This document discusses a study that examined the relationship between mathematical reasoning ability and attainment in mathematics.
The study involved 240 students who completed tests of mathematical reasoning ability and attainment. Structural regression modeling showed that four measures of mathematical reasoning ability (class, variable, order, and classification) predicted success on a test of mathematics attainment.
The findings suggest that developing students' mathematical reasoning ability could help improve their attainment in mathematics. Teachers should implement intervention programs to strengthen students' ability to classify, recognize variables, identify orders, and make classifications, which may ultimately boost mathematics learning.
1) The purpose of this study was to examine the relationship between visual static models and students' written solutions to fraction problems using a large sample of student work.
2) The results indicate that common student errors relate to how students interpret the given model or their own model of the situation. Students' flexibility with visual models is related to successful written solutions.
3) Researchers hypothesize that exposure to varied mathematical representations influences students' ability to flexibly use static visual representations. Students need a solid understanding of real-world situations to successfully create and interpret visual models.
Dr. M.THIRUNAVUKKARASU
Research Associate
Department of Education
Bharathidasan University,
Tiruchirappalli - 620 024, Tamil Nadu, India
E-mail: edutechthiru@gmail.com
Dr. S. SENTHILNATHAN
Director (FAC),
UGC - Human Resource Development Centre
(HRDC)
Bharathidasan University
Khajamalai Campus
Tiruchirappalli - 620 023
E-mail: edutechsenthil@gmail.com
The document provides an overview of the K to 12 Mathematics curriculum guide for the Philippines. It outlines the conceptual framework, which identifies critical thinking and problem solving as the twin goals of mathematics education. It also describes the five content areas - numbers and number sense, measurement, geometry, patterns and algebra, and probability and statistics. Additionally, it provides learning standards and competencies for each grade level from grade 1 to grade 10.
This document outlines the adoption of new K-12 mathematics standards for the state of Washington. It discusses the process by which the standards were developed, including input from national experts, state educators, and thousands of stakeholders. The state superintendent officially adopts the revised standards, noting they set more rigorous expectations and provide clarity to support student understanding and problem solving abilities across grades and subjects. The standards are considered the first step in improving mathematics learning statewide.
This study applied a content analysis method to compare how the algebraic topic of using symbols for unknown quantities is presented in elementary school mathematics textbooks from Taiwan (Nani), Singapore (My Pals Are Here!), and Finland (Laskutaito). Specifically, differences in question types (purely mathematical, verbal, visual, or combined representation), contextual versus noncontextual presentation, and pedagogical content design were compared. The findings showed that (1) fewer visual representations are found in Nani compared with the other textbooks; (2) Taiwan uses more contextual problems than the other countries; and (3) the content design in Taiwan focuses on applying the equivalent axiom to solve for unknown quantities, whereas Singapore and Finland use line segments or divide concepts in geometry graphs. In addition, the Singaporean textbooks teach algebraic simplification, providing this topic earlier than the other countries and enabling students to form connections with junior high school learning. Other implications of this study are discussed, and suggestions for future research are provided.
This document outlines the K to 12 mathematics curriculum for grades 1 to 10 in the Philippines. It discusses the goals of critical thinking and problem solving. It describes the content areas of numbers and number sense, measurement, geometry, patterns and algebra, and probability and statistics. It provides the standards and competencies for each grade level, with a focus on applying mathematical concepts to real-life problem solving. Time allotment for mathematics is 4 hours per week for grades 1 to 6 and 50 minutes daily for grades 7 to 10.
Nature and principles of teaching and learning mathJunarie Ramirez
This document discusses effective teaching of mathematics. It outlines three phases of mathematical inquiry: (1) abstraction and symbolic representation, (2) manipulating mathematical statements, and (3) application. It also discusses the nature and principles of teaching mathematics, including that mathematics relies on both logic and creativity. Effective teaching requires understanding what students know and challenging them, as well as using worthwhile tasks to engage them intellectually. Teachers must have mathematical knowledge and commit to students' understanding.
This document provides an overview of the conceptual framework for mathematics education in the Philippines. It discusses the goals of developing critical thinking and problem solving skills. The framework is built around five content areas: numbers and number sense, measurement, geometry, patterns and algebra, and probability and statistics. Specific skills like knowing, estimating, representing, and applying mathematics are also covered. The framework is supported by theories of experiential learning, constructivism, cooperative learning and discovery-based approaches.
1) The purpose of this study was to examine the relationship between visual static models and students' written solutions to fraction problems using a large sample of student work.
2) The results indicate that common student errors relate to how students interpret the given model or their own model of the situation. Students' flexibility with visual models is related to successful written solutions.
3) Researchers hypothesize that exposure to varied mathematical representations influences students' ability to flexibly use static visual representations. Students need a solid understanding of real-world situations to successfully create and interpret visual models.
Dr. M.THIRUNAVUKKARASU
Research Associate
Department of Education
Bharathidasan University,
Tiruchirappalli - 620 024, Tamil Nadu, India
E-mail: edutechthiru@gmail.com
Dr. S. SENTHILNATHAN
Director (FAC),
UGC - Human Resource Development Centre
(HRDC)
Bharathidasan University
Khajamalai Campus
Tiruchirappalli - 620 023
E-mail: edutechsenthil@gmail.com
The document provides an overview of the K to 12 Mathematics curriculum guide for the Philippines. It outlines the conceptual framework, which identifies critical thinking and problem solving as the twin goals of mathematics education. It also describes the five content areas - numbers and number sense, measurement, geometry, patterns and algebra, and probability and statistics. Additionally, it provides learning standards and competencies for each grade level from grade 1 to grade 10.
This document outlines the adoption of new K-12 mathematics standards for the state of Washington. It discusses the process by which the standards were developed, including input from national experts, state educators, and thousands of stakeholders. The state superintendent officially adopts the revised standards, noting they set more rigorous expectations and provide clarity to support student understanding and problem solving abilities across grades and subjects. The standards are considered the first step in improving mathematics learning statewide.
This study applied a content analysis method to compare how the algebraic topic of using symbols for unknown quantities is presented in elementary school mathematics textbooks from Taiwan (Nani), Singapore (My Pals Are Here!), and Finland (Laskutaito). Specifically, differences in question types (purely mathematical, verbal, visual, or combined representation), contextual versus noncontextual presentation, and pedagogical content design were compared. The findings showed that (1) fewer visual representations are found in Nani compared with the other textbooks; (2) Taiwan uses more contextual problems than the other countries; and (3) the content design in Taiwan focuses on applying the equivalent axiom to solve for unknown quantities, whereas Singapore and Finland use line segments or divide concepts in geometry graphs. In addition, the Singaporean textbooks teach algebraic simplification, providing this topic earlier than the other countries and enabling students to form connections with junior high school learning. Other implications of this study are discussed, and suggestions for future research are provided.
This document outlines the K to 12 mathematics curriculum for grades 1 to 10 in the Philippines. It discusses the goals of critical thinking and problem solving. It describes the content areas of numbers and number sense, measurement, geometry, patterns and algebra, and probability and statistics. It provides the standards and competencies for each grade level, with a focus on applying mathematical concepts to real-life problem solving. Time allotment for mathematics is 4 hours per week for grades 1 to 6 and 50 minutes daily for grades 7 to 10.
