Introduction
This study is about eighth grade
students’ mathematical problem solving skills as a function
of their problem posing experiences. I propose that asking students to
pose problems on a regular basis will improve their ability to solve
mathematics problems, both curriculum-related problems and novel
problems that do not derive directly from the curriculum.
Background
& Setting
In my teaching practice, I
have tried various strategies to improve my students’ problem
solving skills. I have a permanent bulletin board with steps and
strategies prominently displayed. I have assigned the “word
problems” in the textbook for homework and put word problems
on tests and quizzes. I have conducted whole-class lessons on problem
solving strategies. During the 2004-5 school year, I happened across
the Math Trails concept, which involves students creating and solving
their own math problems based on their own home and school environment;
in other words, posing math problems. The National Math Trails website
explains this concept in detail (Skolnick-Palmese, 2002) . Further
reading in the area of problem posing caused me to begin using various
problem posing strategies in the classroom. These strategies seemed to
provide effective tools for assessing the students understanding of
math concepts, and were usually very motivating for the students. This
led me to ask whether using problem posing strategies with students
would lead to improved problem solving skills. Please see Appendix J for
some contemporary illustrations of students creating mathematics.
Problem
It is generally agreed that problem solving,
rather than rote memorization of facts and formulas, should be the
focus of contemporary mathematics education and various methods for
teaching and evaluating student problem solving have been developed
over the years (NCTM, 2000). In addition, the National Council of
Teachers of Mathematics (NCTM) has stated that problem posing
should be an essential component of the mathematics problem solving
curriculum (NCTM, 2000) . Posing problems can be as simple as having
students change one or more components of an existing problem, such as
one in a mathematics textbook. More complex activities may require
students to generate entirely new problems, either based around a
specific mathematical topic or based on exploration of the mathematics
in their own environment (Brown & Walter, 2005) .
Research Question
The study is framed by the research question,
“Will middle school mathematics students become better
problem solvers if they are asked to be mathematical problem
posers?”
Assumptions
I assumed that I would be able to measure
student problem solving ability in a valid and reliable manner by means
of a pre-test and post-test, and that the students in this study had
not been exposed to specific instruction in problem-posing strategies.
Limitations and
Delimitations
This study was limited to eighth grade students
at a suburban middle school in the southeastern United States. These
students are predominantly middle-class and represent a variety of
ethnic and racial groups. Nearly all are proficient English speakers,
although some are not native English speakers. All of the subject
students are enrolled in Algebra 1, at either the regular, honors or
gifted level. The study excludes eighth graders who are enrolled in
pre-algebra or who are in special education mathematics courses. This
study took place during the first nine-week grading period of the
2005-6 school year.
Definition of Terms
According to the NCTM, problem solving means
“engaging in a task for which the solution method is not
known in advance….Solving problems is not only a goal of
learning mathematics but also a major means of doing so”
(NCTM, 2000, p. 52) . George Pólya delineated four steps in
problem solving: understanding the problem, planning a solution,
carrying out the plan, and looking back, or reflecting on, the results
(Pólya, 1957) . Problem posing can be the modification of an
existing problem by changing the assumptions, the attributes, or the
information given, by changing the question that is asked, or it can
involve the creation of an entirely new problem (Brown &
Walter, 2005) .
Researcher’s
Perspective
I am a teacher with six years of experience in
teaching algebra to middle school students, and another twelve years of
experience as a computer teacher and school technology specialist. I
feel that I have some understanding of how students develop problem
solving skills and I have experience in helping students solve math
problems by using technology. I want to find ways to enable my math
students to become better problem solvers, and I feel that problem
posing is one way to do that. By doing a formal study on the
relationship between problem posing and problem solving, I was able to
explore my assumptions about that relationship: that middle school math
students will become better problem solvers if they are asked to be
mathematical problem posers.
Importance of the Study
This problem is worthy of research because both
problem solving and problem posing are important tasks for students
(NCTM, 2000) , but there is insufficient research available on the
connection between the two skills. Several authors have implied
that there is a correlation in their research ( Barnett, Sowder,
& Vox, 1980; English, 1997; Moses, Bjork, & Goldenberg,
1990; Silver, Mamona-Downs, Leung, & Kenney, 1996; Stoyanova,
2003) , but none of them has specifically demonstrated the connection.
Brown and Walter (2005) state that
“problem posing is deeply embedded in the activity of problem
solving” in that one cannot solve a problem
“without first reconstructing the task by posing
new problem(s) in the very process of solving” (p. 2) . They
see many interrelationships between the two processes, and state that
“problem solving may lead to problem posing” (p.
126), but do not seem to have explored the specific link studied in
this research, that practice with specific problem posing strategies
can lead to improved skills in problem solving.
Overview of the
Methodology
This research used quasi-experimental design
with my students and other teachers’ students as subjects.
They took pre-tests on problem solving and problem posing during the
first weeks of school. The experimental classes then engaged in various
problem-posing activities while the control groups were taught with
their teachers’ traditional methods, specifically excluding
problem-posing activities. At the end of the first nine-weeks grading
period, the students took post-tests on problem solving and problem
posing. In addition, selected students were ask to provide written
reflections about problem solving, problem posing and this research
study. The other participating teachers were interviewed about problem
solving strategies that they taught during the course of this study.
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Literature Review
Introduction
Most academic papers do not begin with Sunday
comic strips (Appendix
J), but Ruthie, of "One Big Happy," has been
engaging in some fairly revolutionary mathematics activities. In
addition to making up her own rather complex math problem, she has
created her own number, named “mofe,” and her own
day, Squeeze Day, which comes between Saturday and Sunday. These are
perfect examples of children as mathematicians, a concept that the
National Council of Teachers of Mathematics promotes in the latest
version of the Principles and Standards (National Council of Teachers
of Mathematics, 2000). What do mathematicians do? Instead of solving
problems that someone else has already solved, mathematicians try to
come up with problems that have not yet been solved (Goldenberg, 1993),
they “try to generate questions that no one else has
conceived of and work on problems that no one knows how to solve and
for which a solution method is not mapped out” (Manouchahri,
2001, p. 180). In other words, they pose problems, albeit usually more
sophisticated than Ruthie’s!
Problem posing, or having children
“regularly…formulate interesting problems based on
a wide variety of situations” (NCTM, 2000, p. 258) is
recommended within the problem solving strand of the Standards, but
problem-posing is not often a regular classroom activity. In this paper
I examine the past and current literature on problem posing, discuss
whether research has shown a connection between the two activities, and
briefly explore the relationship between current cognitive science and
problem posing. My question is, “do students become better
mathematical problem solvers if they are asked to be problem
posers?”
Background
Problem solving is an area where students at my
school have been weak on standardized tests. As a teacher, I have tried
to improve their problem solving skills by having students follow the
four steps originated by George Pólya (Pólya,
1945/1957) which are: understand the problem, plan a solution, carry
out your plan and look back to check your work, but with little
success. In the fall of 2004, as I was planning a classroom project, I
came across the Math Trails (Hendry, 2003) concept on the internet and
realized that this idea of students creating, rather than simply
solving, math problems might be worth exploring. This classroom project
led me to a deeper interest in the concept of student-created problems.
The National Math Trails website, http://www.nationalmathtrail.org
(Skolnick-Palmese, 2002), lists many student- and teacher-created Math
Trails, but had any formal research been done on the benefits of
students creating problems?
A search of online educational research sources
for “math trails” provided a wealth of documents.
Documents about math trails throughout the English-speaking world
(Dvorscak, 2001; McAllister, Mealer, Moyer, McDonald, and Peoples,
2003), about the process of teaching teachers to use the concept (Mock,
2001), about the standards underpinning the concept and step-by-step
instructions for a student-created a math trail (Cross, 1997), but no
research on the effects on student learning. An email to the webmaster
and project coordinator of the National Math Trails site elicited an
offer to “ask the mailing list” if any math trails
teachers had conducted formal research (R. Skolnick-Palmese, personal
communication, February 26, 2005), but no replies were received.
Subsequent GALILEO database searches for
“problem solving and mathematics” provided an
overwhelming number of documents and books. At this point, a trip to
the University of Georgia Science Library led me to The Art of Problem
Posing, (Brown & Walter, 1990) among the problem solving books
and I realized that “problem posing” was the
keyword phrase I was looking for. Further perusal of the stacks led me
to books on cognition and mathematics (Campbell, 2005; Sloboda
& Rogers, 1986); perhaps these could help me understand why
problem creation might help students become better problem solvers.
Using “problem posing” as my
search term, I gathered journal articles from a variety of
publications, many of which are published by the NCTM, and tend to
reflect that organization’s current perspective on the
teaching of math. I collected books from the University of Georgia
library, from my school’s professional development
collection, and through inter-library loan. I also purchased a copy of
the newly-released third edition of Brown and Walter (2005), as it was
not yet available at the libraries. A Google search on an
author’s name even led me to a relevant textbook on the beta
version of Google Print (English & Halford, 1995). I examined
over one hundred sources but I still did not find exactly what I was
looking for: evidence that having students pose problems makes them
better problem solvers.
Introduction: Problem
Posing and Problem Solving
Problem-posing is usually discussed in the
context of problem-solving; having students pose problems is often seen
as a sub-category of having students solve problems, and one hopes to
be able to solve the problems one poses.
The undisputed father of modern problem solving
was Hungarian-born mathematician George Pólya who first
published How to Solve it (1957) in 1945. This small but important book
sets forth the four general steps for solving mathematical problems
listed above (understand, plan, carry out, look back) and suggests a
modern heuristic, which “endeavors to understand the process
of solving problems, especially the mental
operations…” (p. 129). His clear writing style and
frequent examples make this book useful to classroom teachers while the
breadth and depth of his understanding of the problem solving process
serve to make this a frequently cited work in academic papers. He
suggests:
By looking back at the completed solution, by
reconsidering and reexamining the result and the path that led to it,
[students] could consolidate their knowledge and develop their ability
to solve problems.…no problem whatever is completely
exhausted.…we can always improve our understanding of the
solution. (Pólya, 1957, p. 15)
Looking back and reexamining a problem is one
short step away from reformulating the problem into a new problem, or
posing a problem (Silver, Mamona-Downs, Shukkwan, Leung, and Kenney,
1996).
