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Chapter 10 provided me with an overview of 

the reform in mathematics education, it is important to make it 

clear that mastery of basic facts remains as essential today as ever. 

A major issue is to help teachers at all levels see the necessity and 

the power of students developing number relationships and the 

connections between addition and subtraction. The message of this 

chapter is that drill is both important and effective. HOWEVER, 

and this is a major caveat to the preceding statement, the drill must 

be drill of an efficient strategy with the emphasis on efficient. 

Efficient strategies should be based on meaning and should not be 

seen as cute or clever tricks, nor should they become a barrier to 

learning meaningful mathematics. The text encourages you to base 

addition facts on number concepts. To that end, you will find an 

alternate strategy called “one- and two-more-than facts,” or facts 

with an addend of either 1 or 2. These main strategies then leave 12 

unaccounted for facts or 6 facts and their commutative partners. 

Several additional strategies (including “counting on”) are also 

suggested.

In chapter 11 there is no doubt that place value provides the 

conceptual foundation for all aspects of whole-number and 

decimal computation. In this chapter a significant focus of place-

value development should be based on the patterns and 

relationships in the number system. By engaging in these pre-

computational activities they are actually learning about place 

value and developing computational flexibility at the same time. 

The chapter illustrates the count-by-ones approach to number 

that most children develop as early as kindergarten. Even those 

children who count quantities to 100 and can read and write these 

numbers, are very likely using a count-by-ones concept to 

understand these quantities. Models for base-ten concepts play a 

major role in the development of these ideas. 

Chapter 12 focuses on developmental steps that 

precedes invented strategies is called direct 

modeling, which is the use of manipulative or 

drawing along with counting to represent radically 

the meanings of an operation four-story problem. 

Students using the method direct molding will soon 

use their ideas to methods that do not rely on 

materials or counting. Generally most text books 

teach standard algorithms, but more than century 

of tradition combined with pressures from others 

who were taught that way may result in the 

thinking that there is only one best approach right 

algorithm. Developing the subtraction algorithm is 

the same as for addition algorithm. 

Chapter fourteen provided me with an overview of algebraic 

thinking. One goal of this chapter is to help develop a useful 

concept of what algebra or algebraic thinking is about at the 

K to 8 level. Algebraic thinking or algebraic reasoning 

involves forming generalizations from experiences with 

number and computation, formalizing these ideas with the use 

of a meaningful symbol system, and exploring the concepts of 

pattern and functions. The equal sign is is one of the most 

important symbol and elementary arithmetic, in algebra all in 

mathematical using numbers and operations. It is important 

that students see functions in all of these representations and 

are able to see how each is a different way of seeing the same 

functional relationship.

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