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 594 pages long and includes a glossary and index.
The student text consists of 12 chapters
Each chapter has a theme that is used in some problem contexts, but this is not done in a way that is invasive - i.e., it does not dictate the mathematics nor is the math necessarily within the context.
Each chapter is divided into two or three sections. The chapter beings with a list of these sections and some text and data that can be used in later problems. For example, the chapter on multiplying whole numbers and decimals has All-American Numbers as the theme. Some problems relate to distances around the U.S.
Each chapter then has a team project at the beginning. This project, if completed, could easily consume one day's math time or considerably more. It appears to function primarily to build interest.
The two or three sections within each chapter typically begin with a fairly non-mathematical interest-building context, but this is not time-consuming. Then there is a small section that addresses a prerequisite skill for review before the major section of the text. While a good feature, these typically only cover a narrow content area.
Next a short list of skills to be acquired is given. However, this list is often too vague to constitute statements of objectives. For example, the last section in the chapter on two-digit divisors with whole numbers and decimals contains three items: divide money, solve problems by making decisions, and explore decimal patterns in division.
Each section is divided into 3 to 8 lessons. There are two common formats for the lessons: learn-check-practice and explore-connect-practice.
The learn-check-practice sequence is more traditional exposition in that a concept or procedure is typically explained and examples are given. However, the explanation is often incomplete and the details are often left unstated. Often, a Talk about it prompt at the end of the learn part of the lesson leaves it up to students to generalize to some element of the overall concept or procedure. The check sequence contains some easier problem types and may ask for explanations or ask probing questions. The practice section typically will present a mix of numeric and context problems, and ask for some explanations and may ask for a journal entry. This is often followed by a small mixed review and test prep section of about a dozen problems covering material presented earlier to end the lesson.
The structure of the explore-connect-practice lessons is similar overall. In the explore phase the material is presented in a discovery model with students typically instructed to work cooperatively. The connect phase is more apt to give information or details or modest explanation rather that provide simple problems. The practice section has a structure similar to the other lesson design above.
Some chapters include an extra stop and practice section where additional practice is most indicated. There are six technology experiences and six games scattered throughout the text.
Each chapter ends with:
The text back matter includes a skills practice page for each chapter, a table of measures, a glossary and an index.
Content Area Evaluations
Multiplication and Division of Whole Numbers [3.7]
Multiplication of Whole Numbers
Multiplication of whole numbers is addressed along with multiplication of decimals in chapter 3.
In lesson 1 students explore the commutative and associative properties of multiplication and multiplying by 10, 100, and 1,000. However, the exploration is not clearly directed at learning these properties. The properties are stated, but how these are useful in simplifying multiplication problems with more than two factors is not explained.
In lesson 2 students estimate products. Students are told, You can use rounding to estimate a product and You can also use estimation to check the reasonableness of an answer. An example of each is given.
In lesson 3 students ... will learn how to multiply by 1-digit and 2-digit numbers. The standard algorithm is presented for multiplying by a 1-digit number in one example which includes one carried digit. The procedure is not explained, thus students would not be expected to learn it here. Instead, this must be considered as review. Then two examples are given for multiplying by a 2-digit number. Explanations within the examples are not sufficient for students to learn this procedure. In the first example, the explanation is Step 1) Multiply the ones. Regroup if necessary. Step 2) Multiply the tens. Add the extra tens. Regroup. Step 3) Add the products. The explanation for the second example is even shorter. Again, these are not sufficient to explain the process in a way that students could reproduce it. There is no attempt to explain the workings of the algorithm.
Student exercises include multiplication problems up to 3-digits by 2-digits. The problem count is sufficient for one lesson, but not sufficient for learning the skill
This is followed by an in-class game that involves 2-digit multiplication. The remaining lessons in the first section of chapter 3 are devoted to the distributive property, choosing a calculation method, patterns with multiples, and an application problem. The lesson on choosing a method of calculation presents one problem solved mentally and with a calculator. It gives a few of the reasons student might use one method or another, but other important reasons are not given. Thus, whole number multiplication is addressed through multiplying by 2-digit numbers but not beyond. It is addressed in a way that might be adequate for review, but leaves much unanswered.
Division of Whole Numbers
Division of whole numbers is addressed along with division of decimals in Chapters 4 and 5.
In Chapter 4, the first lesson reviews the meaning of division by illustrating finding the number of equal groups given the group size and the total or the group size given the number of equal groups and the total. Also illustrated is the use of division when one factor is missing but the product is known. The second lesson extends basic facts by powers of 10 to larger numbers for mental division. The third lesson deals with estimated quotients and compatible numbers.
Lesson 5 provides an interesting introduction to the standard algorithm for division. Working together, students divide $3.75 into two equal shares. First the dollars are distributed, one per share. The extra dollar is regrouped by changing it for 10 dimes. Then, the dimes are distributed and the extra dime is then regrouped by exchanging it for 10 pennies. These are divided equally with one penny left over. Other divisions of money amounts with regrouping into 3, 4, 5, and 9 equal groups are then done.
