I now want to start looking more deeply at some specific questions at various levels, starting with arithmetic, then algebra and geometry.
Students learning arithmetic (and their parents) tend to think in terms of following some rote procedure, just because “that’s the way it’s done”. Modern educators try to focus more on deep understanding, so that what is done is done for a reason. But even after learning the operations well, with a proper focus on meaning, you still have a procedure, and you can wonder: Why this way, and not that? It’s a good question. And if you have been taught in a flexible way, there will be an answer.
Division: Why not right to left?
This sort of issue is exemplified by a question from Sandip in 2015:
Division Direction Addition, subtraction, and multiplication all proceed from right to left. But in the case of division, it is not so. Suppose we want to divide 100 by 4. If we start from the right, then the ones place (0) is divided by 4, and our result is 0. Similarly, in the tens place, 0 divided by 4 again is 0. And finally, 1 can't be divided by 4. Maybe my logic is not so strong, but can division ever start from the right? Compared to the other operations, what makes it exceptional?
I started by introducing one of the methods commonly taught today as a preparation for the traditional “long division” method, which has been called the “scaffold” (or sometimes “pyramid”) method:
The reality is that you can start anywhere! In dividing, we are simply guessing a portion of the quotient, and subtracting to see what is left to be divided. If we start at the left, we can make efficient guesses; other ways will work, but will require extra steps. Consider the example of 941 divided by 7. Here is one method of division that is often taught before the final long-division algorithm: 134 --- 4 30 100 ------ 7 ) 941 -700 ---- 241 -210 --- 31 -28 --- 3 Here, rather than thinking about digits, we just think of pieces of the quotient that we will add up to get the entire quotient. The process amounts simply to subtracting any multiples of the divisor we want (in this case, 700, 210, and 28) until only a remainder is left. We write each quotient at the top, and then add them up at the end.
Why did I start here? Because this method, unlike the traditional algorithm, is very flexible. Once students are familiar with it, they will do it as efficiently as I did above, which is practically the same as the traditional long division (just erase some zeros and collapse the tower on top). But initially, they can just subtract whatever multiples of 7 they happen to see. It does not require all the skills that even many adults find challenging: estimating the “right” digit for the quotient, correcting that estimate if it is wrong, and keeping track of all the “bringing down” and so on that they do without understanding. That process can be scary, because everything has to be done “just right”. The scaffold method puts up a safety net, by not requiring perfection. Just find a way to get the answer, without worrying about doing it the “best” way. That can come later.
But that flexibility also allows you to use this method from right to left, if you want:
If we started at the right, we might do this: 134 --- 9 20 100 5 ------ 7 ) 941 - 35 ---- 906 -700 --- 206 -140 --- 66 -63 --- 3 ... Note that our first step, which gave us 5, was sort of wasted: we ended up changing that digit of the quotient. This will be typical when you work from the right.
That wasted step is my point: working from right to left isn’t wrong, just inefficient. You might say that in division we are dismantling the result of multiplication, and it is natural to take apart first the last parts that were put together, much as we reverse the order of operations in solving an equation:
Another thing worth pointing out is that division amounts to undoing multiplication, which we naturally do in reverse. The division I did above (the first time) is the reversal of the following multiplication process (written in a non-standard way to make more steps visible):
134
* 7
----
28 ------------------------------> subtracted last
210 ----------------> subtracted second
700 --> subtracted first
----
938
+ 3 --> remainder
----
941
That can be one answer to the “why” question; but the best answer, I think, lies in the purpose of the algorithm:
Ultimately, though, we divide left to right simply because that has been found to be the most efficient way to do it. It is the collected wisdom of several centuries, not something that was forced on us by the way numbers work.
One of the benefits of teaching a method such as the scaffold is that we can let children discover for themselves that the method they end up with is not something hard that they are forced to do, but the easiest way to do the task. As I often say, a good mathematician is lazy, always looking for the easy way (even if it requires extra thought up front). And a good mathematician is not just a rule-follower, but a pathfinder, always looking for signs that point to that easy way.
Be sure to read all of this answer (and the further articles it links to), because there is a lot of information there. I especially recommend this page by Dr. Ian, which shows how abandoning efficiency even more than the scaffold method can produce great gains in understanding:
Longer Division ... There are two nice things about starting with 'longer division'. The first is that it's easy to understand - it's just grouping and subtraction, and everything is right there out in the open. The second nice thing about it is that as you get better at guessing, it more or less turns into long division naturally... but you take that step on your own schedule, not when someone else thinks you should.
Adding: Can it be done left to right?
But Sandip stated as given that the other operations are done from right to left. We can raise the same question there: Can they be done from left to right? This, too, is a matter of efficiency, not necessity, as I explained to Tonya in 2003:
Adding Left to Right When I was in school, we were taught to add numbers from right to left. Now my daughter's teacher is teaching her to add left to right. How can I help her to understand both ways? The type of learning is different from when I was a kid. 25 25 15 15 -- -- 40 (my way) 30 10 --- 40 (new way)
This (adding the tens, then adding the ones, then adding the results) is actually a good way to do mental addition, as I explained; but it is less efficient as a written method:
The traditional way (which was developed over centuries as the best method for paper and pencil, since it requires the least writing or erasing) starts at the right. We add the "ones" column, and get 5+5=10. We can't write this as the ones digit in the answer, because it is more than one digit; so we break it up as 1 ten and 0 ones, and write down the 0. The 1 ten is added to the tens column in the problem, giving us 1+2+1 = 4, which we write as the number of tens in the answer.
