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 program comes with a consumable student edition of 379 pages and a teacher's guide. The program also incorporates additional materials for practice, reteaching and extension. The program is organized into 12 chapters containing 9 to 16 lessons each. The chapters include:
At the end of each chapter a Chapter Resources section appears containing:
Within each chapter, one lesson is designated for a Problem-Solving Strategy and another lesson is designated as a Problem-Solving Application. In addition, each chapter includes a chapter theme, a story, and a chapter project. With the full lesson count and the chapter-end material, the number of days to complete the program leaves little room for skipping math on any school day.
The lesson structure follows a fairly regularized pattern. This will be illustrated for Lesson 7 in Chapter 9, titled Dollars and Cents. The teacher guide for the lesson contains a succinct statement of the lesson objective: Count and write dollars and cents. The lesson begins with a Math Minute in which the teacher holds up various bills and coins and asks for the name and value of each. The Introduce phase of the lesson appears next and the teacher guide provides three options to select from. In this case, the choices are:
Identifying and Counting Money (15-20 minutes), a whole class activity. Cards with monetary amounts are matched with coins and bills. Then, some combination of money is displayed and the class is asked for the total. This activity is identified as visual/spacial learning.
Spending Money (25-30 minutes), a whole class activity. Classroom items are tagged with prices in a pretend sales situation. In selecting an item, the price is shown with a collection of money. This activity is identified as kinesthetic learning.
Who's Got the Money? (25-30 minutes), a small group activity. The groups are given a combination of coins and dollars and instructed to find and write the total. This activity is identified as visual/spacial learning.
The Teach phase of the lesson appears next. This consists of students working on two Student Pages from the student edition. The teacher guide provides a paragraph or two of guidance for using each of these pages. One nice feature of the student materials is that the page numbers appear both in numeric form and in words at the bottom of each page. The first student page asks students to count money and write the total in dollar sign and decimal notation (the dollar sign and decimal already appear on the page). The pictures of coins and bills are organized in groups by denomination, and the teacher is to instruct the students to count in this order. Four problems are provided. The second student page asks students to select the combination of bills and coins from the money pictured in order to match the price of a given object. The teacher guide also asks that children discuss why a decimal point is used and the value of a five-dollar bill during this phase of the lesson.
The Wrap-Up phase ends the lesson. In this case, the teacher guide indicates asking students how counting in denominations order from greatest to least is helpful. A Journal Idea is given at the end of this phase in which students are to draw and label combinations of coins equal to one dollar.
For this particular lesson, there is an alternate lesson that is technology-based. In this case, the teacher uses software to display money on the screen and guide students through a discussion addressing the same ideas as presented in the lesson above.
Content Area Evaluations
Addition and Subtraction of Whole Numbers [3.7]
Material for addition and subtraction is concentrated in several chapters throughout the program. The first two chapters introduce these topics. Chapters 5 and 6 progress to the 2-digit cases, and Chapter 11 addresses addition and subtraction for the 3-digit case. Prior to the introduction of the 2-digit case, topics that support this growth appear in the Patterns and Numbers to 100 chapter. Similarly, the Patterns and Numbers to 1,000 chapter appears prior to the treatment of the 3-digit case.
In the introductory material for addition, the first two lessons have students work with models to represent basic addition facts and use these to solve addition problems written horizontally and vertically. The third lesson emphasizes that two addends can be added in either order, and the term addend is used. The next lesson addresses counting-on to do simple additions. The next lesson addresses the doubles facts. The student pages include four problems represented by domino drawings, followed by numeric problems giving the doubles from 1 to 9. The second student page includes 24 numeric problems and 2 word problems. Items on this page include some doubles and some other facts. The sixth lesson presents the near-doubles cases, while Lesson 7 reviews the skills covered to this point in combination. A problem-solving lesson focuses writing number sentences for word problems involving simple addition, followed by a lesson on the facts that sum to 10. Adding 10 is addressed next, followed by adding 9 (near 10). The next lesson is designed to stretch this technique to use for adding 7, 8, or 9. Through this combination of lessons the basic facts have been covered over the course of 12 lessons. The next lesson addresses the sum of three, single-digit addends, followed by a lesson on missing addends to end the first chapter.
Chapter 2 provides a similar, gradual development of subtraction over the course of the 13 lessons provided. The relationship between addition and subtraction is further addressed in this chapter as well. The last lesson of the chapter focuses on choosing addition or subtraction in application problems.
The next chapter provides the background that will be needed for moving to 2-digit addition and subtraction. This includes counting by tens and place value, grouping objects by tens and ones, regrouping, among other topics. The material on place value and especially the material on regrouping will provide good support for the topics to come.
