Methods for Fifth Grade Program Reviews

The fifth grade mathematics program reviews were accomplished by surveying the instructional materials provided by the publisher. The student text, where available, was emphasized so that the review would focus on the material the program presents to students. In cases with no student text, the worksheets and other materials that constitute the presentation were evaluated. Frequently, a great deal of supplemental material is available for a program. It is beyond the scope of these reviews to address each element in a wide range of supplemental materials.

A sampling of topic areas was selected for in-depth evaluation. Ratings were made for each of these topics and for the program as a whole as described below.

Topic Selection

A sampling of topic areas was selected for detailed evaluation. These were selected to cover a range of material including both major and minor aspects of the fifth grade curriculum. The topic selections were also designed to range from material that might be newly introduced in the fifth grade to topics that should be reaching mastery by this level. Together, the topic selections cover a good portion of the material expected in the fifth grade, but it should be emphasized that they are just a sample of the entire curriculum. While the topics are not meant to cover the entire curriculum, they are designed to be sufficient to give a clear impression of the features of presentation and an assessment of the mathematical depth and breadth supported by the program.

Within each topic area, expectations were developed for what should appear in the fifth grade curriculum. In each case, achievement expectations for students were guided by the standards of learning in the Mathematically Correct Standards and the San Diego Mathematics Standards. These provided broad guidelines for what could be expected in the curriculum. However, it should be recognized that standards and curriculum contents may not always align. Since standards of learning mark points at which students are expected to have gained explicit knowledge and competencies, it is often appropriate that topics will appear in the curriculum ahead of their corresponding references in the standards. Therefore, some attention was also given to standards for grade 6 and even grade 7. The details in these cases will be elaborated below.

Topics Reviewed

The topics selected for review are summarized below. In each case, the reason for the selection of the topic is given. The development of the expectations for that topic is summarized and the contents looked for in the review process are noted. These selections are not designed to cover the entire fifth grade curriculum. Instead, the objective was to provide a sample of topics that could address the depth and breadth of content coverage.

The topics selected for review in the grade 5 materials include:

Multiplication and Division of Whole Numbers
Decimal Multiplication and Division
Area of Triangles
Negative Numbers
Powers, Exponents, and Scientific Notation

The summary for each of these topic areas follows.

Multiplication and Division of Whole Numbers

Because this topic is so important to the curriculum, multiplication and division will be discussed separately.

Multiplication of Whole Numbers

This topic was selected for review as it constitutes a major component of the elementary curriculum that should reach mastery by grade 5.

For students to be on track for the study of algebra by grade 8, the development of whole number multiplication should reach mastery by grade 5. This is reflected in the established reference standards that clearly point to mastery in this area.

With respect to the multiplication of whole numbers, the benchmark for grade 5 in the Mathematically Correct Standards is that students will multiply two factors of up to four digits each. The San Diego Mathematics Standards simply expect that students will be able to multiply whole numbers. This is meant as the final, mastery standard.

In evaluating the presentation of multiplication, the primary consideration is that the program design will bring the students to proficiency and accuracy in multi-digit cases by grade 5. Students should understand multiplication and their multiplication processes, and they should use multiplication in application problems. They should use estimates to judge the reasonableness of their solutions. Attention should be given to place value in algorithms, even if this has already been done in earlier grades. The use of the properties of numbers with respect to multiplication is a useful adjunct in the discussion of multiplication, particularly with respect to simplifying multiplication problems for mental solutions. The multiplication of three factors is also a useful topic in this grade level.

Division of Whole Numbers

This topic was selected for review as a major component of the elementary curriculum that should also reach mastery by grade 5. Although the separation of multiplication from division is somewhat artificial, it makes the review process more organized.

For students to be on track for the study of algebra by grade 8, the development of whole number division should be essentially complete by grade 5. The benchmark for grade 5 in the Mathematically Correct Standards is that students will divide dividends up to four-digits by one-digit, two-digit, and three-digit divisors. The standards also expect students to know what it means for one number to be divisible by another. As students will also be working with decimal numbers, the standards expect that students will solve whole number division problems with remainders by rounding a decimal quotient.

The San Diego Mathematics Standards expect that grade 5 students will divide whole numbers. Again, this is meant as the final, mastery standard.

The primary consideration in evaluating the presentation of division of whole numbers is that the program bring the students to proficiency and accuracy in multi-digit cases by grade 5. Special attention to situations with zeros in the quotient can be helpful. Students should understand division and their division processes, and use division to solve problems. In addition, even if it has been covered before, attention to the various treatments of remainders should be given in grade 5. This includes expressing the quotient as a fraction or as a decimal number. Attention should also be given to mental arithmetic cases.

