paper-and-pencil computation

In a previous post, I wrote: Children need practice with a wide variety of basic computation skills.

One of the basic skills listed was: Paper-and-pencil computation.

But which methods of paper-and-pencil computation?

Let’s look at multiplication, as an example.


Most parents I meet learned the multiplication method shown on the right.

Here’s the sequence of steps:

  • 6 times 4 is 24
  • write the 4
  • carry the 2
  • 6 times 2 is 12
  • add the 2 we carried to get 14
  • write the 14
This method has some advantages:
  • every multiplication we do involves only single digits
    • you can multiply very large numbers by only knowing your nines multiplication table
  • it always works the same way
    • regardless of how big the numbers are, the process is always the same
  • it’s concise and efficient
    • if you have a job where you have to do a lot of multiplications every day, and you don’t have access to a calculator, this method is an asset
It also has some disadvantages:
  • it un-teaches place value
    • before learning to multiply, children learn that 24 is 20 + 4 (or that 24 is 2 tens and 4 ones)
    • using this multiplication method, children practice calling numbers by the wrong names
      • for example, when we say “6 times 2 is 12”, the 2 is really 20 and the 12 is really 120
      • similarly, when we say “carry the 2”, the 2 is really 20
  • the method gives the correct answer even if you don’t understand
    • in other words, you don’t have to have good number sense to make it work
    • which also means that practicing this method does not help develop your number sense
    • for example, I once received a package where on it somebody had computed 10 x 12, as shown at right
    • this is why when you ask adults to explain this method, they often can’t: they learned it as a set of rules without conceptual understanding
  • it does not have a mathematical future
    • because the method breaks math rules (calling numbers by their wrong names), you can’t make connections to other math children have to learn (unless that math also breaks math rules the same way)

Because of the conceptual disadvantages, some education jurisdictions are hesitant to teach such paper-and-pencil computation methods to young children.

However, this does not mean that all paper-and-pencil computation methods have these disadvantages.

Let’s look at an example for multiplication:

Multiplication v.2

Let’s multiply 6 x 24 using a different computation method, as shown at right.

Here’s the sequence of steps:

  • 6 times 4 is 24
  • 6 times 20 is 120
  • 24 plus 120 is 144
How this method is different:
  • it reinforces place value understanding
  • it is similar to the way most people multiply in their head
  • it is easy to explain
  • it has a mathematical future, as it correctly uses place value, expanded notation, and the distributive property:
    • 6 x 24 = 6(20 + 4) = 120 + 24 = 144
  • it gets students ready for algebraic processes: 6(y + 4) = 6y + 24
  • it also requires students to know, and practice, how to multiply with multiples of 10 (as in 6 times 20 is 120)
What about bigger numbers?

Don’t we eventually have to carry?

Let’s take a look at 9 x 24.

Here’s the sequence of steps:

  • 9 times 4 is 36
  • 9 times 20 is 180
  • at this stage, if you can’t see that the sum is 216, you can use partial sums:
    • add the hundreds to get 100
    • add the tens to get 110
    • add the ones to get 6
    • altogether, that’s 216

Is there Multiplication v.3?

There are variations to the above method.

Whatever paper-and-pencil methods we decide children should learn, we should ensure that they are understandable and have a mathematical future.

That is:

  • paper-and-pencil computational methods should build on children’s previous knowledge
  • they should prepare children for more complex mathematics

kids need a wide variety of basic computation skills

In another life, I was a district math consultant. In my first month in the job, I received a call from an elementary school principal, inviting me to a parent council meeting for 10-15 minutes to talk about the new curriculum.

I asked, “Would it be possible to have an hour, or an hour and a half, so we can do some math together?” The principal checked with his parent council and we scheduled a math night for parents at the school library.

To make a long story short, word got around and I started receiving invitations for parent math nights across the district. Sometimes the turnout was 20 parents. Sometimes it was over 200, with tables and chairs filling a gym. It was one of my favourite things to do!

I quickly noticed 2 things:

  1. Many of parents feared or even hated math.
  2. Quite a few commented on their children’s basic computation skills.

To get parents thinking about what basic computation skills might be, I did 3 warm-up activities with them:

1. Counting money

I brought some bills and coins and I laid them out on a table: a collection of twenties, tens, fives and coins.

I invited a parent volunteer to count the money. I asked them to touch the bills and coins as they counted.

Then I invited a second parent volunteer to check the first answer.

How would you count the bills and coins?

If you are a typical person (like the parents at the Math Nights), you would first count the $20 bills, then the $10 bills, then the $5 bills, and so on … finally counting the coins, with the smallest last.

I said: “Great! We agree on the answer.”

And I asked: “How many of you would add money this way?” Most parents raised their hands.

“But we have a problem,” I added. “The math is backwards.”

Most parents (if not all) learned in school to add numbers from the right, starting with the smallest place value.

But when asked to add bills and coins, their minds naturally added from the left, starting with the greatest place value.

This is interesting, isn’t it?

Parents noticed:

  • The calculation methods they learned in school are not always (actually, not often) the ones they naturally use in everyday situations.