Chapter 5 provides the material for 2-digit addition. The topic begins by adding 2 multiples of 10 in the first lesson, then adding a multiple of 10 to another number in the second lesson. Estimation enters in lesson 3 by rounding to multiples of 10. The next lesson focuses on identifying when regrouping is required. The student pages for this lesson present 12 problems in which students determine whether or not regrouping is required when adding a single digit number to a 2-digit number. In Lessons 5 and 6, students continue to use models but the presentation now also includes the standard addition algorithm. The emphasis continues to be on regrouping, and although they are still working with models students are now expected to show the solution using the algorithm. The models change to pictures in Lesson 7 and models become optional in Lessons 8 and 9. Lesson 10 is a Problem-Solving lesson that includes patterns in addition and the one-hundred chart. Lesson 11 presents four choices of ways to add - mentally, using models, using a one-hundred chart, or using paper and pencil. Lesson 12 introduces 3 addends in the 2-digit case. Sums do not exceed 2-digits in these cases. The final lesson of the chapter has a problem-solving focus on extracting data from tables. Thus, while many lessons have been devoted to the development of 2-digit addition, and significant attention to regrouping has occurred by this point, models and charts continue to be options available to students.
Chapter 6 provides a similar development for subtraction in the two-digit case. Importantly, this includes a lesson on using addition to check subtraction. However, as with addition, the use of models continues to be an option.
Chapter 9 provides the number sense background for 3-digit numbers. While material appears about patterns of numbers (such as 3-digit numbers increasing by 10), the focus on regrouping that was part of the 2-digit orientation is not found in the 3-digit case.
Chapter 11 builds to the 3-digit case for both addition and subtraction. The development for addition is to start with adding multiples of 100 in the first lesson, introducing addition with models in the second lesson, using models to record solutions in the standard algorithm format in the third lesson, and practicing the addition skill with models optional in Lesson 4. The same development process is used for subtraction in the next four lessons. A lesson on selecting a computational method appears later in the chapter, but students continue to have the use of models available as an option.
In summary, considerable attention is devoted to the addition and subtraction of whole numbers in this program. Attention to basic facts and especially to renaming is very strong. On the other hand, estimation, missing addends, and mental computation tend to focus only on simple cases. Although the program carries these processes through the 3-digit cases, the reliance on models especially for difficult cases may work against mastery in these operations.
Multiplication of Whole Numbers [2.5]
Multiplication is the main focus of the last chapter in the program. This means that classes that do not progress fast enough are at risk of not getting to this material.
The first lesson introduces multiplication as finding the total of equal-sized groups in very simple cases. The second lesson introduces representing repeated additions as multiplication (and vice versa), including the horizontal notation and the terms times and product. The third lesson introduces the representation of multiplication with an array. Lesson 4 continues the array context and introduces the vertical notation, while products are still very simple cases. Lesson 5 focuses on the fact that two factors can be multiplied in either order. Students are not introduced to formal language associated with this, and even the teacher guide refers to the order property of multiplication rather than the commutative property. Lesson 6, the problem-solving focus, introduces drawing pictures to help solve multiplication problems, and Lesson 7 extends these ideas to selecting between drawing a picture, using counters, or using repeated addition to do multiplication. The chapter then changes focus to the introduction of division.
In summary, the program only presents a very basic introduction to multiplication. The most difficult problem is 5 X 5. There doesn't seem to be an expectation that students will learn some of the multiplication facts. The horizontal and vertical notation systems are introduced, and students do write basic number sentences for multiplication.
Time [3.0]
Time is the focus of Chapter 7. The first lesson focuses only on having students get a rough idea of how long a minute is. Rough estimates of time (between minutes, hours, and days) are presented in Lesson 2. The next three lessons focus on telling time to the half hour, five minute, and quarter hour level, respectively, followed by a lesson that combines these three levels of precision. Student work includes writing times in digital format and representing times presented in digital format by drawing hands on clock faces.
Lesson 7 focuses on measuring elapsed time, or representing an end time on a clock face given the start time and the duration. Student pages include only elapsed time in whole hours and, in a few cases, half hours. Cases that wrap around 12:00 are not included. A lesson on reading schedules follows, but this is primarily a function of finding a time in a table. One lesson on calendar use follows, and only basic information about calendars is addressed. Problems such as finding the day of the week some given number of days in the future are not the focus of this lesson.
The attention to working with clock time is probably insufficient for many students to approach mastery in this area, and the attention to working with calendars is even less thorough.
Money [4.0]
Money is the focus of Chapter 4 of this program. In the first lesson, students cover the values and simple sums of pennies, nickels, and dimes. Lesson 2 uses finding the total for each denomination and summing these as a problem-solving approach, although it must be remembered that this work is occurring prior to the chapter on 2-digit addition. Lesson 3 focuses on quarters and other combinations of equal value to quarters. Lesson 4 introduces counting combinations of the four coins that have been presented thus far, and stresses counting from the largest denomination down to the smallest. Lesson 5 then utilizes this counting skill to build to making comparisons between two sets of coins. Lesson 6 introduces half-dollars in a way similar to the introduction of quarters seen in Lesson 3. Lesson 7 then uses the skills acquired so far to have students show different combinations of coins equal to the same amount. The dollar and the equivalent combination of coins is introduced next. The next lesson asks students to say whether or not a pictured collection of coins is sufficient to buy an object at a given price. The final lesson of the chapter asks how much change is given in very simple cases since 2-digit subtraction hasn't been introduced yet. The differences noted on the student pages range from 1 cent to 4 cents.
