Teaching mathematics part IV - laying the foundation

In parts II and III, I discussed why mathematics needs a good teacher, and what that good teaching entails.

Now, let's look at the how.  And let me point out that, at this point, we are mostly concerned about how to teach the subject from the beginning, in the early years.  This is when good teaching is so critical and can make or break a child's attitude toward and understanding of math.  A young math student needs to have a solid foundation upon which to build if he or she is going to be successful in the later years.

Children should not be rushed through mathematics, but should be allowed to take it slowly - slowly enough for each individual child to fully grasp the topics.

"...In arithmetic, above all other studies of the common school course, it is of the utmost importance that one step shall be thoroughly understood before the next is attempted.  The first two years' training is of more importance than all the rest the child receives." (The Eclectic Manual of Methods, p.107) - emphasis mine

Charlotte Mason also understood the importance of laying a solid foundation in the early years.

"Carefully graduated teaching and daily mental effort on the child's part at this early stage may be the means of developing real mathematical power, and will certainly promote the habits of concentration and effort of mind." (Home Education, p.257)

"If the child does not get the ground under his feet at this stage, he works arithmetic ever after by rule of thumb." (Home Education, p.259) 

So where do we start?  How do we lay this so-important solid foundation?  What steps do we take to begin our children in the study of mathematics?

It all begins with numbers - learning numbers as whole quantities and learning all the different combinations that make up that quantity.  This is how the child's mental math ability is built up.  He is taught good number sense, mentally, which at the same time teaches him to think more deeply about mathematics.  A child who does not come away in the first year (or two, if needed) with good number sense has not really been trained to think on a deeper level and understand how math is unfolding.  And that will be detrimental to the rest of his math instruction.  If he doesn't learn to think early on, he'll struggle.  Period.

So, let's go through the method of teaching the first year and we'll see how Ray's Arithmetic mirrors Charlotte Mason's writings on teaching math.

Here we go!

1.  Teach through oral instruction, focusing on the concrete.

"Do not teach the figures in the first lessons, and do not allow the children to do any written work; but teach orally, illustrating every operation, at first, by means of various objects. -- The instruction should be entirely oral, and should deal altogether at first with concrete numbers." (Eclectic Manual of Methods, p.107-108) 

CM also advocates oral instruction and, as we've already discussed, beginning with the concrete.

"Give him short sums, in words rather than in figures..." (Home Education, p.261)

If we begin with written instruction, then we're really beginning with abstract ideas.  Having a child complete a math problem like 2 + 4, on a worksheet or on the board, is an abstract idea, which, as I discussed in an earlier post, is much more difficult for a child to grasp than asking the child how many are 2 beans and 4 beans.  The latter is a concrete idea and is much easier to understand.

2.  Teach through the use of objects to manipulate.

"Begin the teaching of arithmetic, then, with objects, -- blocks, balls, marbles,...etc." (Eclectic Manual of Methods, p.108)

 CM also recommended using objects in the beginning.

"A bag of beans, counters, or buttons should be used in all the early arithmetic lessons, and the child should be able to work with these freely, and even to add, subtract, multiply, and divide mentally, without the aid of buttons or beans, before he is set to 'do sums' on his slate." (Home Education, p.256)

This not only allows the child to understand the concrete ideas first, which will eventually pave the way to understanding the abstract, but will get the student involved and, in turn, get him to grasp the rationale of numbers and how they work himself, (rather than just being spoon-fed by the teacher).

3.  Teach numbering, first with objects, then without.

"The first step is to teach numbering; that is, so to train the child that he can instantly give the number of any group of objects not exceeding ten, at sight, and without counting...Do not allow a child to count by ones to find how many objects there are in a group, but teach him to recognize the group as a whole. -- Teach what three means by repeatedly combining two and one, and one and two, into groups of three apples, three blocks,...etc,...teach four in the same manner...As soon as the class is sufficiently advanced, have the children do the combining and separating of objects for themselves." (Eclectic Manual of Methods, p.108-109)

This teaching of numbering is so important - it is the first step toward training a child to think rapidly and accurately.  It is strengthening a child's mental math ability from the get-go.  The child sees all the different combinations that make up a particular number, which prepares him for addition and subtraction.  He also learns to recognize each quantity as a whole rather than teaching him to rely on counting.  A child who learns to rely on counting is not learning to think mentally.  There is also a good chance that the child will carry this habit of counting over to adding numbers - he'll rely on counting to add instead of his mental ability.

