The choice of the effective curriculum in mathematics is a challenging task that is based on the proper evaluation of proposed curricula’s features and components. From this point, the curriculum evaluation is an important stage before choosing the program for the implementation in the elementary school. Investigations in Number, Data, and Space is a K–5 mathematics curriculum designed by TERC that needs to be evaluated in the context of the project oriented to analyzing the curriculum’s components and to proposing the strategy for implementation of the evaluated curriculum in the elementary school (TERC, 2013). Therefore, the curriculum evaluation needs to be realized in several steps that guide the analysis of the curriculum’s components and include discussion of the evaluation method and criteria; the actual evaluation of the curriculum according to the criteria; the analysis of the curriculum’s alignment with standards; and discussion of the recommendations for the curriculum’s implementation in the elementary school’s environment. The reason for developing and designing the evaluation as well as for presenting the further recommendations is associated with the necessity to guarantee the effectiveness of the curriculum for adding to the students’ achievements, learning, and development of mathematical skills. This paper aims to provide the evaluation of Investigations in Number, Data, and Space in terms of the curriculum’s appropriateness for teaching students the basic mathematical skills according to the core standards.
Features and Components of Investigations in Number, Data, and Space
In order to understand how such a curriculum as Investigations in Number, Data, and Space should be evaluated, it is important to discuss its main features and components that allow distinguishing the curriculum among other programs in mathematics for the elementary school. Investigations is a non-traditional mathematics curriculum designed for the elementary school (K-5) that is oriented to assist students in their understanding of mathematics in terms of operations with numbers, early algebra, geometry, and measurement (TERC, 2013). The curriculum is organized into several units that can be studied during 2-5 weeks for each grade, and these units are divided according to the content area. The curriculum is aimed to develop students’ understanding of mathematics through creating strategies, solving problems, and thinking about mathematical concepts (TERC, 2013). While having the opportunity to participate in different activities and discussions, students use non-traditional materials, and there are no textbooks with set algorithms, mathematical principles, and examples.
The Method of Evaluation
To evaluate the curriculum effectively, it is necessary to choose the appropriate evaluation method. This evaluation method should be effective to present conclusions on the curriculum’s appropriateness to be implemented in the concrete elementary school. In this case, the main focus is on specific objectives and resources of an elementary school where the curriculum is planned to be implemented, on the correlation of the curriculum and the core standards followed in the state, and on the educational accountability associated with the elements of the discussed curriculum (Diamond, 2008, p. 129). To meet such purpose of the evaluation as the selection of the curriculum to be implemented in the concrete environment, it is important to focus on such an evaluation method as review that is associated with the use of the main aspects of the CIPP evaluation model (Glatthorn, Boschee, Whitehead, & Boschee, 2011, p. 361). Therefore, the review of features and components of Investigations is guided by the focus on context, input, process, and product related to the implementation of the curriculum in the concrete school’s environment (Glatthorn et al., 2011, p. 361). From this point, much attention should be paid to answering the questions about the objectives of the curriculum and their alignment with the school’s goals; about the procedures or activities to follow in order to complete the objectives; and about the effectiveness of proposed activities to achieve concrete results and educational outcomes.
The Criteria for the Curriculum Evaluation
The criteria for the evaluation of Investigations need to be formulated depending on the main aspects of the CIPP evaluation model in order to discuss the curriculum from the point of context, input, process, and product and depending on the standards adopted by the National Council of Teachers of Mathematics (NCTM) and on the Common Core State Standards (CCSS) (Principles and Standards for School Mathematics, 2015; The Common Core State Standards, 2015). Following the CIPP evaluation model, the CCSS, and NCTM standards, it is possible to present the following criteria:
- The alignment of the curriculum with the CCSS and NCTM standards.
- The alignment of the curriculum’s objectives with the goals of the elementary mathematics education.
- The balance between the performance-oriented and student-oriented approach.
- The appropriateness of the learning activities, assignments, and assessments for contributing to the students’ understanding of mathematics, development of mathematical skills, and improvement of knowledge.
- The appropriateness of the materials to address the curriculum’s objectives.
- The provision of strategies and opportunities for differentiation within the diverse classroom’s environment.
Evaluation of the Curriculum
In order to evaluate the Investigations according to the formulated criteria and associated standards appropriately, it is necessary to focus on the analysis of the curriculum’s aspects in detail and following separate criteria.
