Numerous sightings were analyzed to determine the instructional criteria ratings for Connected Mathematics. 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  (Adobe PDF document)

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

INSTRUCTION HIGHLIGHTS CHART   (Adobe PDF document)

Overall, analysts rated Connected Mathematics as satisfactory 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 Connected Mathematics.

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.

Connected Mathematics provides the purpose of each unit and communicates it to students through interest-catching questions that start the unit. The purpose is formally presented in a Mathematical Highlights section aimed at students and parents. At the end of each investigation, a Mathematical Reflections section summarizes and reviews concepts or skills and returns to the purpose. Each lesson begins with a relevant story line or situation that relates to the stated purpose. The material often makes the student aware of future lessons that may give them a better understanding of a concept or skill.

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.

Prerequisite knowledge or skills are mentioned in the teacher sections and at the beginning of most investigations. For example, before students make parallelograms from triangles they are reminded of an earlier unit that taught that triangles have rigid structures. Every investigation has a section called Teaching the Investigation that assists the teacher in identifying specific student ideas and offers suggestions on how to address these ideas.

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 use of a variety of contexts is an essential characteristic of Connected Mathematics. The contexts used range from visual models to a variety of symbolic representations of hands-on activities. The number experiences include using fraction strips, cooking, using thermometers and number lines, and exploring consumer issues. The algebra units employ data collection, use of graph paper, and calculators. Area models, grid paper, rulers, and square tiles are used in the geometry units. The firsthand experiences are efficiently used to build formal ideas and skills.

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.

Justification for the importance of the mathematics content in each investigation is provided through a real-world scenario or story line that builds students’ understanding. New terms are presented only as needed and within the context of activities. Connected Mathematics represents ideas through modeling, symbols, and concrete representations that are accurate and comprehensible to students. Appropriate connections are made between the ideas, and the teacher is instructed to ask questions that prompt students to make connections as well. The activities demonstrate to students how to do procedures, and the teaching guide provides suggestions for the teacher on how to explain the concepts or procedures. In addition to exercises presented within a lesson, practice exercises and problems at the end of each investigation require application, connection, and extension of the ideas developed.

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.

Connected Mathematics routinely asks students to explain their answers and to share their answers, ideas, and solutions with the class or group. The Explore section provides the teacher with suggestions on how to engage students in class discussion. Although the Mathematical Reflections section lacks a specific strategy for students to get feedback on their responses or to correct misunderstandings, it allows them the opportunity to summarize what they have learned in the investigation. While there is little specific direction for the students to revise their initial ideas as they progress through the unit, a Self-Assessment page for students is provided at the end of the unit to encourage them to think about what they’ve learned.

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 assessment items are aligned with the benchmark ideas and include a variety of tasks in check-up exercises, quizzes, question banks, unit tests and unit projects, and some alternative tasks for students with special needs. The items require application of the benchmark concepts and skills in a variety of situations both familiar and new. The Mathematical Reflections sections, along with the mid-unit quizzes and check-ups, can be used as embedded assessments. The material makes suggestions to the teacher on how to modify instruction and provides suggestions for special education students and ideas for enhancing instruction for certain concepts and skills.

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.

Connected Mathematics provides notes for the teacher on the mathematics content of each investigation along with considerations for teaching the topic. Teachers are prompted to have the students discuss their ideas with the class. Many investigations and end of unit projects provide opportunities for students to exhibit creativity and use their own strategies to solve problems. The material avoids stereotypes of individuals or groups of people but does not document specific contributions of women, minorities, or other groups. A variety of suggestions and tips are provided for linguistically and behaviorally diverse classrooms.