Why is Understanding Students’ Development of Area Important in Teaching Mathematics?

1. Why is Understanding Students’ Development of Area Important in Teaching Mathematics? Angela Howard

2. The mathematic performance of students in the United States lags behind significantly when compared to their peers in varHow do students develop a conceptual understanding of area? • The answer to this question can be found by studying the Singapore Math framework that is based on developing students’ critical thinking skills, as well as problem solving skills, and not merely on teaching algorithms and root memorization (Chappell & Thompson, 1999).

3. If this is true, what should we do? • This PowerPoint will demonstrate how the Primary Mathematics model that is used within the Singapore Mathematics framework can be integrated with current curriculum activities and correlated with third-grade through fifth-grade Georgia Performance Standards.

4. The research of Dina van Hiele is used to define and identify the levels of thought and understanding that students progress through as they are constructing their own learning about area(Malloy, 1999).

5. To illustrate this model the following activities will span across three different grade levels beginning at the third-grade level and will correlate with each grade’s specific current Georgia Performance Standards (Outhred & Mitchelmore, 2000).

6. Standards • Beginning at the third-grade level, students are required to create an understanding of simple geometric figures such as squares and rectangles that will progress to an understanding of how to measure the area of those shapes. • M3M4 Students will understand and measure the area of simple geometric figures • (squares and rectangles). • Understand the meaning of the square unit and measurement in area. • Model (by tiling) the area of a simple geometric figure using square units • (square inch, square foot, etc.). • Determine the area of squares and rectangles by counting, addition, and • multiplication with models (Education G. D., 2005-2008).

7. Activities

8. Singapore Activity Completion of this problem demonstrates a student’s basic conceptual understanding of the process involved to find the area and perimeter of a simple figure. In addition, a student completing this problem understands that area can be determined by multiplying length x width because an array can be created with the number of square units that make up the length and width inside the shape.This indicates a Level 1 or Analysis understanding.

9. Fourth Grade • Transitioning to the fourth grade level students are required to define and identify the characteristics of quadrilateral figures including parallelograms, squares, rectangles, trapezoids, and rhombi through examination, classification, and construction. In addition, students compare and contrast the relationships among quadrilaterals. • M4G1. Students will define and identify the characteristics of geometric figures through examination and construction. • c. Examine and classify quadrilaterals (including parallelograms, squares, rectangles, trapezoids, and rhombi). • d. Compare and contrast the relationships among quadrilaterals (Education G. D., 2005-2008).

10. Activities Areli, a fourth- grade student was asked to construct a quadrilateral shape and then define and identify its characteristics. Areli chose to illustrate a square and rectangle which indicates a level 0 or concrete understanding. As her understanding progresses, she will advance to other quadrilaterals such as trapezoids and rhombi.

11. This activity illustrates how Areli integrated the components of constructing, defining, and identifying the figures to examining and classifying them when given the following directions: Construct a square and a rectangle. What are some ways that would use to identify each figure? How do they compare and contrast?

12. Singapore Math

13. Fifth Grade • Transitioning to the fifth grade level students will continue to derive their own formulas. Students not only determine the area of various triangles, circles, rectangles, and parallelograms but also derive the formula itself. • M5M1. Students will extend their understanding of area of fundamental geometric • plane figures. • a. Estimate the area of fundamental geometric plane figures. • b. Derive the formula for the area of a parallelogram (e.g., cut the • parallelogram apart and rearrange it into a rectangle of the same area). • c. Derive the formula for the area of a triangle (e.g. demonstrate and explain its • relationship to the area of a rectangle with the same base and height). • d. Find the areas of triangles and parallelograms using formulae (Education G. D., 2005-2008).

14. Activities • Figure 1 shows how John used a formula to find the area of a rectangle. Of course, the unit of area is the 1x1 square on the dot paper, so the John applies the formula to determine the area. Successful completion demonstrates a level 0 or concrete analysis.

15. Figure 2 shows how John then found the area of a triangle using a rectangle. This is known as the partitioning method. “The reasoning behind the partitioning method is that a diagonal of a rectangle partitions the rectangle into two right triangles each of which has area that is half that of the original triangle (citation). Of course, applying the formula would produce the same result.

16. Figure 3 shows how John derived the formula for the area of a parallelogram by constructing a rectangle. By constructing a rectangle that has the same area as a parallelogram, similar partitioning and reasoning can be used to derive the formula for a parallelogram. Successfully completing these activities demonstrates a level 1 – analysis understanding.

17. . Singapore Math • John will apply his understanding to a word problem as presented in the Primary Mathematics curriculum. Successfully completing this example demonstrates a level 2 – analysis understanding.

18. Conclusion • Because, the fundamental understanding of formulas is often sacrificed when students memorize general formulas, it is crucial that current mathematical curriculum and learning environments evolve to emulate Asian countries that have demonstrated high performance. This article has demonstrated how such an evolution can take place by integrating the successful Asian activities that can be used to understand elementary students’ thinking about the specific concept of area with current best practices activities. In addition, by using the research of Dina van Hiele we can define and identify the levels of thought and understanding that students progress through as they are constructing their own learning about area.

19. References • Chappell, M. F., & Thompson, D. R. (1999, September). Perimeter or Area? Which Measure Is It?. Mathematics Teaching in the Middle School, 5(1), 20. • Malloy, C. E. (1999). Perimeter and Area. Mathematics Teaching In The Middle School , 87-90. • Outhred, L. N., & Mitchelmore, M. C. (2000, March). Young Children's Intuitive Understanding of Rectangular Measurement . Journal for Research In Mathematics, 31(2), 144-167. • Education, G. D. (2005-2008). Standards, Instruction, and Assessment. Retrieved April 6, 2009, from Georgia Performance Standards: http://public.doe.k12.ga.us/math.aspx