Transcript Modeling

Use of Modeling to Teach Algebra
EMS 513 Teaching and Learning Algebraic Thinking
Group Members
Matthew Campbell
Breanna Harrill
Carrie Lineberry
Kenny Nguyen
Lesley Percival?
What is Modeling?
 The overarching theme in many definitions that
algebraic modeling is using mathematics to make
sense of some real-world phenomenon which
yields a model of the real-world.
 It is implied that this model is not an exact copy of
the real-world. Depending on the nature of the
problem, there will inevitably be gaps (both
accidental and purposefully) between the realworld and the model.
Development of Modeling
 Black (1962) did not look at the process of “modeling” but did
identified four categories of models: scale, analogue,
mathematical, and theoretical. He introduced a set of
procedures involved in using a mathematical model.
 Identify the relevant variables
 Frame empirical hypotheses surrounding the relevant variables
 Introduce simplifications (sometimes drastic) in order to facilitate
mathematical manipulation of the variable
 Solve the resulting mathematical equations. Failing this, study the global
features of the equations
 Extrapolate to testable consequences in the original field
 Generalize the theory by possibly removing initial restrictions in the
interest of simplicity
Mathematical Modeling
 Confrey and Maloney (2006) define modeling as, “the
process of encountering an indeterminate situation,
problematizing it, and bringing inquiry, reasoning,
and mathematical structures to bear to transform the
situation. The modeling produces an outcome – a
model – which is a description or a representation of
the situation, drawn from the mathematical
disciplines, in relation to the person’s experience,
which itself has changed through the modeling
process.”
Confrey and Maloney’s
Conception of Modeling
Use of Modeling to Teach Algebra
 Modeling can provide a powerful way of connecting vague
mathematical notions and connecting them to the real
world
 Technology such as graphing calculators, digital cameras,
and computer simulations/microworlds are being used to
help with the modeling process or are even models
themselves
 Modeling has been shown in numerous studies to increase
students’ conceptual notions of algebra beyond the
“school-algebra” concepts of solving unknowns
Connections to Teaching (Task 1)
 The task requires students to make a recommendation to
British commanders on whether or not to convoy their
ships during World War I. Using information from Körner
(1996), students had the materials to validate or refute
assertions made by British commanders to not convoy
their ships.
Connections to Teaching (Task 2)
 This task consists of a series activities to teach students
about difference equations by using information about
different prescription and over the counter drugs (amount
of drug taken, half-life of drug, time between doses, etc.)
Connections to Teaching (Task 3)
• This task
involves a
digital camera,
the graphing
calculator, and
Geometer
Sketchpad.
• Students record the image of
the arc of water in a water
fountain and fit a parabola to
that arc.
Connections to Teaching (Task 4)
 This task is a series of activities that model algebra
through the use of algebra tiles.
Connections to Teaching (Task 5)
 In this task students explore exponential models through
the use of M&M’s (or other candy).
 The candy is shaken and then pieces are removed that do
not have writing facing up. The number of pieces of candy
remaining after each turn is recorded in a table.
Comments Regarding Tasks
 Some tasks require students to develop a model to represent and
answer a real world example
 Others go in reverse, where the problem is an abstract
mathematical notion like exponentiation or the form of an
equation.
 The algebra tiles task involves students utilizing existing models
to address abstract mathematics
 Each of the tasks did use, or could have used, technology.
 These tools make accessible different avenues of algebraic
reasoning, providing a dynamic environment with multiple
representations, or merely offering a way to capture a real world
phenomenon.
Conclusions
 Through the use of mathematical modeling,
students were shown to develop a deeper
understanding of the topics that were being
addressed.
 Modeling also provided a link between
mathematics and real-world phenomena and
problems.
Modeling Video
Recommendations for Practice
 Teachers must be aware of several modeling distinctions
and how they relate to a task that they choose for their
own students.
 Teachers and students must remain aware of the goals
and context of a task involving models or modeling.
 Teachers should set high standards for explanation and
justification by students. Justification in the modeling
process has been shown to foster connection building
between a model and a mathematical concept.