Lino and the fundamentals of arithmetic

Lino and the fundamentals of arithmetic

by Anne Flachsmann

cc. Taking her arithmetic lessons as an example, the experienced primary school teacher Anne Flachsmann here shows how, with knowledge and empathy, a child can be encouraged to grow out of a difficult school situation.
    It should be mentioned that Anne Flachsmann, in addition to her rich experience as a teacher allowing her to exactly grasp the learning steps of a child, has extensively addressed pedagogical and psychological issues. Therefore, she succeeds in accurately understanding Lino in his personality and also in deliberately initiating the appropriate next learning steps.
    A teacher’s accurate knowledge of the developmental-psychological and age appropriate order in which methodological learning steps have to be made will allow all children of a year-based class to achieve the learning objectives.
    The high level being expressed in of Anne Flachsmann’s mode of practice – which only a few years ago could be regarded as standard in teacher training and in practice in the Swiss cantons –is being destroyed by the new learning methods and the corresponding teacher training which the curriculum 21 involves. This curriculum ignores 20th century fundamental developmental-psychological findings, and in regard to the didactics it prescribes it falls far back into the time before Comenius. It is imperative to stop this curriculum in the individual cantons and to correct the grave aberrations of the past twenty years. We should continue giving our children the benefit of class teaching with a teacher like Anne Flachsmann. This can easily be achieved given the pedagogical knowledge available today.

“When I met Lino, he had already begun his second year of primary school. They told me that he was very weak in mathematics and that I would always find him where any fooling around was going on. Both things turned out to be true. But I soon realised that Lino is also sensitive and very witty.
I love teaching arithmetic. In the first school year children solve equations in the number range up to 20. At the end of the year they deal with equations such as 19 + 3 or 15 – 9. For these they need to understand the place value of ones and tens, and this is very important for the development of mathematics.
The equations for the second year require an understanding of place value again, but here numbers up to one hundred are used (59 + 3 or 85 – 9).
I started my first lesson with this class, by explaining this context. I demonstrated some calculations on the blackboard and I invited them to follow my processes of thinking continually: “If you are clever, you will do the calculations together with me. In this way you will solve ten calculations, not only the one given specially to you!”

Using the advantages of class teaching

When I had finished the first calculation, Lino raised his hand: “I did not understand that.” Without comment I calculated two more tasks for the class and I clearly felt that the children were following my reasoning attentively. After that every child did a calculation on the blackboard, the order being voluntary, but not so the participation. When about half of the ten children had calculated, Lino raised his hand again: “I have understood it a little.” When it was his turn at last, he managed the task with my help. Then he said: “I think I have almost understood it now.” He was right in his self-assessment, as he would be in other cases. The class remained attentive and concentrated to the very end.
With this approach to learning new subject material I am usually successful. Like this, pupils can follow an arithmetic operation several times and they only raise their hands when they feel able to cope with the task. Since all children know from the beginning that they will also have to do a calculation in front of the others, they normally follow my thought processes or those of their classmates. So they will already have some practice when they have to calculate in their own exercise books later on. The result is a calm atmosphere because all children participate and they can simultaneously learn from each other.
At the end of the arithmetic lesson Lino came to tell me: “Do you know why I like to have arithmetic classes with you? You explain the work several times and you give me time.” This was the beginning of our learning process together.
Indeed, Lino had not mastered the basics of the first school year and he often failed to master the tasks set to him. If he had applied some rule somewhere, he clung to it even when doing calculations that were quite different. “I see,” he explained for example, “you always have to add the last number to the first number.” And so it was extremely difficult for him to let himself in for my explanations. I illustrated my thoughts using few specially selected materials, and took care not to smother him with a variety of solution approaches. And I adapted myself to the fact that he usually grasped only a small part of my instructions. However, a great help in our joint work was his ability to inform me very promptly about how much he had understood.

