12 research outputs found
Applying Math onto Mechanism: Investigating the Relationship Between Mechanistic and Mathematical Understanding
Physical manipulatives are commonly used to improve mathematical understanding. However, it is unclear when physical manipulatives lead to significant benefits. We investigated whether understanding the mechanism of a manipulative would affect mathematical use and understanding. Participants were asked to navigate a physical robot through a maze, and to create a strategy that could navigate differently sized robots through the same maze. Participants with a better understanding of the robotâs mechanism were more likely to utilize complex mathematical strategies during the maze task than participants with lower mechanistic understanding. These participants with higher mechanistic understanding also showed greater understanding of the mathematical relationships within the robot. The study provides evidence for a relationship between mechanistic understanding and mathematical understanding, suggesting that mechanistic manipulatives, upon which mathematics can be applied, may be especially beneficial for fostering mathematical understanding
The impact of educational material use on mathematics achievement: a meta-analysis
As a result of rapid development in technology, utilizing materials in education has become important. To date, researchers have often explored the effects of using educational materials in mathematics instruction on academic achievement. The purpose of this study was to combine the empirical evidence regarding the effectiveness of educational materials in mathematics. For this aim, a meta-analysis method was used in the current research. In line with the aim of the research, 54 experimental studies published between years of 2005 and 2016 were included in the meta-analysis and 58 effect sizes were calculated from these studies. The results of the meta-analysis showed that using materials in mathematics has a positive and high influence on achievement. According to analysis of mediator variables that are related to instructional characteristics, significant differences are found in the variables of mathematics topic, type of material, and application time. However, teaching with materials in mathematics did not seem to differ in effectiveness from teaching without materials, in terms of methodological characteristics of the studies
The effects of using the GoMath program on teaching computation skills for students with learning disabilities
The purpose of this study was to evaluate the effects on teaching math computation skills to students with learning disabilities (LD) using the GoMath program and to examine the teachersâ and studentsâ satisfaction with this program in their teaching and learning. Four, 3rd and 4th graders with LD were taught by one special education teacher in a resource room and participated in learning computation skills for 60 minutes, 5 days per week for 12 weeks, using the Go Math program. A multiple baseline research design with A B phases across students was used to evaluate their performance. The findings indicated that all of the participants increased their addition, subtraction and multiplication computation scores using the GoMath program, and the teachers and students were generally satisfied with the program and its supplemental materials. The results of this study support the use of the GoMath program providing explicit instruction with a multisensory approach to teach math computation skills to students with LD
SOLVING LINEAR EQUATIONS: A COMPARISON OF CONCRETE AND VIRTUAL MANIPULATIVES IN MIDDLE SCHOOL MATHEMATICS
The purpose of this embedded quasi-experimental mixed methods research was to use solving simple linear equations as the lens for looking at the effectiveness of concrete and virtual manipulatives as compared to a control group using learning methods without manipulatives. Further, the researcher wanted to investigate unique benefits and drawbacks associated with each manipulative.
Qualitative research methods such as observation, teacher interviews, and student focus group interviews were employed. Quantitative data analysis techniques were used to analyze pretest and posttest data of middle school students (n=76). ANCOVA, analysis of covariance, uncovered statistically significant differences in favor of the control group. Differences in posttest scores, triangulated with qualitative data, suggested that concrete and virtual manipulatives require more classroom time because of administrative issues and because of time needed to learn how to operate the manipulative in addition to necessary time to learn mathematics content. Teachers must allow students enough time to develop conceptual understanding linking the manipulatives to the mathematics represented. Additionally, a discussion of unique benefits and drawbacks of each manipulative sheds light on the use of manipulatives in middle school mathematics
Using Manipulatives to Investigate ESOL Students\u27 Achievement and Dispositions in Algebra
The purpose of this embedded quasi-experimental mixed methods research was to investigate the effectiveness of concrete and virtual manipulatives on the achievement of English Speakers of Other Languages (ESOL) as they employ them to explore linear and exponential functions in high school Sheltered Common Core Coordinate Algebra. Also of interest were the effects concrete and virtual manipulatives have on their disposition towards mathematics and math class. Another goal was to investigate the benefits and disadvantages of using concrete and virtual manipulatives versus traditional instructional practices.
This was a 5-week study. The control group (N=20) was instructed through the use of mathematics textbooks and Power Points (traditional) and compared to the treatment group (N=19), which was instructed using concrete and virtual manipulatives. One ESOL mathematics teacher implemented this study, teaching both groups by utilizing the sheltered instruction observation protocol (SIOP) (2012) model to integrate content and language.