In 1980, the NCTM declared that problem solving
would be the primary focus for mathematics instruction for the coming
decade, and produced a yearbook on the subject which included
Pólya’s heuristics on the front and back flyleaves
(Krulick, 1980; Osborne & Kasten, 1980). The articles in the
1980 yearbook offer many suggestions for improving students’
problem-solving skills (Branca, 1980; Butts, 1980; DeGuire, 1980;
Jacobson, Lester, and Stengel, 1980), but only mention student
problem-posing briefly (Barnett, Sowder, and Vox, 1980). The four pages
of Barnett’s article that discuss problem posing suggest that
students will be more motivated to solve problems that they find
“interesting and appealing,” (p. 92) and that they
can create some very interesting and appealing problems on their own.
His suggestions are simple to implement in the classroom: give students
a picture and have them tell a story and write a number sentence, give
students numerical data or a specific theme, perhaps related to another
subject area, and have them make up problems. Although
Barnett’s ideas were presented in a concise and very readable
form, and in a book that was widely distributed, it doesn’t
seem as though they took hold.
Stephen Brown and Marion Walter extended the
problem-solving concept to incorporate problem-posing, or having the
students create, or at least extend, mathematical problems (Brown
& Walter, 1983). They have been “heavily engaged in
efforts to place problem posing as a central theme in mathematics
education” since the 1960’s, teaching college
courses, publishing articles and speaking at conferences on the topic
(Brown & Walter, 1993, p. 12). A “What – If
– Not” approach is central to their problem-posing
scheme: take what is given, change one or more attributes and explore
the possibilities of the new problem. These explorations will probably
lead to more questions, or problems posed, leading to a cycling through
the steps. This scheme derives from Pólya’s
“look back” step; it asks that one solve a problem,
then look back and examine the results, then carry the problem a step
further to create a new problem; problem posing is seen as an extension
of problem solving, not as an entirely different process.
At the end of the decade, the NCTM published
Curriculum and Evaluation Standards for School Mathematics (NCTM,
1989), the first national-level standards document published by a
subject-area professional organization. These standards, and the
subsequent revisions, are generally accepted as the standards for
teaching mathematics in the United States, and are often cited by
foreign authors as well. The 1989 Standards state that problem solving
is “the process by which students experience the power and
usefulness of mathematics,” (p. 75) and that students in
elementary school should “formulate problems from everyday
and mathematical situations,” (p. 23) and that middle and
high school should be able to “formulate problems from
situations within and outside mathematics” (p. 75 &
137). To this national organization, problem posing is clearly an
important element of mathematics education.
The NCTM’s 1990 yearbook, Teaching and
Learning Mathematics in the 1990s (Cooney, 1990), focused on taking a
constructivist approach to teaching mathematics, on technology, and on
reaching minority students, but it did not have the problem-solving
emphasis of the 1980 yearbook. In it, Moses, Bjork and Goldenberg
(1990) offered simple strategies for incorporating problem posing into
the K-12 classroom in their chapter entitled “Beyond Problem
Solving: Problem Posing,” following Brown and
Walters’ concepts of modifying existing problems by reversing
the known and the unknown, by eliminating some of the constraints on
the problem, or by changing or expanding the domain of the problem.
Otherwise, problem posing is given scant attention in this
yearbook.Although the NCTM Standards (1989) clearly recommend that
students learn to formulate, or pose, problems, this
organization’s other literature from the same time frame does
not follow through with this concept.
In 1990, Brown and Walter issued a second
edition of what had become a classic in this rather narrow field of
problem-posing (1990). By 1993, they had collected enough literature on
problem-posing to issue a supplemental book, Problem Posing:
Reflections and Applications (Brown & Walter, 1993), with
articles and reflective essays gathered from journals as well as
otherwise unpublished material. In my opinion, this volume is not a
mature collection, but a series of somewhat disconnected pieces that
reflect the various authors’ early struggles with the idea of
students posing problems.
The 2005 third edition of Brown and Walter
expands the discussion on using technology as a problem solving tool,
on the “editorial board” approach described in the
earlier editions and on a “secular Talmud” approach
based on dialogue and storytelling that encourages multiple viewpoints
(Brown & Walter, 2005). In this edition, they also step away
from the problem posing process to reflect more deeply on how problem
posing affects our understanding of the world around us. This edition
also includes an interview with a professor who used the editorial
board approach with in-service teachers in 2003, and who commented that
most of these teachers “immediately saw the benefits [of K-12
students posing problems], but still questioned how they, as teachers,
could promote or even use the problem-posing process. Others worried
about how problem posing could really get them ‘off
track’ in terms of the required material they needed to
cover” (p. 152). This seems to me to be a key factor in the
lack of problem posing in the classroom; those who are aware of the
concept agree that it is worthwhile, but the time demands of a broad
curriculum and extensive standardized testing make in-depth exploration
of a topic difficult. So again, problem posing falls by the wayside
even though it is considered to be very worthwhile.
The 2000 edition of Principles and Standards
(NCTM, 2000) continues to emphasize problem solving and communication.
It also emphasizes algebraic thinking; suggesting that a student who
can understand problems, who can manipulate the knowns, unknowns and
constraints of a problem, and therefore more effectively pose problems,
will be capable of a greater understanding of algebra and the of the
abstract concepts of variables.
In all of this literature, the emphasis is on
moving toward a constructivist view of mathematics education. In a
discussion of radical constructivism and mathematics education, Patrick
Thompson (2004) said, “today, the idea that children
construct their own mathematics is taken for granted in mathematics
education research and in mathematics education at large”
[italics added] (p. 68). For the past fifteen years, the field of
mathematics education has been trying to move away from relying on rote
mastery of algorithms and solving the problems in the textbook to
encouraging students to think like mathematicians, to communicate
mathematically and especially to solve problems. Moses, Bjork and
Goldenberg asked,
Why is problem posing good for us? An
orientation toward posing new problems can be said to be the very heart
of learning mathematics. Learning is a creative act: we learn not by
absorbing but by constructing our knowledge. And we learn mathematics
particularly well when we are actively engaged in creating not only the
solution strategies but the problems that demand them. (Moses et al.,
1990, p. 90)
Mathematics teachers are still grappling with
this; very few mathematics classrooms have students regularly posing
problems, although there are plenty of suggestions on how to create
this inquiry-based, constructivist problem-posing classroom
environments and step-by-step instructions for helping your students
become better problem posers (Bush and Fiala, 1993; English, 1997b;
English, Cudmore, and Tilley, 1998; Gonzales, 1998; Manouchahri, 2001;
Whitin, 2004). Various authors have suggested that having students pose
problems, by creating new ones or by reformulating or modifying
existing ones, can help students understand the structure of math
problems (English, 1997a; Stoyanova, 2003). If one takes the Piagetian
view “that the basic processes through which the structures
underlying formal operations develop are the same as those that
underlie the ability to think mathematically” (Stoyanova,
2003, p. 33), then one could assume that practice with problem-posing
might lead students to become better problem solvers, as I hypothesize,
but this hypothesis does not seem to have been demonstrated
conclusively in the research literature.
Research
An early relevant study was conducted by Lauren
Resnick with French middle school age students (Resnick,
Cauzinille-Marmeche, and Mathieu, 1986). Given a number sentence, such
as 17 – 11 – 4, the students were asked to make up
a story for the given expression The difficulties shown by the students
clarified what Resnick calls “malrules,” or
improper interpretations of the rules of algebra, and brought to light
the fact that “the task of composing stories was so
novel” to these children that their lack of understanding of
the task interfered with the actual purpose of the study (p. 199).
Edward Silver studied problem posing with middle
school students and with pre- and in-service teachers a decade later
(Silver & Cai, 1996; Silver et al., 1996). In his study with
over 500 low-income urban 6th and 7th graders, he found “the
relationship between their problem-solving performance and their
problem posing was examined to reveal that good problem solvers
generated more mathematical problems and more complex problems than
poor problem solvers did”(p. 521). This does not say that
posing problems made them better problem solvers, only that better
problem solvers were able to pose better problems. In the coda to one
study, Silver notes the “largely uncharted wilderness of
mathematical problem posing” and suggests that
“much more research is needed to develop a deeper
understanding of this and related forms of generative cognitive
activity” (Silver et al., 1996, p. 308).
Australian Lyn English has published studies of problem posing with
elementary school students (English, 1997a, 1998), as well as articles
in American NCTM journals directed to K-12 mathematics teachers
(English, 1997b; English et al., 1998). In the 1998 study, English
studied student with average to high number sense and novel
problem-solving ability, asking them to pose problems in both informal
and formal contexts as a pre-test. The test subjects then participated
in a twice-weekly problem-posing program. On the post-test, the subject
children did show significant improvement in ability to generate their
own problems, but their problem-solving ability was not analyzed in
light of their newly acquired problem-posing ability. In the 1997
study, English notes the lack of research on this relationship, saying:
Despite its contribution to
children’s mathematical development, problem posing has not
received the attention it warrants from the mathematics education
community. We know comparatively little about children’s
ability to create their own problems in both numerical and
non-numerical context, and the extent to which these abilities are
linked to their competence in other domains such as number sense or
novel problem solving. (English, 1997a, p. 184)
In another article, English states that
“our classroom activities have also shown how problem-posing
activities can … improve students’ problem-solving
abilities” (English, Cudmore & Tilley, 1998, p. 125),
but the research she cites does not seem to actually make that direct
correlation. English also notes that the Realistic Mathematics
Education curriculum adopted by the Netherlands as the national
curriculum, “places a strong emphasis on students’
problem posing in all mathematics activities” (1997a, p. 184)
and would be a good source of problem posing activities. A recent
Mathematics Teacher feature article provides a detailed picture of
these Dutch reform efforts that have been in place for over a decade
and seem to be working well, even in urban, highly transient
foreign-born populations (Case, 2005).
