Standing in
the Middle
From a Variety of
Perspectives
Tuiren
A. Bratina and Debra
K. Abbott
University of North Florida
Note: The video and audio files require RealPlayer. Read the RealOne
Player instructions to play the audio files.
Children develop their earliest ideas about statistics from hearing
others
use the words in conversation. Young children have a premathematical
sense of
these concepts as indicated by their use of vocabulary words such as
typical,
normal, okay, most, and middle (Watson &
Moritz,
1999, p.18). We created a setting familiar to children (arranging
people according
to their heights) so we could introduce and expand on children's
understanding
of median. We used one animated graphic. Next we facilitated their
increased
understanding of median by building on the scenario depicted by this
learning
object. We believe learning objects can be used to introduce and
expand conceptual
understanding. As you read this article, be thinking of similar
computer experiences
your students can build on to attain understanding of the concept of
your choice.
The concepts of median and mode should be introduced before the mean
(Mokros
& Russell, 1995; Russell & Mokros, 1996; Watson & Moritz, 1999).
Mokros and
Russell explain that "premature introduction of the algorithm for
finding the
mean may cause a short circuit in the reasoning of some children
(p.37)." Their
research reported that middle school students commonly overly rely on
the algorithm,
misapply it or create their own meaningless algorithms, and do not
consider
whether answers make sense in reality (p. 28-31).
Initially teachers should avoid exercises that deal with statistics
in an
abstract way. Instead teachers should have children use learning
tiles, balancing
sticks, play money, or graph construction. Those activities will help
to emphasize
conceptual understanding and inquiry into what constitutes
representativeness
of a variety of data sets (Meyer, Browning, & Channell, 1995). By
exposure to
mean, median, and mode in a variety of contexts through many years of
schooling,
students will not only be able to successfully choose an appropriate
measure
for a specific context, they will eventually be able to use these
measures to
answer more complex questions.
We too believe that the development of children's knowledge of the
concepts
of measures of central tendency should begin with the median. Our case
study
will discuss instructional strategies used to present the concept of
median
to Brock and Taylor, rising fifth graders who have experience using
computers.
A Case Study
In July, Brock and Taylor demonstrated their skills to educators who
were
testing a variety of digital learning objects. This article is an
account of
Brock's and Taylor's actions and interactions as they were presented a
lesson
about the median.
The boys first viewed an animation (Exhibit A at http://www.unf.edu/~tbratina/hec/example/mediananimation.htm)
depicting five young people of unequal heights. This animation is a
series of
frames showing the pictured children being arranged according to their
heights,
from shortest to tallest. The graphic shows the individual standing in
the middle
being singled out. Her height is the median height.
Next the boys were asked to mimic what they saw on the computer
screen. There
were five people in the room. Brock and Taylor positioned themselves
and rearranged
the others in the line-up so that all five stood shortest to tallest.
When asked
how that related to the median, Taylor said, "It's not the shortest
and it's
not the tallest." A probing question directed to Taylor was, "So
Brock's height
is the median? He's not the shortest and he's not the tallest." Taylor
knew
that was incorrect and correctly identified the person in the middle
of the
line-up of persons standing in ascending order of height. She was
asked how
tall she was. Five feet six inches. The statement was made, "Five feet
six inches
is the median height." Exhibit B at http://stream.unf.edu:8080/ramgen/tbratina/medianpt1.rm
is a video of this experience. (To view the video, download the free
RealPlayer
from www.real.com.)
Next one of the adults recited nine numbers, the order being neither
all increasing
nor all decreasing (9, 100, 8, 4, 1, 4, 7, 9, 5).
Brock and Taylor "collaborated" to find the median. The youngsters
used an
algorithm that essentially mimicked the rearrangement procedure they
had performed
to get a line-up of people of ascending heights (Exhibit C at http://stream.unf.edu:8080/ramgen/tbratina/medianpt2.rm).
Brock and Taylor easily moved from semi-concrete (computer
animation) to concrete
(human line-up) procedures. They were also able to apply the
semi-concrete and
concrete experiences to correctly arrive at an abstract process (using
only
symbols).
Fourteen days later (and with no intervention) Brock was asked about
the median.
At first he did not know how to describe the median. So the
interviewer asked
Brock, "Can you tell me the median of the following numbers: 9, 100,
5, 3, 7?"
Brock needed to be reminded that he had studied mathematics problems
two weeks
earlier. Brock beamed as he partially recalled the algorithm. "5" was
Brock's
answer. When asked why, Brock declared five was the middle number.
Pausing,
he then went through what appeared to be a mental exercise. "Seven,"
he recanted.
Brock's justification for stating the median was seven was correct.
