ERIC ED396697: Let's Not Let the Number of Warthogs Be "X."

7 Pages·1994·0.25 MB·English

by  ERIC

DOCUMENT RESUME
ED 396 697
IR 017 873
AUTHOR
Wright, Peter
TITLE
Let's Not Let the Number of Warthogs Be "X."
PUB DATE
94
NOTE
7p.; In: Recreating the Revolution. Proceedings of
the Annual Naticnal Educational Computing Conference
(15th, Bostcn, Massachusetts, June 13-15, 1994); see
IR 017 841.
PUB TYPE
Descriptive (141)
Reports
Speeches / Conference
Papers (150)
EDRS PRICE
MF01/PC01 Plus Postage.
DESCRIPTORS
Cognitive Style; Computer Uses in Education;
Educational Development; *Educational Technology;
Elementary Secondary Education; Foreign Countries;
Information Technology; *Mathematics Education;
*Problem Solving; *Spreadsheets; *Word Problems
(Mathematics)
ABSTRACT
The issue of how to integrate information technology
in the teaching/learning environment remains strongly associated with
the use of the computer as a tool. While technology based tools such
as Logo have been advocated for problem solvers at the elementary
level, spreadsheets have a great deal of potential for use at Loth
the junior and senior high school levels. In this paper, alternatives
to the solution of a variety of math problems, ranging from story
problems to finding the root of equations, are explored. Although not
all problems lend themselves to the use of spreadsheets, such an
approach does add a viable alternative to existing problem solving
strategies. The greater the number of strategies available to
students, the more likely it is that differences in learning styles
can be accommodated. Seven figures depict solutions to the sample
problems. (Author)
Reproductions supplied by EDRS arc the best that can be made
it
from the original document.
*
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Paper (W4-202A)
Let's not let The Number of Warthogs be 'X'
PERMISSION TO REPRODUCE THIS
MATERIAL HAS BEEN GRANTED BY
or
Donella Ingham
enUCAT.0.1
-. !'EPAPTMENT
Peter Wright
Department of Adult Career and Technology Education
E4TPI
Faculty of Education
630 Education Building South
University ofAlherta
TO THE EDUCATIONAL RESOURCES
INFORMATION CENTER (ERIC)
Edmonton, Alberta
Canada T6G 2G5
[phone]
[email protected]
Key words: problem solving, high school, spreadsheet, strategies,
microcomputer
Abstract
The issue of how to integrate information technology into the teaching/learning environment remains strongly associated
with the use of the computer as a tool. While technology based tools such as Logo have been advocated for problem solvers at
the elementary level, spreadsheets have a great deal of potential for use at both the junior and senior high school levels. In
this presentation, alternatives to the solution of a variety of math problems, ranging from story problems to finding the roots
of equations, will be explored. Although not all problems lend themselves to the use of the spreadsheet, such an approach
does
add a viable alternative to existing problem solving strategies. The greater the number of strategies available to students,
the more likely it is that differences in learning style can be accommodated.
Introduction
It is now approximately fifteen years since micromputers appeared in the schools; many interesting phases have been
witnessed since the inception. At first, in the 'euphoric phase', attention focussed on the issues of what type of computers to
buy. how many there should be, and equality of access. Sheingokl (1991) acknowledged this phase quite recently In
remarking that "computer-based technology has been brought into schools during the past decade largely because the
technology was seen as being important in and of itself'. Attention next turned to "but what do we do with them" and In the
absence of quality educational software, a widespread computer programming epidemic broke out thereby marking the
'dawn of reality' phase. Fortunately, during this period, productivity and general purpose software emerged. This eventuality
marked a very significant dovaiturn in the popularity of computer programming and gave rise to the 'exploitation phase'
during which the computer could be employed as a general purpose tool by teachers and students alike. Despite the passage
of time and rapid advances in information technology, the 'electronic education phase' that visionaries had predicted is still
not a penasive reality. The burning issue of how to integrate information technology into the teachingAea ming emironment
thus remains strongly associated with the use of the computer as a toul. This presentation will describe and demonstrate a
number of ways in which the spreadsheet can be exploited in the mathematics classroom and through them, examine the
problem solving strategies available to students. An assumption is made that students have been introduced to spreadsheet
basics.
Of Warthogs and Cockatoos
The first example presented represents the classic story problem an example of which is as follows:
The total number of legs in a group of 14 animals is 38. The group contains only
cockatoos, which have 2 legs each, and warthogs, which have 4 legs each. flow
__
many warthogs are there?
The traditional approach to solving a story problem of this type is to begin by saying "let the number of warthogs be X"
then proceed to establish and solve a set of simultaneous equations. Such a rigorous, analytical strategy can be very appealing
to the mathematically inclined but less so to those who are not. The less mathematically inclined might choose to make an
inspired estimate of the number of animals of each animal type, determine how many legs are implied and then adjust their
estmate until they zero in on the answerthe trial and error method. Both strategies are perfectly valid. The spreadsheet
offers a number of middle-ground alternatives which serve to widen the spectrum of problem solving strategies available to
students. Three potential strategies that students might employ in solving the warthogs and cockatoos problem are described
below in order of increasing level of sophistication.
National F4ucalrnnal Computing Conference I99-f, Boston, Arl
I-1F,ST COPY AVAILABLE
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.
Strategy 1
One of the simplest anticipated solutions might contain three columns as shown in Figure 1. Column 'A' contains an
ti
increasing sequence of natural numbers from one to fourteen (corresponding to the potential number of cockatoos in the
group of animals); the students may either enter these numbers directly or generate them using a simple formula as shown.
contains the corresponding number of warthogs (fourteen minus the number of cockatoos); these numbers may
