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
There is no student text for this program. The Teacher's Manual & Lesson Guide reveals that the program is structured into 120 lessons divided into 12 units:
The Teacher's Manual notes that lessons can usually be covered in a single class period and that teachers need not attempt to start a new lesson every day.
Student work in class is primarily through two soft-cover Journals that total 151 pages and contain a collection of worksheets. These may be supplemented by materials reproduced from the masters in the Resource Book. The masters commonly contain instructions for completing a group activity. An Activity Book contains further material used at times in class. In addition, the Resource Book contains Home Links, which are typically half-page homework assignments.
A typical lesson contains a variety of activities, roughly divided into Instruction and Discussion, Teacher-Directed Activity, and Independent Activities. Lesson 16, Review "Easy" Addition Facts, will be summarized here as an example.
The Instruction and Discussion details the instruction that should occur (sometimes this is partially scripted, but not in this case). First, students are asked to share addition facts that they know and these are written on the board. Teachers then review the use of an addition facts table which is presented on a Journal page. The instruction is to focus on the terms row, column, and diagonal, and illustrations of how to find a few addition facts. Doubles are found from among the facts written on the board. Students are asked to name other doubles and to color them lightly in blue in their Journal page. Next, the plus 0 facts are identified and colored green, and the plus 1 facts are colored red. Teachers are to discuss the plus 0 and plus 1 shortcuts as patterns. Then, 87+0=__ is written on the board. A volunteer solves the problem, and another checks it on the calculator. Several other similar problems are completed. Similar activities are used for the plus 1 shortcut. These are to be done including some 2-digit and 3-digit numbers so that it can be inferred that the shortcuts work with all whole numbers.
The Teacher-Directed Activity is given next in the Teacher's Manual. This involves a Journal page. Teachers are to explain how to use the addition-fact problems (doubles) on the page to find the letters that correspond to the sums and then write the letters in boxes to find the puzzle solution, the word watchdog.
The Independent Activities section is essentially a list of four more activities. The first is for students to fill in some missing values in a number (counting) grid. The second is playing Beat the Calculator. This is a game for groups of three in which one student randomly selects and calls out an addition fact, one student solves the problem with a calculator and a third student solves it without a calculator. The caller decides who wins. Roles change after 10 problems. The third is for students to complete Math Boxes. These are masters in the Resource Book, and the particular set for this lesson contains four problems: A) add 10 to each of four numbers, B) write a 3-digit number and read it to yourself, C) solve four plus 0 problems and four plus 1 problems, and D) write numbers in blanks from 362 (given) to 373 (366 is also given). The final item is a reference to the Home Link to be found in the Resource Book. For this particular lesson, the student instruction says: Explain to someone at home what a doubles fact is. Ask them to give you doubles problems to solve for about ten minutes, or until you know all the doubles facts. As is the standard for these Home Links, a brief Note for Home appears at the bottom including the instruction to return the sheet the next day. In this particular case, there is no instruction for student work to appear on the sheet.
In addition, there are 12 projects described in the Teacher's Manual, and the associated masters are found in the Resource Book. Similarly, there are explorations to be included in the lesson time. For example, unit 8 (Fractions) includes five explorations.
In summary, the lesson structure contains many different activities. The organization is somewhat clumsy for teachers as they must collate a variety of materials to accomplish the lessons. This situation may become more aggravated for more complex lessons. This may prompt some teachers to skip some activities, particularly those listed briefly in the Independent Activities section.
Content Area Evaluations
Addition and Subtraction of Whole Numbers [2.0]
The treatment of addition and subtraction in this program is best understood in light of the program's philosophical stance on mental arithmetic, algorithms, and calculators. Information about the program's perspectives on these matters can be found in the Teacher's Reference Manual. The program places a strong emphasis on mental arithmetic, including ways to simplify mental computations. Consequently, there is also an emphasis on learning the basic addition and subtraction facts well. Secondly, the program's position on calculators is that both students and adults should use them for difficult problems and students should learn about them beginning in kindergarten. The net result is that algorithms as we commonly know them (the standard algorithms for addition and subtraction and closely related variations) are relegated to secondary importance in the program. On the other hand, the program does not shun algorithms. It sees them as valuable learning tools, but does not see the need to master any particular algorithm. It also tends to refer to processes used to simplify mental arithmetic as algorithms. Therefore, by the third grade, this program would like students to experience a variety of algorithms, and it encourages them to develop algorithms themselves and to explore those developed by their classmates.
Consequently, no particular emphasis is given in this program on specific algorithms for addition and subtraction. Several are illustrated in the Teacher's Reference Manual as some of the available methods, but without explicit instruction to teach any particular method.
