Investigations In Number, Data and Space Developed at TERC in Cambridge, Massachusetts Manufactured by Pearson Education Website: http://investigations.terc.edu/index.cfm Copyright: 2008 Cost: Core Curriculum Package with Manipulatives Kit (grade 3) $890

Investigations is a K-5 math curriculum designed to help all children, including those with mild/moderate disabilities. It is aimed to help with the fundamental concepts of number and operations, geometry, data, measurement and early algebra.

Investigations was developed after far-reaching testing and research. Investigations considers the time students need to develop a strong foundation and the time needed to build skills on this foundation. The program is broken down into units. Each unit focuses on an area of in depth content. Units range from 2-5 ½ weeks allowing students to practice ideas in multiple activities and contexts. All Investigations units are designed with all students in mind- girls and boys, members of different cultures, of different ethinicities and different language groups. Investigations was also developed with the wide variety of student strengths, needs, interests and learing styles in mind.

Investigationswas developed with six goals in mind and for that reason the Investigations curriculum is designed to:

Support students to make sense of mathematics and learn that they can be mathematical thinkers

Focus on computation fluency with whole numbers as a major goal of the elementary grades

Provide substantive work in important areas of mathematics—rational numbers, geometry, measurement, data, and early algebra—and connections among them

Emphasize reasoning about mathematical ideas

Communicate mathematics content and pedagogy to teachers

Engage the range of learners in understanding mathematics.

Student working in her student workbook!

Analysis Investigations is a research based math program influenced by many national publications. These publications include Principles and Standards of School Mathematics, the National Research Council’s Adding It Up: Helping Children Learn Mathematics and the Conference Board of the Mathematical Sciences’ The Mathematical Education of Teachers. The program was developed after extensive time spent researching real students and their understanding of numbers and operations. Field-testing was far-reaching. It included documentation of hours in the classrooms, student observations, discussions with teachers and analysis of student work.

Math skills are taught in isolation. Every session in Investigationsinclude Math Focus Points, Today’s Plan, the day’s Classroom Routine or Ten-Minute Math write-up, one to three activities, transparencies and student activity book pages that accompanies the session. Math Focus Points highlight the goals of each session. Today’s Plan provides an overview of the session. It lists materials, amount of time and groupings for the lesson. The session concludes with Session Follow-Up which lists homework and references to pages in the Student Math Handbook.
There are different kinds of activities that make up the sessions. Whole-class discussions allow students to compare methods and share conclusions. During Math Workshop students work on related activities. Assessments include both written activities and observations.
Investigations has been developed developmentally by grade. The grade levels are organized by unites according to the scope and sequence list and pacing information. Benchmarks are assessed at the end of each unit.

Investigations Student Workbooks

Investigations does not provide structured teacher instruction. The basis of Investigations in inquiry. It is an activity based mathematics that encourages students to think creatively and learn from peers. Investigations is intended to help students develop their own problem solving strategies. During math workshop teachers have the opportunity to work with students individually or in small groups to assess and support any struggling learners.

Investigations in Math, Data, and Space offers opportunities for practice and learning to mastery through a variety of activities, such as Ten Minute Math, computer software programs like Shapes and Logopaths, homework, and math games. “Evidence supports our hypothesis that Logo programming environments, properly designed, beneficially affect students' metacognition” (Bonnie Nastasi). The computer software programs encourage students’ metacognitive processing, such as deciding on the nature of the problem and choosing a strategy for that problem. Instructed in topical units, Investigations program includes lessons that are designed to have students acquire skills to develop their “mathematical sense” through computational fluency, hands-on activities, and whole class discussion, thereby providing opportunities for students to practice key math skills.This program takes a multi-sensory approach to learning mathematics mainly on the kinesthetic level. Each complete unit comes with a materials kit that is correlated to the grade levels and units. The materials kit includes block, calendars, buttons, dice geoblocks, coin sets and literally dozens of other manipulatives.

