CM and Math

The past few years, math and writing have been bees stuck in my bonnet.  I'm parking a few math links here for my own help.  Perhaps they will help others too.

Miss Mason:

There is no one subject in which good teaching effects more, as there is none in which slovenly teaching has more mischievous results...a child who does not know what rule to apply to a simple problem within his grasp, has been ill taught from the first, although he may produce slatefuls of quite right sums in multiplication or long division...Care must be taken to give the child such problems as he can work, but yet which are difficult enough to cause him some little mental effort.

(Give the child problems which are difficult enough to cause him some effort.  This is key in understanding so many of CM's recommendations.  CM is gentle, yet rigorous.  CM respects the child, yet challenges the child.)

More from CM on arithmetic:

The copying, prompting, telling, helping over difficulties, working with an eye to the answer which he knows, that are allowed in the arithmetic lesson, under an inferior teacher, are enough to vitiate any child; and quite as bad as these is the habit of allowing that a sum is nearly right, two figures wrong, and so on, and letting the child work it over again. Pronounce a sum wrong, or right––it cannot be something between the two.

On the beauty of mathematics: 

 We take strong ground when we appeal to the beauty and truth of Mathematics; that, as Ruskin points out, two and two make four and cannot conceivably make five, is an inevitable law. It is a great thing to be brought into the presence of a law, of a whole system of laws, that exist without our concurrence,––that two straight lines cannot enclose a space is a fact which we can perceive, state, and act upon but cannot in any wise alter, should give to children the sense of limitation which is wholesome for all of us, and inspire that sursum corda which we should hear in all natural law.

On examination-driven teaching of math:

Arithmetic, Mathematics, are exceedingly easy to examine upon and so long as education is regulated by examinations so long shall we have teaching, directed not to awaken a sense of awe in contemplating a self-existing science, but rather to secure exactness and ingenuity in the treatment of problems.

(I want to awaken a sense of awe in my students.  Math was created by God.  It cannot be denied that mathematical laws exist outside of man-made laws.)

On math and success:

...why should a boy's success in life depend upon drudgery in Mathematics? That is the tendency at the present moment to close the Universities and consequently the Professions to boys and girls who, because they have little natural aptitude for mathematics, must acquire a mechanical knowledge by such heavy all-engrossing labour as must needs shut out such knowledge of the 'humanities' say, as is implied in the phrase 'a liberal education.'... [Mathematics] may not engross the time and attention of the scholar in such wise as to shut out any of the score of 'subjects,' a knowledge of which is his natural right.

(Lord, please help me remember this!  Awaken awe, challenge the student, keep math focus in perspective relative to other subjects.)