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 Fifth Grade Program Reviews.
Student Text Structure
There is no student text.
The curriculum is divided into 9 units with a teacher book for each:
Each unit is divided into 2 to 5 Investigations. Each Investigation is divided into 2 to 10 Sessions. The total number of Sessions is close to 200.
The student work is largely through the use of reproducible Student Sheets (worksheets). There are over 200 worksheets, some to be done in class and some to be assigned as homework. The student work is through these worksheets and through instructions given by the teacher in class. Students do not receive written explanations or examples and do not have a text for reference or a glossary for the definition of terms. Class time is structured around activities, typically in small groups or pairs. There may be more than one activity in a session. Whole class work usually occurs as the teacher introduces an activity and in discussions typically at the end of a session.
Sometimes sessions are independent, but other times adjacent sessions will be grouped together. For example, in Unit 5 (Building on Numbers You Know), the first Investigation is called Exploring Distance Between Numbers. The first Session is called Reasoning About Multiples. There are two Activities that happen to be whole class activities. In the first, the teacher has the class count by 100's. The teacher asks for volunteers for each number, and stops at 1200 to say We're up to 1200 counting by 100. How many students have counted so far? How do you know? Students are to offer explanations that the teacher records on the board. Next, the teacher asks, If we keep counting by 100's, what number will we end with if everyone in the class says one number in the count? Again, the explanations are recorded on the board. The class then counts by 100's in seat order so that the correct answer can be determined. The process is repeated with numbers like 200, 50, 250, and 1000. In the second activity in this Session, the class starts at a different number, such as counting forward or backward by 25 from 1000. The worksheet to accompany this session as homework has three situations counting by 25's, 20's, and 10's. In the third case, the first number counted is 110. In each case, the student is asked how many people are counted to get to 300 and how they know.
By contrast, in Unit 6 (Measurement Benchmarks) the second Investigation is called Measures of Weight and Liquid Volume. The first two Sessions in this investigation are grouped together. In the first Activity, the teacher directs whole class discussion about the markings for weight and liquid capacity on grocery items. In the second Activity, students work in pairs to make a table showing the Metric and U.S. standards weights for some grocery items. In the third Activity the student pairs do the same for the Metric and U.S. standard units for liquid capacity. In the next Activity, each pair of students holds one product container labeled by weight and the pairs line up in weight order. This is repeated for items measured by liquid capacity. Two homework assignments are given for this pair of sessions. In one, students find one product that looks small but has a large weight and one that looks large but has a small weight and record these. In the other, students find the products with the largest and smallest liquid capacities and record those. Related to this is the fact that these methods would provide fruitful ground for addressing the properties of numbers, but they are neither presented nor called upon in step by step justifications. There is a rather casual approach that comes across as 'you can do this' to solve a problem. Steps are often not given explicitly.
Content Area Evaluations
Multiplication and Division of Whole Numbers [1.5]
This program does not teach the standard algorithm for multiplication. If students already know this algorithm, they will still be required to develop other strategies. If they do not know the standard algorithm, the text does not direct them to learn it. Instead, the program encourages students to select their own strategies, but those that are emphasized are often based on partial products. For example, acceptable approaches to 46 x 25 are:
After some exposure to factors of landmark numbers and an introduction to the representation of a product as area, Sessions 2, 3, and 4 of Investigation 3 in Unit 1 introduces multiplication and division clusters. These clusters are designed to lead to partial product solutions. In the first activity, the students are presented with the following cluster:
2x5 20x5 40x5 42x10 42x5
They are asked questions about how knowing one product can help them find another. Students work in pairs and then share a few strategies. Next 18x5 is added to the list and students are asked how 20x5 and 2x5 can help to find 18x5. In the next activity, students work in pairs on various clusters always trying to solve the last product in the cluster. Six problems are represented multiplying by 1-digit and simple 2-digit values. A second, more difficult, student sheet is optional. In the next activity, students work in pairs to list products that could help solve 34x25. Additional problems are optional. The remaining activities address division clusters. As homework, students solve two clusters, one for 65x20 and one for 77x25, and write about how they used the products in the cluster to solve the problems.
Later, in Unit 5 of the curriculum, more material on multiplication is given. Students have done more work finding multiples by this point. In Investigation 2 multiplication and division strategies are mentioned, however, this is really intended as a division solution. In Investigation 3, multiplication clusters reappear. In Student Sheet 20, the clusters relate to solutions for the following products: 22x123, 47x18, 498x9, and 215x72. In student sheet 21, students write about their solution to the cluster for the product 52x21. In session 10 of this investigation, teachers are to evaluate their students on the way they solve 42x51 without being given a cluster of products.
In the final Investigation in this Unit, students solve 253x46. Then, using worksheet 37 students solve four problems multiplying by 2-digit numbers. Another 2-digit problem appears in student sheet 40. In sheet 41, students are asked how one product helps them find others.
Using these various alternatives to the standard algorithm often requires recording so that students do not get lost in the process. While the teacher books recognize the importance of this, there are not specific, regularly applied procedures for recording the process. Instead, the teacher books assert that recording and keeping track of progress is easier than it might seem. This leads to a variety of recordings with varying degrees of clarity. Since recording techniques are variable, these techniques may not be as useful in finding errors or reaching the correct solutions as might be hoped. Related to this is the fact that the manipulations that are used, frequently based on factors and properties, seem to be expressed as an unorganized set of you can situations with unclear rules.
