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 Teacher's Manual contains a brief introduction to the program and the lessons in a few pages. The manual then divides the course into 132 lessons that are not otherwise organized into chapters or units. Each lesson is explained in detail in about 5 to 8 pages of the manual. Assessments are integrated into the program and are part of the lesson count. Thus, the program design allows completion of all material in 33 weeks at a pace of 4 lessons per week. Student material is found in two soft-cover student workbooks, each with about 170 sheets that can be removed. Some are printed on two sides.
The typical lesson structure is divided into four sections - the "meeting," the lesson, class practice, and written practice. The meeting is a brief time designed to begin work in the morning. A variety of topics are covered briefly. For example, in lesson 72, students are asked about the calendar and about patterns and they practice skip counting. They record the current temperature and answer some questions about their weather graph. They count money amounts, answer questions about time, record lunch and attendance counts, and are asked questions about the graphs developed in the classroom. The meeting is designed to be completed in 15 to 20 minutes. The remainder of the lesson is designed to be completed later in the day in about 40 to 45 minutes.
The lesson contains the presentation of new material. This is typically teacher-directed whole-class discussion in which the teacher's presentation is clearly spelled out. For example, lesson 87 teaches subtraction facts for subtracting a single digit number from 10. The presentation, which occupies a little over a page of the teacher's manual, contains instructions to the teacher on how to present the material and scripted statements and questions for the teacher to say to the class. The content of this lesson focuses on "adding on" to do subtraction, for example, "... what number plus seven equals ten." Other times, the teacher directs cooperative student work, but the activity is still clearly defined. For example, in lesson 110 students work in groups of 3 to cover designs with tangram pieces. Worksheets, when needed as part of the lesson, are included in the student workbooks.
The class practice section typically has students completing "fact sheet" problems, typically for one minute but sometimes for five minutes. The fact sheets are typically not related to the content of the lesson for the day, and are often repeated several times. The facts are scored quickly in class, then collected, recorded, and returned.
Finally, the written practice includes a two-sided worksheet. The teacher reads and reviews each problem and assists the students as they work the first side. The items are corrected by the class. The teacher then reads and reviews each of the items on the second side, and the worksheet is sent home with the second side as homework. The material covered on both sides of the worksheet is similar, so that students should bring home problems to work and a completed set of similar problems.
The material covered on the worksheet is not necessarily related to the content of the lesson for the day. Worksheets contain a mixture of items covering several topics that were addressed in various prior lessons. Thus, the worksheets contain a small amount of practice for any one topic distributed over several days. For example, lesson 64 introduces finding the sum of three or more 1-digit numbers. The worksheet for lesson 64 contains 3 of these problems on each side, mixed with a variety of other problems. One or two of these problems then appears on each side of the worksheets for lessons 65, 68, 69, 75, and 76. Although lesson 85 introduces the sum of three 2-digit numbers, the sum of three or more 1-digit numbers continues to appear on worksheets for lesson 87, 93, 102, 107, 112, 115, 117, and 124.
Oral assessments are given every 10 lessons. They are conducted by the teacher with individual students as the others are working on their written work, so the process may take several days to cover all students. Typically the teacher asks the student a few questions and records the responses. These may include asking students to explain each step in their solution.
Written assessments are given every 5 lessons beginning with lesson 10. The written assessments contain a mixture of problems similar to the worksheets on a single side of one page. The problems cover material introduced at least 5 lessons earlier. Teachers are encouraged to discuss errors with students individually.
In summary, the program design is easily implemented by teachers and is planned in such a way that teachers should be able to cover all of the material over the course of the year. Teacher instructions are clear and direct. Lessons focus on a single topic, but practice and assessments of these topics are distributed over time. Lesson topics vary from lesson to lesson rather than covering all aspects of a large topic area together. Student work is thus characterized by a mixture of problem types at all times.
Content Area Evaluations
Addition and Subtraction of Whole Numbers [3.0]
Addition and subtraction with whole numbers is developed gradually over the course of the year. Although other lessons contain topics related to addition and subtraction, 36 lessons focus directly on addition and subtraction.
Considerable attention is devoted to the basic facts in this program. They are built up a little at a time over several lessons. Thus, separate lessons cover adding 0, 1, 2, 9, and 10. Others focus on doubles, doubles plus one, and sums of 10. The remaining addition facts are completed in lesson 58. Because of the relationship between addition and subtraction, slightly fewer lessons are devoted to the subtraction facts. Addition and subtraction facts are also practiced in the class practice section of many lessons.
Addition and subtraction with larger numbers is first introduced by adding and subtracting 10 which is developed in three lessons.