Then I modified the “new” method, showing how it is really the same as the “old” method but with more writing:
We could do the "new way" from right to left as well; the order doesn't matter. Let's compare that with the traditional way:
new: old:
1 <-- tens from 5+5
25 25
+ 15 + 15
---- ----
5+5 --> 10 0 <-- ones from 5+5
2+1 --> 3 4 <-- tens from 2+1, plus 1
----
40
What I was showing here is just like the scaffold method for division: the “new” method, by focusing on meaning rather than efficiency, is flexible enough to allow going in either direction.
I think it's clear that this method is less efficient with pencil and paper, because there is more writing. And working from left to right in the traditional method would require erasing, because you would write 1+2 = 3, but then have to add 1 and change it to a 4 after you added 5+5. That was done in earlier days, because erasing is no trouble on a sand table or chalk board or abacus; it was only with the introduction of arithmetic on paper that the right-to-left method was needed.
Once you have learned the basic method, you can discover the shortcuts for yourself, or with just a little encouragement.
Very interesting answer to my question (why multiplication from right but division from left?). But if division goes from the opposite direction of multiplication because it is the ‘dismantling’ of multiplication, why doesn’t subtraction (the dismantling of addition) go most efficiently from the lefthand side too?
That’s a good question! To tell the truth, I never really answered the ultimate “why” question, did I? The immediate answer is that it’s for efficiency; but why is left to right more efficient for division, while right to left is more efficient for subtraction? It’s not ultimately the mere fact that we are “dismantling” something; it’s how interrelated the digits are. So let’s look into that (keeping in mind that “why” questions don’t always have answers as clear as we’d like).
One thing that will help is to read the whole original answer, which I only excerpted in this post: Division Direction. In particular, a trio of linked answers on how and why long division works illustrates what I mean by “dismantling” (or, in the original, “unpacking”).
Now, look at my example of the multiplication we are dismantling when we divide. Each partial product (one digit multiplied by the multiplier) may affect many digits of the product, so we can’t look at one digit of the dividend (product) and work backward to a single digit of the quotient (multiplicand). Moreover, the rightmost digit is primarily influenced by the remainder, so it says nothing at all about the quotient.
In contrast, in addition, each place is largely self-contained, only affecting the place to the left when a carry is needed, occasionally rippling all the way to the left. A digit in the sum is never affected by digits to the left. So reversing addition can start at the right, with “borrowing” rarely proceeding more than one digit to the left. (It is not really too hard to subtract left to right, if you think of “borrowing” as “lending”, where before subtracting in one column you check whether the column to the right needs assistance.)
So I think it is the complexity of multiplication, with places being largely interdependent, that makes it so much more efficient to work left to right.
You can see some other relevant ideas by comparing polynomial division; consider this page: Why Does the Algorithm for Polynomial Long Division Ignore Smaller Terms?.
I’m not entirely satisfied with this; I’ll keep pondering it.
My impression is that all of the 4 arithmetic operations can be done from the right to the left — and is probably more intuitive when done that way. The scaffolding method introduced here is basically the partial quotient method (or the chunking method), which I think is pedagogically advantageous as it explains how the iterated subtraction aspect works in principle.
For a more higher-math approach to integer long division, this guide might help. It explains what the division actually does at the algebraic/symbolic level, and introduces some alternative methods to long division as well.
Certainly one of my points was that all the operations can be done in either order; the interesting question is why only division seems to work most efficiently from left to right. I’m not convinced that dividing strictly right-to-left would be more intuitive; but clearly what you call “chunking” is – largely because it is not a specific algorithm, but requires thinking. And working generally left-to-right, but with more flexibility, leaves you with reasonable efficiency as well.
One thing I notice in your page is a comment that the traditional long division is a greedy algorithm; in effect, that’s another way to say “left-to-right”. But I don’t see that you ever actually carried out a division right-to-left by any method.
Taking your example of 7) 941
and knowing that
7*1=7
7*2=14
7*3=21
7*4=28
7*5=35
7*6=42
7*7=49
7*8=56
7*9=63
Applying that knowledge from the right, we might do this:
134 r 3 ---------------- 1 we are left with a 10 100 After adding 210 we have a 7 in the hundreds column -30 Because we have a 5 in the hundreds column and need at least 200 60 Because we have a 2 in the tens column and 7*60 gives us a 2 there (7*6 = 42) 3 Because we have a 1 in the ones column and 7*3 is the only one that gives us a 1 ------ 7 ) 941 - 21 ---- 920 -420 --- 500 210 --- 710 -700 --- 10 - 7 --- r 3Hi, Nikolay.
That is an interesting thought, working right to left focusing on the rightmost digit. It might be a good idea if you expected a remainder of zero, because that can help to determine the rightmost digit of the quotient. Unfortunately, if that is not the case (as in your example), the work you put into choosing that rightmost digit (which might be a lot harder in other cases) is essentially wasted, as the rightmost digit of the dividend ultimately determines only the remainder, as I pointed out in my reply to the March 27 comment.
And in fact, if you try your method for an exact multiple of 7 (I chose 2002), it is as efficient as can be!
286 ---- 200 80 6 ----- 7)2002 -42 ---- 1960 -560 ---- 1400 -1400 ---- 0But part of what made that work so well was that there was only one choice at each step. If the divisor had a common factor with 10, we would have choices, and only the right choice would lead directly to the answer. For example, consider this, dividing by 6:
389 ---- 300 50<- another 0 30<- could be 3 or 8 5<- last digit will be 0 4<- could be 4 or 9 ----- 6)2334 24 ---- 2310<- next digit odd 30 ---- 2280 180 ---- 2100<- next digit odd 300 ---- 1800 1800 ---- 0You could turn this into a rather nice method -- but as soon as you enter the real world (where answers are not so neat), most of the learning would be useless.
But you would have discovered some interesting bits of number theory along the way, if you were observant!