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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 3rd 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.
Chapter 11: Too often students are presented with questions to answer and the data with which to answer them. Although the questions maybe of interest from the standpoint of data analysis, they are not especially interesting to students or relevant. Van de Walle and Lovin state that students be given the opportunity to generate their own questions, decide on appropriate data to answer their questions, and determine methods of collecting data. We tend to gather data simply to make a graph. When students ask their own questions, data collection has more value. Teachers focus on how to graph, whereas the focus needs to be on how are graphs constructed and what information is relevant to constructing one. Attention is given to several kinds of graphs that are commonly used in the marketplace (pages 331-336). Terms such as mean, median, and mode in relation to 3rd through 5th grades should be discussed and developed. Activities such as "Leveling the Bars" and "The Mean Foot" (page 326) help students investigate these concepts. The term "mean" or "average" should be laid down in 3rd grade to lay the foundation for median and mode later.
Chapter 12: This chapter emphasizes the references to probability that are all around us, from weather forecasting to medical research. The basic concept the "chance" has no memory is important to realize. "Design a Bag" (page 342, activity 12.2) will work well extending probability lessons in Saxon math. Van de Walle and Lovin stated that "percent" and "fraction" language should not be used at this time. There are many activities that provide opportunities for students example to discover the ranges of chance from impossible (0) to certain (1) with 1/2 being an equal chance of an event happening. The concepts that events are either "independent" and "dependent" are important for students to explore and understand.
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