The Alignment of the Curriculum with the CCSS and NCTM Standards
The Common Core State Standards (CCSS)
The implementation of certain curricula in the school environment usually depends on the results of the curriculum’s evaluation according to the Common Core State Standards that are referred to as the basic educational standards in the United States (The Common Core State Standards, 2015). Focusing on Investigations, it is important to note that the curriculum and actual lesson plans designed for Investigations include references to certain CCSS because the curriculum is standard-based. However, the problem is in the fact that not all CCSS are initially covered in the content of the K-5 curriculum (Porter, McMaken, Hwant, & Yang, 2011, p. 104). Thus, the authors of Investigations needed to develop additional lesson plans and units in order to cover the CCSS content in the program and address the main comments of the experts on the appropriateness of the curriculum to be used in the K-5 setting (TERC, 2013). In this context, the main content of the curriculum covers not all CCSS, and there are units in Investigations that are not correlated with the basic learning standards directly. The authors of Investigations note that the curriculum is directed by the CCSS, but the content area covered in the program is wider and the main principles are adapted according to the NCTM standards (TERC, 2013). As a result, the quality of the mathematical content proposed in the curriculum can be discussed as questionable.
The National Council of Teachers of Mathematics (NCTM) Standards
The Investigations program is designed as a non-traditional curriculum that aims to address the NCTM standards proposed for the area of mathematics. The NCTM standards are formulated for the reformed mathematics education, and they include statements regarding the development of students’ skills in number and operations, algebra, geometry, measurement, data and analysis, probability, and problem solving (Principles and Standards for School Mathematics, 2015). Investigations addresses the standard on number and operations while accentuating the activities important to stimulate the students’ understanding of numbers. The standard related to algebra is addressed in the “Early algebra” program. The standards related to geometry and measurement are directly addressed in the curriculum. Data and analysis, probability, and problem solving skills are developed with references to stimulating the students’ progress in designing algorithms for solving equations and following non-traditional approaches to analyzing the data (Saxe et al., 2010, p. 434). In this case, the mathematical content of Investigations is non-traditional and responding to the recent trends in the sphere of the mathematical education stated by the National Council of Teachers of Mathematics.
The Alignment of the Curriculum’s Objectives with K-5 Objectives and Goals
The implementation of the curriculum in the concrete school’s environment depends on the alignment of the curriculum and school’s goals. The main objectives of the elementary mathematics education are to develop the students’ understanding of whole numbers; to use four operations with whole numbers; to solve problems and analyze patterns; to understand fractions; to understand principles of measurement; and to understand shapes, lines, and angles (Kling, 2011, p. 82). In Investigations, the main focus is on ensuring that students understand fundamental ideas of number and operations as well as algebra, geometry, data, and measurement. The major goals of the curriculum include the development of mathematical thinkers; development of computational fluency; development of skills in working with rational numbers and concepts of algebra, geometry, measurement, and data; and development of reasoning related to mathematical ideas (Appendix A; TERC, 2013). As a result, the focus is on developing mathematical thinkers who can operate numbers and concepts of algebra, geometry, measurement, and data creatively and on accentuating relationships between concepts and procedures. Although the basic goals of the elementary mathematics education are aligned with the goals declared by the authors of the Investigations curriculum, such approach is often not appropriate for using in districts where the curricula for Middle and High schools are directly based on the CCSS because of the lack of alignment with expected outcomes.
The Balance between the Performance-Oriented and Student-Oriented Approach
The Investigations curriculum can be discussed as a student-oriented program in contrast to performance-oriented programs. The main difference between the approaches is the accentuated focus on students’ needs, individual ways of thinking, and learning styles typical for the student-oriented programs and the accentuated focus on achieving the concrete outcomes as a result of completing certain tasks and developing or improving knowledge in certain areas typical for performance-oriented programs (Polly et al., 2013, p. 12-13). The content presented in Investigations should be used to create the environment of the active learning in the classroom when students are engaged in numerous games, life-related activities, and “Discussion” sessions. The teacher is expected to stimulate the creative and logical thinking of individual students to make them create their own approaches to solve mathematical problems during different types of mathematical activities.
Scope and Sequence of the Curriculum, Learning Activities, Assignments, and Assessments
Following Investigations, instructors are expected to provide 40-60 minutes of mathematics instruction for Kindergarten students and 70-75 minutes of mathematics instruction for students studying in Grades 1-5 daily. 10-15 minutes are planned to be spent for additional and individual work. Lessons are taught according to the units organized within the certain content area (TERC, 2013). As a result, the lessons are interchanged and taught in sequence. Much attention is paid to organizing a lesson during a day because certain types of activities are performed daily and during the certain amount of time. Usually, the lesson is divided into four parts where the first part is the mathematics activities, during which students are taught mathematical operations and involved in hands-on activities. The second part is “Discussions”, during which students discuss mathematical concepts and specific methods used in a lesson. The third part is the mathematics workshop, during which students work separately, in pairs, in groups, and participate in games. The final part of the lesson is the assessments, during which students’ success and knowledge are evaluated with the help of written and oral assessments (Jitendra et al., 2013, p. 22). Teachers are also expected to instruct students regarding their homework activities. Different activities for homework often require the participation of parents in preparing the tasks that are usually creative in their character.
