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
The student edition is produced in consumable format, which is essentially a collection of problems such as one might find on worksheets, with very little in the way of exposition included. The roughly 400 pages are organized into 160 lessons across 4 units:
The lesson topics within each unit are varied, not necessarily restricted to the content suggested by the unit title. Nearly all lessons are designed for presentation in a single day if a traditional schedule is followed.
The basic lesson structure is consistent throughout the text and consists of three parts: warm up, teach, and wrap up. The warm-up is meant to take 5 minutes and consists of a problem of the day and a small problem set. The problem of the day is meant to provide an interesting problem that may stimulate class discussion of problem-solving approaches. The small problem set is designed for mental solution practice. The warm-up activities are not necessarily related to the main content of the lesson.
The teach part of the lesson may contain a variety of activities and thus the design will vary from lesson to lesson. For example, lesson 12 calls for the teacher to demonstrate sums that are near doubles so that knowing the doubles can help with the solution. Similarly, using plus-10 facts is shown as a way to help with the plus-9 facts. Students then complete the lesson 12 pages in the student text, which is a consistent component of the teach part of the lesson. The class is then to work on problems from a Thinking Story.
The wrap-up portion of the lesson is typically a summary of the learning for the day. In lesson 12, students are asked to tell their strategy to find 8+6 mentally. An alternative assessment is consistently offered in the Teacher's Guide for use in the wrap-up. In this case, it is an opportunity for students to write about an addition strategy for their portfolio.
Among the variety of activities that may be included in the lessons, one of the most common is games. In Unit 1, for example, 17 of the 46 lessons include games. Six of the lessons in this unit contain a Thinking Story for the teacher to read to the class and discuss the questions it presents. These problems are then frequently referenced in other lessons. One lesson per unit is devoted to a mid-unit review, one to a unit test, and one to extending the concepts presented in the unit. A cooperative student project is also included for each unit.
Supplementary materials available for each lesson include reteaching directions and practice and enrichment pages. These are generally consistent with the material in the lessons, and this review will focus primarily on the instruction in the teach part of the lessons and the student work. It should be noted, however, that the exposition provided by the teacher is not scripted in the Teacher's Guide. In fact, it tends to be described rather briefly. This may mean that there is variability from teacher to teacher in how well this program is implemented.
Content Area Evaluations
Addition and Subtraction of Whole Numbers [5.0]
Several blocks of lessons are devoted to addition and subtraction. The first block of these lessons will be described in some detail as an introduction to the style of this program.
Prior to addressing addition and subtraction directly, students practice finding sums of money by counting. Adding and subtracting 0, 1, 2, or 3 by counting is covered in lessons 6 and 7 with sums and differences that are mostly 2-digit numbers. Lesson 8 is the first of 10 lessons that review addition facts. It presents a subset of addition facts - those where both addends are 5 or less, doubles facts, and those where one addend is 10. Lesson 9 focuses more specifically on doubles. The teach portion of the lesson uses number cubes, fingers, or counters for practice. The student pages present 30 problems that include doubles and other addition facts. An additional Mixed Practice page at the end of the text is available for this lesson (32 of the lessons reference one of these pages). Lesson 10 presents the nines addition facts by relating them to the tens facts. Lesson 11 then presents the commutative property. Although the property is named as the title to the student pages for this lesson, there is no specific dictate to teach the name of the property. The instruction teaches the property by example and the teacher is to draw a definition of the rule from the class. This is followed by 8 pairs of problems. The lesson also introduces the Function Machine that will appear in many lessons. Students solve for outputs in some cases, but for the rule in others which will develop missing addend and related problem types. A Thinking Story is also presented in this lesson. Lesson 12 gives more exposure to the near double facts and the plus 9 facts. The student pages include 36 problems. Lesson 14 introduces the addition facts table and uses it to identify facts that are known by most students and those that remain to be learned effectively. Student pages include 19 numeric and 5 application problems. Lesson 15 uses the function machine and application problems to review addition facts. Lesson 16 presents 15 application problems on the student pages. The second of the two student pages asks questions about which items can be purchased with $17. Finally, Lesson 17 completes the series of 10 lessons on addition facts. This lesson serves as a Checkpoint for addition facts. Students are told they will be timed, given some oral drills, and work in pairs with flashcards in preparation. Then, students are assessed on the completion of 18 facts in three minutes. Apart from this activity, students also work on 7 application problems.