Programs may vary in their coverage of these topics as a function of where they expect students to be in their progress as they enter grade 5 and how much review the program authors feel is indicated. In getting to the goals for grade 5, programs may therefore vary in the topics they present - essentially they may assume different starting points. However, not all of the introductory topics are required or even indicated. What is critical is that the grade 5 objectives be well-supported.

Decimal Multiplication and Division

As in the whole number case, decimal multiplication and division will be addressed separately.

Decimal Multiplication

This topic was selected for review as it constitutes a large component of the curriculum that should be nearing mastery by the end of grade 5.

For students to be on track for the study of algebra by grade 8, the development of decimal multiplication should be reaching mastery by grade 5. This is reflected in the established reference standards that clearly point to mastery in these areas.

The benchmarks for grade 5 in the Mathematically Correct Standards expect students to multiply two decimal numbers through thousandths by grade 5, and to estimate decimal products by rounding.

The benchmark for grade 5 in the San Diego Mathematics Standards is essentially the same. Students are expected to estimate and find the product of two numbers expressed as decimals through thousandths.

In evaluating the presentation of multiplication of decimal numbers, the primary consideration is that the program design will bring the students to proficiency and accuracy in multi-digit cases by grade 5. This essentially comes down to understanding the procedures for multiplication and for the placement of the decimal in the product, and placement both by estimation and by counting the places in the factors is best. Special attention to the need to add leading zeros in some cases is indicated and for understanding the estimates of the products. Rounding the decimal product is an important skill as well.

Decimal Division

This topic was selected for review as it constitutes a large component of the curriculum for grade 5. However, the expectations do not reach mastery in the case of division of decimals by decimals. On the other hand, work through the division of a decimal by a whole number should be at mastery.

The benchmarks for grade 5 in the Mathematically Correct Standards expect students to estimate and find the quotient given a dividend expressed as a decimal through ten-thousandths and a whole number. Students should also solve division problems with remainders by rounding a decimal quotient.

The benchmark for grade 5 in the San Diego Mathematics Standards also expects mastery of the division of a decimal (through ten-thousandths) by a whole number.

This is not to say that the curriculum should not begin the work with the division of decimals by decimals in grade 5, as this process should reach mastery by grade 6. Indeed, the newly adopted standards for California call for mastery in grade 5. Thus, it is advisable that this topic enter the curriculum in grade 5.

In evaluating the presentation of division of decimal numbers, the primary consideration is that the program design will bring the students to proficiency and accuracy in dividing multi-digit numbers by whole numbers in grade 5. Special attention should be given to rounding the decimal quotient, even in the case of the division of two whole numbers. Attention should also be devoted to leading zeros in the quotient and to the need to append trailing zeros to the dividend in some cases. A brief introduction to the case of repeating decimals would be reasonable at this grade level, since it will show up in student work. It is also desirable for programs to introduce the division of decimals by decimals in grade 5.

Area of Triangles

This topic, although typically a small portion of the curriculum, is selected for review because it presents a unique opportunity in the development of logical thinking that can be introduced in grade 5. Specifically, students in grade 5 should be able to see why it is necessarily the case that A = ½ bh for any triangle, and relate the derivation of this formula to the formula for the area of a rectangle. The derivation can be shown in figures and with the algebraic manipulation of simple equations. It can rely on givens, such as congruent figures having equal areas. In short, this provides an opportunity for grade 5 students to approximate the logical thinking of mathematical proof.

Although programs may vary, programs with good development toward geometry and the symbolic representations of geometric quantities will develop the area of triangles by grade 5. This refers to the area as computed from the base and height, not problems in which the area is computed given side lengths.

The Mathematically Correct Standards expect grade 5 students to find the area of a triangle. The San Diego Mathematics Standards expect students to derive and use the formula for the area of a triangle in grade 4.

In reviewing this topic, students need to be acquainted with finding the base and height of a triangle. Importantly, this should include heights that fall outside of the triangle and the fact that the base need not be at the bottom of the figure. The formula should be developed in a way that students can see that it is necessarily the case. The solution for right triangles may be developed first, but it must be generalized to other cases. Applications for the use of areas of triangles are helpful. Finally, students should combine areas of triangles with areas of rectangles to find areas of other figures that are built from these shapes.

Negative Numbers

This topic is selected for evaluation because it is one that can move earlier into the curriculum than is often the case. While it has been known for decades that children can understand and work with negative numbers at a fairly young age, the topic is often held off until later years. An effective curriculum could go far in advancing this topic by grade 5. In determining what the curriculum at grade 5 might include, it is important to consider the development of this topic from grade 3 to grade 6 in the reference standards.

