Methods for Second Grade Program Reviews

The second grade mathematics program reviews were accomplished by surveying the instructional materials provided by the publisher. The student text, where available, was emphasized so that the review would focus on the material the program presents to students. In cases with no student text, the worksheets and other materials that constitute the presentation were evaluated. Frequently, a great deal of supplemental material is available for a program. It is beyond the scope of these reviews to address each element in a wide range of supplemental materials.

A sampling of topic areas was selected for in-depth evaluation. Ratings were made for each of these topics and for the program as a whole as described below.

Topic Selection

A sampling of topic areas was selected for detailed evaluation. These were selected to cover a range of material including both major and minor aspects of the second grade curriculum. The topic selections were also designed to range from material that might be newly introduced in the second grade to topics that should be reaching mastery by this level. Together, the topic selections cover a good portion of the material expected in the second grade, but it should be emphasized that they are just a sample of the entire curriculum. While the topics are not meant to cover the entire curriculum, they are designed to be sufficient to give a clear impression of the features of presentation and an assessment of the mathematical depth and breadth supported by the program.

Within each topic area, expectations were developed for what should appear in the second grade curriculum. In each case, achievement expectations for students were guided by the standards of learning in the Mathematically Correct Standards and the San Diego Mathematics Standards. These provided broad guidelines for what could be expected in the curriculum. However, it should be recognized that standards and curriculum contents may not always align. Since standards of learning mark points at which students are expected to have gained explicit knowledge and competencies, it is often appropriate that topics will appear in the curriculum ahead of their corresponding references in the standards. Therefore, some attention was also given to standards for grade 3. The details in these cases will be elaborated below.

Topics Reviewed

The topics selected for review are summarized below. In each case, the reason for the selection of the topic is given. The development of the expectations for that topic is summarized and the contents looked for in the review process are noted. These selections are not designed to cover the entire second grade curriculum. Instead, the objective was to provide a sample of topics that could address the depth and breadth of content coverage.

The topics selected for review in the grade 2 materials include:

Addition and Subtraction of Whole Numbers
Multiplication of Whole Numbers
Time
Money
Measurement of Length, Weight, Volume, and Temperature
Perimeter

The summary for each of these topic areas follows.

Addition and Subtraction of Whole Numbers

Addition and subtraction of whole numbers was selected for evaluation as it constitutes a major segment of the curriculum in grade 2. The skills and knowledge of addition and subtraction should be approaching mastery by this grade level.

The related Mathematically Correct Standards for grade 2 include the recall of basic facts. Addition of 3-digit numbers is only expected without regrouping, as is the addition of three 2-digit numbers. Estimation of sums and differences for 2-digit numbers is expected, as are simple addition and subtraction problems given in words or based on data from charts or graphs. Students are expected to know about the inverse relationship between addition and subtraction and be able to solve simple "missing addend" problems.

The related San Diego Mathematics Standards are more ambitious. They expect the recall of basic facts as well, but also expect addition and subtraction with 3-digit numbers, including problems requiring regrouping. Likewise, sums of three 2-digit numbers without regrouping are expected. Estimates of sums and differences are also expected in each case, as are solutions of word problems. Students are expected to know about the inverse relationship between addition and subtraction and be able to solve simple "missing addend" problems. They should also use addition to verify subtraction and vice versa. Finally, they should use and describe both the commutative and associative properties of addition.

The differences between these standards are a function of their design. The Mathematically Correct Standards were designed to implement clear objectives that did not increase expectations dramatically in the early grades. The San Diego Mathematics Standards attempt to stimulate learning increases in the early grades. Obviously, the more effective programs will support the more ambitious standards.

The evaluation for this topic area is therefore based on the expectation of developing addition and subtraction at least through 3-digit numbers. The curriculum must provide extensive support to achieve this goal. This includes student understanding of their methods, the representations of place value by the digits used in the standards algorithm and in the renaming (carrying and borrowing) processes, and the estimations of sums and differences. The program should also include problems with missing addends, minuends, and subtrahends at least through 2-digit cases. The curriculum should also include sums of multiple addends, at least through three 2-digit numbers. Overlooked in both sets of standards are the sums of several (4 or more) numbers, at least in the single-digit case. These will be important in other aspects of the curriculum and in the greater development of mental arithmetic. The program should also cover the use of addition to verify subtraction and vice versa.

