At that time my neighbour, Agnes, was eleven years old and was in a special class at school for children with mild learning difficulties. One day when she was visiting us, I called to her from the sitting room to tell me the time from the kitchen clock. She said, “I can’t. I can’t read the numbers on your clock”. I came into the kitchen and looked at the clock which has Roman numerals and realised that when I tell the time I don’t read the numbers, I look at the pattern made by the two hands. As long as I can identify the top of the clock, I don’t need any numbers to be marked at all. I concluded that Agnes, in fact, was unable to tell the time and asked her if she would like to learn. She said she would, so I got out my Cuisenaire rods and we sat down at the kitchen table.
I made a clock with a red rod at each hour position and two red rods at the 12 position. I put white cubes to represent the minutes between the red rods. I took a long orange rod to make the minute hand and a shorter black rod to make the hour hand and I centred the whole clock on a small flower on the oilcloth which covered the kitchen table.
I put the minute hand at the 12 position and moved the hour hand to the 3 position and said, “Three o’clock”. I moved it to the 6 position and said, “Six o’clock”. Then I moved the hour hand back to the 3 position and invited Agnes to speak. She said “Three o’clock”. I moved it to the 6 position and she said, “Three o’clock”. I said, “Listen again” and repeated what I had said and done the first time but beginning with the six o’clock position. Then I placed the hand at 6 and this time she said “Six o’clock”. I placed the hand at 3 and she said, “Three o’clock”.
I went on in the same way with the 9 and 12 positions. She quickly learned to recognise the 6 and 12 o’clock positions but tended to confuse 3 and 9 o’clock. Tending to confuse left and right myself, I have developed techniques to eliminate the “sameness” of left/right symmetry by focusing on an object on my right and saying to myself, for example, “Right is doorside”. As I was sitting on Agnes’s right, I asked her each time she made a mistake, which was near me, the 3 or the 9, until she could identify the 3 as “Glenys’s number” and 9 as the other one.
Then I picked up the two “hands” and gave them to her and said “Six o’clock”. She was able to put one end of the black rod near the 6 position but the other end of the rod was nowhere near the centre of the clock and she didn’t know where to put the orange minute hand. Her actions gave me the feedback that though she had been giving correct answers, she had not paid attention to certain essential elements in the situation. So I took the hands and put them in the 3 o’clock position. Agnes told me the time correctly. Then I asked her,
“Where is the orange hand?”
“On the twelve.”
“Where is the other end of the orange hand?”
“On the flower.”
“Where is the black hand?”
“On the three.”
“And the other end?”
“On the flower.”
Of course, she didn’t answer neatly like this the first time and I had to ask her other questions to make her pay attention to the flower and the fact that one end of each hand was always on the flower.
We went through the other times she already knew how to say in the same way. Then I gave her the hands again and said, “Nine o’clock”. This time she was able to put both hands in their correct position on the edge of the clock but their other ends did not meet in the centre on the flower, but when I asked her, “And what about the flower?” she moved the other ends of the hands so that they were no longer in the correct position on the edge. It took her some time and several attempts to position the hands correctly. She enjoyed being the one to position the hands even though it was, to me, surprisingly difficult for her to do so. I was also surprised that though she had already noticed, before the lesson began, that the hands moved in relation to the numbers on the edge, she had never noticed that the two hands met in the centre.
Remembering that children are in the absolute of action and that I had started off with what was an exercise in perception but not action for her, I tried some simpler action exercises, “Touch the 3”, “Touch the flower”,” Touch the orange hand”, “Touch the flower end of the orange hand”, and so on.
I picked up the hands again and said, “Six o’clock”. This time, she inverted the minute and hour hands. I picked up the hands and, appealing to her power of evocation, I asked her to touch the hand that moved and she touch the black one. “So which one didn’t move?” She touched the orange one. “Where did it stay all the time on the clock?” She placed it with one end on the 12 but the other end was not in the centre. I reminded her of the flower and she repositioned it correctly. I said, “Make the two hands make a straight line” and she was able to adjust them correctly.