Math Investigations addresses strategy instruction through modeling, demonstration, and feedback. Guiding and teaching strategies, such as mnemonic devices for learning number combinations and peer mediation is stressed in this program (Gersten, 1985; Lerner, 1997; Mastropieri et al., 1991). The TERC website advises teachers to “use projects and games to help the teacher guide learning, rather than relying solely on ‘telling’” (Baroody, 1999). The traditional sequence of direct teacher explanations, strategy instruction, relevant practice, and feedback and reinforcement is often effective, but the potential of students to learn through problem solving should not be ignored. Too often, direct instruction approaches squeeze out other possibilities. Use direct instruction only when students are unable to invent their own strategies. In all cases, help them make strategies explicit (Kame'enui & Carmine, 1998).

Connection to Math & Literacy
The Investigations in Number, Data, and Space relates to literacy and mathematics curricula by stressing students' development, investigation, and articulation of mathematical ideas. Listed sequentially by grade level, Investigations focuses on students' overall knowledge and understanding of mathematical concepts that are developmentally appropriate. Also, learning to communicate about mathematical ideas is a major objective of the program. Through class discussion and writing, students develop their mathematical mathematical reasoning and communication skills. TERC instructs teachers to facilitate discussion of key math concepts by:

Focusing class discussion to mathematics

Asking questions that engage student thinking

Connecting discussion to work that precedes and follows it

Math Investigations directly intersects reading, writing, and mathematics in discussion, communication, and exploration of mathematical concepts in the classroom, enabling students to be mathematically literate.

Assessment Investigations incorporates methods of computer-basedc assessments of solving arithmatic problems. A variety of item types provided distinct information that judged the level of understanding demonstrated by different responses. Such systems are able to:

Provide adaptive assessment and instruction

Analyze and interpret a student's performance beyond correctness

Make sophisticated analyses of multiple step problems

Incorporate techniques of dynamic assessment

Allow assessment of higher-order thinking

Circumventing the need for beginners to automatize skills

Maintain a dynamic model and representation of the student's knowledge and skill

Access a data base of mathematics relevant to the curriculum

Include applications outside of the school context

Provide progress maps depicting the student's learning over time

Suggest possible areas for additional study

Map out the "landscape" ahead of the knowledge to be gained

Include points of difficulty and make instructional suggestions

"Doing Mathematics"

To assess student's ability to fully "do mathematics," the state of the art still requires subjective assessments that include performance on extended tastes (Frederiksen & Collins, 1989). Computer traces of students work processes can help document such performance. However, the computer must be integrated into a larger assessment effort. Further, the computer can make additional contributions: they can be used to create simulations and micro worlds that are rich source of problems. Work with Logo and other environments has provided "windows to the mind" for many teachers and researchers (Chazan, 1991; Clements & Meredith, 1993; Weir, 1987). Weir, for example, showed that children's first choices of number and input to the Logo "forward" command revealed much about their number sense. Teachers similarly learn much about their students exploring mathematics concepts, including fraction, with Turtle Math (Clements & Meredith, 1994).

Third grade materials kit

Materials ·Teacher resource packages ·Geoblocks ·Pattern blocks ·Snap cubes ·Machine tape ·Plastic coins ·50 cm rulers ·Red centimeter cubes ·Square color tiles ·Colored inch cubes ·Power polygons ·Measuring tape ·Meter sticks ·Card packages ·Teacher resource binder ·Answer keys ·Calendars ·Student activity books ·Recording sheets for math games

Students using Investigations polygon tiles

Examples of items/tasks -Design and carry out an experiment to determine which paper bridge design will hold the most weight, and then graph and analyze the results. (5th grade)
-Graph the height of the Fastwalker (an animal from the mythical planet of Rhomaar) and write a rule describing its growht (5th grade).
-Solve number puzzles (e.g., "What number(s) is odd, greater than 50, a multiple of 15, and less than 100?) using their knowledge of number properties (odd, even, prime, square) and number relationships (factor and multiple).

