![]() It is very easy for students not used to mathematical logic to get confused by our sometimes-inconsistent non-mathematical spoken language. In the end, I suspect that we all need to be very careful with all of our language about classifying (or defining) all mathematical objects (geometric shapes, functions, properties, etc.). I think David’s “classifications” may be what I’m calling “inclusive definitions”. I ‘blogged on this a year ago ( ) and got some pretty tough pushback from some readers who were perfectly happy to call a square a rectangle while denying vehemently that a parallelogram could ever be a trapezoid.ĭavid Wees elegantly captured in his comment above the point I’m trying to make when he noted the difference between classifying and defining. Some define trapezoids as quadrilaterals with EXACTLY one pair of parallel sides (a la Euclid, an exclusive definition) while others define trapezoids as quadrilaterals with AT LEAST one pair of parallel sides (an inclusive definition). On my point of exclusive vs inclusive definitions, consider Trapezoids. There are some people and textbooks who define rectangles (explicitly or implicitly) as right-angled quadrilaterals with opposite sides congruent that aren’t congruent to each other–from this perspective, squares are NOT rectangles. Now that I have learned to be explicit about this, I rarely encounter this problem anymore.Īt a deeper level, I believe this boils down to exclusive vs inclusive definitions in mathematics, and unfortunately for our students, we teachers are not at all consistent in our usage. Worse, I used to say that the standard english interpretation was wrong, which of course is completely confusing, it is a different convention but not at all wrong. We just say “quadrilateral with two sides equal” and expect students to catch on. The problem is that many of us teachers don’t make this explicit, because we are so used to it ourselves. ![]() ![]() But of course in math “quadrilateral with two sides equal” means “quadrilateral with AT LEAST two sides equal”. Given four cats of the same color, if someone said “two of these cats are the same color”, in standard english logic everyone would agree the statement is false, because it means “two AND ONLY TWO of these cats are the same color”. In standard english logic, “quadrilateral with two sides equal” means “quadrilateral with EXACTLY two sides equal” – this usage is consistent and sensible. Standard english logic is pretty much self-consistent, just different (and not so convenient for math). This is a mistake I have made teaching for years: not recognizing and explicitly explaining that mathematical english logic and standard english logic are different. ![]()
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