Using Well CRAFTED Questions to Develop Early Math Skills

• Early Math Skill Development: Developing strong mathematical skills in early childhood is crucial for both academic success in STEM and everyday quantitative reasoning, shaping how children think and solve problems.
• Brain Areas in Math Learning: Specific brain areas like the intraparietal sulcus and the ventral temporal-occipital cortex are instrumental in mathematical cognition, evolving from general processing in young children to specialized functions as they grow.
• Shift in Cognitive Processing: As children age, there's a noticeable shift from reliance on memory and cognitive control systems to increased reliance on specialized functional networks in the brain, which enhances mathematical processing and understanding.
• Impact of Targeted Math Training: Effective math training can significantly change brain activity and connectivity, especially in the hippocampus, underscoring the importance of well-designed instructional strategies. This is particularly beneficial for children with mathematical learning disabilities.
• Application and Transfer of Math Knowledge: Learning and transferring mathematical knowledge involves both shared and distinct neural mechanisms, indicating that education should not only focus on knowledge acquisition but also on its application in varied contexts.

Designing of Learning

We developed this series of articles with two primary objectives: 1) to highlight some of the key decisions we have made and features we have included that we believe demonstrate the uniqueness of our app and how they support student success, and 2) to share the research supporting the decisions we have made and the features our app includes.

Many of these articles focus on our learning experience design decisions, such as the strategies we have implemented for badging, customization, feedback, social-emotional learning, color and font selection, and more. These strategies provide essential support for student success, both individually and together. Indeed, much of the research on app design focuses on these sorts of elements, highlighting the effectiveness of features such as multiple attempts, immediate feedback, developmental progression, and overall game design in improving student outcomes [1].

Designing for Learning

At the same time, in the research into gamification in mathematics education, very little attention is given to the actual construction of the learning interactions. In other words, while current research has a lot to say about developing gamified learning environments—and we have leveraged these insights in the development of our app—it has essentially nothing to say about the science of mathematics teaching and learning within those gamified environments.

As developers of a game-based, gamified math learning app, we consider effective learning experience design foundational to our app development and our overall approach to supporting math learners.

And, as expert mathematics educators, we have pushed beyond game design by prioritizing and operationalizing research-backed strategies on the science of teaching and learning mathematics.
Leveraging new insights about neurodevelopment in mathematics education and combining many well-known and several less common mathematics teaching ideas and strategies, our learning interactions—the questions we ask and the way we ask them—are designed to build math skills incrementally, resulting in exponential development.

Neurodevelopment in Math

New insights into the neurodevelopmental pathways of mathematical learning underscore both the complexity of the process of mathematics learning as well as the importance of well-designed instructional strategies to support skill development. Let’s look at some of the most recent ideas [2].

Importance of Early Mathematical Development

Developing strong mathematical skills in early childhood is crucial. These foundational skills are not just important for academic success in STEM areas, but they also play a significant role in everyday quantitative reasoning. This early skill acquisition is shaped by the development and specialization of functional brain networks.

Neurocognitive Systems in Math Learning

Mathematical learning is supported by specific neurocognitive systems. These include systems for processing symbolic (like numbers) and non-symbolic (like an array of dots) quantities. The intraparietal sulcus (IPS) in the posterior parietal cortex and the ventral temporal-occipital cortex (VTOC) are key areas involved. These areas form the core building blocks for higher-level mathematical cognition.

Interactive Specialization in Development: As children grow, there's a notable shift in how their brain processes math. Younger children rely more on memory and cognitive control systems, but as they develop, there’s increased reliance on specialized functional networks, particularly in areas like the IPS and VTOC. This reflects a transition from general to more specialized processing.

Role of Medial Temporal Lobe: The medial temporal lobe, especially the hippocampal region, is pivotal in transitioning from less efficient procedural strategies to more efficient arithmetic fact retrieval during the early school years. This suggests a developmental shift from a reliance on working memory to a more automatic retrieval of math facts.

Math Training and Cognitive Development: Effective math training can induce significant changes in brain activity and connectivity, especially in the hippocampus. This plasticity demonstrates that targeted training can lead to improvements in mathematical problem-solving, highlighting the importance of well-designed instructional strategies. This is particularly true for children with mathematical learning disabilities, where appropriate interventions can lead to the normalization of brain activity and connectivity in key areas responsible for numerical problem-solving.

