# A Process to Address Unfinished Learning in Middle School Math

Using the "Understand-Diagnose-Take Action" model to address unfinished learning related to 6.NS.C.7.a

Many middle school students haven’t yet mastered the critical prerequisite skills and concepts leading up to their grade-level content. As a result, math educators regularly confront cases where pieces of the conceptual foundation are shaky or missing. However, students must learn grade-level standards to be college- and career-ready. So, what is an equitable response?

In this six-part series, we are going to provide examples and ideas for how math educators might approach the teaching of grade-level standards when students have unfinished learning from previous grade levels. As former math teachers ourselves, we are committed to helping current teachers find ways to “bridge the gap” in order to provide students the meaningful, grade-level instruction they deserve, while ensuring they receive support to access it.

Many educators have found this three-part framework to be a helpful starting point, so throughout the series, we will follow the Understand-Diagnosis-Take Action process as a guidepost. Together, we worked through the process with six middle school standards, selecting standards for two specific reasons: (1) students commonly have unfinished learning with prerequisite skills and concepts relating to these standards, and (2) the unfinished learning presents a barrier to entry for grade level content.

Here’s what the process looks like in action:

Part A: Understand

1. Unpack the standard – Identify the aspect of rigor, key concepts, skills, and vocabulary.
2. Solve a task – Choose a problem or question you will teach in an upcoming lesson, aligned to the grade-level standard. Solve the task yourself, being metacognitive along the way.
3. Identify the prerequisites – Ask yourself, “What prerequisites are required for students to access this task?”

Unpacking 6.NS.C.7.a: Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret -3 > -7 as a statement that -3 is located to the right of -7 on a number line oriented from left to right

Part B: Diagnose

1. Narrow skills/concepts – Choose 1-2 of the most critical prerequisites to hone in on. Identify the aspect of rigor for each prerequisite: Is students’ unfinished learning about concepts, procedures, or both?
2. Determine current understanding – Ask yourself, “What evidence of students’ understanding do I already have?” and “What additional information do I need?”
3. Give a low-lift assessment – Use a simple method to collect diagnostic data. Some ideas:
1. Add 1-2 questions at the end of an existing assessment. (Let students know they won’t be scored on these items!)
2. Use a 1-2 question diagnostic in place of an “exit ticket.”
3. Weave 1-2 questions into a “marker board practice” session on another topic. Tell students you want to see what they already know to help you plan for next week’s lesson. Jot down anecdotal notes about students’ responses.

Low-lift Diagnostic for 6.NS.C.7.a: Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret -3 > -7 as a statement that -3 is located to the right of -7 on a number line oriented from left to right

Part C: Take Action

1. Determine if the task you selected in Part A:
1. includes scaffolds, allows students to access the content in multiple ways, and provides an opportunity to address students’ unfinished learning, and
2. Based on the results of the diagnostic, engage one, several, or all students in the task.
3. Ask probing questions and provide support to help students make sense of the math and address any misconceptions they may have.

Taking Action on 6.NS.C.7.a: To address students’ unfinished learning with fractions, we adapted the Fractions on the Number Line task from Illustrative Mathematics that aligns to the grade-level standard. Specifically:

• Add a “part 0” to the task, beginning with unit fractions and/or more familiar fractions such as ⅔ or ¾. Ask “Is ½ or -¼ farther from 0? How do you know?” or “Is ⅔ or -¾ farther from 0?”
• Ask students to share their answers and explain reasoning. Take the opportunity to emphasize the following: equal parts, the meaning of the numerator and denominator, and how to break “1” into parts based on the number in the denominator.
• Then, to help clarify where to place fractions greater than 1 and less than -1, ask,  “Where would 4/3 go?” Confirm the location and rationale before assigning students parts a, b, and c.