1. Use of Modeling to Teach Algebra Group Members Matthew Campbell Breanna Harrill Carrie Lineberry Kenny Nguyen Lesley Percival? EMS 513 Teaching and Learning Algebraic Thinking

2. 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. 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 real- world and the model. 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 real- world and the model.

3. 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. 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 Identify the relevant variables Frame empirical hypotheses surrounding the relevant variables Frame empirical hypotheses surrounding the relevant variables Introduce simplifications (sometimes drastic) in order to facilitate mathematical manipulation of the variable 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 Solve the resulting mathematical equations. Failing this, study the global features of the equations Extrapolate to testable consequences in the original field Extrapolate to testable consequences in the original field Generalize the theory by possibly removing initial restrictions in the interest of simplicity Generalize the theory by possibly removing initial restrictions in the interest of simplicity

4. 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 (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.”

5. Confrey and Maloney’s Conception of Modeling

6. Use of Modeling to Teach Algebra Modeling can provide a powerful way of connecting vague mathematical notions and connecting them to the real world 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 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 Modeling has been shown in numerous studies to increase students’ conceptual notions of algebra beyond the “school-algebra” concepts of solving unknowns

7. 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. 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.

8. 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.) 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.)

9. Connections to Teaching (Task 3) This task involves a digital camera, the graphing calculator, and Geometer Sketchpad. 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.

10. Connections to Teaching (Task 4) This task is a series of activities that model algebra through the use of algebra tiles. This task is a series of activities that model algebra through the use of algebra tiles.

11. Connections to Teaching (Task 5) In this task students explore exponential models through the use of M&M’s (or other candy). 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. 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.

12. Comments Regarding Tasks Some tasks require students to develop a model to represent and answer a real world example 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. 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 The algebra tiles task involves students utilizing existing models to address abstract mathematics Each of the tasks did use, or could have used, technology. 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. 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.

13. ConclusionsConclusions Through the use of mathematical modeling, students were shown to develop a deeper understanding of the topics that were being addressed. 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 also provided a link between mathematics and real-world phenomena and problems. Modeling Video

14. 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 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 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. 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.