Use of Modeling to Teach Use of Modeling to Teach AlgebraAlgebra

Group MembersMatthew CampbellBreanna HarrillCarrie LineberryKenny NguyenLesley Percival?

EMS 513 Teaching and Learning Algebraic Thinking

What is Modeling?What is Modeling?

The overarching theme in many definitions The overarching theme in many definitions that algebraic modeling is using that algebraic modeling is using mathematics to make sense of some real-mathematics to make sense of some real-world phenomenon which yields a model of world phenomenon which yields a model of the real-world.the real-world.

It is implied that this model is not an exact It is implied that this model is not an exact copy of the real-world. Depending on the copy of the real-world. Depending on the nature of the problem, there will inevitably nature of the problem, there will inevitably be gaps (both accidental and purposefully) be gaps (both accidental and purposefully) between the real-world and the model.between the real-world and the model.

Development of Development of ModelingModeling

Black (1962) did not look at the process of Black (1962) did not look at the process of “modeling” but did identified four categories of “modeling” but did identified four categories of models: scale, analogue, mathematical, and models: scale, analogue, mathematical, and theoretical. He introduced a set of procedures theoretical. He introduced a set of procedures involved in using a mathematical model.involved in using a mathematical model. Identify the relevant variablesIdentify the relevant variables Frame empirical hypotheses surrounding the relevant Frame empirical hypotheses surrounding the relevant

variablesvariables Introduce simplifications (sometimes drastic) in order to Introduce simplifications (sometimes drastic) in order to

facilitate mathematical manipulation of the variablefacilitate mathematical manipulation of the variable Solve the resulting mathematical equations. Failing this, Solve the resulting mathematical equations. Failing this,

study the global features of the equationsstudy the global features of the equations Extrapolate to testable consequences in the original fieldExtrapolate to testable consequences in the original field Generalize the theory by possibly removing initial Generalize the theory by possibly removing initial

restrictions in the interest of simplicityrestrictions in the interest of simplicity

Mathematical Mathematical ModelingModeling

Confrey and Maloney (2006) define modeling Confrey and Maloney (2006) define modeling as, “the process of encountering an as, “the process of encountering an indeterminate situation, problematizing it, indeterminate situation, problematizing it, and bringing inquiry, reasoning, and and bringing inquiry, reasoning, and mathematical structures to bear to transform mathematical structures to bear to transform the situation. The modeling produces an the situation. The modeling produces an outcome – a model – which is a description or outcome – a model – which is a description or a representation of the situation, drawn from a representation of the situation, drawn from the mathematical disciplines, in relation to the mathematical disciplines, in relation to the person’s experience, which itself has the person’s experience, which itself has changed through the modeling process.”changed through the modeling process.”

Confrey and Maloney’s Confrey and Maloney’s Conception of ModelingConception of Modeling

Use of Modeling to Teach Use of Modeling to Teach AlgebraAlgebra

Modeling can provide a powerful way of Modeling can provide a powerful way of connecting vague mathematical notions and connecting vague mathematical notions and connecting them to the real worldconnecting them to the real world

Technology such as graphing calculators, digital Technology such as graphing calculators, digital cameras, and computer simulations/microworlds cameras, and computer simulations/microworlds are being used to help with the modeling are being used to help with the modeling process or are even models themselvesprocess or are even models themselves

Modeling has been shown in numerous studies Modeling has been shown in numerous studies to increase students’ conceptual notions of to increase students’ conceptual notions of algebra beyond the “school-algebra” concepts algebra beyond the “school-algebra” concepts of solving unknownsof solving unknowns

Connections to Teaching Connections to Teaching (Task 1)(Task 1)

The task requires students to make a The task requires students to make a recommendation to British commanders on recommendation to British commanders on whether or not to convoy their ships during whether or not to convoy their ships during World War I. Using information from Körner World War I. Using information from Körner (1996), students had the materials to (1996), students had the materials to validate or refute assertions made by validate or refute assertions made by British commanders to not convoy their British commanders to not convoy their ships.ships.

Connections to Teaching Connections to Teaching (Task 2)(Task 2)

This task consists of a series activities to teach This task consists of a series activities to teach students about difference equations by using students about difference equations by using information about different prescription and information about different prescription and over the counter drugs (amount of drug taken, over the counter drugs (amount of drug taken, half-life of drug, time between doses, etc.)half-life of drug, time between doses, etc.)

Connections to Teaching Connections to Teaching (Task 3)(Task 3)

• This task This task involves a digital involves a digital camera, the camera, the graphing graphing calculator, and calculator, and Geometer Geometer Sketchpad.Sketchpad.

• Students record the image of the arc of water in a water fountain and fit a parabola to that arc.

Connections to Teaching Connections to Teaching (Task 4)(Task 4)

This task is a series of activities that model This task is a series of activities that model algebra through the use of algebra tiles.algebra through the use of algebra tiles.

Connections to Teaching Connections to Teaching (Task 5)(Task 5)

In this task students explore exponential In this task students explore exponential models through the use of M&M’s (or other models through the use of M&M’s (or other candy).candy).

The candy is shaken and then pieces are The candy is shaken and then pieces are removed that do not have writing facing up. removed that do not have writing facing up. The number of pieces of candy remaining The number of pieces of candy remaining after each turn is recorded in a table.after each turn is recorded in a table.

Comments Regarding Comments Regarding TasksTasks

Some tasks require students to develop a model to Some tasks require students to develop a model to represent and answer a real world example represent and answer a real world example

Others go in reverse, where the problem is an Others go in reverse, where the problem is an abstract mathematical notion like exponentiation or abstract mathematical notion like exponentiation or the form of an equation.the form of an equation.

The algebra tiles task involves students utilizing The algebra tiles task involves students utilizing existing models to address abstract mathematicsexisting models to address abstract mathematics

Each of the tasks did use, or could have used, Each of the tasks did use, or could have used, technology.technology.

These tools make accessible different avenues of These tools make accessible different avenues of algebraic reasoning, providing a dynamic algebraic reasoning, providing a dynamic environment with multiple representations, or environment with multiple representations, or merely offering a way to capture a real world merely offering a way to capture a real world phenomenon.phenomenon.

ConclusionsConclusions

Through the use of mathematical Through the use of mathematical modeling, students were shown to modeling, students were shown to develop a deeper understanding of develop a deeper understanding of the topics that were being addressed.the topics that were being addressed.

Modeling also provided a link Modeling also provided a link between mathematics and real-world between mathematics and real-world phenomena and problems.phenomena and problems.

Modeling VideoModeling Video

Recommendations for Recommendations for PracticePractice

Teachers must be aware of several modeling Teachers must be aware of several modeling distinctions and how they relate to a task that they distinctions and how they relate to a task that they choose for their own students.choose for their own students.

Teachers and students must remain aware of the goals Teachers and students must remain aware of the goals and context of a task involving models or modeling.and context of a task involving models or modeling.

Teachers should set high standards for explanation and Teachers should set high standards for explanation and justification by students. Justification in the modeling justification by students. Justification in the modeling process has been shown to foster connection building process has been shown to foster connection building between a model and a mathematical concept.between a model and a mathematical concept.