id: 06448941
dt: j
an: 2015d.00543
au: Foster, Colin
ti: Doubly positive.
so: Math. Sch. (Leicester) 44, No. 2, 34-35 (2015).
py: 2015
pu: Mathematical Association (MA), Leicester
la: EN
cc: F40 H20 H40
ut: laws of arithmetic; signs; symmetry; addition; multiplication; negative
numbers; integers; directed numbers
ci:
li:
ab: From the text: We weren’t working explicitly on directed numbers at the
time, but during a mathematics lesson a pupil suddenly asked me,
seemingly out of the blue, ‘If two minuses make a plus, why don’t
two pluses make a minus?’ The pupil’s question seemed to
demonstrate a deep commitment to symmetry ‒ the idea that what works
for negatives should work in the same way for positives. This seems
quite reasonable. In a multiplicative context, positive numbers and
negative numbers do behave differently, and there is no reason to
suppose that two positives should multiply to make a negative just
because two negatives multiply to make a positive. We have a different
sort of symmetry, of the $\mathbb Z_2$ kind ‒ ‘like numbers
positive; unlike numbers negative’. The symmetry operates not at the
level of individual numbers but at the level of pairs of numbers. Is
this a satisfying response? Thinking about possible ways of responding
to this pupil’s question led me to consider the multitude of
different models available for working with directed numbers. To what
extent do these models complement one another, or appeal to different
pupils, and to what extent do they unhelpfully clash? Are pupils better
off the more models they know about?
rv: