Science of math learning

The educational framework incorporated into the Levebee app is rooted in scientific research. As a result, the app can effectively help children who struggle with maths for various reasons. The methodology is guaranteed by PaedDr. Renata Wolfova, who has been helping children with reading, writing and math difficulties for over 35 years.

Table of contents:

  1. 1Cognitive load theory
  2. 2The importance of foundational knowledge
  3. 3Structured, systematic and demonstrative teaching is the most effective
  4. 4Dividing the curriculum into the smallest possible units
  5. 5Knowledge can enhance skills, and skills can broaden knowledge
  6. 6Instant feedback
  7. 7Implementing scaffolds to enhance learning
  8. 8The crucial role of executive functions
  9. 9Don’t confuse fun with learning
  10. 10Automation through independent practise
  11. 11Adaptive learning
  12. 12Formative assessment

1. Cognitive load theory

Understanding how the human brain processes information is fundamental to the cognitive load theory and serves as the building block of the Levebee app. Over the course of our evolution, we have gained the innate ability to process certain information, such as counting, without explicitly needing to learn it. While children are not born knowing how to count, they possess the inherent ability to acquire counting skills on their own. However, we have to study more sophisticated concepts (e.g. the decimal system, numerical operations, etc.). 

To accomplish this, a combination of working memory and long-term memory is required. When coming across new information, we rely on our working memory. Due to its limited capacity of 5-7 pieces of data, it can only process 3-4 new pieces of information at a time. Therefore, it is essential to transfer information into our long-term memory as soon as possible, as it has an almost infinite capacity. What's more, long-term memory is available to the working memory without any limitations when solving problems 1 2 .

The animation was inspired by the lecture “Embedding Explicit Direct Instruction” by InnerDrive.

By linking knowledge together into unified structures, working memory can manage it as a single item, thus creating space for the learning of new information 3. For example, which password is easier to remember URGEAP or PRAGUE?

To picture working memory, one can imagine an octopus using its tentacles to hold pieces of information taken from long-term memory or newly acquired information from the surroundings. Nevertheless, its number of tentacles is limited. By adequately connecting long-term memory knowledge, the octopus can retrieve it as a whole with just one tentacle. As a result, other tentacles are available to process additional information. The image was inspired by illustrations from the book Learning How to Learn by Barbara Oakley. 

Therefore, effective learning means not overwhelming the individual with new information and effectively linking information in long-term memory into unified structures. Automated knowledge does not burden working memory. Long-term memory can store a vast amount of complex information and is key to problem-solving and thinking ability. The sooner the curriculum is stored in long-term memory, the sooner working memory is freed to solve new, more complex problems. 

🐝 Levebee is designed to provide each student with an optimal cognitive load that is tailored to their specific abilities. A task completion rate of around 80% is typically considered ideal 4. Accordingly, the exercises are structured to be challenging yet achievable for the students, inspiring them to continue making progress.


1 Cognitive Load Theory, John Sweller, 1988
2 Cognitive Load Theory in Action, Oliver Lovell, 2020
3 Learning How to Learn, Barbara Oakley, 2018
4 Teaching Functions, Barak Rosenshine and Robert Stevens, 1986


2. The importance of foundational knowledge

The early maths concepts acquired before entering school form the basis of a child's foundational mathematical knowledge. Research suggests that a child's progress in maths between the age of 5 and first grade significantly impacts their future success or failure in the subject at ages 12-15, even after accounting for early reading skills, cognitive abilities, family characteristics, and individual traits 1. By providing suitable pedagogical guidance, we can shape a child's mathematical development. Generally, investing in education during early childhood yields much greater benefits than later attempts to improve learning outcomes 2.

🐝 Levebee provides a systematic set of exercises that help children develop skills in grouping, arranging, comparing, and counting. These skills form the basis for future comprehension of number series, decimal systems, numerical operations, and their practical application in real-life situations. Levebee helps preschool children to develop a solid foundation in maths and gives school-age children the chance to revise and improve any fundamental skills they may be missing.


1 Whaťs past is prologue: Relations between early mathematics knowledge and high schooI achievement, Watts, T. W., Duncan, G. J., Siegler, R. S. & Davis-Kean, P. E. 2014
2 Policies to Foster Human Capital, J. Heckman, 2000


3. Structured, systematic and demonstrative teaching is the most effective

A person can acquire knowledge through independent discovery or indirectly through the guidance of others. In discovery-based learning, when students have reached the limits of their existing knowledge, they may resort to taking random steps that are not necessarily effective 1. However, a child can systematically acquire knowledge by receiving information through another person or an application.Studies have shown that clear and well-structured teaching is more advantageous and effective than exploratory learning, particularly for beginners 2. When children are actively involved in the learning process and the teacher guides their activities towards productive goals, they tend to learn better 3.


