Ainsa, T. (1999).  Success of using technology and manipulatives to introduce numerical  problem solving skills in monolingual/bilingual early childhood classrooms.  Journal of Computers in Mathematics and Science Teaching, 18(4), 361-369.        

This study tested the effectiveness of using manipulatives and technology to teach numerical problem solving skills, including counting, identifying shapes, matching colors and numbers, addition, and subtraction.  Children (ages 4-6) in five early childhood classrooms were studied using M&M candies and Skittles (for those allergic to chocolate) as manipulatives.  Of the 101 students, 41 were monolingual and 60 were bilingual.   The candy was used as a hands-on tool to supplement the M&M Counting Book (McGrath, 1995), which was used as part of the mathematics curriculum.  The software used in this study included:  KidsMath (Great Wave Software), Stickybear’s Math Town (Optimum Resources, Inc.), and Stickybear Shapes (Optimum Resources, Inc.). 

Students were given M&M candies (or Skittles) and a counting sheet.  The teacher read the M&M Counting Book and made observations as students performed the activities presented in the book.  Students counted aloud, displayed answers on the sheets, and made shapes using the candy.  After completing the activities, students used the computers to practice the concepts covered in the book.  Teachers documented data while observing students.  In this study, pre- and posttests were not administered.  Success was measured through frequency of correct student responses, observations, and teacher report.  The frequency of successful responses indicates that the combination of manipulatives and technology leads to higher student achievement, regardless of linguistic background.

Baker, J. D., & Beisel, R. W. (2001). An experiment in three approaches to teaching average to elementary school children. School Science and Mathematics, 101(1), 23-31.

Summary: The researchers studied 22 students in grades 4-6, in three multiage groups. The students were attending a mandatory summer session at the Indiana University of Pennsylvania lab school. Three instruction styles were used to teach the concept of average. The three styles were traditional lecture, concrete/hands-on activities, and visual/spreadsheet activities. Using various methods of comparing pre- and post-test data, and interviews, the researchers concluded that there are some advantages to the visual/spreadsheet approach.

Official Abstract: The types of experiences children should encounter to best understand average were investigated in this study. Using a traditional approach with problem solving, a concrete approach with manipulatives, or a visual approach with computer spreadsheets, similar lessons on the arithmetic mean were taught to 22 children in grades 4-6, in three multiage groups. Differences among pretest, posttest, and interview performances suggest some advantage in the use of a visual instructional style. Continued gains in performance were found after 4 months without further instruction. An algorithmic-like definition of average corresponded to better long-term performance than less precise definitions. Collaborative deliberations resulted in positive implications for the researchers' teaching.

Balka, D. S. (1983). Mathematics manipulatives in a pre-vocational program: Teacher inservice and classroom research. (ERIC Document No. ED237739). Washington, DC: U.S. Department of Education.

This article presents the results of action research conducted by four high school teachers who volunteered for this study.

Participants: The teachers collecting data chose a total of 42 students (Sowell, 2) to participate in the study. It was noted that the entire class participated in the activities. Additionally, all of the students were high school level and classified as “mildly mentally handicapped” (Sowell, 10).

Manipulatives: “Try-A-Tile cards”, “Math Match cards”, “Tangle Tables”, “Pathway Activities” (Sowell, 4) tangrams, and polyhedra dice were used in this study.

Math concepts taught:  Manipulatives were used daily (10-15 min.) to reinforce concepts of place value and basic computational skills.

Methodology: Quantitative methods were used as a pre-test, post-test design with the data from each teacher analyzed separately.

Results: Increased computational achievement in whole numbers was reported in three of the four data sets. The analysis on the fourth data set showed not gain or loss in achievement. Overall, the data produced significant results (p<.05) in division and total gains on three of the data sets and on multiplication in one data set. The researcher suggested that although some results are considered significant, they “do not lend support to the notion that the use of mathematics manipulatives with slow learners can improve computational skills.” (Sowell, 8).

Ball, S. (1988). Computers, concrete materials and teaching fractions. School Science and Mathematics, 88, 470-475.

Ball (1988) found that fourth-grade students using both virtual and physical manipulatives scored significantly higher on conceptual understanding of fractions than students using no manipulatives.

Barclay, Jennifer.  (1992).  A study of a manipulative approach to teaching linear equations to sixth grade students.  Unpublished master’s thesis, Texas Woman’s University, Denton, TX. 

The seven-week study conducted by sixth grade teacher, Jennifer Barclay investigated the effectiveness of using the algebra manipulative, Hands-on-Equations to teach the concept of solving linear equations.  Ninety-two of the students were randomly divided amongst 4 different heterogeneously grouped sixth grade classes.  The remaining 31 students were enrolled in a gifted and talented class.  The ages of the students ranged from 11 to 13 years.

All groups received a pretest; approximately one week of instruction with the manipulative; a posttest; three-week retention test; and a six-week retention test.  The results from the test assessed the level of concept mastery and retention.  The range of scores for the tests follow: pretest 23.66% to 35.5%, posttest 93.6 to 96.8%, 3week retention 87.8 – 96.8%, and 6 week retention 92.1 – 98.3%.

The results of the study reveal that students appear to have successfully mastered the concepts of solving linear equations taught in level one of this program.  Recommendations for future studies suggest using three different methods of teaching solving linear equations:  pictorially, with manipulatives, and the traditional symbolic approach. 

Battista, M.T.  (1999).  Fifth graders’ enumeration of cubes in 3D arrays: conceptual progress in an inquiry-based classroom.  Journal of Research in Mathematics Education, 40 (4), 417-448.

This is a case study that involves observing and interviewing three pairs of fifth graders over a 4-week teacher-directed instructional unit where students are involved in constructing an understanding of volume of rectangular solids.  The purpose of this study was to examine the cognitive connections that developed and how those connections change in an inquiry-based problem-centered based classroom.  Participants in the study used cubes as a physical manipulative in an instructional task designed allow students to develop and refine their mental models about finding the volume of a rectangular solid through enumeration of the cubes required to fill the box.  This study was shaped by the premise that change occurs as an accommodation to a perturbation and that perturbations arise through interactions with the physical world and in communications with other people.

The task or problem is, given a picture of a net for a box on grid paper, the picture of a box built from grid paper, or the verbal description of a box predict how many cubes it would take to fill the box.   Participants worked collaboratively in pairs to build the box from the given information and check their prediction about how many cubes it would take to fill the box with the actual cubes.   The discrepancy between the participants’ prediction and the actual number of cubes required to fill the box provided an opportunity for the pairs to reflect on their thinking or enumeration schemes and to make adjustments in their method.  This process was carried out over six iterations.  An interviewer sat with each case-study pair to monitor the evolution of their thinking through observation the pairs and by asking them clarifying questions.  Pre and post-treatment interviews of the case-study pairs as well as transcriptions of the video-tapings of each observation session were also used.  The findings from this study are generalizable based on the comparisons of the findings in the three case-study pairs to the similar findings based on the observations and notes made by the teacher and another researcher on other members of class. 

Immediately after the treatment the only one student was unable to use a mental model to enumerate the number of blocks it would take to fill a box.  On a post-interview four months after the treatment all of the participants except the one who did not attain a good mental model during the treatment were still able to use their mental model to determine the number of cubes required to fill a box.