This is very similar to CM's method.

"He may arrange an addition table with his beans, thus --

OO   O          = 3 beans

OO   OO       = 4 "

OO   OOO    = 5 "

and be exercised upon it until he can tell, first without counting, and then without looking at the beans, that 2+7=9, etc." (Home Education, p.256) - emphasis mine

The place where the two methods differ is when to have the students learn to recognize the combinations of imaginary objects and eventually abstract numbers.

"From combining and separating objects they can see, lead them to combine and separate groups of objects that they cannot see, but can readily imagine, such as animals, houses, trees, tools, toys, or any objects with which they are familiar." (Eclectic Manual of Methods, p.110-111)

So the Manual of Methods recommends introducing imaginary objects after mastering the combinations of objects before their eyes; then abstract numbers after mastering imaginary objects.

"When you are satisfied with the results of the work thus far, take the next step by simply dropping the names of the objects, and teach the abstract digital numbers orally in the same order that you taught the concrete." (Eclectic Manual of Methods, p.111)

It seems that CM, however, recommends teaching with imaginary and abstract along with using objects that the children can see.  Continuing with the above CM quote from p.256 in Home Education:

"Thus with 3, 4, 5, - each of the digits:  as he learns each line of his addition table he is exercised upon imaginary objects, '4 apples and 9 apples,' '4 nuts and 6 nuts' etc.; and lastly, with abstract numbers - 6+5, 6+8." 

4.  Teach addition and subtraction together.

"The one is the reverse of the other, and when taught together they help the child to understand each process more readily than if they were taught separately." (Eclectic Manual of Methods, p.110)

"A subtraction table is worked out simultaneously with the addition table.  As he works out each line of additions, he goes over the same ground, only taking away one bean, or two beans, instead of adding, until he is able to answer quite readily, 2 from 7? 2 from 5?  After working out each line of addition or subtraction, he may put it on his slate with the proper signs, that is, if he has learned to make figures." (Charlotte Mason, Home Education, p.256-257) 

5.  Teach the written characters last.

"...when the children are thoroughly proficient in the preceding steps, teach them the written characters that stand for the numbers which they have learned to use orally." (Eclectic Manual of Methods, p.111)

It seems that CM would agree with this as well - mastering the material mentally through oral teaching before any written work.

 "...the child should be able to work with these freely {objects}, and even to add, subtract, multiply, and divide mentally, without the aid of buttons or beans, before he is set to 'do sums' on his slate." (Home Education, p.256)

Note, however, that this does not mean to teach the children to write the numbers themselves.  It's just teaching them to recognize the written numbers.

"It is granted that much greater apparent advance can be made at this time, and that children can be taught the names of numbers as high as a hundred or more, and to write the figures representing them; but the learning of names and the making of figures do not of themselves imply the gaining or developing of ideas, and classes forced too rapidly over the preliminary ground without thoroughly understanding each step as they advance, will sooner or later show the bad effect of this method of teaching." (Eclectic Manual of Methods, p.113-114)

All that is required in the first year of mathematics is mastering groups of numbers, and addition and subtraction of numbers, through 10.

