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Abstract
Teaching
mathematics meaningfully involves providing students with experiences
that enable them to make sense of mathematical ideas. Now more than
ever, technology can assist teachers in their endeavors to provide students
with rich experiences that enable them to better understand mathematics.
To illustrate the potential benefits of using technology and, in particular,
the benefits of incorporating advanced statistical software in the K-12
mathematics curriculum, this article provides a series of sample lessons
that utilize the statistical program Fathom. Included among these lessons
is a detailed description of an on-site demonstration that was used
to help two in-service and two pre-service elementary teachers better
understand how statistical software programs could be incorporated into
the elementary mathematics curriculum. Reactions from the elementary
students and their teachers are given. Additional lessons are also provided
which are appropriate for use in middle and secondary grade levels.
Now more than
ever, technology can assist teachers in providing students with rich
and meaningful mathematical experiences. Whether it be through the internet
or through software programs, students can use technology to problem
solve, to engage in simulations, and to process and analyze data in
new and exciting ways. As demonstrated in the following statements,
professional groups have recognized the importance of incorporating
technology in the mathematics curriculum.
- “Students
should be able to access, gather, store, retrieve and organize data,
using hardware and software designed for these purposes” (National
Research Council 1996, 145).
- “Technology
is essential in teaching and learning mathematics” (National Council
of Teachers of Mathematics 2000, 24).
Despite the recognition of the importance of technology in mathematics
instruction, the impact of technology on teaching and learning mathematics
has clearly not reached its potential. Technology has yet to be an integrated
part of daily instruction. Many teachers, although aware of the value
and the availability of technology, have not been able to fully transform
their instruction by it. Mathematics teachers continue to face many
barriers in their quest to incorporate technology in their classrooms.
These barriers are not limited to but include: limited materials, time
considerations, pedagogical issues, and support with planning, maintaining,
and coordinating the use of technology (Flores 2002).
In an effort to address such barriers, this article provides sample
lessons that demonstrate one way to incorporate statistical software
in mathematics instruction at a variety of grade levels. Included among
these lessons is a detailed description of an on-site demonstration
that was used to help two in-service and two pre-service elementary
teachers better understand how statistical software programs could be
incorporated into the elementary mathematics curriculum. Lessons for
upper elementary, middle grades, and secondary classes are discussed
in the article as well. Although all of these lessons utilize Fathom
1.1 (Finzer 2001) the activities described can be conducted with most
statistical software packages and/or spreadsheets. After a brief description
of the statistical program Fathom, these lessons will be described.
Fathom
Fathom
is a statistical software package specifically designed to support instruction
in data analysis. Although typically marketed to secondary and post-secondary
educators, its unparalleled user-friendly features combined with data
analysis capabilities make Fathom an appropriate software program for
the elementary school setting as well. By using Fathom in earlier grades,
students should be better prepared to use technology for data analyses
later in their curriculum. Exposing students to software that can be
used across grade levels can promote the integration of technology in
the K-12 curriculum and improve student utilization of technology as
a tool for data exploration.
Using Fathom, elementary students can create various types of graphs
including histograms, bar graphs, dot plots, line plots, and modified
box plots with the click of a mouse. Fathom displays on one page all
calculations and graphs for a data set as well as the data set itself.
Students can manipulate data by dragging points within graphs and examine
the effect of this change on simple statistical measures. The dynamic
aspects of Fathom allow students to observe simultaneously the impact
dragging data points on one graph has on accompanying graphs, charts,
and tables. By seeing these multiple representations of the data simultaneously,
students can: draw from the representation that best fits their learning
style, make connections to other representations that might not have
been considered, and compare and contrast the various representations.
Utilizing graphical, numerical, algebraic, and verbal representations
can be an effective tool in promoting a deeper understanding of data
(Friedlander & Tabach 2001).
The use of programs like Fathom is important in developing students’
ability to collect and represent data and to “propose and justify
conclusions and predictions that are based on the data” (NCTM
2000, 176). Difficulties in sorting and organizing data hinder young
students’ abilities to represent data. Research has shown that
technology is helpful in overcoming such difficulties (National Research
Council 2001). With this in mind, classroom instruction should be designed
that utilizes technology as described in the lessons that follow.
Using Statistical Software to Explore Data with Third Graders
To demonstrate the usability of statistical programs with elementary
students, the following lesson was designed and implemented within a
third grade class. The class was comprised of a heterogeneous group
of 11 females and 5 males ranging in age from 7 to 10 years old. The
on-site lesson was conducted to demonstrate to participating teachers
how statistical software packages such as Fathom can enhance teaching
and learning within their individual classrooms.
