Conceptions of and approaches to learning mathematics

Learning approaches explain cognitive development through observable learning processes and the ways experiences shape behavior. During infancy, learning can happen through classical conditioning, operant conditioning, and modeling. These processes allow infants to form associations, respond to consequences, and learn by observing others.Classical and Operant ConditioningClassical conditioning happens when an infant associates a previously neutral stimulus with another stimulus that naturally...

In psychology, concepts can be divided into two categories: natural and artificial. Natural concepts are formed through direct or indirect experiences. For example, consider the concept of snow. If you live in a place with regular snowfall, such as Essex Junction, Vermont, you know snow through direct experiences. You’ve seen it fall, touched it, shoveled it, and played in it. You recognize its texture, appearance, and even its smell. In contrast, if you live on an island like Saint Vincent in...

Cognitive learning is based on purposive behavior, incidental learning, and insight learning.
E. C. Tolman's theory of purposive behavior emphasizes that much behavior is goal-directed. He argued that to understand behavior, we must look at the entire sequence of actions leading to a goal. For instance, high school students study hard, not just due to past reinforcement but also to achieve the goal of getting into a good college.
Tolman introduced the idea that behavior is influenced by...

Piaget’s preoperational stage, from about ages 2 to 7, is characterized by rapid growth in symbolic thinking. Children increasingly use words, images, and objects to represent ideas, although their reasoning remains largely intuitive. During this period, number sense develops from memorized counting toward an understanding of quantity and basic arithmetic.Counting and CardinalityChildren younger than age 3 may recite number words without fully understanding their numerical meaning. During the...

During Piaget's concrete operational stage, from ages 7 to 11, children exhibit a marked increase in logical thinking skills, specifically in relation to tangible, real-world events. This stage is characterized by the development of several essential cognitive concepts, including conservation, reversibility, and classification, all of which support the child's evolving capacity for structured thought.
Conservation and Constancy of Quantity
A significant cognitive milestone in the concrete...

Mathematical modeling transforms real-world scenarios into mathematical expressions, allowing for structured problem-solving and analysis. This process involves defining the situation, assigning variables to measurable quantities, selecting an appropriate model, and solving the resulting equation. Such models are invaluable in finance, providing precise methods to evaluate investments, loans, and repayment structures.A widely used example is the calculation of fixed monthly payments on a loan,...