Is 4 plus 3 the same as 3 plus 4?
were reasonably confident about the commutative property under addition.
However, Year 8 students were noticeably more precise and more convincing
in their explanations. Many Year 4 students were unable to provide
a model to justify their conjecture.
• What about 4 minus 3 and 3 minus 4?
Are they the same? Students appeared much less assured about the
commutative property for subtraction. They took longer to answer,
and many students altered their original standpoint. Many Year 4
students modelled the equation directly rather than operated on
the number relationship. Most Year 8 students could demonstrate
the importance of number order, but were more likely to change their
conjecture as they talked through the demonstration.
• Does 2 times 5 give the same answer as 5 times 2?
Students were more comfortable with commutativity under multiplication
than under subtraction and just slightly more than they had been
under addition. While most could provide a model to represent the
problem, the understandings of Year 4 students evolved from often
ill-founded ideas and inappropriate explanations. Those for Year
8 students demonstrated a more advanced conceptual understanding.
No student exhibited sophisticated multiplicative thinking through
the use of an array structure. Commutativity of multiplication was
often modelled by way of additive reasoning.
Is there a number you can add to or take away from this number
 but the number still stays the same?
Just over half of the students could identify zero as the identity
for addition and subtraction and were able to provide some explanation.
While most Year 8 students confidently discussed the identity for
multiplication, very few Year 4 students could recognise and explain
it. A small number of students at both year levels were familiar
with the identity for division.
• What about multiplying or dividing? Is there a number
you can multiply (or times) this number by or divide it by, so that
the number stays the same? Tell me what it is and how this works.
Many students appeared to be unfamiliar with the task of justifying
a conjecture using cubes. While many confidently offered a conjecture
and could provide a verbal explanation for the problems, they were
not at ease when asked to model their thinking.