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.
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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?
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.
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
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.
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
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 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.
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.
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
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.
1 Principles of Instruction, Barak Rosenshine, 2010
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.
1 Myths That Undermine Maths Teaching, Sarah R. Powell, Elizabeth M. Hughes, and Corey Peltier, 2022
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.
1 A Thousand Brains: A New Theory of Intelligence, Jeff Hawkins, 2021
2 Embedded formative assessment, Dylan Wiliam, 2018
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.
1 The Use of Scaffolds for Teaching Higher-Level Cognitive Strategies, Barak Rosenshine and Carla Meister, 1992
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.
1 Executive Functions, Adele Diamond, 2013
2 Cognitive Load Theory in Action, Oliver Lovell, 2020
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.
1 What Will Improve a Student’s Memory?, Daniel T. Willingham, 2008
2 Flow: The Psychology of Optimal Experience, Mihaly Csikszentmihalyi, 2008
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.
1 Principles of Instruction, Barak Rosenshine, 2010
2 Myths That Undermine Maths Teaching, Sarah R. Powell, Elizabeth M. Hughes, and Corey Peltier, 2022
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.
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
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.
1 Embedded formative assessment, Dylan Wiliam, 2018