Problem-Solving Strategies in the Applied Mathematics Classroom
Deborah Roberts
November 4, 2003
Committee Members:
Dr. Richard Tondra, Major Professor
Dr. Irvin Hentzel
Sarah Lubienski
Table of Contents
Chapter One
Introduction
3
Chapter Two
Literature Review
8
Chapter Three
Modifications
20
Chapter Four
Analysis and Results
25
Chapter Five
Discussion
29
Appendix
32
Bibliography
43
2
CHAPTER ONE
INTRODUCTION
Ever since I began my teaching career, the question that arose most often
in my classroom was: “Where will I ever use this?” It has often been my
response that whether or not the student uses a particular skill, the logic and
problem-solving skills they are learning will have lifelong applications. Eventually
I began to take a closer look at my glib answer and to analyze just how serious I
was about my comments. Was I really providing my students with the skills that
would help them to be successful in the future? I began to examine more closely
how mathematics education may have changed over the past thirty years, and
how this relates to the general focus of my curriculum and teaching style.
Background
My first recollections of my own math education experiences are positive
ones. I remember races on the board to see who could do multi-digit
multiplication problems the fastest, or verbal quizzes where again speed was the
key. Since these things came easily for me, I continued to gain confidence in my
math abilities. In recent years, I find myself reflecting on the effect these types of
competitions had on the class as a whole. The same two or three students were
always vying for the championship, and I wonder how those who understood the
algorithmic process perfectly, but were unable to work at that top-level speed,
viewed these whole proceedings.
By the time I tackled algebra and geometry in high school, I found that my
mathematics courses took more effort than most of my other studies. However, I
3
was willing to work hard enough to get that ‘A’. I began to regard math as my
most challenging subject and envied those for whom it seemed to come more
‘naturally’. My last two years of high school, I had an instructor who would get
very animated when explaining a new concept, and I enjoyed his excitement in
the subject, but still felt somehow that I was missing a piece of the puzzle.
Although I enjoyed finding the pieces to make everything work out logically, I still
felt that I was barely grasping the basic essentials and was always on the verge
of being lost.
When I began to look at career choices, I found that education seemed
like a good fit for me. However, I didn’t even consider the field of mathematics—
after all, it was my hardest subject! Instead, I considered English, social
studies—there were so many interesting topics out there. Along the way, I was
required to take a mathematics class, College Algebra and Trigonometry (precalculus), as part of my class work. I was rather surprised to find myself intrigued
all over again. I enjoyed that class so much that I decided to tackle calculus, as
well as to consider picking up a minor in mathematics.
I had several lengthy discussions with my college professor about this,
and she gave me a piece of advice that I’ve always remembered. She said that
for most mathematicians, (those who aren’t incredibly naturally talented),
knowledge and problem-solving skills come as a result of being a ‘plodder’. This
idea began to simmer in my mind as I considered the implications. Was it
possible that I had been looking for memorizable skills and handy little algorithms
that I could apply at will rather than digging to the heart of problem-solving
4
concepts? If I had some of those ‘plodder’ tools, would I be able to tackle more
difficult concepts in ways that would allow me to break them down into something
more familiar or relate them to something I already knew? Did I possess the
skills to learn the language and logic of mathematics in a way that would enable
me to look at this as a viable career choice?
I began to pursue my mathematics studies in earnest and decided to
make this my major degree area. Calculus seemed fairly logical, as did discrete
mathematics and most of my other classes. There were a few, like abstract
algebra, that really taxed my self-confidence. Fortunately, it seemed like most of
the class was in the same quandary, and we found ourselves meeting almost
daily to discuss homework assignments or new concepts. I began to see a real
benefit in the sharing of ideas and approaches as we struggled with the more
difficult material.
About this time I took a course on teaching methods for mathematics. I
am embarrassed to admit that I was totally unfamiliar with the National Council of
Teachers of Mathematics or the National Math Standards. Suddenly, the
emphasis was not just on the curriculum itself, but also on the thinking processes
involved in attaining mathematical knowledge, as well as how students could
apply this knowledge to real-world applications! One of the basic principles of
the NCTM standards (p. 21) states: “Students must learn mathematics with
understanding, actively building new knowledge from experience and prior
knowledge.” How much more exciting it would be if students were confident that
they had the tools to not only master the skills presented to them in a math
5
classroom, but also to apply these same skills in their everyday life. This was
what my college professor was trying to explain to me about being a ‘plodder’. I
vowed to be the kind of teacher who challenged and inspired her students to new
levels of achievement.
Overview
I am currently in my tenth year of teaching high school mathematics and
science. Somewhere along the way, the real world has intruded and I find myself
weighed down with six different classes to prepare for, papers to grade, and
other life responsibilities all demanding attention. I have begun to take a closer
look at my teaching styles as well as the strengths and weaknesses of my
students, and am determined to prioritize a little better.
For several years now, I have taught classes in applied mathematics.