Reflecting on the
Students' Work
There are two main points we would like to make relative to Brock's
and Taylor's
learning experiences. First, a symbiotic relationship between digital
and non-digital
resources seems to have been operating. There are so many discussions
that revolve
around technology versus other instructional strategies. Graeme
Wilson
(2001) states, "In their enthusiasm for using technology, educators
often overlook
non-digital activities that work just fine to convey learning and can
perhaps
better adapt to different learning situations." This learning instance
illustrates
that using computers in conjunction with other instructional
techniques
provides a solid foundation for the two children's understanding of
median.
This does not just happen. We encourage teachers to help students make
these
connections by using a variety of complementary instructional tools.
Teachers
need to point out to the learners that the experiences are simply
different
representations of the "same" problem.
Secondly, teachers need to plan ahead for the student to move toward
concept
attainment. The digital learning object was selected because there
were an odd
number of children pictured in the animation. The identical number of
humans
was used in the line-up. An odd number of elements in the set of
numbers were
presented for the abstract portion of the tasks. Caution was also
taken that
there would be no ties. During initial concept formation the most
basic situation
is presented.
Notice that the follow-up was truly action research! The noise in
the background
suggests that this was a natural setting. Actually the noise did not
bother
Brock or the interviewer because they have grown accustomed to it.
Therefore,
no criticism will be made about the setting. However, there is a
criticism regarding
the selection of the numbers in the follow-up session with Brock.
There was
nothing mathematically improper about choosing "5" as one of the
numbers in
the set, but pedagogically speaking, that number should not be used
because
there were five numbers in the set. It frequently causes confusion.
When someone
says "five," it is difficult to detect when the person is talking
about the
number of elements in the set or the element itself.
We want to add that after we concluded our "case study," Brock
and Taylor
reported that during the first two weeks of their fifth grade year,
the teacher
taught the class about the median. The boys told us that they helped
the teacher
explain how to find the median! We believe we established a
"readiness"
for learning this concept in their school setting.
Extensions
The same processes could be used for concept attainment of the other
measures
of central tendency. For example, youngsters could have the concept of
average
(Exhibit D at http://www.unf.edu/%7Etbratina/hec/example/averageanimationelemschool.htm)
presented to them at a semi-concrete level. They could then be given
pennies
to distribute to five humans exactly as the pennies in the animation
were distributed.
The physical actions to build the algorithm for finding the average
would be
acted out by the students. After students go through the semi-concrete
and concrete
operations, they can be presented a set of several numbers (the sum of
which
is divisible by the number of elements in the set). The teacher needs
to enunciate
the connection among the concrete, semi-concrete, and abstract
episodes. It
is not normally readily apparent to the learners.
The teacher must also design initial experiences so that
participants receive
an equal number of pennies and there are no remaining pennies. The
teacher should
advise children that they will encounter more complicated scenarios as
they
master the basic understanding of mean and can correctly compute the
mean.
The scenario is reusable for the third measure of central tendency.
They could
view animated pictures on a computer screen. The animation would show
more than
one person of the same height. Then the students could select the mode
of the
heights of the real-people in a line-up. After the semi-concrete and
concrete
phases, a set of numbers would be presented and the students would
have to produce
the mode from that set. The teacher again needs to carefully select
the elements
of the set (be they digital figures, or humans, or numbers) so that
the students
are introduced to the basic notion of mode. Once grasped, the teacher
can help
the children transition to a more complex set of elements.
Challenges
This article is not about statistical indices. It is about using a
variety
of instructional strategies in stimulating and effective ways for
developing
understanding of a concept, procedure, and application.
The teacher and learner(s) are the central components of the
experience. The
teacher must identify the learning objects that can and should be
connected
to construct the meanings of the topic(s) to be studied. The teacher
must also
determine the order in which to present the instructional strategies.
For example,
we chose to do the semi-concrete level (pictures on the computer)
before the
concrete level (human line-up). This was a practical decision. There
was little
need for direction after Brock and Taylor saw the computer animation.
They easily
copied the process. Teachers should analyze the situation to help
determine
what the best order appears to be.
So it is evident that the teacher's role is crucial in this process
and cannot
and will not be replaced by non-human objects that the teacher and
student(s)
use to move toward concept, procedural, or skill attainment.
References
Meyer, R., Browning, C. & Channell, D. (1995). Expanding students'
conceptions
of the arithmetic mean. School Science and Mathematics, 95(3),
114 -17.
Mokros, J. & Russell, S. (1995). Children's concepts of average and
representativeness.
Journal for Research in Mathematics Education, 26(1), 20-39.
Russell, S. & Mokros, J. (1996). Research into practice: What do
children
understand about average? Teaching Children Mathematics, 2(6),
360-364.
Watson, J. & Moritz, J. (1999). The development of concepts of
average. Focus
on Learning Problems in Mathematics, 21(4), 15-39.
Wilson, G. (September, 2001). The promise of online education: El
Dorado or
fool's gold? From Now On: The Educational Technology Journal,
21(1).
Available online: http://www.fno.org/sept01/online.html
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