Involving the parents

Two months later, I got to know his mother when reviewing his school report. I showed her that Lino’s results in mathematics were barely adequate. I told her, however, that in other subjects I had noticed his contributing many interesting thoughts which would testify an alert intelligence. In contrast, in arithmetic, he didn’t think for himself fluently. But I was sure that he could succeed in this as well, as there was nothing lacking.
When I described him in this way, his mother felt that I understood her son so well that she became just as open and spontaneous in her response to me as he was. She told me that as a child, she had also failed in mathematics, and that her father had had no understanding for this. Although she now held a challenging position in the area of social work, respect for numbers and figures still gripped her to the marrow. When learning with her son she got nervous. I could understand this and we decided that the boy’s elder sister would take this over in the future.
In this conversation I succeeded to form a working alliance with Lino’s mother. We would be working jointly with him from now on.
When greeting me the next morning Lino said right away: “When my mother came home yesterday, she was very proud of me.” She had not been able to praise him for his barely sufficient math score of course, but she had let my encouraging view be known to her son.

Strengthening the children

After three months, we started work on the multiplication table. I hoped Lino would find a better access to mathematics with this. The subject did not include any topics treated but not quite understood in the first year, and many pupils love these tables, because they can get good results by learning them by heart.
But not so Lino. The two times table seemed to be no problem for him, since the steps are simple and well known. But when I asked him what 2 times 2 was I got no answer. He had apparently not yet decided to think seriously about numbers.
Without letting myself be impressed by that I learned one series after the other with my class. The whole class developed an increasing enthusiasm and they enjoyed learning, Lino as well, but very slowly. There were lessons when he did no more than two calculations – talking or dawdling the rest of the time. Also he was still quite actively involved in ragging.
One day I returned a math test to the class. Everybody was happy as they had good marks. Lino was the only one with an inadequate score. He sat there with a cloudy face and watched the cheerful bustle. He knew the tables reasonably well by then, but he was much too slow. Although I felt sorry for him, I did not say one word of comfort to him. It was not possible to overturn reality.
And yet Lino never lost his humor. Again and again he was, for example, convinced that 6 minus 6 equal 1. I explained to him: “But look, if you have six jellybabies and I take away six, then you have none left.” He looked at me briefly and replied: “Then I’ll tell my mother!”
Seven minutes after starting the next test, he was already back, with the test in his hand. This was nothing out of the ordinary, he often did this. With shining eyes he asked me: “What do you think? Am I coming to ask you something, or because I’m done?” I replied: “In your eyes I can see that you’re done.” “It’s true,” he exclaimed,” I’m finished!” He had fulfilled his dream also to be among the quick ones for once. I was curious about the test result and, in fact: All calculations had been solved correctly!
Lino could barely believe this the next day. The whole class shared his joy! Obviously he had been so annoyed about his recent defeats that he had decided to make progress also in mathematics.
After seven months my substitution period ended, and I saw the class off for the last time. It was the beginning of the summer holidays. The small cloakroom was crowded. Everybody was packing and chattering excitedly. Lino stood in front of me in the turmoil and had a last request: “Would you test me the six times table?” He had memorised it, but I had not checked it yet. Summarily I asked him in every which way. He did all calculations without errors and at a brisk pace. He said goodbye cheerfully and happily.
I regret that I will no longer be able to teach Lino. His development is the result of cooperation between Lino, his mother and me.
Like all children, Lino wanted to be good. He also has the convenient gift of expressing himself clearly and he is active and enjoys learning from adults.
I brought a lot of teaching experience into the process and also – and that was crucial – the absolute certainty that there was nothing wrong with him and he was perfectly able to understand mathematics. And I was able to understand him. At school I had also been weak at mathematics. Later, when I wanted to train as a primary school teacher, there was no other way left to me than to tackle the problem. At that time, a friend and teacher showed me the way and taught me with great calm and confidence. He passed his enjoyment of mathematics on to me. I hear him still enthusing: “Today we will begin to tackle the theory of probability, Anne, you’ll love it!”
Today I bring this safe and quiet mood to work with my pupils. I guide the children towards mathematical thought with little use of didactic frippery – merely with crayons, a blackboard and a lot of humor.
The joy I experience when I am able to help a pupil like Lino is a key reason for why I still enjoy teaching.     •
(Translation Current Concerns)

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