These students are for the most part non-college-bound individuals. Many go on
to attend vocational or technical schools, and some will probably enter the work
force directly after high school. I’m concerned that these students are often
under-achieving in their academic pursuits and have low motivation for scholastic
endeavors. Several of them are resourced for learning or behavior disabilities.
Short attention spans for more formal types of instruction are common and the
students tend to prefer active types of learning activities.
Problem
My curriculum includes many word problems that are applications of
vocational fields: business, agriculture, health/medicine, or mechanical. I began
to ask myself several questions:
6
What mathematics skills do high school students not interested in a fouryear college degree need before entering the workplace?
What skills are our local industries looking for in their employees?
Is the applied/practical mathematics curriculum we offer in Corning High
School successful in teaching these skills?
As I researched this topic, I began to see some consistent themes emerge
from interviews and studies on the issue. One of the concerns of many
employers was that new staff often lacks the skills to deal competently with
problem-solving situations that arise on the job. They felt that high school
mathematics preparation for job skills should provide students with the ability to
“use logic to draw conclusions from available information. Students must
problem solve, apply rules and principles to new situations, and use questioning,
inquiry, and the scientific process” (Day, 1997, p. 37).
I began to seriously assess my teaching curriculum and techniques to see
how I could find new ways to motivate my students while also enhancing their
problem-solving skills. This is a rather broad scope, and so I will not try to
encompass all of these aspects of my classroom in this paper. More specifically,
I have decided to focus on the issue of teaching problem-solving skills,
recognizing at the same time that part of the battle comes from sparking student
motivation.
7
CHAPTER TWO
Assessing the Need
Before deciding what types of curriculum changes to implement into the
classroom, it would be in order to assess more specifically the concerns and
needs of the business community. There seems to be a common consensus
from many sources that skill requirements in the labor market have escalated
much faster than the schools have improved (Murnane and Levy, 1997). The
problem is not so much that workers are less educated than they have been in
previous generations, but that the demands of the current workplace are more
sophisticated than in the past. To compete effectively today, American workers
must employ skills at a ninth-to-twelfth-grade level (Scott and Span, 1996). The
concern here is that there is increasing disparity between the skills students learn
in the classroom and those that are applicable on the job site. There is also
evidence to show that those jobs requiring a higher level of math skills also tend
to yield a higher financial return as opposed to salaries provided by lower-skilled
jobs. So what are the math skills that would better prepare our high school
students for success in their future career choices?
The SCANS report issued by the U.S. Department of Labor in June 1991
specifically dealt with the question of what skills they wanted schools to provide
their students with for success in the work place. This commission spent twelve
months talking to business owners, public employers, and the people who
manage employees daily in an effort to answer this question. Their first
assessment was that good jobs depend on people who can put knowledge to
8
work. New workers must “be creative and responsible problem solvers and have
the skills and attitudes on which employers can build” (SCANS, 1991, p. 3).
Traditional jobs are changing and new jobs are created everyday. High paying
but unskilled jobs are disappearing. Employers and employees share the belief
that all workplaces must "work smarter" (SCANS).
Although the SCANS report included the need for basic competencies in
reading, writing, arithmetic and mathematics, speaking, and listening, they also
reported the need for solid thinking skills. These skills include thinking creatively,
making decisions, solving problems, and reasoning. One of the problems, as the
writers of the SCANS report see it, is that employers have never clearly told
educators what students need to know and be able to do in order to succeed.
Therefore, this commission strove to put together a report that would define the
skills needed for employment, propose acceptable levels of proficiency as well as
ways to assess these proficiencies, and develop a strategy to implement these
skills into our nation’s school curriculum. The commission maintained that
SCANS research verifies that what we call workplace know-how defines effective
job performance today. This know-how has two elements: competencies and
foundation (SCANS).
The basic competencies that would be expected from any high school
degree program would include the ability to perform basic computations. The
problem is that students may have difficulty transferring the algorithms taught in
the classroom to real-world problems in the workplace. Employers expect
students to also have the ability to solve practical problems by choosing from a
9
variety of mathematical techniques. The concern arises as to whether schools
have focused too much on the algorithms, and not enough on the more
fundamental issues of quantities and their relationships (Bracey, 2001).
This leads to an assessment of thinking skills, and a look at whether
schools are equipping their graduates with the ability to combine ideas and
information in new ways in order to achieve goals. One of the thinking skills that
the SCANS report specifically targets is that of problem solving. The commission
suggested that students’ problem-solving skills needed to include “the ability to
recognize problems and devise and implement a plan of action to resolve the
problem” (SCANS, p. 35).
The National Council of Teachers of Mathematics has also targeted the
issue of problem solving as worthy of being included in their standards that were
published in 2000. They suggest that teachers should “enable all students to
apply and adapt a variety of appropriate strategies to solve problems while
monitoring and reflecting on the process of mathematical problem solving”. The
goal is to equip students with the knowledge and tools that enable them to
approach and solve problems beyond those they have studied in their curriculum.
