Teaching mathematics part II - good teaching

In part I, I broke down Charlotte Mason's writings on the reasons for studying the subject of mathematics.  I also introduced a problem in our current educational system - that math is pretty much overinflated above the other subjects of study which not only does a disservice to our youth, but creates a disdainful attitude towards math and, I would argue, towards learning in general.

I will be honest and say that I'm not really familiar with the way math is taught in the early grades, but from what I saw by the time the kids came to me around the ages of 13 or 14 -- well, in the words of Miss Clavel*, something was not right.

(*From the Madeline books by Ludwig Bemelmans)

"...As for the value of Arithmetic in practical life, most of us have private reasons for agreeing with the eminent staff officer who tells us that, --

'I have never found any Mathematics except simple addition of the slightest use in a work-a-day life except in the Staff college examinations and as for mental gymnastics and accuracy of statement, I dispute the contention that Mathematics supply either better than any other study.'

We have most of us believed that a knowledge of the theory and practice of war depended a good deal upon Mathematics, so this statement by a distinguished soldier is worth considering.  In a word our point is that Mathematics are to be studied for their own sake and not as they make for general intelligence and grasp of mind.  But then how profoundly worthy are these subjects of study for their own sake, to say nothing of other great branches of knowledge to which they are ancillary!"  (Charlotte Mason, Volume 6, Towards a Philosophy of Education, p.232)

What the staff officer pointed out - this is what my students understood to be true and they used it as a crutch to complain about having to learn math.  We're never going to use this stuff in real life, they would claim.  But that's not the point, I would tell them.  I tried to call attention to the beauty and logical nature of mathematics, and how it weaved in and out of the other sciences, but by that point, the students just didn't care.  It was as if they were too far gone.  For the most part, though, we really can't blame the kids for this disrespect for mathematics (we can, though, for their lack of work ethic which, from my experience in the classroom, seems to be pretty profound - sorry, just had to throw that in there!).

So what's the problem?

It's interesting how CM makes such a case for mathematics requiring a good teacher from the beginning.

"There is no one subject in which good teaching effects more, as there is none in which slovenly teaching has more mischievous results..." (Home Education, p.254)

"The success of the scholars in what may be called disciplinary subjects, such as Mathematics and Grammar, depends largely on the power of the teacher, though the pupil's habit of attention is of use in these too." (Towards a Philosophy of Education, p.7)

This point of a good math teacher is brought up in the Eclectic Manual of Methods as well.

Math "is a subject in regard to which the young teacher is liable to make serious errors of judgment, both as to the method adopted and the manner of conducting recitations under that method." (p.105)

So why is it so important that mathematics have a good teacher?

"Mathematics depend upon the teacher rather than upon the text-book and few subjects are worse taught; chiefly because teachers have seldom time to give the inspiring ideas, what Coleridge calls, the 'Captain' ideas, which should quicken the imagination.  How living would Geometry become in the light of the discoveries of Euclid as he made them!" (Charlotte Mason, Towards a Philosophy of Education, p.232)

Oh how true this is!  We all know how the math that's taught in the U.S. today is "a mile wide and an inch deep."  There's no time.  No time to linger, contemplate, understand, revel in the beauty that is math.  It's all jam-it-down-your-throat-and-move-on-so-we-can-cover-the-material-before-the-big-test.  Lather, rinse, repeat.  Every year.  And because there's so much material to cover before the school year is over, math really cannot be taught properly - resulting in a lack of the necessary depth and deep understanding and reverence.

I know.  I've been there.  I tried many times to give my students time to discover and think deeply about the material.  But two things hindered that attempt:  1) I couldn't give the students the time they needed to delve into the material because we would eventually have to move on to the next topic (see above paragraph), and 2) The kids struggled, big time, because they were not used to thinking.  And there were probably two main causes for this.

First, it was obvious that they had never been given, in their earlier years, the time or the opportunity to think for themselves about mathematics.  And as a result, they were not solid foundationally, even in numbers.  And that's where it all starts - numbers!  It was AMAZING to observe how little number sense my students had, even my 8th grade Algebra students who were considered more gifted mathematically than my 9th grade Algebra students.  The majority of them could not do simple calculations in their head.

Second, there is an issue that is so common in schools today and presents big problems:  Every child moves on.  I taught at three different public schools, one of them being one of the largest and most prestigious public schools in our state, and no child was held back until they reached high school - 9th grade.  So for 9 years, even if a child failed a subject, he still moved on to the next grade!  How is a child supposed to learn Algebra when he doesn't even know the basics?  It's setting them up for frustration and failure.

So how is this remedied?  What are the steps that a good teacher should take to encourage deeper understanding and a more respectful attitude toward mathematics?