A study with college algebra students found
benefits of “an enhancement of student reasoning and
reflection and a heightened level of engagement,” but, again,
no substantive evidence of an improvement in problem solving skills
(Cunningham, 2004, p. 83). This brief article describes one specific
problem-posing activity in detail, and discusses the implications, but
it does not describe a formal research study.
Towards the end of my research, I found Elena
Stoyanova's work (2003). She notes that at a 1993 Psychology of
Mathematics conference, Silver “observed that the most
frequently cited motivation for curricular and instructional interest
in problem posing is its perceived potential as a way for assisting
student to become better problem solvers” (p. 33). But even
his later research does not directly address this question.
Stoyanova’s own 1998 doctoral thesis at Edith Cowan
University (Australia), cited in the above-mentioned article and
entitled, “Extending and Exploring Students’
Problem Solving via Problem Posing,” reported that
“students exposed to problem-posing and problem-solving
activities outperformed students exposed only to problem-solving
activities” (p. 33), but in this article she does not specify
the types of mathematical activities on which the subjects outperformed
the control group.
Technology and problem
posing
How does technology fit into problem posing?
Almost as soon as computers were introduced in schools, the capability
of exploring “what if?” scenarios became apparent
with spreadsheets, student-written programs and mathematical
exploration software such as Green Globs and Graphing Equations,
Geometric Supposer, Geometer’s Sketchpad and Cabri Geometry.
The LOGO programming language allowed even very young children to be
computer programmers. Software designed around the microworlds model
allows students to explore various mathematical concepts such as
manipulation of variables, vectors and coordinate planes within a
closed environment (Moses et al., 1990; Schwartz, 1994).
Today’s graphing calculators and other technologies only make
this easier to implement in the classroom. Websites such as the Texas
Instruments Activity Exchange and Geometer’s
Sketchpad®: Sketchpad Links offer hundreds of activities and
explorations, most created by teachers, showing that using calculators
and computers for problem solving, not just for arithmetic and word
processing, is enthusiastically embraced by teachers and students
(Texas Instruments, 2005; KCP Technologies, 2005). In order to
encourage schools to purchase and use graphing calculators, Texas
Instruments commissioned a study that synthesized the findings of more
than 40 peer-reviewed research articles; they concluded
“students with access to handheld graphing technology engaged
in problem solving and investigations more often and were more flexible
in their solution strategies than students without access”
(Burrill, 2002). In my opinion, as schools move more and more toward
one-on-one computing models, with individual laptop computers rather
than classroom sets of graphing calculators, the possibilities for
technology-enhanced student problem posing expand exponentially.
Digital photography and video further expand students’
abilities to explore and examine the problems they create (Traylor
& Osteen, 2004).
Cognitive Science and
Problem Posing
How can current literature in cognitive science
help us to understand the problem posing and problem solving processes?
In the Handbook of Mathematical Cognition (Campbell, 2005), James Dixon
discusses the idea that our experiences are “contextually
bound” but the fact that mathematical representations are
“completely portable and abstract” (Dixon, 2005, p.
379) seems to contradict the basic premise that knowledge is
constructed from our experiences. He argues that mathematical problem
solving incorporates schemas that develop as an individual solves
problems and accumulates a body of knowledge, and that mathematical
operations are also represented by the problems the individual has
already solved with that operation (Dixon, 2005). This would imply that
posing problems and therefore adding to one’s accumulation of
knowledge might help one develop more sophisticated problem solving
skills, but there is no proof of this.
From another perspective, Lakoff and
Nuñez argue that mathematics does not exist outside of the
human brain; that one cannot prove that a Platonic mathematics exists
any more than one can prove the existence of God. They are concerned
not with single aspects of mathematics, but with the very fundamental
question of what is mathematics and what cognitive mechanisms help us
organize and make use of mathematical ideas (Lakoff &
Nuñez, 2000, 2005). I will need to explore their work more
deeply before I can draw my own conclusions.
Alan Schoenfeld addressed the broad topic of
Mathematical Thinking and Problem Solving by publishing the results of
a 1990 conference which brought cognitive scientists, mathematicians,
math educators and classroom teachers together for perhaps the first
time (Schoenfeld, 1994). Judah Schwartz, professor at MIT and Harvard,
and designer of early microworlds math software, commented on
Schoenfeld’s discussion of problem-solving courses:
When I listen to your description of your
problem-solving courses…it seems to me that they are at
least in part mislabeled. That is, because they are, in some
substantial measure, problem-posing courses. They are courses that help
people learn how to pose problems. This in some sense is harder, and
it’s certainly a different task. (Henkin & Schwartz,
1994, p. 73)
Again, the relationship between problem solving
and problem posing is brought out, but not fully clarified.
From what I can gather, cognitive scientists are
still working on the field of problem solving, but most agree that
constructing problems is a worthwhile activity for helping students
understand the structure of problems and, on a larger scale, the
overall structure of that abstract construct we call mathematics.
Summary
In the process of researching and creating this paper, I have traveled
from the concrete world of student-created Math Trails to conjectures
about the inner workings of the human brain. I have discovered and read
a substantial body of work on problem solving, ranging over a
fifty-year period and encompassing several major shifts in math
instructional programs. I have read a significant body of work on
problem posing, including works describing how to implement it in
classrooms ranging from elementary school to university-level, but
nothing I have read specifically addressed my question, “do
students become better mathematical problem solvers if they are asked
to be problem posers?” Individuals and professional groups
such as the NCTM have recommended that problem posing be an important
component of mathematics instruction, but the latest Georgia
Performance Standards require only that students use a variety of
methods for solving problems, and be able to communicate
mathematically. No mention is made of problem posing (Georgia
Department of Education, 2005). Many authors make an implied or
explicit link between the two functions, but none of them offer
substantive evidence that this link exists in the direction leading
from problem-posing to problem-solving, although some have shown the
converse: that good problem solvers make good problem posers. I hope
that my own future research will be able to clearly show that adding
regular problem posing activities to the middle school mathematics
classroom will lead to an increase in problem solving ability. Again
quoting Pólya, a teacher who “challenges the
curiosity of his students by setting them problems proportionate to
their knowledge, and helps them to solve their problems with
stimulating questions, he may give them a taste for, and some means of,
independent thinking” (1957, p. v), and isn’t
“independent thinking” what education is all about?
Methods
This study asks whether eighth grade
students’ mathematical problem solving skills can be improved
if they are exposed to a variety of problem posing experiences. Many
authors have examined problem solving and problem posing, but few if
any have specifically addressed whether having students pose problems
will improve their problem solving skills. I specifically addressed
this question with my study.
Design
To determine whether asking students to pose
problems leads to an improvement in their problem solving skills, I
used a quasi-experimental design with eighth grade students from a
suburban middle school in the southeastern United States as subjects.
All of the students in the experimental group are students in my
Algebra 1 classes; the control group included students from one of my
classes as well as students from two other math teachers’
classes. All of the students took pre-tests on problem solving and
problem posing during the first two weeks of school. The experimental
classes engaged in various problem-posing activities within the context
of the algebra curriculum, while the control groups were taught the
algebra curriculum with each teacher’s usual methods,
specifically excluding problem posing activities. At the end of the
first nine-weeks grading period, all subject students took post-tests
on problem solving and problem posing. In addition, selected students
were asked to reflect in writing about problem solving, problem posing
and this research study. The other participating teachers were
interviewed about problem solving strategies that they taught during
the course of this study.
Participants
The participants were drawn from three
teachers’ eighth grade Algebra 1 classes. The table below
gives the numbers of participants in each group.
|
Experimental
Groups
|
Control
Groups
|
|
My gifted algebra
class: 16 students (T1)
|
Ms. Al’s
gifted class: 16 students (A1)
|
|
My honors algebra
class: 16 students (T4)
|
Ms. Ap’s
honors class: 12 students (A4)
|
|
One of my regular
algebra classes:
14 students (T1)
|
My other regular class: 14 students (T3)
Ms. Ap’s regular class: 10 students (A3)
|
|
Total
46 students
|
Total
52 students
|
Table 1, Distribution
of study participants
For the sake of brevity, I refer to each class
by its teacher’s initial and class period, as shown above.
All of my classes are preceded with “T,” while the
other two teachers’ classes are preceded with
“A.” All of the “A” classes and
T3 are control groups; all other “T” classes are
experimental groups.
In this school district, algebra is the
standard eighth grade mathematics course, and is available at three
levels: regular, honors and gifted. The honors and gifted classes
follow the same curriculum and use the same textbook; the regular
classes have a less rigorous curriculum and use a different textbook.
Students are placed in the gifted classes if they have met the state
and school district’s requirements for receiving gifted
education services, or if, in the opinion of the teacher, the student
is capable of doing gifted-level work. Students are placed in honors
classes if they achieve a minimum score on an algebra placement test
and have the recommendation of their seventh grade mathematics teacher.
All other students who earned a passing grade in seventh grade
pre-algebra and who passed the mathematics portion of the seventh grade
Gateway test are placed in regular algebra. Students who are repeating
the eighth grade may be placed in regular or honors classes according
to the recommendation of their previous year’s teacher.
Students who do not meet these requirements are placed in pre-algebra;
none of the pre-algebra students will be included in this study, nor
will any students who are in special education mathematics classes. The
student and parent consent form is attached as Appendix A.
The school in this study is one of twenty middle schools in a large
suburban school district. The school district serves more than 142,000
students in Kindergarten through 12 th grade (Gwinnett County Public
Schools, 2005). This school serves approximately 1,700 middle class
students in grades six through eight. The demographic distribution is
illustrated in Figure 1 (State of Georgia, 2004 and 2005) .
Figure
1. School ethnic composition since 2001,
with free and reduced meal recipient data overlaid.