Nature and principles of teaching and learning mathJunarie Ramirez
This document discusses effective teaching of mathematics. It outlines three phases of mathematical inquiry: (1) abstraction and symbolic representation, (2) manipulating mathematical statements, and (3) application. It also discusses the nature and principles of teaching mathematics, including that mathematics relies on both logic and creativity. Effective teaching requires understanding what students know and challenging them, as well as using worthwhile tasks to engage them intellectually. Teachers must have mathematical knowledge and commit to students' understanding.
This document provides an overview of the conceptual framework for mathematics education in the Philippines. It discusses the goals of developing critical thinking and problem solving skills. The framework is built around five content areas: numbers and number sense, measurement, geometry, patterns and algebra, and probability and statistics. Specific skills like knowing, estimating, representing, and applying mathematics are also covered. The framework is supported by theories of experiential learning, constructivism, cooperative learning and discovery-based approaches.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
The Effect of the Concrete-Representational-Abstract Mathematical Sequence o...Janet Van Heck
This document describes a study that examined the effects of using the concrete-representational-abstract (CRA) teaching sequence with explicit instruction to teach addition skills from 0 to 9 to kindergarten students struggling in math. The CRA sequence begins by using manipulatives, then representations like pictures, before moving to abstract problems. Three kindergarten students identified as needing math support through RTI screening were given scripted CRA lessons and their test scores were measured. The study found the CRA sequence improved students' conceptual understanding and performance on addition facts from 0 to 9 when delivered with explicit instruction.
A research on teachers’ beliefs about mathematics teaching and learning betwe...Alexander Decker
This document discusses a research study that examined the beliefs about mathematics teaching and learning held by primary school teachers in Malaysia. A survey was administered to 174 teachers from Sekolah Kebangsaan (SK), Sekolah Jenis Kebangsaan Cina (SJKC), and Sekolah Jenis Kebangsaan Tamil (SJKT) to determine their beliefs regarding student learning, stages of learning, and teaching practices. The results indicated that most teachers' beliefs fell in the mid-range between formal and informal beliefs. Statistical analysis found significant differences in beliefs between the teacher groups, with SJKC teachers favoring more informal beliefs. The conclusions suggest teacher professional development should address fundamental beliefs to promote more
Impact of nce mathematics trainee and graduate teachers onAlexander Decker
This study examined the impact of NCE trainee and graduate mathematics teachers on the achievement of JSSII students in geometry in Katsina-Ala, Benue State, Nigeria. 320 students from 8 secondary schools were sampled using random sampling. A pre-test, post-test design was used where intact classes were randomly assigned to trainee and graduate teachers. Students taught by NCE graduate teachers achieved higher in geometry than those taught by trainees. The study also found that female students achieved at the same level as males, regardless of whether they were taught by trainees or graduates. It was recommended that more attention be paid to training NCE trainee teachers and supporting female students in geometry.
The purpose of this study is to determine the mathematics teacher performance category in building high order thinking skills (HOTS) of students. The study was conducted on 560 students taken randomly from ten junior high schools and eight high schools from eight districts in North Sumatra Province. Data collection techniques and instruments are carried out by giving questionnaires to students which contain a number of questions about students' assessment of the mathematics teacher's performance in constructing the HOTS indicator. Based on descriptive analysis, it was found that the performance of mathematics teachers built HOTS indicators, namely (1) understanding of concepts, (2) mathematical communication, (3) creativity, (4) problem solving, and (5) reasoning is enough category. The results of analysis of variance show that teacher performance builds (1) understanding of concepts, (2) mathematical communication, (3) creativity, (4) problem solving, and (5) reasoning significantly influences students' abilities.
The document provides the curriculum guide for mathematics from grades 7 to 10 in the Philippines. It outlines the conceptual framework, content standards, performance standards, and learning competencies for patterns and algebra in grade 8. Specifically, it covers factors of polynomials, rational algebraic expressions, linear equations and inequalities in two variables, and systems of linear equations and inequalities in two variables. It also provides the time allotment and describes the expectations for student understanding and demonstration of key concepts in this strand.
This document outlines the K to 12 curriculum guide for mathematics III. It includes 18 performance standards covering a range of math concepts from numbers and operations to measurement, geometry, patterns, and data analysis. It also provides examples of strategies and games to help students learn the concepts, including number hunt scavenger games, a "Go Fish" card game focused on addition facts, an activity practicing money skills, and a card game for learning division skills.
The document is the K to 12 Curriculum Guide for Mathematics from Kindergarten to Grade 10 in the Philippines. It outlines the conceptual framework, brief course description, learning area standards, key stage standards, grade level standards, and sample grade level content for Grade 1. The goals of mathematics education are developing critical thinking and problem solving skills. Key concepts include numbers, measurement, geometry, patterns and algebra, and probability and statistics. The curriculum is based on theories of experiential learning, constructivism, and inquiry-based learning.
The document is a curriculum guide for mathematics from grades K to 10 in the Philippines. It outlines the goals of mathematics education as developing critical thinking and problem solving skills. It describes the conceptual framework, which includes 5 content areas: numbers and number sense, measurement, geometry, patterns and algebra, and probability and statistics. It also discusses learning theories that support the curriculum and standards for different grade levels. The guide provides an overview of the mathematics curriculum and expected learning outcomes for students from kindergarten through 10th grade.
This document outlines the conceptual framework and curriculum guide for mathematics education in the Philippines from Kindergarten to Grade 10. The goals of mathematics education are developing critical thinking and problem solving skills. The curriculum covers 5 content areas - numbers, measurement, geometry, patterns and algebra, and probability and statistics. It is grounded in theories of experiential learning, reflective learning, constructivism, cooperative learning, and discovery learning. The curriculum guide describes the learning standards and expectations for each key stage of learning.
The document provides an introduction to the Common Core Georgia Performance Standards for Mathematics. It discusses the focus of the Georgia mathematics curriculum on developing conceptual understanding through problem solving, representation, reasoning and communication. It emphasizes applying math concepts in authentic contexts. The standards are designed to increase coherence and focus from grade to grade to improve math achievement. They define what students should understand and be able to do at each grade level.
An Exploratory Study Of A Story Problem Assessment Understanding Children S ...Angie Miller
This study explored how students use their number sense when solving story problems by analyzing their strategies and reasoning. Researchers interviewed three 8- and 9-year-old students and had them solve story problems. Through qualitative coding, the researchers analyzed how the students' number sense was reflected in their problem-solving approaches and explanations. The results provided insights into how students coordinate different number sense constructs and showed flexibility in problem-solving. The study aimed to understand number sense in a holistic way rather than by assessing individual constructs in isolation.
Enhancing Pupils’ Knowledge of Mathematical Concepts through Game and Poemiosrjce
The study investigated the effects of game and poem enhanced instruction on pupils’ knowledge of
mathematics concepts in mathematics (Fractions and decimals, Volume of, cylinder; triangular prisms and
sphere; Capacity and Weight). A total of 344 pupils from twelve (12) public primary schools of Ogbia and
Yenagoa Local Government Areas of Bayelsa State, Nigeria were involved in the study. A pretest-posttest
control group quasi-experimental design was adopted in the research. The moderating effects of gender were
also examined on the independent and dependent variables. Using ANCOVA statistics, the results revealed that
pupils exposed to game and poem enhanced instruction have better mean score in knowledge of mathematics
concepts than the control groups. Poem was found to be more effective in improving pupils’ knowledge of
mathematics concepts than game and a non-significant gender difference existed in the mean scores in
knowledge of mathematics concepts. There was no significant interaction effect of gender on pupils’ knowledge
of mathematics concepts. The implication of the findings was discussed and appropriate recommendations were
made.
