quick ways, sure ways and shortcuts

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Quick Ways, Sure Ways and Shortcuts Author(s): Hilary Evens and Jenny Houssart Source: Mathematics in School, Vol. 33, No. 5 (Nov., 2004), pp. 22-23 Published by: The Mathematical Association Stable URL: http://www.jstor.org/stable/30215745 . Accessed: 05/10/2013 11:34 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access to Mathematics in School. http://www.jstor.org This content downloaded from 131.252.96.28 on Sat, 5 Oct 2013 11:34:07 AM All use subject to JSTOR Terms and Conditions

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Page 1: Quick Ways, Sure Ways and Shortcuts

Quick Ways, Sure Ways and ShortcutsAuthor(s): Hilary Evens and Jenny HoussartSource: Mathematics in School, Vol. 33, No. 5 (Nov., 2004), pp. 22-23Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215745 .

Accessed: 05/10/2013 11:34

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access toMathematics in School.

http://www.jstor.org

This content downloaded from 131.252.96.28 on Sat, 5 Oct 2013 11:34:07 AMAll use subject to JSTOR Terms and Conditions

Page 2: Quick Ways, Sure Ways and Shortcuts

Quick Ways, Sure Ways and 3hortcuts by Hilary Evens and Jenny Houssart

Introduction

The work described here is a small part of a joint project between the Centre for Mathematics Education at the Open University and the Mathematics Test Development team at the Qualifications and Curriculum Authority. The project started by looking at children's answers to selected questions from Key Stage 2 tests. We were particularly interested in questions which we thought might encourage children to show evidence of algebraic thinking. The next phase of the project was to devise further questions to be tried out with children aged 10-11. Many of these were designed to encourage children to explain their reasoning. This article looks at children's responses to one of these questions, which is shown below. The question was part of a test paper given to 364 children in a range of schools. The children doing the paper were encouraged to explain their answers as fully as possible.

Here is a sequence of numbers.

1, 4, 7, 10, 13 ....

They go up by the same amount each time.

The sequence continues.

Below is a list of numbers.

Circle each number which will be in the sequence.

% 41, 42, 43, 44, 45, 46

Explain how you got your answer

Summary of Responses

The vast majority of children gave an answer to the first part of the question, which required them to circle numbers, with over half circling the two correct numbers. A summary of responses to the first part of the question is given below.

Summary of Answers (364 scripts) Number Percentage of scripts

Correct answer 203 56 Incorrect answer 156 43 No response 5 1

As these questions were written for research purposes rather than as an actual test, we did not classify answers to the second part of the question as correct or incorrect. Instead we looked first at children who had circled the correct numbers and sorted their explanations. Then we sorted explanations given by children circling incorrect numbers in a similar way. In this article we will look mainly at the explanations given by children circling the correct numbers.

Quick Methods

Of the 203 children circling the correct numbers, all but four offered an explanation. Although the question was not designed to encourage children to find or state a rule, 15 did so. Some examples are given below. Like all the children's answers given in this article, they are reproduced using the original spelling and punctuation.

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The answer is 1 more than the 3x table.

The no in the sequence is in the 3x tables and add one on to it.

Of those children who said that the numbers in the sequence were one more than multiples of three, two also said that they could test for multiples of three by adding the digits.

Well if it's the 3 times table you add up the digits and the sequence is 1 number above the 3 times table I took away 1 and all the numbers which were in the 3 times table I circled.

Another child used a related method, adding the digits in the numbers on the list to find the digital roots.

I added both the digits in the numbers and for example (43) 4 + 3= 7 7 is one of the numbers in the sequence.

Mathematically this is correct. The digital roots of all the numbers in the sequence are 1, 4 or 7. This raises the question of how, or even whether, the child knew that this was the case. Without interviewing the child we can't know. However, investigating this answer provided an interesting mathematical interlude to us as researchers. It could also be used as a follow-up opportunity in the classroom to find out which sequences have digital roots that are also in the sequence themselves.

Sure Methods

The idea of 'making sure' was raised by one of the children as shown in the first answer below. These two answers show awareness of both a 'quick method' as described in the earlier section and a 'sure method' based on counting or continuing the sequence.

To make sure I used the sequence but I could have found the multiples of 3 and +1.

The difference is 3 so if you keep adding 3 you will eventually get 43 and 46, (the sequence no's are the one after the 3x table).

This fits in with something we have found in other parts of the project which is that some children will use what they consider to be a safe strategy in a test situation, even if they are aware of a more sophisticated strategy.

By far the most common type of explanation given by children circling the correct numbers, involved counting on in threes or continuing the sequence. Of those circling the correct numbers, 124 explicitly stated they were adding 3:

I worked out my answer by going up add 3.

Keep on adding three each time.

It goes up in 3's from 1. How I got my answer was that I counted on three each time until I reached the number that was on the bottom list.