Prior to the implementing the lesson, the teachers designed a survey
to allow students to collect data for further analysis (see fig. 1).
The survey questions allowed for various types of data to be collected,
including numerical data, such as the number of steps to the door, as
well as categorical data, such as right-handed or left-handed. These
data were then entered into a Fathom database (see fig. 2).
Figure 1. Teacher-designed survey used to collect data.
Figure 2. Data set of student responses to teacher-designed
survey.
The lesson began with a discussion of graphs that included having students
identify where they see graphs in their everyday lives. Students reported
seeing graphs in their social studies and math textbooks, in the library,
and in their parents’ job-related paperwork. Students were also
asked about the usefulness of graphs in understanding data.
Following the discussion of graphs, students were asked to close their
eyes and make predictions about the number of boys and girls in the
class. Individual predictions were recorded at the board for the group,
and trends in their predictions were discussed. While answers varied,
most students predicted correctly that there were more girls than boys
in the group. Next, students were asked to create a bar graph with paper
and pencil representing their individual predictions. Upon completing
their graphs, students were directed to turn their attention to the
computer display for the purpose of exploring the survey data using
Fathom.
Data exploration began by having students answer questions based on
the survey data displayed by Fathom (see fig. 2). On an item-by-item
basis, questions about the students’ responses to the survey were
asked to help familiarize students with the information given in the
table. Sample questions included:
- Are there more
right-handed or left-handed students?
- What might the
“3” represent under the hand column?
- What was the least
number of steps needed to reach the door?
- What was the most
numbers of steps needed to reach the door?
Next, the efficiency of this technology was demonstrated by asking students
to recall how long it took them to create their own bar graphs. Fathom
was then used to create a similar bar graph showing the actual count
of males and females in the class (see fig. 3). The ease and speed of
the production of this graph amazed both the students and the teachers.
Students enthusiastically compared their bar graphs to Fathom’s
bar graph to check the accuracy of their predictions. When asked for
the true number of girls in the class, one student readily said, “Eleven
girls.” When asked how this answer was obtained the student said
she counted the number of girls on the data list. The group was then
asked, if there was another way to find out how many girls there were.
A different student pointed out that the bar graph could also be used
to determine the number of girls. To show students a third way that
Fathom can represent the data set, the cursor was placed on the bar
in the graph and the information was displayed at the bottom of the
screen.
Figure 3. Bar graph of the number of boys and girls
in the class.
Following the examination of the bar graph, two new types of graphs
were introduced, the dot plot and the histogram. First, Fathom was used
to create a dot plot of the “number of steps” variable (see
fig. 4). Because students reported that they had not seen a dot plot
before, a brief explanation was given as to the type of information
that was being displayed by the dot plot. Students were readily able
to interpret information provided by the dot plot and began answering
questions, such as the following:
Figure 4.
Dot plot of the number of steps required to get to the door.
Figure 5. Histogram of the number of steps required
to get to the door.
In
continuing the exploration of this data, the dot plot was turned into
a histogram (see fig. 5). Features of the histogram were explored
using Fathom. For example, when the cursor was placed on a bar of
the histogram, the program revealed the range of values represented
by that bar as well as the number of cases that fell within that range.
Since the students seemed to feel comfortable with the histogram,
the question was posed, “Do boys tend to take more steps than
girls when going to the door?” The histogram was split to show
the breakdown of the number of steps by gender (see fig. 6). By examining
the graph, students determined that boys in general neither took more
or less steps than girls when walking to the door. In other words,
the boys neither sat farther from the door or closer to the door than
did the girls.
Figure 6. Histogram of the number of steps required
to get to the door
split according to gender.
Figure 7. Dot plot of the number of steps required
to get to the door split
according to age.
Before the next question could be posed, a girl raised her hand and
suggested that the difference in the number of steps had something
to do with age. To explore her conjecture, a dot plot of the number
of steps taken was created and then split according to the “number
of years until age 20” variable (see fig. 7). A quick examination
of the graph allowed students to conclude that as was the case with
gender, there was no connection between age of a student and his or
her distance from the door.
As demonstrated with this student’s question as well as in the
previously posed questions, this software readily allowed students
to see multiple representations of the data and translate among these
representations to answer questions. Within a very short period of
time, these third grade students were exposed to four representations
of their survey data and were able to answer higher-order questions
using those representations. After observing this lesson, teachers
stated that students benefit from seeing multiple representations
of the data and can use any one of those representations to draw conclusions.
