Introduction
This is part of a series of second, fifth, and seventh grade Mathematics Program Reviews. This review includes a summary of the structure of the program, evaluations of a selected set of content areas, and evaluations of program quality. Ratings in these areas were made on a scale from 1 (poor) to 5 (outstanding). The overall evaluation was made using the traditional system of letter grades. For details of the methods used in this evaluation see Methods for Fifth Grade Program Reviews.
Student Text Structure
The student text is 570 pages divided into 140 Lessons. Most lessons address a concept or skill, but a few address more than one and a few appear as multiple parts in more than one lesson.
Sometimes, the sequence of topics across lessons jumps from one topic to another, for example:
Other times, the topics across lessons are highly related, as in:
Lesson structure is consistent throughout the text. The lesson begins with a boxed set of activities - Facts Practice, Mental Math, and Problem Solving. The Facts Practice is just a reference to a worksheet to be completed at the outset of the day's class. These are often repeated for several days. The Mental Math section is an assortment of problems which presents a short list of items in varied content areas that students should be able to solve mentally. The Problem Solving section presents a single application problem. These two sections are intended to be done orally in a whole-class situation at the outset of the day's lesson.
Typical lessons then consist of 1 to 3 pages of exposition. The exposition takes a variety of forms, although it is typically some explanations followed by examples and their solutions. Other times, it contains situations and activities.
The exposition is followed by a short Practice section presenting roughly 3 to 10 problems on the content of the lesson. Then, the problem set for the lesson is given. A typical problem set runs about 2 pages. The problems are mixed, with only a few applying to the current lesson and most rehearsing the material from prior lessons. Thus, the coverage on any single topic extends well beyond it's lesson.
The back of the book contains an appendix with supplemental practice problems for over 50 of the lessons, followed by a glossary and an index.
Content Area Evaluations
Multiplication and Division of Whole Numbers [4.2]
Multiplication of Whole Numbers
Topics related to the multiplication of whole numbers appear in 8 lessons scattered throughout the first half of the text.
Lesson 13 reviews the meaning of multiplication as repeated addition and the use of the times sign. Lesson 15 reviews the patterns found in the multiplication table, the use of the rows and columns to find a product, and the terms factor and product. Lesson 17 reviews the standard algorithm for multiplying by a single digit. The carrying process and the place values of the digits are clearly explained.
Lesson 18 makes it clear that the process of multiplying three factors is to multiply two of them and then multiply the product by the third factor. It also notes that any pair can be selected first, and that the order selected can influence how difficult the problem is. This helps students choose the easiest method, but no reference to properties is given. Lesson 30 extends single digit multiplication to multiplying by multiples of 10 (or 100). By relating to the prior work on three factor multiplication and the factors of numbers the lesson shows why the extra zeros can just be appended to the product.
Lesson 57 extends the standard algorithm to 2-digit numbers by reference to the factors of the two digit number and the material on multiplying by multiples of 10 given earlier. It is illustrated as two separate multiplications first, and then the steps for recording the process in the standard algorithm are given. The term partial products is introduced in the process, but properties are not referenced. Some of the practice problems are money amounts in decimal notation. Again, the explanation is very clear and the place values of digits in the procedure are noted.
Lesson 62 extends multiplication to multiplying by 3-digit numbers. The exposition is similar to that used for the 2-digit case. There are 5 problems in the Practice section. Two of the problems in the Problem Set are of this type, mixed with 23 problems reviewing prior work. Most of the lessons in the rest of the book contain one problem of this type. This is the spiral approach followed throughout the text.
Lesson 63 explains how the process may be simplified when one of the factors contains a zero digit. There are 8 problems of this type in practice and another two in the problem set.
Thus, the text covers multiplication from 1-digit through 3-digit numbers. In the 3-digit case, both factors are 3-digits. The standard algorithm is explained clearly, as are techniques for simplifying the process for multiples of 10 or for factors containing zeros. The explanations of these procedures do not make reference to the properties of numbers, but are otherwise very clear. The distributed practice appears to give students sufficient exposure to master the material presented.
Division of Whole Numbers
Topics related to the division of whole numbers appear in 14 lessons distributed throughout most of the text.
Lesson 19 introduces division as searching for a missing factor (a topic already addressed), the division box notation, and fact families for multiplication and division. Lesson 20 introduces three notation systems for division. Lesson 22 introduces the case where there is no whole number answer, the concept of a remainder, and the notational convention for writing remainders. It notes that how we deal with remainders depends upon the question that is asked, but does not explain further at this point.
