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July 30, 2013

Keep Calm and Carry One

Posted by Tom Leinster

Here’s a short diversion. My friend and mathematical big sister Audrey Tan, who’s a professional maths tutor in her native New Zealand, has been waging a campaign for the NZ government to return column addition to the early primary school curriculum. Primary school seems to be ages 5–12, so I guess “early primary” means kids aged 5–8 or so.

What’s column addition? It’s the way I, and probably you, add up natural numbers with pen and paper:

example of column addition

(Demonstration here, on the off-chance that you have no idea what this picture means.) For reasons I don’t understand, the NZ government has decided that this isn’t worth teaching in the first few years. Audrey thinks that’s madness, and is trying hard to put it right.

(Since this has caused some confusion, let me clarify: in NZ, column addition is taught, but much later than you might expect — not until the kids are about ten and have, supposedly, already digested the concept of place value.)

I’m curious to know: is this the method you learned when you were very little? Or did you learn to add some other way?

By the way, I take no credit for the truly inspired pun: that is Audrey’s alone.

imitation poster

Posted at July 30, 2013 5:48 PM UTC

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Re: Keep Calm and Carry One

I’m okay with this algorithm being phased out, despite it being what I learned and what I would probably default to given a series of sums and pen and paper. In my personal opinion, “carry the one” is not the most mathematically transparent algorithm, especially to very young students.

I’m purely speculating here, but it seems to me that the column addition algorithm became prominent despite sacrificing mathematical transparency because the gain was speed and efficiency of written space used.

With the prevalence of calculators available, I don’t think the trade-off is worthwhile anymore. I think everyone should be taught pen-and-paper algorithms for two reasons: 1) the unlikely event that a calculator is not available, and (more importantly) 2) as a foundational step towards mathematical fluency. I think that column addition is a clumsy step that deserves to go the way of short division and multiplication by adding logs: as a curious but outdated method.

Posted by: Travis on July 30, 2013 6:49 PM | Permalink | Reply to this

Re: Keep Calm and Carry One

From your answer, it sounds like you know some other pen-and-paper algorithm for addition. What is it?

Posted by: Tom Leinster on July 30, 2013 6:57 PM | Permalink | Reply to this

Re: Keep Calm and Carry One

Posted by: Travis on July 30, 2013 7:18 PM | Permalink | Reply to this

Re: Keep Calm and Carry One

There’s something about the second part of the partial sums “algorithm” that makes me feel uneasy. Is it assumed that the second lot of additions will be easy enough to do in ones head?

Consider 909 + 991. Are children expected to find 1800 + 90 + 10 trivial? It could just be familiarity with the traditional method, but I find myself wanting to go from least to most significant digit with carrying there.

Or are they implicitly expected to repeat the process to turn 90 + 10 into 100? What happens when one of them asks “what if it keeps going forever and never gets easier”?

Posted by: Ciaran McCreesh on July 30, 2013 8:19 PM | Permalink | Reply to this

Re: Keep Calm and Carry One

Yes, I had the same uneasiness. I wondered if the children were just meant to fall back on some native sense of number. That doesn’t sound too reliable. In fact, it has echoes of the NZ government’s idea (as I understand it) that the kids should just come up with a strategy themselves. But maybe there’s a part of the algorithm I don’t know about.

Posted by: Tom Leinster on July 30, 2013 8:27 PM | Permalink | Reply to this

Re: Keep Calm and Carry One

Since I was quite young I’ve always added numbers in my head using “partial sums” (I think of it as adding from the outside in). For me it is both much quicker and much less error prone than any other way of doing addition in my head, although I was taught the column sum way of adding.

However, I think that some algorithm should be taught. The partial sums approach only works after one understands what addition is doing. At first most children are learning formal rules, which they will understand later. Some want to reform everything so that children will understand everything before they learn any formal rules (perhaps even obviating the need to learn formal rules), but unfortunately children don’t work that way.

Posted by: Bobito on August 1, 2013 12:20 PM | Permalink | Reply to this

Re: Keep Calm and Carry One

I don’t suppose anyone has the time to describe the “partial sums method” in words, for those of us who don’t have time to watch videos about it?

