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Modeling > Introducing
Distribution |
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Introducing
Distribution |
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| Lesson
1A: Body Measure |
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Preparing for
the
Lesson / Lesson
Activities / Students'
Ways of Thinking/
Assessment
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| Prerequisites and Mathematical Concepts |
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Prerequisites
The activities in the lesson rest on a firm grasp of principles of measurement. Prior to the lesson, students should have opportunities to consider the nature of units of measure. For example, students could try to measure a distance in the classroom (or outside) using whatever tools they invent (but not standard tools, like a ruler). Activities like these help students consider the nature of units and how units are accumulated for measure. The theory of measurement progress map in the Assessment System describes these conceptual attainments in greater detail.
Mathematical Concepts
Students investigate variability by designing displays of trends that they notice in a collection of their measurements of the same person's body part. Comparison of different displays supports the development of meta-representational competence, meaning that students are encouraged to understand how every representation highlights some features of the data and backgrounds other features. Hence, representations are trade-offs. This quality of trade-off is a characteristic of all mathematical systems. We aim to help students better understand mathematical representation.
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| Overview
of the lesson |
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1.
Multiple
Measures of
the Same Part
of the Body,
with Two Different
Tools
Multiple
measures are
taken of the
body parts of
several students
and/or of the
teacher. Stations
are set up around
the room, and
everyone measures
the head circumference
of one student,
the armspan
(the length
of a both arms,
horizontal extension)
of another,
and perhaps
the area of
a third person's
hand. For example,
everyone measures
Timothy's wingspan,
Amy's head circumference,
and the area
of Mr. Bill's
hand. Each measurement
is conducted
two times: once
with a crude
tool, and then
again, with
a more precise
or better-suited
tool. For example,
the arm-span
might be measured
with a 15 cm.
ruler and with
a meter stick.
The ruler involves
more iterations
(laying end-to-end)
and hence the
chances for
error are greater
when compared
to using the
meter stick.
If
this is the
introduction
to the unit,
the teacher
probes students'
thinking about
why the measurers
of the "same" arm-span
or head circumference
should result
in such different
measures.
If the lesson
is a model
extension,
then after
collecting
the data,
students
develop distributions
and note similarities
and differences
among these
distributions.
Because some
students believe
that measurement
is only imprecise
if performed
indirectly,
as in the
measures
of height
of the flagpole
in lesson
1B, the distributions
that result
are often
surprising
to students,
and hence
serve as a
crucible for
further discussion
of measurement
and sources
of (random)
error.
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2.
Designing a
Display
After
each person
has measured
each part of
the body with
the tools provided
(at least once),
the measures
obtained are
collected. Students
work in small
groups to design
a display that
shows, without
words, all of
the data and
any of the trends
about the data
that they notice.
(If students
have already
participated
in previous
lessons, then
they simply
enter their
data.)
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3.
Comparing Displays
Students
post their
displays and
classmates,
but not the
designer, attempt
to interpret
their meaning.
The pedagogical
intention is
to explore
different senses
of the "shape" of
the data. Student
inventions
are often idiosyncratic
but the variability
of student
designs is important
for grounding
discussion
about qualities
of data display.
Hence, the
teacher employs
a language for
fostering meta-representational
competence
by asking the
class to consider
which features
of the data
are highlighted
by a particular
display and
which it makes
less obvious
or even hides.
The pedagogical
intention is
to foster the
development
of meta-representational
competence
(Thinking about
representations
as trade-offs,
not as clear
roads to "correct" ways
of showing
data.). After
discussing
qualities of
displays,
the lesson
concludes by
soliciting
students' ideas
about sources
of these qualities.
What about
the measurement
process might
lead to the
shapes of
the data observed?
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Last Updated:
April 19, 2006
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