Introduction
and Pedagogical Framework
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In February 2007 I
approached the Centre for
Learning Innovation to discuss an authentic project for this masters
unit. They suggested that I work on
Financial Mathematics because this was a unit they needed but lacked the
resources to create the course. I chose
to focus my project on Investing and Borrowing which addresses the NSW
Mathematics Years 7-10 Syllabus, Stage 5, Consumer Arithmetic: Solves consumer
arithmetic problems involving simple interest, compound interest and
depreciation.
The course is designed as an online resource for
distance education. It is designed for
implementation from within the Moodle
e-learning environment. Moodle's
features such as Forums, Blogs, Wikis and Chat facilitate a pedagogical approach based on constructivist and social constructionist learning theories.
The instructional style adopted by this is in
line with the following recommendations (Everybody Counts, cited in Schoenfeld,
1992. p. 4):
• Seeking solutions,
not just memorizing procedures;
• Exploring
patterns, not just memorizing formulas;
• Formulating
conjectures, not just doing exercises.
Consequently this course is
styled on the conception that
“a degree of control over their own
learning can provide challenge, motivation and engagement for a wide range of
student groups.” (Hennessy et al, 2007. p.140)
This course provides
opportunities for students to exploit the interactive affordances offered by
the Moodle platform. Students are
encouraged to explore, participate or manipulate various tools to become
actively involved in the construction of individual and collaborative artifacts. This includes self assessment by means of
quizzes and peer tutoring for remediation.
Wikis are used for collaborative writing for the construction
of knowledge. Discussion Forums are used for asynchronous discussion and
collaboration. Chat rooms are used for synchronous lessons or discussions. Blogs
are used for individual reflection. Emails are used for sending scanned
handwritten work done by the student and general asynchronous communication.
In addition to the ICT
resources afforded by Moodle, this course on Consumer Arithmetic utilizes Spreadsheets as Mindtools
to engage learners in critical thinking. The use of spreadsheets as a dynamic
modeling tool enables the student to free themselves from the tedious and time
consuming task of multiple calculations required for compound interest so that
they can concentrate on higher order thinking: analysis, evaluation, synthesis,
elaboration, decision making, designing solutions and solving complex
problems. The spreadsheet allows the
student to pose “What if?” questions, to make hypothesis and to test their
ideas.
Worked examples and assignments provide a skill base
and resource for the students within a Problem
Based Learning (PBL) framework.
“Mathematics instruction should provide students
the opportunity to explore a broad range of problems and problem situations,
ranging from exercises to open-ended problems and exploratory situations.”
(Schoenfeld, 1992. p. 32)
Slavery and Duffy (1995) suggested the following principles
for PBL which guide the delivery of this course:
- Anchor all learning activities to a larger task
or problem.
- Support the learner in developing ownership for
the overall problem or task
- Design an authentic task.
- Design the task and the learning environment to
reflect the complexity of the environment they should be able to function
in at the end of learning.
- Give the learner ownership of the process used
to develop a solution.
- Design the learning environment to support and
challenge the learner’s thinking.
- Encourage testing ideas against alternative
views and alternative contexts
- Provide opportunity for and support reflection
on both the content learned and the learning process.
The framing problem for this
unit provides an “open inquiry learning environment; solving problems in
ill-defined contexts, and cooperative learning” (Lewis et al., 1998). The act of posing problems is integral to the
process of solving problems.
“Finding or posing problems is a
quintessentially creative endeavor.
Whether the found problem is the presence of a hole in the ozone, or
that people who smoke are prone to cancer, those keen enough to call our
attention to them distinguish themselves by their astuteness. They help set community agendas that lead to
discoveries and invention that help make the world better.” (Lewis et al., 1998)
The posing of problems can
occur prior to, during, or after the act of problem solving. By keeping the problem open, the students in
this course can “frame and reframe” their perspective on a problem so they
reach “goal clarity”. Educational
psychologists have discovered that solving a problem is a back-and-forth
(recursive) process, not a linear one (Pea, 1985). In this way the student is able to take
ownership of the problem and see it as their own problem thus enhancing
motivation, responsibility and learning.
This course approaches mathematics
education through solving environmental economics problems at a personal and
social level. Although a framing problem
is provided to the student, the student groups actually need to pose the specific
perspective they will take to the problem. The students become the designer and active
creator of a collaborative knowledge
building process.
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References
Hennessy, S., Wishart, J., Whitelock, D., Deaney, R.,
Grawn, R., la Velle, L., McFarlane, A., Ruthven, K. and Winterbottom, M.
(2007) Pedagogical approaches for technology-integrated science teaching. Computers & Education 48 (2007) 137-152
Jonassen, D. H. (2000) Computers as Mindtools for Schools, Engaging Critical thinking
Second Edition.Prentice hill.
Lewis, T., Petrina, s. and Hill, A. M.
(1998) Problem Posing-Adding a Creative
Increment to Technological Problem solving.
Journal of Industrial Teacher Education, Vol 36 No 1 Fall 1998.
Retrieved from http://scholar.lib.vt.edu/ejournals/JITE/v36n1/lewis.html
on 23 March 2007
Pea, R, D. (1985)
Learning to Think Mathematically
retrieved from http://www.stanford.edu/~roypea/RoyPDF%20folder/A24_Pea_85c.pdf
15 April 2007
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem
solving,
metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook
for Research on Mathematics Teaching and Learning (pp. 334-370).
Savery, J. R. Duffy, T. M. (1995 ) Problem
Based Learning: an instructional model and its constructivist framework. EDUCATIONAL
TECHNOLOGY -SADDLE BROOK NJ- 1995, VOL 35; NUMBER 5, pages 31 Retrieved from http://www3.uakron.edu/edfound/people/savery/papers/sav-duff.html
on 20 March 2007