The learning consultant to whom I report SpiderGirl's learning activities seemed rather impressed that she was learning about integers in Kindergarten. It's understandable, I suppose, given that integers usually aren't broached until middle school here. After all, I haven't told anyone else, lest they think I am "pushing" her.
I offered the concept upon request for something new in math to explore. We had already introduced simple fractions. This seemed to be the next logical concept to introduce. Yet, SpiderGirl hadn't yet mastered subtraction or time or grouping. What would make me think of such a thing?
Was there method to my madness?
I'll be honest. There wasn't; it was intuitive. After all, she wasn't interested in refining skills. And who can blame her? In the words of Anne of Green Gables (L. M. Montgomery), there is just no scope for imagination in repetitive exercises -- and being five is so much about imagination! SpiderGirl wanted a new concept to play with, new characters for her world of math.
Was there method to my madness?
I'll be honest. There wasn't; it was intuitive. After all, she wasn't interested in refining skills. And who can blame her? In the words of Anne of Green Gables (L. M. Montgomery), there is just no scope for imagination in repetitive exercises -- and being five is so much about imagination! SpiderGirl wanted a new concept to play with, new characters for her world of math.
The idea of 'zero' caused mathematicians far more angst than either fractions or integers ever did...
However, a bit of insomniac contemplation and a little research supports my early introduction of both fractions and integers. Believe me, I have thought a lot about fractions and integers. They cause more problems in algebra than algebraic ideas themselves do. In my experience, I have noticed that it is easier to teach an 8 year old about fractions and integers, and basic operations on them, than it is to teach a 13 year old. Why? Is it a difference in the type of student who gets sent to me to learn these things? Is it because the younger ones have not yet learned to fear math? I will argue at least one of the reasons lies with brain development.
According to work by Jean Piaget, before age 6 (or so), the brain is in a phase of accumulating mass and matter. (1) Pruning of less useful grey matter begins around age 6 - 9. At this time, children begin to want to practice and refine skills. (2) By the time adolescence hits, pruning is in full force and children are interested in synthesizing what they have learned, analyzing it, and reasoning with it. While Piaget's theories are somewhat outdated in that the stages are overly simplistic -- the areas of the brain operate on separate timelines and development is affected by environmental and cultural factors -- the sequence of "accumulate then prune" remains true for all areas of the brain and the age ranges are similar. (3)
If we are to maximize learning, it seems reasonable to work with brain biology. Introduction of radically new concepts surely is at odds with a time when the brain is trying to prune and refine itself. A brand new type of number is indeed a radically new concept -- far more so than repeated addition, or sharing equally, or even taking a number apart into its component factors. Instead, perhaps we ought to be introducing fractions and integers in the primary grades, right along with the whole numbers. Introduce operations where there are logical connections rather than grouping them by number type. For example, adding integers fits in very nicely with subtracting whole numbers, making change, reading thermometers, and borrowing money.
...there is just no scope for imagination in repetitive exercises -- and being five is so much about imagination!
"But aren't negative numbers too abstract for minds as young as 4, 5, or 6?" you ask. Chew on this: The idea of 'zero' caused mathematicians far more angst than either fractions or integers ever did; yet, we teach 'zero' to our preschoolers and they understand it just fine.
Realistically, I know that this change in curriculum would be extremely difficult to implement in the current school system. There are too many math phobic teachers at the lower elementary level, and many more who would need a deeper understanding of fractions and integers. But for those of us whose children learn academic subjects to any extent at home, I think it is worth considering.
(1) http://projects.coe.uga.edu/
(2) http://www.
(3) http://www.nimh.nih.gov/health/publications/teenage-brain-a-work-in-progress-fact-sheet/index.shtml#.ThMP8I9Wlko.mailto and http://www.edge.org/q2008/q08_1.html (Howard Gardner)
No comments:
Post a Comment