Poll: the sum(7,9)

[Martin Baxter]

>To clarify:  7 and 9 were arbitrary numbers.  It could have been any
>two single digit addends.  I didn't mean to focus on the manipulating
>the 9 factor.  3 and 5, 6 and 8, what process do you use to instantly
>know, or are there calculations involved?
>
>Mark
>
>On Mar 14, 2005, at 8:38 PM, Mark Swindell wrote:
>
>> It's this:  How do you mentally process simple addition/subtraction
>> facts?  What actually happens in your brain to elicit 16 when you hear
>> 7+9? (for example)
>

For 7 and 9, I do the same as Sarah. I "know" that the answer will be
between 10 and 20. So I take the value needed to turn 9 into 10 (1) and
subtract it from 7 giving (6). 10 + 6 is a pattern that "matches" the
decimal representation 16 (or perhaps 1 in the "tens" column and 0+6 in the
"ones")

3 and 5 are both single digit numbers, so this is different. Single digit
numbers have very strong "personality" as if they were family members. The
personality of 8 is that in different circumstances it may present itself
as 1+7, 2+6, 3+5, 4+4, 2+2+4, 3+1+4 etc etc. In other words I "know" that
3+5 is one of the "faces" of 8.

8 + 6 brings out my personal early training, because I do the same thing I
did with 7 + 9 except that initially I use the duodecimal system. i.e. I
"know" the answer is more than 12 but less than 24, so I take 4 from 6 to
turn 8 into 12, leaving the remainder 2. I "know" that 12 + 2 is 14
(perhaps because I know 2+2 = 4 and 4+10 matches pattern 14.)
I Expect this results from growing up in a country that (at that time) used
duodecimal money (12 pence to a shilling). In that system 8 is 2/3 of 12,
which in my mind is a "stronger" relationship that 4/5 of ten, probably
because the component fraction is larger. So in this case, seeing 8, I
start my thought process with a comparison to 12.

Generally my simple mental arithmetic seems to follow well-trodden paths of
"pattern recognition", operating in a framework reminiscent of set theory.
More complex tasks require the combination of a number of simpler
"component" startegies. The analysis of the problem into simpler parts does
not necessarily aim at greatest efficiency, but tends towards the most
familiar procedures. I suppose that "familiarity" here means that I can
have confidence in the result without double checking it as you might want
to do with a novel or uncommon procedure. I minimise the number of novel
procedures in order to reduce uncertainty in my result. If the level of
uncertainty rises beyond a certain point, I feel I need pen and paper
confirmation.

It's interesting when you look at it to realise that in mental arithmetic,
subtraction can be a key step in the process of addition.

Martin Baxter