In other words, teachers should strive to help students “develop a broad
repertoire of problem-solving (or heuristic) strategies” (NCTM Standards, 2000,
p.335).
Many researchers have concurred with the SCANS report’s stand on the
importance of problem solving in today’s workplace. Although a few will warn
against teaching problem-solving skills at the expense of a solid knowledge base
10
(Sweller, 1989), most commentaries support the agenda that the
mathematician’s main reason for existence is to solve problems (Halmos, 1980).
Lawson breaks problem solving strategies into three main categories:
task orientation strategies which influence the dispositional state of the student,
executive strategies which are concerned with planning and monitoring cognitive
activity, and domain-specific strategies with include heuristics and other
procedures developed for organizing and transferring knowledge (Lawson,
1989). Lawson maintains that there is evidence to suggest that training in the
use of different types of general problem solving strategies will positively affect
performance in both mathematics and other curriculum areas. He then goes on
to suggest that this training will also lead to a greater ability to transfer similar
problem solving strategies to other areas. In summary, Lawson advises that
general problem-solving strategies can have a powerful influence on an
individual’s success in a variety of situations as long as that individual is also in
possession of a well-organized knowledge base.
Problem-Solving Strategies as Related to Learning Taxonomies
Before developing problem-solving strategies for the classroom, it would
be advantageous to review some of the more popular theories on learning
taxonomies. An understanding of how students learn would provide an instructor
with a better foundation upon which to base an instructional problem-solving
strategy.
Piaget divided a child’s learning into stages of development, regarding the
student’s intellectual development as an outgrowth of their physical age. He
11
maintained that by the age of 11 to 15, a child was ready to begin the period of
formal operations. During this time, he determined that children begin “to reason
realistically about the future and to deal with abstractions…or ideas about
qualities and characteristics viewed apart from the objects that have them”
(World Book Encyclopedia, 1986). Since the age of students commonly found in
the high school classroom ranges from about 14 to 18, for the purposes of this
paper, we will assume the students have all reached the period of formal
operations.
Bloom’s taxonomy divides learning into three domains: cognitive, affective,
and psychomotor. The cognitive domain looks at an individual’s knowledge and
intellectual skill development, and is divided into six major categories. Bloom
maintained that each category must be mastered before an individual could
move on to the next level. These six categories include: knowledge,
comprehension, application, analysis, synthesis, and evaluation (Bloom, Mesia,
and Krathwohl, 1964). Many of the concerns put forth from employers is that
while workers may have the knowledge and comprehension of a concept, they
often fall short of the ability to apply this concept to a new situation, or even
more, to analyze the scenario, making inferences from the present situation to
use in future situations. This reinforces Lawson’s concern about transfer of
knowledge.
The SOLO taxonomy (Structure of Observed Learning Outcomes) is
somewhat similar to Bloom’s taxonomy. It was developed by Biggs and Collis in
1982 and is also based on the concept that students learn with advancing
12
degrees of complexity, dividing this learning framework into five stages. The
problem-solving competencies that we are concerned about would fall under the
last two stages. The first of these is the rational level, where students are able to
see the significance of the parts of their knowledge base as it relates to the entire
concept. The last stage is the extended abstract level. This is the most
important stage in terms of competencies in the modern workplace as it deals
with an ability to make connections and transference of previously learned ideas
and principles to new situations that arise (Atherton, 2002).
All of the aforementioned taxonomies have a common theme. It seems
obvious that the concerns expressed by the SCANS report could be narrowed
down to a concern that we are not preparing our students to enter the world of
work at the more advanced end of the learning spectrum. Rather than stopping
with rote memorization of common mathematical algorithms, we need to be
focusing on guiding students to the upper levels of learning, which allows for the
synthesis of ideas, and the transfer and application of a sound knowledge base
to new situations and problems.
Span and Overtoom-Corsmit performed an investigation to assess the
ways that gifted children solved mathematical problems as opposed to the
average student. Their conclusion was that the more talented students
immersed themselves in the details of the problem, using their analytical abilities
to form hypotheses and a systematic approach to finding the solution (Span &
Overtoom-Corsmit, 1986). Average students took a more random approach to
looking for possible solutions, and seemed to lack the tools to tackle the problem
13
in a higher-level manner. The question becomes whether there are specific
strategies than can be taught to students that will enable them to use some of
these higher-level skills that more advanced students already possess.
Problem-Solving Strategies
I began to study various approaches on implementing problem-solving
strategies into the classroom. Burton (1987) suggests the use of an
investigational, enquiry-based style of teaching. Even in my college methods
classes I recalled the emphasis on using open-ended questions and similar
discussion tools to generate higher-level thinking processes. What I was looking
for, however, was a comprehensive set of strategies designed to encourage
students to synthesize prior knowledge with their own creativity in order to allow
them to successfully find solutions to new problem situations. In other words, I
wanted to help provide my students with the tools and confidence to deal with
new situations that might occur in future real-world situations, whether in the
workplace or otherwise.