Qualitative research methods, teacher interviews, recorded field notes, studentsâ work samples and artifacts were utilized. Quantitative data analysis techniques were used to analyze departmentalized Linear and Exponential Functions Summative Assessments (pretest and posttest) to measure mathematics achievement. The one-way ANOVA uncovered no statistically significant difference between the control group and treatment group as they explored linear and exponential functions. The Quantitative Understanding: Amplifying Student Achievement and Reasoning Students Disposition instrument (pre-questionnaire and post- questionnaire) measured dispositions about mathematics and math class. The one-way ANOVA indicated no statistically significant difference between the control and the treatment groupâs dispositions about mathematics and math class
Equivalent Fraction Learning Trajectories for Students with Mathematical Learning Difficulties When Using Manipulatives
This study identified variations in the learning trajectories of Tier II students when learning equivalent fraction concepts using physical and virtual manipulatives. The study compared three interventions: physical manipulatives, virtual manipulatives, and a combination of physical and virtual manipulatives. The research used a sequential explanatory mixed-method approach to collect and analyze data and used two types of learning trajectories to compare and synthesize the results. For this study, 43 Tier II fifthgrade students participated in 10 sessions of equivalent fraction intervention. Pre- to postdata analysis indicated significant gains for all three interventions. Cohen d effect size scores were used to compare the effect of the three types of manipulativesâat the total, cluster, and questions levels of the assessments. Daily assessment data were used to develop trajectories comparing mastery and achievement changes over the duration of the intervention. Data were also synthesized into an iceberg learning trajectory containing five clusters and three subcluster concepts of equivalent fraction understanding and variations among interventions were identified. The syntheses favored the use of physical manipulatives for instruction in two clusters, the use of virtual manipulatives for one cluster, and the use of combined manipulatives for two clusters. The qualitative analysis identified variations in studentsâ resolution of misconceptions and variations in their use of strategies and representations. Variations favored virtual manipulatives for the development of symbolic only representations and physical manipulatives for the development of set model representations. Results also suggested that there is a link between the simultaneous linking of the virtual manipulatives and the development of multiplicative thinking as seen in the tendency of the students using virtual manipulative intervention to have higher gains on questions asking students to develop groups of three or more equivalent fractions. These results demonstrated that the instructional affordances of physical and virtual manipulatives are specific to different equivalent fraction subconcepts and that an understanding of the variations is needed to determine when and how each manipulative should be used in the sequence of instruction
Using the Concrete-Representational-Abstract Sequence to Connect Manipulatives, Problem Solving Schemas, and Equations in Word Problems with Fractions
Students with learning disabilities or learning difficulties in mathematics often have difficulties solving word problems with fractions. These difficulties limit students\u27 abilities to solve everyday math problems and develop the skills necessary for higher level mathematics. Prior research on problem solving indicates that direct instruction on problem schemas can improve problem solving performance. Previous research also suggests that instruction using the concrete-representational-abstract (CRA) sequence and instruction with virtual manipulatives can enhance understanding of mathematical concepts. However, a CRA sequence that incorporates virtual manipulatives has not been combined with schema-based instruction to help students solve word problems with fractions. The purpose of this study was to examine the effects of using an intervention that combined the CRA sequence with virtual manipulatives and schema-based instruction to improve the problem solving performance of students with learning disabilities or learning problems in mathematics on word problems with fractions. This sequence of instruction was combined with a mnemonic strategy called the LISTS strategy to help students remember the steps in the problem solving sequence. Using a single-case multiple baseline across participants design, the researcher provided an intervention to five students in the fifth grade that included instruction in three problem schemas for addition and subtraction (change, compare, and group). Results indicated that all students made some gains in performance on problems similar to those presented during the intervention, but the three students who were able to make connections between problem schemas and equations demonstrated significant gains in performance. The concrete models and virtual models used in the CRA sequence enhanced understanding of fraction word problems for some, but not all, students. Additionally, analysis of student performance on pre- and post-tests of problems with novel features indicated that students made only small gains in performance on fraction word problems that included difficult vocabulary, irrelevant information, or information that required different conceptualizations than those presented during the intervention
Thinking Outside the Box: Enhancing Science Teaching by Combining (Instead of Contrasting) Laboratory and Simulation Activities
The focus of the present work was on 10- to 12-year-old elementary school studentsâ conceptual
learning outcomes in science in two specific inquiry-learning environments, laboratory
and simulation. The main aim was to examine if it would be more beneficial to combine
than contrast simulation and laboratory activities in science teaching. It was argued
that the status quo where laboratories and simulations are seen as alternative or competing
methods in science teaching is hardly an optimal solution to promote studentsâ learning and
understanding in various science domains. It was hypothesized that it would make more
sense and be more productive to combine laboratories and simulations. Several explanations
and examples were provided to back up the hypothesis.