The Mathematically Correct Standards introduce negative numbers in grade 3, where they expect students to recognize the addition of a negative number as the subtraction of a positive number and to locate zero, positive, and negative whole numbers on a number line. The grade 5 standard focuses on number sense for negative numbers in comparing the value of two negative or positive decimals through ten-thousandths. However, by grade 6 the Mathematically Correct Standards expect that students will be competent with the four basic operations with positive and negative numbers - including decimals and fractions. The grade 6 standards also expect students to plot points on a coordinate plane using ordered pairs of positive and negative whole numbers. To achieve these goals, the curriculum would necessarily include work on negative numbers in grade 5 so that mastery of these many objectives would be possible by grade 6.

The San Diego Mathematics Standards contain a somewhat more elaborate outline for this topic. The standards also begin the introduction to negative numbers in grade 3 with their representation on the number line and use of the number line to solve subtraction problems (two positives) with differences in the range of -10 to -1. In grade 4, the San Diego Standards continue the development including the concept that a whole number and its opposite add to be zero and recognize the addition of a negative number as the subtraction of a positive number. Moreover, the San Diego standards expect that students will graph and name ordered pairs on a four-quadrant coordinate grid in grade 4. For grade 5, the San Diego standards expect students to find the sum of a positive and a negative integer or two negative integers and compare positive and negative whole numbers and decimals. Like the Mathematically Correct standards, the San Diego benchmarks expect students to be competent with the four basic operations with positive and negative numbers by grade 6, and this again is including fractions and decimals. Again, the implication is that the curriculum should begin work in these areas if mastery is to be achieved by grade 6.

In evaluating the treatment of negative numbers, a basic level of coverage might address the meaning of negative numbers (the use negative integers on number line and in the context of temperature, elevation, and owing), ordering and comparisons involving negative numbers, the concept of the additive inverse (even if this term is not used), and that a negative number results when a larger value is subtracted from a smaller one. Some of this, however, could be accomplished prior to grade 5. Better content development would also include use of the four-quadrant coordinate grid and addition and subtraction with negative and positive numbers, including applications of these operations. Programs that effectively introduce multiplication and division with negative integers will put students at the greatest advantage for further achievement in grade 6.

Powers, Exponents and Scientific Notation

This combination of topics was selected as a way to look at content that may be just be beginning in grade 5. Some earlier uses of exponent notation may be found in representing area units with squared notation and volume units with cubed notation, but this usage could be treated more as a notation system for units than a real introduction to powers and exponents. One of the earliest entry points for exponent notation is often in writing the factors of numbers. Another is the introduction of the positive whole number powers of 10 that serve as the first introduction to scientific notation.

To understand the development of these topic areas, some of the expectations for several grade levels must be studied.

The Mathematically Correct Standards introduce perfect squares and the squared exponent in grade 3. By grade 5 students are expected to describe and extend patterns of perfect squares and patterns formed by powers of 10. The standards do not call for reading, writing, and evaluating numerical expressions with exponents until grade 6, when scientific notation is also included. By grade 7 (the pre-algebra year), students will be expected to perform operations with exponents, including operations with positive, negative and fractional exponents. The student should also be able manipulate numbers expressed as powers of 10 and in scientific notation.

The San Diego Mathematics Standards expect students to be able to write a cube number as repeated multiplication by grade 4. Students are expected to read, write, and interpret whole number powers of 10 and to describe and extend patterns formed by powers by grade 5. They are also expected to identify and describe square and cubic numbers. Although students find factors of numbers in grade 5, their representation with exponents is not explicitly stated. In grade 6, the standards expect students to incorporate exponents into their problem-solving strategies and explain the relationship of exponents to repeated multiplication. Scientific notation is not mastered until grade 7 when students also use the laws of exponents to solve problems, extract roots, and determine whole number powers of positive rational numbers.

Thus, by grade 5, the understanding of exponents as representing repeated multiplications should be in place. Students should have some experience with squared and cubed numbers, and should have some exposure to positive integer powers of 10 as an extension of place value and a prelude to scientific notation. They may use exponent notation to show multiples of a factor in factoring numbers. Further development in these topics may be introduced and should be considered as pre-teaching in preparation for topics to come. The reference standards indicate further progress to be made by the end of grade 6, and tremendous growth in this area by the end of grade 7, so that some exposure in grade 5 could be useful.