What this does is present the concrete ideas of mathematics which the child can understand, as opposed to presenting mathematics abstractly, which he cannot.  Instead of beginning with the abstract idea of 2 + 1 = 3, for example, we begin with the child combining and separating 2 beans and 1 bean.  And in allowing the child time to manipulate and work freely with these concrete ideas, he develops an understanding of how mathematics works - its logical nature and the truth and beauty and exactness of it.  Using this method to teach subsequent topics - addition and subtraction with larger numbers, place value, multiplication, division, fractions, weights and measures, etc. - will hopefully develop in the child an appreciation and respect for the subject as an immutable law of the universe, worthy to be studied deeper.

~~~

So, we can see how Ray's Arithmetic so far seems to be very conducive to the way CM taught mathematics.  I think that as long as these general steps are taken early on, allowing the child enough time to fully understand each step, we'll see our children begin to flourish in the subject.

One last question!  How will we know that our children are progressing the way they should be?

"Accuracy and rapidity are the important aims, and the children should be drilled until they can give the answers to all the possible combinations and separations instantly, and apparently without stopping to think." (Eclectic Manual of Methods, p.111)

 "...excite him in the enthusiasm which produces concentrated attention and rapid work.  Let his arithmetic lesson be to the child a daily exercise in clear thinking and rapid, careful execution, and his mental growth will be as obvious as the sprouting of seedlings in the spring." (Charlotte Mason, Home Education, p.261)

Accuracy and clear, rapid thinking from the child is how we'll know that they're on the right track, and that they're building that solid foundation so important to future success in mathematics.

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

Teaching Mathematics part III - good teaching continued

Last time I discussed two steps that math teachers should take in helping students pursue math as a delightful subject to study.  Let's talk about two more important things teachers should do.

3.  Demonstrate the hows and whys.

This is very much related to what I mentioned in my previous post about introducing the concrete before the abstract.  In doing so, these concrete ideas must be demonstrated to the children.

“The next point is to demonstrate everything demonstrable. The child may learn the multiplication-table and do a subtraction sum without any insight into the rationale of either. He may even become a good arithmetician, applying rules aptly, without seeing the reason of them; but arithmetic becomes an elementary mathematical training only in so far as the reason why of every process is clear to the child. 2+2=4, is a self-evident fact, admitting of little demonstration; but 4x7=28 may be proved."  (Charlotte Mason, Home Education, p.255-256) - emphasis mine

I remember as a teacher, teaching that "mile wide and inch deep" curriculum in public schools, really only having enough time to just tell my students what to do in order to solve a problem because, like I said in my previous post, we had to cover a certain amount of material before the end of the year (really, before the big test).  But this method only allows for a "monkey see, monkey do" kind of learning, with the students just memorizing steps, not gaining deep understanding.  And this results in children quickly forgetting the material.

I needed time to give my students the opportunity to go behind the scenes, so to say, and discover the how and why of the material.  And sometimes - very rarely - we would have a little time to do that.  But, honestly, it would not go over well.  My students (remember they were 8th and 9th graders) either 1) did not care about the how and why and just preferred me to tell them what to do, or 2) got lost - the demonstrations and proofs of the mathematical processes would completely go over their heads.

So that told me that they were not used to this - to getting underneath the surface and grasping the rationale of the material.  And this not only results in a lack of understanding the material, but a lack of understanding math for what it is.

For example, if children could see how multiplication works it will provide for a much greater understanding than just memorizing the multiplication table will ever do.  And a firm understanding of the early concepts and how they work will have a domino effect - it will make it easier for the child to comprehend more and more mathematical processes because he is being trained to think, not memorize.

"In introducing these new processes of multiplication and division, the principle of developing ideas before words should control, and the knowledge already gained should be used as a stepping-stone to the acquisition of the new ideas.  For example, the child already knows that 2+2+2=6; also, that 3+3=6.  With this knowledge as a basis, it is very easy to show him that three times 2 are 6, and that two times 3 are 6.  Thus, the child's knowledge of addition is used in teaching him multiplication.  This seems much more rational that at once plunging into the multiplication table, and, by dint of incessant repetition, memorizing the combinations of a host of factors and products.  By this first method, the child learns how the product is formed, and why 3x2 or 2x3=6.  By the second method, he simply remembers - if he can remember - the formula."  (Eclectic Manual of Methods, pg.124) - emphasis mine