Lesson 26 introduces the steps in following the standard division algorithm carefully. The place values of the digits being used in the examples are attended to. The lesson also introduces multiplying and adding the remainder to check the calculation. The lesson does make clear that there is a division in each step even when the answer for that division is zero. However, this topic is further extended in Lesson 35 where the steps are illustrated for this situation.
Lesson 43 introduces pictures of mixed numbers (such as 2 ½ pies). This is used to introduce showing the remainder as a fraction. The explanation is clear and the practice problems use only simple fractions.
Lesson 45 covers short division. Since the standard algorithm has already been introduced, the treatment of short division is easily grasped. Lesson 47 extends the discussion of showing the remainder of a fraction to show what this might mean in practice, such as sharing 14 pizzas among 3 classes means giving 4 2/3 pizzas to each class as opposed to just giving 4 to each class and having 2 left over.
Lesson 61 begins to extend the division algorithm by moving to division by multiples of 10. Some of the dividends and quotients are money amounts in decimal notation. Lesson 64 focuses on terminology, introducing the terms divisor, dividend, and quotient. Lesson 65 extends the discussion of fractions in quotients, showing clearly where the numerator and denominator come from.
Lessons 102, 103, and 104 extend the division algorithm to 2-digit divisors. Lesson 102 begins with an application illustrating the use of this process, and gives the steps in a simple example using the standard algorithm and remainder notation. Lesson 103 introduces one way to estimate the first digit of the quotient. Lesson 104 shows another technique, and extends this to estimates for the second digit as well.
Thus, the text covers division by 1- and 2-digit divisors. The steps in the standard algorithm are clearly explained such that students could learn them from the text. Short division is covered, as is the use and meaning of representing the remainder as a fraction. Checking division using multiplication is covered, but the text falls short of using 3-digit divisors.
Decimal Multiplication and Division [4.3]
Decimal Multiplication
Decimal multiplication is addressed beginning with Lesson 119 in the latter part of the text. The first lesson on this topic introduces the concept by asking about and explaining a tenth of a tenth. The explanation is given in the text with illustrations. The solution, one hundredth, is illustrated. The process is annotated both as the multiplication of fractions and the multiplication of decimals. Exposition continues to deal with the procedures for the example - computing as in the whole number case and then placement of the decimal point.
In an interesting approach, students are instructed to copy and study three worked out examples. The text then explains the procedure again, this time with the mnemonic multiply, then count. The steps are reviewed. A small chart is given for the methods in addition, subtraction, and multiplication of decimals. The chart serves more as a reminder than to clarify. Eight practice problems are given, some with two decimal numbers and some with a decimal number and a whole number.
The second lesson for this topic extends the multiplication of decimals to cover the situation where zeros must be added before the significant digits in the product. The method is illustrated first, then explained further by reference to fractions. Another example is shown worked out with explanation, and 10 practice problems are given.
The third lesson on this topic deals with multiplying decimal numbers by powers of 10 including 10, 100, and 1000. To insure that the meaning is clear, the text presents the topic as moving the digits to the left to indicate greater place values. Only after this is explained is shifting the decimal point to the right introduced. One example is used throughout most of the exposition. A second example is worked briefly. Nine practice problems are given.
In summary, multiplication of decimal values is covered in three lessons. The explanations and examples are clear. The practice problem sets are on the small side as is typical in this program. These will only be followed by a small number of practice problems in later lessons since this topic appears near the end of the text. The operations do not extend beyond the number of significant digits used in whole number operations. Rounding a decimal product is not addressed.
Decimal Division
Division of decimal numbers begins in Lesson 129. The lesson addresses the division of a decimal by a whole number. The exposition simply instructs the student to use the standard algorithm and place the decimal in the quotient directly above the decimal in the dividend. No further explanation is given at this point. Two worked-out examples are given followed by 9 practice problems.
The next lesson extends division of decimals by whole numbers by covering the case where leading zeros must be added between the decimal point and the significant digits of the quotient. The first example is done with $0.12 divided by three, so that students can see that $0.04 is the correct answer, and thus that an extra zero is required. Two examples are given worked out with annotations. This is followed by 8 practice problems.