2. All you need is a calculator (?)

Next, I pointed out that calculators were abundant and inexpensive. (Today, every cellphone and tablet has a built-in calculator. Or just type the question in your browser’s address bar and it will find you the answer.)

So I asked parents: “What do you think about not teaching children how to add, subtract, multiply and divide and just let them use a calculator?”

Their answer was that this would not be a good idea.

They explained that kids need to understand the math. With the calculator, they just push buttons.

So, I handed out calculators and asked them to solve 12 x 25.

I added, “But there’s a catch: you’re not allowed to use the #2 key. No cheating by doing this in your head. You have to get the answer using the calculator.”

Here are 3 of the calculator methods parents used.

  • 3x4x5x5 (multiply the factors of the numbers)
  • 6×50 (half the fist number, double the second one)
  • 6x5x5 + 6x5x5 (a combination of the two methods above)

Can you think of a different method?

Parents noticed:

  • They wanted their children to understand.
  • A calculator can be a tool to think-with.

3. On paper & in your head

Last, I presented parents with the problem, 16 x 24.

I asked 2 parents at each table to solve this using pencil and paper and the rest to solve it in their heads.

Before you read further, try calculating 16 x 24 in your head.

I asked the parents who used paper and pencil to share the methods they used. They hesitated to volunteer. When a few did share, we noticed that they used the same method.

When asked to explain the paper-and-pencil method, they had a very difficult time. They could describe how, but could not explain why. For example, “Why did you indent the second row?” Some parents became frustrated and said: “It’s just a rule!”

The atmosphere changed when I asked parents to share the mental methods they used to solve the same problem. There was a palpable energy in the room. Parents were eager to share their personal solutions. They praised others who came up with different methods. They expressed delight at methods that surprised them.

Here are some of the mental methods parents shared:

  • They multiplied 10 x 24, 6 x 20 and 6 x 4, and then added the three products.
  • Some multiplied 20 x 16 and then 4 x 16.
  • Some multiplied 16 x 25 and then subtracted the extra 16.
  • Some multiplied 4 x 4 x 4 x 6.
  • In some rare cases, parents used algebraic structures:
    • 16 x 24 became (20–4)(20+4) = 400 + 80– 80 – 16
    • or, more simply, (20–4)(20+4) = 400 – 16.

Parents noticed:

  • When using a mental method, they understood what they were doing and could easily explain it to others.
  • The paper-and-pencil method was more like using a calculator: pushing buttons to get an answer without much understanding.

Basic computation skills?

I’m old enough to know that I don’t have all the answers.

I do spend numerous days each year in elementary school classrooms, working with students and teachers. Here’s my opinion:

[bctt tweet=”Children need practice with a wide variety of basic computation skills.” username=”georgegadanidis”]

Some basic computation skills:

  • Paper-and-pencil computation.
  • Mental computation, with ability to think flexibly, creatively and playfully about numbers and operations.
  • Solving number-based puzzles with and without tools (like calculators, concrete materials, games, coding, etc).
  • Making sense of numbers and operations in big idea contexts, such as the one below from Grades 1-2 classrooms.

The lyrics in this music video come from parent comments, after children shared their learning at home. From a project supported by KNAER.

In search of discovery learning

And now, as adults, we want the same for our children, which I think is really another issue (having to do with the “good old days” and the older generation underestimating the younger generation), but here are some of my thoughts on all the talk on “discovery learning”:

  1. Ontario education. Over the last 10 years I’ve spent on average 40-50 days each year in elementary classrooms, and I haven’t seen “discovery learning”. I’ve seen guided investigations, where the teachers prompt and scaffold student thinking to help them understand concepts in new ways. And I’ve seen a variety of other approaches too.
  2. Discovery learning. Years ago, Constance Kamii, who studied with Jean Piaget, did a lot of research on students discovering solutions rather than being told how to solve math problems. In one study, she compared Grade 3 children from three different classrooms, who learned how to add sums like 7 + 52 + 186, in three different ways: 1) by learning the addition algorithm you and I learned in school; 2) by not being told how to add and having to discover how to add by themselves; and 3) using a mix of methods 1) and 2) (for example, see this paper). Interestingly, on standard paper-and-pencil tests, the “discovery math” children did the best and the “let’s learn the algorithm” children did the worst. It was also interesting that when the “discovery math” children gave wrong answers, their answers were close, while the wrong answers of the “algorithm” children would sometimes be off by thousands (showing that they had not developed any number sense).
  3. Right vs left. You wouldn’t want to push analogies too far, but if we stereotype a bit, Kamii’s discovery learning approach can be seen as belonging to the right of the political spectrum. Like building a business from scratch, she asks children to construct their own, individual understanding of what addition is. The way I was taught math in school was more like a hand-out: my teacher gave us “the” way to add, ready-made.
  4. Some of this, and some of that. I think we need a balanced approach. Some of this, and some of that. Even if we think we know the “best” way (which I don’t). Children spend a lot of time in math class and need variety, across the spectrum of “tell” and “discover”. Most of all, we need to appreciate the wonderful minds of young children.

If you want to see some Ontario math classrooms in action, take a look at these lesson studies on Repeating Patterns in Grades 1-3.