Elaborating in responding to comments to my recent blog piece, “When I Stopped Rewarding My Son for Good Behavior”, I expressed the opinion that most kids could readily learn to read and do basic arithmetic, even mostly on their own, if they were not required to learn at a set externally mandated standard age, but instead undertook the effort on their own internal developmental timetable when they were ready and interested in acquiring that skill set. One of my thoughtful commenters took issue with my position, saying…
You think a kid is going to learn long division on her own? Why would she? How could long division ever be interesting enough to typical children that it would at any moment be the most interesting thing they could be doing with their time?
Very pertinent questions I think, because thoughtful people could well come down with either a yes or no answer to my commenter’s first question based on their experience, study, and resulting world view. Along with much required academic math learning in general, long division is an interesting case in the discussion of internally versus externally directed education.
My commenter’s questions beg a range of other important questions.
1. With calculators so ubiquitous these days, do people still have the need to do long division in their own heads?
2. Is knowing how to apply the iterative algorithm that constitutes long division help you with a larger understanding of the power of iterative processes in general, perhaps even beyond the world of mathematics? Is long division a critical building block to other knowledge?
3. Are some skills so fundamental that young people should be required to learn them even against their will if necessary?
I recall my mom (who loved math) teaching me long division when I was seven, before my classmates learned it in school. She presented it as a useful tool that was not hard as long as you stuck to the couple basic rules of the process. Being intrigued by numbers and their manipulation, I enjoyed learning the technique even outside any practical context. But since it was the 1960s and before inexpensive hand-held calculators, I soon had the occasion to use the technique to do some everyday calculation or another.
Now in my case I don’t recall asking my mom to teach me this skill, she offered to show me and I accepted, having an interest in most math-related topics. Short of initiating the wish to learn oneself, assenting to a suggestion to be taught something is a good dynamic for an educational experience. Less good in my thinking is being required to learn something before you understand its value or are otherwise ready to learn it.
On the other hand, neither of my kids (now both young adults) were that interested in math and were taught the algorithm in school as part of mandatory math learning with high stakes consequences (bad test scores leading to bad grades) if the skill was not learned and demonstrated. Our daughter, who at the time wanted the approval of teachers and other adults and was a bit of a “trained seal”, learned the algorithm so she could do good on the tests and get the needed kudos from her teacher that she was a “top student”.
Our son in contrast, was always a self-directed learner who was totally intrinsically motivated and did not care how adults evaluated him. He grudgingly learned the algorithm but hated being commanded to do worksheet after worksheet of practice, worksheets that had to be turned in for a high-stakes grade. Having several “drill and kill” math teachers in his older elementary and middle-school years, he became completely math phobic by seventh grade.
So life went on and both our kids are now young adults out basically on their own with jobs and bills to pay. Our daughter works as the part-time manager of a small bakery restaurant as her “day job” and is a budding science-fiction writer on her own time. Our son ran the operational side of a small start-up business for three years (done in by the Great Recession) and is now working as a video engineer.
If I asked either one of them to do a long division problem I don’t think either one could. I don’t think either of them has had the occasion to need use the technique. Again both of them have had bad experiences with academic math, though they both are comfortable with the kind of working with numbers they have had to do for their various jobs. Our daughter has to cash out at the end of her work days and our son had to do a range of number crunching (particularly in his work as the operational manager for the start-up business) including doing budgets, inventory, purchasing and even payroll.
So my take from my experience, which I can’t necessarily universalize but carries a lot of weight in my thinking, is that I was better served for math learning in general for learning the long division algorithm by choice, and my kids would have probably been better served if they had not been coerced to learn it. As the math subject matter they were both required to learn in school became more and more abstract (and less applicable to their real lives), the whole concept of mandatory school and required learning became more problematic and onerous to them. They both were interested in so much that they wanted to learn, but not so much what they were being taught in school.
With those experiences behind me, and having read various unschool and free school thinkers like John Holt, Pat Farenga, John Taylor Gatto and Daniel Greenberg, I am becoming convinced that an adult or youth who has developed their natural knowledge acquisition skills, when faced with a need to use long division, could probably go on the Internet (or have someone show them) and learn the technique in maybe an hour.
So if that is really true, why make it part of a high stakes exercise in required and even coerced knowledge acquisition with unpredictable results in the students whole dynamic and love of learning? And is perhaps the whole practice of requiring kids to learn any number of skills on some external standard timetable rather when they see the need or are otherwise “ready” counter productive and highly inefficient?
That said, I want to acknowledge that some families want to enroll their kids in institutions that will provide those kids with a standard set of knowledge and skills that are (or at least were) generally acknowledged as being important. That learning option should always be available to those that want it.
But the circumstances around when, where, from whom and under what conditions we each learn things like long division should be a significant consideration in everyone’s educational path.
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You talk about long division and a process to be learned. It is not, it is a concept to be understood. There are different ways to understand the concept, one of them is by learning the process of doing long division, but it is not the best way, by far.
I have taught children long division for years and, until they understand the concept, most students find it boring or irrelevant. Yes, you can use calculators but tat is giving power to a system you have not created. Understanding the concept of long division is neither difficult not complicated and enables you do transfer this skill to a lifetime of situations.
Long division rules!!!
Dr. Porter… I agree that all learning is more real, more lasting when the learner understands the concepts rather than just performs the algorithm like a trained seal!
The bigger question that I’m curious of your thoughts on is if teaching math and other concepts outside of a context of the learning wanting to know or appreciating the value of the particular concept is a good idea. What about the whole idea of “when the student is ready the teacher will come”. It seems our system of mandated learning on an externally derived state timetable is more like the opposite is true, and with high stakes testing and ranking to coerce compliance.