Column
also be entered directly or generated by a formula. Column 'C' uses a formula to calculate the total number of legs for the
fourteen animals (two times the number of cockatoos plus four times the number of warthogs). While this formula could be
entered into each cell, students should be expected to be familiar with the simpler concepts of copying formulas between
cells.
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Warthogs and cockatoosstrategy 1
The answer to the problem is obtained by scanning down column 'B' to locate the cell where the total number of legs is
38 (row 11 in this example); the answer to the problem is then derived from the corresponding cell in columns 'A' i.e. there
are 5 warthogs.
Strategy 2
The solution shown In Figure 2 reflects an entirely different way of thinking about the same problem. As with solution 1,
column 'A' is Bled with the natural numbers from one to fourteen to represent the number of animals. In column 'B', each
animal is given two legs because each animal type represented in the group has at least two legs (thereby accounting for the
first 28 legs). This column can be filled manually or using a simple incrementing formula. The "leftover legs" are next
allocated two at a time in column 'C' thereby "creating" the four legged animals. Cell 'Cl7' contains a formula which
calculates and displays the running total of the legs allocated. When this total equals thirty eight, one simply counts the
number of animals which have an extra pair of legsthis will be the animal number read from column 'A' corresponding to
the last entry in column 'C'.
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311
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Figure 4.
Using a opreadsheet to estimate the roots of an equation
Using the graphical capability of the spreadsheet to plot the first iteration yields the graph shown in Figure 5. It is clear
from the Iteration table and reinforced by the graph that one root lies between "1" and "2" and the other lies between "-2"
and "-3".
-4
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Figure 5.
Graph of Y=X2 +X-4, first iteration
From the graph of the first iteration, one of the roots is estimated to be x=1 .5. If this root is required to halter
and
precision, a finer iteration can be tarried out. The graph of such an iteration is shown in Figure 6the start point
increment have been adjusted to zero In on the root.
343
"Recreating the Remlution"
III III II III
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
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1.1
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Figure 6.
Graph of Y=-X2 +X-4, second iteration
From the second iteration, this root can now be estimated to two decimal places (x=1.56).
Towards Winning the State Lottery
This example, which is more broadly based than the previous two, has been desdbed by Wright (1993). A student's
desire to be successful in games of chance could be exploited to evoke an interest in random numbers. A gene-al problem
which might be assigned would be to develop an automated method of picking four natural numbers between one and ten.
This problem, which lends itself well to a spreadsheet-based solution, can be approached with variing degrees of
sophistication according to the extent to which the solutions address the question of repeated number selection. Figure 7
shows two potential solutions.
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0
0
O
I I
1
1
1
a'
a
b
F
I
I
12
0
0
I
1
13
------
----
14
0
15
2 1 9
Nvnbors arI
7
16
4
i
I
17
I
...=IRSFS15=0.A130
InU101t.ndo)
=111A10=A11.1.0)
=Sum(PI 0.F131
..M ,,(A L0.D10)--
0
=11(A 1D=A I 2.1.0)
Figure 7.
Choosing four natural numbers between one and ten
The simpler of the two solutions is in cells Al
to 'A4'. All this solution does is to employ the spreadsheet's random
'
number generator to pick a number between one and tenthe prospect of selecting repeated numbers is not dealt with at
all. The more sophisticated solution (in the block of cells 'A8' to .G16') also does not avoid the selection of repeated
numbers but It does check for their presence and will not print the selection In the dark-bordered box unless the four
numbers are unique. Students may come up with one of many minor variations on the more sophisticated of the two
solutions. An even more sophisticated solution might employ the use of macros to deal with repeated number selection.
National Educational Computing Conference 1994, Boston, MA
344
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Discussion
This paper has described three diverse problem solving contexts within which the spreadsheet might be employed to
considerable advantage--many others could have been presented. It could be argued that the degree of spreadsheet
competency required in the case of strategy 3 for the warthogs and cockatoos problem supplants the complexity of the
traditional, analytical approach. It could be argued, however, that the spreadsheet approach offers two benefits, notably: that
It is more visual (less abstract) and therefore easier to relate to, and that it reflects an appropriate use of technology in an
information technology age.
In the "roots of equations" example, the spreadsheet can also be employed to great advantage by the teacher to
demonstrate various properties of equations. In so doing, the teacher is provided with a valuable opportunity to role model
the effective use of information technology (Wright, 1993).
While technology based tools such as Logo have been advocated for problem solvers at the elementary level
(e.g.Maddux, 1989), spreadsheets have a great deal of potential for use at both the junior and senior high school levels
Alhough not all problems will lend themselves to the use of the spreadsheet, such an approach does add a viable alternative
to existing problem solving strategies. Thr greater the number of strategies available, the more likely it is that differences in
learning style can be accommodated.
References
Maddux, C. D. (1989) Logo: Scientific Dedication Or Religious Fanaticism in the 1990's, Educational Technology, 29(2), pp.
18-23.
Sheingold, K. (1991) Restructuring for Learning with Technology: The Potential for Synergy, Phi Delta Kappan, 73(1), pp.
17-27.
-
of Information Technology for Teacher Education,
Wright, P. W. (1993) Teaching Teachers About.Computers, Journal
2(1), 37-52.
Technology and
Wright, P. W. (1993) Computer Education for Teachers: Advocacy is Admirable but Role Modeling Rules,
Teacher Education Annual,
374-380.
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