Next, this process is illustrated in the text for dividing $4.36 by 3. At each step, the quantity assigned to each of three groups and the quantity remaining is illustrated. Next to this the standard algorithm notation is give for the same step, and an explanation of the step next to the notation. Using this method, the digits in the quotient and in the computational steps can be referred to as the dollars, dimes, and pennies. In this way, the place value of every digit in the standard algorithm can be understood and followed, and the regrouping remains conceptually the same as and exchange of dimes for a dollar or pennies for a dime.
This is followed by 4 practice problems where the standard algorithm is indicated but some digits are missing at each step for the student to fill in. Then 5 numeric division problems are given.
Although this exploration involves decimal notation since money is used, it is positioned where the introduction to the division of whole numbers would be anticipated and works well for that purpose. In fact, it is through the used of the monetary context that the place values of each digit in the standard algorithm are easily followed.
The next lesson addresses dividing by 1-digit divisors. Estimation is used to help determine where to place the first digit in the quotient. Each step is indicated with moderate explanation, including a check by multiplication. A moderate number of numeric and application problems follow. Then, lesson 6 presents an application problem with division. In this application, one more quotient unit is needed to account for the remainder (You will need 4 cartons of 8 books per carton to have enough books for each member of a class of 29 students). Other application problems follow.
Lesson 7 specifically addresses the placement of the first digit. Three more worked out examples and a moderate number of exercises are given, still only dividing by 1-digit numbers. Lesson 8 addresses zeros in the quotient to guard against the potential for misplaced digits and place value errors. A special section of extra practice on division follows, including 12 problems where students are to check for errors and correct them. The section of the chapter ends with Lesson 9 which deals with computing the mean (a division application). The remainder of Chapter 4 deals mostly with decimals.
Chapter 5 introduces dividing by two-digits. The first lesson addresses the simple cases where powers of 10 are involved and the quotients can be found mentally. Lessons 2 and 3 deal with estimation of quotients. Lesson 4 returns to the standard algorithm for 2-digit divisors. Examples and problems deal with the case where the quotient is a single digit with a remainder. Lesson 5 extends this to larger numbers so that the quotients are also 2-digit numbers with remainders. Examples and ample problems are provided. Lesson 6 addresses choosing a calculation method. Lesson 7 returns to the issue of zeros in the quotient and extends the range of the dividend to 5 digits. The remainder of the chapter addresses decimal values.
Thus, there is reasonable coverage of whole number division through 2-digit divisors. Material on estimated quotients is adequate, as is attention to zeros in the quotients. A stronger treatment of remainders could be provided. However, the program does not move students beyond the 2-digit divisor case.
Decimal Multiplication and Division [3.5]
Decimal Multiplication
Decimal multiplication is addressed near the end of chapter 3. In the first lesson, students work together to shade a grid twice, once for 0.5 and once for 0.3. They count the grid squares that are shaded twice and express the result as a decimal. They solve 6 more products with grids and colored pencils, then talk about the patterns they see. The text then illustrates 0.6 of 0.7 in a similar manner. Students find products of two single-digit decimals with grids in the practice problems. They write about the process in their journals.
In the second lesson, the standard multiplication algorithm is used in the examples. In the first case, the annotation indicates that students Multiply as with whole numbers. In step 2, the annotation indicates that The number of decimal places in the product equals the sum of the decimal places in the factors. Two more examples are given, checked by estimation. Students are told to talk about it and to give a rule for placing the decimal. Since the products are money amounts in two of the examples, the values are also rounded to the nearest cent. Then, 10 problems are given for the placement of the decimal point, 34 problems ask for products, and a few application problems are given. This lesson is also followed by a stop and practice section which gives even more product problems.
The next two lessons extend the learning about decimal products. The first of these presents finding high and low estimates for decimal products to give more accurate checking. The next presents the case of adding extra zeros as needed to represent the product. An example is given illustrated with a 10 by 10 grid in which the product is 0.08. Students are told, Sometimes you have to write zeros after the decimal point to get the right answers. Two more examples follow and then the practice exercises.
In summary, the multiplication of decimals is addressed in four lessons. The first uses models and the second uses the standard algorithm. The next two lessons provide extensions. The clarity of the explanations is fair and the examples and problems are adequate. Refined estimates are addressed. However, the level of the presentation is generally low.
Decimal Division
Division of decimals by whole numbers is addressed in two lessons in Chapter 4. The standard algorithm is used. In the first lesson, students divide money amounts by whole numbers. The first example divides $75.45 by 3. The solution is worked out in 5 annotated steps followed by a check. The decimal is placed by estimation in the first step. The second example is worked in less detail. The third example is solved with a calculator to yield 3.6683333. Students are instructed to round the quotient. Numerous problems follow, often with large dividends but with single digit divisors.