Successful implementation of this program involves math coaches, leaders, teachers, and principals, to be strong and consistent in communication. Teachers must talk amongst themselves and also to the students’ families. This is to ensure that all are on the same page and working towards the same goal. Another area to address in terms of implementing this program is professional development. Teachers, leaders, coaches, administrators, and anyone else involved with the program should receive proper training of this program. Teachers can attend various workshops and institutes to further education on using the material in the classroom. This will allow teachers to feel confident as they use the material in their classrooms. Weaknesses Because the program does not specifically use algorithms for most math concepts, it is hard for parents to help their children with homework. Parents' participation in their children's learning of mathematics would require knowledge about Investigations program, which many parents do not have. There have been complaints that because of the way the teacher’s part is scripted, it is almost frowned upon for teachers to encourage regrouping, long division, or other skills used with certain math concepts.

Strengths Investigations is a good program because it incorporates very hands-on activities, including computer software. Students are engaged with class discussion as they are actively learning mathematical concepts. Furthermore, students are able to engage in math areas as they play games. Games are there for students to have fun while learning. Also, Investigations allows teachers to be able to assess the students while they play the game in the specific area. The program includes software that allows teachers to monitor progress of their students in mathematical skills. Finally, this program also takes into account diversity of learners, including students with a language difference (i.e., students in ESL classrooms who speak Spanish at home) as well as students with learning disabilities. Investigations comes with a Spanish teaching companion, very helpful for students who do not speak English at a level of proficiency. This program also provides curriculum adaptations, such as worksheets with fewer problems and number keys, to help students of a variety of learning abilities. This program could be used in an inclusive classroom as well as self contained and resource rooms. The teacher could divide the students by ability and have centers using the different content areas of math investigations. The teachers in the classroom would have to make sure to rotate to each of the groups to make sure the students were staying on track. Also, this program comes with a Spanish teaching companion so that teachers can provide the same information to the ELL/ESL students in the classroom.

Reflection
As first year special education teachers we would not use the Investigations mathematics program in our classroom. We feel that although this program has its strengths, a major weakness is the lack of direct teacher instruction. We feel that students with special needs would benefit from at least some direct instruction from teachers in the area of mathematics. Although students would benefit greatly from some aspects of the program, particularly the hands-on approach, we feel there should be more time devoted to teaching skills. Students with mild/moderate disabilities, and even those without, should have the opportunity to learn mathematics in a variety of ways. Group work and self-discovery should be a part of this learning but it should not be the only method of learning the students are exposed to. This program is also very expensive considering that in most cases it would have to be supplemented with further teaching and materials. A kindergarten and first grade self-contained classroom teacher who uses the Investigations program in her classroom does not feel strongly about it. She says that her students need more support than the program offers. She often has to supplement the program with further instruction. Also, using this program our first year of teaching would be difficult to implement on our own if it was not already being used because there is extensive training involved in using this program. This program also requires a high level of cooperation on the part of parents with things like homework. While this would normally be very feasible, many parents have expressed concerns about the methods by which Investigations employs. Instead of focusing on traditional means of teaching mathematics, which emphasize automaticity in computational skills, Investigations looks at the instruction of mathematical skills in an entirely new way that parents are unaccustomed to. This way, which emphasizes "mathematical experience" over concrete algorithms, is a controversial method, something that parents may not be comfortable with. Quotes from Professionals “It is a good program for student based learning, but we need to focus on basic number facts” (Mrs. Jones, the Best School in America).
“They do have good games, but kids need to practice other ways of practicing facts” (Ms. Jackson, the Second Best School in America). Video http://investigations.terc.edu/library/resources/schooldistweb.cfm http://www.pps.k12.or.us/curriculum/quicktime/counting_jar.html http://standards.nctm.org/document/eexamples/chap5/5.1/part2.htm

Investigations In Number, Data and SpaceDeveloped at TERC in Cambridge, MassachusettsManufactured by Pearson EducationWebsite:http://investigations.terc.edu/index.cfmCopyright: 2008Cost: Core Curriculum Package with Manipulatives Kit (grade 3) $890