Personally, I feel like not teaching a guaranteed algorithm for adding “because people will always have smartphones” is like not teaching kids to read “because everything will be on YouTube”. The benefits of pen-and-paper literacy of all sorts (including mathematical) go far beyond the obvious applications.

Posted by: Mike Shulman on July 31, 2013 4:04 AM | Permalink | Reply to this

Re: Keep Calm and Carry One

The idea is to break down a sum of two numbers into a sum of three or more numbers by breaking off smaller pieces that are easier to compute. As Ciaran pointed out above, it’s not always clear how to split up the sum to get something convenient so the “algorithm” isn’t totally decidable to a young student.

If we take something like 815+249, I know this is the same as 800+200+15+49. If I need to go further down to the ones place, I can do this, but I will still probably need to use a method that involves “carrying a one” somewhere. Seeing the most efficient way to split up the sum takes some experience.

Posted by: Patrick Durkin on July 31, 2013 5:32 AM | Permalink | Reply to this

Re: Keep Calm and Carry One

Mike wrote:

The benefits of pen-and-paper literacy of all sorts (including mathematical) go far beyond the obvious applications.

I think this is a very good point. To me, a strong argument in favour of teaching column addition as early as possible is this:

Everyone needs to know what an algorithm is.

In this increasingly technologized world, it’s more and more important to have a basic understanding of how computers function. If you’ve never grasped what an algorithm is, you’re destined to spend your life confused about everything to do with information technology. This is already depressingly common: see, for example, this awfully muddled piece in the generally admirable Guardian (and some equally muddled comments in response). Column addition is probably the first algorithm that it’s sensible to teach, and the earlier it’s done, the more likely it is that lesson will sink in.

I should clarify that the powers that be haven’t abolished column addition entirely (hence the “early primary” bit). The practice is not to teach it until they’re 10, by which time they’re supposed to have acquired a deep understanding of place value (units vs. tens vs. hundreds etc). I don’t know how they’re supposed to do that.

Also, Audrey tells me that partial sums are, essentially, one of the strategies that the kids are supposed to learn, except that the numbers aren’t aligned vertically as in the video that Travis linked to: everything is written horizontally, on a line. This hardly seems designed to encourage an understanding of place value.

Posted by: Tom Leinster on July 31, 2013 7:58 AM | Permalink | Reply to this

Re: Keep Calm and Carry One

I see, thanks. I reckon that partial sums method is something like what I do when I’m adding in my head. On the other hand, I think the issue under debate is not so much which algorithm to learn as whether to learn an algorithm at all.

(It’s funny watching those videos. That demonstrator has such a teacherly manner. I was transported straight back to primary school.)

Posted by: Tom Leinster on July 30, 2013 7:26 PM | Permalink | Reply to this

Re: Keep Calm and Carry One

I’d be inclined to agree with the government there. There is literally no point in being able to do exact 3-digit arithmetic - by the time these kids grow up, smartphones and wearable computing will be so ubiquitous that you will never be more than 5 seconds away from Wolfram Alpha or Google calculator; in fact by the 2020s you’ll probably never be more than 2 seconds away. (“OK glass, what is 46201*362828”)

The time would probably be better spent on honing estimation skills; that is the ability to look at some numerical claim and spot whether it is “dodgy”.

Posted by: Rationalist on July 30, 2013 6:54 PM | Permalink | Reply to this

Re: Keep Calm and Carry One

So the idea is that everyone will learn to use calculators to do things like 112 + 128, and people who become deeply interested in math will learn the theory of how to compute these sums in college, or on their own? That would be an interestingly different world. Personally I only feel I know what it means to estimate a sum because I know what it means to compute one; this allows me to understand how I can cut corners and get a roughly correct answer, and how much error there will be. But I can imagine someone who treats estimating as a bit like throwing a dart and trying to hit the target: something you learn by doing it a lot.

This makes me uneasy, but perhaps it’s only because I learned math by doing vast numbers of calculations on paper or in my head—and still do. I like knowing how things work.

Posted by: John Baez on July 31, 2013 1:27 AM | Permalink | Reply to this

Re: Keep Calm and Carry One

If I were the god of maths education, I would make sure that kids could

  • do 1- and 2-digit exact addition, including carry
  • do estimation for addition of numbers up to, say, 10,000,000,000
  • know how to use a computer or smartphone to do exact calculations for 3 or more digits

so, for example, you might approximate 128 as 130, approximate 112 as 110 and do the 2-digit addition 11+13=24, and then reason that the exact answer should be about 240.