A Pair-Wise Analysis Of The Cognitive Demand Levels Of Mathematical Tasks Use...Joe Andelija
This study analyzed the cognitive demand levels of 66 paired mathematical tasks used for classroom instruction and homework assignments in an 8th grade classroom over one school year. The study found that approximately two-thirds of the time, the cognitive demand levels of the mathematical tasks assigned for homework differed from those used during classroom instruction. The implications for student learning, teaching practices, homework, and further research are discussed.
Assessing Algebraic Solving Ability Of Form Four StudentsSteven Wallach
This study aimed to assess Form Four students' algebraic solving ability using linear equations across four content domains: linear patterns, direct variation, concepts of functions, and arithmetic sequences. Students took a pencil-and-paper test consisting of items assessing different levels of algebraic understanding based on the SOLO model. Clinical interviews were also conducted. Results showed that 62% of students had less than 50% probability of success at the relational level. Most students were at the unistructural or multistructural levels. High-ability students could identify patterns and relationships between variables more easily than low-ability students. The study provided evidence that the SOLO model is useful for assessing algebraic solving ability in upper secondary school.
Algebraic Thinking Of Grade 8 Students In Solving Word Problems With A Spread...Holly Fisher
The document summarizes a study that examined how 8th grade students solved two word problems using a spreadsheet. The problems involved algebraic thinking through inequalities. Students' representations in the spreadsheet allowed researchers to analyze the algebraic reasoning involved. The spreadsheet helped students treat problems formally impractical for their algebra knowledge through its computational logic.
Understand addition through modelling and manipulation of concrete materialsAlexander Decker
1) The document discusses a case study that integrated mathematical modelling with the manipulation of concrete materials to help 9-year-old students with learning difficulties develop conceptual understanding of addition.
2) Observations revealed that when teachers used explicit instruction in modelling activities it hampered students' conceptual understanding and process skills, but when students initiated modelling themselves they showed improvement.
3) Therefore, the document recommends that teachers should guide students with learning difficulties to participate in and initiate modelling activities rather than just following procedures demonstrated by the teacher.
Analysis of gender related differential item functioning in mathematics multi...Alexander Decker
The document analyzes gender-related differential item functioning in mathematics multiple choice items administered by the West African Examination Council (WAEC) in 2011 in Nigeria. The study found that 39 of the 50 test items showed statistically significant differences in performance between male and female examinees, indicating those items were functioning differently based on gender. The results suggest the need for examination bodies like WAEC to develop test items that do not show such differential functioning and are fair to both male and female examinees.
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
The Effect of the Concrete-Representational-Abstract Mathematical Sequence o...Janet Van Heck
This document describes a study that examined the effects of using the concrete-representational-abstract (CRA) teaching sequence with explicit instruction to teach addition skills from 0 to 9 to kindergarten students struggling in math. The CRA sequence begins by using manipulatives, then representations like pictures, before moving to abstract problems. Three kindergarten students identified as needing math support through RTI screening were given scripted CRA lessons and their test scores were measured. The study found the CRA sequence improved students' conceptual understanding and performance on addition facts from 0 to 9 when delivered with explicit instruction.
A research on teachers’ beliefs about mathematics teaching and learning betwe...Alexander Decker
This document discusses a research study that examined the beliefs about mathematics teaching and learning held by primary school teachers in Malaysia. A survey was administered to 174 teachers from Sekolah Kebangsaan (SK), Sekolah Jenis Kebangsaan Cina (SJKC), and Sekolah Jenis Kebangsaan Tamil (SJKT) to determine their beliefs regarding student learning, stages of learning, and teaching practices. The results indicated that most teachers' beliefs fell in the mid-range between formal and informal beliefs. Statistical analysis found significant differences in beliefs between the teacher groups, with SJKC teachers favoring more informal beliefs. The conclusions suggest teacher professional development should address fundamental beliefs to promote more
Impact of nce mathematics trainee and graduate teachers onAlexander Decker
This study examined the impact of NCE trainee and graduate mathematics teachers on the achievement of JSSII students in geometry in Katsina-Ala, Benue State, Nigeria. 320 students from 8 secondary schools were sampled using random sampling. A pre-test, post-test design was used where intact classes were randomly assigned to trainee and graduate teachers. Students taught by NCE graduate teachers achieved higher in geometry than those taught by trainees. The study also found that female students achieved at the same level as males, regardless of whether they were taught by trainees or graduates. It was recommended that more attention be paid to training NCE trainee teachers and supporting female students in geometry.
The purpose of this study is to determine the mathematics teacher performance category in building high order thinking skills (HOTS) of students. The study was conducted on 560 students taken randomly from ten junior high schools and eight high schools from eight districts in North Sumatra Province. Data collection techniques and instruments are carried out by giving questionnaires to students which contain a number of questions about students' assessment of the mathematics teacher's performance in constructing the HOTS indicator. Based on descriptive analysis, it was found that the performance of mathematics teachers built HOTS indicators, namely (1) understanding of concepts, (2) mathematical communication, (3) creativity, (4) problem solving, and (5) reasoning is enough category. The results of analysis of variance show that teacher performance builds (1) understanding of concepts, (2) mathematical communication, (3) creativity, (4) problem solving, and (5) reasoning significantly influences students' abilities.
The document provides the curriculum guide for mathematics from grades 7 to 10 in the Philippines. It outlines the conceptual framework, content standards, performance standards, and learning competencies for patterns and algebra in grade 8. Specifically, it covers factors of polynomials, rational algebraic expressions, linear equations and inequalities in two variables, and systems of linear equations and inequalities in two variables. It also provides the time allotment and describes the expectations for student understanding and demonstration of key concepts in this strand.
This document outlines the K to 12 curriculum guide for mathematics III. It includes 18 performance standards covering a range of math concepts from numbers and operations to measurement, geometry, patterns, and data analysis. It also provides examples of strategies and games to help students learn the concepts, including number hunt scavenger games, a "Go Fish" card game focused on addition facts, an activity practicing money skills, and a card game for learning division skills.
The document is the K to 12 Curriculum Guide for Mathematics from Kindergarten to Grade 10 in the Philippines. It outlines the conceptual framework, brief course description, learning area standards, key stage standards, grade level standards, and sample grade level content for Grade 1. The goals of mathematics education are developing critical thinking and problem solving skills. Key concepts include numbers, measurement, geometry, patterns and algebra, and probability and statistics. The curriculum is based on theories of experiential learning, constructivism, and inquiry-based learning.
The document is a curriculum guide for mathematics from grades K to 10 in the Philippines. It outlines the goals of mathematics education as developing critical thinking and problem solving skills. It describes the conceptual framework, which includes 5 content areas: numbers and number sense, measurement, geometry, patterns and algebra, and probability and statistics. It also discusses learning theories that support the curriculum and standards for different grade levels. The guide provides an overview of the mathematics curriculum and expected learning outcomes for students from kindergarten through 10th grade.
This document outlines the conceptual framework and curriculum guide for mathematics education in the Philippines from Kindergarten to Grade 10. The goals of mathematics education are developing critical thinking and problem solving skills. The curriculum covers 5 content areas - numbers, measurement, geometry, patterns and algebra, and probability and statistics. It is grounded in theories of experiential learning, reflective learning, constructivism, cooperative learning, and discovery learning. The curriculum guide describes the learning standards and expectations for each key stage of learning.
