Much of this is based/taken from Making Math Stick: Classroom Strategies that Support the Long-Term Understanding of Math Concepts

What a year! The 2020-2021 school year was unprecedented. There was lots of variation amongst jurisdictions during the year in how instruction unfolded (face-to-face, virtual, and hybrid). Regardless of your location, grade level, and method of instruction, we can agree that all teachers felt significant stress in ensuring students engaged meaningfully with mathematics concepts. Some districts carried on with the full curriculum/standards and did their best in working through it. Other districts adjusted the curriculum/standards to prioritize key concepts.

Coming from these varied experiences is the anxiety that teachers, administrators, and districts feel regarding student learning. There is a belief that the past school year has caused significant learning gaps for students. There is a belief that students are falling behind, if not already behind, due to the interruptions they experienced during the 2020-2021 school year. I reference this as a belief because much debate exists as to whether there are learning gaps. The 2020-2021 school year was traumatic for all students across the world – we were, and still are, in a global pandemic.

To address the belief that students have experienced a learning gap, many provinces/states, districts, and/or schools are planning to start the upcoming school year differently than years past. I have heard many in education talking about the need to start the upcoming school year by reviewing math concepts from the previous school year (i.e., students in grade 4 would begin their year by working with concepts from grade 3, students in grade 8 would begin their year by working with concepts from grade 7, etc.). The rationale is that this would address the learning gaps and would ensure students are where they need to be. I want to caution people with this stance.

In such a situation, review is more like cramming that reviewing. Mathematical concepts from the previous grade level are introduced to students in a short period of time, students are given practice questions on singular concepts, and then the teacher moves onto the next concept. Such an approach to instruction and learning seems to focus on dispensing information instead of supporting students in understanding, recalling, and applying the mathematical concept.

Such an approach to education is blocked practice. In blocked practice, students are not required to consider which strategy to apply as this is already inherent in the questions and tasks assigned – all questions and tasks apply only to the single concept that was the focus of the lesson. This removes the requirement that students select a strategy to apply to any given question or task. This one-and-done structure of mathematical instruction does a significant disservice to students. Omitted form this approach to instruction and learning are the cognitive processes that students would use in determining an appropriate strategy. These processes are crucial to mathematical understanding (Costello, 2021).

When we go into a school year with a plan to review, or in this context cram, we are not using student learning as the compass for instruction. We are painting all students with the same instructional brush and thinking they all need the same next steps. Doing this will only benefit some, if any, in the short term. We are not focusing on student learning, we are focusing on curriculum/standard coverage. Coverage and learning are not the same.

A Plan to Move Forward

Let us step back for a moment and think about how students learn. Too often, learning is thought of as only being the acquisition of knowledge. Such a perspective on learning is too narrow. We need to go beyond this. From the perspective of cognitive science, learning is the acquisition of knowledge and skills, and having these readily available from memory to support future meaning-making opportunities (Brown, Roediger III, & McDaniel, 2014). When viewing learning from this broadened perspective, we can appreciate how our instruction must to go beyond ‘getting information into the heads of students.’

So, you may be wondering what can we do to support student learning. Well, I want to offer retrieval practice. Retrieval practice is an approach to instruction and learning that involves students in recalling/retrieving previous learning over an extended period of time. The practice of retrieving concepts from memory multiple times helps individuals learn – the more times we retrieve something, the stronger our memory of it becomes (Brown, Roediger III, & McDaniel, 2014; Costello, 2021).

Instead of cramming at the beginning of the school year, consider planning opportunities throughout the year for students to retrieve mathematical concepts from memory. This provides students multiple opportunities to explore and apply a concept to solve problems. The more encounters they have with the concept, the stronger their understanding becomes, as does their ability to retrieve it.

Retrieval practice is about working smarter, not harder. It is about using questions, tasks, resources that we already have at our fingertips more effectively. It is about focusing instruction on how students learn. Consider the following five options as ways to structure the school year using retrieval practice as the structure. I will briefly explain each option in isolation, but any option can be used in combination with another.

Year Plans

Traditionally, outcomes and standards are divided into units of study. Within such a year plan, the teacher delves into concepts over a continuous block of time. Students can finish exploring these concepts before moving on to the next block. A drawback of this approach to year planning is that mathematical concepts are addressed only once during an entire school year. And addressing any concept just once does not bode well for retention.

Instead of the units-of-study year plan, why not consider a spiralled curriculum? In a spiralled curriculum year plan, concepts are addressed several times throughout the year. During each revisit of the concept, the teacher and students explore the concept a little more deeply. By following a spiralled curriculum year plan, you constantly review previously taught material instead of always broaching new concepts in a sequential order. Constantly returning to concepts allows students to approach problems with confidence because, if students do not understand the concept the first time they learn about it, they have ample opportunity to engage with it more successfully later in the year. Students engage more competently with the concept with each visit.

Daily Cumulative Review

It is important to give students daily opportunities to review important mathematical skills and concepts. At the foundation of daily cumulative review is mixed review –questions on a concept are spread across many review sessions, and each review session addresses a variety of concepts.

