Does the instruction in Mathematics: Applications and Connections provide an opportunity for students to learn the benchmark ideas and skills?

Numerous sightings were analyzed to determine the instructional criteria ratings for Mathematics: Applications and Connections. 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 Mathematics: Applications and Connections. It does so by showing the average score Mathematics: Applications and Connections 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 Mathematics: Applications and Connections 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 Mathematics: Applications and Connections.


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.

The text makes an attempt at setting a purpose through Chapter Openers, but this is not carried out through the lessons. The objectives within each chapter are not necessarily related, and there are few attempts to show how benchmark ideas connect to one another. Some of the connections are legitimate but are so subtle that students will need guidance to see them. A rationale can be inferred from the introduction to each chapter; however, there is no justification provided.


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.

In some lessons, there are prompts in the margins that attempt to address prerequisites but these focus only on procedures. Two features of the material, Error Analysis and Math Hints, could alert the teacher to commonly held ideas, but neither addresses the benchmark ideas usefully on a regular basis. Questions at the start of problem sets claim to be designed to assess students’ understanding of ideas as they are presented in the material, but the only strategies suggested for addressing student difficulties in understanding are to repeat and re-emphasize instruction. There are few strategies to build conceptual thinking and no suggestions for correcting procedural errors. The material asks a few questions about what students know but does not help teachers use this information.


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.

Mathematics: Applications and Connections provides experiences that target number benchmark ideas but does not provide variety beyond the usual paper and pencil exercises, except for some work with spreadsheets. Experiences provided through pictures are sometimes of poor quality. The material provides firsthand experiences that get at the meaning of the geometry ideas through manipulatives and simulations. Mini-labs allow the use of measurement instruments, games, fraction bars, and base ten blocks, but the focus for number ideas is on drill and practice exercises. Several hands-on experiences related to geometry concepts are presented in optional activities for re-teaching only. Across the three grade levels, the material generally provides experiences that give meaningful connections; however, there are few that are truly firsthand and many are repetitious.


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.

Overall, there are some suggestions for helping students develop a sense of the importance of the mathematical procedures but no apparent cases built for the validity of the ideas. Mathematics: Applications and Connections is careful to limit the terms and procedures being introduced, but there is rarely an engaging activity to accompany the terms or procedures. In the case of the number concepts benchmark, there are limited representations of a/b. While most representations are accurate and comprehensible, there are not an appropriate number or variety of representations. Connections to other benchmark ideas are rarely made and where they do occur, they are not noted as such in the material. Although the material does a credible job of demonstrating or modeling skills benchmarks, there is little modeling of concepts; overall, the material lacks commentary on procedures. There are pages of practice exercises, but they are primarily computational and provide little opportunity for practice on the meaning of benchmark ideas.


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.

The text encourages students to express their ideas related to benchmark concepts or skills, explain these ideas in their own words, or explain relationships among benchmark ideas, but there are no occasions where students are encouraged to justify or clarify their ideas. Teachers are advised to work through exercises with students; however, these exercises are limited mainly to drill on the concept or procedure that is presented, and students are not expected to think through procedures. Students are told what they should be able to do on a particular test after they finish the chapter, but no activities explicitly include a process of reflection.


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.

Assessments are of the traditional form and require application of a rule or skill. The material does contain at least one assessment item that addresses each benchmark idea at the appropriate sophistication level. Tests contain many items that are content-matched to the benchmarks. Applications items are primarily practice with skills, with little attention to conceptual benchmarks. Embedded assessments are designed to monitor student progress but lack specific suggestions about using the information to make instructional decisions.


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 little help given to teachers to improve their own mathematics knowledge. Students are rarely encouraged to ask questions and engage in thinking creatively. There are projects and activities at the beginning of chapters that may stir curiosity, and some discovery activities allow students to work in pairs and small groups to encourage questioning and avoid classroom rigidity. The material avoids stereotypes and recognizes the contributions of women and minorities with a variety of pictures. The text attempts to provide options for a variety of learning levels, even though it uses traditional worksheet activities for the most part.

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