Over the past four years, this
school’s African-American population has increased while the
White population has correspondingly decreased. The Asian and Hispanic
populations have grown very slowly. The percentage of students with
limited English proficiency is small; in 2004-5, only 2.2% of the
students received ESOL services. In 2004-05, 12.2% of the
school’s population qualified for gifted education services;
this percentage has been declining in recent years. The percentage of
students receiving free and reduced meals is also shown in Figure 1
(State of Georgia, 2005). Although this percentage is relatively high,
the school is not in a high-poverty area. Most of the students live in
single-family houses in suburban cul-de-sac neighborhoods that have
been developed within the past 20 years. Some students live in
apartments or mobile home communities and a few live in an
extended-stay motel.
Data Sources and
Instruments
Data on student problem solving ability was
collected from several sources, including standardized test data, pre-
and post-tests as described below as well as written reflections.
I used the participants’ scores on
the Problem Solving subtest of the Mathematics CRCT (Criterion
Referenced Competency Test), a statewide standardized achievement test
given the previous April. I computed a correlation coefficient between
the pre-test scores and the students’ spring 2005 CRCT scores
on the problem solving strand to attempt to determine the validity of
the pre-test as a measure of student problem solving ability.
For this study, I have created three different pre-tests and three
different post-tests, which are attached as Appendices B, C, D, E, F,
and G. Due to time constraints, these tests were not pilot-tested.
Each student will take an individual pre-test
which, for the regular algebra students, consists of five open-response
questions adapted from the state’s CRCT review website (The
Riverside Publishing Company, 2005). Using CRCT-type questions will
allow me to validate this test against the students’ seventh
grade CRCT scores (see Appendix
B). The honors and gifted students will take a three-question
test with questions adapted from a problem solving curriculum (Cohen,
1991, see Appendix
C). My past experience has shown that most
honors and gifted students score extremely well on the CRCT, so their
pre-test questions are more difficult, providing more room for
improvement. At the conclusion of the study, each student took an
individual post-test with slightly different questions than the
pre-test (see Appendices D and E).
The students also participated in a group problem solving pre-test in
their classroom. Students worked in small groups to solve three
problems adapted from Cohen (1991) and to pose two problems based on
information provided. This test was the same test for all categories of
students. Collaboration was encouraged, both within and between groups;
I computed an aggregate class score for this activity in order to
determine each class’s overall problem solving and problem
posing ability. At the conclusion of the study, the students
participated in a group post-test with slightly different questions
than the group pre-test (see Appendices F and G).
The pre- and post-tests were scored using a
rubric (see Appendix
H). As I scored the tests I developed more
specific criteria for each scoring point based on commonalities among
the students’ answers. The problem posing questions were
scored on a 5-point scale as follows:
| 0=Problem
not attempted |
1=Problem
attempted, does not match requirements |
2=Problem
attempted, partially meets requirements |
3=Problem
complete, almost meets requirements |
4=Problem
complete, fully meets requirements |
5=Problem
complete, exceeds requirements |
At the conclusion of the study, selected
students were asked to reflect on problem solving, problem posing and
this research study, in writing.
Procedures
During the first nine-week grading period, the
experimental classes were asked to pose problems in various situations,
including during class lessons and discussions, in their homework
assignments, and as assessments, in place of more traditional test and
quiz questions. The control students that I teach were given similar
tasks requiring problem solving, but not problem posing. The other
participating teachers taught their classes using their usual methods,
but specifically refrained from using problem-posing activities for the
duration of this study.
The experimental students were asked to pose
problems related to the algebra curriculum that they are studying at
the time. For their first problem posing activity, students were asked
simply to create a word problem using one of the words representing the
four basic operations that we had gathered during class that day. A few
weeks later, they were asked to create a problem involving positive and
negative numbers, and to delineate the four problem solving steps (See,
Plan, Solve, Check). After learning how to solve equations with
variables on both sides, they were asked to create a problem that could
be solved with an equation of the form Ax + B = Cx + D, to solve their
equation and to demonstrate the accuracy of their solution by making a
table of values comparing the left and right sides of the equation.
In-class problems were done informally, using
personal whiteboards and dry erase markers, for quick teacher review,
and formally, as written journal entries, and evaluated using a
four-point scale, where an “E” was and Excellent or
Exemplary paper, “A” was Acceptable,
“R” needs Repair or Revision and
“N” is Not acceptable. Papers with grades of R or N
could be revised and resubmitted. This was based on a scoring system
developed by Stutzman and Race (Stutzman & Race, 2004).
Sometimes, the student who posed the problem was asked to solve that
problem, and sometimes they exchanged boards and solved another
student’s problem. Collaboration was encouraged for in-class
problem posing. Many of these activities were spontaneous, created as
the students gained a level of understanding about a topic, and as
classroom time was available. The more formal journal activities and
test/quiz problems were generally be planned in advance. All of the
pre- and post-tests were given in the regular classroom setting.
Role of the researcher
In this study, I was an active part of the
study. I was the algebra teacher for all of the experimental groups and
for one of the control groups. I scored all of the pre- and post-tests
to ensure consistency of scoring.
I have successfully used the problem posing
strategies outlined above in previous years, but had not previously
done any formal analysis to determine if they improve student learning.
My informal observation found that students were highly motivated by
in-class, informal problem posing, and that these activities allowed me
to assess the students’ level of understanding quickly and
easily. Gifted students did very well with more formal problem posing
activities done in written math journals. Students of all ability
levels enjoyed solving problems created by their peers; in past years I
have even given a final exam composed solely of student-created
questions.
Analysis
I compared pre- and post-test scores, looking
for significant differences between the means of the various
experimental and control groups. The experimental classes at each level
(regular, honors, gifted) were compared to the corresponding control
classes. In addition, the regular algebra class that I teach was
compared to the regular class taught by Ms. Ap. I analyzed the
individual and the group data using means, medians, standard deviations
and Student’s t-test. I computed correlation coefficients
between the student’s CRCT scores and their individual
pre-test scores.
The student’s written reflections, which were done
anonymously within the classes, were examined for insights into the
numeric data. Interview data from the other teachers will provide
context for the student responses to the problem solving questions.
Summary
In this study, I hoped to demonstrate a
cause-and-effect relationship between asking students to pose
mathematical problems on a regular basis and these students’
problem solving skills. Using a quasi-experimental design with
pre-tests, post-tests and a control group, I hoped to be able to draw
valid conclusions. As with any study of educational systems, I was
unable to conclusively demonstrate a cause-and-effect relationship, but
I hoped to be able to shed some light on the relationship between the
two processes.
return to top
Results and Discussion
Introduction
As I observed my students grapple with posing
and solving math problems, and as I read their journals and their
written discussions of their math problems, I learned quite a bit about
how well, or poorly, they understand the mathematical concepts we try
to teach. For example, when asked to create a “distance
equals rate times time” (d = rt) problem, I realized that
many of them could not distinguish between the rate and the distance.
They did not have a clear idea that a rate of speed is the distance
traveled divided by the time spent traveling and they did not always
realize when they needed to convert units of measurement, such as hours
to minutes. I assumed that they had mastered these skills in earlier
grades, but it seems they only learned them well enough to pass the
chapter test, not well enough to use in future math problems, let alone
in real life. I also found that many of them could pose very creative,
interesting problems, even though most had seldom if ever been asked to
create a math problem on their own. However, they could not always
solve the problems they created.
Based on the quantitative results, this study
did not conclusively answer the research question. This may be due to
the treatment’s lack of influence on the students, the lack
of validity or reliability of the pre- and post-tests, or other factors
beyond the scope of this research.
Comparison of pre- and
post-test scores
The pre- and post-test scores for each student
do not support a conclusive answer to the research question . On the
individual pre- and post-tests, the regular algebra control
groups’ mean scores increased while the experimental
group’s mean scores declined (Figure 2); this is the opposite
of what I expected to find. The mean scores for all the honors and
gifted groups all increased, but the control groups increased more than
the experimental groups, again, not what I had hoped. T-test results
are shown in Table 2. These would indicate that there is no significant
difference between groups T1 and T3, and between groups T2 and A1, but
that there is a significant difference between the honors groups, T4
and A4, and two of the regular groups, T1 and A3.
On the group problem solving task, nearly all
groups’ scores declined; the exception was the gifted control
group, which increased (Figure 3). The problem posing scores also
declined from pre- to post-test (Figure 4), but the students’
problem posing activities in class show definite improvement in problem
posing skills, as evidenced by examination of their problem posing
journals, which contradicts the test scores. The honors algebra control
class (A4 Honors Control) either did not complete the group pre-test or
their papers were misplaced. They did complete the group post-test, but
without pre-test scores, the post-test scores have little meaning, so
they are not reported here.
Figure 2.
Mean scores on the individual pre- and post-tests.
|
T1 Regular Algebra Experimental
|
. |
|
T3 Regular Algebra Control
|
t-test T1 and T3= 0.015
|
|
A3 Regular Algebra Control
|
t-test T1 and A3= 0.155
|
|
.
|
.
|
|
T4 Honors Experimental
|
. |
|
A4 Honors Control
|
t-test T4 and A4= 0.645
|
|
.
|
.
|
|
T2 Gifted Experimental
|
. |
|
A1 Gifted Control
|
t-test T2 and A1= 0.013
|
Table 2.
t-test results on the gain/loss scores for each pair of classes, alpha
= .05.
Figure 3.
Mean scores for all classes on the group problem solving pre- and
post-tests.
(A4 Honors Control class did not participate in this activity).
Figure 4
. Mean scores for all classes on the group problem posing
pre- and post-tests.
(A4 Honors Control class did not participate in this activity).