The document provides an introduction to the Common Core Georgia Performance Standards for Mathematics. It discusses the focus of the Georgia mathematics curriculum on developing conceptual understanding through problem solving, representation, reasoning and communication. It emphasizes applying math concepts in authentic contexts. The standards are designed to increase coherence and focus from grade to grade to improve math achievement. They define what students should understand and be able to do at each grade level.
An Exploratory Study Of A Story Problem Assessment Understanding Children S ...Angie Miller
This study explored how students use their number sense when solving story problems by analyzing their strategies and reasoning. Researchers interviewed three 8- and 9-year-old students and had them solve story problems. Through qualitative coding, the researchers analyzed how the students' number sense was reflected in their problem-solving approaches and explanations. The results provided insights into how students coordinate different number sense constructs and showed flexibility in problem-solving. The study aimed to understand number sense in a holistic way rather than by assessing individual constructs in isolation.
Enhancing Pupils’ Knowledge of Mathematical Concepts through Game and Poemiosrjce
The study investigated the effects of game and poem enhanced instruction on pupils’ knowledge of
mathematics concepts in mathematics (Fractions and decimals, Volume of, cylinder; triangular prisms and
sphere; Capacity and Weight). A total of 344 pupils from twelve (12) public primary schools of Ogbia and
Yenagoa Local Government Areas of Bayelsa State, Nigeria were involved in the study. A pretest-posttest
control group quasi-experimental design was adopted in the research. The moderating effects of gender were
also examined on the independent and dependent variables. Using ANCOVA statistics, the results revealed that
pupils exposed to game and poem enhanced instruction have better mean score in knowledge of mathematics
concepts than the control groups. Poem was found to be more effective in improving pupils’ knowledge of
mathematics concepts than game and a non-significant gender difference existed in the mean scores in
knowledge of mathematics concepts. There was no significant interaction effect of gender on pupils’ knowledge
of mathematics concepts. The implication of the findings was discussed and appropriate recommendations were
made.
A Pair-Wise Analysis Of The Cognitive Demand Levels Of Mathematical Tasks Use...Joe Andelija
This study analyzed the cognitive demand levels of 66 paired mathematical tasks used for classroom instruction and homework assignments in an 8th grade classroom over one school year. The study found that approximately two-thirds of the time, the cognitive demand levels of the mathematical tasks assigned for homework differed from those used during classroom instruction. The implications for student learning, teaching practices, homework, and further research are discussed.
Assessing Algebraic Solving Ability Of Form Four StudentsSteven Wallach
This study aimed to assess Form Four students' algebraic solving ability using linear equations across four content domains: linear patterns, direct variation, concepts of functions, and arithmetic sequences. Students took a pencil-and-paper test consisting of items assessing different levels of algebraic understanding based on the SOLO model. Clinical interviews were also conducted. Results showed that 62% of students had less than 50% probability of success at the relational level. Most students were at the unistructural or multistructural levels. High-ability students could identify patterns and relationships between variables more easily than low-ability students. The study provided evidence that the SOLO model is useful for assessing algebraic solving ability in upper secondary school.
Algebraic Thinking Of Grade 8 Students In Solving Word Problems With A Spread...Holly Fisher
The document summarizes a study that examined how 8th grade students solved two word problems using a spreadsheet. The problems involved algebraic thinking through inequalities. Students' representations in the spreadsheet allowed researchers to analyze the algebraic reasoning involved. The spreadsheet helped students treat problems formally impractical for their algebra knowledge through its computational logic.
Understand addition through modelling and manipulation of concrete materialsAlexander Decker
1) The document discusses a case study that integrated mathematical modelling with the manipulation of concrete materials to help 9-year-old students with learning difficulties develop conceptual understanding of addition.
2) Observations revealed that when teachers used explicit instruction in modelling activities it hampered students' conceptual understanding and process skills, but when students initiated modelling themselves they showed improvement.
3) Therefore, the document recommends that teachers should guide students with learning difficulties to participate in and initiate modelling activities rather than just following procedures demonstrated by the teacher.
Analysis of gender related differential item functioning in mathematics multi...Alexander Decker
The document analyzes gender-related differential item functioning in mathematics multiple choice items administered by the West African Examination Council (WAEC) in 2011 in Nigeria. The study found that 39 of the 50 test items showed statistically significant differences in performance between male and female examinees, indicating those items were functioning differently based on gender. The results suggest the need for examination bodies like WAEC to develop test items that do not show such differential functioning and are fair to both male and female examinees.
An Analysis Of Knowledge In STEM Solving Algebraic ProblemsAmy Isleb
This study assessed undergraduate students' abilities in solving algebraic word problems and their attitudes towards STEM. 63 undergraduate mathematics education students participated. Students took tests on algebraic knowledge and applying algebra in science contexts, and a questionnaire on STEM attitudes. Results showed most students had low levels of algebraic knowledge and applying algebra to science. However, most students had positive attitudes towards STEM. The study aimed to understand the relationship between algebraic abilities, STEM attitudes, and performance.
Assessing Multiplicative Thinking Using Rich TasksMaria Perkins
The document describes the development of a learning and assessment framework to evaluate students' multiplicative thinking skills in Years 4 to 8. Researchers designed rich assessment tasks and administered them to nearly 3,500 students. The tasks were based on an initial hypothetical learning trajectory identified from the literature. Student response data were used to refine the tasks, scoring rubrics, and a final learning and assessment framework consisting of nine levels of increasing complexity in multiplicative thinking. Sample assessment tasks, such as a multi-part "Butterfly House" problem, and corresponding scoring rubrics are provided as illustrations.
This document discusses frameworks for children's mathematical thinking and the importance of experiences in developing conceptual understanding of mathematics. It argues that experiences should illustrate concepts in multiple contexts to build representations and connections. Concrete materials can help students understand concepts but must be closely linked to the intended concept. The teacher's role is to provide engaging experiences that emphasize conceptual understanding and a positive attitude towards math.
READING COMPREHENSION AND PROBLEM SOLVING SKILLS OF GRADE SEVENSTUDENTS: A MI...AJHSSR Journal
ABSTRACT: The purpose of this study was to determine the relationship between the extent of students‟
reading comprehension and problem solving skills and identify teaching strategies that would address the
problem in teaching problem solving in Mathematics. The research utilized mixed explanatory design. The
subject consists of 189 grade 7 students who were part of the general section enrolled at Davao City National
High School. Purposive sampling was used in identifying the respondents taking the reading comprehension test
and problem solving test while random sampling was used in identifying participants for the key informant
interview. The result of the study revealed that students reading comprehension and problem solving skills were
at developing level. Moreover, reading comprehension skill was a predictor of problem solving skill. This
means that students‟ problem solving skill is dependent on their reading skills. Results also showed from the
conducted focus group discussion that students gave importance to vocabulary and main idea in learning
problem solving. Furthermore, using differentiated instruction was the identified best teaching strategy to
understand problem solving.
This document discusses the development and testing of an algo-heuristic model to improve statistics learning among elementary school teacher education students. It involved three phases: 1) expert evaluation of the model, 2) testing with lecturers, and 3) pre-experimental testing with students. Test results found a significant increase in student test scores after using the model, indicating it can effectively improve statistics learning. The document also reviews various literature on algo-heuristic learning theories and algorithms.
This document summarizes the results of a longitudinal study on introducing algebraic concepts to elementary school students. The study involved over 70 students from grades 2-4 participating in weekly algebraic activities over 2.5 years. Key findings include:
1) Students were able to understand elementary algebraic ideas such as thinking of arithmetic operations as functions, using the number line to represent numbers and operations, and accepting negative numbers.
