Review of the Literature
Introduction
Joyce & Weil, (2000) emphasize that learning to think (learning to learn) is what
school is all about. If this is the case, then our education system has significant changes to
consider. “In mathematics, the problem encountered by students can be described as
‘rigidly applied algorithms.’ Students memorize formulas and can then plug numbers
appropriately into those formulas. But in the absence of some trigger that a particular
formula is wanted, they prove unable to marshal it” (Gardner, 1999, p.120).
Research shows that students memorize strings of formulas in an attempt to solve
a problem in mathematics. Gardner implied that students who do this never really
understand the formula. “The formula was just a syntactic string that had been committed
to memory” (Gardner, 1999, p.120). American schools have focused primarily on the
teaching of facts and formula (Keefe & Herbert, 1992) . The immediate concern of the
students is to memorize the formula to be able to apply it to an assignment. Educators
reinforce this behavior by simply continuing to give students such assignments. It is true
that they are able to apply the formula appropriately in the short term. However, the
retention of the concept, if it is ever understood, only exists for the short term.
Problem Solving
If educators want to attempt to resolve this issue, we must try another approach.
One such possibility to address this issue is to focus on problem solving. There are two
main views of problem solving. Jinfa Cai (National Council of Teachers of Mathematics,
2001) summarizes the first as a process that increases a student’s ability to teach himself
thinking skills, to deal with unfamiliar situations, and this process magnifies his
persistence. The other view is that problem solving is, by itself, a context for students to
learn and understand mathematics (National Council of Teachers of Mathematics, 2001).
With either view, the ability to problem solve increases the student’s ability to learn. And
of course, that is the true intent -increasing students ability to learn. Educators need to
help students make the necessary mathematical connections. Problem solving is one such
way of accomplishing this objective (Polya, 1957).
“But I shall take pains to state in clear words the rules and ways of investigation
which are followed by all able men, who in most cases are not even conscious of
following them” (Bolzano cited in Polya, 1957, p.57). Some students acquire the ability
to problem solve, while others that have had the same experiences do not acquire these
skills. The student should be able to work with a set of predefined steps to solve a
problem whether they work at a conscious or unconscious level. Paul A. Forester, the
author of both Algebra I: Expressions, Equations, and Applications and Algebra and
Trigonometry: Functions and Applications which have been adopted at several California
high schools, breaks down the problem into these steps. However, Forester does not make
the students aware of these steps.
Instructors, authors, and mentors, should not break the problem down, but rather
give the students tools: an independent blue print from the problem, general steps or
questions to solve problems. “Moreover, when the teacher solves a problem before the
class, he should dramatize his ideas a little and he should put to himself the same
questions which he uses when helping the students” (Polya, 1957, p.5). Without this, the
student does not acquire the skills to problem solve; rather the student goes through the
instructor’s, the author’s, or the mentor’s problem solving skills, not their own.
Consequently, the student does not develop these skills--he does the work through
someone else’s problem solving skills (Polya, 1957).
Research has shown that the active involvement of progressing through each of
the individual steps is crucial (Wheatley, 1991). The learner must find himself engaged in
activities that help them handle the new knowledge in an efficient manner. Once
engaged, knowledge is no longer passively received, but is built by the learner (Wheatley,
1991). An example of this set of steps would be to define the variables, write the
equation, and apply the equation to a couple of circumstances (Polya, 1957). Using the
ideas of Polya and others, instructors can lead students into their investigation by asking
questions such as, “How do you answer each of the individual parts, rather than
attempting to solve the whole problem at once?” By giving the student a problem and an
independent set of steps to solve problems in general, the student has to ask himself,
“Where am I in the process?” and “What am I going to ask myself to get to the next
step?”
