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How Claire's Teacher Guided Her to Persevere Effectively

by Dr. Margaret Taplin
Institute of Sathya Sai Education, Hong Kong
In the circles, arrange the digits 1 to 7 so that the sum of the digits is the same number in any three boxes in a row.

IMAGE

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My goal is to put numbers in the boxes so that they add up to the same total in each direction.

These are the clues I have:
I can use each number from 1 to 7 once.
I have to find a total for each row
Row 1, row 2 and row 3 must add up to the same total.

One way I can do this is to guess. [Put numbers in randomly.]

That didn't work, so I'll try putting in some different numbers. [Repeat]

I'll try again. [Repeat]

I've tried that way three times and I don't think it is working. I'll try something different. I'll pick a total - 13 will do - and try to make the numbers add up.
2+4+7=13.  I'll put 4 in the middle.
3+4+6=13
That leaves 1+4+5. That's not 13.

I'll try again. Pick 11.
1+3+7=11. Put 3 in the middle.
6+3+2=11
That leaves 4=3+5. That's not 11.

I've tried that idea twice and it doesn't seem to be working. I think I'd better try something different. I'll put 4 in the middle because that's the middle number. Then I'll match the highest and lowest numbers.
The lowest is 1 and the highest is 7, so 1+4+7=12.
That leaves 2 as the lowest and 6 as the highest, so 2+4+6=12.
That leaves 3 and 5, so 3+4+5=12.
That's it.
I found the numbers, but I had to try it three different ways.

We can see that this student was demonstrating the values of patience, fortitude, and was remaining unruffled by ups and downs, successes and failures. The student tried an idea for long enough to give it a chance to work, but knew when it was a good time to try a different approach.

Now, let us see how Clare's teacher helped her to persevere successfully with the problem on which she had been overpersevering:

In this second attempt, Clare began by using the strategy of changing numbers at random. After four attempts I suggested that she should try a different strategy. She did not have any ideas, so I suggested that she might look at the problem column by column. She started with the units column and used the strategy "I have to get rid of 4, so change 3 and 1." She continued this strategy into the tens column, "21 - get rid of 6 so change 5 and 1" and the hundreds column, "27 - need 11 - get rid of the bigger numbers, 9 and 7".

Question 2 (with prompting):

In this magic square, each row, each column and the two long diagonals must each add to the same total and each of the numbers from 1 to 25 is used once and once only. Find the missing numbers.

tab.gif (50 bytes) tab.gif (50 bytes) 25 18 11
3 21 tab.gif (50 bytes) 12 tab.gif (50 bytes)
tab.gif (50 bytes) 20 13 tab.gif (50 bytes) 4
16 14 tab.gif (50 bytes) tab.gif (50 bytes) 23
15 tab.gif (50 bytes) 1 tab.gif (50 bytes) 17
At first Clare had some problems with this question. She did not read the question before starting. She started on the right track and knew that all rows and columns had to add up to 65. She did not read that each number could only be used once. After she changed numbers three times I suggested a fresh start. Clare selected a different row, but again it was one with two gaps and she encountered the same problems. After another three attempts I intervened. We discussed the problem and Clare worked her way across the square pointing out that they all had two or more missing numbers. It was at this point that she found that one column only had one gap. Clare went on to solve the question with no further difficulty.

Question 3 (with prompting):
IMAGE
Clare was keen to begin this problem after the confident note on which we had finished the previous question. She began by randomly selecting 12x9, an approximate guess. She then tried verbalising multiplication facts:
10x9=90 - not high enough
11x9=99 - too high.
She then looked unsure of what strategy she should adopt. I provided a hint, that "having one on the end is hard isn't it, not many multiplication facts have a one on the end". This did not prove to be helpful so I suggested a change of approach, looking at the subtraction part of the question. This seemed to help Clare as she began to list subtraction facts beginning with 20 and giving an answer of 6:
20-14
19-13
...
13-7
She then said, "Now I'll go back and see if any of these multiplied equals 91". Thus she was able to select her own strategy and was successful.

Question 4 (without prompting):

What is my mystery number? If I divide it by 3 the remainder is 1. If I divide it by 4 the remainder is 2. If I divide it by 5 the remainder is 3. If I divide it by 6 the remainder is 4.

In solving this problem, Clare randomly chose a number and divided it by 3, 4, 5 and 6. When this did not produce the desired answer, she tried it two more times, using different random numbers. As these attempts were both unsuccessful, she decided to change her strategy. Clare thought of using a strategy which had been suggested to her in a previous problem solving activity, that of developing a system in her choice of numbers. As the mystery number had to have a remainder of 3 if divided by 5, it had to either end in 8 or 3, as each multiple of 5 ends in either 5 or 0. Clare then wrote down all the numbers that, when divided by 6, had a remainder of 4, as this was the largest of the numbers in the clues and therefore would require going through fewer figures. Of these she looked for the numbers ending in 8 and 3 and divided them by the various numbers in the clues. On her third attempt, she found that 58 was the mystery number.
As can be seen from the above examples, Clare showed increasing courage, self-confidence and independence in her ability and willingness to face the challenges of the problems. At first she did not use the strategy instinctively. On problem 2, it was necessary for the teacher to intervene several times, to prompt her to change strategies and to suggest some alternative strategies. The need for teacher intervention had decreased by problem 3, and by the time she reached problem 4 she was able to recognise for herself when it was appropriate to change strategies.

The following procedure can be useful for teachers to follow, in teaching students how to persevere effectively with a task.


Recommended procedure for
introducing problem solving management model


  1. Give the student a preliminary problem to solve without guidance. Observe whether the student instinctively used the model.
  2. If the model was not instinctively used, introduce it and work through a second problem, demonstrating how to use it.
  3. Ask the student to repeat the first problem while you guide him/her to use the model, i.e. prompt the student to change to a different approach after a maximum of three repetitions of the previous strategy.
  4. Give two more problems, monitoring the strategy pattern and reminding students, when necessary, to follow the model.
  5. Give a fifth problem and ask the student to try to follow the model, changing approach when appropriate, without any prompting from you.

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