JayChesser

     Chapter 9 focuses on developing measurement concepts.  Teachers should think about three areas when thinking about planning measurement instruction.  First, students must understand the attribute to be measured.  Second, that students should understand how filling, covering, matching, or making other comparisons of an attribute with measuring units produces a number called a measure.  Third, students will use common measuring tools with understanding and flexibility.    Students should also be exposed to informal units of measure as well as standard units.  Experimenting with informal units allows students to make comparisons and predictions.  The use of estimation is also encouraged because it helps students focus on the attribute being measured and the process required to measure it. 

     Chapter 10 is on algebraic reasoning.  The basis of algebraic reasoning begins in Kindergarten and should move from a very concrete toward symbolism and functions in the upper grades.  In the third grade, algebraic reasoning is still concrete, but extending into logical thinking and making predictions about patterns and relationships.  It is suggested by the authors that teachers should extend the task of repeating patterns by having students predict what element will be in the 15th or 17th position before starting.  Then have students extend the pattern to verify.  The use of grid configurations (3-square, 4-square, and 5-square wide) allows students to see what patterns develop with a recurring pattern.  "Grid Patterns" on page 292 addresses this type of extention.  Patterns with numbers can go beyond merely skip counting to more involved patterns such as double the previous number, or the next square  number (What's Next and Why).

     Chapter 11 addresses exploring data analysis.  Too often our students are presented with questions to answer and the data with which to answer them.  Although the questions may be of interest from the standpoint of data analysis, the questions are not necessarily of interest to our students.  "Students should be given opportunities to generate thier own questions, decide on appropriate data to help answer these questions, and determine methods of collecting the data." according to this chapter.  We tend to gather data simply to make a graph.  When students generate thier own questions to ask, the data that they collect has more value to them.  How they organize the data and represent it have more meaning.  As teachers, we spend too much time discussing the "how-tos" of graphing.  The issues of analysis and communication should be a teacher's goal.  Not how the graph was constructed.  "In the real world, technology will take care of details of graph construction."  As stated throughout the book, students should be able to express their thoughts about their graphs in writing.  Attention is given to several types of graphs that are used in the marketplace.  These graphs are located and discussed on pages 331 - 336.  In addition, time is spent discussing terms like mean, median,  and mode in relation to third through fifth grades.  These terms should be investigated by students through activities like "Leveling the Bars" and "The Mean Foot" (page 326).  For third grade students, it is suggested to start with the term "mean" or average.  After the ground work is laid for the mean, the other terms can be investigated.

     Chapter 12 emphasizes the reference to probability all around us.  We see probability in the weather forecast to medical research.  The overwhelming concept appears to be that chance has no memory.  It is important for students to understand this concept.  Saxon Math has a series of lessons dealing with probability ranging from impossible to certain.  Several activities in this book extend these concepts in Saxon Math to a different level.  I will probably take a few of the activities to enhance what Saxon does.

     Chapter 5 deals with developing Fraction Concepts.  Manipulatives are strongly advised.  Three types of models are discussed:  region/area models, length models, and set models.  Activities like 5.1 and 5.2 may be helpful.  Students have difficulty when models show fractional parts that are not equal.  Working through these activities allows students to see the difference and have a chance to vocalize or write an explanation of how the parts are equal or are not equal.

     Chapter 6 is on Fraction Computation.  The development of fraction concepts should follow the same path as the development of whole numbers.  Manipulatives to explore and answer simple contextual tasks is paramount.  Just as a number represents a quantity or value, fractions likewise must be constructed and understood by the learner.  Rules should be abandoned until students can bridge the gap between concept and application of the rules.  Begin simple and build from there.  The concepts of multiplying and dividing fractions again begins with informal exploration beginning with simple contextual tasks.  Relating multiplying fractions to whole number multiplication helps students.  Traditional algorithms are not to be taught initially.  Time must be allocated for invented strategies by students.  The most important thing for the teacher to remember is that the strategy must be constructed by the student and that the student must be able to explain his/her strategies and answers.

