Admittedly, the Standard Algorithm for Double-Digit Multiplication does assume a general knowledge of decimal place (ones, tens, hundreds, thousands, etc.), but other than this assumption it depends wholly on principals of addition and multiplication which should be firmly entrenched by the end of the third grade. The first TERC method illustrated seems to assume the notion that children will have been properly drilled in fractions before dealing with double-digit multiplication. I'm turning 31 this month, so things may have changed since I was in elementary school, but I recall learning double-digit multiplication before covering fractions. Does this hold true for anyone else?
"In TERC Investigations, no algorithms are taught; instead, students are encouraged to reason through problems with something called 'cluster problems'. Let me show you an example right out of the teachers manual."
Well, this sounds okay, but…
Oh, wait. One definition first: an "algorithm" is a systematic method of solving a certain kind of problem.
How is the first TERC example not systematic? I would personally just truncate it to (2031)+(631) because honestly, if you can't figure out how to multiply single digit numbers by double digit numbers, you're stuck counting on rote memorization to handle the multiplication tables from one to twelve. Please, please tell me they still teach multiplication tables one through twelve.
Okay, the second example: Partial products was something that I recall having gone over once in elementary school, more than twenty years ago. Once. It wasn't focused on, because it is more difficult. For teachers. If a student is able to understand decimal place (ones, tens, hundreds, thousands, etc.) why the hell shouldn't they understand the partial product algorithm? Personally, I didn't use it because it meant more writing, which is a pain. (I remember when having to write-off a phrase or paragraph x number of times was a form of punishment.)
Lattice Method? Ugh. I vaguely recall some such thing from High School Algebra, because my instructor covered every possible method to learn how to solve problems. I'd listen to the lesson for five minutes, sleep for twenty, wake up for the last five to catch the lesson plan, then study the book at home that night. (Musty old tomes of True Mathematical Learning. Ah, how I treasure that knowledge!) And you know what? I still made a straight A in that class for two and a half months- 'til my class participation grade started to catch up to me.