In order to test whether learning with the combination of laboratory and simulation
activities can result in better conceptual understanding in science than learning with laboratory
or simulation activities alone, two experiments were conducted in the domain of
electricity. In these experiments students constructed and studied electrical circuits in three
different learning environments: laboratory (real circuits), simulation (virtual circuits), and
simulation-laboratory combination (real and virtual circuits were used simultaneously). In
order to measure and compare how these environments affected studentsâ conceptual understanding of circuits, a subject knowledge assessment questionnaire was administered
before and after the experimentation. The results of the experiments were presented in four
empirical studies. Three of the studies focused on learning outcomes between the conditions
and one on learning processes.
Study I analyzed learning outcomes from experiment I. The aim of the study was to
investigate if it would be more beneficial to combine simulation and laboratory activities
than to use them separately in teaching the concepts of simple electricity. Matched-trios
were created based on the pre-test results of 66 elementary school students and divided
randomly into a laboratory (real circuits), simulation (virtual circuits) and simulation-laboratory
combination (real and virtual circuits simultaneously) conditions. In each condition
students had 90 minutes to construct and study various circuits. The results showed that
studying electrical circuits in the simulationâlaboratory combination environment improved
studentsâ conceptual understanding more than studying circuits in simulation and laboratory
environments alone. Although there were no statistical differences between simulation
and laboratory environments, the learning effect was more pronounced in the simulation
condition where the students made clear progress during the intervention, whereas in the
laboratory condition studentsâ conceptual understanding remained at an elementary level
after the intervention.
Study II analyzed learning outcomes from experiment II. The aim of the study was to
investigate if and how learning outcomes in simulation and simulation-laboratory combination
environments are mediated by implicit (only procedural guidance) and explicit (more
structure and guidance for the discovery process) instruction in the context of simple DC
circuits. Matched-quartets were created based on the pre-test results of 50 elementary school
students and divided randomly into a simulation implicit (SI), simulation explicit (SE), combination implicit (CI) and combination explicit (CE) conditions. The results showed that
when the students were working with the simulation alone, they were able to gain significantly greater amount of subject knowledge when they received metacognitive support
(explicit instruction; SE) for the discovery process than when they received only procedural
guidance (implicit instruction: SI). However, this additional scaffolding was not enough to
reach the level of the students in the combination environment (CI and CE). A surprising
finding in Study II was that instructional support had a different effect in the combination
environment than in the simulation environment. In the combination environment explicit
instruction (CE) did not seem to elicit much additional gain for studentsâ understanding of
electric circuits compared to implicit instruction (CI). Instead, explicit instruction slowed
down the inquiry process substantially in the combination environment.
Study III analyzed from video data learning processes of those 50 students that participated
in experiment II (cf. Study II above). The focus was on three specific learning
processes: cognitive conflicts, self-explanations, and analogical encodings. The aim of the
study was to find out possible explanations for the success of the combination condition in
Experiments I and II. The video data provided clear evidence about the benefits of studying
with the real and virtual circuits simultaneously (the combination conditions). Mostly
the representations complemented each other, that is, one representation helped students to
interpret and understand the outcomes they received from the other representation. However,
there were also instances in which analogical encoding took place, that is, situations in
which the slightly discrepant results between the representations âforcedâ students to focus
on those features that could be generalised across the two representations. No statistical differences were found in the amount of experienced cognitive conflicts and self-explanations
between simulation and combination conditions, though in self-explanations there was a
nascent trend in favour of the combination. There was also a clear tendency suggesting that
explicit guidance increased the amount of self-explanations. Overall, the amount of cognitive
conflicts and self-explanations was very low.
The aim of the Study IV was twofold: the main aim was to provide an aggregated overview
of the learning outcomes of experiments I and II; the secondary aim was to explore
the relationship between the learning environments and studentsâ prior domain knowledge
(low and high) in the experiments. Aggregated results of experiments I & II showed that
on average, 91% of the students in the combination environment scored above the average
of the laboratory environment, and 76% of them scored also above the average of the
simulation environment. Seventy percent of the students in the simulation environment
scored above the average of the laboratory environment. The results further showed that
overall students seemed to benefit from combining simulations and laboratories regardless
of their level of prior knowledge, that is, students with either low or high prior knowledge
who studied circuits in the combination environment outperformed their counterparts who
studied in the laboratory or simulation environment alone. The effect seemed to be slightly
bigger among the students with low prior knowledge. However, more detailed inspection
of the results showed that there were considerable differences between the experiments regarding how students with low and high prior knowledge benefitted from the combination:
in Experiment I, especially students with low prior knowledge benefitted from the combination
as compared to those students that used only the simulation, whereas in Experiment
II, only students with high prior knowledge seemed to benefit from the combination relative
to the simulation group. Regarding the differences between simulation and laboratory
groups, the benefits of using a simulation seemed to be slightly higher among students with
high prior knowledge.