From the implementation of the study and the results of the treatment on the students several conclusions were drawn.  The first is that in a constructivist classroom students’ construct, refine and revise their conjectures to accommodate conflicts that arise from discrepancies carrying out a task or through communication.  However, students’ theory building is incremental and there is a need for multiple and varied opportunities for students to build an understanding of difficult concepts and the process needs to be mediated by teachers and by instructional material.  The researcher expressed his belief that the treatment would not have been so effective if students had not have been able to self-check their predictions the manipulative. I believe the most powerful statement from the study and its’ implications for preservice and inservice teacher education is that, “Only by thoroughly understanding the pedagogical approach and the usual paths students take in learning particular mathematical ideas---including stumbling blocks and learning plateaus---can teachers know when to intervene.”  

Battista, M.T., Clements, D.H., Arnoff, J.Battista, K., & Van Auken Borrow, C.  (1998).  Students’ spatial structuring of 2D arrays of squares.  Journal for Research in Mathematics Education, 29 (5), 503-532.

 This is a case study carried out over one year that involves interviewing twelve second graders, a third grader, and a fourth grader.  The purpose of this study was to examine in detail how students structured and enumerated 2D rectangular arrays of squares in order to gain a better understanding of the mental processes being used. In this study spatial structuring is defined as the “mental operation of constructing an organization or form for an object or set of objects.” In a prior study, Battista and Clements found that before students could count or enumerate the number of blocks accurately in a 3D rectangular solid the student needed a mental model or spatial structuring in order to organize the information for counting.  In the earlier study the most effective spatial structuring model was found to be when a student held a mental model which allowed them to see the rows and columns as well as the layers that made up the solid.  In the results of the earlier study it was noted that students often had difficulty in their mental model of a single layer.   Spatial structuring with a 2D rectangular array is important in the development of the concepts of area and multiplicative thinking and led this research team to further study. 

Pre-study interviews were carried out with students to establish the protocol for interviewing students and to establish distinctions in the way they went about the task.Using this information the research team developed descriptions for the three levels of sophistication in student’s structuring of 2D arrays as well as descriptions of student’s thinking during the activity.  All students were interviewed at the beginning of the year and the end of the year.  Three times during the year 4 of the original group of twelve were interviewed again.  Particular attention was given to students as they transition from one level to another.  Students received no instruction on counting squares in a rectangular array during the year that while study was being conducted.  The interviews were videotaped, transcribed, and analyzed.  The task used in the interview was made up of seventeen different rectangular shapes.  Each shape was scored with squares in a different way and two of the seventeen were blank on the inside.  Students were asked to predict how many squares it would take to cover the rectangle and then they were asked to draw in the squares then predict again how many squares it would take.  Participants were given tiles with which to cover the rectangles.

One conclusion drawn from the study was that students progressed to a more sophisticated level of when they experienced perturbations or difficulties in predicting the correct number of squares required to cover the square.  This study informs curriculum and instructional designers that the mental model for the row-by-column matrix arrangement is not in the array itself, students must construct a mental model for this arrangement.  Therefore, before an array can be used as a tool to develop concepts such as area and multiplication we must ensure that student’s have attained an understanding of row-by-column structure of the array.

Belcastro, F. (1993).  Teaching addition and subtraction of whole numbers to blind students: a comparison of two methods.  Focus on Learning Problems in Mathematics, 15 (1), 14-22.

Belcastro conducted an experimental study of five blind first-grade students.  The students were split into a group of three and a group of two.  The larger group used Belcastro rods to study addition and subtraction of whole numbers, while the smaller group used buttons and other traditional materials.  The Belcastro rods are similar to Cuisenaire rods in size and shape, but replace the colors of the Cuisenaire rods with specific textural clues such as longitudinal grooves, horizontal grooves, and holes.  Rods of base length 2, 4, and 8 have longitudinal grooves.  Horizontal grooves are found on rods of base length 3, 6, and 9.  Rods that represent 5 and 10 have holes drilled through them.  Rods for 1 and 7 have no markings but are distinguishable because of the large difference in their lengths.  The Belcastro rods were employed in a way similar to the way Cuisenaire rods are employed to teach whole number addition and subtraction.

In the fall of 1990, the five students were given a verbal pretest.  The next day, intervention began in the form of instruction utilizing either the Belcastro rods or traditional materials.  Cuisenaire rods were not used as they had been previously found to fail with blind students.  Once the instruction was completed, the children were administered a posttest.  All instruction and testing were concluded by January 1991. 

All students missed all the questions on the pretest so only the posttest was considered.  Students using the Belcastro rods did better.  Their mean score on the 10-question test was 9.67 while the traditional group's mean was 7.5. 

The author concedes that the sample size was too small to make any generalizations to all blind students.  He suggests, however, that they are sufficiently promising to warrant additional testing of the rods with more blind students as well as sighted students.   In addition to the small sample size, I wondered about the author's bias, given that he was the inventor of the manipulative that he was testing.

Berlin, D., & White, A. (1986). Computer simulations and the transition from concrete manipulation of objects to abstract thinking in elementary school mathematics. School Science and Mathematics, 86, 468-479.

Berlin and White (1986) found no statistically significant differences between second- and third- grade students using physical manipulatives, virtual manipulatives, and both treatments on measures of spatial sense and patterning.

Bishop, Joyce Wolfer. (1997, March). Understanding of Mathematical Patterns and Their Symbolic Representations. Paper presented at the Annual Meeting of the American Educational Research Association, Chicago, IL. (ERIC Document Reproduction Service No. ED 410 107)

This study explores seventh- and eighth-grade students’ thinking about mathematical patterns. Interviews were conducted in which students solved problems about sequential perimeter and area problems modeled with pattern blocks and tiles, generalized the relationships related to the patterns and represented the relationships symbolically, identified other valid symbolic expressions of the pattern, and encountered equation-evoking situations. Research questions pertained to the strategies middle school students use to reason when solving pattern problems, symbolic representations the students develop, the students’ interpretations of equation-evoking situations. The results of this study support the use of mathematical patterns to promote algebraic reasoning and provide descriptions of middle school students’ reasoning as they engage in solving a specific type of pattern problem. Findings also suggest that experience exploring the relationships in sequential perimeter and area patterns may help students develop an appreciation for the meaning of expression. Contains 16 references.

Video and audio taping, and examples of student work were collected and coded for strategies, accuracy of outcomes, and implications for student understanding.  Students used 5 distinct strategies that were not modeled for them; 3 main strategies for identifying alternative symbolic expressions; and 8 different strategies for equation-evoking situations.

Three clusters of students were identified: Verbal and Single-Operational Expressions and Equations, Transition from Verbal and Single-Operational to Symbolic Expressions, and Equations, and Symbolic Expressions and Equations.

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Chassapis, Dimitris. (1998-1999). The Mediation of Tools in the Development of Formal Mathematical Concepts: The Compass and the Circle as an Example. Educational Studies in Mathematics. 37(3), 275-93.

This study focuses on the process by which children develop a formal mathematical concept of the circle by using various instruments to draw circles within the context of a goal-directed drawing task. Particular attention was given to the transition from using tracers and templates to using a compass for drawing circles and to the extent to which the use of different drawing instruments may contribute to the formation of a formally defined mathematical concept of the circle. The critical difference considered in the study is that the compass, in contrast to circle-drawing tracers or templates, induces by its ph7sical structure and its functional use the generative features of formal mathematical concepts of the circle, that is, the centre and the radius. Analysis of the empirical data indicates that the use of the compass in circle drawing structures the circle-drawing operation in a radically different fashion than circle tracers and templates, and brings into play an action-bound practical thinking. Such thinking has an overall positive influence on the construction of analytical concepts by children that are analogous to the formally defined mathematical concepts of the circle.