Although the proposed activities, assignments, and assessments are good for teaching students regarding mathematical ideas in the context of mathematical realities because there is correlation between teaching concepts and proposing tangible activities, the content of the curriculum cannot be easily adopted in districts where the performance-oriented approach is followed in Middle and High schools. The proposed instructions and activities are effective to stimulate the students’ logical thinking and problem solving, but they are not appropriate to teach students how to work with simple algorithms which are important to be used in other grades (Diefes-Dux, Zawojewski, & Hjalmarson, 2010, p. 808; Doabler, Fien, Nelson-Walker, & Baker, 2012, p. 202). The other weak point is the homework proposed in lessons plans and oriented to tangible and creative activities. Although this approach is effective for Kindergarten, it is time-consuming and ineffective for Grades 1-5.
The Appropriateness of the Materials to Address the Curriculum’s Objectives
Investigations in Number, Data, and Space (K-5) is designed as a non-traditional curriculum. Therefore, there are no textbooks proposed for using by students. Instead, the curriculum is supported with “Student Math Handbook” containing the used and discussed math vocabulary and directions for different activities and “Student Activity Book” for performing activities in a written form. The curriculum is supported with a set of materials to be used by teachers to prepare instructions and to organize activities in the class. The additional sets of manipulatives, cards, and software are effective to contribute to students’ development of basic mathematical skills (TERC, 2013). The materials are appropriate for the students’ age. However, the absence of the traditional textbook can be discussed as a weakness of the curriculum because it limits possibilities to adopt the non-traditional program in a wide range of schools (Bieda, Ji, Drwencke, & Picard, 2013, p. 4). Furthermore, the handbook does not provide algorithms for working with numbers and mathematical problems, and it does not represent the meaningful mathematical content.
Strategies and Opportunities for Differentiation within the Class
Investigations provides many opportunities to differentiate the instruction in the classroom because the lesson plans include activities appropriate for the differentiated instruction, and additional resources are published to increase the number of activities for the diverse student population in the classroom (TERC, 2013). Multiple solutions to different mathematical problems proposed during the lesson sessions are effective to be used for diverse learners who can use the algorithms developed independently, according to their needs and learning styles (McGee, Wang, & Polly, 2013, p. 18; Polly, 2012, p. 79). In this case, the curriculum is effective to be used in highly diverse classrooms.
Recommendations for Using the Curriculum
The implementation of Investigations in Number, Data, and Space (K-5) in the elementary school is associated with a range of barriers because the curriculum is non-traditional in spite of being standard-based. The curriculum can be implemented in the certain school district because it is effective to respond to the issue of diversity and development of mathematical thinking in students (Polly et al., 2013, p. 14). In this context, it is important to provide recommendations for the effective implementation of the discussed curriculum in the district that uses traditional programs in Middle and High schools:
- Investigations can be implemented in the traditional classroom not in all grades of the elementary school, with the main focus on integration of the curriculum in Kindergarten and Grade 1.
- The program of Investigations for Kindergarten and Grade 1 can be implemented without changes in the content, activities, and assessments because of the orientation to tangible activities appropriate for the diverse class.
- The program of Investigations for Grades 2-3 should be altered and improved to fit the requirements associated with the CCSS. More attention should be paid to using traditional algorithms for solving mathematical equations supported with strategies independently developed by students during their activities.
- In Grades 4-5, the materials used for Investigations can be referred to as additional materials for games, for the work in pairs and in groups.
- In Grades 4-5, textbooks are needed to be used in addition to the handbooks provided according to the Investigations curriculum in order to make students become aware of principles of using standard computational methods in algebra.
- Formal assessments should be used in addition to proposed non-formal and non-structured assessments.
- Homework assignments should be revised and changed according to the needs of the class.
In spite of the fact that Investigations in Number, Data, and Space (K-5) developed by TERC is discussed as a non-traditional program proposed for the reformed curriculum, it can be adopted in different elementary schools when certain assignments and strategies are changed according to the needs of the specific group of students. The curriculum can be discussed as effective to stimulate the students’ interest in mathematics, and it is appropriate to be used in Kindergarten and Grade 1. However, in order to implement the complete curriculum in the elementary school, it is necessary to pay attention to reviewing the curriculum’s components and to add the materials appropriate for the age of students studying in Grades 2-5. Furthermore, the curriculum should be checked and revised in terms of alignment of its objectives and activities with the CCSS.
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Appendix A: Goals of the Investigations Program