Content Area Evaluations

For each of the selected topics, a rating was made to reflect the breadth and depth of mathematics learning supported for that topic area. These ratings considered the level of the mathematics presented with respect to the criteria established in each topic area above. However, the rating is meant to reflect the expectation for student learning supported by the instructional materials for that topic. Thus, more than the mere presence of the identified content areas was considered. Other features of the program for the topic area were also taken into account. Specifically, the mathematical depth, the quality of the presentation, and the quality of the student work were also considered. More details on these dimensions are given below as the dimensions used for evaluating the entire program are discussed.

Program Quality Evaluations

From the study of the instructional materials, and especially from the in depth study of the selected topics, ratings were made on several dimensions for each program. Ratings ranged from 1 (poor) to 5 (outstanding) on each of these dimensions.

Mathematical Depth

A single rating for mathematical depth was made based on program contents. This primarily reflects the extent of the mathematics covered relative to the expectations for each of the topic selections. However, the depth of coverage within particular achievement levels was also considered. This insures that programs would not benefit from a superficial coverage of a wide range of achievement targets.

Quality of the Presentation

A single rating for the quality of the presentation was generated based on several considerations. Together, these survey program aspects that can promote student learning. The factors considered in evaluating the quality of presentation include:

Clarity of Objectives

Making lesson objectives clear to students, as well as to their teachers and parents, supports student learning. In the ideal case, clear statements of objectives are consistently presented.

Clarity of Explanations of Concepts and Procedures and Definitions of Terms

To support student learning, explanations and definitions need to be mathematically correct but also need to be clear and understandable to students. Long explanations are not necessarily better, since they may strain student attention or overload students with details. Crisp, clear, and accurate explanations and definitions are desired.

Quality and Sufficiency of Examples

Explanations in mathematics should make effective use of good examples to promote student understanding. Often times, multiple examples are required to illustrate a topic, particularly when different examples illustrate different aspects of the topic. However, in most lessons a few examples will be sufficient. More important is the quality of the examples and their annotations. The examples must complement other explanations and clearly illustrate the important points to be covered in a way that students can easily grasp them.

Efficiency of Learning

This dimension is designed to account for the fact that student time for the study of mathematics continues to be at a premium. To make the greatest achievements over the course of the fifth grade year, there must be a reasonable degree of efficiency supported by the instructional materials. There are several ways that programs can promote efficient learning. Obviously, the program should be free from irrelevant and distracting content. It is also important to maintain a focus on mathematics throughout each lesson. Programs that focus learning activities on lesson objectives and structure lessons in efficient patterns will also be more successful. Efficiency is reflected in programs that support a high level of mathematics achievement in a reasonable expenditure of student time. Some programs are characterized by a fluctuation between efficient instruction and distraction. The sequence of lessons could impact on this rating. Likewise, excessive instruction in the use of calculators, addressing the mechanics of these machines rather than the mathematical focus, could impact negatively on efficiency.

Quality of Student Work

The work that students are expected to complete is critical to an effective mathematics education. The evaluation of student work was based on two considerations:

Quality and Sufficiency of Student Work

In part, the quality of student work depends on the amount of student work required for a particular topic. While student work can be carried to an excessive length, too little practice is an obvious and serious limitation. Mathematics education should not aim for a casual exposure to a great variety of topics, but rather should lead to mastery in selected areas. Student practice must be sufficient to support this development. This does not necessarily mean massed practice on a different and narrow topic area each day. In fact, frequently revisiting a topic after it is introduced can be a good feature.

Beyond quantity, the quality of student work was also considered, and should reflect both adequate symbolic or numeric cases and adequate application problems. Methods that varied to fit the context were also a positive feature. For example, a few problems that address renaming (borrowing and carrying) specifically or problems that ask students to detect errors were positive features.

Range of Depth and Scope of Student Work

As distinct from the sheer quantity of student work and the quality of the exercises, attention was given to the range of depth and scope of the student work. Within any topic area, students should be competent across a range of difficulty or complexity of work. The student work would naturally be expected to build from the easy and simplex to the difficult and complex cases. The consideration therefore focused on the coverage of a range of depth across the student exercises.

Overall Evaluation

Finally, an overall evaluation was generated for each program. These ratings were based both on the topic evaluations and the overall ratings for mathematical depth, quality of presentation, and quality of student work. Among the topic areas, greater weight was given to the major topics - multiplication and division with whole numbers and with decimals - than to the other topics.

Thus, the ratings generated for each program included five ratings of specific topic areas and ratings of mathematical depth, quality of the presentation, and quality of student work along with an overall rating of the program.