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| Home > Data
Modeling > Introducing
Distribution |
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Introducing
Distribution |
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| Inventing
Measures |
| Task |
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Invent
a measure of
what the data
seem to suggest
is the best
estimate of
the height of
the flagpole.
Then invent
another measure
of how precise
or accurate
the measurements
were. Be sure
to describe
as completely
as you can how
to calculate
each measurement.
Each measurement
should have
a method for
getting it,
so that other
people could
follow your
method. Tinkerplots
should be available
to students.
Teacher
note : You
may wish to
break this
into two parts
and have students
work first
on inventing
their measure
of the best
guess of the
height of
the flagpole.
There is a
worksheet
that accompanies
this task,
appended to
the end of
this lesson.
Students can
work in pairs,
individually,
or in small
groups. You
may wish to
introduce
students to
TP formula
feature as
a way of expressing
their method
so that others
can use it
too.
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| Purpose |
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When
students
create
measures,
they
are confronted
with
the need
to think
hard
about
qualities
of the
distribution.
For example,
they
may have
an intuition
that
best
estimates
of height
are somehow
in the
middle.
But how
should
middle
be quantified?
Similarly,
students
might
think
about
accuracy
as "how
close" the
measurements
are or
how "compacted" they
are.
But how
can these
intuitions
be quantified?
We are
setting
the stage
for familiar
statistics,
such
as indicators
of central
tendency
(mean,
median,
mode),
as well
as indicators
of variation,
such
as average
deviation,
mid-50
range,
and the
like.
Some
may be
invented
by students.
Others
will
be introduced
later,
but students
will
have
a better
sense
of what
the statistics
indicate.
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| Whole
Group Conversation
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Students
present the
method developed
by other students
and they guess
about what aspect
of the data
led the person
to invent that
particular method.
At the end of
the elicitation,
the teacher
asks students
to group methods
that are alike
in their notebooks,
and to tell
how they are
different from
other methods
in other groups.
Students share
their groups
and explain
how they view
similarities
and differences.
Teacher
note: Be
sure that
if students
say "average" that
there is
some directed
exploration
of qualities
of averages,
including
their susceptibility
to extreme
values.
Students
present the
method developed
by other students
and they guess
about what aspect
of the data
led the person
to invent that
particular method.
At the end of
the elicitation,
the teacher
asks students
to group methods
that are alike
in their notebooks,
and to tell
how they are
different from
other methods
in other groups.
Students share
their groups
and explain
how they view
similarities
and differences.
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| Students'
Ways of Thinking |
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Students
often appeal
to modal values
when reasoning
about best estimates
of height under
the premise
that identical
measures signals
convergence
to the true
height. Other
students prefer
the mean, because
it represents
all cases, and
still others,
the median,
because it splits
the data in
the middle.
However, students
often make very
sensible proposals
that blend these
intuitions.
For example,
some suggest
finding a modal
clump of values
(the stems in
a stem-and-leaf
display) and
then finding
the mean of
the leaves.
Others have
a sense of neighborhood
of values, especially
around the center,
and first define
a middle group,
and then define
an indicator
of center in
that middle
group. It is
important to
acknowledge
the legitimacy
of these different
ways of thinking.
Students
generally find
reasoning about
precision or variation
more challenging.
What does it mean
to be closer or
more compact or
to agree more
often? One solution
that we have seen
invented repeatedly
is to think of
agreement as measured
by distance. For
example, a student
suggested taking
the difference
between the mean
and the extreme
values and adding
these to form
an index. Upon
further reflection,
this student
decided that the
precision of any
point could be
represented by
the difference
between it and
the mean (or median),
and by extension,
that the precision
of the whole group
could be measured
by finding the
average difference.
This led to a
surprise. The
average was zero.
This provided
an opportunity
for the teacher
to introduce the
absolute value
function in a
context where
it could be put
to use. Other
students focused
less on individuals
and more on center
clumps. With
guidance, many
of these students
generated a mid-range
estimate of precision.
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Last Updated:
April 13, 2006
All Rights reserved. |
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