Then I said, “Nine o’clock. Put the hand that doesn’t move first”. Though she had problems centring the hands, she didn’t confuse them.
When we did 12 o’clock, she wanted to put the two hands side by side, though that wasn’t of course the way I’d done it when I’d showed her that time initially. Instead of doing it myself again, I took my kitchen clock off the wall and said, “Look what my clock does” and turned the hands to show 12 o’clock on the kitchen clock.
“Oh! It goes underneath!”
“Do the same with your clock. Which one goes underneath on the kitchen clock, the long hand or the short one?”
“The short one.”
“Do the same on your clock.”
We continued for a little while alternating between me placing the hands and her saying the time, and me saying the time and her placing the hands for these four times.
This first lesson took about an hour. What I have not brought out in the above description is the fun we both had. Every time Agnes realised something new she bounced up and down in her chair and laughed with pleasure. She remained attentive and interested the whole time, making me realise that it is possible to be a good student though extremely slow. She also made me realise how many different awarenesses are involved in learning to tell the time because I had to do something specific to force each one for her.
As Agnes was getting up to leave, Christian, who had passed through the room a few times during the lesson, asked Agnes, “One o’clock”. Of course she didn’t know how to place the hands for this and I was slightly annoyed with Christian for asking her.
Agnes came again the next day and showed that she had spent some of her night integrating the previous day’s work because she was able to do everything she had done before but faster and with more confidence. Her only problem was remembering to make the two hands meet in the middle. So we went on showing and saying the other “hour” times, i.e. “one o’clock”, “two o’clock” and so on. I continued to “place” the hand and not to “turn” to avoid the distraction of having to turn the minute hand at the same time and to focus on the position of the hands rather than on their movement. Agnes, anyway, already knew that the hands turned and in which direction. When she needed to find the name of an “hour position” she knew where she had to begin counting the red rods and in which direction to turn. These were awarenesses she had already had and which I did not need to force. On the other hand, she did not at first realise that for the positions of, for example, “ten o’clock” she did not need to start counting from the beginning but could start from “nine o’clock” which she already knew. When I saw this, I started asking her to show me “one hour before six o’clock”, “two hours after nine o’clock”, etc. It was a surprise to her to see that, for example, “one hour after six o’clock” was the same as “two hours before nine o’clock”. I could see she did not understand why this was so, it was just a happy coincidence, but I did not choose to work on that problem as it was not essential to learning to tell the time.
I did work a lot on “opposite” pairs such as “two o’clock” and “ten o’clock” as she continued to make left/right confusions. I got her to sit slightly to the left of the clock so that the “big” numbers could become “Agnes’s numbers” and, as I was sitting on the right, “small” numbers became “Glenys’s numbers”.
I also asked her questions such as:
“How many red positions are there?”
She counted them and said, “Twelve”. (I had wondered whether the two red rods on the “12” position would cause a confusion, but she accepted them as a convention to know where to begin counting.)
“How many hours does this clock show?”
As the lesson took place in French, this transfer was easy for her because in French “twelve o’clock” is “douze heures”, literally “twelve hours”.
However, when I asked her how many hours there are in a day, she was unable to answer. So I asked her what time she had lunch – at about eight o’clock or about twelve o’clock? She knew she had lunch at about twelve o’clock. I asked her more questions about what time she did various things. She was unable to give times for many things but she did know whether she did them in the morning, the afternoon, the evening or at night. So I drew a vertical line on a piece of paper to represent her day and first marked on it large intervals representing morning, afternoon, evening and night. Then we wrote in some of the things she did in a day: got up, had breakfast, went to bed, slept, etc. Then I wrote in the numbers of the hours from one o’clock in the morning to twelve midday. With this she was able to say that she got up at about eight o’clock in holiday time and had breakfast at about nine, that at four o’clock in the morning she was asleep. I pointed at the kitchen clock and asked her what time it was now. She was able to say, correctly:
“In the morning or the afternoon?”
“In the afternoon.”
