Questions 4 and 5 asked children to identify the identity elements for addition and subtraction and multiplication and division with reference to the number 7. About 10% of children exhibited some confusion about what the question was asking them to do. In some cases children appeared confused by the wording of the question, or their confusion was a result of weak arithmetic skills. In many cases the interviewer did not follow up with the probe for an explanation as to ‘how this works’, and in quite a few cases accepted an answer for one of the required operations without probing or clarifying as to whether this was true for the paired operation.

Confusion about basic arithmetic

Confusion about what was required in the question
Confusion about what was required was evidenced in several different ways:

• Students thought the objective was to establish 7 as the answer:
  
  Yes, 9 minus 2 equals 7, and 5 plus 2 equals 7. (Y4)
  
  Yes, 14 take away 7, take away 14. No, 14 take away 7. Yes, 7 take away 14 equals negative 7. (Y8)
  
  Yeah, 49 divided by 7 equals 7. (Y8)
  
  
• Using one number but using it several times:
  
  No, you could take away and then plus again, or you could plus then minus again. (Y4) [generalisation of a – b + b = a or a + b – b = a ]
  
  Because you can take away 3 from it and it will be …um, it will be 4 and if you put the 4 back to the 3, it will be 7 again. [As above but with a specific case]
  
  
• Creative approach using literal sense of number:
  
  You can say 10 times 7 and then it would equal 70, and if you took away the zero it would be 7 again [10 x 7 = 70 = 7]. (Y4)
  
  If you add 10 it will become 17, one ten and one seven. (Y4)
  
  Like 7 plus 10 equals 17, so you’ve still got the 7, is that what you mean? (Y8
  
  Draw it nicely and then put an equals sign on and then do nothing! (Y4)
  

Explanations
Many students merely restated the conjecture rather than justified or explained their truth statement and in most cases these restatements were accepted by the interviewers as explanationsæno further probing occurred. For example:

Yes, you can have 7 take away zero equals 7 and you can have zero plus 7 equal 7. (Y4)

The most common explanation for the identity element for addition or subtraction involved the notion of adding nothing or taking away nothing.

Well, if you have 7 and you like plus nothing and that stays the same number, and same for take away. If you have 7 and take away nothing it’s still the same number. (Y4)

Well, you write down 7 and you put a minus there and you put a zero and there’s no number. You don’t take away a number. (Y4)

Yes, because you just keep the number 7 and don’t take away any numbers, and you still have 7. (Y4)

Zero’s nothing. So it doesn’t count as nothing. You can’t put any more numbers into it. (Y8)

Others focused on the notion that zero ‘doesn’t do anything’ or zero is ‘not a number’:

Well, I’ll explain, cos zero isn’t actually a number it doesn’t do anything. (Y8)

You can add zero, or take away zero, cos nothing happens. (Y8)

It’s zero, cos you can add zero onto 7 cos it’s not a number. (Y8)

7 minus zero, you can’t do, so it just becomes 7. (Y8)

However, the weakness of this form of explanation lay in its incompleteness; when zero is involved in multiplication or division, such as 0 x 6 = 0, it is an important part of an equation and does in fact ‘do something’! Students who supported the notion that zero ‘does nothing’ were more likely to claim also that zero was the identity element for multiplication and division, though not in those terms. The everyday language they used for the number zero included a number of interesting descriptors:

A none number, so even if you add on to this number it isn’t even counting (Y4)

Zero is no number (Y4)

Zero’s worth nothing (Y8)

Zero’s nothing, so you can’t do anything to it (Y8)

A few students provided an explanation that included counter-examples. These counter-examples served as warrants for the statement that zero was the only answer possible:

Zero, because if you put 7 plus 1 or 7 take away 1, it would equal 6 or 8. And any more would equal more. So you just leave it, zero, just take away zero. It would be none, would leave it, take it back to 7. (Y8)

One student tried to locate the situation of taking or giving nothing into a real context (she approached her explanations for the earlier questions of the study in a similar way):

Yes, zero. You have 7 cards and you put zero more, and somebody gives you zero more cards, so zero plus 7 is still 7…and if you had 7 apples and at school zero people took zero, you would still have 7 because there were no people taking no apples from you. (Y4)

This explanation is somewhat more complex than required for a satisfactory response as the student over-compensates with zero people, possibly trying to model the equations 7 + 0p = 7.