The operations for sums of two-digit numbers leading to the standard algorithm are covered in 7 lessons. The first four of these lessons use the context of dimes and pennies. In this way, the place value associated with each digit in the problem is clear. In the first two of these four lessons, problems are selected so that there is no need for trades (carrying). The third lesson introduces the need to rename (trade or carry), and the fourth provides more practice with trades. These are followed by three lessons on the standard addition algorithm. The third extends the trades to the case where the sum is greater than 100. The explanations of all procedures used are clear.
Sums of three or more numbers are developed before the standard algorithm for subtraction is introduced. The sum of three or more single-digit numbers is addressed in one lesson. The sum of three 2-digit numbers is addressed in two lessons, first with sums less than 100 and then with sums greater than 100.
The introduction of the standard algorithm for subtraction follows a similar course over four lessons. The first two develop the process in the context of dimes and pennies. Trades (borrowing) are included from the outset. The second two lessons cover two-digit numbers in the standard algorithm outside of the monetary context.
Thus, the treatment of addition and subtraction is very thorough from basic facts through sums and differences with two digit numbers. Considerable attention is given to trades (carrying and borrowing) along the way so that students should understand why the standard algorithms work. Sums of three 2-digit numbers are also well covered.
Along the way, considerable attention is given to the recognition of addition and subtraction situations in context problems. These are called "some, some more" and "some, some went away" stories in this program, and their recognition is repeatedly practiced. Attention is also given to writing number sentences, especially as related to these context problems. In this way, attention is given to regularly including units in these expressions. Thus, the relationship between applications and symbolic forms is well established.
On the other hand, this program does not extend to addition and subtraction with 3-digit numbers. Although rounding is addressed, estimation of sums and differences is not specifically covered. Although the opposite nature of addition and subtraction is addressed and some subtractions are treated as "adding on" problems, the processes of using addition to verify subtraction and vice versa are not covered. There are not special sections that address mental arithmetic, the commutative and associative properties, or missing addend/subtrahend/minuend problems.
Thus, the treatment of the addition and subtraction of whole numbers appears very solid but at a level below what is sought in this review.
Multiplication of whole numbers [4.3]
The multiplication of whole numbers is addressed in about 10 lessons spread across about the last third of the year. The first conceptual introduction is through the use of equal groups (how many altogether) stories. The concept gradually transitions to an array representation of multiplication. Equal groups are first introduced in two lessons, then, children draw graphic representations of the equal groups in a third lesson. This skill is then simplified to drawing an array. The symbolic representation of these stories progresses right along with the stories themselves, until students are writing number sentences with units to represent the multiplication.
Other lessons, not part of the lessons counted here as multiplication lessons, address skip counting by various values, and skip counting or multiples are also referenced in dealing with multiplication and learning some of the basic facts. The students practice multiplying by 1, 10, 100 and 2, 3, 4, and 5 (not in that order) across several lessons.
Thus, the basic introduction to multiplication sought in this review is well accomplished in this series of lessons. A good subset of multiplication facts is covered, with the exception of multiplying by zero. Both horizontal and vertical notation are introduced. The highlight of the program is probably the emphasis on understanding multiplication and the relation to problem situations, although more attention could be given to differentiating between situations that can and cannot be solved by multiplication and the terms "factor" and "product" are not used. The program does not go beyond the expectations of the review to cover topics such as missing factors or combinations of operations.
Time [4.7]
Much of the coverage of time is covered in the "meeting" part of this program, reducing the need to devote many lessons to this topic. Numerous problems related to time appear on the student worksheets as well.
Time appears in the "meeting" part of the program on a consistent basis. The meeting for lesson 90 provides a typical case. With respect to dates and days of the week, students spell the current month and the student of the day writes the date. Then students are asked for the date and day of week some number of days ago or hence, what month corresponds to a given month number, and what the last and next month are. With respect to time, the student of the day sets the clock to some given time (to five minutes). Students are asked what time it is on the clock, what the time was an hour ago and what it will be in one hour and how to write the time in digital notation.
In addition to this daily practice related to time in the "meeting" part of each lesson and the workbook problems, there are 7 lessons dealing with time specifically - 5 on clock times or intervals and 2 on dates or days of the week.
Specific lessons on clock time start early in the year as students learn first to tell time to the hour, to identify the numbers on the clock face, and to determine elapsed time of one hour. Still early in the year, precision is increased to telling time to the half hour. About mid-year, a.m., p.m., noon and midnight are covered. A few weeks later precision is increased to reading and writing time down to five-minute intervals.