So I showed her on the line of her day how after lunchtime we start counting the hours again from one to twelve to finish in the middle of the next night. I asked her about times she did things in the afternoon and evening and she situated them on the line. I asked her how many hours there were before lunchtime and after lunch. She was able to answer 12 in each case but when I asked her how many hours there were altogether in a day, she was unable to answer because she was unable to add 12 and 12. So I put a red rod next to each hour on the line and she was able to count them and say there were 24 hours in the day.
I asked her if she knew another way to say “twelve o’clock” when it was lunchtime; she did, it was “midday”. She also knew that “twelve o’clock” in the middle of the night was called “midnight”. But when I asked her how she knew when she looked at the clock and it showed 12 o’clock if it was midday or midnight, she couldn’t answer. So I said: “If you look out of the window and it’s dark, is it midday or midnight?” “Oh! Midnight!” she answered, emphasising the part “-night” and with a look of triumph on her face and I realised she was hearing the “night” in midnight” for the first time. (In French, the word for midday “midi” is not transparent in the same way so we couldn’t have fun with it.)
This second lesson took about an hour and a half and at the end Christian again teased me by asking Agnes to show the time “half past four”. I say teased me and not Agnes because she was not in the least bothered by the question, she didn’t know, she knew she didn’t know and was not at all upset by her ignorance.
On the third day, I set up the clock without hands and with a sheet of A4 paper covering the left half of the clock along the 12-6 line.
“Agnes, can you see all the clock?”
“How much can you see?”
I took away the paper.
“How much can you see now?”
“All of it.”
I covered the right half of the clock and asked her the same questions. She had no hesitation in answering any of them.
Then I gave her the paper and asked her to cover half the clock. This was more difficult for her to do physically and it took some time before she could place the paper exactly on the 12-6 line.
For the next step, I used two sheets of A4 paper and placed one as in the first exercise and the second covering the bottom half on the 9/3 line so that only a quarter of the clock was visible. The word “quarter” did not come spontaneously from Agnes as “half” had, but when I asked her “Can you see all, half or a quarter of the clock?”, she had no hesitation in choosing “A quarter”.
I continued by showing her the other 3 quarters and then by alternating between showing halves, quarters and the whole clock. She made no mistakes. When I gave her the pieces of paper and asked her to show me “a quarter”, “A different quarter”, “A half”, etc., she found it more difficult but the problem seemed to be one of lack of physical dexterity rather than not understanding what was needed. I say this because whenever I asked her to say if the way she had put the papers was correct, she never said it was so when it was not.
Then I gave Agnes the “hands” and asked her to put them in the twelve o’clock positions, which she did. Then I slowly turned the minute hand and stopped it on the 6 position and asked her now much of the clock it had travelled over. She was unable to reply. So I covered the left half of the clock with a sheet of paper and did the exercise again. This time she was able to say “Half” or to be exact, she said in French, “La moitié”, the word she had used in the previous exercises. I asked her if she knew the word “demi” and whether it was the same or different in meaning from “moitié”. She knew they had the same meaning and she even knew that “demi” was the word used for telling the time. So when I put the hands to the twelve o’clock position and then to the half past twelve position, she was able to say the time correctly in each case. I gave her the hands and asked her to show me successively “Twelve o’clock” and “Half past twelve”, which she was able to do after reminders to “Centre on the flower!”
We went on with “half past” the other hours in the same way and then with “quarter past”, which posed no special problem. “Quarter to” naturally took more time as it involves a different way of looking at the clock – counting backwards and not forwards from the 12 position. As she had started off already knowing the direction the hands turned, she had a solid criterion for “backwards” and “forwards” and by using these words, she was able to distinguish between a “quarter before” and a “quarter after” a given hour. Again she already knew the conventional way of saying these times and had no trouble switching from the words which more clearly express the mental operation to those conventionally used.