Lynn Steen addresses this question in an article in Educational
Leadership in which he discusses the necessity for a broader curriculum focus
that puts greater emphasis on patterns and relationships, spatial reasoning,
observation and conjectures, and genuine problems, rather than spending so
much time on more common algorithms like fractions, graphing by hand, or twocolumn proofs. He pushes for a larger picture of mathematics from kindergarten
through high school, with more experiences on the themes of “chance and
change, shape and dimension, and quantity and variable” (Steen, 1989, p.19).
14
Halmos suggests that the answer is in a more in-depth approach to curriculum
work, emphasizing the quality of the exploration over the quantity of material
covered (Halmos).
A change in curriculum focus would necessitate a change in teaching
practices. Steen suggests the use of more active learning processes that
incorporate real-life problem-solving situations presented with instructional
variety, as well as some means of assessing the success of the various
approaches (Steen). Susan Forman suggests that while real-life problems are
usually fairly concrete, they are not always as straightforward as one might think.
She also concurs that the use of this type of problem can challenge students to
use higher-level thinking skills while making connections between the
mathematics they’ve previously learned and the world around them (Forman,
1995-96). She reminds us that workers in what used to be called “blue-collar”
jobs are now expected to have the skills to use their own problem-solving
strategies while working in an environment that incorporates a high level of
technology dealing with large amounts of data.
Benander, Cavanaugh, and Rubenzahl developed a group of problemsolving activities for their classrooms at a community college in Massachusetts.
Their first focus was on developing a sense of cooperation and confidence
among the students in approaching problems. They used both group-building
exercises as well as initiative problems to accomplish these goals (Benander,
Cavanaugh, & Rubenzahl, 1990). These activities included games that involved
the students physically while at the same time developing reasoning skills. The
15
activities also included student self-evaluations of both their success and their
sense of confidence at the end of the activities. They reported that students
expressed a reduction in anxiety levels in approaching math problems, a higher
confidence level in their abilities, and a more positive attitude towards math
education.
Barba and Rubba approach the incorporation of problem-solving
strategies as a procedural task analysis whereby tasks or problems are broken
down into pieces. These pieces are then analyzed to determine which bits of
information are the most important and what the relationship is between them
(Barba & Rubba, 1992). They suggested the use of audio or videotape so that
the teacher can then analyze the student’s declarative, procedural, and structural
knowledge base. The teacher can then look for misconceptions or areas in the
student’s problem-solving skills that need remediation.
The Workplace Literacy Project funded by the U.S. Department of
Education specifically targeted lower-achieving math students. Rather than
concentrating on applied math problems only, however, this program “blended
conceptual approaches to mathematics with problem-solving exercises that were
anchored in the students’ world” (Woodward, 1999, p.75). One of the instructors
involved in the pilot project, Terry Wilson, focused on using common
technologies such as calculators and software programs to collect and analyze
real data, rather than pencil and paper skills more often seen in the mathematics
classroom. The students used simple calculators for all the fraction work, and
instead put their time into learning to use spreadsheets and computer graphics to
16
make pie charts and other pictorial representations to display their results. The
students then presented their results orally to the class, verbally expressing their
understanding of the problem and the results. Students were excited to
approach their math in a manner that they felt closely mirrored skills they would
use in the workplace later on.
Another discrepancy in approaches to problem-solving strategies is
whether we are teaching students particular rules or algorithms that they learn by
rote, or whether we encourage them to take commonsense approaches to
analyzing new problems. The challenge appears to be how to integrate these
two facets, so that students have a strategy in mind for where they are headed in
their problem-solving efforts, and then have the algorithms in their repertoire of
skills that they can use to finish the job. Booth suggests that if we teach students
how to switch back and forth between strategies, so that they are comfortable
implementing the most practical system for the problem rather than the most
basic or elementary strategy they know, we can help them increase their level of
cognitive functioning greatly (Booth, 1981). She maintains that many students
cling to their earliest problem-solving strategies basically because they are
comfortable with them and haven’t compared them to the efficiency of more
recent and advanced methods they now know. If we can use the students’ own
valid commonsense approaches and integrate them with more ‘mathematical’ or
logical approaches, we can effectively broaden their problem-solving capabilities.
Ruth Parker maintains that a good portion of the difficulties of students
(and later adults) in mathematics is that too often teachers view mathematics as
17
a “collection of abstract skills or concepts to be mastered” rather than regarding
mathematics as an understanding of patterns and relationships (Parker, 1991, p.
443). She suggests that we must make a clear connection between classroom
mathematics and real world mathematics. In other words, as the NCTM
standards state: students must see mathematics as “an integrated whole,
explore problems and describe results using graphical, numerical, physical, and
algebraic and verbal mathematical models or representations” (NCTM
Standards, p.84). Ms. Parker encourages teachers to develop classrooms where
students learn to work both cooperatively and independently on real-world based
problems, stressing that students should be assured that there are many
approaches to any problem rather than just one right strategy. In effect, the math
classroom should resemble the environment one might encounter in the
workplace.