The use of circle tracers and templates, providing regulation and control of the human-hand movement in doing the same practical actions as those a freehand circle drawing, (thus not qualitatively transforming the circle-drawing operation), seems to influence although not to radically change the children’s spontaneous concepts of the circle. On the other hand, the use of the compass, which structures the circle-drawing operation in a radically different fashion than circle tracers and templates, creates the preconditions which may give rise to concepts constructed in the realm of action-bound practical thinking, because it is a functional meaning of circle-drawing that emerges when using a compass.

Chattin-McNichols, J.  (1992).  Montessori Programs in Public School.  (Report No.  EDO-PS-92-7).  Champaign, IL:  University of Illinois at Urbana-Champaign Children’s Research Center.  (ERIC Document Reproduction Service No. 348165).

Montessori was one of the pioneers of manipulative use.  The Montessori program relies on student participation in different activities.  Teacher presentation is minimal.  Students “create” their own learning.  Students work individually or in small groups for three to four hours each day.  Students cooperate rather than compete with each other.  Montessori programs show increased achievement test data.  Regrettably, most children do not have access to Montessori education due to lack of money, class spaces, teacher training, and program availability.  Traditional programs that use manipulatives can take the general Montessori philosophy and cater it to meet the needs of their curricula and classroom restraints.  Manipulatives, classroom interactions, and student-centered learning combine to create a beneficial learning environment for children both socially and academically.

Chester, J., Davis, J., & Reglin, G. (1991). Math manipulatives use and math achievement of third-grade students. Charlotte, North Carolina: University of North Carolina at Charlotte.

Summary: The researchers studied two third-grade classes in western Iredell County, North Carolina. Each class contained 26 students. Manipulatives (types were not specified) were used with the experimental group to teach a geometry unit. The same unit was taught to a control group using only the text and traditional lecture-style instruction. The study was conducted over a period of two weeks. Using analysis of covariance, the researchers concluded that the experimental group scored significantly higher on the posttest than the control group.

Official Abstract: Recent reports indicate that although 17-year-old high school students know some basic addition and subtraction facts, few of the students are capable of solving multi-step mathematics problems. A non-equivalent pretest-posttest control group design study examined the effects of a teaching method emphasizing manipulative use on the mathematics achievement of third-grade students. Two third-grade classes with 26 students each were selected to participate in the study. Reported demographic data indicated that the control group class from western Iredell County was composed of 10 (38%) white male students, 3 (12%) black female students, and 13 (50%) white male students, and that the experimental group class from southern Iredell County was composed of 10 (38%) white male students and 16 (62%) white female students. A 2-week geometry unit from the Silver Burdett textbook was administered in both classes. The experimental group teacher used mathematics manipulatives to teach the concepts presented in the unit, and the control group teacher used only drawings and diagrams to teach concepts. Analysis of covariance revealed that the experimental group using mathematics manipulatives scored significantly higher in mathematics achievement on the posttest scores than the control group. Further study is recommended to see if this finding is generalizable beyond the two classes studied or the subject of geometry. The pretest and the posttest are attached.

Cobb, P. (1995).  Cultural tools and mathematical learning: a case study.  Journal for Research in Mathematics Education, 26 (4), 362-385.

The investigator did a case study of four pairs of second graders who were beginning to learn about place value, specifically tens and ones.  The researcher was interested in the transition that children make from counting by ones to counting by tens and ones.  The manipulative used was the hundreds board, and multi-link cubes in bars of ten were also available to the children.  Videotapes were made of all math lessons in the entire class for one whole year.  They were also made of the eight children in the study over a ten-week period during that year.  In the classroom, typically there was small group problem-solving followed by whole class discussion.  One camera was used for the latter, while two were focused on the small group work.

It was found that children's use of the hundreds board did not support their transition from counting by ones to counting by tens and ones.  However, the hundreds board did appear to support their ability to reflect on their mathematical activity once they had acquired the concept.  The investigator observed that there seemed to be an all-or-nothing quality to this ability, as if the children made a sort of quantum leap to it.

The author suggests that the hundreds board does not facilitate the acquisition of the concept of counting by tens and ones, because of its specific prestructure.  He suggests that an empty number line might be better, especially if children are encouraged to discuss their solutions to well-selected tasks that facilitate rich imagery. 

The author poses an interesting analogy between architecture and math.  An architect of a building organizes our experience, physical and otherwise, within the building.  The architecture of our math notational system, including the manipulatives that we choose to use to convey it, organize our math experience, both constraining it and supporting it in ways that we are often unaware of. 

The author also used the investigation to look for evidence to support learning theories.  Specifically, he contrasted the constructivist point of view emblemized in Piaget with sociocultural theory emblemized in Vygotsky.  While constructivists emphasize individual diversity, the sociocultural theorists emphasize homogeneity within the cultural group. Piaget focuses on conceptual reorganization while Vygotsky would emphasize the need to enculturate children into established math practices.  The sociocultural theorists would propose that the tools we pick drive the concepts that we teach, while the constructivists would say that concept construction precedes symbols.  The author, though a constructivist, found evidence in the study that the two theories are complementary.  My own interpretation of his finding could be stated thus: Act locally to link into the global mathematics community.

Conroy, L. M., Tracy, D. M., & Eckart, J. A. (1994). The differential effects of Miras and mirrors on eighth-grade females' and males' ability to learn principles of plane mirrors. School Science and Mathematics, 94(8), 395-400.

Summary: The researchers studied 101 eighth-grade physical science students at a Midwestern, suburban, upper-middle income, junior high school. There were five classes total. To teach the five principles of plane mirrors, two classrooms used Miras only, two classrooms used mirrors only, and two classrooms used Miras and mirrors. The same unit was taught to all five classrooms. The study was conducted over a period of 18 weeks. Using analysis of variance methods, the researchers concluded that both male and female students benefited from instruction using both Miras and mirrors. Other findings were that male and female ability to learn these concepts may be differentially effected by the manipulatives used.

Official Abstract: Study of (n=101) eighth-grade physical science students learning principles of plan mirrors using mirrors and Miras found that males scored significantly higher than females on a chapter test, but that all students benefited when both Miras and mirrors were used throughout the learning process.

Cotter, J. A. (2000). Using language and visualization to teach place value. Teaching Children Mathematics, 7(2), 108-114.

Summary: The researcher studied 32 first-grade students at a rural Minnesota elementary school during the 1994-95 school year. There were two classes of 16 students each. To teach place value, the experimental classroom used the "Asian" method, using language patterns and visualization with abacuses and base-10 blocks, while the control classroom used a traditional approach. Using interviews with the two teachers and the students, the researcher concluded that the students taught in the "Asian" method exhibited a better understanding of place value.

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Dairy, L. (1969).  Does the use of cuisenaire rods in kindergarten, first and second grades upgrade arithmetic achievement (Report No. PS002132)?  Colorado Springs, CO:  Department of Research and Special Studies.  (ERIC Document Reproduction Service No. ED032128)

A three-year study was conducted on students in kindergarten, first grade, and second grade to determine the usefulness of incorporating Cuisenaire rods into the mathematics program.  The study comprised control groups from Columbia School and experimental groups from Whittier, both schools comparable in demographics.  Number of participants varied due to enrollment changes over the three-year period.  In the experimental groups, kindergartners received individual instruction on the use of the rods, first grade students completed teacher-created worksheets based on the use of rods in conjunction with the Laidlaw workbooks, and second grade students performed tasks directly from the workbooks using Cuisenaire rods.  Use of Cuisenaire rods was ongoing throughout the three-year period, except during the geometry and measurement unit when geoboards were utilized and when money was used to introduce money concepts.