The presentation with respect to the addition and subtraction is consistent with the above positions. Considerable attention is given to learning the addition and subtraction facts. However, the program begins without specific attention to them. Part of four early lessons includes playing a card game requiring basic facts. Notation for applied problems is introduced next. At this stage, the notation system consists of a number model for a problem and a unit box in which the units of addition or subtraction are written. Students practice translating a few simple stories into this notation. Then, they begin to practice the facts in games competing against a calculator. The facts table is introduced in lesson 16, and students are taught how to use it. Practice in basic facts continues in games and other activities for many lessons. Students study the fact table and learn groups of related facts over time. Fact triangles and flash cards are also used. Students are assessed on a few facts at the end of Unit 2. Some simple extensions to the basic facts are covered in Unit 3, such as adding or subtracting multiples of 10. More extensions are provided in Unit 4 which begins to stress mental arithmetic. Addition and subtraction diagrams are introduced, which are basically boxes in which to write the known values and one for the unknown. Sums and differences involving multiples of 10 continue to be important, but other 2-digit addition problems also become evident at this stage. Students are also encouraged to estimate solutions to be sure their answers are reasonable. Addition and subtraction receive little attention in Unit 5. Unit 6 returns to these topics. Addition with several addends is covered first. Students are introduced to adding three numbers in any order, finding the easiest way to obtain the solution. Four problems are discussed. Students find the sum the measures (single digits) of sides of a triangle and of a rectangle on a journal page. Parts and Total diagrams are introduced as a way to record information from a simple context involving the addition of two or more addends. Compare diagrams (with boxes for quantity, quantity, difference) are introduced in the next lesson. Students use these two methods for extracting a symbolic representation from a simple context problem. Change diagrams (starting value and ending value known, change value unknown) are introduced in the next lesson as a way to record missing addend and missing subtrahend problems. Unit 7 has a lesson involving multiple operations, such as adding 5 and subtracting 10. Repeated addition and subtraction using a calculator is the subject of another lesson. Further work with sums and differences involving multiples of ten is covered in another lesson in this unit, this time using the calculator. Sums of multiple addends appear again as the subject of a lesson in this unit.
To this point, addition and subtraction problems are generally simple instances, with the 2-digit cases usually being designed to be of an easy variety but a few more difficult cases appear. The remainder of the course does not focus on addition and subtraction, except in a money context. Even there, the process tends to be handled by treating the cents and the dollars separately and correcting (making trades or renaming) to reach the final solution. Thus, an interesting case appears as an Extension (optional component) to lesson 100. Here, the sum of 2354 and 4667 is shown to be like the sum of $23.54 and $46.67, so that the solution can be obtained by adding the dollars and adding the cents and combining the totals. This is not to suggest that addition and subtraction of 4-digit numbers are a realistic component of this curriculum. Even though treatment in money amounts involves dollars and cents, this is likely to be achieved as two processes and not as mastery of addition and subtraction with 3-digit numbers. The addition of three digit numbers is part of a lesson in Unit 8, but the experience is very limited.
In summary, the program places lots of attention on basic facts and on mental arithmetic in simple cases. Attention to trades or renaming is given in this context, not as a component of the standard algorithm or a close variation of it. Sums of multiple addends are practiced, but usually only in simple cases. There is a lot of attention to isolating the known values and the unknowns from simple situations. Units are given a lot of attention, but since the same units apply to all terms in addition and subtraction problems, the units are written once in a box. The attention to basic facts occurs in many contexts, including games and competition against a calculator. The treatment beyond basic facts, on the other hand, is not given regular procedures. Instead, the methods and their understandings are expected to grow from the work of the students themselves. Consequently, finding best methods and watching for common error situations is a matter for the expertise of the teacher and not part of a regularized procedure. Consequently, the degree of student learning beyond basic facts may be variable and highly dependent on the expertise of the teacher or even of classmates. However, it must be noted that the student work on difficult problems is limited.
Multiplication of Whole Numbers [4.5]
The same philosophical stance with respect to algorithms noted above continues with respect to multiplication in this program. However, a partial products algorithm will be introduced in grade 3. Nonetheless, this review only expects introductory material with respect to multiplication in grade 2, and thus the use of an algorithm is not anticipated anyway.
In fact, the coverage of multiplication is fairly extensive for a grade 2 program. The bulk of this material appears in the context of two series of related lessons, one given about mid-year and one toward the end of the year.
Although skip counting and other related material may occur earlier, the topic begins in earnest in Lesson 54 of Unit 6. Students are presented with a problem situation - 3 packs of gum with 5 sticks per pack. A discussion of multiples of equal groups is to follow. The multiplication symbol and the terms multiplication, times, and multiplied by are introduced. A few more simple number stories are addressed in the lesson. The next lesson supplies the array context for multiplication. The template multiplication diagram used to record the translation of a problem situation has boxes for rows, ___ per row, and ___ in all. The language (e.g. a 2-by-6 array) is introduced as well. Several simple situations are introduced, translated, and solved. Pairs work with arrays of counters and record their work on journal pages. More of this material appears in the next lesson. Patterns of multiples (by 2, 3, 4, 5, and 10 in simple cases) are the topics of two lessons in Unit 7.