Smarter kids could do 3-digit exact addition as an extension exercise.

Maybe my own personal biases are coming in here, but I remember being really turned off by maths at the age of about 10 because I had a book of about 1000 3-digit additions and subtractions to do (this was in the 1990s). I can imagine that a smart 10-year old today would be really bored and question the relevance of her maths class when she realized that she was doing pointless work that she would obviously never use.

Why not give the kids more fun work on probability theory? Show them the Efron dice.

Maybe teach them about prime numbers and how every number can be broken up into them.

Teach them how to do Fermi calculations, like how to estimate the number of leaves on all the trees in their city?

For weaker students, why not teach them how to solve practical maths problems using google or using their smartphone/a cheap school smartphone? Maybe show them how to query Wolfram Alpha?

There are plenty of interesting and useful things to do in mathematics education; it seems that 3-digit manual addition is not one of them.

Posted by: Rationalist on July 31, 2013 5:47 PM | Permalink | Reply to this

Re: Keep Calm and Carry One

I think there’s a danger of setting up a false dichotomy between learning to do exact computations and learning to estimate. I don’t think it’s an either/or.

Practising exact computation can help you learn how to do rough estimation, as John says. At least, that’s true as long as your exact algorithms are reasonably transparent. I’d say that column addition is quite transparent: you can easily see why it works, and it helps to instil that all-important sense of place value. (I’d say the long division algorithm is much less so.)

I suspect the education people in the NZ government have their hearts in the right place here. No one wants to train a generation of automatons, who can compute accurately but have no conceptual understanding. But what’s possibly being overlooked is that calculation can often be a stepping stone to understanding. This is apparent at all levels of mathematical learning: we do some formal calculations before being able to justify them.

Another thing that’s often forgotten is the pleasure of calculation. Once you know you can do it right, carrying out an algorithm can be a thing of joy: it’s like running a perfectly-tuned machine, or even being one, and it’s almost meditative. After doing it enough times you get bored, but by then you’re on top of it — and your confidence has grown.

Posted by: Tom Leinster on July 31, 2013 12:34 PM | Permalink | Reply to this

Re: Keep Calm and Carry One

Hear, hear; both this and your other comment. Not only is it essential to understand what an algorithm is in the modern world, I think it’s also essential to understand the difference between an exact answer and an estimate, and indeed what an “estimate” really means. How can you comprehend that if you aren’t taught how to get exact answers, so that you can compare them with your estimates?

There is some beautiful mathematics behind the long division algorithm, but I agree that it’s less transparent — I’m not even sure how well it can really be explained until you have some algebra. Same for the manual square root algorithm, which already in my day was no longer taught in schools (I only learned it as part of training for a math competition).

Posted by: Mike Shulman on July 31, 2013 8:19 PM | Permalink | Reply to this

Re: Keep Calm and Carry One

Is there some other way to add natural numbers with pen and paper?

Posted by: Mike Shulman on July 30, 2013 6:56 PM | Permalink | Reply to this

Re: Keep Calm and Carry One

What is being taught as an alternative? I had my doubts when told that schools in England are now teaching a different way of doing long multiplication:

http://en.wikipedia.org/wiki/Grid_method_multiplication

But it makes sense, and doesn’t require learning a new method for multiplying x + 2 by x + 3. Is something similar going on with addition, or are they simply not teaching children how to add large numbers at all?

Posted by: Ciaran McCreesh on July 30, 2013 6:58 PM | Permalink | Reply to this

Re: Keep Calm and Carry One

Hi Ciaran — nice to hear from you.

As I understand it, what’s currently taught to these kids is to come up with their own strategy for adding numbers. So, no algorithm at all. Is it working? Well,

According to the New Zealand Herald, New Zealand 9-year-olds finished last-equal in maths among peers in developed countries in an international survey published in December 2012.

A letter from famous Kiwi Vaughan Jones points out that not to use column addition is to fail to take advantage of the excellent innovation that is the decimal number system. (“Would one try to teach addition and multiplication using Roman numerals? Not a chance.”)