The next lesson makes the observation that division of decimals is not usually expressed with a remainder, but usually the quotient is solved to the point that there is no remainder. The case of repeating decimals is not mentioned. Two worked out examples and 9 practice problems follow.
Dividing decimals by 10, 100, or 1000 is addressed next. As was the case with multiplication, the discussion makes a point of noting that the digits are changing place value rather than to simply say that the decimal point is moving. Two examples follow that are worked out and explained in detail, followed by 12 practice problems.
Finally, the text addresses the case of the division of a decimal number by a decimal number. The distinction between this case and division of a decimal by a whole number is pointed out first. Then, the technique - moving the decimal in the divisor and the dividend the same number of places in the same direction until the divisor is a whole number - is given. The reason this works is explained in detail by representing the division as a fraction and multiplying by an equivalent of 1, e.g. 10/10. Again, this is added to a table used to help remember operation techniques with decimals. Another example is then worked with annotation, followed by 9 practice problems.
In summary, division of decimals is covered in five lessons. The explanations and examples are clear. The practice problem sets are small given the spiral nature of this program. However, these will only be followed by a small number of practice problems in later lessons since the topic appears near the end of the text. The operations do not extend beyond the number of significant digits used in whole number operations. Repeating decimals are not addressed, nor is rounding of the decimal quotient.
Area of Triangles [1.0]
This topic is either not present or appears to such a minimal extent as to be effectively not present.
Negative Numbers [4.0]
Negative numbers are introduced at the end of the text. First, negative numbers are introduced on the number line. Their representation with a leading negative sign is noted. Next, the negative result when a larger number is subtracted from a smaller one is noted. A reference to temperatures below zero is then given as one use for negative numbers.
Next, an example of a subtraction problem (two positives) with a negative answer is given. Using the number line and counting down to the negative solution is illustrated. Six simple subtraction problems with negative solutions are then given. Since this is near the end of this text, students will not have opportunity to rehearse this material in subsequent lessons this year as is the style for this text.
The final lesson expands the coordinate grid to four quadrants and uses the new knowledge of negative numbers to locate and plot points in all four quadrants.
Thus, negative numbers are introduced as a concept and represented on the number line. The negative result of subtracting a larger (positive) number from a smaller one is covered. The addition and subtraction with positive and negative numbers is not covered. The use of negative numbers to produce the four quadrants of the coordinate plane is introduced.
Powers, Exponents and Scientific Notation [1.0]
Although square units notations are mentioned, this topic appears to such a minimal extent as to be effectively not present.
Program Quality Evaluations
Mathematical Depth [3.6]
The mathematical depth supported by this program is reasonable. Understanding and skills are well supported for whole number multiplication through 3-digit factors and for whole number division through 2-digit divisors, although 3-digit divisors would be preferred. Treatment of remainders and their meaning is well supported. Multiplication and especially division of decimals is well supported for this grade level. However, these topics tend to fall toward the end of the year and will need reinforcement in the next grade level. The treatment of negative numbers is adequate as an introduction but does not extend to operations with negative numbers. Area of triangles and powers and exponents are not covered in this grade level.
Quality of Presentation [4.4]
The presentation of material is fairly unique and follows a relatively consistent pattern from lesson to lesson. The examples and explanations are strikingly clear and the instructional design should lead to efficient learning. The consistent structure and clarity of presentation should combine to produce a program that is easily implemented in the classroom. Despite the rather stark nature of the student text, attention to conceptual understanding is provided. Although usually clear from lesson titles and contents, statements of objectives are often not explicit.
Quality of Student Work [3.9]
The student work is presented as a small set of problems on lesson content plus a mixed set of problems covering many areas that appeared in prior lessons. In this way, students experience distributed practice of content areas many times over the course of the year. In this way, the student work is of reasonable quantity for most topics. Those topics that appear late in the year, however, will not have the opportunity for distributed practice and will thus have to be reinforced in the next grade level. The range of student work is not limited to the lowest achievement levels as in many programs, although support for learning at the highest levels is lacking.
Overall Program Evaluation
This program provides a clear presentation and reasonable degree of student work that should support achievement to moderate levels. More advanced topics, such as areas of triangles and powers and exponents, are simply not addressed, although a moderate introduction to negative numbers does appear. Thus, although achievement at the high levels expected in this review is often not supported, the support for more moderate achievement levels is strong.
| Prior | Contents | Next |