning, it isn’t surprising that he lacks a repertoire of basic math strategies to draw upon. We can assume that this is a persistent issue with him, and worthy of an IEP Goal.
In general, Student K is fluent in his basic math facts, and can compute whole numbers and decimals accurately. His decoding and comprehension of grade-level text, including word problems, is adequate. He works independently, and learns new material quickly. We can feel that, with the proper supports, he will be well able to master grade-level content.
The Standard and the Student
Now, let’s return to our standard, and figure out what parts of the standard he might have trouble with:
- Procedures for computing fractions: dividing by multiplying the reciprocal, representing the problem with a visual model
- Identifying and representing the problem to be solved in a real-world situation or word problem involving mathematics: restating the problem in his own words; representing the problem using numerical (equations) and visual (area) models.
- Starting a task on time
- Completing a task within the allotted time
Of these three areas, only one is specific to the standard (#1), but, since fractions represent an area where so many students have difficulty, and since grasp of fractions by 6th grade is important for later math courses, it is worth focusing more on in his IEP goals.
Our Standards-based IEP Goals
To address the specific learning barriers we mentioned above, we write the following goals (the specially-designed instruction is in italics):
- Goal #1: Given 2 fractions with 2, 3, 4, 5 or 10 as the denominator, Student K will accurately compute the fractions, using numerical and visual models, with 75% accuracy.
- Objective #1.1: Given 2 fractions with 2, 3, 4, 5 or 10 as the denominator, Student K will accurately divide the fractions, by multiplying the first fraction by the reciprocal of the second, with 75% accuracy.
- Objective #1.2: Given 2 fractions with 2, 3, 4, 5 or 10 as the denominator, Student K will accurately divide the fractions, using an area model, with 75% accuracy.
We have limited the types of fractions Student K needs to use to demonstrate understanding of the standard, and have specifically named one new visual model we want him to use when computing.
- Goal #2: After reading a word problem at the 5th grade reading level, Student K will explain the problem to be solved, and represent the problem using numerical and visual models, with 75% accuracy.
- Objective #2.1: After reading a word problem at the 5th grade reading level, Student K will verbally restate the problem to be solved, in his own words , with 75% accuracy.
- Objective #2.2: After reading a word problem at the 5th grade reading level, Student K will represent the problem to be solved, using an appropriate equation , with 75% accuracy.
- Objective #2.3: After reading a word problem at the 5th grade reading level, Student K will represent the problem to be solved, using an area model , with 75% accuracy.
Again, we have controlled the level of the text to focus K’s energy on the content, have let the student explain verbally (rather than in writing), and have specified a new procedure (the area model) to add to his repertoire.
We noted areas not based on this standard, but which would definitely affect Student K’s performance of this (and many other) tasks. We will also include IEP goals on transitioning appropriately from one area to another, and on starting a task within 5 minutes, and will add accommodations that Student K has a reduced number of items to finish per task, and/or extended time to complete tasks.
Summing up our Standards Work
As previously stated, you would not unwrap every standard for a grade level. However, simply choosing one standard in a problem area (use the reports from your standardized testing to help you choose), and doing this exercise will help you focus in on what your student’s most important obstacles to learning are.
Have you ever unwrapped standards?