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 Second Grade Program Reviews.
Structure
This book program comes with student book, a teacher's book and ancillary extra exercise and overhead materials. The book contains twenty-eight chapters plus an unnumbered "Getting Ready" introductory unit. The majority of chapters contain four or five lessons, a chapter test/review and test prep materials.
The teacher's manual recommends one day for most lessons, but a significant fraction of lessons are recommended for two days. Coupled with chapter reviews and assessments, the book easily approaches 170 days book work or related activities. As most of the topics covered are important at this age, it becomes critical for teachers to keep on track. Math must be taught every day, it cannot be skipped just because there is an assembly or a special activity like the Halloween parade.
Although there are exceptions, the book stays admirably clear of calculators and computer games. In general, the book is complete, without much reliance on technology. The pages are clean, without a lot of distracting photos. There is a relatively heavy use of manipulatives even into late in the course. Teachers might be well advised to tailor the use of these to the needs of individual children, allowing those who have mastered the skills to skip the counters, mats, pictures, etc., and even advance on to the enrichment sheets provided in the teacher's manual.
Lessons generally follow the same pattern. The teacher may use one or both of two optional group activities as a getting started activity. The teacher then teaches the content of the lesson and goes through some problems with the students. Students follow the lesson by doing the two pages of exercises in their textbooks. After this, the teacher works with the class, wrapping up the lesson and assessing their understanding and skill informally. Additional exercises are available for reteaching, extra practice or enrichment. Within these guidelines, the teacher is largely left on his/her own as to the exact presentation of the topic. There is relatively little guidance on when or how to use the optional introductory or follow up activities.
A series of activities related to math are given to fill some of the gaps in the day, such as lining up time. Children do things like skip count, practice number facts, determine how many groups of two there are and so on.
Content Area Evaluations
Addition and Subtraction of Whole Numbers [4.4]
Addition and subtraction are largely introduced in two blocks, with a reasonable sequence within and between the blocks. Mathematical vocabulary is introduced appropriately and words such as "sum", "difference", and "addend" are used from the start.
The introductory unnumbered chapter introduces some simple concepts and properties of addition and subtraction. Chapter 1 covers addition up to 18 and Chapter 2 handles the related subtraction facts. In these chapters addition starts with doubles rules and making ten and builds up through all the facts. Number sentences are written from the start and fact families are described. Although there are problems in the book and some extra drill exercises in class, one does not see any assessment that asks children to demonstrate mastery of all the sums to 18, nor is it clear that there are enough interspersed activities to practice these to mastery.
Chapter 3 introduces the concept of place value with finding groups of ten in a picture with more than ten similar objects. This leads to using ones blocks and tens sticks to represent numbers, to regroup more than 10 ones into a tens stick and the left over ones, and to modeling word problems with blocks and numbers. Chapters 10 and 11 cover two digit addition and chapters 12 and 13 cover two digit subtraction. Although the lessons begin with manipulatives, there is clear emphasis that borrowing and carrying (regrouping, renaming, trading) are the ways to deal with these problems. Chapter 26 presents addition and subtraction with larger (up to 999) numbers. This may be a bit late in the book, especially since, in the gap between chapter 13 and chapter 26, there is little opportunity to apply the arithmetical skills learned up through chapter 13, although there is some review and cumulative test problems.
Because of the importance at this age of mastering addition and subtraction of multidigit numbers, a detailed description of the scope of adding such numbers follows. Subtraction follows a parallel path, usually a chapter later.
As noted above, chapter 5 introduces the initial concepts of place value for ones and tens. Chapter 10 starts from this, and connects the basic number facts to the concept via blocks, for example recognizing that 7 ones + 8 ones = 15 ones = 1 ten and 5 ones, which is represented both with blocks and tens sticks and with words as above. The next section builds on this by modeling 2 digit plus one digit addition, with extra ones blocks being grouped into a tens block, and numerical representations of the addition problems being given. This leads into a similar exercise on 2 digit plus 2 digit addition, and then to a lesson in which numbers are written in a vertical column, with special emphasis on alignment of similar places, although the actual addition is still done with blocks. Finally, in chapter 11, the formal carrying algorithm appears with the presence of a box above the tens place in which students write the carried digit. At the same time, students continue to model the addition with blocks and sticks. The procedure is first introduced with two digit plus one digit addition then two digit plus two digit in the next lesson. Only in the third lesson of chapter 11 do students actually carry without using blocks. This lesson also introduces larger numbers (total up to 99) and sums of three two digit numbers (total up to 99). The next two lessons continue to practice these skills, and are followed with a set of word problems invoking these skills. Chapter 26 reprises these skills in the context of sums up to 999, with a relatively quick presentation.