The next block of lessons addresses subtraction facts. The first focuses on facts with subtrahends or differences of 0, 1, or 2. The next involves missing addends and includes a game related to this. The next lesson uses the Function Machine approach and also presents addition facts with sums of 11 and 14 and the corresponding subtraction facts. The mid-unit review follows that covers an assortment of problems on the material covered thus far. The next lesson continues a focus on missing addends and subtraction, and includes 5 application problems and another Thinking Story. Use of the addition table to do subtraction is addressed next, followed by a lesson that continues a mix of addition and subtraction problems. The next four lessons focus more specifically on subtraction, including application problems. This is followed by a Checkpoint similar to that given for addition facts.
To this point, two blocks of lessons have focused on addition and subtraction facts. The next lesson presents addition and subtraction on a calculator, followed by two lessons covering more addition and subtraction and their applications. The next lesson is on three addends, but some problems involving an addition and a subtraction are included so that students can learn that order doesn't matter for three addends but that it does matter when a subtraction is involved. The lesson includes 16 numeric and 6 application problems. The next three lessons involve reading various types of graphs and using the data to answer questions. Two lessons on place value then appear followed by counting by tens which also begins the extension of addition to two-digit addends. The balance of the first unit focuses mainly on measurement. Thus, many lessons in the first unit have addressed addition and subtraction facts in many different ways.
The second lesson in Unit 2 continues work with place value in preparation for two-digit addition. Seven lessons on 2-digit addition then follow. The first three of these lessons rely heavily on various place value models to introduce the subject, and include two games. Regrouping (renaming or carrying) is included in some problems from the outset. The next lesson explains the use of the vertical notation and the algorithm that includes students noting the regrouping (carried numbers). The place value of the digits used is addressed. The next lesson focuses on word problems, and the final two lessons in the block present a mixture of numeric and word problems.
A single lesson reviews the subtraction facts covered earlier, followed by a lesson that covers only regrouping tens as ones, a subskill needed in subtraction. Nine lessons on the subtraction of two-digit numbers are then presented. The first addresses multiples of 10. The next covers subtraction from a multiple of 10, and thus focuses on renaming (borrowing). The explanation clearly indicates how this process relates to the vertical notation and algorithm. The next lesson continues subtraction from multiples of 10 and includes a game. The next lesson illustrates regrouping for more general cases. The next three lessons gradually build on this skill and include many application problems. The next lesson presents 16 completed subtractions and asks students to indicate which of the solutions is wrong. This is an important feature, and the instruction is meant to show that addition can be used to check subtraction. The next lesson serves as an (untimed) check on student 2-digit subtraction ability. Next, three lessons and a mid-unit review address addition and subtraction of 2-digit numbers together. The remainder of Unit 2 addresses a variety of other topics, although practice of the addition and subtraction skills reappears periodically.
The focus of Unit 3 is measurement, but a few lessons that address addition and subtraction should be noted. The first lesson of the unit involves comparisons, and includes comparisons of sums and differences. The next lesson addresses the sum of three addends. This is shown both as two separate additions and as column addition with the standard algorithm. A few lessons later a Keeping Sharp lesson appears that reviews addition and subtraction, and the next two lessons extend column addition to four addends. Lessons on odd and even numbers also build to a lesson in which students learn about the sums of two evens, two odds, and an even and an odd number. Another lesson focuses on shortcuts to use in mental addition, followed by a lesson on using rounding to estimate sums and differences.