Investigationsis a K-5 math curriculum designed to help all children, including those with mild/moderate disabilities. It is aimed to help with the fundamental concepts of number and operations, geometry, data, measurement and early algebra.Investigationswas developed after far-reaching testing and research.Investigationsconsiders the time students need to develop a strong foundation and the time needed to build skills on this foundation. The program is broken down into units. Each unit focuses on an area of in depth content. Units range from 2-5 ½ weeks allowing students to practice ideas in multiple activities and contexts. AllInvestigationsunits are designed with all students in mind- girls and boys, members of different cultures, of different ethinicities and different language groups.Investigationswas also developed with the wide variety of student strengths, needs, interests and learing styles in mind.Investigationswas developed with six goals in mind and for that reason theInvestigationscurriculum is designed to:Support students to make sense of mathematics and learn that they can be mathematical thinkersFocus on computation fluency with whole numbers as a major goal of the elementary gradesProvide substantive work in important areas of mathematics—rational numbers, geometry, measurement, data, and early algebra—and connections among themEmphasize reasoning about mathematical ideasCommunicate mathematics content and pedagogy to teachersEngage the range of learners in understanding mathematics.AnalysisInvestigationsis a research based math program influenced by many national publications. These publications include Principles and Standards of School Mathematics, the National Research Council’s Adding It Up: Helping Children Learn Mathematics and the Conference Board of the Mathematical Sciences’ The Mathematical Education of Teachers. The program was developed after extensive time spent researching real students and their understanding of numbers and operations. Field-testing was far-reaching. It included documentation of hours in the classrooms, student observations, discussions with teachers and analysis of student work.Math skills are taught in isolation. Every session in Investigations

include Math Focus Points, Today’s Plan, the day’s Classroom Routine or Ten-Minute Math write-up, one to three activities, transparencies and student activity book pages that accompanies the session. Math Focus Points highlight the goals of each session. Today’s Plan provides an overview of the session. It lists materials, amount of time and groupings for the lesson. The session concludes with Session Follow-Up which lists homework and references to pages in the Student Math Handbook.There are different kinds of activities that make up the sessions. Whole-class discussions allow students to compare methods and share conclusions. During Math Workshop students work on related activities. Assessments include both written activities and observations.

Investigations has been developed developmentally by grade. The grade levels are organized by unites according to the scope and sequence list and pacing information. Benchmarks are assessed at the end of each unit.

**Investigations in the classroom!**Investigationsdoes not provide structured teacher instruction. The basis ofInvestigationsin inquiry. It is an activity based mathematics that encourages students to think creatively and learn from peers.Investigationsis intended to help students develop their own problem solving strategies. During math workshop teachers have the opportunity to work with students individually or in small groups to assess and support any struggling learners.See

Investigationsat work!Investigations in Math, Data, and Spaceoffers opportunities for practice and learning to mastery through a variety of activities, such as Ten Minute Math, computer software programs likeShapesandLogopaths, homework, and math games. “Evidence supports our hypothesis that Logo programming environments, properly designed, beneficially affect students' metacognition” (Bonnie Nastasi). The computer software programs encourage students’ metacognitive processing, such as deciding on the nature of the problem and choosing a strategy for that problem. Instructed in topical units,Investigationsprogram includes lessons that are designed to have students acquire skills to develop their “mathematical sense” through computational fluency, hands-on activities, and whole class discussion, thereby providing opportunities for students to practice key math skills.This program takes a multi-sensory approach to learning mathematics mainly on the kinesthetic level. Each complete unit comes with a materials kit that is correlated to the grade levels and units. The materials kit includes block, calendars, buttons, dice geoblocks, coin sets and literally dozens of other manipulatives.Math

