Tuesday, September 13, 2011

The Abstracts of Math

Remember I mentioned MATH is one of those subjects that is truly a transition for me or maybe it would be better to say that the way we approach math has to change.  I recently read through the Arithmetic section of Volume 1 and once again I left it feeling I had more clarity on the subject.  It is always nice to have your instincts validated.

Our approach to math (oh, when I say we or our I mean my husband and I) has been the very traditional school system way.  Memorize math facts move on to more memorization of times tables and if anything is missing add more drill page problems to daily routines.  Well, in the early years you can certainly get away with this because lets face it:  it is basic math.  There comes, in a few years, a time when all these basics have to be extracted in order to understand new concepts requiring certain knowledge.   By now some of that information is at a very abstract level.  Enters Miss Charlotte Mason with her infinite wisdom:
"The chief value of arithmetic, like that of the higher mathematics, lies in the training it affords to the reasoning powers, and in the habits of insight, readiness, accuracy, intellectual truthfulness it engenders....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." pg 254, Volume 1 Home Education
At a certain point I realized that our oldest was not understanding new concepts of math because she just simply didn't have a real grasp on her math fundamentals.  Mind you, we aren't even talking higher math like Algebras or Geometry (although the argument can be made that it is the beginning of those) .  We were just dealing with fractions and unit values.  She just kept digging herself trying to compute new things without really knowing where the idea even began.   We stopped and just took it slow.  She is better about it now but it is still a subject she handles with care.

My next train of thought was how do I approach this with the younger ones, so they don't dig the same.  These are a few things I discovered on approaching math:
"...[D]emonstrate 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." pg 255-256 Volume 1 Home Education
She goes on to explain how the child will gradually move from using tangible items to imaginary ones and last arrive at the idea of abstract numbers:
"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.
He may arrange an addition table with his beans...and be exercised upon it until he can tell, first without counting, and then without looking at the beans...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." pg 256 Volume 1 Home Education
Hmm, I can do this. I've tried to incorporate some math programs which use this philosophy in teaching math in early years.  I've even considered learning more about Montessori since I know some very smart CM moms whom think it is a great way to approach this philosophy (that might be my next very big reading assignment to tackle after CM's six volumes ; D )  Mind you she doesn't say that this is a subject to be taught with a living whole book on an idea like history would; but what I understand is that the child should relate to numbers in very real every day situations. So in  a sense we can say that the approach  should be a living approach that can easily transition the child from real tangible objects to the more imaginative forms which will ultimately get them to understand the black and white of abstract concepts. 
"It is quite true that the fundamental truths of the science of number all rest on the evidence of sense; but, having used eyes and fingers upon ten balls or twenty balls, upon ten nuts or leaves, or sheep or what not, the child has formed the association of a given number with objects, and is able to conceive of the association of various other numbers with objects.  In fact, he begins to think in numbers and not in objects, that is, he begins mathematics. " pg 262 Volume 1 Home Education

Now, here is something that really struck me about our approach to math.  One of those thoughts that you need to hear, even though it is tough to hear, even though your instincts tell you it should be so:
"Arithmetic is valuable as a means of training children in habits of strict accuracy...That which is wrong must remain wrong: the child must not be let run away with the notion that wrong can be mended into right  The future is before him:  he may get the next sum right, and the wise teacher will make it her business to see that he does, and that he starts with new hope.  But the wrong sum must just be let alone.  Therefore his progress must be carefully graduated; but there is no subject in which the teacher has a more delightful consciousness of drawing out from day to day new power in the child.  Do not offer him a crutch; it is in his own power he must go.  Give him short sums, in words rather than in figures, and excite in him 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." pg 260-261 Volume 1 Home Education.
That is a lot to digest, but just so powerful.  We want our kids to feel good so we just keep going on with the next problem ... the next page ... the next concept ... the next book ... and before you know it your child just feels overwhelmed and your frustrated.    This was us.  I'm so glad that I went with my instincts and slowed down before really hitting those upper level mathematical concepts.  I've appreciated reading this small section on math and getting a clear understanding of how and why to approach math in a gentle manner.  I might not have a natural mathematician who can just see all the abstracts in her head from the get go; but I know I do have a logical thinking child that with time will process this subject appropriately.  I will leave you with one last wonderful quote:
"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.  The reason why mathematics are a great study is because there exists in the normal ind an affinity and capacity for this study; and too great an elaboration, whether of teaching or of preparation, as I think, a tendency to take the edge off this manner of intellectual interest." pg 264 Volume 1 Home Education


...they call me mommy... said...

WOW. A lot to think over. This is definitely an area, I'm still learning (like every other area ;) )! Thank you for sharing your thoughts and Miss Mason's quotes!!!!

Eva said...

This is so true for the early grades, but it's also important to not loose touch with in the higher grades. What math program(s) are you using? After going back and forth between quite a few, I have finally settled on a combination of Miquon, Singapore, Key to, and NEM. We also use "Live Education" to have the connection to life and some math picture books.

Grace'n'Chaos said...

You're welcome Amy.

Eva, I think you're right. You would hope that by following this notion the young student will be ready to deal with the abstracts of math that I know will be there. For my sixth grader I've combined MUS Zeta with Life of Fred Decimals. I'm giving myself the year to consider a pre-algebra program for her. She needs a lot of hand holding on this subject.

Eva said...

I looked at the MUS program many years ago, but I didn't like that you had to watch videos. I haven't used any of the "Live of Fred" series. How do you like them? My children don't like it when I help them with math. Maybe because it has always been one of my weak subjects. I'm not quite sure what I will do with my 6th grader after she is done with the last Singapore Math book. My son has taken up NEM, but I'm not sure if that will be the right choice for her. We'll see.

Sharon Lovejoy said...

"strict accuracy" I like that concept or that view of it.

Thanks for this thoughtful posting.

All joys,

Sharon Lovejoy Writes from Sunflower House and a Little Green Island

Grace'n'Chaos said...

Eva, she is enjoying LoF. And she is like your kids doesn't like much help from me. The MUS dvd's give her that confidence, each lesson is pretty short and she can visualize the concept before she tackles it on paper. We alternate programs to give her that mental break she needs!

Sharon Lovejoy said...


My newest book, Toad Cottages & Shooting Stars has nature, cooking, art, traditions, gardening...and I am happy that homeschoolers, Montessori, and Waldorf are all using it.

Thanks for your sweet words!

Sharon Lovejoy Writes from Sunflower House and a Little Green Island