Parker had several recommendations to implement these concepts into
the mathematics curriculum. First of all, she suggested presenting open-ended
problems or situations that would encourage the students’ independent
mathematical reasoning without leading them to a particular solution. Secondly,
she asked students (and collaborative groups) to share their ideas with others in
the class, verbalizing their thoughts and reasoning. Lastly, she encouraged
students to try more than one approach to the problem, and stressed that there
could be several correct ways of solving a problem (Parker).
Parker also had specific suggestions for staff development. She
recognized that it is difficult to change the goals in mathematics curriculum
18
without providing teachers with new methods to do so. She encouraged
professional development projects that would last from three to five years and be
available to all mathematics faculty. She also recommended that new
assessments be written that would allow teachers to evaluate the broader goals
of the current NCTM standards. According to the NCTM (1991, p. 1),
“Mathematical power includes the ability to explore, conjecture, and reason
logically; to solve non-routine problems; to communicate about and through
mathematics; and to connect ideas within mathematics and between
mathematics and other intellectual activity. Mathematical power also involves the
development of personal self-confidence and a disposition to seek, evaluate, and
use quantitative and spatial information in solving problems and in making
decisions.”
It would seem imperative, therefore, to implement specific strategies into
the curriculum that would encourage and develop these problem-solving skills.
Ina Miller suggests some techniques in an article for Lifelong Learning that
summarize many of the ideas presented previously by other authors. These
include acting out the problem, constructing a table, making a model, or drawing
a picture, looking for a pattern, making an organized list, working backwards, and
writing an equation to represent the problem (Miller, 1986). I decided to try some
of these specific strategies with the students in my classroom to see if I could
increase their level of comprehension and competency in working with problem
solving.
19
CHAPTER THREE
Classroom Strategies
The class that I had targeted for my research was my Applied
Mathematics I classroom. I chose to specifically strive to improve the students’
self-reliance and competency as problem-solvers. As mentioned in my
introduction, these students often lack self-confidence in their mathematics
abilities and are sometimes difficult to motivate. If the research I had studied
holds true, it seems of utmost importance to provide these students with both the
confidence and the skills to enable them to be successful in their future
workplaces. I decided to implement the changes over a period of two units of
study. I would then compare the results from the unit tests from this year’s class
to those from last year’s to see if there were measurable signs of improvement.
Unfortunately, my class sizes range from fifteen to twenty students, so it may be
difficult to call this a viable sample group.
The units that I focused on for my classroom implementations were
entitled “Working with Shapes in Two Dimensions” and “Working with Shapes in
Three Dimensions”. These materials are part of a CORD series for applied
mathematics. The first unit dealt with identifying common geometric figures and
calculating their perimeters and areas. The students also were challenged to find
solutions to work-related problems that involved these figures (Appendix A). The
second unit worked with geometric shapes in three dimensions: cylinders,
rectangular solids, cones, and spheres. They learned to calculate the surface
area and volume of these figures as well as solve problems involving them.
20
The strategies that I implemented into my classroom came in a couple of
different forms. The first strategy I used was to challenge the students to do
some problem-solving activities in an informal setting or as more of a game-type
activity. The purpose was to encourage them to use a variety of problem-solving
strategies in a non-threatening environment. The second approach I used was to
give them a list of strategies to help them break down problems into workable
pieces, and then to require them to demonstrate the steps that they followed
when they used these strategies as part of their required work.
While looking for activities to supplement into my curriculum, I came
across a text written by Dolan and Williamson entitled Teaching Problem-Solving
Strategies. It contained several fun activities that were designed to challenge
and strengthen students’ logic abilities with more informal activities. One of the
activities that I used was entitled “Selecting a Model” (Appendix B). This activity
involved using either drawings or actual physical objects to come up with models
for the situations presented. I gave this activity to my students individually, but
encouraged them to discuss options or ideas openly within their table groups.
Assessment for this activity was pass/fail. I wanted to see the results of their
brainstorming regardless of accuracy. I observed that the students were quite
involved and animated in their discussions and efforts on this activity, and
several times we stopped to discuss questions or approaches as a class.
The second activity I used from this text was called “Regular Polygons in a
Row”, and was designed to help students explore the concept of perimeter
(Appendix C). I had the students work in pairs on this activity to encourage
21
further team building, and again graded this activity on a pass/fail basis. I was
surprised by how intense the students got by the end of the activity, as they tried
to find the correct pattern that would give them an algorithm for finding the
perimeter of any number of polygons placed end to end.
Another adaptation I made to my curriculum was to write a list of
strategies on the board that I left up for the entire time that we spent on these two
units (several weeks). These strategies were designed to help the students
become comfortable with organizing their own logical thinking processes as new
problems or situations arose. These strategies included:
1)
2)
3)
4)
What does the problem want to know?
What information do you have?
Can you draw a picture to represent the problem?