At the end of each year, Test 5 (Numbers) of the Metropolitan Readiness Test was administered to both kindergarten groups.  First graders took the same test in the fall of the last two years of the study. Both first and second grade students completed the Metropolitan Upper Primary Test (Arithmetic) each spring.  Consistently, all three experimental groups (K-2 at Whittier) performed at a higher level than the control groups (Columbia).  Using end-of-year norms for the final year’s testing, the scores were all above the 80%ile, indicating that the utilization of Cuisenaire rods does enhance mathematical achievement of primary students.

Davis, B. and Shade, D.  (1994).  Integrate, Don’t Isolate!—Computers in the Early Childhood Curriculum. (Report No. EDO-PS-97-14).  Champaign, IL:  Children’s Research Center.  (ERIC Document Reproduction Service No. ED 376991).

This study looks at the effectiveness of integrating curricula in computer labs and with computers in the classroom.

Advances in technology make integrated computer use in the classroom possible.  Unfortunately, computers are mostly used as an isolated tool for specific skill repetition and pre-made “quests”.  Children need to take charge of their learning while using computers as a resource tool, organizational instrument, and presentation device.  When computers are used as a drill and practice tool, they negate the positive attributes computers can provide. 

If computers are kept in a computer lab, it further isolates the impact of technology on education.  Children may only go to the lab for one hour once a week.  Computers become a separate, disjointed part of education.

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Ernest, Patricia S.  (1994, Nov).  Evaluation of the effectiveness and implementation of a Math manipulatives project.  Paper presented at the Annual Meeting of the Mid-south Educational Research Association, Nashville, TN.  Available:  Eric Document 391 675

The study consisted of 40 high school teachers from 26 schools.  The teaching experience ranged from 1-42 years with a mean of 17.5 years.  The teachers taught the following courses:  Math 7, Math 8, General Math, Pre-Algebra, Consumer Math, Technical Math, Algebra I, Algebra II, Geometry, Trigonometry, and Pre-Calculus.  The teachers attended a weeklong intensive training workshop in the use of manipulatives, implemented the teaching strategies discussed during the workshop in their classroom instruction during the following year, then attended a follow up session to discuss strategies and problems identified during the implementation phase of the study.

The manipulative utilized for this study was the Mathematics Manipulatives Kit consisting of dice, polyhedra dice, spinners, Pattern Blocks, circular counters, color cubes, attribute blocks, geoboards, fraction bars, Algebra Tiles, protractors, compasses, geometric models, graphing calculators.

Data was gathered to evaluate the weeklong teacher training workshop and the implementation of manipulatives in classroom instruction.  On-site observations were conducted to record utilization by course and manipulative, student participation, student attitudes toward the manipulatives, and interaction with the content.  Evaluation of the workshop revealed that the teachers found the quality of instruction to be excellent to very good.  Evaluation of the Math Manipulative Observation lessons revealed that students enjoyed using the manipulatives and that “on task” involvement was very high.  Students exhibited confidence, eagerness, and a desire for other experiences. They often employed discovery and problem solving strategies beyond the assignment.  Teachers reported that students enjoyed and were more interested in assignments when manipulatives were used.  Teachers also reported that more planning time and class time was needed for lessons involving manipulatives.

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Flanagan, Robin. (1996, April). Unintended Results of Using Instructional Media: A Study of Second- and Third-Graders. Paper presented at the Annual Meeting of the American Education Research Association, New York, NY. (ERIC Document Reproduction Service No. ED 394 514)

Much of the research on classroom use of educational media has been hampered by difficulties in isolation a single element of the medium—television programming, for instance—that influences behavior in a reliable way. Still, each medium facilitates a particular type of learning environment, and the collective characteristics of those environments must be examined for possible effects. The learner in the television-based learning is often passive, and some experts would suggest that such learners exhibit learned helplessness. This refers to behavior observed in situations where a person’s actions have no effect on outcomes. This report describes a study which updates the author’s previous work in this area. This study tries to replicate an earlier finding that 15 minutes of a mediated learning experience, like a math video, would more often lead to less persistence or propensity for challenge, than a more active learning environment would. The study focused on 90 second- and third-graders in four classrooms from three different schools. Students in two of the classrooms were from a small city in upstate New York. One of these classes was bilingual. Two of the classrooms were from suburban New York. Using tangram puzzles of varying difficulty, the researcher found that students who viewed a video gave up on hard puzzles and opted for easier ones sooner than students who has previously been engaged in more active treatments of the same topic. Five figures and three tables illustrate the results.

• second- and third-grade students
• 90
• tangrams
• problem solving
• five sessions for 40 minutes
• 2x2 matrix: video or non-mediated activity as one dimension of the matrix and subject matter as the other dimension, either scale models or mental arithmetic.

The students answered a questionnaire following the initial activity. (Very hard- Very easy; Very fun- Very Boring). Students watching 15 minutes of television would be less persistent in working on hard math puzzles than they would be following 15 minutes of an activity on the same topic.

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Garrity, C. (1998).  Does the Use of Hands-On-Learning, with Manipulatives, Improve the Test Scores of Secondary Education Geometry Students?  An Action Research Project Submitted at Saint Xavier University (Chicago, Illinois).  Available ERIC Document ED 422 179

This study documented the difficulty of high school students to visualize and understand geometry problems and sought to improve this ability by implementing a constructivist approach which included manipulatives, cooperative learning, and real-life problem solving.  The study was conducted with 47 sophomore students enrolled in two high school geometry classes.  One group was considered the control group and was initially taught using the traditional teacher lecture method and the second group was the experiential group which was taught using manipulatives, cooperative learning groups, and real-life problem solving situations.  Research methods included students and parent surveys, teacher created quizzes and tests, teacher observations, and interviews.  Specific concepts and manipulatives used to teach the classes were listed.  They included:  geoboards (points, lines, segments), graphing calculators (angles), plastic straws (lines, transversals, polygons), and toothpicks (diagonals).  The researcher concluded after the initial part of the study that the scores of the experiential group were higher than those of the control group, thus, the traditional teaching method is less effective than using manipulatives, cooperative groups, and real-life examples.  The researcher also noted students favored group learning and real-life problems and exhibited positive changes in attitude and enthusiasm.

Gibson, H. L., Brewer, L. K., Magnier, J. M., McDonald, J. A., & Van Strat, G.A. (1999, April). The impact of an innovative user-friendly mathematics program on preservice teachers’ attitudes toward mathematics. Paper presented at the Annual Meeting of the American Educational Research Association, Montreal, Quebec, Canada.

The Impact of an Innovative User-Friendly Mathematics Program on Preservice Teachers’ Attitudes Toward Mathematics is a study that was a collaborative project between the University of Massachusetts at Amherst, and Springfield Community Technical College. It was conducted at the University of Massachusetts School of Education in 1998 by researchers Dr. Helen L. Gibson, Laura K. Brewer, Jean-Marie Magnier, James A. McDonald, and Dr. Georgena A. Van Strat. The primary goal of this study was to determine if a constructivist approach to learning of preservice teachers in their college mathematics program would improve their attitudes toward mathematics. The researchers wanted to see how they could enhance preservice teachers’ attitudes towards math as the progressed in the college level mathematics course sequence.  This document promotes instructional strategies that use manipulatives and hands-on learning experiences in order to explore real-life situations that relate to students’ everyday life.

Participants: The participants of this study were 52 paraeducators enrolled in the UPDATE program. 

Manipulatives: A variety of manipulatives were used in this study. The examples that were given were Cuisenaire rods and pattern blocks (Gibson et. al, 14).