In summary, this is a relatively complete presentation of second grade addition and subtraction, especially if care is taken that students actually master their number facts. There is some concern about the large gap in which there is relatively little application of practice of the skills learned, and about the extended use of manipulatives.
Multiplication of Whole Numbers [2.6]
At this level, multiplication is largely an exposure topic rather than one targeted for final mastery. On the other hand, understanding of the concept should be built and some facts can be learned and applied.
In preparation for this topic, students do skip counting exercises once as a lesson and 20 times as part of the daily routine. The topic itself is covered in 4 lessons (5 class days) of Chapter 27, the penultimate chapter.
The first lesson contains adding equal groups with blocks (for example, groups of 3). This is a standard starting point, but it is well done. Lesson 2 then builds from this to talk about adding equal groups, and the idea that repeated addition can be thought of as multiplication (2 +2 + 2 = 6 is the same as 3 x 2 = 6). It also introduces the "x" symbol for multiplication, the word "product", writing multiplication equations in a horizontal form and introduces a number of facts related to 2 times and 5 times, but not all of them. Lesson 4 builds in some of the 3 times and 4 times facts, but not all, and the final section deals with problem solving.
This book deals well with the repeated addition of equal groups as a way of understanding multiplication, but does not contain an array/area representation that makes some ideas clearer. Initial facts are introduced for a few numbers, but only as they come up in a lesson and not with systematic mastery of any set of facts. This is a start toward preparing students for multiplication, but falls below what this review expects for top marks.
Time [3.7]
Chapter 8 presents concepts of time. The clock face is described and the concept of counting by 5's to find the minutes after the hour is explained. In the first four sections students read and write times to the nearest 5 minutes. One page of problems gives the starting time for an activity as well as the length of the activity (minutes or hours), asking students to compute the final time. Alternative methods to stating the time (minutes before, quarter past, quarter to, half past) do not appear in the text.
Chapter 9 has two sections on using a calendar. The number of days in a week and months in a year are reviewed and an entire year of calendar pages is presented. Questions regarding dates, days of the week, number of days in a month, etc. follow. The next three sections of Chapter 9 return to telling time with sections of "early or late," sequencing events, and reading a schedule. The latter section has some problems requiring the students to determine duration of events, to the nearest half hour. Given the starting and ending times.
Although some opening the day calendar and time exercises are suggested in the teacher's edition, none of these really focuses on stating the date accurately and locating it on a calendar.
Money [3.6]
Most of the money applications occur in Chapters 6 and 7. Chapter 6 contains 4 sections dealing primarily with counting coins. Section 1 presents pennies, nickels and dimes, section 2 adds quarters, section 3 has counting "collections," and section 4 brings in half dollars. Chapter 7 moves into applications. The first two units give "hands on" practice with 1) combining coins two different ways to make a given value, and 2) using the smallest number of coins to obtain a value. Units 3 and 4 have the children compare amounts to prices (Do I have enough to buy the truck? How much can I buy with the money I have?) and then make change. The amounts involved are all under $1.00.
Addition and subtraction with coin amounts under $1.00 are presented in Chapter 13, logically after adding and subtracting with regrouping has been taught. The students do a page of word problems, with coin amounts, in which they must choose the operation (add or subtract) and then perform it. Addition and subtraction of dollar amounts is presented briefly at the end of the book (Chapter 26), after 3 digit sums have been discussed. The two pages of word problems show addition and subtraction with regrouping, but the decimal point never gets a real explanation. The students are reminded that one dollar is the same as 100 cents, but the concepts of wholes and decimal parts is not explained. The point just appears.