A form of daily cumulative review that I have found to be effective is a mixed review consisting of three to five somewhat brief questions that target multiple concepts (Leinwand, 2017). You can set this review to occur every day at the beginning of math class. Students would need approximately five minutes to solve the three to five questions. You would need an additional five minutes to review correct responses, either by modelling the process or by calling on students to share their thinking. If you check responses, you can provide students with feedback on which strategy they chose to solve each problem and how they applied the strategy.

By engaging students in daily cumulative review, you not only reinforce the material, you also give students a chance to clarify any misunderstandings they may have about the concepts.

Cumulative Assessments

Cumulative tests represent a shift in how we construct tests. Cumulative tests focus on spacing and mixing key concepts. Not all curriculum outcomes and standards have the same impact on student learning. Key concepts are outcomes or standards that not only are significant in grade level achievement but also pivotal for future learnings. The curriculum for each grade includes six to eight key concepts. After teaching one key concept you would provide students with a test addressing that key concept. You would do the same thing for the second key concept you teach. But you would also address the first key concept. This process of tagging on previous key concepts to tests supports students in that the key concepts are being spaced out and mixed on tests. Students also receive feedback on their thinking in relation to all key concepts on every test.

The cumulative test approach focuses on key concepts for a few reasons. First, it is more manageable for you and your students to return repeatedly to a subset of concepts. Second, key concepts by definition, support the learning of all curriculum outcomes and standards. Narrowing the focus on key concepts will support students in achieving all curriculum outcomes and standards.

Homework

Homework can be a polarizing topic. While some schools choose to assign homework others adamantly oppose it. For those schools that assign homework, I would like to offer two options.

The structure of the assigned task can help you make use of homework as a learning strategy. Leinwand (2017) provided a template for homework that involves a range of tasks.

• First are two problems requiring a new skill. These two problems are enough for students to approach the concept while avoiding the possibility of reinforcing a misconception.
• Second, are four problems requiring recall of concepts learned in previous lessons. These four problems serve the purpose of mixing the review, whereby students need to retrieve previous learning from memory.
• Third, students are provided two problems requiring extra work or explanation. These two problems serve the purpose of affording students the opportunity to problem solve.

Another structural option for the homework plan is a cumulative review approach. You would provide students with three to four problems that focus on three to four concepts you have identified as being crucial to student mathematical understanding (that is, key concepts). You could choose key concepts that were explored previously in the year and concentrate all homework on these key concepts. You could focus on the same key concepts throughout the entire year, or you could cycle through various key concepts during the year using bi-weekly or monthly blocks.

Writing-to-Learn

Writing-to-learn activities help students learn, reflect on, and synthesize what they are learning. Such writing is evidence of students thinking, making connections, and investigating possibilities. Writing-to-learn is a process whereby students strengthen previous learning, add layers of details to new learning, and make connections among concepts. It is a place for them to make their thinking visible to themselves.

During writing-to-learn tasks, students are writing for themselves, to develop their mathematical understanding. Consider writing-to-learn tasks as a written form of a think aloud.

Examples of writing-to-learn are:

• Free Writes
• 30 Words or Less
• Plus/Minus Chart

Consider offering students opportunities for writing-to-learn throughout the year. Provide students with a concept to explore (one from earlier in the year) so that they have to retrieve a memory, reconstruct it and then communicate their understanding of it.

Summary

Retrieval practice goes beyond short-term performance. Retrieval practice expands the perspective of learning and is something to consider if your goal is to support student mathematical understanding. By revisiting mathematical concepts throughout the year, whether this is through year plans, a writing prompt, homework, etc., students will become more independent problem solvers and will strengthen their ability to retain and recall previous learning to solve novel situations.

Retrieval practice isn’t additional work for teachers. It is using what we already have in a manageable and sustainable way that has tremendous value added in terms of student learning.

Reach out to me on email (david@costellomath.com), Twitter (@dr_costello) and/or my website (www.costellomath.com) to discuss retrieval practice or any of the approaches shared in this post.

References

Brown, P. C., Roediger, H. L. III, & McDaniel, M. A. (2014). Make it stick: The science of successful learning. Cambridge: The Belknap Press of Harvard University Press.

Costello, D. (2021). Making math stick: Classroom strategies that support the long-term understanding of math concepts. Markham: Pembroke Publishers Ltd.

Leinwand, S. (2009). Accessible mathematics: Ten instructional shifts that raise student achievement. Portsmouth: Heinemann.

Leinwand, S. (2017, April). 10 Instructional tweaks every mathematics leader needs to advocate for and be able to model. Paper presented at the 49th National Council of Supervisors of Mathematics Annual Conference, San Antonio, Texas.

Roediger, H. L. III, & Karpicke, J. D. (2018). Reflections on the resurgence of interest in the testing effect. Perspectives on Psychological Science, 13, 236–241.

Rohrer, D. (2009). The effects of spacing and mixing practice problems. Journal for Research in Mathematics Education, 40(1), 4–17.