What mathematical processes will you need to do to get where you
want to go?
5) Can you write a formula to present your problem?
6) Solve!!!
We spent quite a lot of time modeling these strategies by doing problems as a
class using this approach. I was amazed to realize that many times students
were off track because they got caught up in the language of the problem and
forgot to look at what exactly the problem was asking for. I encouraged them to
write down information as they encountered it, making a list of all data whether
they knew if it would be relevant to the solution or not. I also strongly
recommended a picture wherever it was applicable. When I stopped to help a
student with a question, I often responded first with: “Where’s your picture?”.
Then we coud begin a discussion of what they were trying to do.
22
We also spent a great deal of time discussing the nuances of language
and what that means in terms of mathematics. If they are to find the price per
square foot of carpet, what mathematical operation does that represent? If they
only sell one-third of their tickets, what does that mean in terms of numbers?
What clues will help them distinguish between perimeter, area, or volume
problems?
Once they had assessed what they needed to do, I asked them to write
down the formula or expression that they will use. Many in this class struggle
with pencil and paper mathematics, but I wanted to emphasize the problemsolving and logic aspect of the process rather than the ability to crunch numbers.
Therefore, I allow full use of calculators, while also spending time discussing how
to decide if an answer is reasonable. I encourage the students to look at the
calculator as a tool to supplement their own logic and skills. Every time they
reach a solution, I ask them to look at it and decide if it seems reasonable. If
lumber costs $3.30 per board foot, would it make sense that fifteen board feet
would cost $.22? Once the students got in the habit of checking their own work
over, it was amazing how many times I would hear them say: “Well, that can’t be
right!”, and back to the beginning they would go.
Another change I made was in the way I presented some of our lab
activities (Appendix D). I began to ask more specific strategy-oriented questions
on the lab papers that I handed out at the beginning of the lab and we would go
through the first few questions as a class. This seemed to help the students get
23
a better overall understanding of what the activity was about so that they could
develop their own plan of how they wanted to solve the problem.
Finally, I adjusted the way I assessed these activities. Although for my
final research analysis I went back to unit test scores for comparison, as we
engaged in some of these new activities I found myself using a more holistic
approach to assessing the students’ work. I found a text published by the
National Council of Teachers of Mathematics entitled How to Evaluate Progress
in Problem Solving that outlined some suggestions. I tended to use a
generalized holistic approach that was less time-consuming than some other
forms of assessment and took into account my considerable class load. The
holistic approach allowed me to take into consideration other aspects of the
students’ solutions than just the correct answer, while still being efficient enough
to be manageable on a day-to-day basis. Since I wasn’t using a more specific
rubric, I made it a point to make comments or suggestions on students’ papers
when applicable. I also revisited the more challenging problems in class
discussions, both during and after homework assignments.
Overall, I made a concerted effort during this time to conduct my
classroom in a manner that continually encouraged the students to be creative,
but yet logical, in their problem-solving approaches. I made a conscious effort to
use open-ended questions in our discussions, to encourage multiple approaches
to the same problem, and to provide an atmosphere where the students felt
comfortable sharing their thoughts and ideas.
24
CHAPTER FOUR
Assessing the Modifications
For a formal comparison of the effect of the modifications I implemented
into my curriculum for this quarter, I used only the unit test scores as compared
to test scores from the same unit from the class the year before. The second unit
was rather long, so in the past I have tested over this unit once mid-chapter, and
then a final time at completion of the unit.
I took a look at comparisons of both the test scores from the two units in
which I had made modifications, as well as the previous six unit tests (Figure 1).
In this manner, I could better ascertain whether class score discrepancies were
related to the differences in teaching methods or just different student abilities.
Class Test Average
Two-Year Comparison
100
90
80
70
60
50
40
30
'01-'02 Class
02-'03 Class
20
10
0
1
2
3
4&5
6
7
8A
8B
Unit
Figure 1
100
At first glance, it would appear that there was some improvement after the
80
60
40
20
0
modifications were implemented. Although the 2002-03 class also scored better
on the unit 6 test, which was before I had implemented the modifications, I was
encouraged by the fact that they consistently continued to score better than the
25
class from the year before on the remaining three unit tests, the ones after I’d
implemented the modifications. This class had scored slightly better than the
previous year’s class on the first four unit tests on the graph, but the discrepancy
between the two classes scores increased in the later units, and continued to rise
until the final unit test showed an improvement of over ten percent from the
previous year.