Investigationsaddresses strategy instruction through modeling, demonstration, and feedback. Guiding and teaching strategies, such as mnemonic devices for learning number combinations and peer mediation is stressed in this program (Gersten, 1985; Lerner, 1997; Mastropieri et al., 1991). The TERC website advises teachers to “use projects and games to help the teacher guide learning, rather than relying solely on ‘telling’” (Baroody, 1999). The traditional sequence of direct teacher explanations, strategy instruction, relevant practice, and feedback and reinforcement is often effective, but the potential of students to learn through problem solving should not be ignored. Too often, direct instruction approaches squeeze out other possibilities. Use direct instruction only when students are unable to invent their own strategies. In all cases, help them make strategies explicit (Kame'enui & Carmine, 1998).Connection to Math & LiteracyThe Investigations in Number, Data, and Space relates to literacy and mathematics curricula by stressing students' development, investigation, and articulation of mathematical ideas. Listed sequentially by grade level, Investigations focuses on students' overall knowledge and understanding of mathematical concepts that are developmentally appropriate. Also, learning to communicate about mathematical ideas is a major objective of the program. Through class discussion and writing, students develop their mathematical mathematical reasoning and communication skills. TERC instructs teachers to facilitate discussion of key math concepts by:

Focusing class discussion to mathematicsAsking questions that engage student thinkingConnecting discussion to work that precedes and follows itMath Investigations directly intersects reading, writing, and mathematics in discussion, communication, and exploration of mathematical concepts in the classroom, enabling students to be mathematically literate.AssessmentInvestigationsincorporates methods of computer-basedc assessments of solving arithmatic problems. A variety of item types provided distinct information that judged the level of understanding demonstrated by different responses.Such systems are able to:Provide adaptive assessment and instructionAnalyze and interpret a student's performance beyond correctnessMake sophisticated analyses of multiple step problemsIncorporate techniques of dynamic assessmentAllow assessment of higher-order thinkingCircumventing the need for beginners to automatize skillsMaintain a dynamic model and representation of the student's knowledge and skillAccess a data base of mathematics relevant to the curriculumInclude applications outside of the school contextProvide progress maps depicting the student's learning over timeSuggest possible areas for additional studyMap out the "landscape" ahead of the knowledge to be gainedInclude points of difficulty and make instructional suggestions## "Doing Mathematics"

## To assess student's ability to fully "do mathematics," the state of the art still requires subjective assessments that include performance on extended tastes (Frederiksen & Collins, 1989). Computer traces of students work processes can help document such performance. However, the computer must be integrated into a larger assessment effort. Further, the computer can make additional contributions: they can be used to create simulations and micro worlds that are rich source of problems. Work with Logo and other environments has provided "windows to the mind" for many teachers and researchers (Chazan, 1991; Clements & Meredith, 1993; Weir, 1987). Weir, for example, showed that children's first choices of number and input to the Logo "forward" command revealed much about their number sense. Teachers similarly learn much about their students exploring mathematics concepts, including fraction, with Turtle Math (Clements & Meredith, 1994).

Materials· Teacher resource packages

· Geoblocks

· Pattern blocks

· Snap cubes

· Machine tape

· Plastic coins

· 50 cm rulers

· Red centimeter cubes

· Square color tiles

· Colored inch cubes

·

Power polygons· Measuring tape· Meter sticks

· Card packages

· Teacher resource binder

· Answer keys

· Calendars

· Student activity books

·

Recording sheets for math gamesExamples of items/tasks-

Design and carry out an experiment to determine which paper bridge design will hold the most weight, and then graph and analyze the results. (5th grade)-Graph the height of the Fastwalker (an animal from the mythical planet of Rhomaar) and write a rule describing its growht (5th grade).

-Solve number puzzles (e.g., "What number(s) is odd, greater than 50, a multiple of 15, and less than 100?) using their knowledge of number properties (odd, even, prime, square) and number relationships (factor and multiple).