The second lesson is on the division of decimals by whole numbers outside of a money context. In the first example, the decimal is placed by estimation. In the second and third examples, a math tip notes that the decimal in the quotient should be placed directly above the decimal point in the dividend. The problem set includes many 5-digit dividends, but the divisor continues to be a single-digit whole number.
The extension to 2-digit divisors does appear in a lesson in Chapter 5 on dividing money, although the 2-digit divisors that do appear are generally simple ones like 50 or 12.
Thus, division of decimals by whole numbers is covered to some extent, but the division of two decimals is not addressed. In the work that does appear, most divisors are single-digit whole numbers. For the material covered, the number of problems is sufficient, but the difficulty level is low.
Area of Triangles [3.4]
The area of triangles from base and height is addressed in two lessons near the end of chapter 10. In the first lesson, students explore the area of rectangles cooperatively using a geoboard. They form rectangles and record the length, width, and area of each in a table. They form two triangles
by placing a rubber band between two opposite corners. They check that the two triangles are the same size. They record the area of the triangles (they are supposed to discover that the area is half that of the rectangle). They are asked to describe patterns in their data tables and then state how to find the area of a triangle that is half of a three-unit by two-unit rectangle.
Next, students are told that they ... can use the base and height to find the area of a right triangle. The terms base and height are highlighted, but not defined. The glossary entry for base defines it as, the bottom of a polygon or solid. This definition will lead to difficulty in finding the area of triangles. Students are then given a worked-out example that includes the formula A=1/2 X (b X h). The example works out the substitution numerically, but does not explain the process further. Students are given 11 problems represented graphically, one word problem, and a journal writing assignment to end the lesson.
The second lesson will generalize this result to the case for any triangle, not just right triangles. Again in a cooperative exploration, students draw a triangle inside a rectangle. One entire side of the rectangle must correspond to the base of the triangle. Then, they cut out the triangle and the unshaded parts of the rectangle and are supposed to discover that the unshaded parts will just cover the triangle.
Next, students are told that A = 1/2 X (b X h) can be used to find the area of any triangle. Two examples are worked out numerically without further discussion. A note in the margin indicates that the height is always perpendicular to the base.
This section is followed by 11 problems presented graphically, two word problems, and a journal writing assignment. In one of the graphic problems, the side selected as the base must not be on the bottom of the triangle if the line that represents the height is to fall inside the triangle. The drawing identifies the two measurements needed to correctly apply the formula, but doesn't attempt to clarify why a certain side was selected as the base to show the height measured inside the triangle. Based on the text, students will miss this point, and will likely not realize that a measure outside the triangle could represent the height. They will also not be likely to see that two rectangles can be formed for non-right triangles and that the area of the triangle will be the sum of half of the areas of the rectangles. These lessons do not present examples of the use of knowing the area of triangles, although the next two lessons use the area of rectangles and triangles in finding the areas of other polygons.
Negative Numbers [1.0]
This topic is either not present or appears to such a minimal extent as to be effectively not present.
Powers, Exponents and Scientific Notation [1.5]
The only discussion of exponents and powers appears in an exploration of place value in chapter 2. The powers of 10 expressed as factors and in exponent notation are covered here. Students first explore some numbers that are powers of 10 and answer questions about them. A chart is then given that shows the powers of ten as numbers, as repeated multiplications, and in exponent form. Arrows point to the exponent and base. There is no real explanation. Students answer further questions about powers of 10 in exponent notation. Thus, the coverage is only a very minimal treatment of the topic in question. This is a very basic introduction to the notation and meaning of exponents, and only in the context of powers of 10.
Program Quality Evaluations
Mathematical Depth [3.2]
The mathematical depth of this program is modest. Student mastery of the multiplication and division of whole numbers cannot be expected and will have to continue into the next grade level. Multiplication of decimals could also be further extended. Division of decimals is restricted to instances of division by simple whole numbers. The area of triangles is reasonably well covered, but the more advanced topics of negative numbers and exponents are either missing or not seriously attempted.
Quality of Presentation [3.6]
The quality of presentation is adequate for the level of mathematical depth of the program. Especially noteworthy are the frequently clear statements of objectives. Examples and explanations are also reasonably clear, although some material is left up to students to interpret or explain themselves.
Quality of Student Work [3.0]
The student work provided in this program is adequate for the level of mathematical depth attempted. The quantity of work is generally sufficient, although the range of depth and scope of student work is restricted, reflecting the generally low level of mathematical depth.
Overall Program Evaluation
The overall rating for this program is modest. In general, the level of mathematics supported is below expectations and student achievement will be limited accordingly. The presentation and student work are also modest, so that student achievement is moderately well supported to the level of mathematics presented.
| Prior | Contents | Next |