As educators, we need to understand if our math instruction was successful in reaching students. If it’s not, we need to adjust as soon as we can. The key to being able to nimbly adjust our instruction is strong assessment. Some curricula provide great assessment options, while other curricula, even well-aligned curricula, provide weaker options. What should a teacher do then? What should teachers do if they know they have unaligned curriculum with gaps or incorrect focus? The assessment items that come with that curriculum would only reinforce the problem. Where should teachers turn when students simply need more practice?

One solution to these questions is to supplement, strategically, with well-aligned items. In this blog post you’ll hear from two educators who helped design a new mini-assessment focused on conceptual understanding of fractions—a topic often overlooked or poorly addressed in existing instructional materials. Conceptual understanding of fractions requires that students are able to first position fractions within the number system, both as an extension of their earlier work with whole numbers and as a subset of the full rational number system that they encounter in middle grades. You’ll also see how these teachers integrated the newly designed mini-assessment into their existing programs: *Go Math!* and *Eureka Math.*

Raven Redmond is currently a fifth-grade lead instructor who teaches all subjects at Memphis Delta Preparatory Charter School in the heart of South Memphis, Tennessee. She has served in education for four years with various roles. | |

Jennifer Ahearn is in her seventh year as the Math Coach at Lowell Elementary School in Teaneck, NJ. She has worked in the district for over 14 years, starting her career as the RtI teacher for grades 5-7 and as a seventh-grade math educator. |

*Q:*** Were you familiar with Student Achievement Partners’ (SAP) mini-assessments prior to beginning this project? What role do you see these assessments serving within a broader formative assessment process?**

*A: Raven*: When I was asked to join this project, I was not familiar with SAP’s mini-assessments, and it has been a pleasure to dive into the process of creating this mini-assessment. This mini-assessment, like curriculum-based assessments, is aligned to standards. However, curriculum-based assessments often focus too heavily on the procedural aspects of standards or attempt to cover all standards equally, sometimes missing important emphases or big ideas. I see these mini-assessments as a useful tool for teachers, providing tangible reteaching opportunities grounded in evidence of student thinking and reasoning elicited by strongly aligned assessment questions.

*A: Jennifer*: Prior to beginning this project, I was unfamiliar with SAP’s mini-assessments. Now, I frequently refer to the mini-assessments while coaching. Not only do they provide opportunities for teachers to gain knowledge about assessing the conceptual understanding, procedural skill and fluency, and application demands of the standards, but they are also a great reference tool for achieving the necessary level of cognitive complexity in your classroom. These mini-assessments give teachers the opportunity to target their instruction to supporting or deepening students’ understanding of the standards being assessed.

*Q:* If teachers are considering using a mini-assessment within an existing curriculum, what should they consider to ensure they’re maximizing its effectiveness?

*A: Raven*: I think it is important to understand how the assessment can be integrated with the curriculum for a specific purpose (e.g., to measure student understanding of standard(s) after instruction). One reason a teacher might use a mini-assessment instead of a curriculum-embedded assessment is to better understand any potential gaps or alignment issues with the curriculum itself. For example, the *Eureka* modules on grade 3 fractions invest a lot of time on area model representations of fractions. The mini-assessment on this topic is more aligned to the idea that a fraction is a number, and therefore might help teachers understand how to invest their time differently or where to go deeper with students.

*A: Jennifer:* Teachers should first ensure that their curriculum is well aligned to the standards—the assessment can actually help with this analysis. Unfortunately, many textbooks do not meet the depth of the skills required by the standards. *Go Math!*, for example, is lacking the strong foundation in understanding fractions as numbers needed for students to be successful on this assessment. The fraction lessons in *Go Math*! chapters 8 and 9 do not allow enough opportunities to engage with the number line representation, which is useful in helping students understand how fractions extend their earlier work with whole numbers. In particular, the area model representation might be overused in the curriculum which in and of itself isn’t problematic, but it does mean there is less time for students to spend strongly grounded in the goal of understanding the concept of the fraction as a number. A circle partitioned into 4 equal sectors with 3 shaded doesn’t neatly transfer to understanding ¾ as a number between 0 and 1 that is closer to 1.

Question 9 on the mini-assessment will likely be particularly difficult for students who have only been exposed to the *Go Math!* curriculum, because, in this program, when students are asked to practice representing fractions as points on a number line in Lesson 8.5, every example shows the unit fractions as “one length away from zero.” This signals that *Go Math*! users should consider incorporating supplemental lessons into the curriculum to focus on the use of number lines and representing the value of a fraction on the number line.