Brain Network Plasticity in Math Learning: Mathematical cognition involves distributed brain systems. Analysis of these systems reveals that mathematical learning and proficiency are supported by the organization and reorganization of large-scale brain networks, which can be modified through learning and intervention.

Transfer of Mathematical Learning: Understanding how children transfer mathematical knowledge to new contexts is crucial. Recent studies show that learning and transfer involve both shared and distinct neural mechanisms. This indicates that effective math education should focus not only on knowledge acquisition but also on the application of this knowledge in varied contexts.

Let’s Simplify This

That was a lot of information—good information, but a lot! To help make sense of all that, we offer a simplified version that will set up an overview of how we operationalize all these insights into practical strategies in our app through our CRAFTED question framework.

• Learning math early is important because it helps kids think better about numbers and solve everyday problems.
• There are special parts in the brain that get really good at understanding numbers and amounts.
• As kids grow, their brains get better at using these parts, making math easier.
• A part of the brain called the medial temporal lobe helps kids remember math facts quickly.
• Practicing math can actually change the brain, making it better at solving math problems.
• This brain change is especially helpful for kids who find math hard, as it can make things easier for them.
• Overall, learning math is not just about knowing numbers; it is also about being able to use this knowledge in different situations.

Developing Neurodevelopmental Connections through Well CRAFTED questions.
What all this tells us is that effective instructional strategies need to support the organization and reorganization of large-scale brain networks in an intentional and progressive manner.

In our app, we accomplish this through our well CRAFTED question framework. In this approach, we have intentionally designed every learning interactions with a combination of insights of neurodevelopment and specific mathematics teaching practices.

C: Concrete-Representational-Abstract (CRA)

CRA involves a progression from using concrete objects, to visual representations, and finally to abstract symbols and numbers [3]. This progression is foundational to all the learning interactions we utilize, as it mirrors the neurodevelopmental pathways in the brain's construction of numerical cognition. It specifically engages areas like the intraparietal sulcus, which is pivotal for quantity representation, and the ventral temporal-occipital cortex, which is essential for processing visual number forms. This alignment with neurodevelopment suggests that as students move from concrete to abstract mathematical concepts, they are also tracing the natural development trajectory of their brain's numerical cognition systems.

R: Relatability

We focus on relatability by connecting mathematical concepts to familiar and relevant contexts for the learner. For example, emphasizing familiar number sets like 5s and 10s leverages working memory and declarative memory systems in the brain, engaging regions responsible for the manipulation of quantity representations [4]. It capitalizes on the familiarity and innate understanding of one's own body, such as in finger counting and the ubiquity of base-10 systems in our environment. This strategy enhances mathematical learning and retention by tapping into neurocognitive pathways that are already well-established and frequently used in everyday life, thus making the learning process more intuitive and relatable.

A: Aggregability

By aggregability, we mean teaching concepts through strategies that allow students to break down complex mathematical problems into smaller, more digestible parts. For example, by chunking information and looking for symmetry and balance [5] amongst quantities, students utilize areas of the brain responsible for cognitive control and working memory, like the dorsolateral prefrontal cortex and the intraparietal sulcus. This strategy aligns with the brain's ability to organize and manipulate information, specifically engaging cognitive control processes. This approach not only makes mathematical problems more manageable but also mirrors the brain's natural process of categorizing and processing complex information.

F: Flexibility

Through all our questions, we aim to teach specific concepts in a way that allows for cognitive flexibility that empowers the transfer of math skills to novel challenges. For example, utilizing strategies such as doubling and halving in multiplication, encourages diverse problem-solving approaches [6]. This method involves the cognitive control system of the brain, integrating different functional circuits for numerical problem-solving. This flexibility is managed by regions such as the insula and the dorsolateral and ventrolateral prefrontal cortex. By promoting different mathematical techniques, this strategy demonstrates and enhances the brain's natural adaptability and capacity for diverse cognitive processing in solving mathematical problems.