Math concepts taught:  Algebra

Methodology:  Between June 1998 and December 1998, two questionnaires were administered. Students completed the Revised Teacher Attitudinal Survey (RTAS) and Instructional Strategies Survey. The revised survey contained 44 statements to which the students responded on a scale of 1-5 with 1 correlating to strongly agree and 5 – strongly disagree. The 44 items were used to compute four subcategories: “Views about Mathematics, Being Good at Mathematics, Learning Mathematics, and Teaching Mathematics” (Gibson et. al,  9) that were intended to measure attitudes about mathematics. The two surveys provided both qualitative and quantitative information about the program. The RTAS was administered twice per course.  The Instructional Strategies survey was only administered once at the end of the course.

Results: The results indicated that the attitudes toward mathematics did not change during any of the three courses that were taken. The qualitative data indicated that the methods used helped them more than a more traditional approach would have with the added benefit was that the subjects now understood the use of manipulatives.

Gresham, G., Sloan, T., & Vinson, B. (1997).  Reducing Mathematics Anxiety in Fourth Grade “At-Risk” Students.  Available ERIC Document ED 417 931.

This paper examined whether fourth grade mathematics anxiety could be decreased by employing mathematical instructional strategies based on National Council of Teachers of Mathematics Standards (NCTM).  The study was conducted for six months with 17 fourth grade students and one teacher.  Pre- and posttest anxiety scales were given to the students and a journal of instructional strategies was kept by the teacher.  Instructional practices included cooperative learning, real-life problem solving, manipulatives, calculators, and computers.  Specific class activities which included numeration and number sense, geometry and measurement, as well as computation and estimation are listed.   Also included are manipulatives that were used during these activities such as geoboards, rulers, pattern blocks, and computers.  The researcher concludes that students anxiety is decreased when instructional methods are implemented based on the NCTM Standards.

Groves, Susie. (1994, April). Calculators: A Learning Environment to Promote Number Sense. Paper presented at the Annual Meeting of the American Educational Research Association, New Orleans, LA. (ERIC Document Reproduction Service No. ED 373 969)

The Calculators in Primary Mathematics Project in Australia was a long-term investigation into the effects of the introduction of calculators on the learning and teaching of primary mathematics. The Australian project commenced with children who were in kindergarten and grade 1 in 1990, moving up through the schools to grade 4 level by 1993. Children were given their own calculators to use when they wished, while teachers were provided with some systematic professional support. Over 60 teachers and 1,000 children participated in the project. This paper describes some critical number sense and reports on the results of interviews with 4^th-grade children (n=58), approximately half of whom had long-term experience with calculators. Children with long-term experience with calculators performed better on the 12 mental computation interview items overall, the 24 number knowledge items overall, and the 3 estimation items taken individually. Overall, their performance was better on 34 of the 39 items, with the greatest differences in performance in mental computation generally occurring on the most difficult items. Their pattern of use of standard algorithms, left-right methods, and invented methods for mental computation items did not vary greatly from that of the non-calculator children.

A written test, a test of calculator use and two different interviews were used. The first interview focused on different aspects of children’s understanding of the number system, together with their choice of calculating device for various computational tasks and their solutions to “real world” problems based on division and multiplication. The second interview, which focused on number sense, was designed to complement the two tests and the first interview.

This paper and other (Groves, 1993a; submitted) show that children with long-term experience of calculators performed better than children without such experience on a range of computation and estimation tasks and some “real world” problems; exhibited better knowledge of number, particularly place value, decimals and negative numbers; made more appropriate choices of calculating device; and were better able to interpret their answers when using calculators, especially where knowledge of decimal notation or large numbers was required.

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Hartshorn, R. and Boren, S. (1990).  Experiential Learning of Mathematics:  Using Manipulatives.  (Report No. EDO-RC-90-5.  ) Charleston, WV:  Appalachia Educational Laboratory.  (ERIC Document Reproduction Service No. ED 321967).

This study is a compilation of results from previous statewide studies on the use of manipulatives in the classroom.  Using data from surveys, Hartshorn and Boren found: Primary teachers generally accept the use of manipulatives Manipulatives are useful in the transition from concrete to abstract when taught in steps (semi-concrete, semi-abstract) Experienced teachers use manipulatives less than inexperienced teachers Teaching with manipulatives is effective only when the proper manipulative and activity is used Long-term use of manipulatives is more effective than short term use. Teachers’ training influences the effectiveness of manipulatives Manipulatives are used infrequently at the secondary level even though many students need ideas introduced at the concrete level.

Hatfield, M. M.  (1994).  Use of Manipulative Devices: Elementary School Cooperating Teachers Self-Report.  School Science of Mathematics, 94, 303-309.

The article discusses the use of manipulatives in the elementary setting (K-6).  87 teachers were obtained for the research based on a survey that was mailed to 106 (K-6) teachers with 5 or more years in teaching.  Those who responded to the survey were used in the study.  This quantitative study shows the familiarity, availability, and use of eleven different manipulatives.  The manipulatives used in this study were: pattern blocks, cuisenaire rods, geoboards, flexi-counters*, base 10 blocks, ropygrams* number/math balance, bundleable materials, tangrams, fraction bars, and attribute blocks.  Note: those manipulatives marked with an * are not manipulatives but were used to determine response bias.  23.8% of those that responded said they were familiar with the flexi-counters and 1.2% said they were familiar with the ropygrams but that neither of the manipulatives were available.

The results of the study show that there is a decline at the intermediate grades (4-6) in terms of use of manipulatives.  It further shows the need for universities to have more say as to where and with whom their pre-service teachers will conduct their experience.

Haughland, S. (2000).  Computers and Young Children.  (Report No. EDO-PS-00-4).  Champaign, IL:  University of Illinois.  (ERIC Document Reproduction Service No. ED 438926).

“Computers have an impact on children when the computer provides concrete experiences, children have free access and control the learning experience, children and teachers learn together, teachers encourage peer tutoring, and teachers use computers to teach powerful ideas.”

Although theory suggests a constructivist philosophy for children’s computer use, most teachers use technology in traditional ways (basic skills and instructional games).  When computers are used effectively, children have significantly greater developmental gains than children without computer use.  Computer use enhances children’s self-concept.  Young children demonstrate increased levels of spoken communication and cooperation when working with a group and discussing their experiences while using the computer.  Teachers with proper computer training (defined by practical experience, workshops, models and mentors, and supervisory follow-up) effectively integrate computers into their lessons when provided with adequate classroom resources.

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Johnson-Gentile, K., Clements, D.H., & Battista, M.T.  (1994).  Effects of computer and noncomputer environments on students’ conceptualizations of geometric motions.  Journal Educational Computing Research, 11 (2), 121-140.

This is a quantitative study that included interviews, which involved 223 fifth and sixth grade students from 9 different teacher’s classrooms during the spring semester.  The teachers were veteran teachers in both urban and suburban settings.  Participants were using either Miras or the Logo MIRROR program to identify lines of symmetry and paper or acetate sheets or the Logo Geometry MOTIONS microworld to determine congruence, slides, flips and turns.  The purposes of this study examine students’ conceptualization of geometric motions and the effects of presenting the curriculum via a computer with computer-based manipulatives or via paper and pencil with hand held manipulatives.  Possible impact caused by gender differences and students’ levels of thinking, based on the van Hiele taxonomy, in the domain of geometric motions were also investigated.