Mental arithmetic should be supported at least through sums and differences with 2-digit numbers. Techniques to simplify mental computations should be taught, which may refer to the commutative and associative properties.

These processes should be accompanied by the use of addition and subtraction to solve applied problems. To accomplish this, the program will need to address knowing when to add and when to subtract, as well as writing simple number sentences for problem situations. Thus, a variety of application problems should be provided, and attention should be given to correctly recording units in the solution.

These are ambitious content expectations for the curriculum and can only be achieved in the most carefully designed programs. These curriculum contents represent what can be expected in the programs with the greatest mathematical depth. Programs that cover such topics in a casual way that will not really support student learning of these topics should not be considered as effective in this regard.

Multiplication of Whole Numbers

Multiplication of whole numbers was selected for evaluation as it is a key element of the curriculum that is only at an early stage by grade 2. While some introduction to division is expected in conjunction with this topic, the review will focus only on multiplication.

The Mathematically Correct Standards expect grade 2 students to recognize the multiplication sign, know what the terms factor and product mean in multiplication, and understand that multiplication represents repeated addition. It also expects students to know some of the multiplication facts, to include multiplying single digit numbers by 0, 1, 2, and 10. The expectations in the San Diego Mathematics Standards are the same.

In evaluating the curriculum, attention is therefore given primarily to material that introduces multiplication. The process should be modeled in various ways, such as with repeated addition, arrays, area, and skip-counting or multiples.

While there is no expectation of mastery for the multiplication facts through 10 times 10, a subset of the facts should be covered. This should include mastery for multiplying by 2 and 10, and perhaps other facts. Also, the special cases of 0 and 1 should be addressed clearly. The curriculum should present multiplication notation, including the X (times) symbol and both the horizontal and vertical forms, along with the terms factor and product. Students should be able to translate simple application problems into number sentences which they can solve, and they should record units in solutions. To understand this process, programs should provide opportunities for students to learn when problems can and cannot be solved by multiplication. Programs that go further in this topic area may have students solve more difficult problems, relate multiplication to division, introduce problems with missing factors, or combine multiplication with other operations.

Time

This topic area is selected as one that is unique to the early grades and is a critical topic in measurement with important applications in all walks of life. Work with time should be progressing well by grade 2.

The Mathematically Correct standards expect students to tell time to the quarter hour using analog and digital clocks and to write the time to the same degree of precision, including the use of a.m., p.m., noon and midnight. Students are also expected to solve simple problems involving elapsed time. They are also expected to use a calendar to identify the date, day of the week, month, and year and to write the date using words and numbers and only numbers.

The San Diego Mathematics Standards contain similar expectations. In addition, students are expected to order events by time sequence, identify equivalent periods of time (weeks in a year, days in a month, minutes in an hour), determine past and future days of the week, and identify specific dates on a calendar.

Accordingly, the evaluation should expect the curriculum to support learning about time at least to the levels of these standards. This should include telling time at least to the quarter hour using both analogue and digital clocks and writing the time at least to the same degree of precision. The curriculum should also address the use of a.m., p.m., noon, and midnight. With respect to the duration of events, neither set of standards offers much input. However, the curriculum should begin to introduce this topic, perhaps as finding the duration of an event in whole hours given the start and end times and as using a clock to measure small time intervals (e.g., 5 minutes). The curriculum should also include reading calendars and writing the date. The day of the week is an interesting topic for grade 2 because of the cyclic nature, providing an early introduction to modular arithmetic.

Money

We are fortunate to have a monetary system that uses a base-10 system and includes decimal notation. Consequently, work with money is a very important part of the early curriculum. Furthermore, the arithmetic of money has clear connections to real life that none can deny. Thus, work with money was selected as a topic for evaluation.

The Mathematically Correct Standards expect students to count, compare, and make change, using a collection of coins and one-dollar bills and to recognize the relative value of penny, nickel, dime, quarter, and dollar. Students are also expected to read and write amounts of money using cents symbol and with dollar signs and decimal points. Finally, students are expected to be able to show different combinations of coins that equal the same amount of money.