At the end of this lesson, Christian yet again asked her something she could not do, “Ten past eight”. I snapped at him angrily, “But that’s for tomorrow!” and then I suddenly saw the method in his teasing questions. They were not just any question that Agnes was unable to answer about telling the time; they always concerned the very next step. For someone reading this account, this was probably obvious long ago but it was not to me at the time. I had never taught anyone to tell the time before and had never even thought about it much so when I started, I had no idea what the steps would be, of what the “right way” to go about it was. Up to the end of the third lesson, I felt I was just being guided by what Agnes could and couldn’t do. Christian’s intervention made me realise that I (and he too, of course) did have, on the intuitive level, criteria for ordering the different exercises I suggest to Agnes. Up until then, though I had seen that for other teachers (Maurice Laurent in particular in maths seminars) it could be obvious which awarenesses had to be forced and in which order to master a given skill, Gattegno’s description of the “temporal hierarchies” in learning had remained an intellectual notion for me. I was as pleased and excited at being capable of recognising an example of the temporal hierarchies actually functioning in a learning situation as Agnes was at learning to tell the time.
It is perhaps not necessary to describe the following lessons in detail – Agnes learned to tell the time in about 6 hours spread over a week. From then on, every time she came to the house, I would ask her the time and though at first it took her a minute or two to find the answer, she seldom made a mistake. The mistakes she did make were to confuse the left and right sides of the clock or the minute and hour hands. There was one further important awareness about telling the time. This was that to be able to tell the time, a single position on the edge of the clock has to be able to trigger up to five different numbers, for example: nine (o’clock), quarter (to), forty-five (minutes past), fifteen (minutes to), twenty-one (o’clock). Each different number is part of a specific numbering system which implies a different way of perceiving and interpreting the clock.
On a different level is an awareness I didn’t have and haven’t had yet: which awareness I had to force for Agnes to realise that the two hands were joined at the centre of the clock. Each time she set up the clock, she had to be reminded to “centre the hands on the flower”, which showed that it never became obvious for her.
Encouraged by our success with the clock, I continued working with Agnes two or three hours a week all that summer, using the Cuisenaire rods to work on basic arithmetic. This was a quite different experience from working with the clock. I could quite easily find individual exercises involving addition, subtraction, multiplication or division for Agnes to resolve using the rods but I had little idea of what order to do them in or, having finished one exercise, what to do next. The temporal hierarchies in this area were not (and are still not) evident to me.
This was no surprise to me. I think I may say that telling the time is a skill I fully master – it is something I do quickly, correctly and effortlessly on every occasion. This is not true for the basic operations of arithmetic. I get by in every day life but with a lot of effort and many mistakes in anything to do with calculation and numbers.
What did surprise me was that this made no difference to Agnes. She was just as happy to come to these disjointed arithmetic lessons as she had been to learn to tell the time. It made me realise the difference between children and adult students. Agnes had no expectations that I should be a quick or efficient maths teacher, that the lesson should have any particular objective or that as a teacher I should not make mistakes. Several times I would have to stop in the middle of an exercise and say to Agnes, “I don’t know how to go on from here. Can we try something different?” and she’d happily gather up all the rods, put them away and wait for a new challenge. This lack of social pressure meant that I could devote all, or nearly all, my energy to the pedagogical work in hand. It is an experience which has helped me since then to identify when I am putting my energy into social consideration in the classroom when I’m working with adults, and through being able to identify it, to control and reduce a use of energy which is extraneous to the essential teaching/learning situation.
I realised that if I compared myself to an experienced primary school teacher with a thorough grasp of the temporal hierarchies involved in learning arithmetic, I was very inefficient in helping Agnes gain arithmetical know-how in exchange for her time. This consideration was, however, outside Agnes’s realm of concerns. As long as I was present to her learning needs on the microsecond by microsecond level, she was motivated to continue.
At the time, Agnes made me realise that a slow learner can nevertheless be a good learner in the sense of being effortlessly present to the challenges of a situation for a considerable period of time. As long as I did my job right, Agnes was never fidgety, naughty or distracted during our lessons. Writing this article has made me see that in one way I was a better maths teacher (that was my one and only attempt) than I am an English teacher (my usual occupation): having little knowledge in the field and no experience, I was forced to be continually and totally present to my student and her needs.
© Glenys Hanson, The Science of Education in Questions, N° 13, September 1995, Besançon, France.
“Agnes and the Temporal Hierarchies” by Glenys Hanson is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.