The teacher cannot control the act of learning, but rather he can increase the probability
of specific behaviors through the promotion and support of various activities (Gagnè,
1985). If the teacher tends to ask guiding questions that lead their students through the
teacher’s instruction, even the student that is an active participant in the lecture and in the
review of challenging problems from the homework isn’t necessarily gaining any
problem solving ability from the instruction. The student gains little or none of these
skills, because the student has not internalized a set of problem solving steps. The timeappropriate questions demonstrate the teacher or author’s problem-solving ability. Again,
the questions do not represent the student’s ability to problem solve. The presentation of
the problem solving process is often too subtle for the student to pick up on in a lecture
format. The student needs to have significant opportunities to acquire problem-solving
skills, while the instructor helps the students discreetly (Polya, 1957).
Scaffolding through the acquisition of the skills of problem solving is key. “In
other words, the heuristics should be taught as concepts, and named, with the objective
that, for example, the term ‘making a plan’ evokes in the students mind a range of well
defined plans which he has used for different problems” (Bell, Costello, & Kuchemann,
1983, p.209). A student’s ability to ask himself and others the time-appropriate questions
of the problem solving process assists in the success of the problem (Polya, 1957).
Examples of such questions are as follows: What is the problem and do I understand it?
What am I trying to solve? What does that mean? Have I seen a problem similar to this
that I have solved before? If a student’s time is filled with repetitious exercises and
routine operations, his intellectual development is squandered. As indicated above, if
educators can help students to solve their questions with questions, then we stimulate
their interest in a means of independent thinking (Polya, 1957).
Cooperative Projects
Using cooperative grouping is an option to assist students in the ability to problem
solve (Davidson, 1990). Members of a cooperative group have the ability to learn from
the teacher as well as from one another. The learners have a collective unit that generates
discussion rather than isolation. “Also, shared responsibility and interaction produce
more positive feelings towards the tasks and others, generate better inter-group relations,
and result in better self images…”(Joyce & Weil, 2000, p.34). Ultimately, the collective
group is greater than the sum of its members. The teacher, who organizes activities so
that students work in pairs or large groups, share responsibility, and help one another,
increases the mastery of material by all individuals within the class. Talking through
math problems with classmates, the student understands how to solve problems correctly
(Davidson, 1990). In such cases, the student demonstrates and explains his reasoning to
others with mathematical terminology. Davidson (1990) highlights the five components
to cooperative groups:
1.
The teacher clearly structures positive interdependence within each group.
2.
The student engages in face-to-face interaction during the assignment.
3.
The teacher structures the activity to ensure individual accountability and the
learning of all in the group.
4.
The student utilizes interpersonal and group skills.
5.
The teacher creates opportunities for groups to evaluate their collective
effectiveness (Davidson, 1990, p. 105-106).
The teacher taking an active role in the above components increases the likelihood of
success for any group within the class. The teacher’s role is to facilitate groups, intervene
as needed to direct groups in the appropriate educational goals, and supervise the group’s
activities.
The student will need the ability to problem solve in their future, both in the
workplace and at home (Thornburg, 2000). He will need to be able to communicate,
collaborate, and critically think to be successful. Besides being promoted through
cooperative groups, these abilities are inherent in project-based learning. Thornburg
(2000) contended that in such a learning activity, the student engages his mind fully in
the educational process, not only gaining mastery of the information he needs, but
exercising his creativity in ways that keep open the doors for lifelong learning. Thus, the
student immersed in cooperative, project-based learning will benefit from the ability to
transfer these skills from the classroom to the workplace. The most intelligent individuals
in the world who are unable to express their ideas clearly to others and lack the social
skills needed to work in collaborative teams are unemployable (Thornburg, 2000).
Cooperative project-based learning needs to become an integral component of
instruction, not just an opportunity for the student to be involved in a supplemental
activity. Rather than just having the student solve two-step equations or evaluating
expressions, the student should be afforded the opportunity to develop cooperative and
team-oriented behavior as this will benefit him in his ability to be a life-long learner and
to solve challenging problems. In cooperative groups and project-based learning the
student does this; the student connects relevant concepts to one another through
interaction and shared responsibility as the project is completed. The student learns to
appreciate the ability of others within the group; he is willing to give assistance to other
group members, as well as being willing to ask for assistance from other group members
(Riel & Fulton, 2001). The student utilizes the ability to problem solve in order to
produce a meaningful product and he will need this ability in his future. November
(2001) has stated that communication, collaboration, and critical thinking are among six
skills needed to solve real problems. Having a deep understanding of a concept enables
the student to internalize and used the concept to problem solve.
