Learning arithmetic

My attitude towards learning and education has been shaped by my own experience as well as by authors like A.S. Neill and John Holt, who represent what might be called the “free school” (“free” as in freedom) approach. Taking a cue from software, this might also be called the “free/open school” approach, an approach that basically gives students much more freedom to choose by themselves what to learn, how to learn, when to learn, and even where to learn. Teachers play a more passive role in the background as advisers. Traditional schools, in contrast, are so regimented that John Holt has called this traditional system a virtual prison for children and youth.

As an example of the free school approach, I’d like to quote the first few paragraphs of Chapter 1 (And ‘Rithmetic) from the book Free At Last (The Sudbury Valley School) by Daniel Greenberg:

Sitting before me were a dozen boys and girls, aged nine to twelve. A week earlier, they had asked me to teach them arithmetic. They wanted to learn to add, subtract, multiply, divide, and all the rest.

“You don’t really want to do this,” I said, when they first approached me.

“We do, we are sure we do,” was their answer.

“You don’t really,” I persisted. “Your neighborhood friends, your parents, your relatives probably want you to, but you yourselves would much rather be playing or doing something else.”

“We know what we want, and we want to learn arithmetic. Teach us, and we’lll prove it. We’ll do all the homework, and work as hard as we can.”

I had to yield to them, skeptically. I knew that arithmetic took six years to teach in regular schools, and I was sure their interest would flag after a few months. But I had no choice. They had pressed hard, and I was cornered.

I was in for a surprise.

My biggest problem was a textbook to use as a guide. I had been involved in developing the “new math,” and I had come to hate it. Back then when we were working on it — young academicians of the Kennedy post-sputnik era — we had few doubts. We were filled with the beauty of abstract logic, set theory, number theory, and all the other exotic games mathematicians had played for millennia. I think that if we had set out to design an agricultural course for working farmers, we would have begun with organic chemistry, genetics, and microbiology. Lucky for the world’s hungry people that we weren’t asked.

I had come to hate the pretensions and abstruseness of the “new math.” Not one in a hundred math teachers knew what it was about, not one in a thousand pupils. People need arithmetic for reckoning, they want to know how to use the tools. That’s what my students wanted now.

I found a book in our library, perfectly suited to the job at hand. It was a math primer written in 1898. small and thick, it was brimming with thousands of exercises, meant to train young minds to perform the basic tasks accurately and switfly.

Class began — on time. That was part of the deal. “You say you are serious?” I had asked, challenging them; “then I expect to see you in the room on time — 11:00AM sharp, every Tuesday and Thursday. If you are five minutes late, no class. If you blow two classes — no more teaching.” “It’s a deal,” they had said, with a glint of pleasure in their eyes.

Basic addition took two classes. They learned to add everything — long thin columns, short fat columns, long fat columns. They did dozens of exercises. Subtraction took another two classes. It might have taken one, but “borrowing” needed some extra explanation.

On to multiplication, and the tables. Everyone had to memorize the tables. Each person was quizzed again and again in class. Then the rules. Then the practice.

They were high, all of them. Sailing along, mastering all the techniques and algorithms, they could feel the material entering their bones. Hundreds and hundreds of exercises, class quizzes, oral tests, pounded the material into their heads.

Still they continued to come, all of them. They helped each other when they had to, to keep the class moving. The twelve year olds and the nine year olds, the lions and the lambs, sat peacefully together in harmonious cooperation — no teasing, no shame.

Division — long division. Fractions. Decimals. Percentages. Square roots.

They came at 11:00 sharp, stayed half an hour, and left with homework. They came back next time with all the homework done. All of them.

In twenty weeks, after twenty contact hours, they had covered it all. Six years’ worth. Every one of them knew the material cold.

That’s the free school approach.

A well-known school that uses this system is Summerhill School in the U.K. The school was set up by A.S. Neill, who wrote about his experiences in his book Summerhill: A Radical Approach to Child Rearing. The books How Children Fail and How Children Learn by John Holt are also illuminating.


  1. Posted February 3, 2009 at 2:24 am | Permalink

    I totally agree that we need to wait for the students to ask the question before we give them the answer. I discovered that for myself when i was teaching, i thought to myself, ‘we are giving them the answers before they ask the questions’.

    But what alternative is there? we can’t really wait for them to ask the ‘right’ questions and we do need to prepare them for University (for those who want to go there)


  2. Roberto Verzola
    Posted February 3, 2009 at 8:10 am | Permalink

    Hi Serge,

    There are alternatives.

    You can simply wait. The same book I quote in the post cites a student who did nothing for years but go fishing in the school pond. He learned everything about fishing but that’s all he did. His father was very worried but since his son looked very happy, he let his son keeping going to school. Eventually, the boy expanded his fishing interest and learned other fields as well, with the same single-mindedness as he did with fishing.

    You can also give students the whole syllabus for an entire year, or an entire phase of education. Here in the Philippines, the first six years of formal education are called elementary grades, followed by four years of high school, and then four or more years of college.

    Our system allows out-of-school children (or even adults, actually) to take a qualifying exam and be considered elementary school graduates. They can take another set of exams and be high school graduates after passing it. There may also be year by year exams, but I’m not sure about this.

    So, children can be reminded that if they want to “graduate” like their friends in the formal school system, they must cover the areas in the syllabus, but at their own pace and time. They can also ask for an exam anytime, if they want to, to check their own progress. Between this and the standard approach of telling students they must learn a particular topic at a particular date and time in a particular room, I think it is obvious who will learn better.

    The free/open approach may take longer or shorter, depending on the child. But what does a year or two matter? They will be much happier. They don’t have to wake up early, they can play as much as they want, they can focus on what truly interests them. The whole learning process will be a JOY, as it should be.

    If they are not interested in getting a piece of paper called a diploma, but know exactly what they want to become, they can even forego the formal exams and just keep learning.


  3. Posted February 3, 2009 at 1:31 pm | Permalink

    I agree but as always, this method is not for everyone. Some students like structure while others don’t. Ideally, we would need to cater for the way students like to learn, which means doing some sort of test to determine how they best learn and allow them to learn that way.

    I have the same issue with teaching. At my old school, we teachers were effectively told we had to teach according to the one particular method. I was furious with the directive as teachers also have prefered ways of teaching. It would have made more sense to find them out, help teachers become even more effective at their natural approach while strengthening their weak areas of teaching.

    Going even further, match the way students like to learn with the prefered way of teaching of teachers.


  4. Roberto Verzola
    Posted February 3, 2009 at 7:55 pm | Permalink

    I suppose it depends on what you mean by “structure”. I was referring more to the highly regimented system of class schedules for lectures and exams. I cannot imagine a child actually wanted to be told to go to a room at this time, and learn about one topic and move to another room and learn another topic, and so on, the whole day, day after day. Well, a few maybe. Especially if they have become so used to being told what to do, that they don’t have an idea what to do when given the freedom to do so. That would still be so unlike a child. I suppose they would welcome occasional scheduled lectures, but as I rule I doubt it. Anyway, I do agree with you that everyone has his/her own way of learning and it should be respected.

    The books that I liked most about this topic were How Children Learn and How Children Fail, both by John Holt. He calls the traditional school a prison for children.

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