Sorry Brett, as a professional mathematician, I have to disagree. While it is quite true that we think strategically it is also true that we learn best procedurally. That is why in study after study when “strategic” systems for teaching mathematics are placed along systematic, step-by-step systems that teach organized methods — such as Project Follow Through, the largest educational comparative study in history — the systematic approach beats the “strategic” systems that pretend to teach “number sense”. And it isn’t even close.
But as we’re talking matters of principle here let us talk about mathematics and what it means to learn it. I happen to be currently teaching a course in ALGORITHMIC discrete Mathematics to third-year engineers. Guess what? If they don’t learn procedures — they fail to learn that course. It’s about algorithmic methods, which is a significant portion, perhaps the greater part, of what mathematics is about.
In one instance a student put up his hand, perplexed in the middle of our midterm; and stuck. You see, I had forbidden calculators because all arithmetic on the exam was simple enough. To get into this program and succeed for three years one has to be smart and capable. And this studentshowed a great capacity to learn in this course, able to do all I had taught. But got stuck at a certain point in a problem where, in order to complete the work, they would need to find the remainder when 2015 is divided by 7. He told me “I never learned long division”.
You see, the student’s early teachers apparently were of the view that teaching algorithms didn’t matter. Put another way they thought DON’T help them to stand on the shoulders of giants (as Newton put things) … make them into giants all on their own. A fool’s errand, akin to asking someone to reinvent the wheel before they are allowed on a bicycle. Or to build their own bicycle. Even our giants are giants because they have learned and mastered the excellent methods developed by their predecessors. We are not cave men — yet our minds are no more developed than those early ancestors. The difference is that we can pass down the learning of millennia so that our children are launched from their education a more advanced level. Your child is a 21st Century Child not because their brain is wired differently from those in ancient Greece but because of the wealth of accumulated knowledge (of things such as optimized and elegant procedures developed by the best minds of the past).
Now I know the reply my edu-wonk friends make to this kind of example: “But they don’t NEED long division! It’s EASY to get that remainder using a ‘strategy’ — here let me show you …” They needn’t show me. I could produce, with little thought, a bucketload of simple tricks to milk 2015 for that remainder without doing long division. But the reason I … and you … are so adept at doing so is not because we’re smart, or good “strategic thinkers” is that we are FAMILIAR with the processes because we have been taught systematically and given sufficient practice.
It is true that humans think strategically. But it is a lie to suggest they can do so in a procedural vacuum. Working memory has so much capacity for juggling symbolic material … about the same as a single 7-digit phone number. Yet, in working memory is where all such processes take place. So … yes, students can develop simple processes of a couple of steps by themselves, that combine methods they already know (have committed to long-term memory) to a desired end. But anything more sophisticated than 2-digit arithmetic quickly clogs the works.
Educationists who get excited when primary grade students “explain” how to arrive at 6+8=14 in “three different ways” seem to believe this proves what a great way this is to learn. Actually what teaching — and REQUIRING — students to do this is training (yes, training!) them in mental habits that will bog them down for life, and focus attention at the wrong level of meaning.
We are told that the purpose of holding back on content is to teach understanding. But this is a misunderstanding of understanding. Understanding cannot exist in a vacuum of knowledge. You understand what you KNOW. Further, all permanent learning comes from attaching new information to that which is already established in long-term memory. By limiting the amount of direct-taught information a student is to put in long-term memory before being asked to “understand” you are strictly limiting their capacity to understand. The most effective teaching provides content and context that helps understanding so that students get the best of both worlds: Learning the great heritage passed down from previous generations, which is the Prime Directive of education, and the chance to engage with that knowledge meaningfully, developing understanding of it as it is learned. If someone knows how to squirt understanding into childrens’ brains without directly teaching them knowledge, they are keeping it a secret — such has never been shown to exist in any controlled experiment. It’s a pipe dream, built on rhetoric, slogans … and TED talks.
Finally, it is dishonest to pretend that Common Core demands this kind of teaching. People who make that claim infuriate the writers of the math standards. Both Jason Zimba and Bill McCallum have declared, in no uncertain terms, that their intention was NOT to transform teaching methodology. It is true that the wording of the standards in places has lent itself, unfortunately, to such interpretation by educational ideologues eager to see their favourite pedagogy-du-jour implemented by fiat system-wide. But the founders deny that this is the purpose of the standards. Now CCSS has many problems besides these silly edu-speak dog whistles, but taken simply as a set of content standards CCSS is not bad for the most part, up to about Grade 7, with some caveats. It is the eager ideologues who are wrecking what might have been an improvement (in most states, at any rate) over what had been in place before. (Here I leave aside the political problems with CCSS which are another question altogether.)