# Text Reflections

### Ch. 14, 15, 16, 17

## Chapter 14

** Algebraic Thinking: Generalizations, Patterns, and Functions**

1. Algebra is a useful tool for generalizing arithmetic and representing patterns and regularities in our world.

2. Symbolism, especially involving equality and variables, must be

well understood conceptually for students to be successful in

mathematics, particularly algebra.

3. Methods we use to compute and the structures in our number system can and should be generalized. For example, the generalization

** Algebraic Thinking**

There are three strands of algebraic reasoning, all three infusing the central notions of generalization and symbolization:

1. Study of structures in the number system, including

those arising in arithmetic (algebra as generalized

arithmetic)

2. Study of patterns, relations, and functions

3. Process of mathematical modeling

**The Meaning of Variables**

Expressions or equations with variables are a means for expressing patterns and generalizations. When students can work with expressions involving variables without even thinking about the specific number or numbers that the letters may stand for.

## Generalization with OperationsYoung students explore addition families and in the process learn how to decompose and recompose numbers. The monkeys and trees problem provides students a chance not only to consider ways to decompose 7 but also to see generalizable characteristics, such as that increasing the number in the small tree by one means reducing the number in the large tree by one. | ## Generalization Through Exploring a PatternOne of the most most valuable methods of searching for generalization is to find it in the growing pattern represented with visual or concrete materials. One method of identifying the pattern is to examine only one growth step and ask students to find a method of counting the elements without simply counting each by one. The problem below is an example of a growing pattern. | ## Importance of the Equal SignThe equal sign is one of the most important symbols in elementary arithmetic, in algebra, and in all mathematics using numbers and operations. At the same time, research dating from 1975 to the present indicates clearly that = is a very poorly understood symbol. It is important for students to understand the concept of the equal sign. |

## Generalization with Operations

Young students explore addition families and in the process learn how to decompose and recompose numbers. The monkeys and trees problem provides students a chance not only to consider ways to decompose 7 but also to see generalizable characteristics, such as that increasing the number in the small tree by one means reducing the number in the large tree by one.

## Generalization Through Exploring a Pattern

One of the most most valuable methods of searching for generalization is to find it in the growing pattern represented with visual or concrete materials. One method of identifying the pattern is to examine only one growth step and ask students to find a method of counting the elements without simply counting each by one. The problem below is an example of a growing pattern.

## Importance of the Equal Sign

The equal sign is one of the most important symbols in

elementary arithmetic, in algebra, and in all mathematics

using numbers and operations. At the same time, research dating from 1975 to the present indicates clearly that = is a very poorly understood symbol. It is important for students to understand the concept of the equal sign.

## Chapter 15

## Developing Fraction Concepts

1. For students to really understand fractions, they must experience fractions across many constructs, including part of a whole, ratios, and division.

2. Three categories of models exist for working with fractions, area (example: 1/3 of a garden), length (example 3/4 of an inch), and set or quantity (example: 1/2 of the class).

3. Partitioning and iterating are ways for students to understand the

meaning of fractions, especially numerators and denominators.

4. Students need many experiences estimating with fractions.

Meanings of Fractions

Fractions are a critical foundation for students, as they are used in measurement across various professions, and they are essential to the study of algebra and more advanced mathematics.

** Fraction Constructs**

Understanding fractions means understanding all the possible concepts that fractions can represent. One of the commonly used meanings of fraction is part‐whole, including

examples when part of a whole is shaded.

**Part‐Whole:** Using the part‐whole construct is an effective starting point for building meaning of fractions. Part‐whole goes well beyond shading a region. For example, it could be part of a group of people (3/5 of the class went on the field trip), or it could be part of a length (we walked 3 1/2 miles).

**Measurement: **Measurement involves identifying a length and then using that length as a measurement piece to determine the length of an object. For example, in the fraction 5/8, you can use the unit fraction 1/8 as the selected length and then count or measure to show that it takes five of those to reach 5/8.

**Division: **Consider the idea of sharing $10 with 4 people. This is not a part‐whole scenario, but it still means that each person will receive one‐fourth (1/4) of the money, or 2 1/2 dollars. Division is often not connected to fractions, which is unfortunate.

