the following 10 rules
which are based on decades of research
aim to demonstrate that the standard way
to teach
children math is counterproductive
because it focuses
on symbols to be able to think
mathematically however
means not to think in symbols but to
learn to think
in relationships
math can't be taught just like we can't
learn ping pong from watching videos
children can't learn math from reading
textbooks or listening to a teacher
instead they learn math by doing math
ideally with real objects because only
when they do math
relationships are constructed right
where math happens
in their heads
it happens in the head whatever is on
paper is merely a representation of
mathematical thinking
that happens in the brain just like
musical notes
what is on paper is just a
representation of music
that actually happens when someone plays
the piano
[Music]
to be a good musician it's not enough to
be able to read the notes
we also need to practice a lot the same
is true for math
which is why practicing mental
arithmetic is so important
math needs years of practice this
becomes clear when we look at how
children
learn to understand a number say eight
not the symbol eight but the idea of the
quantity of eight
to internalize this seemingly simple
idea
children need a lot of practice in two
skills
first they need to learn how to create
order
and then later on how to create
hierarchical relationships
let's look at order first
construct order when four-year-olds
learn to count
most have trouble ordering objects in
their heads
if the things they count are unevenly
distributed
sometimes they skip objects then they
count the same ones twice
to do it right children have to learn
how to construct
order in their heads this seems easy
but actually takes our brains a lot of
practice
once children learn to order objects in
their heads
they can put them in relationships
hierarchical relationships as children
construct
order they count the objects as follows
one two three four
five six seven and eight
as they do that the number eight
represents the eighth
place in the order in other words eight
always includes
one two three four five six seven
the idea of eight is therefore a
hierarchical
relationship between the eighth object
and all those preceding it
if we don't learn to do this sort of
abstraction by doing lots of math in our
heads
we won't be able to form a solid
foundation for arithmetic
after building them children need to
learn to break relationships apart again
we can see how hard this is when we
present a five-year-old an image of six
dogs and two cats and then ask are there
more dogs or more
animals while most adults who see the
full picture
find this question odd a five-year-old
typically just answers
more dogs when you ask further
more dogs than what the child replies
than
cats in other words if you ask are there
more dogs or more animals
the child hears are there more dogs or
more cats
at age 5 most kids didn't practice
enough math to break hierarchical
relationships apart
while still remembering the whole
this happens because once the child has
to cut the hole into parts
for them at that moment the whole no
longer exists
they have not yet constructed the
concept of eight
without thinking of it as a sum of its
parts
so when they divide the animals into
cats and dogs
all they can think of are two parts of
which one looks larger
the idea of eight is then forgotten
to also think about all animals would
require
two opposite mental actions first
divide the whole and then put it back
together
a mental process that most five-year-old
children
precisely can't do only by age
seven most children can see the whole
and keep its abstraction in their heads
and still divide the sum in its parts
experiences precedes language
as we demonstrated it takes a child a
lot of mental training
and hands-on experiences to form the
concept of a number
at the age of five we can build a simple
row of
eight later form eight square then eight
root only once we have constructed
number concepts inside our heads
can we effectively learn how to express
them with images
symbols and language
math can be expressed in different
languages
a hundred thousand years ago we used
objects to express our mathematical
thinking
later we used images around one
thousand years ago we began to reduce
images to arabic
numeral symbols in future we might
replace symbols with bits
or express math in graphic simulations
or
games in other words while math thinking
always happens in our heads the language
that represents our thinking
is evolving
most people don't have math but language
problems
we know for example that eleven-year-old
unschooled street vendors
are often highly proficient in complex
money transactions
but incapable of doing paper and pencil
arithmetic
this phenomenon known as street
mathematics
shows that when smart kids struggle in
school they often just can't express
their thinking in symbols their brains
can do
math but have language problems
one way to solve this is to do it your
way
just like nobody ever learned to speak a
language just by learning the rules of
grammar
nobody learns math by memorizing the
rules of how to arrange
numbers and symbols in order to find the
right answer to a problem
whenever we do that we stop constructing
fundamental principles inside our heads
to get better and confident children
should be encouraged to find their own
path
and use their own language to express a
solution
which brings us back to rule one
math can't be taught it has to be
constructed
if we want to learn math we have to do
math in our heads
ideally with real life experiences
later we replace the objects with
abstractions
such as language symbols or whatever the
future might bring
the ideas presented in this video are
based on the work of jean piaget
constance comey keith devlin georgia de
clark
and jerome bruner who all contributed
immensely to the body of work and
research on
how children and adults learn math
if you want to get better at math today
join keith devlin from stanford
university
and over a hundred thousand students
from all around the world
in his free course on thinking
mathematically
see the descriptions below for more
details and links for the research
sprouts videos are published under the
creative commons license
that means our videos are free and
anyone can download
edit and play them for personal use and
public schools
governments and non-profit organizations
can also use them for training
online courses or designing new
curriculums
to help us stay independent and support
our work you can join our patrons and
contribute
just visit patreon.com sprouts
even one dollar can make a difference
you