Thursday, May 21, 2015

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



  1. Love this post! Looking forward to the series. We also use Ray's, and I just love it. My appreciation for, and now love of, math has grown by leaps and bounds since I began teaching my kids. It's really a beautiful subject!

    1. Glad you enjoyed it! And, I'm glad to hear you like Ray's. It doesn't seem that many use it, but it sure does look promising...especially the first several years.

  2. Excellent post, Angela. Thanks so much for your insight into the least favorite subject in our homeschool.
    A little background.... We began homeschooling our oldest 2 with A Beka Math because it's what was used in the Christian school that our children attended before we began homeschooling them. They hated it. I hated it. 100 problems per day (including all the drills and the lesson)? No thanks!
    So we switched to Saxon. My older children did great with it, even though they are not Math lovers.
    We are also using Saxon with our younger 5. They don't like Math, either, but do well (for the most part). My 16-yr-old son just finished Saxon Calculus and is now in a business Math course. My other 4 are 2 years above grade level in Math, too.
    Once we began implementing the CM philosophy in our homeschool, I really wanted to change Math curriculum to something more CM-ish, but was advised by other AO moms not to, because of what they called "curriculum gaps". But Saxon Math is so dry!! And the children simply do not like. It is a battle sometimes just getting through a lesson.
    So I would like your opinion as a Math teacher. My children that are still in Saxon are ages 14, 13, 11, and 8. Do you think it's a bad idea to change curriculum at this point?.

    1. Oh wow, I'm sorry none of your family likes math! That's interesting to me, though. I remember as a teacher so many parents of children who didn't like math saying that they themselves never liked it either. And I would be thinking, "You're not helping the situation!" I'm not saying your like or dislike of math is an influence (you said you didn't like A Beka, but you didn't state if you actually enjoy math or not!), but it's interesting how much attitudes can rub off on others, either from parents or older siblings, etc.

      As for your question, I would say when a child is ready for Algebra, and that would be around 13 or 14 years old, switching curriculum should (in theory!) be okay, because all the foundational math needed to be successful in Algebra *should* be in place. For the younger grades, switching would be doable, but would require some research on your part. You would need to understand what all your children have learned through their current curriculum, and compare that with the new. That way you can see where to start in the new curriculum and where you may need to supplement for any gaps. It could be a lot of work, or not, depending on how alike the curriculums are. If you're really serious about switching, then I say go for it, but again, do the research :)

      I wish I could help you more! I'm just not familiar with any of the other math curricula, really, except for Ray's and I've looked at MEP a little, too.

    2. Oh, I hope my previous comment didn't come off rude! I was just trying to point out that the way in which we approach things has an effect on how those we influence approach things. So if your older kids disliked math, that may have rubbed off on your youngers so that your younger kids went into it at the very beginning with the expectation that they wouldn't like it. Or it could just be that old saying "To each his own." We don't all love the same things :)

    3. You comment wasn't rude at all! Math is not one of my favorite things, but I do ok in it. I would just rather identify parts of speech, read Charles Dickens, or study Nature. :)
      It is possible that my dislike of it has rubbed off on my children, even though my hubby loves Math...he is not with them all day like I am so he isn't around them enough to have his love of it be contagious.
      Thanks for your advice. I am considering changing our Math curriculum for next year, but not without some research first, just to make sure it's doable. It will be worth the time and effort of researching - if it improves their attitude. :)

    4. I was also going to say that I've got a few more posts on this topic, and in the next two I'm discussing some steps that a good math teacher should take in order to encourage a more respectable attitude toward math and a deeper understanding. And most of it is based, again, on CM's writings. So maybe those could help in choosing your curriculum. I'll probably post my next one tomorrow or Saturday.

    5. I will be eagerly awaiting those posts. :)