A further 35 children circling the correct numbers appeared to have added 3 but did not state this explicitly:

I just carried on the sequence.

I just went up in the same pattern and found the answers.

Mathematics in School, November 2004 The MA web site www.m-a.org.uk

This content downloaded from 131.252.96.28 on Sat, 5 Oct 2013 11:34:07 AMAll use subject to JSTOR Terms and Conditions

Page 3: Quick Ways, Sure Ways and Shortcuts

We have used the term 'sure methods' for this section, based on a child's written comment, but would concede that this method may not always be that reliable. In fact 33 children circling incorrect numbers also claimed to have added three or continued the sequence. In same cases these children showed evidence of counting on and some answers, like the one below, contained counting errors at tens boundaries.

All the numbers is in the 3 times table 16 19 21 24 27 30 33 36 39 41 44 47.

Shortcut Methods Some children used what we refer to as shortcut methods. These tended to be based on counting on, but avoided simply counting on three at a time. For example, some children added 30 to the sequence rather than repeatedly adding three.

I figured it went up in 3's so if10 was one 40 would be one so that makes 43 one and 46.

40 is 30 more than 10 and 43 is 30 more than 14 and 46 is 30 more than 16.

There were other examples of shortcut methods which were less straightforward and led us to think carefully about what the children had done and why.

Because the number 4 was in the sequence so I knew 40 would be. I then went up in threes from 40.

I timesed it by 4 and kept taking away threes 52 - 3 = 49 49 - 3 = 13.

These two answers led us to explore the mathematics behind what the children had done and consider why it worked. Both children had multiplied two numbers and expected the product to be in the sequence. In each case the numbers multiplied were themselves in the sequence (4 and 10 in the first case, 13 and 4 in the second case) though the children did not mention this.

A closer look confirms that the product of any two numbers in this sequence is also in the sequence. It can be confirmed algebraically that this is true for this sequence and for any other which starts with one and goes up in equal steps. It seems highly unlikely that the children knew this, so perhaps they made lucky guesses, or were acting as a result of a misconception. Possibly the first child thought that if a number was in the sequence then 10 times that number would be as well due to experience with easier sequences such as multiples of two, five and ten. Perhaps they had noticed that 1 and 10 were both in the sequence and hence assumed that as 4 was 40 would be as well. Perhaps the second child saw 13 as the end of the first 'chunk' of the sequence and thought that multiplying 13 by 4 would give another number in the sequence.

As well as leading us to do some mathematics ourselves, these two answers reminded us of issues which were raised elsewhere in our research. The first is an important reminder that written answers can only tell you so much about what the child was doing and thinking. There were many occasions in our research when we wished we had been there when the children had written the answers and could ask them to tell us more. The other issue is what constitutes a correct method or a correct explanation. Both these children used a method that worked and explained what they had done. Although it would seem harsh to call

their method incorrect or their explanation inadequate, it is quite possible that they actually misunderstood but were lucky. This raises the perennial problem that questions which encourage interesting and diverse responses are often the hardest to mark.

Using Questions in the Classroom

When reading children's responses to this question we were very aware of the limitations of working only with written answers and for this reason our project also contains classroom observation. We believe that questions such as this one also have potential for use in classrooms beyond practice for tests. In this case the children might be introduced to the sequence of numbers before being shown the second part of the question. Questions and prompts used by the teacher might include the following:

What can you tell me about this sequence? What is the pattern?

Children may be encouraged to be more precise in their answers both by follow-up questions from the teacher or by listening to each other's answers. This may lead them to state the rule or pattern they see more explicitly, rather than just talking about 'adding on' or 'one more'.

Questions to accompany the second part of the question might include the following:

What are you thinking about when you attempt this part of the question?

What are you going to do/doing?

Why does that work?

Is there another way? How did you check your answer?

Answers given to this question also have potential for use in classrooms both at Key Stage 2 and with older children. Pupils can be asked to consider answers given by others in their class, or they could be shown some of the answers shown in this article and asked to comment.

Some of the shortcut methods shown earlier could prove particularly interesting if used in this way. Students are likely to have their own views as to whether they consider these explanations to be correct and why. They can also be challenged to look more closely at methods such as finding digital roots or multiplying two numbers in a sequence and deciding which sequences these methods will work for.

Perhaps our most interesting finding was that some of the 11-year-olds answering this question produced answers which challenged us and we believe could provide food for thought for children and teachers at Key Stage 2 and above. We would be very interested to hear if any readers try this question out with their classes and have similar experiences. M

Keywords: Explaining; Sequences; Assessment.

Authors Hilary Evens and Jenny Houssart, Centre for Mathematics Education, The Open University, Walton Hall, Milton Keynes MK7 6AA. e-mail: [email protected]

Mathematics in School, November 2004 The MA web site www.m-a.org.uk 23

This content downloaded from 131.252.96.28 on Sat, 5 Oct 2013 11:34:07 AMAll use subject to JSTOR Terms and Conditions