@Mathowitz I don't either. Particularly when students don't even know what is happening. I taught a group of pre-service teachers how to use models alongside the standard algorithm as they divided ... And, every one of them said they had no clue why it worked until that moment!

@techknowmath @Mathowitz And yet a child with a huge bag of candy understands partitioning and doesn't end up frustrated if they don't use the biggest divisor first.

@Mathowitz @LaneWalker2 @techknowmath This may be the disconnect. You see value in getting the right answer. I see the value in understanding a complete and robust framework. In a world of calculators, computing won't take you far. But being able to internalize rigorous complex abstractions does indeed take you far.

@Mathowitz @LaneWalker2 @techknowmath So yes, this is where we depart. I'm not so invested in getting a computational result. The value lies in the ability to learn and understand abstract systems. You have to provide access and instruction on the *complete* system to deliver lifelong value.

@Mathowitz @LaneWalker2 @techknowmath You are suggesting that understanding the components of a system are central to understanding the whole system. Yes! Numeric fluency includes understanding algorithm particulars. You cannot understand the decimal representation of 1/7 if you don't understand long division.

@Mathowitz @LaneWalker2 @techknowmath This is a great question! The relationship between decimal representations and fractional representations is a core aspect of the systematization of mathematics: Reals vs. Rationals. You can argue that we don't need to know this. That same argument holds for all math.

@Mathowitz @LaneWalker2 @techknowmath That's the most direct way. What other way do you have of generating the decimal expansion of 1/7?

@Mathowitz @LaneWalker2 @techknowmath Which algorithm is the best for computing the decimal expansion of 1/7? I'm trying to imagine doing it with partial quotients, and it feels like it would be very easy to get lost, and miss the fact that the decimal expansion is a repeating seven digits.

@andrewprock @Mathowitz @techknowmath True: we don't see the repetition until 7 place vals, tedious either way. But not choosing greatest divisor once messes up the entire trad algorithm. I am thinking one may get lost when they are fixed on one way of decomposing