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Effects of Perceived Mathematics Connection on Mathematics Motivation: Mediating Role of History of Mathematics Concepts
Abstract
The present study assessed the mediating effects of history of mathematics concepts on the relationship between mathematics connection and mathematics motivation. The study was a descriptive survey carried out on 390 students randomly selected from Asuoso Senior High School in the Offinso-North District of Ghana’s Ashanti region. Data was collected using Likert type structured questionnaires. Reliability analysis was computed in SPSS (V. 23) and confirmatory factor analysis was computed in AMOS (V. 23). In addition, average variance extracted (AVE), convergent validity, and discriminant validity were computed in Microsoft Excel from factor loadings. Also, composite reliability was calculated at https://www.thestatisticalmind.com/composite-reliability/. Structural Equation Modelling (SEM) was computed in Amos (V. 23) to determine direct, and mediating effects among the main constructs. The study found that mathematics connection positively and significantly influenced mathematics motivation of senior high school students. The indirect effects of mathematics connection on mathematics motivation through history of mathematics concepts was positive and significant. The study recommended that mathematics teachers should desist from merely topic explanations and rote memorization of solution procedures. They should rather make mathematics connections and history of mathematics concepts imminent in their lesson planning and delivery. To implement this, mathematics topics should be linked to real-life scenarios, other subjects, and students’ lives, and the history behind such concepts are explained in plain language. Not only teachers, NaCCA should ensure that the mathematics syllabus and textbooks for senior high schools have incorporated real-life scenarios as well as the historical background of concepts to ensure learners develop the interest and motives for studying mathematics.
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Purpose
One popular method to assess discriminant validity in structural equation modeling is the heterotrait-monotrait ratio of correlations (HTMT). However, the HTMT assumes tau-equivalent measurement models, which are unlikely to hold for most empirical studies. To relax this assumption, the authors modify the original HTMT and introduce a new consistent measure for congeneric measurement models: the HTMT2.
Design/methodology/approach
The HTMT2 is designed in analogy to the HTMT but relies on the geometric mean instead of the arithmetic mean. A Monte Carlo simulation compares the performance of the HTMT and the HTMT2. In the simulation, several design factors are varied such as loading patterns, sample sizes and inter-construct correlations in order to compare the estimation bias of the two criteria.
Findings
The HTMT2 provides less biased estimations of the correlations among the latent variables compared to the HTMT, in particular if indicators loading patterns are heterogeneous. Consequently, the HTMT2 should be preferred over the HTMT to assess discriminant validity in case of congeneric measurement models.
Research limitations/implications
However, the HTMT2 can only be determined if all correlations between involved observable variables are positive.
Originality/value
This paper introduces the HTMT2 as an improved version of the traditional HTMT. Compared to other approaches assessing discriminant validity, the HTMT2 provides two advantages: (1) the ease of its computation, since HTMT2 is only based on the indicator correlations, and (2) the relaxed assumption of tau-equivalence. The authors highly recommend the HTMT2 criterion over the traditional HTMT for assessing discriminant validity in empirical studies.
Factor analysis is particularly suitable to extract few factors from the large number of related variables to a more manageable number, prior to using them in other analysis such as multiple regression or multivariate analysis of variance. It can be beneficial in developing of a questionnaire. Sometimes adding more statements in the questionnaire fail to give clear understanding of the variables. With the help of factor analysis, irrelevant questions can be removed from the final questionnaire. This study proposed a factor analysis to identify the factors underlying the variables of a questionnaire to measure tourist satisfaction. In this study, Kaiser-Meyer-Olkin measure of sampling adequacy and Bartlett’s test of Sphericity are used to assess the factorability of the data. Determinant score is calculated to examine the multicollinearity among the variables. To determine the number of factors to be extracted, Kaiser’s Criterion and Scree test are examined. Varimax orthogonal factor rotation method is applied to minimize the number of variables that have high loadings on each factor. The internal consistency is confirmed by calculating Cronbach’s alpha and component reliability to test the instrument accuracy. The convergent validity is established when average variance extracted is greater than or equal to 0.5. The results have revealed that the factor analysis not only allows detecting irrelevant items but will also allow extracting the valuable factors from the data set of a questionnaire survey. The application of factor analysis for questionnaire evaluation provides very valuable inputs to the decision makers to focus on few important factors rather than a large number of parameters.