The diagram was inspired by the lecture Embedding Explicit Direct Instruction by InnerDrive.

The path to understanding begins with methodically distributing the curriculum in a structured manner that avoids overloading the working memory of a child. This method helps the child to keep up with the material and facilitates a smooth learning experience, allowing knowledge to be acquired effortlessly in a state of consciousness called flow 4.

Additive skills (i.e., addition and subtraction) predict unique variance in multiplicative skills (i.e., multiplication and division); multiplicative skills predict unique variance in fraction arithmetic; and fraction skills predicte unique variance in algebra 5 .

This structured approach for preschool and first-grade children is based on the early development of mathematical concepts, such as comparison, counting, numerical symbols, and other related skills, following the spiral structure of mathematical knowledge. Additionally, it fosters an understanding of the number series, including the decimal system as in the positional value of digits in a number. It also helps develop essential mathematical skills, such as addition, subtraction, multiplication, and division. By mastering these foundational skills, children are equipped with the tools they need to tackle more complex arithmetic problems, including word problems.


Math Spiral Diagram by Renata Wolfova, Chief of Methodology at Levebee

The way a task is presented is also significant. Mental modelling is a fundamental method of learning. Visual stimuli can aid in the understanding of problems, enabling children to grasp concepts more quickly. The process gradually becomes automated and internalized, shifting from "physical" actions to mental processes and developing abstract thinking skills. For instance, when first learning to count, a child may rely on their fingers to keep track, but with practice, they no longer need to. 

Incorporating multiple senses is key to enhancing knowledge acquisition and promoting lasting retention. This approach will help to expedite the understanding of deep structures. It helps to stimulate, motivate, and perceptually develop children, ultimately enabling them to independently apply newly acquired skills 5.

🐝 The Levebee app arranges exercises in a way that first enhances the child's understanding of problems, with a visual approach serving as the foundation for success. Once a concept is grasped, the app provides enough opportunities for systematic practice. The application uses a 10x10 grid of numbers. Students arrange bars on a grid, later they work solely on the grid without the bars and eventually perform mental calculations. The aim is to develop a strong mental representation of numbers from 0-100 and equip students with the skills needed to perform addition and subtraction operations. This provides the foundation for solving more advanced problems, including complex arithmetic and word problems.


1 Why Inquiry-based Approaches Harm Students’ Learning, John Sweller, 2021
2 Does Discovery-Based Instruction Enhance Learning?, Alfierit L., Brooks, P. J., Aldrich, N. 3., & Tenebaum, H. R., 2011
3 Should there be a three-strikes rule against pure discovery learning? The case for guided methods of instruction, Richard E Mayer, 2004
4 The Hierarchical Relations Among Mathematical Competencies, Chang Xu, Sabrina Di Lonardo Burr, Jo-Anne LeFevre, 2023
5 Flow: The Psychology of Optimal Experience, Mihaly Csikszentmihalyi, 2008
6 Cognitive Load Theory in Action, Oliver Lovell, 2020


4. Dividing the curriculum into the smallest possible units

Breaking the curriculum into small units 1 not only helps to minimize cognitive overload for each child but also enables educators to identify any gaps in their understanding of mathematical concepts. Children can focus on one new concept at a time, which allows them to discover specific areas where they may be struggling. 

🐝 The Levebee app features hundreds of exercises, with each one designed to correspond to a single mental step. The goal is to introduce students to fundamental mathematical concepts, ensure their comprehension, acquisition, and facilitate the transfer of this knowledge into long-term memory. The final stage is the readiness of information when tackling problem-solving scenarios, such as word problems. 


1 Principles of Instruction, Barak Rosenshine, 2010

5. Knowledge can enhance skills, and skills can broaden knowledge

When new information is introduced, it is first processed by working memory and then transferred to long-term memory. The information can then be applied in problem-solving activities. Conceptual knowledge and procedural skills are mutually reinforcing, with each supporting the development of the other. Understanding the concepts facilitates the acquisition of procedural skills while practising those skills improves comprehension of the concepts 1. For example, by arranging bar models which represent numbers from 1 to 10, children gain an understanding and verify their knowledge of decomposing numbers from 0 to 10.