Table 3 and Figure 5 show the range of
individual student percentage gains or losses between the pre-test and
the post-test scores. All except the T1 Regular Experimental classes
had ranges between 23 and 31 percentage points; the T1 class was
extreme with a range of 53 points and the only decrease in mean and
median scores. The gifted and honors experimental groups, T2 and T4,
showed the lowest gains, followed closely by the honors control group,
A4, while the A3, T3 and A4 control groups showed the largest gains.
|
Individual % Gain/Loss summary
|
n
|
Mean
|
Median
|
Std Dev
|
Range
|
|
T1 Regular Exp
|
14
|
-2.50%
|
-2.50%
|
13.04%
|
53
|
|
T3 Regular Control
|
14
|
5.50%
|
10.63%
|
8.74%
|
28
|
|
A3 Regular Control
|
10
|
7.29%
|
5.63%
|
10.63%
|
38
|
|
T4 Honors Exp
|
16
|
2.60%
|
2.08%
|
9.14%
|
31
|
|
A4 Honors Control
|
12
|
4.17%
|
2.08%
|
8.24%
|
29
|
|
T2 Gifted Exp
|
16
|
0.13%
|
0.00%
|
6.43%
|
27
|
|
A1 Gifted Control
|
16
|
6.38%
|
8.33%
|
6.91%
|
23
|
|
Table 3.
Range of student gain or loss between individual pre- and post-test,
with standard deviation and range.
Figure 5.
Comparison of mean and median gain or loss between individual pre- and
post-test.
return to top
Test Scoring
The tests were scored using the attached rubric
(Appendix H).
Each test problem was awarded zero to four points based in each of four
categories. For example, in scoring the pre-test question,
“Jan could walk 4 miles in 30 minutes. How long will it take
her to walk 6 miles?” a student who set up a proportion, or
demonstrated how they used multiplication and division to solve the
problem, was awarded three points for “problem solving
strategies.” If the student included a variable in the
proportion, or wrote another type of equation that included a variable,
they were awarded 4 points. If they carried out their plan and arrived
at a solution, they were awarded three points in the “problem
solving steps” category; if they clearly showed that they
checked their work, for accuracy and/or reasonableness, they earned
four points. Surprisingly, very few students showed their checks, even
though most of them had just completed a chapter on solving equations
in which checking your work was required, and checking answers for
reasonableness had been discussed extensively in class. A totally
correct solution was worth four points in the “correct
solution” category; a solution that included a minor
arithmetic error was worth three points, solutions with major logic or
computational errors were worth one or two points. If the student
showed the computations used and simply stated the answer, they were
awarded two points for “work shown, solution
explained.” To earn four points in this category, they needed
to explain how or why they did the given computations in the context of
the problem. Simply stating that “I divided 30 by 4, then
multiplied by 6” was worth only three points; stating
“I divided 30 minutes by 4 miles to find out the rate in
minutes per mile, then I multiplied by 6 miles,” was worth
the maximum four points. Even incorrect solutions could earn maximum
points in this category if the student clearly explained how they
arrived at their solution. They would, however, lose points in the
“correct solution” category.
Qualitative Results
The research question, which asked if middle
school mathematics students become better problem solvers if they are
asked to be mathematical problem posers, was not supported by the
quantitative results of this research. In fact, pre- and post-test
scores indicate just the opposite: that problem posing practice leads
to lower scores on problem solving tasks with regular algebra students,
and only slightly increased scores with honors and gifted students.
Examination of the student’s problem
posing journals shows growth in their problem posing skills, in the
quality and complexity of the problems posed and in the number of
problem solving steps shown and explained. For example, the first
problems posed by students tended to be one-sentence basic arithmetic
problems. The students were unsure how to check their work, other than
with a calculator. By mid-September, one student was able to create the
following problem using addition of positive and negative numbers:
“I once had 5 ducks. Each duck had 2
ducklings. 5 ducklings died. Then 3 ducks had 4 ducklings each and 2
ducks had 3 ducklings each. 15 ducklings died. 5 ducks had 5 ducklings
each. 2 ducklings died. 2 ducklings had 15 ducklings each. How many
ducks do I have altogether?” He then correctly wrote and
solved the mathematical expression and checked his work for accuracy
and for reasonableness.
Another student’s problem involved
shopping and debt:
“On your credit card, you have $100.
You want to buy a pair of pleather boots costing $35, a silk shirt that
is $20, two cotton shirts costing $15 each, and a pair of leather
gloves that is $15. You also need to buy some groceries which will
total up to $53. How far in debt are you?” She also correctly
solved and checked the problem.
In the T2 gifted class, students were asked to
create problems that could be solved with an equation like ax
+ b = cx + d. They created problems involving renting
everything from cars to cats, with a “membership
fee” and daily rental rates, as well as travel problems like
this:
“Libby and Melanie are each driving
758 miles from Atlanta, Georgia to Tampa, Florida. Libby leaves first,
driving 104 miles per hour. She is 64 miles from Atlanta when Melanie
leaves. She is driving 120 miles per hour. How far do each of them
drive before they are driving next to each other.”
These students were asked to provide a verbal
model similar to one in their textbook and solve and check their work.
Informal observations show that these students were very willing to
create problems if they could share them with their peers, and perhaps
challenge their peers to solve a difficult problem.
When students were asked to reflect about their
participation in this research project through anonymously written
responses, the responses ranged from “problem solving is
lame” and “boring” to “fun and
exciting” and “they get your brain
pumping.” When categorized as positive, negative or neutral
responses, about one-third of the responses fell into each category.
When asked why they felt the group scores declined, the only concrete
answers were that the students knew it didn’t count for a
grade, so they did not put their best effort into it.
The two other teachers whose students
participated in the study were interviewed at the study’s
conclusion. The interview questions are attached as Appendix I. Ms. Ap (A3
Regular Control and A4 Honors Control classes) stated that she does
teach specific problem solving strategies, including working backwards,
drawing a picture or diagram, creating a graph or table, working a
simpler problem and guess-and-check. Samples of her students’
problem solving work are displayed in the hall outside her classroom.
She has successfully used these strategies in the past. When asked
whether her students enjoy problem solving activities, her answer was,
“only if food is involved!” Ms. Al’s
class of gifted students (A4 Gifted Control class) solves challenging
multi-step problems at the end of each chapter, but she does not teach
specific problem-solving strategies. This approach seems to work for
these students, though, as their scores on the group post-test showed a
mean gain of over eight points, and this class was the only class that
showed a gain. The group pre- and post-test problems were similar to
the problems she assigns to her students.
In summary, the quantitative results of this
study show some students problem solving test scores increasing and
some decreasing, in both group and individual settings. On a written
problem posing test, all groups of students earned lower scores on the
post-test than on the pre-test. The qualitative results were also
mixed, showing improvement in the students’ problem posing
journals, but mixed reactions to problem solving as a classroom
activity.
return to top
Discussion
There is extensive literature on problem
solving, beginning with Pólya in 1945 (Pólya,
1957), and continuing through the 1980’s, a decade that the
NCTM declared would be dedicated to teaching problem solving (Krulick,
1980). The idea of K-12 students posing problems is also introduced
(Barnett, Sowder, and Vox, 1980) . By 1989, the NCTM was suggesting
that students of all grade levels formulate problems (NCTM, 1989), but
this suggestion has not been widely incorporated into state mathematics
curricula. In the newest Georgia Performance Standards for mathematics,
middle school students are only asked to formulate questions or pose
problems during the 6 th and 7 th grade units on data analysis and
probability (Georgia Department of Education, 2005). Brown and Walter
(1983, 1990, 1993, 2005) are the most active proponents of problem
posing, although they focus on college mathematics courses.
Problem-posing is usually discussed in the
context of problem-solving; having students pose
problems is often seen as a sub-category of having students solve
problems, and one hopes to be able to solve the problems one poses. In
my teaching practice I have found that students enjoy creating their
own math problems, and I had hoped that this study would show a
positive correlation between having students pose mathematics problems
on a regular basis and their ability to solve problems.
Having read Resnick’s study of French
middle schoolers (Resnick, Cauzinille-Marmeche, and Mathieu, 1986) , I
expected that my students would need some help in creating their first
problems due to their unfamiliarity with the concept. By their third
attempt, my students knew what was expected of them, and most were able
to pose math problems as requested.
As Silver found in the 1990’s, the
students in this study who were better problem solvers were also better
problem posers ( Silver & Cai, 1996; Silver et al., 1996) . In
this study, the two gifted classes scored best in both areas. Yes,
these students are in gifted classes because they
have higher scores in mathematical skills, including problem solving,
but the correlation between problem solving and problem posing skills
can be seen from the group activity data, where all groups completed
the same problems.
Silver (1996) , English ( 1997a) and Stoyanova
(2003) have all called for more research into the relationship between
problem posing and problem solving. I hoped to add to this body of
knowledge, but my research has been inconclusive.
Educational research is always subject to
variables outside the control of the researcher. When dealing with
human subjects, especially 13-year olds for whom many non-academic
factors strongly influence student achievement, it is difficult to say
with certainty that the treatment provided did or did not cause the
effect that was observed and recorded.
When I attempted to assess the validity of the
pre-test instrument, I found that the correlation computed between
students’ CRCT scores and the pre-test scores was not strong.
The regular algebra classes’ correlation was .42; for the
honors and gifted classes, only .12. The difference between the two
groups’ scores can probably be explained by the fact that the
regular algebra pre-test questions (Appendix B) were derived from
questions in the CRCT practice test bank (Riverside Publishing Company,
2005), while the honors/gifted questions (Appendix C) were more
challenging questions derived from another source (Cohen, 1991). The
CRCT questions are all multiple choice and scoring is based solely on
the correctness of the answer and the number of questions completed,
whereas the pre-test questions were open ended and scoring was based on
problem solving strategies used and demonstrated, not just the
correctness of the answer. This might explain the relatively low
correlation of .42 between the two sets of scores.
What caused the quantitative results of this
study to appear opposite of what was predicted? It is my opinion that
the students’ actual ability to solve mathematical problems
was not accurately reflected in their pre- and post-test results. At
the start of a new school year, when these students participated in the
pre-test, students are eager to learn and eager to please their
teachers. By the end of the first marking period, when they completed
the post-test, they have figured out what activities really count
toward their grades and are not willing to put much effort into
something that may not “count,” or that they
consider boring. When the students were asked why they
thought scores had decreased, these are the two factors that were most
often cited. Test fatigue may also factor into the unexpected scores.