Keywords: factor analysis, Kaiser-Meyer-Olkin, Bartlett’s test of Sphericity, determinant score, Kaiser’s criterion, Scree test, Varimax
Integration of history of mathematics (HOM) in mathematics education is emerging as frontier endowed with utility to stem decline in uptake, enhance performance and alter the negative perceptions about the subject. This case study sought to explore the influence of history of mathematics on grade eleven girls' perceptions of the Theorem of Pythagoras in particular and mathematics in general. In this exploratory study, the genetic approach of HOM was used in teaching the Theorem of Pythagoras. Findings indicate that the inclusion of aspects of the history of mathematics catalysed positive learner perceptions of the Theorem and mathematics in general. Notable changes in learner perceptions included affirmations of an increased level of motivation and confessed improvement in readiness of grade eleven learners in tackling challenging tasks involving the Theorem of Pythagoras. Learners also alluded to increased confidence in dealing with proofs, enjoying making their own discoveries and solving mathematical problems in general.
In this action study, instructional environments were enriched with activities related to the history of mathematics in order to deepen students‟ beliefs about mathematics and reveal for them fun, interesting, and useful activities. The study enrolled twenty-four 8th-grade students. Data were collected by using multiple data collection tools. The data regarding the students‟ beliefs about mathematics were collected via written opinion forms and semi-structured
interviews. Prior to the study, the students perceived mathematics as a branch of science that is closed to development and did not know why or how it flourished. After the study, they stated that mathematics is open to development and is used to solve everyday problems. The study revealed a decrease in students‟ absolutist beliefs about mathematics, and students found math fun and interesting as a result of engaging in activities that promote active problem-solving. In the future, other action studies involving the history of mathematics may be used to teach different topics at different grade levels.
The orientation of education nowadays should be in line with the fulfillment of 21st century demand which encourage learners to perform the originality of human resources quality. This research aims to know whether there is effect connection mathematics ability toward student’s mathematics learning achievement and to know the contribution of connection mathematics ability toward student’s mathematics learning achievement at the elementary school. Respondent in this research were 35 students at Watu Weri state elementary school. Proportionate stratified random sampling technique was applied in collecting sample. The research design was ex-post facto. Data were collected through test and documentation. Data were analyzed by linear regression. The research result shows that there was significant effect mathematics connection ability toward student’s mathematics learning achievement. The contribution of mathematics connection ability toward students mathematics learning achievement was 21,9% . This result indicates that high or law student’s mathematics learning achievement significantly affected by their mathematics connection ability.
Initial ability is ability to determine the success of students in learning. High or low initial ability can cause differences in solving mathematics problems, particularly in connecting ability in solving mathematics problems based on initial ability of the students. This study is descriptive research with qualitive approach.the researcher is the main instrument guided by test of mathematics initial ability, test of mathematics problems problem solving, anda valid guided interview. Data were coffected through test anda verified through interview. The research subjects were the students of grade IXA at SMPN 10 Bulukumba which consisted of two subjects who have high initial ability anda low initial ability. The research process were conducted in the following steps (a) formulating mathematical connection indicators, (b) formulating supporting instrument, test of initial ability, test of problem solving, and guided interview which is valid anda reliable, (c) choosing research subjects based on initial ability, (d) conducting data collection to reveal mathematical connection ability of the students using test of mathematics problem solving. (e) conducti g triangulation to obtain valid data, (f) conducting data analysis, (g) conducting data analysis, and (h) conducting conclusion drawing based on the results of the study. The result of the study reveal that (a) the subject with high initial ability can complete test of mathematics problem solving based on mathematical connection inndicators, (b) the subject with low initial ability cannot solve the test of mathematical problem solving because there is no knowledge to support the problem solving.
The students use computers, smartphones and other electronic devices everyday for their own needs. The success in Mathematics is getting lower and lower, both on local level in Macedonia and on global level also [6]. A lot of the students are having difficulties with learning Mathematics, and with that also the motivation for learning Mathematics decreases. The difficulties are shown in both, mastering the material in school and solving the exercises at home. Also, when the students solve real-life problems, they are not connecting what was learned on Mathematics classes with those problems. They almost don't use the Mathematics and the electronic devices at all for solving the real-life problems. If we succeed in encouraging the students in using these electronical devices and softwares for: examining, investigating, visualizing and solving real-life problems, we will get students that are motivated for learning mathematics, and with that also we will get increased level of knowledge of mathematics. In this project, for increasing the motivation for learning mathematics and increasing the level of knowledge, we made a web application http://mathlabyrinth.azurewebsites.net The problems that are put on the web application are connected with real-life problems the students have, who need knowledge from secondary education mathematics to solve them.