Posted by: Tom Leinster on July 30, 2013 7:08 PM | Permalink | Reply to this

Re: Keep Calm and Carry One

There could just as well be a problem with the survey than with the teaching of math in NZ. If teachers in NZ focus on being able to do estimations, and the survey measures the ability to do exact calculations, then obviously NZ is going to end near the bottom.

Posted by: Sjoerd Visscher on July 30, 2013 10:33 PM | Permalink | Reply to this

Re: Keep Calm and Carry One

I expected to find a disaster at the other end of your link, but you’re quite right: “grid multiplication” is actually rather pretty. It really drives home the distributivity of multiplication over addition, and it also makes good use of the geometric picture of multiplication as an area computation.

I think one could also use it to teach approximate calculations and estimation, by leaving out the lower-order grid cells.

Posted by: Neel Krishnaswami on July 31, 2013 9:18 AM | Permalink | Reply to this

Re: Keep Calm and Carry One

Re, “is there another way”: well, yes and no.

Let’s agree for the moment that it’s inhuman to require adding a column more than about 15 multidigits (and so also to multiply two 15-digit numbers), and so suppose instead that the place-column sums are at most two digits. Rather than carry the second digit to the head of the next column (where I will forget it, or forget what it means when I get there, because that’s the kind of poor computer I am), let the sum digits be written on two lines, the “carry” digits on the second line; So that

  9737
  8934
+  736
------
  7397
+12010
------
 19307
+00100
------
 19407

Typically, in the second round of summation all the cary digits are 0 or 1 and there are no cary digits the third round. I’m sure you can cook up examples where the second round carries a 1 into a string of five nines, but we can be smarter about that, too. It’s basically the same addition algorithm, but with legibility in mind rather than saving space. It also has the (dubious) advantage that you can write it, in order, in an i.r.c. or similar medium.

Posted by: Jesse C. McKeown on July 30, 2013 10:48 PM | Permalink | Reply to this

Re: Keep Calm and Carry One

Of course another option is to stick with carrying, but get the children to understand what’s ‘really’ going on ;)

Being an abelian group, every delooping n-groupoid B n 10\mathbf{B}^n \mathbb{Z}_{10} exists.

Carrying is a 2-cocycle in the group cohomology, hence a morphism of infinity-groupoids

c:B 10B 2 10. c : \mathbf{B} \mathbb{Z}_{10} \to \mathbf{B}^2\mathbb{Z}_{10} \,.

Posted by: David Corfield on July 31, 2013 7:24 AM | Permalink | Reply to this

Re: Keep Calm and Carry One

I like this one!

Posted by: Bruce Bartlett on September 4, 2013 12:46 PM | Permalink | Reply to this

Re: Keep Calm and Carry One

Audrey tells me that the idea behind the government’s strategy is this:

Children will become so numerate that they don’t need to write down anything at all.

If you think this sounds like a caricature — that no one could possibly believe this — check out this now-deleted sentence from the Ministry of Education’s website:

Q: “When should I start teaching the written form?”

A: “Teachers should debate whether they will introduce the written form at all. […]”

Posted by: Tom Leinster on July 31, 2013 8:06 AM | Permalink | Reply to this

Re: Keep Calm and Carry One

Anyone who wants to read more discussion of this issue can check out the 96 comments on my Google+ post on this topic, inspired by Tom’s post here. There’s really nothing like the teaching of basic math to stir people’s passions.

Posted by: John Baez on August 1, 2013 4:08 AM | Permalink | Reply to this

Re: Keep Calm and Carry One

Some misimpressions in my G+ post are corrected here.

Posted by: John Baez on August 4, 2013 3:53 AM | Permalink | Reply to this

Re: Keep Calm and Carry One

I learned this method (column addition with carry) in 1st grade (Germany 1970) but I do not think I have used it often since then. It probably made things more obvious when it came to adding in base 2 or 16 later on, or to understand the carry bit in assembler programming.

Nevertheless, I think the method is nice to introduce to first-graders because it can easily explained to them why it actually works. Whether one should train pupils to proficiency in this method is a different question; there it easily ends up in boredom. So it is important, not to ride a particular horse to dead with mindless computations.

