Borrowing, Carrying, Understanding
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Borrow 1, make the 7 into 17, then subtract 9.
Borrowing and carrying.
Simple math right?
We've all done it. In math class, or balancing our checkbooks (pre-online banking).
But what does it mean to borrow 1?
A friend of mine, not a math lover, is preparing to become an elementary school teacher. She's currently taking a methods class in teaching math in elementary school, and she had an epiphany. About borrowing and carrying.
"Borrowing and carrying don't explain the importance of place value. We always just borrowed and carried because that is what we were told. We never knew why."
Now, I'm pretty sure that as a functional adult, my friend can do what we long called borrowing and carrying. And I don't know whether some 4th grade math teacher actually did explain the relevance of place value to the mathematical process of subtraction to a brain that wasn't ready to process it yet. But I know that my friend, an adult preparing to become a teacher, understands arithmetic better today than she ever has, and she is ecstatic about her epiphany.
The language arts version of the borrowing scenario is the "You can't start a sentence with because" rule. You can start a sentence with because, but the brain of a child just beginning to write sentences might not be ready for the complexity of dependent and independent clauses. So we don't let them do it; we let them develop the skills of competent simple sentence writers. But often they get to 11th grade and are still holding on to the "you can't start a sentence with because" rule, because that is what they have been taught.
I'm sure there is a science version, and a world language version, and a social studies version of, "You just need to know this. It's going to be on the test."
There is no question that as brains develop they become capable of grasping ideas and processes of greater complexity. Part of really grasping the them demands an understanding of the concepts underlying them. And there is a joy that comes with that understanding. Learning that you can start a sentence with because opens a whole new range of possibilities for writing good, strong, interesting sentences. Because it gives us a stronger grasp of the arithmetic, understanding the concepts behind what we call borrowing in math makes math seem less of a mystery.
The purpose of all of our education, formal and informal, is to help us make greater sense of the world, so we can get a better handle on our place in it. That's big stuff, and every little bit helps.
Being able to mimic a process or an idea is not the same as understanding it. When explaining why a student should know something, "because it's on the test," or "this is just what you do," are never sufficient responses for a mind hardwired to try to understand the world it inhabits.
Recently, my twin 8 year-old boys struggled with their homework, which called for them to use mental math to solve double-digit subtraction problems. Frustration led one of them to ask "Why can't we just use a calculator?"
ReplyDeleteThose adhering to Einstein's philosophy of never having to remember information which can easily be referenced might agree with my kid.
However, I think we, as an American society, need to remember a few things. First, America is the new kid on the block in regards to education, coming well after the ancient civilizations of Asia and Europe. Thus, it shouldn't surprise anyone that American student outcomes trail the Asian and European countries. But, we're America. We were built on competition and the entrepreneurial spirit. We believe we should be the best at everything. Second, the American educational system has often mirrored other influences over the past century. Mirroring the Ford motor company, and the success of the assembly line, American schools were built under the premises of most efficiently educating students by putting 25-30 kids in a classroom and teaching all of them the same core material so that in 4 years, each kid can come out of there with the basics to produce in an American society. Around the same time, philosophers and researchers were calling these core knowledge components the 7 Cardinal Principles of Education. Things didn't change much. When Kennedy wanted to Race to Space, an emphasis was placed on math and science curricula. When George W. Bush wanted No Child Left Behind, educators focused on individual student needs, especially the needs of previously underserved student populations. And the list goes on. Remember Ebonics? Madeline Hunter's ITIP? Howard Gardner's Multiple Intelligences? Today, the buzz words are Differentiated Learning, PLCs, and Varied Instruction. Third, the heavy reliance on technology hasn't helped educators' efforts in teaching mastery of basic concepts. No child is far from some type of calculator application, considering the built-in applications to cell phones and computers. So, it makes sense to me that a kid might prefer a calculator to paper and pencil. Using a calculator makes solving math problems easier and more convenient.
In my opinion, only when our educational system stops mirroring outward business and political influences to form educational movements du jour will we have a system which can move forward in a cohesive way; a system in which my kids won't have to practice mental math without first having to master the concepts of borrowing and carrying. How best to do that? I have no idea. Where's my calculator?
Let's not forget Bloom's Taxonomy. Knowledge (or regurgitation of facts) is the most basic level. The next step is Comprehension, which, in other words, is understanding why. More advanced levels follow but addressing those in any forum is sometimes depressing because shamefully educators fail to move past the most basic levels because of NCLB and the pressure of meeting state standards. Teachers often teach to the test and nothing more. In my opinion, we need to meet the standards and allow students to reach the level of self-actualization, the level at which they can truly learn to think creatively and critically.
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