Musings on Math

This post is just an unloading of thoughts onto a page. I've been thinking too much, and sometimes the only way to steer a clear path through all these thoughts is to write about it. So I warn you, this is going to be long, intense, and possibly tangled, and I might change my mind on some things in the future. This "thinking love" we have for our children is a long-term process. I fully expect to make mistakes and learn from them.

Let me tell you about my son SA. He is five and a half, and he has a special love for numbers and math. He seems to see the world through math-coloured glasses. 

- When we first began our poetry teatimes half a year ago, he spent weeks (months?) creating math equations out of the food on his plate. The strips of cheese were made into plus and minus signs, the slices of apples and crackers were added and subtracted. Similarly, if he has a stick at the beach, or sidewalk chalk on a sidewalk, he is making math problems for himself to figure out.

- He is always asking math questions and making math observations. My approach to this has always been to ask questions back to him to see if he can figure things out himself. He often can. If the question seems beyond the math building blocks he's already assimilated, I'll answer truthfully and make some comments about the process I used to get to the answer. 
For example, he was in the back seat of the car one day when I handed Stephen three $20's. 
"Here's $60," I said. 
SA had noticed the $20's. 
"How many 20's makes 60?" he wondered out loud.
"How many do you think?" we asked him. 
"Three," he said.
He thought for a couple more minutes. "60 is 20 3's," he said next.
Then his mind seemed to go to the number 23, (it sounds like 20 3's) because his next announcement was, "23+27=60."
"No, that's not right," we said. "Can you try again?"
It took him a few minutes, but he figured it out. 23+37=60.
This type of conversation is not untypical with him.

- He almost always understands the math concepts I've taught him immediately, without frustration. Now don't be thinking he's a genius...I still mean easy elementary school concepts. Borrowing and subtracting was the latest one. But hey, I distinctly remembering having considerable frustration with that concept...in grade two.

Charlotte Mason's insistence on a method, not a system, is making more and more sense to me. (See Volume 1, Home Education, p. 6-10) What is the difference, you ask? 

System implies:
- following a path of definite rules in order to achieve precise results.
- does not take the child into account as a self-acting, self-developing being.
- focuses on the development of a skill set, or subject matter, rather than the whole person.
- easier (and lazier) to contemplate for the parent.

Method implies:
- a way to an end
- step by step progress in that way
- begins with a vision of the end result
- is natural and easy, yet careful and watchful use of every circumstance in the child's life to bring him to the goal.

There is always the danger that a method (even a good method) can become a system.

Practically, this concept has meant more to me in this (seemingly) gifted area of my son's life than in any other area. 
- It means that I make my goals for his math development, and every curriculum, every game, every math activity I use serves that end.
- It means that I use the curriculum in ways that serve my son, even if that means using it differently than it was intended, or speed through things that he already knows intuitively. 
- It means that math is taught consecutively, in a logical order, but once it is mastered, we do not drag it out. 
- It means that I try to provide activities that give him the space to allow him to have his own "aha" moments, his own connections.

So far, in Kindergarten, this has been fairly "natural and easy." I introduce a new math game or activity every Monday. I do not choose these in advance, but try to either match them to something he's interested in at the time, or to something in his regular lessons. From Tuesday to Thursday, I aim to make our daily Miquon math lesson about 10 minutes long. We use manipulatives to introduce any concept, but I do not insist on their use for every problem if he doesn't need them. I do not necessarily have him do worksheets for every lesson. Friday, for variety, we do a worksheet from MEP Math (a program that emphasizes mental puzzles/problem solving, rather than manipulatives.). He also often does lessons from Khan Academy during his daily "screen time." I'm not sure yet how I feel about that (and yes, it's my fault he discovered it). Positively, it gives him a sense of how BIG the world of math is, and how much there is to learn. Also positively, it motivates him to become more fluent at the things he already knows. (It's practice drill, in other words.). 

It sounds like I'm very sure of myself, doesn't it? But I'm actually questioning myself all the time.
- Am I holding him back? I feel this especially in our daily math lessons, which are all very easy for him. My intuition tells me not to skip any foundational steps. But when he started playing with Khan Academy, within a day he had discovered adding and carrying, which before then I had no plans to teach him for a while yet. (I made sure to demonstrate with Cuisenaire rods.) As with all his math so far, he understood it immediately, and made no mistakes in his practice problems. Does this mean I should skip ahead?
- How much practice work should we do to gain fluency? When does it become useless busywork?
- Am I challenging him enough? I've read a lot on math, and one of the great values of math is supposed to be training to "try, try again," learning to persist even when problems are hard. But nothing has been hard yet. Does that even matter in Kindergarten? Will he eventually hit something that's challenging for him, or do I need to move through the material faster so we get there sooner?

And then I question myself because I'm questioning myself. I mean, lady, your boy likes math. What a problem to have! He's in love with it. He enjoys everything, even the things that are too easy. He is not bored (yet.).

I think all my struggling with this is because I'm a planner. I would like to be able to see precisely the steps and the speed and the direction I'm going. I like systems a bit too much! :) Lazy ol' me would like to find the perfect math program that I could plug him into and let him go. I have some goals, though. They may yet change, but they give direction to my methods for now.

At the end of all his learning at home, I want SA to have:
- A sense of awe and wonder in mathematics as the language God built into the universe. (I'm thinking of this quote from Galileo: "Philosophy is written in that great book which ever lies before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.")
- The foundational tools he needs to build on to figure out any math problem he wants to figure out.
- An attitude that is conducive to problem solving, especially persistence and determination.
- A big picture, an understanding of the historical context of the study of math. (Please note that I understand that my goals may be less lofty for another child, but because of the intense interest SA has in math, I feel this goal would be important for him.)

There. I feel a little bit better now that my thoughts are written down. I feel a little hesitant about posting this. Strangely, it would be much easier to write about real problems my child is having. But I've decided to put it out there in case anyone has any wisdom or experience to share with a young homeschool mom. Believe me, non-problem that this is, it has consumed a lot of my thoughts and energy in the last year and a half.

(Edit. I should make clear that Charlotte Mason would not have recommended math lessons this young. I'm just in a situation where withholding the knowledge his mind craves would be unkind. This child too is a born person. Charlotte Mason was also quite accommodating to mothers who wanted to begin teaching reading at a very young age, as long as it was done gently and naturally (no pressure), so I don't feel she was inflexible. Second, she was critical of the concept of Kindergarten. When I speak of him being "in Kindergarten," I'm simply referring to his age of five and a half.)