Even though these two quotes are about multiplication, the idea still applies to any mathematical topic.  Taking the time to demonstrate and allow the students to discover the hows and whys of these mathematical processes is so important in deepening their understanding and developing an appreciation for the logical nature of mathematics.

4.  Focus on real, contextual problems.

Mathematical drill is important.  Students need to practice being accurate and quick in their calculations.  However, too much focus on drill will not give us the results we want to see in our students.  Instead of encouraging deep understanding and a sharpening of a child's reasoning abilities, it only encourages rote memorization.

"Multiplication does not produce the 'right answer,' so the boy tries division; that again fails, but subtraction may get him out of the bog.  There is no must be to him; he does not see that one process, and one process only, can give the required result.  Now, 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."  (Charlotte Mason, Home Education, p.254)

So how do we really train a child's ability to reason and problem solve, and encourage deeper understanding?

“How is this insight, this exercise of the reasoning powers, to be secured? Engage the child upon little problems within his comprehension from the first, rather than upon set sums. The young governess delights to set a noble 'long division sum,'––, 953,783,465/873––which shall fill the child's slate, and keep him occupied for a good half-hour; and when it is finished, and the child is finished too, done up with the unprofitable labour, the sum is not right after all: the last two figures in the quotient are wrong, and the remainder is false. But he cannot do it again––he must not be discouraged by being told it is wrong; so, 'nearly right' is the verdict, a judgment inadmissible in arithmetic. Instead of this laborious task, which gives no scope for mental effort, and in which he goes to sea at last from sheer want of attention, say to him–– 

'Mr. Jones sent six hundred and seven, and Mr. Stevens eight hundred and nineteen, apples to be divided amongst the twenty-seven boys at school on Monday. How many apples apiece did they get?'

Here he must ask himself certain questions. 'How many apples altogether? How shall I find out? Then I must divide the apples into twenty-seven heaps to find out each boy's share.' That is to say, the child perceives what rules he must apply to get the required information. He is interested; the work goes on briskly; the sum is done in no time, and is probably right, because the attention of the child is concentrated on his work. 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.” (Charlotte Mason, Home Education, p.254-255) - emphases mine

In addition to being able to add and subtract, for example, children need to be able to apply these processes in order to solve problems.  This not only strengthens a child's reasoning skills and builds up his ability to do mental math (which is so important in training the child to think), but also keeps him interested in the work.

“That he should do sums is of comparatively small importance; but the use of those functions which 'summing' calls into play is a great part of education.” (Charlotte Mason, Home Education, p.254)

As usual, let me reminisce!  When I would assign problems to my students, which ones do you think 90% of them would skip?  That's right...the real world, contextual, word problems - the problems in which they had to think and figure out how to apply what they knew.  It was disheartening.  And it wasn't just my students - this was the norm, at every school and for every math teacher I ever worked with.  The kids claimed they had no idea how to go about solving problems like these.  They wouldn't try.  Many of them wouldn't even read the problems (I'm sure there was some laziness at play here as well).  It was apparent that this was not something they were used to.  They were used to being drilled, not to think.

"Consider how mathematics was introduced to children a few years ago.  To pupils of Junior School age only arithmetical ideas were presented.  Algebraic and geometrical ideas were thought to be beyond the understanding of young children.  Great emphases was laid on mechanical work and learning by rote." *

Let me stop here and say that this is what I'm talking about.  Our children are not given the time to think and discover the ideas of mathematics, and so we're left with teaching rote memorization.

Continuing on...

"...What was the result of this preoccupation with mechanical 'sums'?  It was true that a few pupils attained an almost machine-like accuracy and came to enjoy the subject in which they could achieve such perfection.  However, the majority managed only partially to understand the elementary mathematical concepts.  There were some, indeed, who gained no real knowledge of the subject; who remained completely mystified by the language of number and size.  The unfortunate pupils in this latter category usually developed an early antipathy toward mathematics.  Everyone who has been involved in the teaching of 'Secondary Stage' maths will have encountered the child, otherwise able, who has no understanding of Number." *