The group of lessons near the end of the year begins with Lesson 102 in Unit 11. This unit introduces the terms factor and product and students solve a few simple problems. In Lesson 104 students practice a few simple multiplication facts (by 2, 5, and 10), and supply a row (or column) of facts of their choosing. In the next lesson, students are introduced to the products table. This is like the multiplication table for 0 through 10, except that the rows and columns are not identified. Instead, the full multiplication notation is written in each box and the products are missing except for the diagonal. Students are told that the squares of a number appear in the table already. As a class, rows of the table are completed. Turn around facts are used to complete the table as needed (commutative property or related facts). The next lesson expands on related facts or fact families, and uses fact triangles for this work.
In summary, this program provides the introductory material for multiplication that is expected in this review. Students represent multiplication in various models, learn some of the multiplication facts and the notation, and learn the terms factor and product. They are given reasonable practice in translating between simple situations and multiplication notation and solve some simple problems.
Time [2.5]
The program begins the treatment of this topic in Unit 1 with material that suggests students come into this program with considerable knowledge in this topic area, although teachers may use this to assess the incoming abilities of students. In an activity done separately, students read clocks to the half-hour and draw clock hands for times to the quarter hour, they also complete a calendar for the current month.
The topic resumes in Unit 3 where telling time is addressed. The terms minute hand, hour hand, clock face, analog clock, and digital clock are introduced. Reading the clock face is practiced, first with just the hour hand. The items will appear in problems given over the next several lessons in the Math Boxes part of the Independent Activities and in some Journal pages. Students again draw clock hands to show times in Unit 4, only this time they are showing the time a given amount of time later, such as 15 minutes after 10:30. The treatment of time also contains material on recording the time including the a.m. and p.m. notation, and reading time to 5-minute increments is the expectation.
Work with calendars includes daily activities such as writing the date on their papers and references to calendars and birth-dates. Some material about clock time and dates is reviewed late in the year, and several items practice reading clocks at this point.
In summary, the treatment of time in this program addresses the material that is expected in this review. However, it does so in a rather casual, almost haphazard manner. Actual instruction in these topics is sporadic.
Money [4.0]
The use of money amounts is fairly common in this program. This begins in Unit 1 where students report the value of a few collections of coins, report the value of a collection of $1, $10, and $100 bills in the context of place value, and are given one problem in making combinations of coins to equal 25 cents.
In Unit 3, students practice paying exact amounts with coins, practice coin exchanges, find the cost of two items, and practice with change in simple contexts. Unit 4 contains 3 lessons in the context of shopping focusing on total cost, estimated monetary sums, and making change in whole dollar amounts. This includes finding the value of a set of bills and coins, and showing the same amount with different coins. This includes the notation systems for dollars and cents.
Little further material with money occurs until Unit 10 where money is used in the further discussion of place value and the introduction of decimals. This begins with a review of money concepts and then uses money as the introduction to decimal representations. Two lessons then address the methods needed to use calculators with money amounts. One lesson briefly addresses the uses of estimated versus exact costs. Next, Lesson 94 addresses making change to $10.00. However, this is in the context of a detailed procedure in which calculators are used to find the total and check the change returned. At least, however, the person playing the salesclerk role must count out exact change.
Money also is prominent in Unit 11. Addition and subtraction with dollars and cents are the topics of the first two lessons of the unit. In addition, problem totals are more than $1 and less than $10. As is typical for this program, the methods used are selected by the students and then shared. Differences in prices are treated in a similar manner. Another lesson even addresses finding estimates of 10% or 15% of a money amount in the context of tips, but again the methods are developed by the students and shared.
In summary, the content expected in this review is well covered by the material in this program. Explicit methods and techniques are frequently not provided and are expected to be solicited from the students along with their explanations. However, the level of this material is consonant with the expectations in this review.
Measurement of Length, Weight, Volume, and Temperature [5.0]
Activities related to the measurement of length, weight, volume (capacity), and temperature are scattered throughout the text. For length, activities address the use of rulers, meter and yardsticks, measuring height, measuring perimeters, using non-standard and standard units, relating inches to centimeters, measuring in feet and meters, estimating linear measurements, using correct units, and some linear measure conversions.
For weight, activities include weighing students and various objects, using both pounds and kilograms, estimating weight and weight differences, using a balance, and using appropriate units.