2) Students could work with algebraic notation like mappings and expressions, represent relationships involving unknown quantities, and solve problems using multiple representations like tables, graphs, and equations.
3) By the end of 4th grade, students in the study group significantly outperformed a control group on tasks involving equations, functions,
Assessing Algebraic Solving Ability A Theoretical FrameworkAmy Cernava
This document presents a theoretical framework for assessing algebraic solving ability. It discusses three phases of algebraic processes: 1) investigating patterns in numerical data, 2) representing patterns in tables and equations, and 3) interpreting and applying equations to new situations. It also reviews different approaches for developing algebraic solving ability in students, such as game-based learning, building conceptual understanding of variables/functions, and representing problems in multiple ways. The theoretical framework is based on these algebraic processes and how they align with levels of the SOLO model for assessing student understanding.
REALITY – BASED INSTRUCTION AND SOLVING WORD PROBLEMS INVOLVING SUBTRACTIONWayneRavi
This study was conducted to determine the effect of reality based on the solving word problems involving subtraction. Descriptive-Comparative research design using paired sample T-test was used to utilized in the study. The study was carried out in Tibungol Elementary School to Fifty student of Grade Three section 1. Results revealed that there was a significant difference on the pretest and post test scores of pupils in reality based approach. Further, the reality based approach is effective in improving the performance of student.
A scenario based learning of electrical circuitsAlexander Decker
This document presents a framework for teaching electrical circuits using scenario-based learning. The framework consists of 5 activities where student teams use simulation software to explore series, parallel, and combination circuits. Through guided discovery and discussion, students learn the concepts of voltage, current, and resistance in different circuit types. The goal is for students to develop problem-solving skills and gain a conceptual understanding of electrical principles. Assessment involves discussion as students compare their predictions and findings. The scenario is designed using constructivist and cooperative learning principles to actively engage students in building knowledge through hands-on exploration and peer collaboration.
This study investigated the effects of using a graphical calculator (Microsoft Math Tool) on students' performance in linear functions. Ninety-eight students from two schools participated, with one group receiving instruction using the graphical calculator (experimental group) and the other receiving traditional lecture-based instruction (control group). Pre-tests showed no significant difference between the groups. Post-test results revealed significantly higher scores for the experimental group compared to the control group, suggesting that using graphical calculators can improve students' understanding of linear functions and performance on related tasks. The study recommends integrating graphical calculators into mathematics instruction.
The document provides an introduction to the Common Core Georgia Performance Standards for mathematics. It discusses how the standards focus on applying mathematical concepts in authentic problems, problem solving, reasoning and representation. It emphasizes concentrating early learning on numbers, measurement and geometry with less emphasis on data analysis and algebra. The standards aim to present mathematics concepts clearly and specifically to improve focus and coherence.
A General Analytical Model For Problem Solving Teaching BoSDereck Downing
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Modelling the relationship between mathematical reasoning ability and mathematics attainment
1. Journal of Education and Practice www.iiste.org
ISSN 2222-1735 (Paper) ISSN 2222-288X (Online)
Vol.4, No.17, 2013
54
Modelling the Relationship between Mathematical Reasoning
Ability and Mathematics Attainment
Benson Adesina Adegoke
Institute of Education, University of Ibadan
E-mail: doctoradegoke@yahoo.com
Abstract
In this article, the author seeks to explain the indicators of mathematical reasoning ability and examines the
relationship, using structural regression modeling technique, between mathematical reasoning ability and
students’ attainment in mathematics.
The sample consisted of 240 Senior Secondary School One Students (Age 14-16 years) who were randomly
selected from four senior secondary schools in Isokan and Irewole Local Government Areas of Osun State,
Nigeria. A 24-item Mathematics Reasoning Ability Test (MRAT) and a 36-item Attainment in Mathematics Test
(AMT) were constructed to explain the indicators of mathematical reasoning ability and to assess how far these
indicators enter into success in school mathematics work. Students’ scores in MRAT and AMT were analysed by
using maximum likelihood estimates of LISREL version 8.88. Results from the structural regression model
showed that four fundamental notions (viz: class, variable, order and classification) were measures of
mathematical reasoning ability and two, success in mathematics reasoning ability reliably predicted success in
mathematics attainment. These findings suggest the need for mathematics teachers to mount intervention
programmes that will help students develop and improve their mathematical reasoning ability and ultimately
improve their attainment in mathematics.
Keywords: Mathematical reasoning ability; Attainment in Mathematics; Mathematics learning in Nigeria;
Structural regression modeling
1. Introduction
Mathematics is an excellent vehicle for the development and improvement of a person’s intellectual competence
in logical reasoning, spatial visualization, analysis and abstract thought. Students develop numeracy, reasoning,
thinking skills, and problem solving skills through the learning and application of mathematics. These are valued
not only in science and technology, but also in everyday living and in the workplace. The development of a
highly skilled scientifically- and technologically `based manpower requires a strong grounding in mathematics.
The Federal Government of Nigeria realizes that she needs citizens who can demonstrate adequate knowledge
of Mathematics in order for the country to cope with increasing complex nature of economic activities in the
world of today. This perhaps suggests why mathematics is made a compulsory subject for all students at the
secondary school level in Nigeria. This is reflected in the Nigerian National Policy on Education (Federal
Government of Nigeria, [FGN], 2004). No doubt, an emphasis on functional mathematics education will ensure
that Nigeria has an increasingly competitive workforce to meet the challenges of the 21st century.
However, despite the importance of mathematics, secondary school students’ level of achievement, continued
to be low (Uwudiae, 2012). On the average, between 2009 and 2012, less than 50% of the students who sat for
Mathematics in the Senior Secondary School Certificate Examination (SSSCE) conducted by West African
Examination Council (WAEC) and National Examination Council (NECO) obtained a minimum of credit pass in
Mathematics. The poor trend in achievement in mathematics among secondary school students continues to
attract attention from the major stakeholders (parents, researchers, teachers, and examining bodies) in education
in Nigeria (Uwadiae, 2012).
In their effort to improve students’ level of achievement in mathematics, researchers (Adegoke, 2011; Awofala,
Awoyemi, Fatade, & Nneji, 2012) have made several suggestions. Among such suggestions include the adoption
of integrative teaching method, counseling strategies and use of indigenous language in the teaching of
mathematics (Adegoke, 2011). Despite all these suggestions, little or no improvement has been observed. In the
on-going search for ways of improving success in mathematics, a look at the influence of mathematical
reasoning ability on students’ achievement success in mathematics may suggest intervention programmes that
may be embarked upon by mathematics teachers. In fact, some studies such as Choudhury and Das, 2012; Heng
– Yuku and Sullivan (2000) and Nunes, Byrant, Barnes, and Sylva (2012) have suggested that there is a link
between mathematical reasoning ability and attainment in mathematics. Specifically Nunes, Byrant, Barnes, and
Sylva (2012) in their study found out that mathematical reasoning ability reliably predicted students’
achievement in mathematics. Similarly, Choudhury and Das, 2012 in their study in Malaysia found out that
geometrical ability (ability to reason with spatial figures) was a good predictor of students’ achievement in
mathematics. Results of these studies have suggested the need for changes in the curriculum of mathematics.