For both the teachers and the students, a new way of viewing and utilizing
technology was realized as a result of this lesson.
Student and Teacher Reactions
After the lesson, the teachers were asked to provide written feedback
concerning the use of this statistical software with their third grade
classes. The following comments were offered.
Overall,
teachers viewed this software as being a useful, teaching tool that
they would like to incorporate in their teaching.
The third graders were asked to provide written comments about the lesson
as well (see fig. 8). Like their teachers, comments indicated that students
were impressed by the speed and efficiency of the program as well as
the visual representations it produced. The use of the technology generated
enthusiasm among many students and appeared to be a motivating factor
of the lesson.
Figure 8. Student reactions to the use of Fathom.
Statistical
Software and the Middle and Secondary Mathematics Curriculum
According to the NCTM’s Data Analysis and Probability Standard
for middle grades students should be able to “select, create,
and use appropriate graphical representations of data” (NCTM 2000,
248). Students should also be able to use their observations of data
to develop and test conjectures. Statistical software can be an effective
tool in facilitating students’ exploration of data in these ways.
The following sample fifth grade integrated math and science activity
demonstrates both the dynamic nature of statistical software packages
and how such software can be readily incorporated in the middle school
mathematics curriculum. For this activity, adapted from a web lesson
at: www.educate.si.edu/resources/lessons/currkits/oceans/secrets/proced
Students
create a profile of the ocean floor, as it exists between Florida and
Dakar, Senegal.
Figure 9. Data set for ocean floor depths.
Within
the data set (see fig. 9), the variable “d” represents the
distance from Florida and the variable “depth” is the depth
(in kilometers) of the ocean floor relative to the ocean surface. By
plotting “d” along the horizontal axis and “depth”
along the vertical axis, a profile of the ocean floor is created (see
fig. 10). Students can then label the different areas of the floor,
for example the continental shelf, and answer questions about the profile
they have created. Using this same set of data in a math class, students
can begin to explore the concept of average by trying to estimate the
average depth of the ocean for this region using the visual of the ocean
floor.
Figure 10. Line scatter plot of the ocean floor depths
that includes the average depth.
Fathom
can calculate and display the average ocean depth on the graph, as indicated
by the red line. Students can make conjectures concerning the effect
of changes in the ocean floor on this average. These conjectures may
be explored by dragging points, in essence changing the ocean floor,
and observing the effect on the average depth. Investigations of this
type are conducted with ease using Fathom. To have students produce
such calculations and corresponding graphs by hand would be very time
consuming and meaningful connections could be lost by focusing on calculations
versus timely data observations.
| Middle grades
students do not have to be limited to interpretations of graphs.
Eighth grade, beginning algebra students can use statistical software
to explore linear functions with data. In the activity “Batter
Up!” (Erickson 2001), students explore the relationship between
the number of times at bat for each National League baseball player
and the number of games played. Students are provided with a scatter
plot (see fig. 11) of the data and asked to guess the equation of
the line along the upper edge of the points. |
Figure
11. Scatter Plot of baseball data. |
In addition, students are asked to predict the slope of the line, its
intercept, the difference between players that lie on the line and those
that lie below the line, and ascertain why that line exists. Investigative
questions of this nature help students see how algebra is useful in
describing relationships in their world. This line of questioning engages
students in discussions about their thoughts and ideas pertaining to
the mathematics at hand. Once predictions are made, students then explore
the data set and its scatter plot using statistical software. Students
are instructed to place a moveable line on the scatter plot and examine
the actual equation of the line, which is provided by the software (see
fig. 12). Note that the equation is not given in terms of x and y but
instead uses the variable names. This allows students to focus on the
relationship between the variables that they are exploring. The position
of this line can be changed and students can observe the effects on
the equation. Students then record this equation and compare it to the
equation previously guessed.
Figure 12. Scatter Plot with movable line and equation.
In
this activity, the use of statistical software can better enable teachers
to develop algebraic thinking among diverse student populations by not
requiring a large amount of time to be devoted to computation. As a
result, more class time can be spent engaging the students in making
conjectures, posing and answering questions, recognizing trends in the
data, all of which can lead to deeper conceptual understanding. This
type of technology also enables students to perform tasks in investigating
questions that otherwise could not be done, such as using the movable
line.