Posted by: Marc Olschok on August 2, 2013 12:59 AM | Permalink | Reply to this

Re: Keep Calm and Carry One

Tom, your link to your mathematical genealogy seems to be broken, but here is another that shows Martin Hyland’s progeny.

Posted by: Todd Trimble on August 3, 2013 4:34 AM | Permalink | Reply to this

Re: Keep Calm and Carry One

Huh, strange: it worked before, but now it doesn’t. I’ve changed it to the one you give (which is identical). Thanks.

Posted by: Tom Leinster on August 3, 2013 8:54 AM | Permalink | Reply to this

Re: Keep Calm and Carry One

There are several distinct questions here.

1. Should the standard algorithm be taught at all?

2. Given an affirmative answer to 1, when should it be taught?

3. Given an affirmative answer to 1, should it be the only algorithm which is taught?

For me the answers are:

1. Most certainly

2. Whenever the student can actually comprehend what is going on.

3. Definitely not.

Here are my reasons:

**1 - should the algorithm be taught at all?**

It is incredibly important to understand algorithms. This algorithm is very efficient, and is one of the most useful algorithms I can think of. Personally, I think that algorithms should be stressed a lot more than they are currently. In particular, I think that computer programming should become just as standard as mathematics is now. It is just as useful, builds careful reasoning, and educates people about how computers really work, which is fundamental to how the world works.

I also think that the natural numbers are really fundamental to mathematics, and understanding how the basic operations on natural numbers work in a place value system is pretty important. Knowing an algorithm for addition helps with this a lot.

**2 - When should it be taught?**

It is possible to train a student to perform the addition algorithm without understanding what is going on at all.

This has certain advantages. They are able to perform well on certain kinds of tests. Knowing how the algorithm works might help them to later understand why the algorithm works.

The major disadvantage is that it gives the impression that mathematics is mindless rule following, and this is done at such an early age that children might never recover. This is enough of a disadvantage for me that I am okay waiting, and approaching this in a slow and cautious manner. I think that as prerequisite to being taught the algorithm the student should

a. Understanding how to count.

b. Understand what addition is: they should understand that 3+4 is the number of things you get if you combine 3 and 4 things together.

c. Understand the place value system. They should know how many objects the symbol 10 represents. They should know that 57 means 5 10s and 7 ones. They should understand that 100 means 10 10s. Without understanding these basic things, even writing the number 53 has no real meaning. So we should not expect them to do 53+89.

d. Struggle with the problem on their own a bit. Children should realize that the standard algorithm is fantastically clever, and that it solves an important problem. Spending some time struggling with this problem on their own is needed to have an appreciation for the problem. Some children will only be able to solve addition problems in an ad hoc way, and will get inconsistent results. Other students might develop algorithms of their own, which is wonderful because they get to feel the thrill of mathematical creation.

e. At this point the algorithm can be introduced. It will have meaning to the students, and they can understand both how and why it works.

**3. Should it be the only algorithm which is taught**

I think it is essential to teach more than one algorithm. If only one is taught, it seems like the algorithm defines addition, rather than just being one way to compute a sum. This is also a great early opportunity to discuss the advantages and disadvantages of one algorithm over another. I still think that the standard algorithm should be the one which is drilled most often, but discussing other algorithms is a useful way to force students to think about why these algorithms work.

Here is an algorithm with a lot of merit:

568
+479
——
17 <— sum of the 1s
130 <— sum of the 10s
900 <— sum of the 100s
——
7 <— sum of the 1s
40 <— sum of the 10s
1000 <— sum of the 100s
——
1047 <— final answer

This algorithm is a lot windier than the standard algorithm, but has some pedagogical advantages. Instead of thinking about adding 6 and 7, with this mysterious 1 that somehow pops over from the 1s column, here you think of adding 60 to 70 (realizing the meaning of those 6 and 7), and you can clearly see that the “carry” is an extra 10 coming from the sum of the 1s. A fun thing to think about is how many iterations are needed for the algorithm to terminate.

I actually think that this is a better “first algorithm” to teach. The standard algorithm could be developed as a “shorthand” for this one.

Posted by: Steven Gubkin on August 15, 2013 8:13 PM | Permalink | Reply to this

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