2. Journal of Education and Practice www.iiste.org
ISSN 2222-1735 (Paper) ISSN 2222-288X (Online)
Vol.4, No.17, 2013
55
In order to place this study in a very clear perspective, it is important to distinguish between ability (potential
success) and attainment (actual success) in the study of any subject. According to Lee (1967) ability to succeed
in a subject was defined as the power of an individual to grasp and manipulate the fundamental notions required
for its study while attainment in the study of a subject, on the other hand, was taken as the acquired degree of
mastery over the complexities of its conventional structure, starting from the simplest beginnings and constantly
proceeding to more difficult work. Under this definition, ability is related to school work and is certainly
necessary for achievement in it. This is so, because, some grasp of fundamental notions (though not explicitly
demanded) is implicitly contained in the study of any subject. Lee further emphasized that attainment in a school
subject entails a sequence of work, which rests on a prior acceptance of certain basic elements, concepts, and
thought. At the school level, it therefore, seemed probable that an assessment of the student’s success in grasping
these basic notions might prove a useful pointer to his or her likely success in the study of the subject and hence
of an estimate of his or her ability in the subject itself.
One of the earliest works on mathematical reasoning ability was carried out by Brown in 1916 (Adegoke,
2003). However, Brown’s (1961) work had, in reality, judged mathematical attainment and not mathematical
reasoning ability. Most of the items contained in Brown’s work reflected normal class work on mathematics. The
later works of Hamley in 1934 (Adegoke, 2003) provided a basis for mathematical tests which were unrelated to
school work. A succession of small pieces of work (e.g. Blackwell, 1940, Bennett, 1948; Jenkins, 1939) and
which had followed Hamley’s work suggested that such tests could function as mathematics reasoning ability
tests.
Hamley (Adegoke, 2003) hypothesized that three stages could be distinguished in the process of mathematical
reasoning. These are;
a) Dividing the given material into classes, each determined by some unifying characteristics.
b) Detecting a prevailing order within each class.
c) Seeking out correspondence of relationship between the members of two or more classes.
Hamley felt that the ability to succeed in the study of mathematics could correspond to the ability to carry out
these three processes of classification, order, and recognition of correspondences in dealing with the simple
materials which is basic to mathematics, namely, arithmetical numbers, algebraic symbols and spatial figures.
Jenkins (1939) introduced an additional stage into this scheme of mathematical reasoning, when he suggested
that the ability to recognise some constant variable within a group preceded the identification of that group as a
class. From the analysis of the batteries of tests constructed and administered to groups of children aged 13 to 15
years in London, Lee (1967) showed that there was sufficient evidence to suggest that the tests actually
measured mathematical ability rather than attainment. Choudhury and Das (2012), Berrett and Williams (1997)
Fischbein and Nachieli (1998) and Adegoke (2003) used the ideas contained in Hamley’s and Jenkin’s works to
develop their mathematical reasoning ability tests.
The works of Hamley and Jenkins (Adegoke, 2003) rest on the assumptions that the fundamental
notions of mathematics are of two kinds:
a) The notions of arithmetical numbers, algebraic symbols and spatial figures. The study of mathematics
cannot proceed at all without these, for they provide the material with which mathematics work. They also
designate the subject field called mathematics.
b) The notions of variable, class, order, and correspondence. These notions play a large part in
mathematical work, since most problems are solved by recognizing classes amongst given data, creating orders
within these classes and picking out correspondences to lead to unique conclusions.
A variable is a quantity which can take different values although its basic structure remains the same
throughout; an even number is a variable, since there are many such numbers all having the property of exact
divisibility by two. A class is a group of quantities having a common characteristics, a group of even numbers is
thus a class, the common characteristics being the even-ness of the numbers or the property that each is exactly
divisible by two. An order is obtained when quatities are arranged in sequence according to a fixed rule or law;
thus even number can be arranged in ascending or descending order. A correspondence is obtained when two
ordered classes of quantities are placed side by side in such a way that each pair is bound by a law which can be
enunciated, thus, the sequence of odd and even numbers show a correspondence in which each member of one
class differs from the corresponding member of the other class.
The notions of variable, class, order and correspondence can be seen in every phase of mathematical work,
and hence can be regarded as the fundamentals required for the study of mathematics. Together with the notions
of arithmetical numbers, algebraic symbols and spatial figures, they provide the basis for tests of mathematical
ability (Lee, 1967). From the psychological angle, this is equivalent to the hypothesis that four stages can be
distinguished in the dynamic process of mathematical reasoning, namely those of recognizing variable,
classifying, ordering and recognising correspondences when solving school mathematical problems. The first
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objective of this study was, therefore, to determine the extent to which these four dynamic processes are
indicators and sufficient measures of mathematical reasoning ability.
In Nigeria, the curriculum of secondary school subjects is normally prepared by the Nigerian Educational
Research and Development Council (NERDC). The mathematics curriculum as prepared by NERDC in 2005
consists of four major sections. These are Number and Numerations, Algebra, Geometry, and Statistics. Scrutiny
of the curriculum shows that within these four sections, a total of about fifty distinct topics are taught including
percentages in arithmetic, quadratic equations in algebra, circles theorems in geometry and measures of central
tendency, dispersion, and probability theories in statistics. These are the fundamental concepts of mathematics
which Nigerian students must learn during their secondary school education. Students’ level of attainment in
mathematics is the extent to which they are able to master these fundamental concepts of mathematics. In this
study emphasis was, however, on three sections, namely, Number and Numerations, Algebra, and Geometry.
This is because, this study centers on senior secondary school one students (Ages 14 - 16). In Nigeria, usually,
in-depth study of statistics starts with students in senior secondary two (Ages 16 -18).
In this study, therefore, the second objective was to present structural model of the linkages existing between
the fundamental notions of mathematical reasoning ability and attainment in mathematics. In order to study the
relationships between the fundamental notions of mathematical reasoning ability and attainment in mathematics
the author developed a Structural Regression model (SR) (see Figure 1). SR is a variant of structural equation
models. The other examples are Path Analysis (PA) and Confirmatory Factor Analysis (CFA) (See Adegoke,
2012; Kline, 2005 for more detailed information on Structural Equation Modelling). An SR model results from
the synthesis of path model (PA) and CFA measurement models.
Figure 1: Structural regression model of Mathematics reasoning ability and Mathematics Achievement.
As in PA, SR allows test of hypotheses about direct and indirect causal effects. However, unlike PA, these
effects incorporate a measurement component that represents observed variables as indicators of underlying
factors as in CFA. In SR models, the researcher can test hypotheses about structural and measurement relations
within a single model (Adegoke, 2012; Kline, 2005).
In figure 1, MRA and AM are factors or latent traits and they are not directly measured but rather assessed
indirectly using scores from the indicators. For example, MRA is assessed through students’ ability to recognize
the four processes of recognition of variable (VAR) classification (CLA), order (ROO), and recognition of
correspondences (COR) in dealing with the simple materials which is basic to mathematics, namely, arithmetic
number, algebraic symbols, and spatial figures. AM is in turn predicted by MRA, a second factor, which is
assessed through scores in number and numeration (NNU), algebra (ALG), and geometry (GEO). In the figure
Es represent errors in the measurement of observed variables and Ds represent Disturbance (variance) in the
latent traits (or factors). Because the causes of exogenous variables are not represented in the models, they are
free to vary and covary (Kline, 2005). However, note that the Ds associated with latent traits in the figure reflect
omitted causes of latent variable rather than observed variables.
The third objective of this study was to determine the fitness of the two-factor model of Mathematical
reasoning ability and attainment in Mathematics and determine the extent to which the indicators of
mathematical reasoning ability predict attainment in mathematics. These objectives were realized through the
assessment of the maximum likelihood parameters of the hypothesized model, and the relationships between the
indicators and their corresponding latent traits and the relationship between the latent traits.