## Area ModelsIn sharing, all of the tasks involve sharing something that could be cut into smaller parts. The fractions are based on parts of an area. This is a good place to begin and is almost essential when doing sharing tasks. There are many good area models, but circular fraction piece models are the most commonly used for area. One advantage of the circular model is that it emphasizes the part‐whole concept of fractions and the meaning of the relative size of a part to the whole. | ## Length or Measurement ModelsThe number line is a significantly more sophisticated measurement model (Bright, Behr, Post, & Wachsmuth, 1988). Many researchers in mathematics education have found it to be an essential model that should be emphasized more in the teaching of fractions. | ## Set ModelsIn set models, the whole is understood to be a set of objects, and subsets of the whole make up fractional parts. |

## Area Models

In sharing, all of the tasks involve sharing something that could be cut into smaller parts. The fractions are based on parts of an area. This is a good place to begin and is almost essential when doing sharing tasks. There are many good area models, but circular fraction piece models are the most commonly used for area. One advantage of the circular model is that it emphasizes the part‐whole concept of fractions and the meaning of the

relative size of a part to the whole.

## Length or Measurement Models

The number line is a significantly more sophisticated

measurement model (Bright, Behr, Post, & Wachsmuth,

1988). Many researchers in mathematics education have found it to be an essential model that should be emphasized more in the teaching of fractions.

## Chapter 16

** Developing Strategies for Fraction Computation**

1. The meanings of each operation with fractions are the same as

the meanings for the operations with whole numbers. Operations with fractions should begin by applying these same meanings to fractional parts.

●For addition and subtraction, the numerator tells the number of parts and the denominator the unit. The parts are added or subtracted.

●Repeated addition and area models support development of

concepts and algorithms for multiplication of fractions.

●Partition and measurement models lead to two different

thought processes for division of fractions.

2. Estimation should be an integral part of computation development to keep students’ attention on the meanings

** Conceptual Development Takes Time**

It is important to give students ample opportunity to develop fraction number sense prior to and during instruction about common denominators and other procedures for computation.

The Common Core State Standards suggests the following developmental process:

Grade 4: Adding and subtracting of fractions with like denominators, and multiplication of fractions by whole numbers.

Grade 5: Developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit

fractions).

Grade 6: Completing understanding of division of fractions.

** A Problem‐Based Number-Sense Approach**

Students should understand and have access to a variety of ways to solve fraction computation problems. In many cases, a mental or invented strategy can be applied, and a standard algorithm is not needed. The goal is to prepare students who are flexible in how they approach fraction computation.

The following guidelines should be kept in mind when developing computational strategies for fractions:

1. Use contextual tasks.

2. Explore each operation with a variety of models.

3. Let estimation and informal methods play a big role in the

development of strategies.

4. Address common misconceptions regarding computational

procedures.

** Computational Estimation**

Estimation is one of the most effective ways to build understanding and procedural fluency with fractions. There are different ways to estimate fraction sums and differences :

1. Benchmarks

2. Relative size of unit fractions.

## Chapter 17

** Developing Concepts of Decimals and Percents**

Decimals are critically important in many occupations. For nurses, pharmacists, and workers building airplanes, for example, the level of precision affects safety for the general public. Because students and teachers have been shown to have greater difficulty understanding decimals than fractions, conceptual understanding of decimals and their connections to fractions must be carefully developed.

**Extending the ****Place-Value System**

Before exploring decimal numerals with students, it is advisable to review ideas of whole-number place value. One of the most basic of these ideas is the 10-to-1 multiplicative relationship between the values of any two adjacent positions.

**A Two-Way Relationship:** The 10-makes-1 rule continues indefinitely to larger and larger pieces or positional values.

**Regrouping:** Even at this stage, students need to be reminded of the powerful concept of regrouping. Flexible thinking about place values should be practiced prior to

exploring decimals. Students should revisit not just making 1 ten from 10 units, but thinking about regrouping 2,451 into 24 hundreds, 245 tens or 2,451 ones.

**The Role of the Decimal Point:** Students must know that the decimal point marks the location of the ones (or units) place. On many calculators, when there is a whole number answer no decimal point appears—only when the ones place needs to be identified will the decimal point show in the display. Students also need to see that adding zeros to

the left of a whole number will have no consequence and adding zeros to the right of a decimal fraction will not change the number.