These students spent nearly two weeks in late September taking the
CogAT (Cognitive Abilities Test) and the ITBS (Iowa Test of Basic
Skills), and are inundated with pre- and post-tests in all of their
academic subjects. By the time they took this study’s
post-test in October, they had very little interest in a test that
“didn’t count.” Examination of their CRCT
scores next spring, when the scores do “count,” as
they must pass the CRCT to earn promotion to the ninth grade, may show
a more significant difference.
When examining the regular algebra
students’ individual pre- and post-test scores, it is clear
that many students’ skills and strategies improved in solving
the types of problems presented, at least in the T3 and A3 control
classrooms. Why did the T1 experimental class’ scores
decline? From my observations as the teacher in that classroom I would
attribute the decline to attitudinal issuesa lack of effort on the test
rather than to the problem posing or problem solving activities
completed during the study. This particular class of students has
developed into an unruly, overly social group. They profess to like
working in groups, but they do not meet the goals set for the groups.
Even when given a second class period to complete their group
post-tests, (they had only 20 minutes available on the first day), only
a few groups added to their rather incomplete answers, even when they
were encouraged to look at the problem solving steps illustrated on the
bulletin board. Several groups did not even attempt some of the
problems although they were happy to spend the allotted 15 minutes
“working” with their group. Of the 14 students in
this group, only four showed an increase in scores, one remained
constant and nine declined. In the A3 class, only three students had
declining scores; the other seven gained points. In the T3 class, only
two students declined, two remained constant and 10 improved their
scores.
In my opinion, the group activity scores
declined for most groups due to a lack of interest in the activity, as
discussed above. The A1 gifted control improved in problem solving,
perhaps due to that teacher’s frequent use of challenging,
multi-step problems in her classroom.
return to top
Conclusions
What conclusions can be drawn from this data?
With regard to the relationship between problem posing and problem
solving, this data did not add significantly to the literature. This
experimental design and conditions did not conclusively demonstrate any
relationship between the two factors studied.
A longer-term study of integrating
problem-posing into the mathematics curriculum might give different
results; the NCTM recommendations for including problem posing and
problem solving are certainly not intended to be limited to one quarter
of one school year. If I continue to have my experimental group
students engage in problem posing on a regular basis, a comparison of
next spring’s CRCT scores might show a significant
difference. If so, this would lend support for the research question.
In an effort to take a comprehensive picture of
the relationship between problem solving and problem posing, this study
was perhaps too broad, including three different levels of student
ability and three different types of activities. Perhaps narrowing the
focus to one ability level, but across a wider span of classes, would
yield more conclusive results. Due to student schedule changes, some of
the students who participated in the pre-tests were no longer in the
study classes at the time of the post-test so the composition of the
groups for the group activities was not constant. This should not have
been significant, as only a class summary score was generated for the
group activities, but perhaps it made a difference.
Significance and
recommendations
This study has minimal educational significance,
due to the inconclusiveness of the results. It has, however,
illustrated the need for more specific instruction in problem solving
strategies if our students are to become proficient problem solvers.
Further research might lend more support to my hypothesis. This
study could be repeated with fewer variations between the pre- and
post-tests, and using the same pre-and post-tests for all groups. A
more controlled testing environment, with specific time limits, would
also provide better data. Although I had hoped the group problem
solving activities would provide additional support for the results
obtained with the individual student activities, this was not the case.
While the mean scores on the individual activity generally increased,
the mean scores on the group activities generally declined. More
controls over the group activities, such as time limits and stricter
controls over the composition of the groups, might change this.
A
longer term study might also give more conclusive results. If the
problem posing curriculum were continued throughout the year, and CRCT
scores compared from the previous year to the current year, rather than
use the pre- and post-tests, the data would more accurately reflect
student progress toward the original goal of improving CRCT scores in
problem solving.
In summary, this study attempted to determine
if students who are asked to pose mathematical problems on a regular
basis would become better problem solvers. The results were
inconclusive, due perhaps to the design of the study and to factors in
the affective domain that were not controlled by the study. Further
research is needed if this relationship is to be conclusively
demonstrated.
return to top
References
Barnett, J. C., Sowder, L., & Vox, K.
E. (1980). Textbook problems: Supplementing and understanding them. In
S. Krulick (Ed.), Problem solving in school mathematics
(pp. 92-103). Reston, VA: National Council of Teachers of Mathematics.
Branca, N. A. (1980). Problem solving as a
goal, process and basic skill. In S. Krulick (Ed.), Problem
solving in school mathematics (pp. 3-8). Reston, VA:
National Council of Teachers of Mathematics.
Brown, S. I., & Walter, M. I. (1983).
The art of problem posing (1st
ed.). Hillsdale, N.J.: Lawrence Erlbaum Associates.
Brown, S. I., & Walter, M. I. (1990).
The art of problem posing (2nd
ed.). Hillsdale, N.J.: Lawrence Erlbaum Associates.
Brown, S. I., & Walter, M. I. (2005).
The art of problem posing (3rd
ed.). Mahwah, N.J.: Lawrence Erlbaum Associates.
Brown, S. I., & Walter, M. I. (Eds.).
(1993). Problem posing: Reflections and applications.
Hillsdale, N.J.: Lawrence Erlbaum Associates.
Burrill, G. et al. (2002). Handheld graphing
technology in secondary mathematics: Research findings and implications
for classroom practice. Retrieved May 2, 2005 from http://education.ti.com/downloads/pdf/us/CL2872.pdf
Bush, W. S., & Fiala, A. (1993).
Problem stories: A new twist on problem posing. In S. I. Brown
& M. I. Walter (Eds.), Problem posing: Reflections
and applications (pp. 167-173).Hillsdale, N.J.: Lawrence
Erlbaum Associates.
Butts, T. (1980). Posing problems properly.
In S. Krulick (Ed.), Problem solving in school mathematics
(pp. 23-33). Reston, VA: National Council of Teachers of Mathematics.
Campbell, J. I. D. (2005). Handbook
of mathematical cognition. New York, NY: Psychology Press.
Case, R. W. (2005). The Dutch revolution in
secondary school mathematics. Mathematics Teacher, 98(6),
374-384.
Cohen, S. R. (1991). Figure it out:
Thinking like a math problem solver (Vol. 4 & 5).
North Billerica, MA: Curriculum Associates, Inc.
Cooney, T. J. (Ed.). (1990). Teaching
and learning mathematics in the 1990s. Reston, VA:National
Council of Teachers of Mathematics.
Cross, R. (1997). Developing maths trails. Mathematics
Teaching (158), 38-39.
Cunningham, R. F. (2004). Problem posing: An
opportunity for increasing student responsibility. Mathematics
and Computer Education, 38(1), 83-89. Retrieved March 1,
2005 from WilsonWeb database.
DeGuire, L. (1980). Pólya visits
the classroom. In S. Krulick (Ed.), Problem solving in
school mathematics (pp. 70-79). Reston, VA: National Council
of Teachers of Mathematics.
Detorie, R. (2005, March 27). One big happy.
Atlanta Journal Constitution. Retrieved April 3, 2005 from http://www.creators.com/comics_show.cfm?next=6&ComicName=obh
Detorie, R. (2005, March 20). One big happy.
Atlanta Journal Constitution. Retrieved April 3, 2005 from http://www.creators.com/comics_show.cfm?next=6&ComicName=obh
Dixon, J. A. (2005). Mathematical problem
solving: The roles of exemplar, schema and relational representations.
In J. I. D. Campbell (Ed.), Handbook of mathematical
cognition (pp. 379-395). New York, NY: Psychology Press.
Dvorscak, B. (2001). Virtual math
trail. Retrieved January 18, 2005 from http://www.iit.edu/~dvorber/lesson1.htm
English, L. D. (1997a). The development of
fifth-grade children's problem-posing abilities. Educational
Studies in Mathematics, 34, 183-217. Retrieved March 1, 2005
from WilsonWeb database.
English, L. D. (1997b). Promoting a
problem-posing classroom. Teaching Children Mathematics, 4,
172-179. Retrieved March 1, 2005 from WilsonWeb database.
English, L. D. (1998). Children's problem
posing within formal and informal contexts. Journal for
Research in Mathematics Education, 29, 83-106. Retrieved
March 1, 2005 from WilsonWeb database.
English, L. D., Cudmore, D., &
Tilley, D. (1998). Problem posing and critiquing: How it can happen in
your classroom. Mathematics Teaching in the Middle School, 4(2),
124-129. Retrieved March 1, 2005 from WilsonWeb database.
English, L. D., & Halford, G. S.
(1995). Mathematics education: Models and processes.
Mahwah, NJ: Lawrence Erlbaum Associates. Retrieved April 2,2005 from http://www.google.com/print
Georgia Department of Education. (2005). Georgia
performance standards: Mathematics. Retrieved May 2, 2005
from http://www.georgiastandards.org/math.asp
Goldenberg, E. P. (1993). On building
curriculum materials that foster problem posing. In S. I. Brown
& M. I. Walter (Eds.), Problem posing: Reflections
and applications (pp. 31-38). Hillsdale, N.J.: Lawrence
Erlbaum Associates.
Gonzales, N. A. (1998). A blueprint for
problem posing. School Science and Mathematics, 98(8),
448-456. Retrieved April 1, 2005 from WilsonWeb database.
Gwinnett County Public Schools (2005). About
us. Retrieved November 13, 2005 from http://www.gwinnett.k12.ga.us/gcps-mainweb01.nsf/pages/Home-AboutUs1~MainPage
Hendry, D. R. S. (2003). Apple
learning interchange: Teaching practice: The national math trail.
Retrieved September 12, 2004 from http://www.ali.apple.com
Henkin, L., & Schwartz, J. L. (1994).