For volume, this includes the use and simple equivalence of units such as cup, pint, and quart, using appropriate units and fractional units, measurement in relation to recipes, and capacities of irregularly shaped containers,
For temperature, this includes reading temperatures in both Celsius and Fahrenheit scales, indicating temperatures on models of thermometers, and using negative numbers in the context of temperature. Comparisons of the two scales are made and students estimate and compare temperatures.
Although scattered throughout the text, much of the material appears in the measurement unit of the program, Unit 9, especially for linear measurement topics. Here, students use the rules for translating between yards and feet, meters and centimeters, and meters and decimeters. They estimate distances in feet and measure to check their estimates. They learn about miles and kilometers and solve problems comparing travel distance. Fractional units, such as half-inchs and half-centimeters, are also addressed. Several capacity equivalencies within measurement systems are presented.
In summary, the expectations for this review with respect to the measurement topics are well covered by this program. Sufficient activities and experiences address this material.
Perimeter [5.0]
Although not presented completely until Unit 9, finding a perimeter actually appears before mid-year in Lesson 49. In the context of addition with several addends, two problems are given where students sum the side measures of figures in centimeters - a triangle and a rectangle. The term perimeter is not used in this context.
Unit 9, the measurement unit, addresses perimeter. The instruction includes a discussion of ways to measure the distance around an object, which includes measuring the edges and finding the sum. Students are asked to think of applications. Students then find a few perimeters. A Language Arts Connection in the Teacher's Manual notes that perimeter comes from the Greek metron (measure) and peri (around). The next lesson is designed to emphasize that perimeter and area are different, and point to the different measures used for each.
Thus, this program covers the review expectations with respect to perimeter. Students are provided with the concept, the term, and the technique required.
Program Quality Evaluations
Mathematical Depth [3.1]
The mathematical depth of this program varies from topic to topic, largely reflecting the philosophical stance taken by the program. Topics that are new at this grade (e.g., multiplication), and are therefore treated conceptually but superficially, contain adequate depth for this grade level. This is also true for the more experiential measurement topics. On the other hand, the resistance to algorithms and mastery and the reliance on calculators seem to work together to restrict the depth of the material in addition and subtraction. Attention is given to basic facts and mental arithmetic, the latter a key to understanding, avoiding algorithms restricts student experience in solving more difficult problems. Although teachers are encouraged to discuss solution methods and student-developed algorithms, the lack of a regular, defined presentation of this material means that a weak presentation is a risk in this approach.
Quality of Presentation [2.0]
The program design is such that teachers will usually communicate the objectives of their lessons to the students. The explanations of concepts and procedures provided by the program are likely to be quite mixed in quality. Sometimes, the details are clearly explained in the Teacher's Manual in ways that are likely to be communicated to students. However, in many cases teachers are expected to lead discussions where methods and reasons are supplied by students. This requires great skill and direction on the part of teachers and may not be effective in many cases. The program is sometimes very efficient in covering a particular content area, as in the case of Perimeter given above. Other times, however, the program is likely to utilize inefficient methods. At least, there is little non-mathematical material outside of problem contexts that would serve as a distraction. Calculator use is a bit excessive and may inhibit student learning as there is a tendency to rely on the calculator for more difficult cases.
Quality of Student Work [2.0]
The student work provided by the program is extensive in that students engage in a variety of activities throughout the course of each lesson. The extent of homework is limited and could be used more effectively, even in grade 2. The student work contains distributed practice throughout the course, which helps with the development of the skills addressed. On the other hand, the problem counts are minimal as concepts and procedures are introduced.
The biggest drawback with respect to student work is that the range of difficulty is often very limited. This limitation could mean that students may not be challenged by this program as there are not opportunities provided for them to engage in more advanced work. Items requiring extensive practice, especially addition and subtraction with larger quantities but even perhaps in telling time, are also too limited.
Overall Program Evaluation
The overall evaluation of this program is mediocre, but that is far from telling the whole story. It is unusual in that some topics, such as perimeter and measurement, are covered quite well while other topics, notably addition and subtraction of whole numbers and to a lessor extent the work with time, are given a fairly weak treatment. The result appears to be related to the overall program philosophy which chooses to emphasize ideas and calculators and even mental arithmetic but de-emphasizes matters that require extensive practice and the use of algorithms. Multiplication, which is in the early stages in grade 2, is covered well given that the expectations are mostly conceptual and not intending to lead to mastery (or even close to it) at this level.
Despite good coverage of some topics, it may be difficult to identify a situation where the use of this program is very appropriate. If expectations are high, then the program seems to be inappropriate due to the lack of support for the mastery of central topics. For situations with lower expectations, the program may contain too much attention to the higher-level topics and not enough attention to support success with addition and subtraction. Thus, it is difficult to recommend this program despite the circumstances.
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