Web-Based Technology
“Imagine the possibility of our students providing a web page that is a genuine service to
the Internet community, something the students care about, and something that is needed”
(Ryder, 1996). The implication from Ryder’s comment is that technology can easily be a
tool to assist in the project-based learning. The student can use a computer first as a tool
that improves his productivity. Technology and media extend the student’s capabilities to
see, hear, and learn (Ryder, 1996). As the student becomes less conscious of the use of
technology, he uses it as an extension of his own intentioned will (Ryder, 1996). As an
individual becomes more efficient, he has the freedom to focus on less mundane activities
and more time to reflect on higher order thinking (November, 2001). The more reflective
students are, the more likely they are to improve their products.
Teaching the student about computer applications for productivity purposes is not
the ultimate goal. “Educational technology is about educating students, serving our
communities, and improving our institutions and society” (Burniske, 2001,p. 527). The
teacher does not teach computer technology for the sake of learning about computers, but
he teaches computer technology to increase student achievement. Burniske has indicated
that computer technology integration in content areas should be a consequence of the
learning goals for our students.
Meaningful learning can results from the desire to interact with the Web
environment. “The force toward meaningful learning will be curiosity, puzzlement, and a
desire to understand fuzzy aspects of the social and intellectual world encountered on the
Internet and the World Wide Web” (Geisert & Futrell, 2000, p. 96). Students are more
likely to persist in attaining and utilizing technology if the technology encompasses
meaningful learning to the students. Distance learning is equally or more effective than
traditional instruction when the implementation of methods and technologies is
appropriately used to the educational activity, when either there is student-to-student
interaction or there is timely teacher-to-student feedback (Barron, 1999). The Internet and
the World Wide Web collaborative conferencing capabilities (a discussion board, for
example) enhance instructor-student and student-to-student interaction (Rafaeli & Ravin,
1997). Hence, these collaboration tools complement student learning (Dwyer, Barbieri, &
Doerr, 1995).
In order to integrate technology into the classroom, cooperative, project-based
learning offers significant opportunities for teachers to acquire technological skills.
“Teachers need an ability to manage the use of many technologies in the classroom
without having to know the technical details” (November, 2001, p. 39). In other words,
teachers do not have to pretend that they have knowledge and understanding of
everything. Students respect teachers more when the teacher looks for assistance to solve
technical problems.
November (2001) stated, in reference to technology, “Do not worry about learning the
skills. It is more important to learn what the students can do with the skill that might add
value to their class work” (p. 39). If the instructor has a general idea what the application
can do for him, then the students will problem solve and use their fellow students to
direct their way through to a solution. The teacher should be willing to ask a student
questions similar to: How did you accomplish creating that object? What were the
keystrokes in making the two lines perpendicular to one another?
Again, technology is a means for assisting learning. Educators should not be
hesitant to partake in integrating such technologies. The desire to understand aspects of
the social and intellectual world encountered on the Internet and the World Wide Web
promote learning (Geisert & Futrell, 2000).
Summary
“Cooperative learning implies a group of students working toward a common
learning goal or accomplishment of an allocated task. This is a realistic emulation of the
working world, where business goals commonly are reached through a number of people
working together.”(Geisert & Futrell, 2000, p. 244) Collaboration, problem solving and
technology are all important tools for the future of education. Training in these areas need
to become a priority, but not as a teacher assignment only, but to have some relevance to
the student. “…it is important for students to share in identifying the problem. Students
must have a sense of owning the problem rather than seeing it as an assignment from the
teacher.”(November, 2001, p. 50) The job before us now is how to facilitate these
different skills. New technologies are opening up ways to approach the problems found in
education. New ways to enhance the learning experience and allow students to
collaborate with others that are geographically separated are now readily available. The
purpose of our research is to evaluate one of those ways.