Two fifth grade classes and one-sixth grade class were assigned to one of three treatment groups for an eight-day motions unit.  The LOGO group received all of their instruction using the Motions strand of the Logo Geometry curriculum, the non-logo group received instruction in the identical curriculum using noncomputer manipulatives rather than the Logo tasks.  A third nontreatment group participated in the regular mathematics program           including a two-day textbook lesson on symmetry.  A pretest of general achievement in geometry was administered to all students.  Immediately upon completion of the unit a posttest on motion geometry was administered to all students and it was readministered one month later.  Two boys and two girls were randomly selected from each classroom for an individual thirty-minute structured interview.

An ANOVA on the pretest showed not significant differences but the ANOVA on the posttest and delayed posttest showed a significant treatment effect.  Both the Logo and nonLogo posttest scores were higher than the control group.  The immediate posttest did not show a significant difference in the Logo and nonLogo group, however the delayed posttest scores were significantly higher for the Logo group. 

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Karp, K. (1990).  Manipulative materials in the primary level mathematics lesson:  Are there viable alternatives (Report No. SE051515)?  Garden City, NY:  Adelphi University.  (ERIC Document Reproduction Service No. ED320774)

This study investigated student achievement differences between three mathematics programs:  Explorations (Addison Wesley), Mathematics (Silver Burdett), and the Comprehensive School Mathematics Program (CSMP).  Manipulative use varied among the programs, ranging from a highly hands-on manipulative approach (Addison Wesley) to an abstract focus incorporating no manipulatives (CSMP).  Mathematics (Silver Burdett) was a combination of both approaches.  Five elementary schools in a predominantly white middle-class district were the subjects for this study.  After sampling the three series, 18 teachers were voluntarily assigned the pilot programs:  five working with Addison Wesley resources, five incorporating Silver Burdett materials, and eight using the CSMP.

The three mathematics programs were implemented in first grade, with an average class size of 21 students, over the course of one school year.  Pre-testing was administered during a one-day window in October and post-testing was completed in the spring during a two-day period.  Data from the tests, teacher questionnaires, and structured interviews were used to determine effectiveness of the programs.  An analysis of covariance (ANCOVA) was completed to evaluate whether students performance was above or below the expected scores.  Results show the CSMP as the most effective in raising students to a higher achievement level.  The Explorations cohort showed the least gains.  Teachers in this program reported the following concerns:  excessive time spent creating resources, complexity in managing the classroom, and the need for extra time to complete lessons.

Kieran, C., & Hillel, J. (1990). ?It?s tough when you have to make the triangles angle?: Insights from a computer-based geometry environment. Journal of Mathematical Behavior, 9, 99-127.

Kiernan and Hillel (1990) found that sixth-grade students using the computer-microworld virtual manipulative made significant gains in understanding the nature of isosceles triangles.

Kim, S. (1993). The relative effectiveness of hands-on and computer-simulated manipulatives in teaching seriation, classification, geometric, and arithmetic concepts to kindergarten children. (Doctoral dissertation, University of Oregon, 1993). Dissertation Abstracts

International, 54(09), 3319.

Kim (1993) found no statistically significant differences between kindergarten students who viewed or used physical manipulatives and those using virtual manipulatives on measures of addition, geometric classification, and counting skills.

Kjos, Ruth, Long, K. (1994).  Improving Critical Thinking and Problem Solving in Fifth Grade Mathematics.  An Action Research Project Submitted at Saint Xavier University (Chicago, Illinois).  Available ERIC Document 383 525

This research describes an intervention to improve the critical thinking and problem solving ability of fifth grade students.  The study was conducted with 171 fifth grade students from two public schools in Illinois.  The research methods included student math autobiographies, teacher created tests, teacher surveys, and student surveys.  The instructional interventions implemented to improve critical thinking skills and problem solving were student journal writing about metacognitive processes, direct instruction to students on how to think critically about and solve problems, and the use manipulatives to improve instruction.  Specific manipulatives that were used were tangrams (shapes and area), unifix cubes (area and perimeter), colored counters (probability), base ten blocks (place value and decimals), pattern blocks (fractions and percents), and calculators (percents).  The study concluded that the implementation of the above mentioned teaching strategies improved student attitudes, increased the students’ ability to write about their own thinking, and increased student problem solving abilities.

Kohler, M., Kohler, E. (1996).  Improving Mathematics Education in Grades 6-9 through the Integration of Content, Technology, and Manipulatives:  Formal Cumulative Evaluation Report.  National Science Foundation Grant ESI-9155296.  Available ERIC Document ED 401 129

This report described the findings of a three-year project in Alabama which focused on improving the teaching behaviors, knowledge, and attitudes of 58 mathematics teachers in grades 6-9.   Research methods used were pre- and posttests, grades, focus groups, questionnaires, interviews, observations, and evaluations.  This paper reported whether or not the participants felt the project was successful rather than describing the actual methods used over the three year study to improve teaching behaviors, knowledge, and attitudes.  Nevertheless, the research concluded that participants felt that their mathematics knowledge was increased and they were more skilled at using manipulatives and computers in their instruction and felt they did so more frequently and effectively after participating in the study.  Teachers also noted improved student performance and attitudes in their classrooms.

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Lackey, B. and Reglin, G.  (1991).  Manipulatives and Achievement of Subtraction Basic Facts for Rural Second Grade Students.  Journal of Research in Education, 1, 53-56.

In this qualitative research, the study investigates the effects of a manipulative instructional approach and traditional instruction on the achievement of subtraction facts for 4 African-American and 26 white second graders in a rural North Carolina public school.  The 30 subjects were below/average, average, and above average in ability.  There was no correlation between the race and ability of the subject.  The 30 students were broken down into two groups.  One group used a traditional approach to the subtraction facts.  The other group used a manipulative approach.  The data was collected through tests, and the ability to communicate their understanding of subtraction.  It was concluded that greater gains in achievement were of subtraction basic facts occurred with the manipulative instruction approach.

Lara-Alecio, R., Parker, R., Aviles, C., Mason, S., & Irby, B. J. (1998). A study of the use of manipulatives in the assessment of mathematics instruction with ESL Hispanic students. Bilingual Research Journal, 22(2-4), 215-235.

Summary: I’m including this reference and abstract because it seems like an ideal research article for our group. Unfortunately, the report is written in Spanish only. If anyone is able to translate, I’d be interested in reading it. The follow abstract was the only part available in English.

Official Abstract: As an alternative form of mathematics assessment for use with limited-English-proficient students, 14 mathematics tasks using manipulatives were administered to 45 Hispanic students in grades 1-3 and readministered 2-3 weeks later. Test reliability and validity, task difficulty, and the relationship among test subscales across grades were examined.

One final note: I found abstracts of dissertations that sounded very interesting and informative on the topic of education research with math manipulatives, but I learned that acquiring doctoral dissertations from other universities is very difficult and/or costly. In some cases, the abstracts included detailed information about the research and findings that could be useful to us in our own dissertation research. If you are interested in investigating this, check the library databases on the GMU library web site, and search in the topic “Education” to find the “Dissertation Abstracts” database.

Leinenbach, Marylin; Raymond, Anne M. (1996).   A two-year collaborative action research study on the effects of a “Hands-on” approach to learning algebra.  Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Panama City, FL.   Available:  Eric Document 398 081

The two-year study consisted of two phases:  phase one involved instruction with algebra manipulatives and phase two was a follow up on participants from phase one.  The first phase consisted of five eighth grade classes, approximately 120 students, age 13.  The second phase was a follow up on the same students regarding their retention of the manipulative “algebra learning strategies” during their 9^th grade math course.  The manipulative used was “Hands-on-Equations” developed by Dr. Henry Borenson.  The manipulative uses pawns, number cubes and a balance to teach the concept of solving linear equations.