Money will appear again later in the course in other chapters. In Chapter 9, lesson 7 covers the use of the dollar sign and decimal notation to read and write money amounts. Finally, Lesson 10 in the chapter on 3-digit addition and subtraction (Chapter 11) builds to the point of having students solve addition and subtraction in dollar sign and decimal notation. Although the other work in 3-digit addition and subtraction leaves the use of place-value models as an option for the student, the use of money or play money is not mentioned on the student pages for this lesson. However, the teacher guide suggests that it is optional for this lesson.
In summary, the program does a good job of covering reading and writing money amounts, making equal-value exchanges, and showing combinations of coins equal to the same value. Work with counting money and making change is present, but doesn't get to very high values. Addition and subtraction of money amounts in dollar sign and decimal notation is included, but it is unclear whether or not students will use money or play money as an aid in this process.
Measurement of Length, Weight, Volume, and Temperature [4.5]
Measurement is the focus of Chapter 8 in this program.
Length is addressed in Lessons 1-5 of the chapter. The first lesson begins with non-standard units. Lesson 2 moves to measurement in inches and feet and Lesson 3 adds yards. Lesson 4 is a problem-solving lesson and deals with estimates and verification by measurement. Lesson 5 addresses meters and centimeters. These lessons include attention to terms, abbreviations, and simple equivalencies.
Weight is the subject of the next three lessons. The treatment is first in non-standard units, then pounds, then grams and kilograms. Estimates and abbreviations are included, but equivalencies are not.
Volume (capacity) is addressed in three lessons. As with weight, the pattern is non-standard first, cups, pints, and quarts second, and liters third. Terms, estimates, and abbreviations are addressed. Simple equivalences are addressed withing the U.S. customary system.
Temperature in both Fahrenheit and Celsius is the topic of one lesson. Student work expects skill in reading a thermometer in 5-degree increments. Some attempts are made to get students to recognize hot, warm, and cold temperatures. Some of the thermometers pictured go below zero, but this is not discussed and negative temperatures are not involved in the student work.
In summary, the content expected in this review is all addressed for these measurement topics. Some of the material must be considered only introductory, however, as the student work is somewhat limited, especially for temperature.
Perimeter [2.5]
Perimeter is the subject of Lesson 6 in Chapter 10. The structure of the lesson, however, may lead to some ineffective learning. The focus seems to be on counting the side lengths rather than summing them. For example, most of the student problems include figures with all sides of length one-inch. Consequently, students may just find the number of sides rather than the sum of their lengths. On some figures, some of the sides are of length two-inches. However, these are indicated as two one-inch segments, so students may not grasp the underlying concept. In the Introduce phase of the lesson, the first option does indicate a sum, but all sides are of length one. In the second option, a three-by-three grid is used, but the solution is to count the squares around the outside rather than find the sum of four 3s. In the third option, students work in groups to find the perimeter of a 3-foot by 2-foot rectangle, but the solution method is up to discovery learning. In the second half of this option, students stand in the shape of squares, triangles, or rectangles, but again the perimeter is the count of students. Thus, this introductory material is at risk of communicating the wrong concept to some students.
Program Quality Evaluations
Mathematical Depth [3.5]
The mathematical depth of this program is moderate with respect to the expectations of this review. In some ways, the program appears to come close to expectations but then to fail to implement them completely. Although the program carries addition and subtraction of whole numbers through the 3-digit cases, for example, the reliance on models may be excessive. Although the program does supply a basic introduction to multiplication, there doesn't seem to be an expectation that students will learn some of the multiplication facts. The attention to working with clock time may be insufficient for many students to approach mastery. Addition and subtraction of money amounts in dollar sign and decimal notation is included, but it is unclear whether or not students will use money or play money as an aid in this process. The material on measurement is fairly reasonable, although attention is limited in the case of temperature. Finally, the topic of perimeter is addressed but the treatment is at risk of conveying the wrong concept to students.
Quality of Presentation [3.1]
The quality of presentation was seen as moderate for this program. In many ways the presentation has effective features. However, the most serious drawback appears to be the reliance on explorations and models that consumes a large part of each lesson. It would be expected that these approaches would appear to a more varied extent depending on the level and content of the lesson.
Quality of Student Work [3.3]
The quantity and quality of student work generally appears adequate to support modest achievement levels. However, the reliance on models even for addition and subtraction never disappears and may threaten student mastery.
Overall Program Evaluation
Students using this program have a reasonable chance of moderate achievement levels. On the other hand, this program is not seen as supporting high achievement levels. It is possible that a skillful teacher could overcome some of the limitations of this program and use it more effectively. The heavy reliance on models and the potential confusion in the treatment of perimeter are examples of areas where an effective teacher could improve upon the student learning supported by this program.
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