The results of the four empirical studies support the hypothesis concerning the benefits
of using simulation along with laboratory activities to promote studentsâ conceptual understanding of electricity. It can be concluded that when teaching students about electricity, the students can gain better understanding when they have an opportunity to use the simulation
and the real circuits in parallel than if they have only the real circuits or only a computer
simulation available, even when the use of the simulation is supported with the explicit
instruction. The outcomes of the empirical studies can be considered as the first unambiguous
evidence on the (additional) benefits of combining laboratory and simulation activities
in science education as compared to learning with laboratories and simulations alone.Siirretty Doriast
A case study : investigating a model that integrates dictionary and polygon pieces in teaching and learning of geometry to grade 8 learners
Considering that geometry is taught according to certain principles that do not encourage creativity, I have decided to employ the mixed methods philosophical framework applying the concurrent transformative design in the form of an exploratory case study. The case study to (i) explore and design a model that influences learning using polygon pieces and mathematics dictionary in the teaching and learning of geometry to grade 8 learners; (ii) investigate if the measurement of angles and sides of polygons using polygon pieces assisted by mathematics dictionary promote learnersâ comprehension of geometry and (iii) investigate how mathematics teachers should use polygon pieces along with mathematics dictionary to teach properties of triangles in order to promote learnersâ conceptual understanding.
Drawing from my research findings a model has been developed from the use of polygon pieces and mathematics dictionary. The model use of mathematics dictionary in teaching and learning geometry is to develop learnersâ mathematics vocabulary and terminology proficiency. Polygon pieces are to enhance the comprehension of geometric concepts.
The quantitative data emerged from marked scripts of the diagnostic and post-intervention tests, the daily reflective tests and intervention activities were analysed as percentages and presented in line and bar graphs. Qualitative data obtained from observation notes and transcribed interviews were analysed in three forms: thematically, constant comparison and keywords in context.
These findings support other research regarding the importance of using physical manipulatives with mathematics dictionary in teaching and learning geometry. They align with other findings that stress that manipulatives are critical facilitating tools for the development of mathematics concepts. The investigations led into the designing of a teaching model for the topic under study for the benefit of the mathematics community in the teaching and learning of geometry, focusing on properties of triangles. The model developed during this study adds to the relatively sparse teaching models but growing theoretical foundation of the field of mathematics.Mathematics EducationPh. D. (Mathematics Education
Proceedings of the tenth international conference Models in developing mathematics education: September 11 - 17, 2009, Dresden, Saxony, Germany
This volume contains the papers presented at the International Conference on âModels in Developing Mathematics Educationâ held from September 11-17, 2009 at The University of Applied Sciences, Dresden, Germany. The Conference was organized jointly by The University of Applied Sciences and The Mathematics Education into the 21st Century Project - a non-commercial international educational project founded in 1986. The Mathematics Education into the 21st Century Project is dedicated to the improvement of mathematics education world-wide through the publication and dissemination of innovative ideas. Many prominent mathematics educators have supported and contributed to the project, including the late Hans Freudental, Andrejs Dunkels and Hilary Shuard, as well as Bruce Meserve and Marilyn Suydam, Alan Osborne and Margaret Kasten, Mogens Niss, Tibor Nemetz, Ubi DâAmbrosio, Brian Wilson, Tatsuro Miwa, Henry Pollack, Werner Blum, Roberto Baldino, Waclaw Zawadowski, and many others throughout the world. Information on our project and its future work can be found on Our Project Home Page http://math.unipa.it/~grim/21project.htm
It has been our pleasure to edit all of the papers for these Proceedings. Not all papers are about research in mathematics education, a number of them report on innovative experiences in the classroom and on new technology. We believe that âmathematics educationâ is fundamentally a âpracticumâ and in order to be âsuccessfulâ all new materials, new ideas and new research must be tested and implemented in the classroom, the real âchalk faceâ of our discipline, and of our profession as mathematics educators. These Proceedings begin with a Plenary Paper and then the contributions of the Principal Authors in alphabetical name order. We sincerely thank all of the contributors for their time and creative effort. It is clear from the variety and quality of the papers that the conference has attracted many innovative mathematics educators from around the world. These Proceedings will therefore be useful in reviewing past work and looking ahead to the future