I decided to run the statistical test for two independent populations to see
if it would support my claim that the test scores had improved. I first needed to
perform an F-distribution test to see if the variances were equal. The
assumptions made for this test are that the populations are independent and
approximately normal. I charted my data for the previous year’s class as well as
the recent year’s class to see if they approximated a normal curve (Appendix E
and Appendix F). I thought the first year’s data was very close to a normal curve,
but I was somewhat concerned with whether the second chart could be
considered normal or if it was slightly skewed to the right. I did perform the equal
variance test on these sets of data. My calculations were as follows:
Ho: variances are equal (s12 = s22)
H1: variances are not equal (s12  s22)
N1 = 15
N2 = 15
x1 = 74.378
x2 = 81.933
s1 = 12.147
s2 = 11.932
F = s12 / s22 = 1.0364
26
In a two-tailed F test with 14 and 14 as my degrees of freedom and using
 = .05, the rejection region is F > 2.9829. Since my test statistic does not fall in
the critical region, I do not need to reject my hypothesis and I can consider my
variances equal. I went on to do a two-tailed t-test analysis. For this test my
data was as follows:
Ho: 1  2
H1: 1 < 2
sp2 = 144.961
t = -1.6602
With the two-tailed t-test with 28 degrees of freedom, I will reject my null
hypothesis if t < -2.048. Since my test value did not fall within the critical region, I
could not reject my null hypothesis and therefore could not statistically support
my assertation that my modified curriculum increased the test scores of my class.
Before I quit with the statistical analysis, I decided to also compare my
modified curriculum’s scores with those from the same class for the first part of
the year before I had implemented the modifications. I again ran an Fdistribution test to see if my variances were equal. My data was as follows:
Ho: variances are equal (s12 = s22)
H1: variances are not equal (s12  s22)
N1 = 15
N2 = 15
x1 = 77.2933
x2 = 81.933
s1 = 11.99862
s2 = 11.932
F = s12 / s2C = 1.0111
27
In a two-tailed F test with 14 and 14 as my degrees of freedom and using
 = .05, the rejection region is F > 2.9829. Since my test statistic does not fall in
the critical region, I do not need to reject my hypothesis and I can consider my
variances equal. I went on to do a two-tailed t-test analysis. For this test my
data was as follows:
Ho: 1  2
H1: 1 < 2
sp2 = 144.961
t = -1.0260
With the two-tailed t-test with 28 degrees of freedom, I will reject my null
hypothesis if t < -2.048. Since my test value did not fall within the critical region, I
could not reject my null hypothesis and therefore, again, I cannot prove that my
class showed improvement.
Due to the fact that I teach in a small school, my test sample was very
small. Even if the statistical analysis had been in favor of my hypothesis, it would
probably not be valid to draw too many conclusions from this. I turned then to
other observations to continue my assessment of the modifications I was
implementing.
28
CHAPTER FIVE
Further Discussion
There were several observations that I think bear mentioning. First of all,
the fact that the data from the scores after the curriculum modifications seemed
skewed slightly to the right indicated to me that there were fewer students in the
lower skills area. In other words, although the overall scores only showed a
slight improvement, the growth in the lower third of the class was more marked. I
would speculate that perhaps those that scored in the top third from the
beginning of the year already possessed competent problem-solving skills,
whereas those who struggled with these concepts reaped greater benefits.
Another observation I would like to point out is that I felt that the class as a
whole exhibited a greater willingness to participate in activities, classroom
discussions, or other challenges presented to them. They were more willing to
‘risk’. I felt they were gaining confidence that they had some tools with which to
break down situations into workable pieces, and did not let a fear of failure keep
them from tackling new situations. I tried to reinforce this behavior by positively
responding to all efforts and by giving credit for exhibiting sound problem-solving
logic rather than just a correct answer.
The modifications that I was making in my applied math classroom carried
over into other areas of my teaching. I also teach chemistry and physics, and I
found many times I would use the same problem-solving approaches to their
curriculum as I did to the applied mathematics. I soon came to the conclusion
that no matter what the level or topic of study, these same skills were important
29
for the students to find success. My upper classmen, juniors and seniors, were
often dealing with the same confidence issues as my freshmen and sophomores.
Many of them will soon be going to college or out into the workplace, and it is my
job to see that they have the skills to be successful.
One of the areas that I would like to explore further is the use of rubrics to
assess these problem-solving skills. I found several interesting approaches in
my research, and I plan to try other methods in addition to the generalized
holistic approach I used most often during my study. This past summer I helped
evaluate ICAM math tests that our area high school juniors had taken as part of
their state assessment requirements. The evaluation involved the use of an
analytical rubric that divided the problem into several phases and assigned points
according to the level of understanding the student exhibited in solving the
problem. I found I did not always agree with the rubric we were required to follow
in our assessment, and thought it would be interesting to devise my own for
some of my classroom testing to see if I could do better!
I find myself recalling one of those defining points in my teaching career
when I had a student who was helping his dad and uncle harvest come excitedly
to class one day. He shared with us that they were hauling the corn to the bins,
and were debating how many more bushels of corn they could get into the bin
where they were currently unloading. They knew about how many cubic feet it
took per bushel, but were estimating how much space was presently left. My
student had just finished studying cylinders and cones, and recognized the
shapes in the bin. He quickly pencil and papered out the correct volume for the
30
remaining space. He was quite proud of making the connection and actually
being able to use something he had learned, and of course, his teacher was
exceedingly proud of him also!