1.  Take our time and let each child progress as fast as he is able.

"Bear in mind that, in the study of Arithmetic especially, one step must be mastered before another is attempted.  Progress is necessarily slow and the golden rule is, 'Make haste slowly.'" (Eclectic Manual of Methods, p.131)

"Mr. Sealey, who has done such excellent work on the teaching of junior school mathematics in Leicester, agrees that much depends on the skill of the teacher...He considers that 'assignment cards' made specially to suit the needs of each child should become the main-spring of mathematics at this stage...If the child is to succeed in becoming 'mathematically literate' the teacher must treat him as an individual.  Different minds do not learn new concepts always in the same way.  Each child takes his own unique path to knowledge.  It is important that each child should be allowed to work at his own pace.  Miss Mason wisely remarked that the tortoise should not be expected to keep pace with the hare."  (G.L. Davies, Knowledge of the Universe, PNEU article) 

Every child is different and learns at a different pace.  It shouldn't be about finishing x number of lessons by such and such date no matter what.  Of course we need to encourage and guide our students to understand concepts in as timely a manner as possible (which really goes back to developing good habits), but we also need to give them plenty of time to master each step before moving on.  Math builds and if previous material is not learned, then the student will struggle later on.

2.  Develop the concrete ideas of mathematics first.

CM and Ray's both understand the importance of beginning with the concrete, not the abstract.  In this way, the students are presented with concrete ideas that mean something to them.  And these ideas are so critical in keeping the children interested in the work and, as a result, teaching them to think.  (Emphases mine in the below quotes.)

"The little child cannot grasp abstract ideas.  It is true you can teach him to repeat, '2 and 2 are 4;' '2 from 4 leave 2;' '2 times 2 are 4;' and '4 divided by 2 equal 2.'  But without the proper preliminary work, these words cannot possibly convey any clear meaning to his mind.  This kind of instruction in a primary class is simply machine drilling on abstract numbers and words which convey no ideas, or at best a mere jumble of ideas to the child's mind.  It is one of the worst, and at the same time one of the most common, faults in the teaching of arithmetic, and it is one which is very apt to disgust pupils with the subject from the outset.  On the other hand, if the proper method of teaching is pursued, which may properly be called the object method, the children are taught to think; they will be interested at the very beginning, and they will be kept interested by this method until they are successfully carried to the point where the object method is no longer necessary, and their minds are ready to grasp the abstract, through careful preliminary drill on the concrete." (Eclectic Manual of Methods, p.107-108)

This is also what CM said - that children should not be presented with the abstract with the intention of applying the material to real situations, but should instead be presented with the real, concrete ideas, resulting in an ability to understand the abstract later on.

"But children should not be presented with the skeleton, but with the living forms which clothe it.  Besides, is it not an inverse method to familiarise the child's eye with patterns made by his compasses, or stitched upon his card, in the hope that the form will beget the idea?  For the novice, it is probably the rule that the idea must beget the form, and any suggestion of an idea from a form comes only to the initiated...The child, who has been allowed to think and not compelled to cram, hails the new study with delight when the due time for it arrives." (Home Education, p.263-264) 

"The fact is that children do not generalise, they gather particulars with amazing industry, but hold their impressions fluid, as it were; and we may not hurry them to formulate...The child whose approaches to Arithmetic are so many discoveries of the laws which regulate number will not divide fifteen pence among five people and give them each sixpense or ninepence; 'which is absurd' will convict him, and in time he will perceive that 'answers' are not purely arbitrary but are to be come at by a little boy's reason.  Mathematics are delightful to the mind of man which revels in the perception of law..." (Towards a Philosophy of Education, p.152) 

This is what a good math teacher does - he (or she) allows the individual student time to discover and manipulate the concrete ideas that can be built upon, guiding them along the way.  This will eventually lead to the understanding of the abstract, and in the process, gain an appreciation for the beauty and truth of mathematics; the logical and rational nature of it.

{On a side note, there is, I believe, another big issue at play:  I don't think every student needs all the upper level math that we currently make them learn, and CM would agree.

"But why should the tortoise keep pace with the hare and 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.'" (Towards a Philosophy of Education, p.232-233)

Not every student has an aptitude for math or will pursue a career requiring math, and because of this, one would think they wouldn't be required to master all this upper level math.  But they are.  And it puts way too much pressure on the student and may, as a result, cause them to neglect the other subjects that provide for a well-balanced education.  But this issue requires a different kind of remedy.}

So now I've discussed two steps toward pursuing mathematics as a delightful subject worthy to be studied.  Next time I'll discuss two more steps in reaching this goal.

Other posts in this series:
Part I - why study math?
Part II - good teaching <---- you are here
Part III - good teaching continued
Part IV - laying the foundation