*Q:*** What was your role in creating the administration guidance for the grade 3 fractions mini-assessment? Why do you believe this is an important topic to assess correctly?**

*A: Raven and Jennifer: *We worked together with Student Achievement Partners on this mini-assessment. We started by analyzing how the *Eureka* *Math* and *Go Math! *curricula assess this topic. We analyzed some questions from the curriculum-embedded assessments as well as other public-facing assessment materials and used this to design an assessment that strongly tied to the idea of a fraction as a number *and* that had a good mix of lower and higher conceptual complexity questions. This mini-assessment serves as an important tool to assess students’ conceptual understanding of fractions at different levels of complexity, allowing for a more complete picture of how to support and/or extend their learning. In creating this assessment, we also looked at fraction questions from multiple grade levels. We knew that this mini-assessment had to serve as a way to assess the foundational understanding of fractions necessary to engage with later work in this area.

*Q:* How does the new third grade mini-assessment map to your curriculum’s lesson(s)? What did you learn about your curriculum during the design of the assessment that might be helpful for educators using it?

*A: Raven and Jennifer: *Through the mapping process, we learned that some lessons are much more critical than others, and also that there may need to be adaptations made to our curricula to fully support the understandings articulated by the grade 3 standards. The following tables show how individual assessment items correlate to lesson content in *Eureka Math* and *Go Math! *and we know that these will be helpful tools as we consider what program adaptations might be needed to support student learning:

AssessmentQuestion |
Corresponding Lesson(s) from Module 5 In Eureka Math |

1 |
Lesson 14: Place fractions on a number line with endpoints 0 and 1.Lesson 15: Place any fraction on a number line with endpoints 0 and 1. |

2 |
Lesson 14: Place fractions on a number line with endpoints 0 and 1.Lesson 15: Place any fraction on a number line with endpoints 0 and 1. |

3 |
Lesson 7: Identify and represent shaded and non-shaded parts of one whole as fractions. |

4 |
Lesson 16: Place whole number fractions and fractions between whole numbers on the number line. |

5 |
Lesson 16: Place whole number fractions and fractions between whole numbers on the number line.Lesson 17: Practice placing various fractions on the number line. |

6 |
Lesson 29: Compare fractions with the same numerator using <, >, or =, and use a model to reason about their size. |

7 |
Lesson 30: Partition various wholes precisely into equal parts using a number line method. |

8 |
Lesson 14: Place fractions on a number line with endpoints 0 and 1.Lesson 15: Place any fraction on a number line with endpoints 0 and 1. |

9 |
Lesson 16: Place whole number fractions and fractions between whole numbers on the number line.Lesson 17: Practice placing various fractions on the number line. |

10 |
Lessons 22–23: Generate simple equivalent fractions by using visual fraction models and the number line. |

11 |
Lesson 24: Express whole numbers as fractions and recognize equivalence with different units.Lesson 25: Express whole number fractions on the number line when the unit interval is 1. |

12 |
Lesson 11: Compare unit fractions with different-sized models representing the whole. |

AssessmentQuestion |
Corresponding Lesson(s) from Go Math! Chapters 8 and 9 |

1 |
Lesson 8.5: Fractions on a Number Line |

2 |
Lesson 8.5: Fractions on a Number Line |

3 |
Lesson 8.4: Fractions of a Whole |

4 |
Lesson 8.5: Fractions on a Number Line |

5 |
Lesson 8.5: Fractions on a Number Line |

6 |
Lesson 9.3: Compare Fractions with the Same Numerator |

7 |
Lesson 9.2: Compare Fractions with the Same DenominatorLesson 9.3: Compare Fractions with the Same NumeratorLesson 9.4: Compare Fractions |

8 |
Lesson 8.5: Fractions on a Number Line |

9 |
Lesson 9.6: Model Equivalent FractionsLesson 9.7: Equivalent Fractions |

10 |
Lesson 9.6: Model Equivalent FractionsLesson 9.7: Equivalent Fractions |

11 |
Lesson 8.6: Relate Fractions and Whole Numbers |

12 |
Lesson 8.3: Unit Fractions of a Whole |

*Q*: For educators who want to undertake a similar mapping process, either on their own or as part of a PLC, what is the first thing you’d recommend they do to get started?

*A: Raven and Jennifer:* The first consideration in this work is to not assume that either the assessment or the curriculum is better aligned than the other without investigating that claim. As an example, the mini-assessment we designed leans heavily toward eliciting evidence of students’ understanding of a fraction as a number, tying closely to the grade 3 cluster heading. So instead of a typical lesson-by-lesson, standard-by-standard analysis of our instructional materials, we might instead ask: How much time is spent building the idea of a fraction as a number? How much time is spent using representations of fractions that aren’t directly supporting this understanding? Which lessons seem most relevant in helping students answer questions like those on the assessment? Even with materials that are fairly well-aligned, we should constantly be examining places to shift emphasis of instruction to support students in reaching the goals of the standards.