🐝 The Levebee app provides enough exercises for children to understand and practise each topic thoroughly. After completing a diagnostic test, the app will adjust the difficulty level of the exercises to match the child's knowledge and skill level, ensuring that each exercise builds upon their current understanding. Therefore, students are able to learn with continuity and are presented with subject matter that is neither too easy nor too difficult. This approach enables children to understand the problem and use the knowledge previously stored in their long-term memory. Simultaneously, the application provides opportunities for the child to practise newly acquired knowledge as needed. By integrating new knowledge into their existing mental structure, the child is able to store it in long-term memory and make it ready for further use. This ensures that the child starts the next cycle with a new level of knowledge and skills.


1 Myths That Undermine Maths Teaching, Sarah R. Powell, Elizabeth M. Hughes, and Corey Peltier, 2022


6. Instant feedback

Instant feedback is a key factor in understanding a concept and is essential for effective learning 1. Identifying the step at which an error occurred and adjusting the procedure is crucial. Feedback should prompt the student to think, it should be precise and relate to learning objectives 2. Consistent feedback in the form of praise, rewards, and visible recognition makes a child’s performance more tangible and supports their ability to evaluate their own progress, especially during the initial stages of learning.

🐝 Levebee allows children to make mistakes, but only to a limited extent to prevent them from becoming too frustrated. It provides positive feedback when they get something right and offers guidance when they make a mistake, allowing them to find the correct solution on their own. Children may be more receptive to feedback from the app than from their parents.


1 A Thousand Brains: A New Theory of Intelligence, Jeff Hawkins, 2021
2 Embedded formative assessment, Dylan Wiliam, 2018


7. Implementing scaffolds to enhance learning

Using scaffolds can be a beneficial approach for improving understanding of information and fostering students’ motivation as well as engagement. Well-designed tasks and proactive support can help children achieve a deeper and more effective comprehension of the subject matter 1. For support to be effective, it must be practical and constantly adjusted to meet the changing needs of the learner. Scaffold learning should be used as a temporary measure to help students maintain their cognitive abilities and then gradually reduced.

🐝 The Levebee app offers children scaffolds through the use of visual aids and models, which are gradually phased out as the child's learning progresses.


1 The Use of Scaffolds for Teaching Higher-Level Cognitive Strategies, Barak Rosenshine and Carla Meister, 1992


8. The crucial role of executive functions

Effective learning involves more than just acquiring knowledge - it also requires the ability to concentrate, use working memory, plan ahead, resist distractions, and apply other cognitive skills collectively known as executive functions. When interpreting diagnostic tests results to assess a child's learning difficulties, it's important to distinguish between the role of knowledge and executive functions. Poor performance may not necessarily indicate a lack of knowledge but could stem from underlying executive function challenges that are not immediately apparent 1.

As educators, it's essential that we understand what we're doing, why we're doing it, how we're doing it, and where we're headed. When solving a problem, it's important for the child to be able to focus fully on the task at hand in order to achieve success. Solving a calculation such as 93 - 67 requires short-term, focused concentration. It is also interesting to note that counting exercises can be a helpful way to encourage children to enter a state of intense concentration. External stimuli can be a source of distraction and can negatively impact working memory. When a child is in the process of developing a basic understanding of a particular topic, it's important to prioritize the content and carefully consider the approach, while minimizing distractions and allowing smooth mental work through a focused and direct approach. Only then can we gradually increase the cognitive load which will lead to a sense of satisfaction and accomplishment 2.

🐝 The Levebee app is designed to follow these key principles of success. The structure of the individual exercises is designed to reflect the full spectrum of knowledge and skills, with each exercise building on the previous one to create a cohesive learning experience. Students progress from acquiring knowledge and practising new skills to applying the knowledge. The Levebee app is intentionally designed to avoid overwhelming children's working memory with graphics or excessive animation and distracting them with side stories or other elements that could disrupt their focus while they are still learning. When practising, it is recommended to gradually increase the emphasis on executive functions. Levebee offers exercises that align with that method. For instance, the “shooting range” type of exercise. The settings help students perform smoothly without unnecessary strain on executive functions. Adjusting the size and type of the font, contrast, audio instructions, choosing between 15 languages of the instructions etc. helps in achieving this goal.


1 Executive Functions, Adele Diamond, 2013
2 Cognitive Load Theory in Action, Oliver Lovell, 2020


9. Don't confuse fun with learning

The process of learning is shaped by our conscious thoughts and focused attention. 1. If a child spends the majority of their time in an app focusing on choosing the colour of their virtual pet's clothing, it will have no meaningful impact on their mathematical abilities. Engaging in non-educational activities for the sake of fun is unproductive and may not motivate children who are struggling with maths. Moreover, students who struggle for various reasons need to catch up quickly and efficiently, not fall behind even further than their peers.

