Why is study of conceptual math important even for those who don't intend to pursue math, science, or advanced engineering fields? Studying conceptual mathematics develops solid logic and critical thinking skills. It trains the mind in important thinking processes for problem-solving and decision-making, both of which affect performance in other areas. Also, the SAT test includes problem-solving oriented math (rather than application of formulas, which is where most homeschool math focuses) through Algebra 2 (Advanced Algebra), and standardized test scores are the most heavily weighted single factor in college admission. Until relatively recently, study of conceptual math has historically been a staple of a good education, including even a "classical" Humanities education, in part because of its role in teaching logical analysis and thinking skills.

Why is a foundation in conceptual math essential for those who might pursue fields like math or science that are based on Calculus and higher math? Formula-based math does not prepare students well for Pre-Calculus and beyond, where the math becomes almost entirely conceptual. This is why math curriculums designed specifically for homeschoolers -- which are all formula-based, so they are easier to study at home -- can't effectively include Pre-Calculus and higher as part of the curriculum.

What makes conceptual math hard to learn and teach well? One major reason is that real math requires both inductive thinking (coming up with

*how*to solve the problem) and deductive thinking (coming up with the

*right*answer from the approach you have chosen). Many math curriculums, including many that are popular for being easier for students to learn and teachers to teach, start with math formulas and focus on how to use them to get answers for specific types of problems. This approach deemphasizes the inductive side, and limits the deductive side to a specific set of recognizable problem types, making "math" easier to learn, but also skipping much of what makes math effective for developing problem-solving and critical thinking skills. (It also misses one of the primary aspects of the math portion of the SAT test.)

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