The first phase consisted of three parts.  The first nine weeks involved instruction taught in a non-manipulative style using the adopted textbook.  The 26 lessons, Hands-on Equation manipulative program were then implemented.  After the completion of the manipulative lessons, the instruction returned to a non-manipulative style with the adopted textbook.

Data collection methods consisted of surveys, student reflections, work samples, test scores and interviews.  Students were encouraged to use manipulatives during quizzes and tests that were designed in a format that paralleled the manipulative instruction.  All students took a mandatory standardized algebra test at the end of the school year.

The results of the first phase revealed that the class averages during the textbook phases were lower than the manipulative phase.  The teacher noted that students were better able to show understanding of algebraic concepts with the manipulatives.     The teacher’s concern was that she had weakened the students’ abilities to work algebraic problems without manipulatives, but the results of standardized exam revealed that 80% of the students scored 60% or better. This far exceeded the expectations of the administration and colleagues and led Leinenbach to believe that she had successfully helped students bridge the gap between concrete and the more abstract algebra.

The second phase consisted of a survey of all students who participated in the 8^th grade study, and only nine responses were received for the second phase of the study.  No results were reported for this phase.

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McClung, Lewis W.  (1998).  A study on the use of manipulatives and their effect on student achievement in a high school algebra one class. Unpublished master’s thesis, Salem-Teikyo University, Salem, WV.  Available:  Eric Document 425 077

 McClung’s nine-week study investigates the use of Algeblocks in a high school Algebra I class.  There are 2 classes, 49 students in total, of sophomore and junior level students, ranging in age from 15-17 years.  The manipulative studied is Algeblocks and the topic of the lessons taught during this nine week study is polynomials.

The study consists of a pretest, treatment and posttest.  The control group, Group A receives a traditional teaching method of lecture, homework, and in-class worksheets. The treatment group, Group B receives traditional teaching method of lecture, homework, but instead of in-class, worksheets the students work with the manipulative Algeblocks.

McClung uses a two-sample t-test to analyze the data.  The pretest data reveal that there is no significant difference between the two groups but the posttest analysis reveals that there is a significant difference in the achievement levels of the two groups.  A comparison of the group means shows that Group A mean = 77 while Group B’s mean = 52.  These results would seem to indicate that the use of manipulatives in algebra at the high school level is not beneficial. 

McClung suggests several key factors that may have influenced the results. The students were out of the range of concrete operational stage and into the formal operational stage of development. The students were not allowed to use manipulatives on the posttest.

The instructor was new to the concept of using manipulatives and did not acquire sufficient knowledge of the manipulatives before the study began which resulted in the manipulatives not being properly incorporated into the curriculum. 

McCoy, L. P.  (1989).  Perceptual Preferences of Mathematically Deficient Elemntary Students: Implications for Instruction.  U.S. Indiana: National Center for Research on Teacher Learning.  (ERIC Document Reproduction Service No. ED305 379)

Subjects in this study were eleven students from two public schools enrolled at a university remedial mathematics clinic.  Another group consisted of eight average/above average students experiencing slight or no difficulties in mathematics.  The students were in grades 3 through 6.  The focus of the qualitative study was on assessing the use of concrete materials in mathematics instruction, comparing the perceptual preferences of mathematically-deficient and average/above average elementary school students, and using the information to make recommendations for instruction. 

Results concluded that the students in the average/above average group preferred an auditory or visual mode of learning, while the remedial students preferred a kinesthetic mode.  There was no difference in preference for tactile mode.  The final conclusion is that the remedial students would benefit from more diverse instructional activities.  The results strongly support the use of concrete manipulatives.    

Meira, L. (1998).  Making sense of instructional devices: the emergence of transparency in mathematical activity.  Journal for Research in Mathematics Education, 29 (2), 121-142.

The investigator explored the idea of transparency, explaining it as an index of the learner's access to mathematical knowledge and activities.  He tried to discern, through this study, whether transparency resides in the manipulative itself, or whether transparency emerges from the user's interaction with the manipulative, given his or her background.

Nine pairs of eighth graders, aged 13-14, participated in the study on a volunteer basis after school.  All investigated the concept of linear functions.  Three pairs each were randomly assigned to use winches, springs, or number machines.  The winches had rollers of different circumferences around which were wound cords with objects tied to their ends.  The springs could hold weights of various sizes.  The number machines were computers with input/output displays. 

The investigator observed the classes of the participants for three weeks prior to their two after school 1 1/2 hr problem-solving sessions, which were videotaped.

The manipulatives that were used were ranked by their epistemic fidelity, that is, by which should inherently show the concept of linear functions most clearly.  They were judged to be ordered as follows: winch, spring, computer display.  The videotapes were analyzed to see if the students found transparency in the same order.

It became clear that it is not the manipulative itself that "contains" the concept, that is, transparency does not reside in the object.  Rather, transparency emerges in the process of the objects being used by students who come to the task with prior knowledge and who participate in discussion that ensues in their use.  It was found that the winch and spring, judged most transparent inherently, were the least transparent to the students.  While these two manipulatives were supposed to make math concepts apparent, the students had to expend much effort, instead, including employing math, to make sense of the manipulatives.  On the other hand, students readily made mathematical inferences about linear functions from the input/output computer display.

Moore, J. L., and Schwartz, D. L.  (1994).  Visual Manipulatives for Proportional Reasoning.  U.S. Tennesse: National Center for Research on Teacher Learning. (ERIC Document Reproduction Service No. ED376 200)

The goal of the qualitative research was to design a learning environment that facilitates a move from implicit to a more explicit understanding of proportionality.  49 high ability sixth grade mathematics students using the Jasper Adventure Series of problems participated in the research. The Jasper Adventure Series was developed by the Cognition and Technology Group at Vanderbilt in 1992.  The research was conducted based on pre-post test and being able to extrapolate and visually prove answers.  Students were more successful using manipulatives.

 It was concluded that the potential of a manipulable  visual representation for highlighting the structural invariances within a proportion and the proportional invariances between domains leads to an understanding that transfers to more complex proportional problems.  

Moyer, Patricia S.  (2001).  Are we having fun yet?  How teachers use manipulatives to teach mathematics.  Educational Studies in Mathematics: An International Journal, 47(2), 175-197.

Teachers often comment that using manipulatives to teach mathematics is “fun!”  Embedded in the word “fun” are important notions about how and why teachers use manipulatives in the teaching of mathematics.  Over the course of one academic year, this study examined 10 middle grades teachers’ uses of manipulatives for teaching mathematics using interviews and observations to explore how and why the teachers used the manipulatives as they did.  An examination of the participants’ statements and behaviors indicated that using manipulatives was little more than a diversion in classrooms where teachers were not able to represent mathematics concepts themselves. The teachers communicated that the manipulatives were fun, but not necessary, for teaching and learning mathematics. 

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Neiderhauser, D.; Stoddart, T. (2001). Teachers' Instructional Perspectives and Use of Educational Software. Teaching and Teacher Education, 17(1).

1.  Age/grade level:  elementary school (K-6) teachers

2.  Number of participants:  1093 teachers

3.  Manipulatives used: educational software

4.  Math concept taught:  basic math skills and open-ended problems

5.  Duration of the study:  survey data

6.  Research methods/procedures:  written survey (questionnaire)

7.  Results:  Eighty-five percent of the teachers surveyed used only skill-based software.  Teacher surveys reflected a learner-centered orientation and a constructivist view of learning, but those values were not reflected in their children's computer use.  Virtual manipulatives, characterized as open-ended software, would promote a constructivist approach to computer learning.  Unfortunately, teachers view virtual learning as different then classroom learning.