Wow.  This whole paragraph - all of it - describes my experience teaching math in the public school system to the T.  The kids had no knowledge or respect for mathematics.  They saw math as confusing.  They disliked the subject.  They didn't know how to think.

"...The basic ideas of mathematics must be understood if we are to be able to follow modern science, but undue emphasis on 'mechanical drill' will not achieve this end." * - emphasis mine

There must be a balance of drill and solving real, contextual problems - aka word problems; problems in which the child must apply what he has learned.

(*Quotes taken from the PNEU article Knowledge of the Universe by G.L. Davies)

~ ~ ~ ~ ~ ~

So to summarize, the successful teaching of mathematics requires, above all else, a teacher who understands how to teach mathematics.  And this good teacher will:

• Give the individual student plenty of time to develop the ideas of mathematics.
• Focus on the concrete before the abstract.
• Get under the surface and allow the student to understand the hows and whys of mathematical processes.
• Include plenty of word problems for the students to solve to encourage deeper thinking.  

These are the things to not only think about and remember as we teach our children math, but also as we search for an appropriate math curriculum.  

And hopefully, as a result, the child will be trained to think, will gain a solid understanding of mathematics, and will develop an appreciation for it as a delightful subject worthy to be studied.

Next time, I'll outline the first year of mathematics instruction.  And you may be surprised to see how much Ray's Arithmetic mirrors CM's thoughts on math.

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

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

Teaching mathematics part I - why study math?

I'm about to begin homeschooling my oldest and, as I mentioned in my post on our curriculum choices for next year, it took me awhile to finally settle on a math curriculum, even though I have a math background and taught high school math in the public school system for five years.  However, figuring out what and how to teach a first grader is a little different!

After the request of a reader to write a little more about teaching mathematics and choosing a curriculum, I've been doing some research as well as some reflecting on my days as a teacher.  As a result, I have a few posts in the making on this subject - a short series.

And I just want to say that I'm so thankful for the request to write more about this topic because I have really enjoyed it and have learned quite a bit in the process!

Most of the research that I've been doing about mathematics comes from Charlotte Mason's writings, a Parent's Review article entitled Knowledge of the Universe by G.L. Davies, and The Eclectic Manual of Methods, which is the manual outlining the teaching methods of the Ray's Arithmetic books - the curriculum that I've chosen for our math studies.  So, part of what I've written about will include how the method of teaching mathematics in the Ray's Arithmetic books not only leads to a solid foundation in mathematics, but also lines up pretty well with what Charlotte Mason (CM) wrote about teaching math.

Let me say, though, that although I have chosen at this point to use Ray's Arithmetic as our primary math curriculum, I'm not saying that Ray's Arithmetic is the best, or the only, curriculum that 1) aligns with CM's methods, and 2) leads children to a solid foundation in mathematics.  I'm sure there are many wonderful math curriculums out there - I've heard many homeschooling moms attest to this fact.  I chose Ray's because I do believe it will lead children to that all-important solid foundation in mathematics AND I like how Ray's is a simple, no frills, straightforward way of teaching the subject.  I think the simplicity of the method allows for deeper understanding in the long run.

My hope is that, even if you use a different math curriculum, you will still glean something useful from this series.

Moving on!

Math is kind of my thing.  That probably sounds conceited, but of course I don't mean it to sound that way.  I've always loved math, I went to college and earned a math degree, and then went on to teach math in the public school system for a few years.  It comes fairly easily to me and I've always thought that it was a beautiful subject.  It's rational, logical - a subject to be revered.

But not everyone would agree with me.  I saw it first hand in the years that I taught 8th and 9th grade students Algebra I.  The attitude towards mathematics was pretty sorry.  The students did not want to do math.  They didn't like it.  They thought it was too hard and just didn't understand why they had to learn this, in their words, pointless subject.

It's a good question to ponder, though.  Why study mathematics?

First, Charlotte Mason (CM) has something interesting to say about why we should not study mathematics.  Something that I've never really thought about but makes perfect sense as I look back on my days as a math teacher.