     Chapter 7 explores the use of Decimals and Percentages.  It begins by discussing ways to help students gain a better understanding of decimals by developing a conceptual understanding between fractions and decimals.  It is suggested that students should discover the relationship between decimals and fractions by reviewing the base-ten system and ultimately learning that this system can be extended to include numbers less than one.  By using the 10 x 10 squares, students can create fractions to represent numbers such as .65.  Students color 65 out of the 100 squares to begin to see the relationship of decimals and fractions, and begin to compare the value.  Teachers should ask questions like "Is this decimal greater than 1/2?"  Students should be allowed to verbalize their answers to ensure concept comprehension.

     Chapter 8 deals with Geometric Thinking and Geometric Concepts.  Students score lower than other countries in the area of geometry.  Because of this, schools are embracing the works of Pierre van Hiele and Dina van Hiele-Geldof.  They believe that students go through five different levels (0 - 4) as they progress through geometric thinking and concepts.  The levels are not age dependent in the sense of developmental stages of Piaget.  Their belief is that geometric experience is the greatest single factor influencing advancement through the levels.  The Hieles believe that the more students play around with the ideas in their suggested activities, the more relationships they will discover.  Teachers are not to tell students about their explorations, but rather facilitate the study.  Teachers should challenge students with questions and allow the students to explain in thier own words what they are doing.  By listening to their answers, teachers can assess student understanding of geometry.

     I think the chapters in this section of reading are very helpful in teaching our third graders.  Content of the activities can be enhanced to challenge students.  So the activities reach all students!

     The foundation for this book centers around the constructivist viewpoint.  This viewpoint emphasizes the fact that all people construct their own knowledge.  The author states "As you read these words, you are giving meaning to them.  You are constructing ideas."  Basically we use ideas we already have to construct new ideas thus developing in the process a network of connections between ideas.  In my undergraduate work, this viewpoint was strongly supported by the college that I attended.  In my short number of years teaching, I have found this viewpoint to be true.

     As teachers we can support this type of teaching in our classrooms by providing an environment that is conducive to the constructivist viewpoint.  They recommend that students be allowed to use manipulatives, symbolism, and even computers as models or tools for learning.  Social interaction with other students is encouraged.  And students should have a set time for reflective thinking and writing.  In the last few math workshops this has been strongly recommended as a way to see if our students truly understand the concepts we are introducing in math.  Students need to realize that there is more than one correct answer.  The writing and social interaction allows for others to see that there is more than one answer.

     This book encourages teachers to make math more meaningful for students.  Problems should be presented in such a fashion that they are within the students' zone of proximal development.  The problems should be concerned primarily with making sense of the math involved in solving the problem.  Students should be able to understand that the responsibility for determining if they are correct or not rests with them.  Students should be able to explain themselves.

      Chapter 2 deals with number sense, a relational understanding of numbers, and operation sense which is a complete and flexible understanding of the operations.  Many activities were included to give the teacher ideas of how to teach from this viewpoint.  I am not sure that I understood all of the fuss about remainders when you are dividing.  "Students should not just think of remainders as 'R 3' or 'left over.'  Remainders should be put in context and dealt with accordingly." 

     Chapter 3 focuses on helping children master the facts.  According to the authors "Mastery of basic facts means that a child can give a quick response (in about 3 seconds) without resorting to nonefficient means, such as counting.  Children simply need to construct effecient mental tools that will help them."  This chapter is full of strategies to help teachers.  Some of these strategies we have implemented before.  Others we will try when we get to that set of facts.

     Chapter 4 presents strategies to help children with whole number computation.  The traditional algorithms that we use in math may not be the most efficient or useful methods of computing.  Flexible methods and invented strategies are encouraged to help children with whole number computation.  Flexible methods involve taking apart and combining numbers in a variety of ways.  Children need a good knowledge of the properties and the operations such as addition to subtraction, addition to multiplication, and multiplication to division.  Invented strategies are those strategies constructed by students.  Success of these strategies requires that the strategy be understood by the one who is using them.  The chapter talks about each of these areas and provides examples of how to use the strategy/method.