Noble, T., Nemirovsky, R., Wright, T., &Tierney, C.  (2001).  Experiencing change: the mathematics of change in multiple environments.  Journal for Research in Mathematics Education, 32 (1), 85-108.

The investigators carried out a case study, observing how two fifth-grade boys explored the concepts of the mathematics of change across several embodiments of change.  The essential concepts are rate of change, which in calculus would be the derivative, and accumulation, which in calculus would be the integral.  The class that the boys were in was participating in a four-week unit on the topic.  The investigators filmed the whole classroom and also focused in on one group of two boys.

The students in the class had taken trips across the classroom, proceeding at different rates, marking their progress by dropping bean bags at specified time intervals.  It was, therefore, possible for them to see that the space intervals between the bean bags differed depending on their rate of movement across the room.

The study focused on the boys' interaction with three additional embodiments of the mathematics of change.  The first embodiment was taking trips with Cuisenaire rods of two lengths along meter sticks.  The second was a similar exploration on a written table of values.  The third was a computer software program called Trips©.

The investigators explored two concerns: 1) where the mathematics reside, that is, whether in the manipulative or in the students, and 2) how students make connections across embodiments. 

On the first point, they argue that the mathematics emerges from the students' process of making the environment into a lived-in space for themselves rather than in the manipulative materials themselves.  While the designer of an activity may have certain expectations for what the student will experience, the way students act and make sense of their actions can vary widely from the designers' expectations.  A space is considered "lived-in" when the students' interactions with it are relational, intentional, and creative.  "Relational" refers to how the changes affect the space as a whole, "intentional" describes a space in which students do things and accomplish purposes, and "creative" spaces are those in which the space is constantly being recreated as it is experienced. 

On the second point about students making connections, the investigators describe students as finding family resemblances among the embodiments together with their own background of experience.  The strength of the concept results from the overlapping of many fibers, as in a thread. 

In the specific investigation of the two boys, the researchers observed that the boys brought to the three embodiments their previously owned concept of racing, even though the curriculum developers had deliberately avoided terminology of racing in the design.  Nevertheless, this allowed the boys to make the space their own, by enabling them to become engaged and to interact with the environments on their own terms.  The boys got similar numerical results in the table and computer environments.  However, because of some difficulty with manipulating the Cuisenaire rods, the numbers in this environment did not match the results in the other two environments.  Nevertheless, the boys were able to criss-cross their experiences and find "family resemblances" among the various embodiments of trips, with their underlying concepts of rates of change and accumulation.  The boys found similarities among the trips, while each trip retained its own identity.

Noss, Richard. Healy, Lulu. Hoyles, Celia. (July 1997). The Construction of Mathematical Meanings: Connecting the Visual with the Symbolic. Educational Studies in Mathematics. 33(2), 203-33.

In this paper, we explore the relationship between learners’ actions, visualisations and the means by which these are articulated. We describe a microworld, Mathsticks, designed to help students construct mathematical meanings by forging links between the rhythms of their actions and the visual and corresponding symbolic representations they developed. Through a case study of two students interacting with Mathsticks, we illustrate a view of mathematics learning which places at its core the medium of expression, and the building of connections between different mathematisations rather than ascending to hierarchies of decontextualisation. 

This is a qualitative case study- observation between a pair of students with one computer. They needed to program a computer to complete the task presented.

The students empirical solution emerged from their expressions of the invariant structures, rather than preceding them. Second, with Mathsticks the means of expressing actions is firmly soldered to the activity.  The students were responsible for placing the matches in such a way that the colour-change occurred, and for establishing the rhythms of action which led to their becoming expressed symbolically.

Nute, N. (1997). The impact of engagement activity and manipulatives presentation on intermediate mathematics achievement, time-on-task, learning efficiency, and attitude. (Doctoral dissertation, University of Memphis, 1997). Dissertation Abstracts International, 58(08), 2988.

 Nute (1997) found no statistically significant differences between fourth-, fifth-, and sixth-grade students who viewed or used physical manipulatives, virtual manipulatives, or both on measures of patterning and geometric transformations. However, all groups scored higher than those students with no manipulative exposure.

Nute, N.  (1997).  The impact of engagement activity and manipulatives presentation on intermediate mathematics achievement, time-on-task, learning efficiency, and attitude.  Dissertation Abstracts International, 58(08), 2988.

This study examined the effect of engagement activities and manipulative-type presentations on students’ math achievement, time-on-task, learning efficiency, and attitude.  The participants were 241 intermediate students (grades 4, 5, and 6). 

Students were randomly assigned to groups.  Six groups received a combination of instructional strategies using manipulatives--both concrete and computer.  One control group had no manipulatives.  Data was collected in three ways:  students took a post-test measuring their achievement of patterns content, completed a time-on-task measurement, and filled out an attitude questionnaire. 

The results indicated that the computer only presentation took more time than the concrete manipulative only presentation.  With regard to grade level effects, time-on-task was equal for fourth and fifth graders.  Efficiency was higher for sixth graders than for fourth and fifth graders.  Manipulative groups showed higher-level recognition and application achievement performances than the control groups.  Overall, manipulative instruction strategies showed more effective for higher-order tasks than did no manipulatives instruction.

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Outhred, L.N. & Michelmore, M.C. (2000) Young children’s intuitive understanding of rectangular area measurement. Journal of Research in Mathematics. 31(5). pp 602-625.

A sample of 115 children was randomly selected from 40 Grades 1 to 4 classes in four schools serving a range of cultural groups in a medium socioeconomic area of Sydney. The focus of this research was to analyze the strategies young children use to solve rectangular covering tasks before they have been taught area measurement.

Research Questions:

1. What strategies do young children use to find the number of unit squares that cover a rectangle?

2. Can children's strategies be classified into a sequence of developmental levels?

3. What operational principles underpin this developmental sequence?

Information concerning the strategies that children used to solve a variety of array-based tasks was collected in individual interviews conducted early in the school year. The interviewer (the first author) inferred children's strategies from a combination of observation and careful questioning as the children worked through tasks involving drawing, counting, and measurement

Children's solution strategies were classified into 5 developmental levels; Level 0: Incomplete covering, Level 1: Primitive covering, Level 2: Array covering, constructed from unit, Level 3: Array covering, constructed by measurement, and Level 4: Array implied, solution by calculation.

Four Principles Underlying Rectangular Covering

Crucial learning leaps occurred when children start thinking in terms of rows. Initially, rows are recognized as geometrically equivalent; the fact that the number of units in each row is constant emerges later. Finding the number of rows is the next problem to be solved; when this problem is solved, a child is only a short step from being able to calculate the total number of units.

An important implication is that students need to link area measurement to both linear measurement, and multiplicative concepts before the area formula can be meaningfully learned.

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Parham, J. L. (1983).  A meta-analysis of the use of manipulative materials and student achievement in elementary school mathematics.  Dissertation Abstracts International 44A, 96.

Park, J.  (1993).  Time studies of fourth graders generating alternative solutions in a decision-making task using models and computer simulations.   Journal of computing in childhood Education, 4 (1), 57-76.

This study is a quantitative study completed with 240 fourth graders from 12 classrooms in two public school districts.  The students all had prior experience using the computer.  The purpose of this study was to determine any differences in the time required to complete a decision making task presented in four different ways.  Participants in the study manipulated bags of real jellybeans as well as images bags of jellybeans on the computer screen. Variables looked at in this study were the time between when the task was given and the student begin to show his or her response, the estimated time-per-move while developing the first response, and the time required if another response could be produced.    The computer kept track of the data generated during the com