Only two lessons are devoted to days and date, as this material is given much attention in the "meetings." Early in the year, a lesson is given on the days of the week. Late in the year, students learn to represent the date using digits.
Thus, the coverage of this topic area is characterized by a modest number of explicit lessons and distributed practice across the entire year. The material expected in this review is well covered by the program, affording students the practice they need together with some internalization of duration.
Money [3.0]
Dimes and pennies are used extensively in this program in the development of 2-digit addition and subtraction as noted above. In support of this, one early lesson is devoted to counting money amounts in dimes and pennies. From that point on, counting money becomes part of the daily "meeting." Another lesson covers trades for dimes and pennies. Obviously, this lesson is also in support of the 2-digit addition and subtraction to come. Another lesson covers counting nickels (counting by fives). In the second half of the year, a lesson is given on reading and writing money amounts in both the dollar sign/decimal notation and in cents notation. This is followed by a lesson that addresses counting quarters (counting by twenty-fives), including writing the sum in dollar sign and decimal point notation. Later, a lesson is given covering counting and showing money amounts in quarters, dimes, nickels, and pennies. Students are regularly asked to count money and show different combinations of coins equaling the same amount in the "meeting."
Thus, the program provides ample exposure to the values of coins, to counting an assortment of coins, to showing combinations of coins equal to the same amount, and to reading and writing money amounts in dollar and in cents notation. Small amounts are repeatedly used in the context of other lessons. However, for all the counting of money, there is little attention to the idea of making change, even to one dollar. Practice working with coins and bills is not stressed either, probably because students are not working with operations using three-digit numbers to any great degree yet. More advanced topics, such as addition and subtraction with dollar sign and decimal point notation, or the clear relationship to the use of decimals in general, are not covered either. Thus, the content coverage of this program falls short of the expectation for this review in that larger money amounts and making change are not well supported. Again, the characterization of the program is one in which the content that is covered is addressed thoroughly, but more advanced material is lacking.
Measurement of Length, Weight, Volume, and Temperature [3.5]
This program devotes 5 lessons to the measurement of length spread across the year. Two lessons each are devoted to weight, volume, and temperature.
The first lesson on length uses the one-inch tiles that are frequently used in this program. The important points in this way of measuring length are stressed: "put the tiles next to each other; put the first tile even with the end of the object; keep them straight along the edge of the object." Students measure several objects with this method, to the nearest inch, and the word "about" appears frequently. In this first lesson, line segments are included among the items measured.
The next lesson on this topic uses rulers to measure to the nearest inch. Students measure objects and a practice sheet contains line segments to be measured (the segments are in varied orientations). The words, "to the nearest inch" appear frequently. Again, the important aspects with respect to the alignment of the ruler and the object are stressed. Students are also introduced to the abbreviation for inches (") at this stage.
A little after mid-year, a lesson is devoted to measuring and drawing line segments to the nearest half inch. They have had some exposure to halves already. The class counts by halves across the image of a ruler, and then the half-inch markings are added. The convention for writing measures as mixed numbers is shown. Students use their rulers to measure and produce line segments to the nearest half inch.
Several lessons later, students learn to measure in feet and inches. Often, these measures are estimated (guessed) before the measurement is taken. Student height measures are made by first marking the height on the wall with a piece of tape and then measuring the height to the tape. The measurement is accomplished by counting the number of feet by iterating the length of a one-foot ruler, then measuring the inches that are "left over."
Later, students measure and construct line segments in centimeters. First, they construct a line segment four inches long and then measure it with the centimeter scale of the ruler. Several different lengths are practiced, drawn in one measurement system and then measured in the other.
Thus, students are given clear instruction in measurement of length in inches, feet and inches, and in centimeters. They are not given explicit instruction in conversion of measures either within or between systems.
The first lesson on weight involves using non-standard units and a balance. Objects around the classroom (scissors, tiles, pencils) are used for this purpose. Instruction in the use of the balance and estimated weights in non-standard units are included. The second lesson on weight occurs late in the year. Students estimate and measure the weight of objects (including other students) in pounds using a scale. Metric units of weight are not covered.
In the first lesson on the measurement of volume, students are introduced briefly to the tools - cups and spoons - for measuring a cup, a half-cup, a tablespoon, a teaspoon, and a half-teaspoon. The abbreviations for some of these units are introduced briefly. This is only a brief introduction to these tools and offers no measurement experience to the students. The lesson is to be based on a recipe selected by the teacher for this purpose. In this lesson, students identify which tool is best for the measurement of each ingredient according to the quantity required in the recipe. A few lessons later, students actually measure and mix the ingredients for the recipe, probably in groups with extra supervision from a volunteer if available. This lesson is unusual for the program in that explicit instructions are not provided in the teacher's manual. The actual activity will depend on the recipe the teacher has selected. Students measure with the tools they discussed earlier and mix the ingredients, following the instructions provided in the recipe. Since this material is covered in the context of a recipe, other measurement units (quart, gallon, liter) are not covered. Equivalent units are not covered.
