As teachers plan instruction for this fall after extended remote learning, there is a lot of discussion about the best way to address students’ unfinished learning. Leveraging the natural coherence of mathematics and following a coherent curriculum can be a first step in tackling this challenge.
What Is Mathematical Coherence?
Coherence is the idea that mathematics domains are connected to one another and should be presented to students in a way that allows them to see those connections. It’s thinking holistically about mathematics. Students learn new ideas, but they also see how to connect and build off the content they've learned previously.
The logical progression of coherence lays the foundations for work in later grades. For example, in fourth grade, students must apply and extend previous understandings of multiplication to multiply a fraction by a whole number. This extends to fifth grade, when students are expected to build on that skill to apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. (AchievetheCore.org has interactive coherence maps for specific grades and domains.)
Coherence is built both vertically (i.e., topics are connected across grades) and horizontally (i.e., topics students learn within a grade level are also logically connected).
"Coherence is the idea that the mathematics domains are connected to one another and should be presented to students in a way that allows them to see those connections."
Why Is Coherence Important (Especially Now)?
Coherence is an important part of mathematics education because, when done effectively, it helps students more deeply understand mathematical ideas while also making complex subjects easier to understand. Given the disruption in learning during spring 2020, a coherent curriculum is even more important this school year than ever before. Teachers can take advantage of the coherence of mathematics by following the learning progressions to uncover and address students’ unfinished learning.
Figure 1 below shows a learning progression from Curriculum Associates’ core mathematics program i-Ready Classroom Mathematics, but the example can be applied to mathematics curricula in general. This particular progression shows how fourth grade mathematics builds upon concepts learned in earlier grades.
Looking at the figure below, you see that to be successful on the lesson “Multiply by Two-Digit Numbers,” students need to be proficient on previously taught fourth, third, and second grade skills. By tracing a standard back to its logical prerequisites, teachers can see how the concepts they’re teaching now were addressed in preceding standards. It also helps them find any content limits, such as what numbers or models were used by students in previously learned standards. From this, teachers can adapt grade-level lessons to determine appropriate places to integrate those prerequisites to support on-grade level instruction. Teachers can incorporate the prerequisites that students are missing and still address grade-level content in the same lesson.
How to Adapt Lessons to Address Unfinished Learning
Consider various way to integrate unfinished learning into grade-level instruction. Some ideas include:
- Adding a warm-up activity that connects to prior learning
- Building prerequisite instruction into a grade-level lesson
- Having a lesson on prior-grade skills before the grade-level lesson in which those skills will be applied
Providing additional support to students by using activities, games, and practice that focus on prerequisite skills and concepts using coherent curriculum materials, such as i-Ready Classroom Mathematics, makes it easy to adapt lessons to address students’ learning needs, including unfinished learning. Below are a few examples of how the coherent structure and great instructional design can help teachers address students’ prerequisite and grade-level needs.
Visual Learning Progressions (such as the ones available in i-Ready Classroom Mathematics and on AchievetheCore.org) help teachers identify the coherence of standards across grade levels and provide a visual representation of how previous concepts are built upon and relate to current grade-level concepts being taught.
Based on the learning needs of their students, teachers can seek out resources to address prerequisite skills and concepts in small groups or with the whole class. If they find that an entire class needs to build its prerequisite knowledge, they should incorporate prerequisite learning into on-grade level lessons in order to keep instruction pacing on track.
Building on Prior Knowledge
Start each instruction session with an activity that engages students in previous learning that is directly related to the problems in the day’s instruction. This intentional choice of content for the Start activities provides students with an opportunity to feel successful as they begin the lesson and to understand how content they learned previously relates to new skills or concepts.
Add time for in-depth exploration and refinement into your instructional design.
Exploration: Give students time to explore new concepts in relation to previously learned concepts. Use rich tasks that provide multiple entry points, and allow all students to access the mathematics being taught in a way that makes sense to them. In-depth explorations are a great way to help students connect what they know with what they are learning.
Refinement: Devote class time to deepening understanding and practicing skills. In i-Ready Classroom Mathematics, we call this the “Refine session” because it’s meant to both reinforce and polish comprehension. Refinement activities and problems help students build their fluency and check for understanding. Cumulative problems are great ways to reiterate the interconnectedness of mathematics and ensure that students don’t see lessons as isolated events.
By following a coherent curriculum, teachers can more easily adapt lesson content to address prerequisites during on-grade level instruction. Addressing students’ unfinished learning this school year is not an easy task, but the coherence of mathematics is an important tool that can help with this work.
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