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What role does consolidation play in a skill-based approach to math instruction?
This skill-based approach emphasizes a deep understanding of mathematical content—such as number sense, statistics, and geometry—while also focusing on essential processes like problem-solving and reasoning.
Building knowledge in a meaningful way is the foundation of strong and lasting learning. However, is this enough? Does it mean that practice is no longer necessary? What about homework? As with everything, balance is key.
This article examines the meaning of consolidation and explains why practice is a crucial aspect of learning.
Contents
What does it mean to consolidate learning in the math classroom?
To develop a deep understanding, students must first build knowledge with a strong emphasis on comprehension. In other words, they need to understand what they are doing and why it works.
However, comprehension alone is not sufficient. Just as it makes no sense to consolidate something that has not yet been constructed, it is equally ineffective to build knowledge without ever consolidating it.
Therefore, once students have explored concepts, constructed understanding through manipulatives, and developed conceptual clarity, it is time to consolidate that learning through practice. This practice helps them build fluency and develop fact fluency and procedures.
The Role of Practice in the Math Classroom
Practice is one of the fundamental pillars of learning consolidation. Through practice, students reinforce the procedures they have developed, which allows them to gain efficiency and focus on more advanced concepts.
This means that when they encounter a problem, students should not let the small steps needed to solve it become a barrier. In other words, they should not face a “problem within a problem.”
Instead, when certain procedures become automatic, students can concentrate on more complex and higher-order thinking tasks. This reduces cognitive load and frees up working memory, allowing complex problems to be broken down into manageable exercises.
Traditionally, practice was limited to endless worksheets filled with calculations and repetitive procedures. This approach created a distorted view of mathematics as a monotonous and rigid subject.
To move beyond this narrow perspective, didactics incorporate diverse forms of practice that enhance learning and promote deeper understanding.
4 key moments for consolidating learning in Innovamat’s classroom curriculum
Throughout the week, we design lessons to build numerical understanding, explore various content domains, and engage in systematic practice.
Each lesson includes moments for students to consolidate their learning through different formats, such as practice, written reflection, and discussion. Let’s delve into these moments in more detail.
Consolidation moments in knowledge-building sessions
During knowledge-building sessions, we allocate time for students to consolidate their learning.
After developing new understandings, students spend a few minutes working individually in their written learning logbooks to reinforce concepts and begin practicing. In elementary school, this moment is referred to as “Let’s Log It,” while in middle school, it is known as “Let’s Practice.”
Additionally, we often begin each session with a review of prior learning and a revisit of key concepts. For instance, if a session focuses on the 4 times table, a great warm-up activity could be to engage students in a short game or routine that reviews the 2 times table and explores the concepts of doubling and halving.
Productive practice sessions
Productive practice involves presenting students with an open-ended question, a real-world context, and a meaningful goal that requires them to practice to “produce” a response. By participating in this type of practice, students apply various mathematical processes in a problem-solving environment.
Each grade band has dedicated moments and sessions for productive practice.
Below are some examples:
Early childhood education
In early childhood education, students engage in activities that help them develop foundational counting skills and additive thinking. One example is the “Number 10” workshop, where students find different ways to make ten using two number cards. This activity provides productive practice with single-digit addition.
Elementary school
In elementary school, basic arithmetic practice is often embedded within a productive learning context. For example, in one activity, we explore multiplication tables by investigating the concept of multiplicative persistence in two-digit numbers. Through challenges and examples, we define multiplicative persistence as the number of times we must multiply the digits of a number together until we get a single-digit result.
Middle school
Middle school is no exception. One example is the “Search for 0” activity, which reinforces addition and subtraction with integers. In this task, students follow specific rules to figure out how many jumps they need to take along the number line to reach 0.
Content closure resources
We offer content closure resources to help consolidate the learning that has taken place.
For example, in elementary school, these resources can be found in the Adventures Toolbox, while in middle school, they are included in the Closure of the Section. Here, we revisit the material covered throughout the section, synthesize key discoveries, and connect them to other sections or even concepts beyond mathematics.
Personalized practice sessions
Once a week, we recommend a session for systematic, personalized practice aimed at developing fact fluency content and procedures. Repeated practice is essential for consolidating understanding. In educational settings, we refer to this type of practice as reproductive practice. Traditionally, it consisted of mechanical exercises without any personalization.
In our curriculum, these sessions occur within an app, which features small applets offering activities tailored to each student’s demonstrated knowledge and what has been covered in class.
How to consolidate long-term learning
A key factor in consolidating long-term learning is promoting practice based on spaced repetition. This approach involves strategically spacing out the presentation of content to enhance knowledge retention before it is forgotten. By doing this, we can flatten the forgetting curve and gradually strengthen long-term memory.
For instance, when students first learn subtraction, they are likely to forget it if they don’t revisit it soon. However, if they engage in a personalized practice session before this knowledge fades, they reinforce that understanding, resulting in a less steep forgetting curve. Additionally, by revisiting content across different sessions and activities—such as productive practice tasks, connections, and a spiral presentation of concepts—we will gradually increase retention in long-term memory. With each review, the forgetting curve flattens further, and learning becomes more solidified.
Own elaboration based on Vlach, H. A., Sandhofer, C. M., & Kornell, N. 2008. The spacing effect in childrenʼs memory and category induction. Cognition, 109, 163-167.:
Through spaced practice, we can also help students support fact fluency of basic math learning concepts, such as multiplication tables. This means that they can recall certain fundamental knowledge quickly, whether by memory or by deducing it from previous results, with minimal effort and in a reasonable time frame.
Recommendations for managing consolidation moments in Innovamat’s classroom curriculum
To conclude this article and assist you in maximizing the consolidation moments we suggest, here are some recommendations for managing them effectively:
1. Utilize the weekly personalized practice session
The personalized practice session is essential not only for consolidating learning but also for developing fluency in procedures. As we noted, building knowledge alone is insufficient for generating learning; practice moments are critical, particularly systematic practice.
Our curriculum incorporates an adaptive app designed to tailor learning based on each student’s performance.
2. Review reports carefully
Following each personalized practice session, teachers receive a detailed progress report for each student. This tool offers an overview of the entire class and helps educators make informed decisions tailored to individual needs.
3. Identify key content for consolidation
We understand that the consolidation of learning is a gradual process that does not occur overnight. However, it is important to identify which content should be reinforced at each grade level to adjust practice sessions and ensure effective progression.
In this ebook, you will find a guide outlining the process for consolidating basic operations as students advance through elementary school. For other content domains, consolidation should take place gradually, and sharing what is being worked on with students will help in this process. These content domains are summarized in the Toolbox.
4. Prepare for productive practice sessions
Productive practice sessions are excellent opportunities to reinforce learning and apply mathematical processes. To maximize their effectiveness, we recommend preparing key questions that stimulate reasoning and create a problem-solving environment.
5. Be attentive to the advice and materials outlined in the guides
The sessions include various practice moments. Pay close attention to the guide’s suggestions to utilize the materials effectively. Some proposals will be designed to cater to the different paces of the class.
6. Make decisions based on your observations
The discussions we initiate at the beginning of the knowledge-building sessions often provide insights into the class’s progress. By asking specific questions, we can identify which concepts have been consolidated and which areas require additional support. Use these moments to make informed decisions.