The point is this: We have to make math education applicable to daily life
and then insure that the students have the skills and confidence to transfer their
mathematics knowledge to these real-life situations. Carmel Schettino (2003) put
this very well in a recent article for the Mathematics Teacher journal. She holds
that “an educator must first commit to the premise that helping students develop
their ability to solve problems independently is the major goal of mathematics
education”. She maintains that this “entails teaching students that they have the
freedom to solve problems with a set of given tools and knowledgeable guidance
and that the goal is to further develop their mathematical toolkit”.
I completely concur with Ms. Schettino’s position. As the National Council
of Mathematics reminds us: “A major goal of high school mathematics is to equip
students with knowledge and tools that enable them to formulate, approach, and
solve problems beyond those that they have studied” (NCTM, p.335). This is the
same theme that appears in the Department of Labor’s SCANS report. My goals
as a mathematics instructor must include the desire to provide my students with
the skills and confidence to meet this challenge. This entire project has
broadened the scope of how I perceive myself as a mathematics/science
teacher. I would now venture to say that the curriculum itself is not the main
focus of my instruction, but rather the desire to provide my students with the skills
and assurance to approach future undertakings with confidence.
31
Appendix A
Area Problem Examples (from the CORD curriculum)
1) Mary trains for cross-country by running around the block 5 times each
evening. The block is a rectangle 500 feet long and 250 feet wide.
a) How far does she run when she runs ONCE around the block?
b) Find the total distance that she runs each evening.
c) Convert the distance she runs to miles. (5280 ft. = 1 mile)
2) A furniture factory builds rectangular tables that are 5 feet long and 2 feet 8
inches wide. They put oak trim all around the edge of the table. This month they
will build 85 tables, and the trim costs 56 cents per foot. How much will they
spend on trim this month? Ask if you need help understanding the steps of this
problem.
3) A biologist is studying insect populations in a field. In a test square 10 feet on
a side, he finds 30 greenbugs. The whole field is 450 feet long and 300 feet
wide. He wants to estimate the total number of greenbugs in the field. Follow
these steps:
a) Find the area of the test square.
b) Find the area of the field.
c) How many of the test squares would fit in the entire field?
d) Approximately how many greenbugs are in the field?
4) A gardener has a rectangular garden that is 120 feet long and 80 feet wide.
She is planting the whole garden with a cover crop of rye grass, and she is
supposed to use one pound of seed for every 1000 square feet. Find the area of
the garden, and then find how many pounds of seed she should use.
5) A map of Ms. Brown’s land shows a large pond (almost circular) inside a
rectangular field. She wants to plant the field to pasture, so she needs to know
the area of the field, not counting the area of the pond.
a) Measure, and use the scale given to find the lengths she needs to
know. Scale: 1 inch = 300 feet
b) Find the area of the field and pond.
c) Find out how many square feet of grass she will plant.
32
Volume Problem Examples (from the CORD curriculum)
1) A loading chute brings grain into a bin as shown in the drawing. The bin is a
cone. How many bushels of wheat will the bin hold? (There are 0.8 bushels per
cubic foot.)
2) A spherical water tank has a radius of 10 feet. It lets the water flow out of the
spigot at 2.5 ft.3/sec. How many minutes will it take to drain the tank?
3) A company has a shipment of 100 mirrors arriving in boxes which are 3 feet
long, 15 inches wide, and 6 inches tall. They want to store them in a space that
is 8 feet long, 6 feet wide, and 5 feet tall. They want to know if all the boxes will
fit. Follow these steps.
a) Change all the units of the dimensions of the box to feet, and find the
volume of one box.
b) Find the volume of all 100 boxes.
c) Find the volume of the storage space.
d) Answer clearly: will all the boxes fit in the storage space?
4) A construction company is pouring a sidewalk that is 120 yards long, 6 feet
wide, and 9 inches thick. Concrete will cost $35 per cubic yard. They want to
know how much all the concrete will cost. Follow these steps.
a) Change all the units to yards.
b) Find the volume of the sidewalk.
c) Find the cost of the concrete.
5) A backhoe has a bucket that holds about 5 cubic feet. The backhoe operator
knows that he can dig about 30 buckets an hour. He needs to dig a trench that is
400 feet long, 6 feet wide, and 2 ½ feet deep. He wants to estimate how long it
will take him to dig the trench.
a) How many cubic feet per hour does he dig if he digs 30 buckets per
hour at 5 cubic feet per bucket?
b) What is the volume of the trench?
c) How long will it take him to dig the trench?
33
Appendix B
34
35
36
Appendix C
37
38
39
Appendix D
Measuring Soft Drink Cans and Cartons
Problem:
How much wasted space is there in a 12-pack carton of pop?
Procedure:
What will you need to know to figure out this problem?
What formulas will you use?
Data:
Calculations:
Solution:
Critical Thinking:
Can you think of a design that would make better use of the carton’s
shape? (Draw a sketch to demonstrate.)
40
Appendix E
41
Appendix F
42
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