In mathematics, synthesis is not a specific field of study but a higher-order cognitive skill. It is the process of combining distinct mathematical ideas, concepts, or procedures to form a new understanding or to solve a complex problem. This skill moves beyond simply knowing individual rules; it involves recognizing and building connections between them to see a larger, integrated picture.
The Process of Mathematical Synthesis
The act of synthesis is a constructive, bottom-up cognitive process. It involves taking individual components, which may have been identified through analysis, and composing them into a coherent whole. Analysis is the complementary process of breaking down a problem into its smaller, more manageable parts.
This constructive process leads to moments of insight, where connections that were not previously apparent become clear. For example, understanding the rules of exponents and the definition of logarithms as separate topics is one level of knowledge. Synthesizing them to understand that they are inverse operations of each other represents a deeper, more flexible comprehension that allows for more advanced problem-solving. This integration of knowledge is what allows mathematical thinking to evolve.
Synthesis in Action: Practical Examples
An example of synthesis is analytic geometry, a mathematical development in the 17th century. This field combines the principles of algebra with the spatial concepts of geometry. Before its development, algebra and geometry were largely separate disciplines. The act of synthesis was to use algebraic equations to describe, analyze, and solve problems involving geometric shapes, establishing a correspondence between curves and equations. An equation like x² + y² = r² is a synthesis, as it uses algebraic symbols to represent the purely geometric concept of a circle on a coordinate plane.
Synthesis also occurs when solving problems that span multiple mathematical and scientific disciplines. Consider calculating the trajectory of a projectile to hit a moving target. This single problem requires the synthesis of several concepts. The laws of physics provide the foundational equations for motion, while trigonometry is needed to resolve forces and velocities into their horizontal and vertical components. Calculus is then used to handle the changing rates, such as the object’s velocity and acceleration over time.
Even specific mathematical procedures can be the result of synthesis. Synthetic division is a streamlined technique for dividing a polynomial by a linear factor like (x – a). It emerged as a synthesis of the logic found in the more cumbersome process of polynomial long division. By focusing only on the coefficients and reversing the sign of the constant in the divisor, the procedure synthesizes the repetitive steps of long division into a faster, more efficient algorithm that uses fewer calculations.
Cultivating Synthesis Skills in Mathematics
Developing the ability to synthesize mathematical ideas is an active process that can be cultivated through deliberate practice. Engaging with problems that are specifically designed to require concepts from different areas of mathematics forces the brain to search for connections. Instead of solving five problems on a single topic, tackling one problem that requires five different topics builds the mental flexibility needed for synthesis.
Another effective method is to actively find multiple pathways to solve a single problem. By solving a problem using one method, and then challenging oneself to solve it again using a completely different approach, a learner builds a richer understanding. This practice reveals how different concepts and procedures can be equivalent or related, strengthening the network of knowledge. Comparing the elegance or efficiency of different solutions further deepens this integrated understanding.
Creating visual representations, such as concept maps, is a direct way to practice synthesis. This involves placing a central idea on a page and then drawing explicit links to other related concepts, labeling each link to describe the relationship. This act of externalizing and organizing knowledge forces a person to articulate the connections between different mathematical truths. Similarly, the act of explaining a complex mathematical idea to someone else requires the speaker to synthesize their own understanding into a coherent and logical narrative, solidifying their grasp of the material.