Medical educators attempt to create reliable and valid tests and questionnaires in order to enhance the accuracy of their assessment and evaluations. Validity and reliability are two fundamental elements in the evaluation of a measurement instrument. Instruments can be conventional knowledge, skill or attitude tests, clinical simulations or survey questionnaires. Instruments can measure concepts, psychomotor skills or affective values. Validity is concerned with the extent to which an instrument measures what it is intended to measure. Reliability is concerned with the ability of an instrument to measure consistently.1 It should be noted that the reliability of an instrument is closely associated with its validity. An instrument cannot be valid unless it is reliable. However, the reliability of an instrument does not depend on its validity.2 It is possible to objectively measure the reliability of an instrument and in this paper we explain the meaning of Cronbach’s alpha, the most widely used objective measure of reliability.
Path analysis was used to test the predictive and mediational role of self-efficacy beliefs in mathematical problem solving. Results revealed that math self-efficacy was more predictive of problem solving than was math self-concept, perceived usefulness of mathematics, prior experience with mathematics, or gender (
N = 350). Self-efficacy also mediated the effect of gender and prior experience on self-concept, perceived usefulness, and problem solving. Gender and prior experience influenced self-concept, perceived usefulness, and problem solving largely through the mediational role of self-efficacy. Men had higher performance, self-efficacy, and self-concept and lower anxiety, but these differences were due largely to the influence of self-efficacy, for gender had a direct effect only on self-efficacy and a prior experience variable. Results support the hypothesized role of self-efficacy in A. Bandura's (1986) social cognitive theory. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
The research presented in this article integrates 3 theoretical perspectives in the field of motivation: expectancy-value, achievement goals, and interest. The authors examined the antecedents (initial interest, achievement goals) and consequences (interest, performance) of task value judgments in 2 learning contexts: a college classroom and a high school sports camp. The pattern of findings was consistent across both learning contexts. Initial interest and mastery goals predicted subsequent interest, and task values mediated these relationships. Performance-approach goals and utility value predicted actual performance as indexed by final course grade (classroom) and coach ratings of performance (sports camp). Implications for theories of motivation are discussed. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
We tested whether classroom activities that encourage students to connect course materials to their lives will increase student
motivation and learning. We hypothesized that this effect will be stronger for students who have low expectations of success.
In a randomized field experiment with high school students, we found that a relevance intervention, which encouraged students
to make connections between their lives and what they were learning in their science courses, increased interest in science
and course grades for students with low success expectations. The results have implications for the development of science
curricula and theories of motivation.
A range of indicators, such as the average variance extracted (AVE), is commonly used to validate constructs. In statistics, AVE is a measure of the amount of variance that is captured by a construct in relation to the amount of variance due to measurement error. These conventional indices are formed by factor loadings resulting from estimated least squares or maximum likelihood regressions. Thus, a new proposition that provides new factor loadings may result in a more informative AVE index. Consequently, this study consists of the improvement of the index by using adaptive regressions. A Monte Carlo simulation study was performed considering different numbers of outliers generated by distributions with symmetry deviations and excess kurtosis and sample sizes defined as n = 50, 100, and 200. The conclusion was that, in formative structural models, the adaptive linear regression (ALR) method showed good efficiency for correctly specified models. The results obtained from the ALR method for models with specification errors showed low efficiency, as expected.
The aim of this study is to investigate high school students self-efficacy beliefs in mathematical literacy (ML) and to explore their views on the connections between mathematics and the real world according to their levels of ML self-efficacy beliefs. Both quantitative and qualitative data was collected from 40 high school students. Data collection tools included an ML Self-efficacy Scale and an interview schedule. Data analysis indicated that high school students had medium levels of ML self-efficacy beliefs. In relation to high school students ML self-efficacy beliefs, participants had similar views on the connections between mathematics and the real world. Although students who had medium and high levels of ML self-efficacy beliefs had positive views on connections between mathematics and the real world, the findings suggested that these were limited to the benefits of connections, to the situations, circumstances of using mathematics in real world and to the mathematical concepts they can use. The findings also indicated that student views were below the preferred, expected levels.