Measurement of Length, Weight, Volume and Temperature [2.9]
Measurement is covered in three short chapters (20, 21, 22) entitled "Length," "Perimeter and area" and "Capacity, weight and temperature." The length chapter opens with nonstandard units (paper clips) and proceeds to inches in the next lesson, using a ruler marked only in inches, not in fractions of inches. A warm up activity asks students to measure from the tip of their index fingers to their wrists and then determine, for example, who has the largest and smallest hands. This is interesting activity for getting kids involved personally with measurement, but public measurement of body characteristics always has the danger of generating unpleasant interactions among people (hey, gorilla hands!!). Similarly, a later practice problem set (20.3) has students measure hands, ears, feet and arms. This has the very real possibility of breaking down into an unexpected and unproductive diversity/tolerance activity.
The fourth lesson introduces the foot and notes that 1 foot = 12 inches. The only foot exercises involve drawing pictures of classroom items and classifying them as less than a foot, same as a foot or more than a foot in length. Paths, consisting of straight segments, are measured with 6 inch rulers. This exercise is followed by an exercise in which curved paths are measured by matching to a length of string and measuring the string.
The second chapter on measurement opens with the introduction of a centimeter ruler. The students draw lines having a given centimeter measure. The next lesson introduces decimeter (a unit seldom used in metric applications) and notes that 1 decimeter = 10 centimeters. Students then measure items in centimeters and decide if they are less, more or the same as a decimeter.
Perimeter is defined and measured for very simple shapes. No fractional units are used. After introducing length of curves, there is no application of this skill to perimeter, although two optional class exercises deal with measuring the perimeter of curves using non-standard units in the least sophisticated possible way: Laying objects along the perimeter and counting them. Area is introduced but not named as such. Students do exercises in which they discover how many 1 inch squares fit on given rectangles.
Chapter 22, the final measurement chapter, introduces cups, pints and quarts, with one set of exercises in which children break a given set of pints or quarts into cups. One lesson is devoted to deciding if various objects weigh about a pound and the final lesson introduces the Fahrenheit thermometer. A final exercise set has the children choose the appropriate tool (ruler, measuring cup, thermometer) for a particular situation.
Overall, this text produces, at best, a basic introduction to measurement. Little is offered for the more advanced students. The problem sets are brief and superficial.
Perimeter [3.0]
As noted above, perimeter is introduced in the chapters on measurement. It is covered briefly, with appropriate definition and direction as to how to calculate it. The problems are not challenging in terms of measurement or arithmetic, nor do they, outside of two possible non-standard units explorations, extend beyond very basic shapes. This is a minimal presentation that just meets the simplest criteria of this review.
Program Quality Evaluations
Mathematical Depth [3.8]
This book has a moderate rating for mathematical depth. It is reasonably strong on addition and subtraction, a major topic at this age, and satisfactory for time and money, also important at this age, but is weaker in measurement and multiplication. A student could probably progress to most third grade programs from this program, but the level of preparation targeted by this book is a not at the top level.
Quality of Presentation [3.5]
The presentation in this book is a little above average, but with a somewhat heavier and extended emphasis on the use of manipulatives. The instructions to teachers are a bit cursory and do require the teacher to carry the primary load for presentation of most topics with only vague directions and the actual student work packets. For top notch, experienced teachers who know exactly what works, this is fine, but newer teachers or those who are weaker in mathematical understanding, more explicit directions as to presentation might lead to an increase in student learning.
Quality of Student Work [3.0]
The amount of work is equivalent to that seen in most other programs. Some of the units cover material at a lower level than might be hoped, while some, such as addition and subtraction, are more reasonable. There are some concerns that the periodic review of topics that have already been covered may be a bit thin to consolidate what is learned.
Overall Program Evaluation
Students using this book have a reasonable chance of being moderately prepared for future mathematics and to use the math they have learned. On the other hand, this program presents a lower level of mathematical depth than desired in this review, and this is also evident in the student work. As a result, the overall evaluation places this program 5th of the 8 programs reviewed for this grade level. Given this, any implementation of this program should strive to insure that all students master the material to the level presented.
| Prior | Contents | Next |