The first lesson on temperature occurs fairly early in the year. Students are taught to read a thermometer marked only with 10-degree increments to the nearest 10 degrees. No negative numbers are shown. No indication of different temperature scales is given. However, student worksheets do show finer degree markings and negative values. With values up to 100 degrees, it is obviously meant to represent a Fahrenheit scale. At this point, recording and charting the ambient temperature (to the nearest 10 degrees) becomes part of the daily "meeting" activity. The second lesson on temperature occurs about mid-year. Students learn to use thermometer calibrations marked in 2-degree increments. They count by 2's to identify the markings in learning about the scale, and read off temperatures to the nearest marking. Fahrenheit and Celsius scales are mentioned, but it is noted that only the Fahrenheit scale will be used in class. Beginning the next day, a new graph is started recording temperature to the nearest two degrees.
In summary, these measurement topics cover a variety of units. The mechanical issues of actually taking measurements are addressed. In some cases estimated measurements are called for, but in others they are not. There is little attention to the metric system. Equivalent units within or between measurement systems receive only minimal attention. Thus, this program provides roughly the measurement experience expected in this review, although quarts, gallons, and some metric values are not attended to.
Perimeter [4.5]
This topic is explicitly addressed in lesson 113 of 132. Students are told explicitly, "To find the perimeter of a shape, we measure all the sides of a shape and add them together." It should be noted that this program has already included the sums of several numbers several times by this point. The application of perimeter is illustrated verbally in one example. Then, the class presentation goes through finding the perimeter for several objects (rectangle, square, triangle) using a worksheet. The presentation is clear and straightforward, but more applications of the use of perimeter would be helpful.
Program Quality Evaluations
Mathematical Depth [3.5]
This program provides very good depth of coverage for the material that is addressed in nearly all cases. However, the level of the mathematics covered is frequently below the expectations for this review. This was especially the case with respect to addition and subtraction but also with respect to work with money. On the other hand, work with multiplication, time, and perimeter generally met expectations. Nonetheless, the limitations in the program with respect to review expectations had some negative consequences for this rating.
Quality of Presentation [4.5]
Irrespective of mathematical depth, this program has many high quality features of presentation. For example, the lesson instructions almost always include clear statements of the lesson objectives to be made by the teacher at the outset. The daily structure is divided into a variety of activities of reasonable duration for second grade students. The program as a whole should be easily completed within the school year. The instructional material is explicit and clear in nearly all cases, and enough specifics are given and worked through completely for adequate learning to occur. Frequent assessments are built into the program so that student progress can be carefully monitored. The instructional style is very efficient in the presentation of each new concept. The topics appear to be carefully arranged throughout the year so that the needed prerequisites are covered in each case.
Quality of Student Work [3.5]
The student work is presented in a consistent manner throughout the course. In each lesson, students complete one side of a worksheet and review their answers while in class. The second side has similar items presented and is given as homework. The worksheets require only a small time commitment from the students at home. The presentation of the worksheets is clear enough for parents to assist their students as needed. The lack of a student text means that there is no glossary to be consulted. Presumably, prior worksheets could be consulted as needed when students need to review. The presentation of student work is structured so that students are consistently working on a variety of problems on each worksheet, and practice on any single content area is distributed over many days. Thus, the quantity of student work in each content area appears to be sufficient. The worksheets are complemented by other student work materials, notably fact sheets completed frequently in class and other instructional sheets that are often distributed as part of the lesson component. The daily "meeting" activity also provides on-going exposure in multiple areas.
The biggest drawback with respect to student work reflects the limitations of mathematical depth in this program. As a consequence, the range of student work is limited in several topics. This limitation could mean in practice that students may not be challenged by this program as there are not opportunities provided for them to engage in more advanced work.
Overall Program Evaluation
The overall evaluation of this program is moderate, but that is far from telling the whole story. It is unusual in that the topics that are covered are generally covered thoroughly, yet more advanced topics are generally missing. This suggests that this program would support learning effectively to a certain level, but beyond that level achievement will be very limited. Thus, the use of this program should be evaluated carefully with respect to expectations for student achievement. If the goals for students are consistent with this program, then the program will provide good support for those goals. If the goals for student achievement exceed the material in this program, then it could only be effective with the addition of supplemental materials.
| Prior | Contents | Next |