A discussion of Alan Schoenfeld's chapter. In A. H. Schoenfeld (Ed.), Mathematical
thinking and problem solving (pp. 71-76). Hillsdale, N.J.:
L. Erlbaum Associates.
Jacobson, M., Lester, F., & Stengel,
A. (1980). Making problem solving come alive in the intermediate
grades. In S. Krulick (Ed.), Problem solving in school
mathematics (pp. 127-135). Reston, VA: National Council of
Teachers of Mathematics.
KCP Technologies. (2005). The
geometer's sketchpad®: Sketchpad links. Retrieved
May 2, 2005 from http://www.keypress.com/sketchpad/general_resources/links.php#classact
Krulick, S. (Ed.). (1980). Problem
solving in school mathematics. Reston, VA: National Council
of Teachers of Mathematics.
Lakoff, G., & Nuñez, R. E.
(2000). Where mathematics comes from. New York:
Basic Books.
Lakoff, G., & Nuñez, R. E.
(2005). The cognitive foundations of mathematics: The role of
conceptual metaphor. In J. I. D. Campbell (Ed.), Handbook of
mathematical cognition (pp. 109-124). New York, NY:
Psychology Press.
Latterell, C. M. (2003, January). NCTM-oriented
versus traditional problem solving skills. Paper presented
at the Joint Mathematics Meeting, Baltimore, MD. (Eric Document
Reproduction Service No. ED 474450)
Manouchahri, A. (2001). A four-point
instructional model. Teaching Children Mathematics,
8(3), 180-185.
McAllister, D. A., Mealer, A., Moyer, P. S.,
McDonald, S. A., & Peoples, J. B. (2003). Chattanooga
math trail: Community mathematics modules: University of
Tennessee at Chattanooga. (ERIC Document Reproduction Service No.
ED478915)
Mock, J. (2001). Math trails. Teaching
Pre K - 8, 31(4), 50-54.
Moses, B. M., Bjork, E., &
Goldenberg, E. P. (1990). Beyond problem solving: Problem posing. In T.
J. Cooney (Ed.), Teaching and learning mathematics in the
1990s (pp. 82-91). Reston, VA: National Council of Teachers
of Mathematics.
National Council of Teachers of Mathematics.
(1989). Curriculum and evaluation standards for school
mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics.
(2000). Principles and standards for school mathematics.
Reston, VA: Author.
Osborne, A., & Kasten, M. B. (1980).
Opinions about problem solving in the curriculum for the 1980's: A
report. In S. Krulick (Ed.), Problem solving in school
mathematics. Reston, VA: National Council of Teachers of
Mathematics.
Pólya, G. (1957). How
to solve it; a new aspect of mathematical method (2d ed.).
Garden City, N.Y.: Doubleday.
Resnick, L. B., Cauzinille-Marmeche, E.,
& Mathieu, J. (1986). Understanding algebra. In J. A. Sloboda
& D. Rogers (Eds.), Cognitive processes in
mathematics (pp. 169-203). Oxford: Clarendon Press.
The Riverside Publishing Company. (2005). CRCT
online. Retrieved June 24, 2005 from https://regiond.georgiacrct.org/servlet/a2l
Schoenfeld, A. H. (Ed.). (1994). Mathematical
thinking and problem solving. Hillsdale, N.J.: L. Erlbaum
Associates.
Schwartz, J. L. (1994). The role of research
in reforming mathematics education: A different approach. In A. H.
Schoenfeld (Ed.), Mathematical thinking and problem solving
(pp. xix, 339). Hillsdale, N.J.: L. Erlbaum Associates.
Silver, E. A., & Cai, J. (1996). An
analysis of arithmetic problem posing by middle school students. Journal
for Research in Mathematics Education, 27, 521-539.
Retrieved March 1, 2005 from Jstor database.
Silver, E. A., Mamona-Downs, J., Leung, S.
S., & Kenney, P. A. (1996). Posing mathematical problems: An
exploratory study. Journal for Research in Mathematics
Education, 27(3), 293-309. Retrieved March 1, 2005 from
Jstor database.
Skolnick-Palmese, R. (2002). The
national math trail. Retrieved March 9, 2005 from http://www.nationalmathtrail.org
Sloboda, J. A., & Rogers, D. (Eds.).
(1986). Cognitive processes in mathematics.
Oxford: Clarendon Press.
Stoyanova, E. (2003). Extending students'
understanding of mathematics via problem-posing. Australian
Mathematics Teacher, 59(2), 32-40. Retrieved March 27, 2005
from http://search.epnet.com
State of Georgia. (2004). 2003-2004
Annual report cards on K-12 public schools. Retrieved June
24, 2005 from http://reportcard.gaosa.org/
State of Georgia. (2005). 2004-2005
Annual report cards on K-12 public schools. Retrieved
November 13, 2005 fromhttp://reportcard2005.gaosa.org/
Stutzman, R. Y. & Race, K. H. (2004).
EMRF: Everyday rubric grading. Mathematics Teacher, 97(1),
34-39.
Texas Instruments. (2005). Activities
exchange. Retrieved May 2, 2005 from http://education.ti.com/educationportal/activityexchange/activity_list.do?cid=us
Thompson, P. W. (2004). Perspective on
"radical constructivism and mathematics education". In T. P. Carpenter,
John A. Dossey, Julie L. Koehler. (Ed.), Classics in
mathematics education research (pp. 68). Reston, VA: The
National Council of Teachers of Mathematics.
Traylor, K. & Osteen, J. (2004). Digital
video for problem solving/decision making. Retrieved May 3,
2005 from http://techintegration.editme.com/digitalvideo
Whitin, P. (2004). Promoting problem-posing
explorations. Teaching Children Mathematics, 11(4),
180-186.
return to top
Appendices
Appendix
A
Consent Forms
Research
Study - Parental Permission Form
(to be completed either by the parents/legal guardians of minor
students)
I agree to allow my child,
________________________________________, to take part in a research
study titled, “Effect of Problem Posing on Eighth Grade
Students’ Mathematical Problem Solving Skills,”
which is being conducted by Mrs. Kathleen Traylor, an 8 th grade
Algebra teacher at Shiloh Middle School who is also enrolled in a
graduate degree program in the Instructional Technology Department at
the University of Georgia (770-736-4321). This study is being conducted
under the direction of Dr. Michael Orey, Department of Instructional
Technology, University of Georgia, (706-542-4030).
I do not have to allow my child to be in this
study if I do not want to. My child can stop taking part at any time
without giving any reason, and without penalty. I can ask to have the
information related to my child returned to me, removed from the
research records, or destroyed.
The reason for the study is to find out if
having students create their own math problems will improve their skill
in solving other math problems.
- The research is not expected to cause any
harm or discomfort. My child can quit at any time. My child’s
grade will not be affected if my child decides to stop taking part in
the research.
- The study will be conducted for nine weeks
during regularly scheduled class.
- Students will take two pre-tests about
problem solving and problem posing, one by themselves and one as a
member of a group. Students in some classes will be asked to create
math problems as part of their normal class lessons, in homework
assignments and on tests and quizzes. At the end of the nine weeks,
they will take two post-tests, again, one individual and one group,
about problem solving and problem posing. These are the
“experimental” classes.
- Students in the
“control” classes will take the same pre- and
post-tests, but they will be taught using the teacher’s usual
teaching methods, specifically not including problem posing.
- Any information collected about my child
will be held confidential unless otherwise required by law. My
child’s identity will be coded, and all data will be kept in
a secured location.
- The researcher will answer any questions
about the research, now or during the course of the project, and can be
reached by telephone at: (770) 972-3224, or by email at
Kathy_Traylor@gwinnett.k12.ga.us. I may also contact the professor
supervising the research, Dr. Michael Orey, Instructional Technology
Department, at (706) 542-4030.
return to top
Signature
Page
I understand the study procedures described above. My questions have
been answered to my satisfaction, and I agree to allow my child to take
part in this study. I have been given a copy of this form to keep.
__I DO give permission to you to reproduce
materials that my child may produce as part of classroom activities. No
last names will appear on any materials submitted by the teacher.
__I DO NOT give permission to reproduce
materials that my child may produce as part of classroom activities.
___Kathy Traylor________
_______________________ __________
Name of Researcher Signature Date
Telephone: 770-972-3224
Email: Kathy_Traylor@gwinnett.k12.ga.us
_________________________
_______________________ __________
Name of Parent or Guardian Signature Date
Please sign both copies, keep one and return
one to the researcher.
Additional questions or problems regarding
your child’s rights as a research participant should be
addressed to Chris A. Joseph, Ph.D. Human Subjects Office, University
of Georgia, 612 Boyd Graduate Studies Research Center, Athens, Georgia
30602-7411; Telephone (706) 542-3199; E-Mail Address IRB@uga.edu
return to top
Research Study -
Student Consent Form
I, ________________________________________,
agree to take part in a research study titled, “Effect of
Problem Posing on Eighth Grade Students’ Mathematical Problem
Solving Skills,” which is being conducted by Mrs. Kathleen
Traylor, an 8 th grade Algebra teacher at Shiloh Middle School who is
also enrolled in a graduate degree program in the Instructional
Technology Department at the University of Georgia (770-736-4321). This
study is being conducted under the direction of Dr. Michael Orey,
Department of Instructional Technology, University of Georgia,
(706-542-4030).
I do not have to be in this study if I do not
want to. I can stop taking part at any time without giving any reason,
and without penalty. I can ask to have the information related to me
returned to me, removed from the research records, or destroyed.
The reason for the study is to find out if
having students create their own math problems will improve their skill
in solving other math problems.
- The research is not expected to cause any
harm or discomfort. I can quit at any time. My grade will not be
affected if I decide to stop taking part in the research.
- The study will be conducted for nine weeks
during regularly scheduled class.
- Students will take two pre-tests about
problem solving and problem posing, one by themselves and one as a
member of a group. Students in some classes will be asked to create
math problems as part of their normal class lessons, in homework
assignments and on tests and quizzes. At the end of the nine weeks,
they will take two post-tests. These are the
“experimental” classes.