T: Tangibility

As often as possible, we link abstract mathematical symbols to more tangible, concrete concepts. We achieve this through the use of digital manipulatives and interactive models that represent mathematical concepts in a more physical or visual form [7]. The strategy aligns with the role of the medial temporal lobe, especially the hippocampus, in binding neural representations across symbolic and non-symbolic quantities. This suggests that making abstract concepts tangible facilitates the hippocampus's role in connecting different forms of numerical representations, thereby enhancing the understanding and retention of mathematical concepts.

E: Exponentiality

Our focus on exponentiality emphasizes the natural human tendency to conceptualize numbers logarithmically rather than linearly [8]. For example, teaching students to think in terms of exponential growth or decay, connects with the role of hippocampal-neocortical circuits in mathematical learning. The hippocampus plays a crucial role in binding mathematical components such as operands and answers. This strategy leverages our innate inclination towards exponential thinking in mathematics, aligning with the natural functioning of the brain's learning and memory systems.

D: Developmental Progression

We embed all of these strategies (and others) within specific patterns and structures [9] that operate at a meta-level to impel a developmental progression in math learning that aligns with the brain's plasticity and its ability to reorganize large-scale networks in response to learning and intervention. This approach involves gradually increasing the complexity of math problems as students' skills develop, mirroring the evolving organization of brain networks crucial for mathematical cognition. By following a developmental approach, this strategy reflects how the brain's networks evolve and adapt, accommodating the growing mathematical abilities of learners and fostering a deeper understanding of mathematical concepts over time.

The Brain Science of Math Learning

Each element of our CRAFTED framework is intentionally linked to specific neurodevelopmental processes and operationalized by research-based teaching strategies. Understanding and leveraging these connections has allowed us to create brain-compatible learning interactions that build math skill effectively and intuitively, resulting in more confident and capable math learners!

[1] Moyer-Packenham, P. S., Lommatsch, C. W., Litster, K., Ashby, J., Bullock, E. K., Roxburgh, A. L., ... & Jordan, K. (2019). How design features in digital math games support learning and mathematics connections. Computers in Human Behavior, 91, 316-332. https://doi.org/10.1016/j.chb.2018.09.036

[2] Menon, V., & Chang, H. (2021). Emerging neurodevelopmental perspectives on mathematical learning. Developmental Review, 60, 100964. https://doi.org/10.1016/j.dr.2021.100964

[3] Gallo-Toong, N. (2020). The extent of use of concrete-representational-abstract (CRA) model in mathematics. International Journal for Research in Mathematics and Statistics, 6(5), 1-25.

[4] Kullberg, A., Björklund, C., Brkovic, I., & Runesson Kempe, U. (2020). Effects of learning addition and subtraction in preschool by making the first ten numbers and their relations visible with finger patterns. Educational Studies in Mathematics, 103, 157-172. https://link.springer.com/article/10.1007/s10649-019-09927-1

[5] Lipka, J., Adams, B., Wong, M., Koester, D., & Francois, K. (2019). Symmetry and measuring: Ways to teach the foundations of mathematics inspired by Yupiaq Elders. Journal for Humanistic Mathematics, 9(1), 107-157. https://scholarship.claremont.edu/jhm/vol9/iss1/7

[6] Downton, A., Russo, J., & Hopkins, S. (2022). Students’ understanding of the associative property and its applications: Noticing, doubling and halving, and place value. Mathematics Education Research Journal, 34(2), 437-456. https://doi.org/10.1007/s13394-020-00351-w

[7] Colgan, L. (2021). Elementary math matters: It’s a bird, it’s a plane . . . it ‘s a math manipulative!. Gazette-Ontario Association for Mathematics, 59(3), 43-45. http://proxy-ln.researchport.umd.edu/login?url=https://www.proquest.com/scholarly-journals/elementary-math-matters-bird-plane-manipulative/docview/2546656046/se-2

[8] Dehaene, S., Izard, V., Spelke, E., & Pica, P. (2008). Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures. Science, 320(5880), 1217-1220. https://www.science.org/doi/10.1126/science.1156540

[9] Mulligan, J., Oslington, G., & English, L. (2020). Supporting early mathematical development through a ‘pattern and structure’ intervention program. ZDM, 52, 663-676.