How I'd like to teach Division

(Not Your Parent's) Long Division:

Dividing requires finding out how many "times" a number "goes into"another. It's really just counting and subtracting. 

Divide 2467 by 14

Start by taking one 14 out of 2467.

2467 - 14 = 2453

And then another,

2453 - 14 = 2439 

and another!

2439 - 14 = 2425

14 has been removed 3 times. The division question, really, is asking if this pattern were continued until no more 14's could be taken out, how many 14's would have been removed in total.

Speed up by taking out ten 14's at once.

 2439 - (14 X 10) = 2439 - 140 = 2285 

Or 100!

2285 - (14 X 100) = 2299 - 1400 = 885

14 has now been removed 113 times! Completing the remaining steps could look like this.

885- (10 X 14) = 855 - 140 = 745

745 - (10 X 14) = 745 - 140 = 605

605 - (10 X 14) = 605 - 140 = 465

465 - (10 X 14) = 465 - 140 = 325

325 - (10 X 14) = 325 - 140 = 185

185 - (10 X 14) = 185 - 140 = 45

45 - 14 = 31

31 - 14 = 17

17 - 14 = 3

Counting the total number of 14's removed: 3 + 10 + 100 + 60 + 3 = 176.

That number leftover, in this case 3, is called The Remainder. 

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Reflections:

Once students grasp the general mechanics of dividing numbers and can see division as a method to split into pieces, group items, etc. They are made to divide larger and larger dividend by larger and larger divisors. Ultimately students (and everyone else) reach for calculators, which is all well and good, but what about when you don't have access to a calculator (rare these days, save for the constraints of very restrictive test environments). Long division is the answer according to math teachers for the last .... years (I don't know when this algorithm came about; I'm curious). 

We shouldn't go as far as to say Long Division is bunk or useless. It's not. It is a pretty necessary algorithm used in high school division of polynomials. BUT it is overly restrictive about format and removes the choice students can make about what multiple of the divisor they want to remove at each step. Student with stronger multiplication skills may take fewer steps, but I don't see any real benefit in a student knowing that 140 goes into 1067 exactly 7 times (which is what Long Division would require here after dividing 2467 by 1 X 14 X 100).

Long Division is a special case of this division procedure. I'm not sure which would be easier to write up a pure algorithm for or describe to a computer. Which is the kind of question I plan to put to my students in regards to algorithms and the kind of thinking I want them to develop alongside, the why does it work and when might it not work.

I think generally it's a lot easier to remove the larger multiples first. Perhaps partly because we have more experience and therefore comfort with smaller numbers. I wonder if starting with removing a single number would slow students down in terms of seeing this.

I think that I would plan to leave it to students to realize for themselves that they can remove multiples that are not powers of 10. I think 2 and 5 are good ones for lots of division problems.

Oh, a game that will lead to the more traditional long division is to challenge students to the same numbers in the fewest possible steps.

Not sure of the easiest or cleanest way to set this up on a page. The important part is that students can easily count the number of divisors removed and have space to compute the subtraction when necessary.