Specifically, five research questions were answered. These were:
1) What is the nature of the relationship among the indicators of each of the latent variables?
AMMRA
B
ALG
GEO
CLA
ECLA
EORE
ECOR
EGEO
DMA
NNU
DMRA
ENUM
EALG
ROO
COR
VAR
EVAR
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2) Does this model fit the observed data?
3) If the model fits the observed data, what are the models fit statistics?
4) What is the nature of the factor loadings of each of the latent traits?
5) To what extent does mathematical reasoning ability predict the attainment in Mathematics?
2. Method
2.1 Participants
The participants were 240 Senior Secondary One (Age 14 – 16 years) students randomly drawn from 10 Senior
Secondary Schools in Isokan, and Irewole Local Government Areas, Osun State, Nigeria. In this study, only the
students in science classes took part. From each school, intact science classes were used. Among the 240
students sampled, 127 (52.9%) were boys, while 113 (47.1%) were girls. Their ages ranged between 14 and 16
years (Mean age = 15.43; SD = 1.79).
2.2 Materials
For this survey, two instruments were used. These are:
a) Attainment in Mathematics Test (AMT)
b) Mathematical Reasoning Ability Test (MRAT)
AMT: This consists of 36 multiple choice items with four options (ABCD). The author of this article developed
these items from three major sections of mathematics (Number and numeration, Algebra, Geometry, and
Geometry) as prescribed by the Nigerian Educational Research and Development Council, (NERDC, 2005).
Initially there were 60 items. These items were subjected to pilot testing among 30 SSII students in Ayedade
local Government Area, Osun State, Nigeria. Twenty-four items whose difficulty indices were either below 0.30
or above 0.71 were deleted. For the remaining 36 items, each of the four major sections consists of nine items.
Test blue print placed under Knowledge, Understanding and Thinking were used to establish the content validity
of the final 36 items. The difficulty indices of each item ranged between 0.38 and 0.75, while the discriminating
indices ranged between 0.43 and 0.51. The reliability indices of each section: Number and numeration, Algebra,
and Geometry were 0.71, 0.76, and 0.69 respectively. This was established by using Kuder Richardson 20
formular. The maximum obtainable score in each section was 12, that is, each item attracted a score of 1 for right
response and 0 for wrong response.
MRAT: The author of this article constructed this instrument. It was constructed by combining the fundamental
processes of recognition of variable, class, order and correspondence with the three basic types of materials
employed in the study of Mathematics, these being arithmetical numbers, algebraic symbols, and spatial figures.
As suggested by Lee (1967), separate tests were designed to estimate capacity for handling each of the four
processes in terms of each of the three types of materials making twelve sub tests in all. Initially under each sub
test, there were four items, that is, the draft copy consisted of 48 items. These items were subjected to pilot
testing among 35 students in Ayedaade Local Government Area, Osun State, Nigeria. In each of the sub tests,
items with low difficulty levels (p < .30) were eliminated. In the final draft, there were two 24 items. The
difficulty and discriminating indices of each item ranged between 0.37 and 0.70; and 0.30 and 0.57 respectively.
The reliability indices of each section ranged between 0.69 and 0.81. The maximum obtainable score in MRAT
was 24, that is, each item correctly answered attracted a score of 1 while wrong responses attracted 0.
Examples of item used in the MRAT include:
1. Classification-Arithmetical Number (CLA-AN): In each of the rows of numbers, there are five numbers on the
left. Four have something in common and one is different. Underline the one which is different. When you have
done this, place a circle on one of the numbers on the right which could best take the place of the one you have
underlined.
Left Right
45, 54, 63, 27, 30 33, 36, 66, 93,
In this problem, on the left hand side, 30 have been underlined, because it is not divisible by 9. On the right hand
side, 36 have been circled because it is the only one out of the four numbers on the right which is divisible by 9
2. Order-Arithmetic Number (ROO-AN). The following rows show the beginnings of series of numbers, and
some numbers are missing. Fill in the blank spaces with the numbers you think should be there.
1, 3, 5, 7, 9, ---, ---
2.3 Procedure
Three research assistants were recruited for this study. They were all graduate students in the Institute of
Education, University of Ibadan, Nigeria. They have received formal training in test administration. However,
the author of this article explained the purpose of the study to them. The author also went round to monitor how
the tests were administered. The data collection lasted four weeks. The administration of each of the tests took
place during the normal time scheduled for mathematics on the official school time table. This was to avoid
disruption to the school programmes. For MRAT, the students completed it on the average, in 50 minutes,
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though the time allowed was one hour. The data collected were analysed using maximum likelihood estimates of
LISREL Version 8.80 (Linear Structural Relations; Jöreskog & Sörbom, 2003).
3. Results
The descriptive statistics (Pearson Correlation coefficient, mean, and standard deviation) were calculated using
LISREL Version 8.80. Also variance-covariance statistics were obtained. Table 1 presents the Pearson product-
moment correlation coefficients between the variables. Mean and standard deviations of each of the six variables
are as presented on the table.
Table 1: Pearson correlation coefficient, mean, and standard deviation
VAR CLA ROO COR NNU ALG GEO
VAR 1.000
CLA .630* 1.000
ROO .537* .535* 1.000
COR .833* .623* .516* 1.000
NNU .860* .633* .540* .705* 1.000
ALG .589* .821* .736* .586* .592* 1.000
GEO .636* .751* .794* .582* .640* .589 1.000
Mean 6.49 5.92 5.13 7.08 6.87 5.21 6.41
SD 2.85 2.57 2.35 2.92 2.83 2.40 2.50
Note *p < .05 Key: VAR = recognition of variables, CLA = classification of given materials, ROO = recognition
of order, COR = recognition of correspondences, NNU= Number and Numeration, Algebra = Algebra, GEO =
Geometry.
Research Question One: What is the nature of the relationship among the indicators of the latent variables?
In this study, as specified in the preceding sections, the hypothesized indicators of mathematical reasoning
ability were the fundamental processes of recognition of variable, class, order and correspondence and for
attainment in mathematics they were number and numeration, algebra and geometry. It is noted from Table 1
that, one, all the variables are moderately and significantly correlated with one another. Two, the relationships
between measures of mathematical ability and measures of attainment in mathematics are moderate and positive.
In fact, a perusal of the relationships between each of the measures of mathematical reasoning ability and each of
the measures of mathematics attainment shows that mathematical reasoning ability is a good predictor of
attainment in mathematics.
The highest correlation (r = .860) is between recognition of variables in a given material, and number and
numeration. The observed high and positive relationship between recognition of variables and number and
numeration suggests that a student with high ability of recognizing some constant variable within a group of
numbers is likely to do well in problems dealing with number and numeration.
The lowest correlation (r = .516) is between recognition of correspondences in a given material and spatial
figures. Though in this study, the correlation coefficient may appear low, it is moderate and more importantly,
statistically significant. The obtained variance-covariance statistics lend credence to the fact that there existed
positive and high relationships among the indicators of numerical and attainment in mathematics.
Research Question Two: Does the hypothesized model fit the observed data?
From the maximum likelihood estimates, the independence model Chi square analysis shows that the variables in
this study are correlated χ2
(13, N = 240) = 352.52, p < 0.05. The value of the minimum fit function Chi Square
is χ2
(21, N = 240) = 635.55, p < 0.05. Although in a very good model fit situation, the hypothesized model fit
function Chi Square statistics should not be significant, other fit indices when examined, suggested reasonable
overall model fit.