In the secondary mathematics curriculum, data analysis software can
continue to serve as a tool for enhancing student learning. According
to the NCTM’s standards, in grades 9 through 12 students should
compute statistics and begin to use simulations in exploring probability
distributions. They should understand the difference between sample
statistics and population parameters and “how sample statistics
reflect the values of population parameters and use sampling distributions
as the basis for informal inference” (NCTM 2000, 324). The following
activity “Exploring Normal Distributions” which can be accessed
at www.keypress.com/fathom/downloads/sample_activities/NormalDist demonstrates
the power of technology in exploring the effects of random sampling
on statistical measures. In this activity, the software allows the student
to sample the population over and over to investigate the changes in
the mean and standard deviation. Once these statistics are collected
for a large number of samples, they are plotted on a graph and students
are asked to describe the distributions. Without the assistance of technology,
conducting investigations of this type would be extremely tedious and
students would likely lose sight of the overall objective of the activity.
In many cases, teachers would choose not to conduct an activity of this
nature at all due to time constraints. When this happens, students miss
out on opportunities to enhance their understanding of statistics and
the sampling process.
Discussion
Technology is changing the world around us. Just as it is changing jobs
and businesses, it also can change the way teachers work to provide
students with rich and meaningful mathematical experiences. The previously
described activities are just a small sampling of the ways statistical
software can revolutionize the way the K-12 mathematics curriculum is
implemented. When technology is integrated into classroom instruction
as described in these lessons, students are given the opportunity to
explore mathematical processes, concepts, and meanings that time and
curricular demands would otherwise not allow.
Understanding statistical data is a particularly important mathematical
skill because students are bombarded with data in their everyday lives.
As adults they will have to make life choices based on this data. It
is more important than ever that our students have a strong understanding
of numbers and the various forms in which numbers are represented regardless
of the level of their computational skills. The lessons contained in
this article reveal how technology such as statistical software packages
supports data investigations by students at all grade levels in new
and exciting ways. When using such technologies teachers are better
able to balance concept development with skill development and reach
diverse student populations because of the speed and accuracy in which
technology produces graphs and computes statistical measures. The integration
of statistical software in the K-12 curriculum can enrich instruction
by also providing a means for exploring data through multiple representations.
As demonstrated in the third grade lesson, seeing multiple representations
of the data simultaneously allowed students to draw from the representation
that best fit their learning styles, make connections to other representations
that might not have been considered, and compare and contrast the various
representations. For the students who participated in this lesson, making
such comparisons served as an effective tool for promoting understanding
and higher-order thinking. Through the use of technology, a discussion
of the data extended beyond observational questions to the production
of inferences and the drawing of conclusions. These types of engagements
are proven to increase student achievement (Marzano, Gaddy, & Dean
2000).
Providing teachers with ideas for better incorporating technology into
the classroom is essential to enhancing mathematics instruction in the
K-12 curriculum (Flores 2002; NCTM 2000). The sample lessons given in
this article can serve as sources of ideas for teachers attempting to
incorporate technology in their mathematics instruction. When teachers
use technology to enhance classroom learning, we all benefit.
References
-
Erickson, T. Data in depth: Exploring mathematics with Fathom.
Emeryville, CA: Key Curriculum Press, 2001.
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Finzer, W. Fathom (Version 1.1) [Computer software]. Emeryville, CA:
KCP Technologies/Key Curriculum Press, 2001.
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Flores, A. “Learning and Teaching Mathematics with Technology.”
Teaching
Children Mathematics, 8 (February 2002):308-10.
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Friedlander, A. & Tabach, M. “Promoting Multiple Representations
in Algebra.”
In The Roles of Representation in School Mathematics, 2001
Yearbook of the National Council of Teachers of Mathematics (NCTM),
edited by Albert A. Cuoco and Frances R. Curcio, pp. 173-185. Reston,
VA.: NCTM, 2001.
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Joyce, B. & Showers, B. Power in Staff Development through
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Training. Alexandria, VA: Association for Supervision and Curriculum
Development (ASCD), 1983.
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Marzano, R., Gaddy, B., & Dean, C. What works in classroom instruction?
Aurora, CO: Mid-continent Research for Education and Learning, 2000.
National Council of Teachers of Mathematic (NCTM). Principles
and Standards
for School Mathematics. Reston, VA: NCTM, 2000.
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National Research Council. Adding It Up: Helping Children Learn
Mathematics.
Washington, DC: National Academy Press, 2001.
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National Research Council. National Science Education Standards.
Washington, DC: National Academy Press, 1996.
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