|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 (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 (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
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
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
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
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
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
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
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.