"The question of Arithmetic and of Mathematics generally is one of great import to us as educators.  So long as the idea of 'faculties' obtained no doubt we were right to put all possible weight on a subject so well adapted to train the reasoning powers, but now we are assured that these powers do not wait upon our training.  They are there in any case; and if we keep a chief place in our curriculum for Arithmetic we must justify ourselves upon other grounds.  We take strong ground when we appeal to the beauty and truth of Mathematics..."  (Towards a Philosophy of Education, p.230-231) - emphases mine

She points out that the purpose of studying the subject of mathematics is not to develop reasoning skills in children - because they already have them!

"Perhaps we should...cease to put undue pressure upon studies which would be invaluable did the reasoning power of a child wait upon our training, but are on a different footing when we perceive that children come endowed to the full as much with reason as with love..."  (Towards a Philosophy of Education, p.151) - emphases mine

Children are born with the ability to reason, just like they're born with the ability to love.  It's something God put in us when he created us.  It's one of the things that makes us human - our intellect.  And as we grow, not only do we mature physically, but also mentally.

However, our society has put much more pressure on the teaching and learning of mathematics more than any other subject (besides language) because it is seen as a means of learning problem solving and reasoning skills.  And this is not conducive to what CM said about how "education should be a science of proportion, and any one subject that assumes undue importance does so at the expense of other subjects which a child's mind should deal with." (Towards a Philosophy of Education, p.231)

"'The mind feeds on ideas and therefore children should have a generous curriculum.'" (Towards a Philosophy of Education, p.111) 

So, why study math, beyond what we would consider practical to our daily lives?  It seems there are two main reasons.

1.  To sharpen reasoning skills.

There's a difference between sharpening a skill and developing a skill.  Developing really means to cause something to come about, but we already know that our reasoning skills are something God gave us when he made us.  So math doesn't develop our ability to reason, but helps to sharpen - strengthen, hone, improve - this ability.  And it also strengthens within us certain habits.

"The chief value of arithmetic, like that of the higher mathematics, lies in the training it affords the reasoning powers, and in the habits of insight, readiness, accuracy, intellectual truthfulness it engenders."  (Charlotte Mason, Home Education, p.254)

But this sharpening of our reasoning doesn't come just from the study of mathematics.

"...Our business is to provide abundant material upon which this supreme {reasoning} power should work; and that whatever development occurs comes with practice in congenial fields of thought."  (Charlotte Mason, Towards a Philosophy of Education, p.151)

So we must first realize that math is not the means to the building up of our reasoning skills and our ability to think and problem solve.  Those things come from taking part of an intellectual feast from many different subjects.  The study of math is important, but it's just one piece of the puzzle.

2.  To develop an appreciation for and a reverence of the beauty and truth of mathematics as one of the natural laws of the universe.

Let me continue the very first CM quote from above (again, emphases mine).

"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."  (Towards a Philosophy of Education, p.230-231)

And another quote.

"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."  (Towards a Philosophy of Education, p.232) 

Mathematics is to be studied for the same reasons we study any other science - for its beauty and truth.  But the problem in schools today is, it seems, a problem that's been around for awhile.

"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." (Charlotte Mason, Towards a Philosophy of Education, p.231)

Could I just bold that whole quote?  Read it again, please.  Seriously.

Whoa.  How true is this today?  Our current educational system is what?  REGULATED BY EXAMINATIONS.  There's way too much emphasis placed on excelling in mathematics (and, again, language), that we just glaze over the wondrous nature of it.  Instead, we push and push and cram and cram the material down the kids' throats, in an attempt to get them to pass the test, and in turn, we create disdainful attitudes toward the subject amongst our students.  The kids are not inspired with "that sursum corda which we should hear in all natural law."  There is no "sense of awe in contemplating a self-existing science."  There is no respect for the subject as what it is.  And on top of that, the other subjects that are worthy of studying and altogether provide a feast of ideas upon which to grow are pushed to the side with the belief that they are not as important.

I've seen it in action and it doesn't seem to be going away anytime soon.  What a disservice we are doing to our youth.

So, mathematics is to be studied because it's worthy to be studied.  It not only sharpens our reasoning skills and strengthens within us useful habits, but it is a natural law of the universe; a self-existing science.  However, we must not neglect the other subjects that make for a well-rounded education.

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

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