I'm old enough that I had a slide rule when I was in school. Everyone did. They were built in to our pencil boxes.
Most of us didn't use them, except the fancy-pantses who also insisted on using mechanical pencils. But we all knew how to use them.
I'm graybeard enough that even through college, electric calculators were prohibited in class because they would give you answers without helping you understand the problem.
I graduated in 2009 and in my engineering courses calculators were disallowed for similar reasons. I have found that courses that disallow calculators generally have the problems set up in a more elegant manner. E.g done properly the results end up in whole numbers or radicalize well rather than just an incoherent string of decimals. Courses that were focused on physical phenomenon like device electronics allowed calculator use because multiplying with Planck's constant by hand gets old quickly.
answers without helping you understand the problem
This is my problem with common core math. It shows how to get to an answer, but I feel it bypasses fundamental understanding on why the answer is what it is.
"Common core" can mean different things to different people. When my kids were in grade school, I found that a few states have put their common core standards in fairly readable form. My impression is that common core is remarkably similar to what my siblings and I learned in K-12 math class. It actually led me to wonder what all of the fuss is about.
The main thing that seems to be lacking in K-12 math today, is proofs. Those were my favorite part of math.
With that said, K-12 math teaching in my generation wasn't all that successful.
I feel that common core focuses more on teaching the various methods than embracing the idea that correct answers and understanding are largely what matter rather than any specific method.
It's natural to lump these together as if they are a unit, but I think it's important to be explicit about the fact (which of course you know) that they are all too easily separable: it is incredibly easy, given sufficiently powerful tools, to get correct answers with little to no understanding. True understanding (not just "oh, I think I get it") that doesn't help to get answers is not nearly as common, but of course it exists, too.
Yes I totally agree with your sentiment that it is vitally important to separate the correct answer from the method by which it is achieved. Honestly understanding mathematics is more important at some level than getting the right answer to a particular problem. That's the whole point, once you understand the grammar and vocabulary of mathematical thinking you can attempt to solve problems that are totally unrelated to things you know and still sometimes get useful results. The overwhelming question of the ages is how do you teach with that goal in mind and take account of all of the different learning pathologies while doing it in a universally accessible and measurable way. I think the problem is intractable as it is currently being approached.
Interestingly that song is apparently about the first time they tried to massively redesign the mathematics curriculum based on on fuzzy math that we were falling behind the Soviets or whatever. My parents tell me they rebased around set theory early etc but it foundered on nearly the same rocks that common core mathematics is now - teacher must be well educated in the pedagogical paradigm and fully understand it. E.g. you can't just take the same set of teachers from math v1.2.1 and slam them into math v2.1.0 or whatever without significant issues the two approaches are not abi compatible. That's why in my with opinion common core we are seeing teachers test finding answers via method 1 specifically or method 2 specifically rather than encouraging the use of either to get a correct answer by an understood method. /rant
That all sounds about right. The song was from around the time I was in elementary school.
I suspect that part of the problem is that teaching math concepts is at least somewhat orthogonal to drilling kids on being able to perform basic arithmetic operations fluently. Of course, these days I assume at least some of the debate probably gets into the fact that being able to do, say, long division quickly has about the same utility as learning Palmer script.
Comments
I'm old enough that I had a slide rule when I was in school. Everyone did. They were built in to our pencil boxes.
Most of us didn't use them, except the fancy-pantses who also insisted on using mechanical pencils. But we all knew how to use them.
I'm graybeard enough that even through college, electric calculators were prohibited in class because they would give you answers without helping you understand the problem.
I graduated in 2009 and in my engineering courses calculators were disallowed for similar reasons. I have found that courses that disallow calculators generally have the problems set up in a more elegant manner. E.g done properly the results end up in whole numbers or radicalize well rather than just an incoherent string of decimals. Courses that were focused on physical phenomenon like device electronics allowed calculator use because multiplying with Planck's constant by hand gets old quickly.
This is my problem with common core math. It shows how to get to an answer, but I feel it bypasses fundamental understanding on why the answer is what it is.
"Common core" can mean different things to different people. When my kids were in grade school, I found that a few states have put their common core standards in fairly readable form. My impression is that common core is remarkably similar to what my siblings and I learned in K-12 math class. It actually led me to wonder what all of the fuss is about.
The main thing that seems to be lacking in K-12 math today, is proofs. Those were my favorite part of math.
With that said, K-12 math teaching in my generation wasn't all that successful.
I feel that common core focuses more on teaching the various methods than embracing the idea that correct answers and understanding are largely what matter rather than any specific method.
It's natural to lump these together as if they are a unit, but I think it's important to be explicit about the fact (which of course you know) that they are all too easily separable: it is incredibly easy, given sufficiently powerful tools, to get correct answers with little to no understanding. True understanding (not just "oh, I think I get it") that doesn't help to get answers is not nearly as common, but of course it exists, too.
Yes I totally agree with your sentiment that it is vitally important to separate the correct answer from the method by which it is achieved. Honestly understanding mathematics is more important at some level than getting the right answer to a particular problem. That's the whole point, once you understand the grammar and vocabulary of mathematical thinking you can attempt to solve problems that are totally unrelated to things you know and still sometimes get useful results. The overwhelming question of the ages is how do you teach with that goal in mind and take account of all of the different learning pathologies while doing it in a universally accessible and measurable way. I think the problem is intractable as it is currently being approached.
That's hardly a new thing with common core. See Tom Lehrer's "The New Math."
"...But in the new approach, As you know, the important thing is to understand what you're doing, Rather than to get the right answer."
Interestingly that song is apparently about the first time they tried to massively redesign the mathematics curriculum based on on fuzzy math that we were falling behind the Soviets or whatever. My parents tell me they rebased around set theory early etc but it foundered on nearly the same rocks that common core mathematics is now - teacher must be well educated in the pedagogical paradigm and fully understand it. E.g. you can't just take the same set of teachers from math v1.2.1 and slam them into math v2.1.0 or whatever without significant issues the two approaches are not abi compatible. That's why in my with opinion common core we are seeing teachers test finding answers via method 1 specifically or method 2 specifically rather than encouraging the use of either to get a correct answer by an understood method. /rant
That all sounds about right. The song was from around the time I was in elementary school.
I suspect that part of the problem is that teaching math concepts is at least somewhat orthogonal to drilling kids on being able to perform basic arithmetic operations fluently. Of course, these days I assume at least some of the debate probably gets into the fact that being able to do, say, long division quickly has about the same utility as learning Palmer script.