The San Diego Mathematics Standards are essentially the same, except that students are expected to show different combinations that equal the same amount of money with combinations of coins and bills.

In evaluating the grade 2 curriculum materials, these programs should therefore support reading and writing money amounts both in cents notation and using the dollar sign and decimal point. The programs should include selecting different combinations to show the same amount of money. Thus, the instruction should include counting money and making exchanges of equal value. Importantly, the curriculum should include making change. More advanced work could introduce students to making change to $10.00. Similarly, a more advanced approach would translate making change from a counting process to one involving written addition and subtraction in the dollar sign and decimal point notation. At some point, the transition from money notation to the more general use of decimals must occur. An early step in this process is the recognition of 0.1 as one-tenth. This is an advanced topic for second grade.

Measurement of Length, Weight, Volume, and Temperature

The early curriculum includes measurement as important skill development, as a connection of mathematics to real-world applications, and as an useful context for the study of various operations and with various bases. Measurement was included in the evaluation because it is so important in the early curriculum.

The Mathematically Correct Standards are fairly explicit in identifying expectations. Length should be estimated in inches and centimeters and measured in feet and inches and in meters and centimeters. Students are also expected to know that one foot equals 12 inches and to know the abbreviations for foot, inch, and centimeter. They are even expected to draw line segments to the half-inch and centimeter. Weight should be estimated and measured in pounds and kilograms, and the corresponding abbreviations should be known. Volume should be estimated and measured in cups, pints, quarts, gallons, and liters. Students should know that a liter is a little more than a quart. Temperature is to be measured and recorded in Fahrenheit to the nearest 2 degrees.

The San Diego Mathematics Standards are similar, but there are some differences. Exact expectations for length measurement are not given, nor is the drawing of line segments of given lengths. Weight is to be estimated and measured in pounds and kilograms. Volume is to be estimated and measured in cups, pints, quarts, gallons, and liters. Temperature is to be measured and recorded to the nearest 2 degrees in both Fahrenheit and Celsius, including the use of the degree symbol. No other details about abbreviations are given. No details about equivalent units (e.g., 12 inches in 1 foot) are given.

In evaluating the curriculum materials, the programs should include material for length, weight, volume and temperature. This should involve estimation and measurement in inches and centimeters, pounds and kilograms, cups, quarts, gallons, and liters. Measurement of temperature in degrees (to the nearest 2 degrees) should be included, but estimation of temperature is not critical (and may not be advisable beyond certain limits). Other units, such as pints or yards, may be added for greater depth. However, more substantive advancement would involve looking at relationships of equivalent units (inches per foot, quarts per gallon, etc). However, not too much of this should be expected in grade 2.

Perimeter

The perimeter of simple polygons is included as a topic for evaluation because, although it would traditionally be considered fairly advanced for grade 2 students, it is well within their ability and understanding level. It also can be addressed fairly briefly and specifically.

The Mathematically Correct standards expect students to measure perimeter to the nearest inch and centimeter in grade 1 and to determine the perimeter by adding the side measures in grade 2. The San Diego Mathematics Standards expect students to estimate and make linear measurements to the nearest centimeter and inch for the perimeter of a polygon. Students are also expected to measure perimeters in inches of squares and rectangles. While this is not explicit, it suggests addition of the side measurements.

In any case, the evaluation for this topic area should focus on the concept of perimeter and finding perimeter by summing the side measures. Actual measurement of perimeter can be accomplished in both direct means (e.g., wrapping a tape measure around an object) or an indirect means (e.g., wrapping a string around an object and then measuring the string). The program may use any techniques with respect to the concept of perimeter. The critical factors are that the program develop the concept in a way that is understood and present adding the lengths of the sides. It is best if the perimeter is not restricted to squares and rectangles.

Content Area Evaluations

For each of the selected topics, a rating was made to reflect the breadth and depth of mathematics learning supported for that topic area. These ratings considered the level of the mathematics presented with respect to the criteria established in each topic area above. However, the rating is meant to reflect the expectation for student learning supported by the instructional materials for that topic. Thus, more than the mere presence of the identified content areas was considered. Other features of the program for the topic area were also taken into account. Specifically, the mathematical depth, the quality of the presentation, and the quality of the student work were also considered. More details on these dimensions are given below as the dimensions used for evaluating the entire program are discussed.

