FOM: Maths Barbie

[Dean Buckner]

Occasionally you read a paper that follows through exactly a train of
thought you had, and adds something to it.  Such is Richard Heck's paper

http://www.people.fas.harvard.edu/~heck/counting.pdf

which I thoroughly recommend anyone interested in the connection between
human understanding of number, and formal theories of mathematics.

Heck's thoughts are just those which would occur to anyone (such as myself)
who have had the task of teaching pre-shool (< 5) children the basic
concepts of counting and arithmetic, and who want to understand the strange
mathematical world that young children inhabit.

His central idea is about Hume's Principle.  Children know how to count, and
can answer questions like "how many", and can understand when there are
"just as many".  But "such children have no, or only a very minimal,
understanding of one-one correspondence" (Heck p. 14).  He cites Isobel who
counts 4 barbies on the table, then puts hat on to match each Barbie.  So,
she understands "just as many hats (as Barbies)".  But then Heck asked "how
many hats", and couldn't elicit an answer (hope I understood his point).
Isobel understands (in some sense) that there are 4 Barbies, that each
Barbie has just one hat, but not that there are 4 hats.

So he concludes "The notion of one-one correspondence is very
sophisticated".  He is implying (i think) that (i) children develop
numerical concepts in stages or layers (ii) the order of development shows
which concepts are "logically prior".  So it looks like Hume's principle is
not logically prior to some basic mathematical concepts, and it is therefore
a mistake to base foundational theories upon it.  My own experience of
teaching very young children (my own) suggests this is right.

Some other interesting ideas in the paper:

* Children don't understand "how many" when asked again.  The number
can change.  Indeed.  This was the hardest thing I found about their world.
You count fingers say, and sometimes get "six" or "four".  It really didn't
matter - as though the number were an empirical property of the hand."
"There are four hats on the table" really does mean something like I ended
with "four" when I counted the hats" says Heck.  Yes.
* "Understanding "145" has something to do with one's understanding of
the decimal system".  Yes.  It's well-known you can get young children to
count up to very large numbers (over 100 anyway).  Then you say "There were
two cows in a field.  Then the farmer took one cow away.  How many cows were
left".  No reply, blank!  So, what are they doing when they recite numerals
in the right order.
* They use "Monday Tuesday Wednesday" instead of 1,2 3 for example.
Connects with an idea I had about fiction.  An author (like Tolstoy or
Balzac) has a very large number of characters in their book.  How do they
ensure that they are not repeating the names of characters.  (i) Keep a list
of every name, and every time you introduce a new name, make sure it is not
on the list.  Think about it: an author must be using this process in their
head at list.  Imagine an author who simply could not remember the name of
any character.  (ii) Devise a method for generating new names, such that
applying the method will not generate a previous name.  Then all you have to
remember is the method, and the last name you used.  Here's an idea: have a
short set of characters to use as initial set of names.  Use them in an
order that you can memorise.  when you run out, simply append another
character, and re-use the old set.  When you run out, add a third, and so
on.  E.g. "1", "2", "3" &c.  When you run out, use "11", "12" and so on and
on.  An inexhaustible supply of fresh names!
* "It is not at all clear what the logical form of, say "two men went
for a walk"  should be taken to be."  Says Heck.  Yes.  It can't be a
million miles from "A man went for a walk.  There was another man going for
a walk with him".  Note, as I have expressed it, there are two sentences,
not one.  In the first sentence, we only grasp there is one man.

And some ideas of my own to add:

* There must some primeval connection between the way which writers
introduce new characters to stories, or tell us which of the existing
characters are in question, and the way we understand number.  You read "a
soldier was returning home.  He met a witch".  Do you really understand the
story if you don't understand there were two characters in it?
* Very young children do grasp stories.  I'm not sure quite how, and
it's a strange world not like ours, but they do grasp them somehow.  They
understand about the three dogs, with the different eyes, and that these are
different from the soldier or the witch.  How could you understand the story
if you thought these different characters were all the same one?
* "when I think that there are two blocks on the table, I do not seem
to be thinking that there are as many blocks on the table as there are
numerals from "one" to "two".  Yes.
* How much of maths, for children, is connected with difficult
grammar?  My daughter (at 6) understood basic subtraction (cows in field
type).  But the school insisted on putting questions like "which number,
when added to five, gives eight?".  Difficult quantifier concept here.  I
tried all different methods of phrasing the question 8-5 =?, and found some
got quick answers, some completely impenetrable.  Shows complex interplay
between grammar and maths.
* Another phenomenon which is well-known by teachers, which I mention
purely for interest.  If you add three to five, you start with five (as a
given) then count up to eight.  You go "six seven eight".  Children under
six insist on counting the five as well.  Thisis called "counting in" as
opposed to "counting on", and is a barrier all children must cross.  The
term comes from Piaget i think.


Dean Buckner
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