Successful implementation of this program involves math coaches, leaders, teachers, and principals, to be strong and consistent in communication. Teachers must talk amongst themselves and also to the students’ families. This is to ensure that all are on the same page and working towards the same goal. Another area to address in terms of implementing this program is professional development. Teachers, leaders, coaches, administrators, and anyone else involved with the program should receive proper training of this program. Teachers can attend various workshops and institutes to further education on using the material in the classroom. This will allow teachers to feel confident as they use the material in their classrooms.WeaknessesBecause the program does not specifically use algorithms for most math concepts, it is hard for parents to help their children with homework. Parents' participation in their children's learning of mathematics would require knowledge about Investigations program, which many parents do not have. There have been complaints that because of the way the teacher’s part is scripted, it is almost frowned upon for teachers to encourage regrouping, long division, or other skills used with certain math concepts.StrengthsInvestigations is a good program because it incorporates very hands-on activities, including computer software. Students are engaged with class discussion as they are actively learning mathematical concepts. Furthermore, students are able to engage in math areas as they play games. Games are there for students to have fun while learning. Also, Investigations allows teachers to be able to assess the students while they play the game in the specific area. The program includes software that allows teachers to monitor progress of their students in mathematical skills. Finally, this program also takes into account diversity of learners, including students with a language difference (i.e., students in ESL classrooms who speak Spanish at home) as well as students with learning disabilities. Investigations comes with a Spanish teaching companion, very helpful for students who do not speak English at a level of proficiency. This program also provides curriculum adaptations, such as worksheets with fewer problems and number keys, to help students of a variety of learning abilities.This program could be used in an inclusive classroom as well as self contained and resource rooms. The teacher could divide the students by ability and have centers using the different content areas of math investigations. The teachers in the classroom would have to make sure to rotate to each of the groups to make sure the students were staying on track. Also, this program comes with a Spanish teaching companion so that teachers can provide the same information to the ELL/ESL students in the classroom.ReflectionAs first year special education teachers we would not use the

Investigationsmathematics program in our classroom. We feel that although this program has its strengths, a major weakness is the lack of direct teacher instruction. We feel that students with special needs would benefit from at least some direct instruction from teachers in the area of mathematics. Although students would benefit greatly from some aspects of the program, particularly the hands-on approach, we feel there should be more time devoted to teaching skills. Students with mild/moderate disabilities, and even those without, should have the opportunity to learn mathematics in a variety of ways. Group work and self-discovery should be a part of this learning but it should not be the only method of learning the students are exposed to. This program is also very expensive considering that in most cases it would have to be supplemented with further teaching and materials. A kindergarten and first grade self-contained classroom teacher who uses theInvestigationsprogram in her classroom does not feel strongly about it. She says that her students need more support than the program offers. She often has to supplement the program with further instruction. Also, using this program our first year of teaching would be difficult to implement on our own if it was not already being used because there is extensive training involved in using this program. This program also requires a high level of cooperation on the part of parents with things like homework. While this would normally be very feasible, many parents have expressed concerns about the methods by which Investigations employs. Instead of focusing on traditional means of teaching mathematics, which emphasize automaticity in computational skills, Investigations looks at the instruction of mathematical skills in an entirely new way that parents are unaccustomed to. This way, which emphasizes "mathematical experience" over concrete algorithms, is a controversial method, something that parents may not be comfortable with.Quotes from Professionals“It is a good program for student based learning, but we need to focus on basic number facts” (Mrs. Jones, the Best School in America).“They do have good games, but kids need to practice other ways of practicing facts” (Ms. Jackson, the Second Best School in America).

Videohttp://investigations.terc.edu/library/resources/schooldistweb.cfm

http://www.pps.k12.or.us/curriculum/quicktime/counting_jar.html

http://standards.nctm.org/document/eexamples/chap5/5.1/part2.htm

Activities**http://standards.nctm.org/document/eexamples/chap5/5.1/index.htm**

Playing Fraction Track **http://standards.nctm.org/document/eexamples/chap5/5.1/index.htm**

Other WebsitesBridges to Classroom-Mathematics

http://www.comap.com/product/?idx=966

Sample Lesson Plan

http://school.discoveryeducation.com/lessonplans/programs/mathinvestigations1/

http://investigations.scottforesman.com/

http://www.youtube.com/watch?v=nclMgD1o7d4

Referenceshttp://investigations.scottforesman.com/

http://investigations.terc.edu/index.cfm

Group MembersSara Fallahi

Kerri Nicoletti

Krissy Philipino