Unit 4 contains several topics, but important lessons in addition and subtraction are also given. One lesson introduces grouping by tens as a means to simplify the addition of strings of numbers. Two lessons review addition and subtraction. Twelve lessons extend addition and subtraction through the 3-digit problems. As with the 2-digit case, these begin with careful illustration using place value models and extend through the use of the standard algorithms, application problems, and estimation. Five lessons extend addition and subtraction through 4-digit numbers including the sum of three addends, application problems, and estimation.
In summary, this program provides attention to addition and subtraction with great depth and breadth, all the way from basic facts through the sums of three 4-digit numbers. Careful attention is given to renaming. Estimation and missing addend problems are covered. The use of the commutative and associative (not by name) property and mental operations are also covered. Much attention is given both to the numeric cases and to applications. Thus, there is substantial attention to addition and subtraction provided in this program. On the other hand, students are only occasionally required to include units in their solutions. Nonetheless, this is clearly a thorough presentation in these topic areas.
Multiplication of Whole Numbers [4.8]
Multiplication of whole numbers receives considerable attention in Unit 4 of this program. Five lessons are devoted to illustrating the meaning of multiplication and the representation of multiplication as repeated addition, area, multiples or skip counting, and arrays. Most of these include illustrations as arrays and repeated additions for the same context. Word problems involving simple multiplication applications are addressed next. This is followed by a lesson that introduces the facts for 2, 3, 5, and 8. A few lessons later, more applications of multiplication are given and missing factors are also introduced as a prelude to division. Lessons toward the end of the unit use Function Machine notation to further show the relationship between multiplication and addition and to introduce multiple operations in simple cases.
Thus, this program covers the introduction to multiplication effectively. Many lessons are devoted to developing the concept in various ways. Students learn some of the multiplication facts and the notation used for multiplication. The terms factor and product are introduced. Some applications of multiplication are included. Basic number sentences involving multiplication are introduced. Multiplication by 0 and 1 are given some attention as special cases. Calculator work in this introductory material is a bit excessive given that students are just learning the concept, and the attention to units in application problems is weak. Nonetheless, this is an effective second-grade introduction to multiplication.
Time [4.0]
The treatment of time is primarily concentrated into two blocks of lessons on telling time. The first block begins with orientation to the face of a clock through telling time in quarter hours over three lessons. This includes students writing time in the digital format and drawing hands on clock pictures and the use of the terms quarter after, half past, and quarter to.
The second block of lessons includes one lesson on telling approximate time to the half-hour, a lesson on telling time to the minute, and a lesson on time in minutes before the hour. In telling time to the minute the program takes advantage of the landmarks (half and quarter hours) that the students already know. This block of lessons also includes a Mastery Checkpoint, although the next lesson includes additional similar problems.
Two isolated lessons also address time topics. The first appears early in the year and gives the basics of reading a calendar. Another occurs late in the year and covers a.m. and p.m. and the days of the week. Calendar use and days of the week appear briefly in a few other areas of the text, notably in some of the Problems of the Day. These provide additional exposure for concept development.
Several other lessons use time as a context for operations, such as addition and subtraction in minutes, days, or years. For example, the sum of the number of days in 3 particular months appears in a lesson on column addition.
In summary, the presentation of topics related to time generally covers the material expected in this review. The program will address telling time again in grade 3, so the program doesn't anticipate completing this topic this year. More attention is given to telling time and solving problems in the context of time than to dealing with calendars or placing events in temporal order. Thus, the coverage is generally adequate but may require reinforcement for some students to be successful.
Money [3.5]
Money appears in a variety of contexts found in many of the lessons in this program. It is used primarily as a helpful context for the instruction and problems that address other topics, rather than working with money being more directly an end-objective itself. For example, money is used extensively in addressing place value, addition, subtraction, renaming, and fractions. Most of these instances, however, do not involve the use of the dollar sign and decimal point notation. Indeed, most of the instances of money amounts involve either some number of cents or some number of whole dollars. On the other hand, the use of the dollar and cent symbols occurs early on in the program and continues throughout, suggesting that most students are expected to know this at the outset of grade 2.