Research Question Three: If the model fit the data, what are the model fit statistics?
The model fit statistics examined include: Root-Mean-Square-Error of Approximation (RMSEA) = 0.33 with the
90% Confidence interval (1.18 – 1.69); Comparative Fit Index (CFI) = 0.71; Normed Fit Index (NFI) = 0.71; and
Parsimony Goodness of Fit Index (PGFI) = 0.44. These values indicate overall satisfactory model fit (See Hu
and Bentler, 1999; Kline, 2005).
Research Question Four: What is the nature of the factor loadings of each of the latent traits?
Reported in Table 2 are the unstandardised and standardized maximum likelihood estimates for all parameters of
the two-factor (Mathematical Reasoning Ability and Attainment in Mathematics) SR model.
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Table 2: Maximum Likelihood Parameter Estimates of the Structural Regression Model
Parameter Standardardised Estimate Unstandardised Std. Error Z
Factor Loadings
MRA VAR 0.55 1.56 0.15 10.39
MRA CLA 0.84 2.17 0.13 16.43
MRA ROO 0.83 1.96 0.12 16.09
MRA COR 0.30 0.88 0.14 06.42
AM NNU 0.53 1.50 - -
AM ALG 0.83 2.00 0.18 10.93
AM GEO 0.83 2.07 0.19 10.91
Measurement error variance
EVAR 0.70 5.70 0.46 12.45
ECLA 0.29 1.90 0.15 12.35
EROO 0.31 1.72 0.14 12.42
ECOR 0.91 7.76 0.68 11.49
ENNU 0.72 5.79 0.47 12.33
EALG 0.31 1.77 0.15 11.92
EGEO 0.32 1.99 0.17 11.96
Also presented in Figure 2 are the standardized solutions of the hypothesized regression model.
Figure 2: Standardized solutions of the hypothesized structural regression model.
From Table 2 and Figure 2, it can be observed that almost all the factor loadings (standardized coefficient) range
from being moderate (0.30, mathematical reasoning ability to recognition of correspondence among given
materials) to being high (0.84, mathematical reasoning ability to classification) and statistically significant, p
< .05. The z-statistic in each case is greater than 1.96. This shows that the four indicators viz: recognition of
variable, class, order and correspondence can validly measure mathematical reasoning ability.
Research Question Five: To what extent does mathematics reasoning ability predict attainment in Mathematics?
From Figure 2, the obtained coefficient of 1.19 (z = 11.25) shows that reasoning ability predicted attainment in
mathematics quite well. The obtained coefficient of 1.19 implies that for every one full unit increase in reasoning
ability, attainment in mathematics increases by 1.19 units.
4. Discussion
The major goals of this study were to explain the four basic indicators of mathematics reasoning ability test and
to develop a model that could help explain the relationship between mathematical reasoning ability and
attainment in secondary school mathematics. Factor loadings of the structural regression analysis showed that the
four fundamental notions of recognition of variable, classification, recognition of order, and recognition of
correspondence, among materials given in mathematical problems, are measures of mathematical reasoning
MRA
B
AM
EORE ORE
0.30
0.84
0.83
ECLA
ECOR
EGEO
ALG
GEO
CLA
NUN
ENUM
EALG
COR
VAREVAR
0.55
0.55
0.83
0.83
0.70
0.72
0.29
0.31
0.91
0.31
0.32
1.19
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ability. These findings were consistent with the works of earlier philosophers such as Lee (1967) and Hamley
(1934).
Empirical evidence from literature (e.g. Adegoke, 2003; Berret & Williams, 1997 Fischbein & Nachieli, 1998;
Heng-Yuku & Sllivan, 2000) suggests that ability to succeed in the study of mathematics could correspond to the
ability to carry out the four processes of recognition of variable, classification, ordering, and recognition of
correspondence. This is because these notions play a large part in Mathematical work, since most problems are
solved by recognizing classes amongst given data, creating orders with these classes, and picking out
correspondence to lead to a unique conclusion. The ability to carry out these processes is needed in dealing with
the simple mathematics which is basic to mathematics, namely arithmetical numbers, algebraic symbols, and
spatial figures.
For mathematics achievement, number and numeration, algebra, and geometry, as the factor loadings of
structural regression model showed are valid indicators of attainment in mathematics. The earlier works of
philosophers such as Hamley (1934) and Jenkins (1939) had shown that arithmetical numbers, algebraic symbols,
and spatial figures are the fundamental notions of mathematics. In fact, the study of mathematics cannot proceed
at all without these, for they provide the material with which mathematicians work. They also designate the
subject field called mathematics. In all Anglo-phone countries (Ghana, Sierra Leone, Gambia, Nigeria and
Senegal) in West Africa, curriculum of secondary school mathematics includes topics in algebra, geometry and
statistics. More over items for the public examinations are usually drawn from these topics (see Mathematics
Syllabus of West African Examination Council, 2011).
Examination of the raw test scores showed that no student obtained consistently low marks for the tests of
mathematical ability while obtaining consistently high marks for the corresponding tests of mathematical
attainment. On the other hand, quite a number of the students scored well above average on the ability tests and
below average on the attainment tests. Analysis of data showed that the two batteries of ability and attainment
tests in Mathematics share a common ground in that success in the former seems to predict success in the latter,
in the sense in which ability and attainment in the study of a subject were defined in the earlier sections of this
article. This finding corroborates the empirical evidence from literature (e.g. Choudhury and Das, 2012; Heng-
Yuku & Sullivan, 2000; Nunes, Byrant, Barros, & Sylva, 2012). For example, in their study, Nunes, Byrant,
Barros, and Sylva (2012) found that mathematical reasoning ability did made independent contributions to the
prediction of mathematical achievement. Similarly, Choudhury and Das (2012) in their study found that
geometrical ability contributed significantly to students’ achievement in mathematics in Malaysia. These
findings show that ability to succeed in the study of mathematics could correspond to the ability to carry out the
four processes of recognition of variable, classification, ordering, and recognition of correspondence. This is
because these notions play a large part in Mathematical work, since most problems are solved by recognizing
classes amongst given data, creating orders with these classes, and picking out correspondence to lead to a
unique conclusion. The ability to carry out these processes is needed in dealing with the simple mathematics
which is basic to mathematics, namely arithmetical numbers, algebraic symbols, and spatial figures.
5. Implications of Findings and Recommendations
The findings of this study showed that students’ level of mathematics reasoning ability plays a major role in their
attainment in mathematics. Therefore it is important that teachers take note of this, and more importantly mount
intervention programmes that can help students develop their reasoning ability. This can be achieved by giving
them cognitive tasks that are not necessary curriculum based. Examples of such tasks include finding missing
numbers in an array of numbers as stated in one of the problems used in this study (See the two examples given
under AMT)
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Ibadan, Nigeria.
Adegoke, A. B. (2011) “Effect of direct teacher influence on dependent-prone students’ learning outcomes in
secondary school mathematics”, Electronic Journal of Research in Educational Psychology, 9, 283 – 308
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Mathematics”. PhD Thesis, University of London Library.
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Benson Adesina Adegoke holds B.Ed (Physics), M.Ed and PhD degrees in Educational Evaluation, of
University of Ibadan, Nigeria. He specialises in research methods as well as in teaching methods of mathematics
and physics. He is a research fellow at the Institute of Education, University of Ibadan. He is a member of
Science Teachers Association of Nigeria as well as American Psychological Association
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