- Students in the
“control” classes will take the same pre- and
post-tests, but they will be taught using the teacher’s usual
teaching methods, specifically not including problem posing.
- Any information collected about me will be
held confidential unless otherwise required by law. My identity will be
coded, and all data will be kept in a secured location.
- The researcher will answer any questions
about the research, now or during the course of the project, and can be
reached by telephone at: (770) 972-3224, or by email at
Kathy_Traylor@gwinnett.k12.ga.us. I may also contact the professor
supervising the research, Dr. Michael Orey, Instructional Technology
Department, at (706) 542-4030.
return to top
Signature
Page
I understand the study procedures described above. My questions have
been answered to my satisfaction, and I agree to take part in this
study. I have been given a copy of this form to keep.
__I DO give permission to you to reproduce
materials that I may produce as part of classroom activities. No last
names will appear on any materials submitted by the teacher.
__I DO NOT give permission to reproduce
materials that I may produce as part of classroom activities.
_Kathy Traylor_________
_______________________ __________
Name of Researcher Signature Date
Telephone: 770-972-3224
Email: Kathy_Traylor@gwinnett.k12.ga.us
_________________________
_______________________ __________
Name of Participant Signature Date
Please sign both
copies, keep one and return one to the researcher.
return to top
Appendix
B
Regular Algebra Individual Pre-Test
|
Name: _______________________
|
Teacher:
________________
|
Date:
_________________
|
Regular Algebra – Individual
Problem Solving – Activity 1
|
Do your best to solve each problem. Show
ALL of your work, including methods that did NOT work. Please also
explain your solutions. This test is to see HOW you solve problems, not
just to see if you can get the correct answer. You may use more paper
if you need more room. No calculators.
|
|
1. Jan
could walk 4 miles in 30 minutes. How long will it take her to walk 6
miles?
|
|
2. In
Naomi's class, 24 of the 32 students went on a field trip. What percent
of the group went on the trip?
|
|
3. On a
test with 20 items, Ellis had a percentage grade of 80%. How many
answers did he get right?
|
|
4. How
many points were scored during the five games?
|
|
5. Jim had
$20.00 to shop for birthday gifts. He purchased an item for $8.75,
another for $3.50 and a third for $6.47. How much money does Jim have
left?
|
Questions
Copyright (c) 2005 by The Riverside Publishing Company. All Rights
Reserved. Selected 6/22/05 from online question bank located at https://regiond.georgiacrct.org/servlet/a2l
and modified to open-response format.
return to top
Appendix C
Honors and Gifted
Students Individual Pre-Test
|
Name: _______________________
|
Teacher:
________________
|
Date:
_________________
|
Honors/Probe
Algebra – Individual Problem Solving – Activity 1
Do your best to solve each problem. Show ALL
of your work, including methods that did NOT work. Please also explain
your solutions. This test is to see HOW you solve problems, not just to
see if you can get the correct answer. You may use more paper if you
need more room. No calculators.
- Karen has a new job selling cars. She sold
4 cars her first week on the job. In the second week, she sold 8 cars.
She sold 14 cars in her third week and 22 cars in her fourth week. If
Karen continues to sell cars at this rate, how many weeks will it take
her to sell 58 cars in a week?
- On how many pages in a 500-page book will
the number 3 appear at least once?
- A swimming pool is 25 feet wide and 40 feet
long. Around the pool is a path that is 4 feet wide. What is the area
of the path?
Problems adapted from Cohen, S. R. (1991). Figure it out.
(Book 5). North Billerica, MA: Curriculum Associates, Inc.
return to top
Appendix
D
Regular Algebra
Individual Post-Test
|
Name: _______________________
|
Teacher:
________________
|
Date:
_________________
|
Regular Algebra – Individual
Problem Solving – Activity 2
|
Do your best to solve each problem. Show
ALL of your work, including methods that did NOT work. Please also
explain your solutions. This test is to see HOW you solve problems, not
just to see if you can get the correct answer. You may use more paper
if you need more room. No calculators.
|
|
1. Cydney
could walk 7 miles in 30 minutes. How long will it take her to walk 9
miles?
|
|
2. In
Juan's class, 6 of the 24 students are in Band. What percent of the
class is in Band?
|
|
3. On a
test with 25 items, Eddie had a percentage grade of 80%. How many
answers did he get right?
|
|
4. How
many points were scored during the six games?
|
|
5. Jung
had $40.00 to shop for birthday gifts. He purchased an item for $10.75,
another for $3.25 and a third for $6.27. How much money does Jung have
left?
|
Questions
Copyright (c) 2005 by The Riverside Publishing Company. All Rights
Reserved. Selected 6/22/05 from online question bank located at https://regiond.georgiacrct.org/servlet/a2l
and modified to open-response format.
return to top
Appendix E
Honors and Gifted
Algebra Individual Post-Test
|
Name: _______________________
|
Teacher:
________________
|
Date:
_________________
|
Honors/Probe
Algebra – Individual Problem Solving – Activity 2
Do your best to solve each problem. Show ALL
of your work, including methods that did NOT work. Please also explain
your solutions. This test is to see HOW you solve problems, not just to
see if you can get the correct answer. You may use more paper if you
need more room. No calculators.
- Kim got 85 pieces of candy on Halloween.
She ate 5 pieces on Halloween (October 31), 7 pieces on November 1, and
then she ate 9 pieces on November 2, then 11 pieces on November 3. If
she follows this pattern, when will she run out of candy?
- On how many pages in a 300-page book will
the number 7 appear at least once?
- A playground is 15 feet wide and 30 feet
long. Around the playground is a sidewalk that is 3 feet wide. What is
the area of the sidewalk?
Problems adapted from
Cohen, S. R. (1991). Figure it out. (Book 5).
North Billerica, MA: Curriculum Associates, Inc.
return to top
Appendix
F
Group Pre-Test
|
Name: _______________________
|
Teacher:
________________
|
Date:
_________________
|
Group Problem Solving – Activity A
|
Do your best to solve each problem. Show
ALL of your group’s work, including methods that did NOT
work. Please also explain your solutions. This test is to see HOW you
solve problems, not just to see if you can get the correct answer. You
may use more paper if you need more room. No calculators.
|
- Stewart bought 40 meters of fencing to make
an enclosure for his dog. If Stewart makes a rectangular enclosure,
what is the largest area it can have?
- How many triangles are in this picture?
- Apples are sold in 5-pound bags and pears
in 4-pound bags. Mrs. Brown bought 27 pounds of fruit. How many bags of
each kind did she buy?
- Be creative! Make up a problem which you
would solve by using this equation, then solve the problem:
- Be creative! Make up a problem involving a
frog and negative and positive numbers, and then solve the problem.
Problems1-3 adapted
from Cohen, S. R. (1991). Figure it out. (Book
5). North Billerica, MA: Curriculum Associates, Inc.
return to top
Appendix
G
Group Post-Test
|
Name: _______________________
|
Teacher:
________________
|
Date:
_________________
|
Group Problem Solving – Activity 2
|
Do your best to solve each problem. Show
ALL of your group’s work, including methods that did NOT
work. Please also explain your solutions. This test is to see HOW you
solve problems, not just to see if you can get the correct answer. You
may use more paper if you need more room. No calculators.
|
- Shawna’s goat needs at least 400
square feet of room in its pen. If she makes a rectangular pen, what is
the shortest length of fence Shawna will need to construct the pen?
- How many rectangles are in this picture?
- Apples are 70 cents a pound and bananas are
50 cents a pound. Jeri spent $2.60 on fruit. How many pounds of each
kind did she buy?
- Be creative! Make up a problem which you
would solve by using this equation:
- Be creative! Make up a problem involving
money and negative and positive numbers.
Problems1-3 adapted from Cohen, S. R. (1991). Figure it out.
(Books 4 & 5). North Billerica, MA: Curriculum Associates, Inc.
return to top
Appendix
H
Evaluation Rubric
Problem
Solving Rubric
|
0 points
|
1 point
|
2 points
|
3 points
|
4 points
|
Score
|
|
What problem solving strategies were
used?
|
Problem not attempted
|
Guess, no check
|
Guess and check
|
Diagram, work backwards, solve a
simpler problem, list or table, pattern, logical reasoning
|
Write and solve and equation
|
. |
|
How many of the problem solving steps
were used?
|
Problem not attempted
|
Examined and analyzed the problem
|
Planned a solution
|
Carried out the plan, found a solution
|
Checked the solution, looked back
|
. |
|
Is the solution correct?
|
Not at all correct
|
Some aspect(s) correct, but major
computation or logic/reasoning errors
|
Partially correct, major computation or
logic/reasoning errors or incomplete
|
Almost totally correct, minor
computation errors or logic/reasoning errors
|
Totally correct
|
. |
|
Is the work shown and is the solution
explained?
|
No explanation, minimal work shown
|
Some work shown but minimal/no
explanation
|
Some/all work shown but minimal or no
explanation
|
All work shown, incomplete explanation
|
All work shown, solution completely
explained, any alternate strategies explained
|
. |
| . |
. |
. |
. |
. |
TOTAL, out of 16 points
per problem
|
. |
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Appendix
I
Interview Questions
Interview
Questions
About the problem solving
process
- What strategies did you use?
- Do you like solving problems with a group?
- Were you able to explain your problem
solving process?
- Did you get the right answers?
- What did you like and dislike about the
problem solving activities?
About problem posing
(these questions are only for experimental group participants)
- Do you think problem posing helped you
with problem solving?
- Did you enjoy posing problems?
- Would you like to continue having problem
posing assignments through the rest of the school year?
- What did you like best and least about
posing problems?
For the other participating teachers
- What problem solving strategies did you
teach during the course of this study?
- Have you used these strategies
successfully in the past?
- Do your students enjoy problem solving
activities?
- Do you have any other questions or
comments about this study?
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Appendix
J
(Detorie, 2005a)
(Detorie,
2005b)
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