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For over 30 years, with NCTM reforms, we have been talking about “teaching through problem-solving”, rather than teaching *about* problem-solving. I am not sure “teaching through problem-solving” is all that well-defined.
In fact, one prominent MATHCOUNTS coach tells us that he spends nearly 50% of his preparation time with his students covering counting and probability topics, because of their importance in MATHCOUNTS contests.
Algebra is often taught as a series of formulas and algorithms for students to memorize (for example, the quadratic formula, solving systems of linear equations by substitution, etc.), and geometry is often taught as a series of exercises that are often done by rote (for example, the infamous ‘Two-Column Proof’).
Indeed, at most universities, a undergraduate-level course in discrete mathematics is a required part of pursuing a computer science degree.
Many students’ complaint about traditional high school math—algebra, geometry, trigonometry, and the like—is The somewhat abstract nature of these subjects often turns off students.
We strongly recommend that, before students proceed beyond geometry, they invest some time Introduction to Number Theory textbooks, or by signing up for our introductory Counting and Number Theory classes—with very little algebra background.