First, I think my general thinking hasn't changed that much from the original post, where I stated:
algorithms, when used as a result and in conjunction with understanding and meaning, can be a good thing, while acknowledging that we have often emphasized the algorithm at the expense of understanding and meaningI appreciated the detailed descriptions and explanations of the various algorithms in the Fuson and Beckmann article and they certainly got me thinking. The authors argue that the CCSS-M includes the meaning and sense-making as part of students' development and understanding of algorithms, which I agree with. They also state that in the past the algorithm was often taught without that meaning and sense-making portion, and I think most of us agree that is not a great approach.
The part I still disagree with the CCSS-M (and maybe Fuson and Beckmann as well, I'm not sure) on is the idea of the standard algorithm. I think the crux of this argument boils down to folks who argue for one specific standard algorithm because it is the most "efficient." I would argue that the most efficient way to do most of these calculations is with Siri or Google Glass or Wolfram Alpha. I'm more interested in students understanding the mathematical underpinnings of the algorithm(s) then in being able to quickly apply them. I want them to have number sense and mathematical understanding, but I don't think that necessarily means being able to "efficiently" compute a four digit by four digit multiplication problem by hand.
So, as long as I can change the standard algorithm to a standard algorithm that makes both mathematical sense and is most helpful to that particular student (which, I admit, doesn't flow quite as freely), then I'm good.
As a side note, this TED Talk happened to get posted just as I was trying to compose this blog post.
I think it's interesting in and of itself, but I also found it interesting how often he refers to "algorithms" and "mathematical modeling." I think this shows the power of algorithms, but also the need for our students to understand the algorithms, and also to understand that algorithms are first and foremost developed by humans and are not always set in stone, as when the algorithms appear to "adjust" or "learn" in the flipping sequence.
If our students can be begin to understand that algorithms don't supersede our understanding, but can help enhance it, then I think we're on the right track.