Does the instruction in Math 65, Math 76, and Math 87 provide an opportunity for students to learn the benchmark ideas and skills?

Numerous sightings were analyzed to determine the instructional criteria ratings for Math 65, Math 76, and Math 87. The following chart provides a typical example of the sightings that were analyzed to determine each criterion rating. Looking at these sightings will provide a picture of the overall instructional guidance provided in the textbook.

TYPICAL SIGHTING CHART  pdficon.gif (224 bytes)(Adobe PDF document)

The graph below depicts major strengths and weaknesses in the overall instructional guidance provided by Math 65, Math 76, and Math 87. It does so by showing the average score Math 65, Math 76, and Math 87 received on each of the 24 instructional criteria, across all six of the benchmarks used for the evaluation.

INSTRUCTION HIGHLIGHTS CHART  pdficon.gif (224 bytes)(Adobe PDF document)

Overall, analysts rated Math 65, Math 76, and Math 87 as unsatisfactory in helping students achieve the number, geometry, and algebra benchmarks used for the evaluation. The following describes the seven instructional categories and their criteria and summarizes the analysts’ justification for their ratings for Math 65, Math 76, and Math 87.

(Note: The teacher’s edition for these textbooks is identical to the student edition, with the addition of answers to exercises and general introductory statements at the beginning of each book. There is no instructional guidance for teachers included in the teacher’s edition.)


Instructional Category I

Identifying a Sense of Purpose
Part of planning a coherent curriculum involves deciding on its purposes and on what learning experiences will likely contribute to achieving those purposes. Three criteria are used to determine whether the material conveys a unit purpose and a lesson purpose and justifies the sequence of activities.

This material is organized by lessons and has no units or chapters. Most lessons open with a brief statement related to the direction of the lesson or what students will do in the lesson. There is rarely commentary for the students or the teacher on how to set up the lesson and almost no information given that would help students to see a relationship between this lesson and lessons in the past or future. Because the presentation of a skill or concept is very systematically sequenced within a lesson or part of a lesson, one can infer the rationale for the sequence of activities, although there is none stated. In some lessons, two unrelated ideas are presented back-to-back, with no rationale for their juxtaposition. Skills and concepts are sequenced in the sense that, somewhere in the text, students may encounter the next step or level of an idea.


Instructional Category II

Building on Student Ideas about Mathematics
Fostering better understanding in students requires taking time to attend to the ideas they already have, both ideas that are incorrect and ideas that can serve as a foundation for subsequent learning. Four criteria are used to determine whether the material specifies prerequisite knowledge, alerts teachers to student ideas, assists teachers in identifying student ideas, and addresses misconceptions.

While there is an occasional reference to prerequisite knowledge, the references are neither consistent nor explicit. In the instances where prerequisite knowledge is identified, the reference is made in the opening narrative that mentions skills that are taught in earlier lessons. The lessons often begin a new skill or procedure without reference to earlier work. There are warm-up activities at the beginning of lessons that provide practice for upcoming skills, but they are not identified as such. There is no guidance for teachers in identifying or addressing student difficulties.


Instructional Category III

Engaging Students in Mathematics
For students to appreciate the power of mathematics, they need to have a sense of the range and complexity of ideas and applications that mathematics can explain or model. Two criteria are used to determine whether the material provides a variety of contexts and an appropriate number of firsthand experiences.

The experiences provided are mainly pencil and paper activities. The material uses a few different contexts such as calculators and fraction pieces, and in grade 6 there are three lessons using fraction manipulatives. In grade 7, students put together a flexible model and later work with the paper and pencil measurements. A lesson in grade 7 on volume of rectangular solids uses a variety of drawings and equations that are right on target with the benchmark, but there is no suggestion that students actually use sugar cubes, as referenced, to build a figure. For the algebra graphs and algebra equations concepts, no variety of contexts is offered. Most firsthand experiences are found in the supplementary materials where students are given a few opportunities to do measurements, work with paper models of figures, collect data, and construct graphs.