Children find maths enjoyable if they’re good at it! Solving puzzles can be a fun activity for kids as long as they are able to successfully complete them. Maths problems are such puzzles. It's important not to underestimate children, as they are eager to learn how to calculate like adults. They only need to have enough knowledge to do that, and fun alone cannot provide it. The aim is not simply the end result, namely the correct answer to a calculation, but rather a well-executed thought process that leads to the correct answer 2.

🐝The Levebee app provides step-by-step guidance and immediate feedback to guide the child towards successful and fluent mental performance. The child can focus entirely on the task at hand without any distractions such as unnecessary graphics or long, unrelated stories. The points earned as a reward are only handled at the end of the task. Even spending the points has a pedagogical purpose from the point of view of financial literacy. The app doesn’t offer activities that are not related to maths, yet children find it highly engaging and perceive it as a game.


1 What Will Improve a Student’s Memory?, Daniel T. Willingham, 2008
2 Flow: The Psychology of Optimal Experience, Mihaly Csikszentmihalyi, 2008


10. Automation through independent practise

Extensive and successful independent practice is necessary for skills and knowledge to become automatic and fluent 1. As basic maths skills become automatic, learners can free up their working memory capacity and improve their problem-solving abilities.

It is crucial to start practising new knowledge without time pressure in the beginning, but as learners progress, gradually increasing the cognitive load and adding time pressure can be beneficial 2. This forms the foundation for retrieving previously learned knowledge from long-term memory and leads to increased fluency even in more challenging tasks.

🐝 Levebee provides an unlimited amount of computer-generated tasks. These exercises are designed to help students practise and integrate knowledge into their existing framework of mathematical understanding. To achieve this, the activities come in different levels of difficulty and may include a time limit, if needed.


1 Principles of Instruction, Barak Rosenshine, 2010
2 Myths That Undermine Maths Teaching, Sarah R. Powell, Elizabeth M. Hughes, and Corey Peltier, 2022


11. Adaptive learning

Personalised learning is crucial, but interpretations of this term can vary, and not all interpretations may be pedagogically beneficial1:

❌ Adapting the educational content to the learning styles of the given student (visual, audio learning preferences, etc.) has already been repeatedly disproved in terms of its benefits 2 3. Choosing an appropriate learning approach from the point of view of the given content, not the individual student, has the greatest influence on the effectiveness of learning.

❌ Research suggests that relying solely on the subjective opinion of the students (what they want to study and for how long) may not be the best method for choosing an appropriate educational strategy (Dunning-Kruger effect 4). 

✅ Therefore, the most effective method of personalization is adaptive learning, which involves adjusting the educational content according to the student's actual knowledge and skills.

🐝 We use the principle of adaptive learning in the Levebee application. Thanks to the detailed structuring of mental steps into a large number of exercises, Levebee offers exercises that are always just at the limit of a child's capabilities. The child has the necessary knowledge and skills to understand problems at a more advanced level. As a result, teaching promotes the child's ongoing development and enables them to smoothly understand and solve problems. Levebee can thus automatically aim at the already mentioned 80% success rate, which saves time for the teacher as they only need to make minor adjustments to the student's learning path in the app if necessary.


1 Teachers vs Tech, Daisy Christodoulou, 2020
2 Do Visual, Auditory, and Kinesthetic Learners Need Visual, Auditory, and Kinesthetic Instruction?, Daniel T. Willingham, 2010
3 Learning styles don’t exist, Carl Hendrick, 2023
4 Unskilled and unaware of it: How difficulties in  recognizing one’s own incompetence lead to inflated self-assessments, Kruger, J. and Dunning, D., 1999


12. Formative assessment

Formative assessment provides valuable feedback to both students and teachers, helping them to adjust their teaching and learning strategies 1. It helps them understand where the student stands in terms of their knowledge and skills, and how best to support them in their education. Mathematical ideas and skills are interconnected and form a coherent structure. In order to tailor teaching strategies to individual students, diagnostic tests must be designed in accordance with this structure.

🐝 Levebee can identify areas of strength and weakness in comprehension, auto maticity, and application of mathematical concepts. With Levebee, teachers can easily identify which material individual children or the class as a whole understand and have sufficiently practised, which areas they have a basic understanding of but lack proficiency in, and what needs to be revised from the ground up. Based on the information gained from the Levebee app, teachers can adjust their teaching methods outside of the app as well.


1 Embedded formative assessment, Dylan Wiliam, 2018