Here is another installment in my series reviewing the NY State Regents exams in mathematics.

It is a true geometric wonder that a triangle’s medians always intersect at a single point. It is a remarkable and beautiful result, and the fact that the point of intersection is the *centroid* of the triangle makes it even more compelling.

This result should absolutely be a part of the standard Geometry curriculum. It important and beautiful mathematics, it extends a fundamental notion of mathematics (symmetry) in new ways, and it is readily accessible through folding, balancing, compass construction, and coordinate geometry.

But here’s what happens when high-stakes testing meets meaningful mathematics.

This wonderful result has been reduced to an easy-to-test trick: *the centroid divides a median in a 2:1 ratio*.

It’s not hard to see how such a fact can quickly become an instructional focus when it comes to centroids: if that’s how it’s going to be tested, that’s how it’s going to be taught. Of course, teachers should do more than just teach to a test, but there’s a lot riding on test results these days, and it’s hard to blame teachers for focusing on test scores when politicians, policy makers, and administrators tell them their jobs depend on it.

This is just one example of many, from one test and one state. This is an inseparable component of standardized testing, and it can be found in all content areas and at all levels. And for those who argue that the solution is simply to make better tests, keep this in mind: New York has been math Regents exams for over eighty years. Why haven’t we produced those better tests yet?

This is trivial pursuit Geometry. Good for you for pointing out this “problem”.

Yes, it is a true geometric wonder that a triangle’s medians always intersect at a single point.

It is a miracle that the medians of a single non-isoseles triangle intersect at a single point.

It is a double miracle that this is true for all triangles.

It is another true geometric wonder that a tetrahedron’s four medians always intersect at a single point.

Also, a true geometric wonder that a parallelepiped’s four diagonals always intersect at a single point.

Jerome: these results are not so surprising. Since the medians are defined affinely, and since all triangles lie in a single orbit of the affine group, both properties of medians (intersection and the 2:1 ratio) follow from what happens in the equilateral triangle.

Here’s an informal version of the above formal argument (someone should make an animation): imagine fixing two vertices of a triangle (that is, the length of one side) and then moving the third vertex in a straight line from the equilateral point to an arbitrary point. The medians are defined by averages, so points on them also move linealy at constant speeds. In particular this will apply to the point 2/3 along the way down any median, so the three medians still intersect, and the point of intersection is still 2/3 down each of them.