DJohnston

Teaching Student-Centered Mathematics: Chapters 9-12

Chapter 9: In developing measurement concepts three areas should be considered when planning instruction.  Students need tol understand the attribute to be measured. They will need to understand how filling, covering, matching, or making other comparisons of an attribute with measuring units produces a number called a measure. Lastly, students should know and use common measuring tools with understanding and flexibility. Van de Walle and Lovin stated the students should be exposed to informal units of measure as well as standard units. Experimenting with informal units allows students to make comparisons and predictions about how such attributes length and area. The use of estimation is also encouraged to focus student attention on the attribute being measured and the measuring process. Activities such as Changing Units (9.1), More Than One Way (9.2), and Fill and compare (9.5) are excellent tasks for lower elementary students to develop measurement concepts.

Chapter 10: Math is filled with patterns and relationships and can involve colors, size, shapes, values, and many more. The basis of algebraic reasoning begins in Kindergarten and should move from very concrete and obvious toward symbolism and functions in upper elementary and middle school. In the third grade, algebraic reasoning is still very concrete but extending into logical thinking, making predictions, and being able to explain relationships and confirm predictions. Have students use different grid configurations to see what patterns develop with a recurring pattern (i.e. A-B-C) in a 3-wide grid, a 4-wide, and 5-wide. There are several questions to go with this activity on page 292 (Grid Patterns 10.2). Growing patterns in 3^rd grade evolves from building growing patterns until comfortable, explore and discover how to extend the pattern in the same or logical manner. And finally, extend a pattern on grid paper recording each step and writing how their extension indeed follows the pattern. Finally, patterns with numbers can go beyond merely skip counting to more involved patterns that follows rules such as "double the previous number," or "add the next square [number]." (What’s Next and Why 10.6). The "Start and Jump Numbers" (10.7) activity is also good for students to explore how patterns change by changing a start number or a jump number and then make comparisons.

Teaching Student-Centered Mathematics: Chapters 5-8

Chapter 5 deasl with the development of fraction concepts based on the use of models and manipulatives. The physical models help students build concepts in their minds of region or area, length, and sets. The "correct shares" activities willl help my students understand that like fractions are congruent, whereas unlike fractions are not equal. Students frequently want to divide a circle into thirds by drawing two straight parallel lines as one would a square or rectangle.

Chapter 6: The development of fraction concepts should follow the same path as the developement of whole numbers. Just as a numeral represents a quantity or value, fractions like wise must be constructed and understood by the student. Haste to teach rules must be abandoned until student conceptualization can bridge the gap between cocept and application of rules that emerge as a natural part of solving problems involving fractions. Van de Walle and Lovin reiterate time and again that traditional algorithms are not to be taught initially. I will need to try all tasks in a variety of ways until I am comforatble and can anticipate strategies and results.

Chapter 7 explores the use of decimals and percentages. Building on the concepts of fractions, students can discover the relationship between fractions and decimals by reviewing the base-10 system. Using the 10 x 10 squares, students can create fractions to represent numbers such as .45. With questions such as, "Is this greater or less than 1/2?" students can begin to construct an understanding of decimals and percentages.

Chapter 8 deals with geometric thinking and geometric concepts. In the work of van Hiele and van Hiele-Geldof, learners progress through 5 developmental stages (not age dependent) as they progress through geometric thinking. Experience with geometric models and activities is the single greatest factor. the more students explore and manipulate objects and models in the suggested exercises, the greater their conceptualization and understanding.

Teaching Student-Centered Mathematics: Chapters 1 through 4

The underlying premise of the book is that students learn best by creating their own knowledge and understandings by being able to manipulate objects and  models to build withing their mind a concept of the world around them, most specifically in this case, mathematics. Van de Walle and Lovin base their text on the constructivist method. One only has to teach in an elementary classroom to see that students constantly comment, "Can I see that?" "Can I hold that?" "Can we make one?" To a child and many adults as well, learning comes best with we can hold an object and turn over and around not only in our hands but our minds as well.

Chapter 2 expounds on developing number and operation sense, the foundation for all mathematics. Without a child being able to grasp that a numeral has a quanity or value attached, there is little on wich to develop concepts of one more, some, all, less than and greater than, half, etc. Number sense used to come in part when children played board games, were engaged in pretend play, and in the magical world of Kindergarten with sand and rice tables and centers filled with a myriad of physical objects to be counted, stacked, and manipulated. Many children now have little opportunity to "play" with simple toys that have to be handled and positioned. I see more and more students who have had 4 to 5 years of school and no sense. How then does opertaion sense emerge? There are several activities in this first chapter that will be immediately implemented in math class to reach those students who are significantly deficit in this area.

Chapter 3 adheres to the concept that all math facts are related and that by building this concept in a physical way through manipulatives helps a child see that math goes beyond symbols and cooresponding rules that must be memorized and time tested. By learning how facts of addition/subtraction and mulitplication/division are related at a basic level, then the understanding and manipulation of larger numbers can be made by taking large numbers apart and solving problems in smaller chunks. The strategies for multiplication facts were taught to us in a math workshop earlier this year. I have already begun to use "doubles" for times two and am working on the "clock facts" for times five.

Chapter 4 presented challenges for those of us who religiously learned our algorithms just as our teachers presented them. If you were lucky, you had a teacher that presented more than one way to do a problem. I learned exactly one way to do each operation and thankfully master more the required ones to get me through high school. I t was not until college that I learned there is more than one way to solve most any problem. The idea of grounding mathematics in everyday life is not new. Math was invented in ever increasing complexity to solve the problems of life encountered everyday. This chapter is rich with expanding concepts of invented strategies that are constructed by the learner and is reliable. the underlying basis is a well developed number and operation sense. Practice by the teacher must be done ahead of time to be able to provide guidance and encourage students to step away from looking at the teacher for the "right" way of performing an operation.