Constructivism is a psychological theory that roots itself in the idea that learning is cummulative. In order to learn and develop, a person must apply both the knowledge that he knows and the skills that he has mastered to the new stimuli and information that he receives from the world around him. The theory focuses on the step-by-step development of a child over the course of his childhood. Despite the long-term nature of the theory, the concepts behind accommodation and assimilation, which are critical components to constructivism, apply to the cumulative nature of mathematics. 

Mathematics, perhaps more than any other subject, is quite evidently cummulative. Children at an early age learn how to count, which is the foundation for mathematics. Without this knowledge, they will not be able to learn all subsequent topics in the subject. In order to count (and actually be able to conceptualize what counting means), young students must use Piaget's processes for learning, accommodation and assimilation. Assimilation is when a child makes "sense of new experiences in terms of what they already know," and accommodation is when a child takes "on board new things as they experience them," (New Learning 203). In this case, assimilation happens as the child applies his developed understanding of object permanence to the idea of differentiating between objects; for example, in order to count oranges, a child must first understand and recognize what an orange is, then he must understand that more than one orange might exist. With this information, he can then begin to count oranges. Accomodation occurs when a child obtains new information to supplement what he already knows. This new information might alter some of his previous assumptions, and accommodation is the ability to apply and contextualize this new information to make it useful. Accomodation in the case of counting might be applying the idea behind counting one-by-one to addition, which is counting, but not necessarily item for item. The child would accommodate the new knowledge that says that 1+1+1+1+1, which is counted as 1, 2, 3, 4, 5 can be grouped into 1+1 = 2 and 1+1+1 = 3, and those numbers 2 and 3 can be added to each other as 2 + 3 to make 5, which can be explained in 2 and 3's most broken down forms as (1+1) + (1+1+1) = 5. Ultimately, mathematics, as a cumulative subject that builds upon previous skills and knowledge, is an example of the constructivist theory that assers that learning happens when children actively apply their former knowledge of the world with their new understandings of the world (New Learning 203).

This concept is incredibly insightful because it emphasizes the importance of establishing a strong foundation for early childhood education, and it also helps to explain why so many students who struggle early on in their academic careers end up dropping out of school or failing to find steady footing. The constructivist theory would argue that students who are in classes that are too advanced beyond their knowledge and skillset will not be able to assimilate and accommodate appropriately in order to learn and ultimately achieve to their mental capacity. This type of philosophy supports the idea behind benchmark testing that ensures that students learn fundamentals well before they advance to new subject matter.

Although the constructivist theory seems to be realistic and logical, it ignores the fact that some subjects can be learned as individual entities, perhaps out of the context of information that might otherwise be deemed fundamental to its acquisition. However, despite the fact that this could happen, it doesn't mean that the learner actually understands why what they're learning is true, which suggests that they haven't truly learned that information. For example, a student might be able to memorize a series of perfect squares, but that doesn't necessarily mean that he understands what perfect squares are, especially if he doesn't understand how to multiply. In other words, a child might know that 5^2 = 25, but he might not be able to explain what a square number is or be able to relate square numbers to multiplication.