While the traditional meaning of connected knowledge is valuable in some
school subjects, it does not address the main activities of knowledge
acquisition in subjects such as physics and mathematics. The goal of
this article is to analyze the relationships between the concepts
“learning for understanding” and “connected
knowledge”, a central theme in feminist epistemology. In learning
for understanding, the learner forms multiple, intricate connections
among the concepts she is studying in school, between school concepts
and her everyday concepts, and between school concepts and their wider
context. Viewing connected knowledge as tightly related to understanding
has several important implications. It brings connected knowledge into
the central learning activities that take place in school science and
mathematics, and gives it a high status. It contributes to our
understanding of gender-related patterns in thinking; and it may form a
unifying theoretical framework for many studies and projects in the
field of gender fair education.
Using different measures of constructs in research to develop robust evidence of relationships and effects is seen as good methodological practice. This assumes these measures possess high convergent validity. However, proxies—alternative measures of the same construct—are rarely perfectly convergent. Although some convergence is preferred to none, this study demonstrates that even modest departures from perfect convergent validity can result in substantial differences in the magnitudes of findings, creating challenges for the accumulation and interpretation of research. Using data from published research, the authors find that substantial differences in findings between studies using desired and proxy variables occur even at levels of convergent validity as high as r = .85. Implications of using measures with less-than-ideal convergent validity for the interpretability of research results are examined. Convergent validities above r = .70 are recommended, whereas those below r = .50 should be avoided. Researchers are encouraged to develop and report convergent validity data.
Robotics naturally lends itself to teaching math, science, technology and engineering in the K-12 classroom. LEGO Mindstorms makes it easy for students even in kindergarten to design and build their own robotic creations. However, the key to bringing engineering into K-12 through robotics is educating teachers on the content, how to use the materials, and how open-ended design problems can be effective in the classroom. This paper details the Tufts University Center for Engineering Educational Outreach's theoretical framework, motivations, and efforts involved in bringing engineering via LEGO robotics into every kindergarten through fifth-grade classroom in one school through the Systemic School Change in Engineering Project. Preliminary results and recommendations are presented.
A general formula (α) of which a special case is the Kuder-Richardson coefficient of equivalence is shown to be the mean of
all split-half coefficients resulting from different splittings of a test. α is therefore an estimate of the correlation between
two random samples of items from a universe of items like those in the test. α is found to be an appropriate index of equivalence
and, except for very short tests, of the first-factor concentration in the test. Tests divisible into distinct subtests should
be so divided before using the formula. The index
[`(r)]ij\bar r_{ij}
, derived from α, is shown to be an index of inter-item homogeneity. Comparison is made to the Guttman and Loevinger approaches.
Parallel split coefficients are shown to be unnecessary for tests of common types. In designing tests, maximum interpretability
of scores is obtained by increasing the first-factor concentration in any separately-scored subtest and avoiding substantial
group-factor clusters within a subtest. Scalability is not a requisite.
We discuss the expectancy–value theory of motivation, focusing on an expectancy–value model developed and researched by Eccles, Wigfield, and their colleagues. Definitions of crucial constructs in the model, including ability beliefs, expectancies for success, and the components of subjective task values, are provided. These definitions are compared to those of related constructs, including self-efficacy, intrinsic and extrinsic motivation, and interest. Research is reviewed dealing with two issues: (1) change in children's and adolescents' ability beliefs, expectancies for success, and subjective values, and (2) relations of children's and adolescents' ability-expectancy beliefs and subjective task values to their performance and choice of activities.
One of the central tenets of classical test theory is that scales should have a high degree of internal consistency, as evidenced by Cronbach's a, the mean interitem correlation, and a strong first component. However, there are many instances in which this rule does not apply. Following Bollen and Lennox (1991), I differentiate between questionnaires such as anxiety or depression inventories, which are composed of items that are manifestations of an underlying hypothetical construct (i.e., where the items are called effect indicators) and those such as Scale 6 of the Minnesota Multiphasic Personality Inventory (Hathaway & McKinley, 1943) and ones used to tap quality of life or activities of daily living in which the items or subscales themselves define the construct (these items are called causal indicators). Questionnaires of the first sort, which are referred to as scales in this article, meet the criteria of classical test theory, whereas the second type, which are called indexes here, do not. I discuss the implications of this difference for how items are selected, the relationship among the items, and the statistics that should and should not be used in establishing the reliability of the scale or index.
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