In some instances, topics of special importance in dealing with money amounts are addressed. For example, at several points throughout the text, students are asked whether or not some quantity of money is sufficient to purchase some given combination of items.
One lesson is devoted to exchanges of money amounts, but this is as a prelude to renaming in addition and subtraction. Later, money (cents or whole numbers of dollars) is used to introduce the place-value context for operations with 3-digit numbers. After that, in Unit 4, the use of the decimal notation and the translation between amounts in dollars (and cents) and amounts in cents receives more direct treatment. In later lessons, a minimal extent of operations in dollar sign and decimal notation appears, but this material is at most a simple introduction to work in this notation.
In summary, the program offers a fairly unique treatment of money-related topics. Money is used more as a useful context for mathematics in general rather than being emphasized as a topic itself. Used in this way, instruction and problems in a money context are numerous throughout the text and this must strengthen the relationship of money problems to mathematics in general. On the other hand, common topics related to money - such as showing different combinations of money that equal the same value or counting up to make change - receive relatively less attention. More of this activity, especially making change as opposed to subtracting monetary amounts, might be helpful in bringing students to mastery of simple monetary transactions. Finally, at the upper level of potential topics for second grade, this program does not cover the relation of dollar sign and decimal notation to decimal numbers and doesn't give great emphasis to addition and subtraction in dollar sign and decimal notation. Yet, the frequent appearance of money in various operations yields a distinct approach to the treatment of this topic area.
Measurement of Length, Weight, Volume, and Temperature [4.5]
The measurement area topics receive considerable attention in this program, particularly length (and distance) measures. A series of five lessons in Unit 1 addresses estimates and measurements in centimeters and meters and in inches, feet, and yards, including stating equivalencies within a measurement system. Two lessons in Unit 3 are devoted to map-reading and working with distances in kilometers. Operations with length measures occur in several lessons, and many abbreviations are introduced although students are rarely required to write the units themselves.
Weight is primarily addressed in a series of three lessons in Unit 3. One lesson addresses kilograms and grams and the conversion of kilograms to grams, including estimating weights in these units. Pounds and ounces are treated similarly, although the abbreviations are not given. In the third lesson in this series students engage in an activity to group craft sticks building up to a weight of one kilogram.
The next two lessons deal with volume (capacity), and present fluid ounces, pints, quarts, and gallons. This includes giving equivalencies and having students do multiple operations to figure such equivalencies as the number of fluid ounces in 2 gallons. The text also notes that a pint of water weighs about 1 pound, and students give approximate weights for other volumes of water. Abbreviations for these units are also not used. The second of these two lessons addresses metric measures of capacity, including milliliters and liters. Importantly, the relation of cubic volume measures (cubic centimeter and cubic decimeter) to the capacity equivalents is given, along with the fact that a liter is a little more than a quart and that a liter of water weighs 1 kilogram. Abbreviations for milliliters and liters are not given.
Only one lesson focuses on temperature. Students are brought into the reading of thermometers by first completing several problems in counting by twos, including counting up from negative numbers. The Teacher's Guide indicates that a number line with negative and positive numbers should be used to introduce this. Then, reading thermometers (marked in 2-degree increments) to the nearest single degree is presented, including a few problems with negative temperatures, including -13 degrees in one item. Although explicit instructions are not given, the answers in the Teacher's Guide include the degree symbol. Curiously, the first answer in this group is marked with a capital "F" while none of the others are and the text doesn't cover the distinction between Celsius and Fahrenheit. Discussion of hot and cold room temperatures is the topic of the wrap-up for this lesson.
In summary, the coverage of these measurement topics generally meets the expectations of this review. Most attention is given to length (and distance), including fairly common usage of this context in operations problems. Least attention is given to temperature. Many of the lessons provide equivalencies, and even important links between different systems. On the other hand, attention to abbreviations is only modest and students are rarely asked to write units in their answers. Nonetheless, this group of topics should provide good preparation for the material to come in the next grade level.