Program Quality Evaluations

From the study of the instructional materials, and especially from the in depth study of the selected topics, ratings were made on several dimensions for each program. Ratings ranged from 1 (poor) to 5 (outstanding) on each of these dimensions.

Mathematical Depth

A single rating for mathematical depth was made based on program contents. This primarily reflects the extent of the mathematics covered relative to the expectations for each of the topic selections. However, the depth of coverage within particular achievement levels was also considered. This insures that programs would not benefit from a superficial coverage of a wide range of achievement targets.

Quality of the Presentation

A single rating for the quality of the presentation was generated based on several considerations. Together, these survey program aspects that can promote student learning. The factors considered in evaluating the quality of presentation include:

Clarity of Objectives

Making lesson objectives clear to students, as well as to their teachers and parents, supports student learning. In the ideal case, clear statements of objectives are consistently presented.

Clarity of Explanations of Concepts and Procedures and Definitions of Terms

To support student learning, explanations and definitions need to be mathematically correct but also need to be clear and understandable to students. Long explanations are not necessarily better, since they may strain student attention or overload students with details. Crisp, clear, and accurate explanations and definitions are desired.

Quality and Sufficiency of Examples

Explanations in mathematics should make effective use of good examples to promote student understanding. Often times, multiple examples are required to illustrate a topic, particularly when different examples illustrate different aspects of the topic. However, in most lessons a few examples will be sufficient. More important is the quality of the examples and their annotations. The examples must complement other explanations and clearly illustrate the important points to be covered in a way that students can easily grasp.

Efficiency of Learning

This dimension is designed to account for the fact that student time for the study of mathematics continues to be at a premium. To make the greatest achievements over the course of the second grade year, there must be a reasonable degree of efficiency supported by the instructional materials. There are several ways that programs can promote efficient learning. Obviously, the program should be free from irrelevant and distracting content. It is also important to maintain a focus on mathematics throughout each lesson. Programs that focus learning activities on lesson objectives and structure lessons in efficient patterns will also be more successful. Efficiency is reflected in programs that support a high level of mathematics achievement in a reasonable expenditure of student time. Some programs are characterized by a fluctuation between efficient instruction and distraction. The sequence of lessons could impact on this rating. Likewise, excessive instruction in the use of calculators, addressing the mechanics of these machines rather than the mathematical focus, could impact negatively on efficiency.

Quality of Student Work

The work that students are expected to complete is critical to an effective mathematics education. The evaluation of student work was based on two considerations:

Quality and Sufficiency of Student Work

In part, the quality of student work depends on the amount of student work required for a particular topic. While student work can be carried to an excessive length, too little practice is an obvious and serious limitation. Mathematics education should not aim for a casual exposure to a great variety of topics, but rather should lead to mastery in selected areas. Student practice must be sufficient to support this development. This does not necessarily mean massed practice on a different and narrow topic area each day. In fact, frequently revisiting a topic after it is introduced can be a good feature.

Beyond quantity, the quality of student work was also considered, and should reflect both adequate symbolic or numeric cases and adequate application problems. Methods that varied to fit the context were also a positive feature. For example, a few problems that address renaming (borrowing and carrying) specifically or problems that ask students to detect errors were positive features.

Range of Depth and Scope of Student Work

As distinct from the sheer quantity of student work and the quality of the exercises, attention was given to the range of depth and scope of the student work. Within any topic area, students should be competent across a range of difficulty or complexity of work. The student work would naturally be expected to build from the easy and simplex to the difficult and complex cases. The consideration therefore focused on the coverage of a range of depth across the student exercises.

Overall Evaluation

Finally, an overall evaluation was generated for each program. These ratings were based both on the topic evaluations and the overall ratings for mathematical depth, quality of presentation, and quality of student work. Among the topic areas, greater weight was given to the major topics - multiplication and division with whole numbers and with decimals - than to the other topics.

Thus, the ratings generated for each program included five ratings of specific topic areas and ratings of mathematical depth, quality of the presentation, and quality of student work along with an overall rating of the program.