A few students who incorrectly stated that there was no possible solution to the addition and subtraction problem based their justification for this conjecture on their existing knowledge of addition and subtraction in relation to these operations respectively making the answer larger or smaller. This tendency to over-generalise properties of numbers often represents a source of confusion for children and clearly illustrates the need for explicit discussion of identity elements and zero within the classroom. Other common examples of over-generalising include the mistaken belief amongst many children that multiplication always makes things bigger and division makes things smaller.

The child’s reliance on prior knowledge to examine a conjecture may have obvious shortcomings particularly when their recall of counter-examples is not immediate or outside of their experience. The benefits of class discussion and sharing and challenging of conjectures by teacher and peers is an example of how children’s knowledge can be challenged and refined (Simon & Blume, 1996).

A summary of all the students’ responses to the questions of identity for addition, subtraction, multiplication, and division is provided in the following table:

This analysis highlights several areas of concern. Firstly, because such a large number of students appeared not to understand the question it could be reasonably inferred that students were not used to discussing properties of numbers in general, and identity elements specifically. For those students who did provide a correct answer, this was often offered in a less than confident manner, and for some there appeared to be some uneasiness in trying to explain what appeared to be a self-evident statement. Resulting explanations were often tentative, with students seeking affirmation that they were ‘on the right track’.

Students included in the category correct, with weak or no explanation often provided a restatement of the fact: Yes, 7 plus zero is 7, or Multiplying by 1 will make it equal 7– rather than providing any warrant to justify the conjecture.

Secondly, students’ understanding about identity element for multiplication and division appear to be noticeably less developed in comparison to addition and subtraction, even at Year 8.

Overall, very few students in our sample were able to articulate a ‘good’ explanation to justify their choice of identity element for any of the operations, especially in relation to the identity element for multiplication and division. A recurring source of confusion for students at both year levels was the nature of zero. Many students overgeneralised the role of zero in addition and subtraction to multiplication and division examples. Examples of ‘good’ explanations included awareness of multiplicative thinking and recognition of the inverse relationship between multiplication and division: 7 divided by 1, there’s only 1 seven in 7.

3.5

Yes, zero, cos it’s like the same – you’ve got 7 and you times, you multiply it with zero and it just stays 7. (Y4)

7 divided by zero, well you divide it into zero but then it just stays the same. (Y4)

No, [possibly meant to say ‘yes’] because if you divide 7 by zero, it will just go, [reference to the 7 doing nothing] it will stay 7. (Y8)

If you divide 7 by zero it’s, it will always be 7 because zero is nothing and you can’t do anything with nothing. (Y8)

Because there’s only one group of 7, so you’re not timesing it by any more because there’s only one. And with dividing, you’re not dividing it by anything apart from, just dividing it by one. So there’s nothing else you can divide it by [indicates an imaginary split with 7 cubes], so it stays 7. (Y8)

You’ve just got one group of 7, and times it by one it will still be the same because there’s no other group with 7. (Y8)

We’ll, you’ve just got 7 cubes and it’s just times one [picks up all 7 and places them down again] and it’s still 7. The answer is 7. (Y4)

And, 7 divided by 1 equals 7 because you go one into 7 and it goes 7 times. (Y8)

7 divided by 1. There’s only one 7 in 7. (Y8)

Well, that, if it was to say like 2 times 4, that equals, you count 2 fours, but because you go 1 times something it just stays the same. (Y4)

It’s just like, if you times any number by 1 it still stays the same. (Y8)

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