In summary, the instruction in multiplication of whole numbers is not learned from a text and thus is highly dependent upon teacher supervision. While the objective is to get students to devise methods that make sense to them, there is little regularity to any particular approach. The number of practice items is very limited, as is the level of difficulty of the products.
Division of Whole Numbers
The treatment of division is similar to the treatment of multiplication in this program. In Unit 1, division clusters are presented. In the worksheets, there are 10 division problems presented in this format.
Unit 5 takes up division to a greater extent. Investigation 2 is on multiplication and division strategies. In the first session, students work alone or with a partner to determine how many 21-cent ringles are in 1 dollar, 2 dollars, and 5 dollars. As homework, they figure out how many 3-cent zennies are in 1 dollar, 2 dollars, 3 dollars, and 10 dollars. They are to show how they found the solution. In session 2 they work alone or in pairs to figure out how many markers each student in the class would get if 1, 2, or 5 boxes of 70 markers were shared equally. As homework, they make up their own coin and determine how many are in 1, 2, 5, and 10 dollars.
A teacher note (p. 47) discusses the use of standard algorithms: When students use the standard algorithms, often they are performing the steps by rote and cannot explain how their notation relates to their understanding of the problem. It goes on to present an anecdote wherein a teacher probes two students who used the standard algorithm to find they could not completely explain it. Both students were asked by the teacher to develop other methods.
Session 3 presents three ways of writing division problems. As homework, students select and solve one division problem from a choice of six. The divisors to select from are 7, 8, 12, 15, 21, and 24. They are instructed not to use either the standard algorithm or a calculator. Session 4 is devoted to the treatment of remainders choosing rounding up, rounding down, or expressing the answer with a remainder. As homework, students show what to do in 4 different situations involving dividing 171 by 5.
In the next investigation, student worksheets contain 10 division problems presented as division clusters. In two other worksheets, they make up situations to match division problems. They also make up their own cluster to help solve 187 / 13. In subsequent worksheets, students solve 10 division problems in different ways (i.e. by starting with different given facts. Divisors in these 10 problems include 2, 3, 5, 7, 8, 9, 12, 15, 24 and 30. Subsequently, they choose a problem which may include division and show two solutions.
A few simple division problems appear later in the unit and a few are given as practice in other units. Problems identified as challenges include 703/17 and 1904/6.
In summary, the lack of a text and the methods employed provide students with no regular and orderly procedures to use. While the program suggest that it strives for understanding, the depth and breadth of the division experience the students gain is minimal.
Decimal Multiplication and Division [1.0]
These topics are either not present or appear to such a minimal extent as to be effectively not present.
Area of Triangles [1.0]
This topic is either not present or appears to such a minimal extent as to be effectively not present.
Negative Numbers [1.2]
The topic of negative numbers is not directly addressed in this program, but negative numbers do occur in another context. Students deal with figures moving left or right on a computer screen across a numbered grid (no negatives). Step sizes and be positive (right) or negative (left), usually no more than negative one. Students can get the idea from this that a step-size of negative one means counting down. Some of the work involves graphing with step size on the y-axis, but this is not the same as using the four-quadrant coordinate grid.
Students do not get the benefits of relating to negative numbers on the number line, or relating to them in other contexts, or ordering and comparing negative numbers. The idea of an additive inverse, or even the negative result of subtracting a larger number from a smaller one is not supported. The use of the step-size parameter should not be considered a treatment of addition and subtraction with negative numbers. It is essentially a signal for a counting direction.
Thus, very little about negative numbers is conveyed by this program.
Powers, Exponents and Scientific Notation [1.0]
This topic is either not present or appears to such a minimal extent as to be effectively not present.
Program Quality Evaluations
Mathematical Depth [1.2]
This program received the lowest rating of Mathematical Depth of the fifth-grade programs in this review. The strongest presentation it offers is in the case of multiplication and division of whole numbers. However, these suffer from several drawbacks. The instruction is not learned from a text and is thus highly dependent upon teacher direction. At the same time, the emphasis on having students to devise their own methods leaves open the possibility that the students will not achieve any regular and reliable approaches. This risk is aggravated by a limited number and range of practice items.
The treatment of the other topic areas in the review is even weaker or non-existent. In short, the prospects for student learning are such that this program cannot be recommended.
Quality of Presentation [2.0]
The quality of presentation for this program also received the lowest rating among the fifth-grade programs reviewed. The lack of a student text by the fifth grade contributes to this as it leaves students without such resources as a glossary or the opportunity to review prior instruction independently. The classroom strategy is of inefficient design that focuses on a more experiential exposure to mathematics concepts. The program provides extensive guidance to teachers but it focuses largely on process and is not sufficiently directive as regards the objects of instruction. This means that any success that might possibly be achieved will be highly teacher-dependent.
Quality of Student Work [1.2]
Although there is a fairly reasonable number of student worksheets, the actual work expected is severely limited in depth and scope and is unlikely to support mastery of content.
Overall Program Evaluation
This program received the lowest overall rating of the fifth-grade programs in this review. The level of achievement supported falls far short of the expectations in this review. This program cannot be recommended for use in fifth-grade classrooms.
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