Instructional Category IV

Developing Mathematical Ideas
Mathematics literacy requires that students see the link between concepts and skills, see mathematics itself as logical and useful, and become skillful at using mathematics. Six criteria are used to determine whether the material justifies the importance of benchmark ideas, introduces terms and procedures only as needed, represents ideas accurately, connects benchmark ideas, demonstrates/models procedures, and provides practice.

There is rarely a justification of the mathematics used in any exercises, activities, or procedures in this material. Terms and procedures are introduced at an easy pace and are comprehensible, but they are presented without the context of an activity and without appropriate application. Representations of number benchmarks are accurate and easy to understand, but they are limited and show no variety. Graphs are sometimes misleading or unclear. Charts are sometimes used to identify connections, such as a fraction-decimal-percent equivalents chart in grade 7, but there is no attempt to engage students in making connections. The material demonstrates procedures and includes a brief commentary. Modeling takes the form of illustrations at the beginning of the lesson. For students who can pick up an idea or procedure the first time they see it, the material is clear and understandable. The material provides one practice exercise for a given part of each skill and procedure introduced in almost every lesson. It then provides a few practice questions in subsequent lessons that ask students to re-do the procedure or answer the same question again. Practice involves little variety and few opportunities to apply knowledge in new situations or problems.


Instructional Category V

Promoting Student Thinking about Mathematics
No matter how clearly materials may present ideas, students (like all people) will devise their own meaning, which may or may not correspond to targeted learning goals. Students need to make their ideas and reasoning explicit and to hold them up to scrutiny and recast them as needed. Three criteria are used to determine whether the material encourages students to explain their reasoning, guides students in their interpretation and reasoning, and encourages them to think about what they’ve learned.

There are occasional opportunities for students to represent their understanding of an idea through a written description. Supplementary material entitled "Writing About Mathematics" is available "…to allow for students’ written expression across the mathematics curriculum," but the tasks do not ask students to express, clarify, or represent their ideas in any way. Instead, students are asked questions that are similar to the practice exercises and are told to write out the descriptions of how to solve the problems. There is no evidence in any of the materials of opportunities in which students are engaged in monitoring their progress toward any of the benchmarks analyzed. They are not asked to think about their ideas nor given the opportunity to revise initial ideas based on what they have learned.


Instructional Category VI

Assessing Student Progress in Mathematics
Assessments must address the range of skills, applications, and contexts that reflect what students are expected to learn. This is possible only if assessment takes place throughout instruction, not only at the end of a chapter or unit. Three criteria are used to determine whether the material aligns assessments with the benchmarks, assesses students through the application of benchmark ideas, and uses embedded assessments.

The items in Practice and Problem Sets sections, and in the tests, are matched to the topic of the benchmarks rather than the central ideas and skills. There are no non-routine items or problems that test students' mathematical understanding or application of the benchmarks. Practice at the end of each lesson is not included as an embedded assessment strategy to monitor students’ understanding or to provide suggestions about instructional decisions. The only reference to following up student performance is in the introduction in grade 8 where teachers are advised to use the supplemental practice problems in the back of the book for remediation of students scoring below 80% on the tests.


Instructional Category VII

Enhancing the Mathematics Learning Environment
Providing features that enhance the use and implementation of the textbook for all students is important. Three criteria are used to determine whether the material provides teacher content support, establishes a challenging classroom, and supports all students.

There is no math content help for teachers; however, teachers with limited understanding can likely work through the material. There are no opportunities for students to express curiosity or creativity and no places in which students are encouraged to ask questions. The only "challenge" referenced is a competitive race to solve short answer warm-up problems. The material easily avoids stereotypes and language that may be objectionable by making few references to real life situations and applications. The material does suggest remediation for addressing students with special needs, including those who score below 80% on tests. However, this remediation means doing more of the same kinds of problems they have been doing with each lesson. There are no alternative formats suggested for instruction and assessment, and no suggestions as to how to modify activities as needed.

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