For the "exercises are important and also build conceptual understanding", see "BASIC SKILLS VERSUS CONCEPTUAL UNDERSTANDING, A Bogus Dichotomy in Mathematics education" by H. Wu.
The big challenge in teaching is not to make the exercises seem lame.
I haven't read this whole article yet (PDF here https://math.berkeley.edu/~wu/wu1999.pdf), but this idea is my #1 problem with Tom Lehrer's "New Math" song. His big complaint at the beginning is that the emphasis is too much on understanding and not on efficiently/correctly calculating an answer. As funny as the song is, I don't think that complaint has aged well at all. Also the old algorithm was just as complicated as the new one, but at least the new one makes it easier to see what's going on, so the whole joke kind of falls apart.
As far as I understand the article, the author (and myself) absolutely wants people to practice longhand addition etc., and is pushing back on the idea that you can teach how to calculate XOR how to understand place value.
If some of the kids go on to study CS, they can then think about the similarities and difference between decimal, hex and binary addition, how half and full and ripple-carry adders work, and how you add bigints. At that point you need both a conceptual and a procedural understanding of digit-wise addition.
If you want to do say 67 + 24, perhaps even in your head, there are more efficient ways to add than the standard algorithm, and I think that's what new math was trying to get at. But at some point you might want to add 25137 + 1486 and then your neat tricks no longer work and you need something that scales.
New math or common core or any approach that tries to center ‘understanding’ over rote procedure is definitely pushing in the right direction. I think that people who have the closed understanding of addition, that there is one true algorithm for doing it and who will start your 25137+1486 problem off by adding six and seven to get three and carry the one… are missing out on a deeper intuition about numbers, because they only think of those numbers as sequences of digits.
But someone who sees that as ‘add fifteen hundred and take away fourteen’ is much closer to understanding what that expression actually represents, as well as being able to produce 26623 almost immediately without writing anything down.
It’s not about ‘neat tricks’, it’s about numbers having shape and feel and flavor.
> 25137+1486 problem off by adding six and seven to get three and carry the one… are missing out on a deeper intuition about numbers, because they only think of those numbers as sequences of digits.
This is precisely the dichotomy that is bogus according to the article.
25137 = 20000 + 5000 + 100 + 30 + 7 and 1486 = 1000 + 400 + 80 + 6, then you add (7 + 6) + (30 + 80) + (100 + 400) + (5000 + 1000) + (20000 + 0) to get the result. The fact that we can do that and combine it all tightly into columns is IMO a very deep insight into what a "number" really is, while also providing a general pen-and-paper algorithm for adding any two numbers. The insight provides an algorithm, and the algorithm leads us to an insight.
Discovering that 1486 = 1500 - 14 isn't a particularly deep insight into numbers either. It's a useful technique and I think it's fine that we teach it, but I don't think it has any particular conceptual merit that the standard algorithm lacks. I certainly don't see how it puts a child any closer to understanding what addition really means.
No but that’s actually exactly what ‘new math’ was about. The thing Tom Lehrer was lampooning was all this talk of the ‘tens place’ and the ‘hundreds place’ rather than just plugging and chugging the digits, you know;
Seven plus six is thirteen carry the one leaves three, four plus eight is twelve carry the one leaves two, two plus four is six, five plus one is six, two six six two three…
Seeing that as a decomposition of multiples of powers of ten and how that makes ‘carrying’ happen is exactly a result of having a deeper understanding of the way the numbers work.
For the student who doesn't understand, one rote algorithm is as boring and stupid as any other. That student is plugging and chugging all the same, whether or not they have heard of a "tens place".
For the student who does understand, the "new" algorithm at least is elucidating and actually makes sense as a direct application of the basic principles of our number system. The "tens place" is in fact a real thing, regardless of what you call it.
I’m not remotely arguing against encouraging kids to discover the mathematical principles themselves. I’m also not advocating teaching swapping + (1500 - 14) into -14 + 1500 by a careful application of the laws of commutativity. I’m saying that having a comfort and confidence with what summation is is way more valuable than learning that addition is a procedure applied to digits.
The big challenge in teaching is not to make the exercises seem lame.