Perimeter [4.5]
Perimeter is addressed in a few distinct lessons in this program. Late in Unit 1 students are introduced to perimeter in the context of a lesson on the measurement of length in centimeters. The related student page indicates that students should add the measures of the sides to find the perimeter. The teacher is to explain this as well. The three problems are all quadrilaterals with side lengths in the single-digit range.
The topic re-appears a few lessons later. The teacher is explicitly instructed to define perimeter. The six problems include an octagon and two triangles. Students are then to find the perimeters of objects in the classroom in both inches and centimeters. The wrap-up for this lesson instructs the teacher to have students write the definition of perimeter.
In Unit 3, students find perimeter in two lessons on column addition. The second of these places perimeter in a context problem involving fencing an oddly-shaped yard.
Thus, the concept and definition of perimeter are supplied in this program. Finding perimeter by the addition of side lengths appears several times. An application that requires perimeter is also given. Interestingly, other ways of finding perimeter are not elaborated. Nonetheless, this presentation does a good job of providing the material expected in this review.
Program Quality Evaluations
Mathematical Depth [4.6]
With but a few exceptions, this program is represented by a noted depth of the coverage of the mathematics topics at this grade level. The material is consistently up to the level of the expectations for this review, and yet the coverage at these levels is typically thorough. In part, this seems to be accomplished by minimizing repeating material from earlier grades. Some of the topics will need further development, or at least reinforcement, in later grades, but this is reasonable in each case. Given the generally high level of the text, the early introduction of decimals in a money context is somewhat weak and the incorporation of units in student answers is minimal. Considering all factors, however, this program provides a STRONG mathematical grounding for a second grade program.
Quality of Presentation [4.0]
This program has many high quality features of presentation. The sequence of presentation builds topics in a logical manner. The program appears to present the material in an efficient manner in order to maximize learning. There are few instances in which teacher or student time appears to be used unwisely. Introductory material for initial student learning appears reasonable. Ample options for teachers are included.
Perhaps the most serious concern with respect to the presentation is that the explicit guidance to teachers for each lesson is brief. This means that inexperienced teachers will have to decide how much time should be devoted to their instruction and what areas will require special emphasis. This even gets to the point that details that might be expected in a presentation may not be addressed in the Teacher's Guide. Since instruction through the student materials themselves is only partial, there will likely be a great reliance on the expertise of the teacher, and some further assistance for teachers would be useful.
Quality of Student Work [4.0]
The student work is presented in a consistent manner throughout the course in that there are pages for students to work and discuss in connection with the lessons. The presentation of the worksheets is clear enough for parents to assist their students as needed, but the program lacks explicit guidance with respect to homework activities. The presentation of student work is structured so that students work primarily on activities or problems related to a single topic in the main part of each lesson. The warm-up each day will typically have a different focus.
Distributed practice is less common in this program. Practice of content over a period of several days is mainly limited to those blocks of contiguous lessons within a single topic area. Otherwise, distributed practice occurs primarily through review or mixed-practice activities, and the use of these may vary from one teacher to the next. The quantity of student work does appear to be sufficient. The range of student work within individual topics is frequently outstanding, especially for topics that are central to the development in this grade level.
Overall Program Evaluation
Overall, this program is outstanding. It is characterized by the presentation of material at a high level for second grade. It is unusual in that a high level is achieved without reducing depth along the way. This may be the result of continued high expectations across grade levels coupled with less review of material from prior grades than other programs may contain. Thus, the topics are generally covered thoroughly but leaving room to bring in the more advanced topics. This suggests that this program could support learning effectively to a high level of achievement.
There are two main concerns in implementation of this program. One is that students that transfer into this program may find it difficult to catch up to the high level of instruction. Re-teaching guidance provided may assist here. The second concern is that the wide range of supplemental materials coupled with the brief guidance given to teachers in the Teacher's Guide, while leading to flexibility in implementation, may not provide adequate direction for less-experienced teachers.
Despite these limitations, however, the potential for supporting learning at the high levels provided by this program is very compelling, and it received the highest overall rating for second grade programs.
| Prior | Contents | Next |