Assessing 10- to 11-year-old children’s performance and misconceptions in number sense using a four-tier diagnostic test

Background: Number sense is a key topic in mathematics education, and the identification of children’s misconceptions about number is, therefore, important. Information about students’ serious misconceptions can be quite significant for teachers, allowing them to change their teaching plans to help children overcome these misconceptions. In science education, interest in children’s alternative conceptions has led to the development of three- and four-tier tests that not only assess children’s understandings and misconceptions, but also examine children’s confidence in their responses. However, there are few such tests related to mathematical content, especially in studies of number sense.

Purpose: The purpose of this study was to investigate children’s performance and misconceptions with respect to number sense via a four-tier diagnostic test (Answer Tier → Confidence rating for Answer Tier → Reason Tier → Confidence rating for Reason Tier).

Design and method: A total of 195 fifth graders (10–11 years old) from Taiwan participated in this study. The four-tier test was web-based and contained 40 items across five components of number sense.

Findings: The results show that (1) students’ mean confidence rating for the answer tier was significantly higher than for the reason tier; (2) an average of 68% of students tended to have equal confidence ratings in both answer and reason tiers; (3) students who chose correct answers or reasons had higher mean confidence ratings in most items (36 out of 40) than those who did not; and (4) 16 misconceptions were identified and most of them were at a strong level.

Conclusion: The four-tier test was able to identify several misconceptions in both the answer and reason tier and provide information about the confidence levels. By using such information, teachers may be better positioned to understand the nature of learners’ misconceptions about number sense and therefore support their pupils’ progress in mathematics.     

Elementary mathematics teachers’ judgment accuracy and calibration accuracy: Do they predict students’ mathematics achievement outcomes?

In this study we investigated whether elementary mathematics teachers’ knowledge of their students, as reflected in both the accuracy and confidence with which they are able to estimate their students’ task-specific performance on sets of mathematics problems, predicted students’ overall mathematics achievement. Thirty-nine teachers made predictions about the performance of a random sample of target students (n = 150) in their classrooms on sets of “easy” and “difficult” multiplication and division problems. Teachers also provided confidence ratings for those judgments. From these data, indicators of teachers’ judgment accuracy, judgment confidence and calibration accuracy (a measure of metacognitive monitoring) were then related to all of their students’ (n = 834) performance on year-end standardized mathematics achievement tests. Multilevel analyses indicate that teachers’ calibration accuracy, but not their task-specific judgment accuracy, significantly predicted students’ mathematics achievement. Implications for future research on teacher knowledge as well as professional development programs are discussed.     

Students’ perceptions of mathematics classroom learning environments: measurement and associations with achievement

In this study, we measured students’ perceptions of mathematics classroom learning environment and investigated their associations with students’ achievement. The Mathematics-Related Constructivist-Oriented Classroom Learning Environment Survey (MCOLES) was developed with seven dimensions and 56 items, using theories surrounding classroom learning environment. For a sample of 423 grade 10 students from five schools in India, we validated the MCOLES by exploratory factor analysis and then by confirmatory factor analysis, which suggested the exclusion of 11 items and yielded an 11-factor solution. For achievement on a topic taught, mainly medium correlations emerged with the learning environment factors, suggesting practical implications for classroom teaching. This study is methodologically significant in proposing and validating the new MCOLES for measuring classroom learning environments in secondary-school mathematics.

     

Gender Differences in Lunar‐related Scientific and Mathematical Understandings

This paper reports an examination on gender differences in lunar phases understanding of 123 students (70 females and 53 males). Middle‐level students interacted with the Moon through observations, sketching, journalling, two‐dimensional and three‐dimensional modelling, and classroom discussions. These lunar lessons were adapted from the Realistic Explorations in Astronomical Learning (REAL) curriculum. Students’ conceptual understandings were measured through analysis of pre‐test and post‐test results on a Lunar Phases Concept Inventory (LPCI) and a Geometric Spatial Assessment (GSA). The LPCI was used to assess conceptual learning of eight science and four mathematics domains. The GSA was used to assess learning of the same four mathematical domains; however, the GSA test items were not posed within a lunar context. Results showed both male and female groups to make significant gains in understanding on the overall LPCI test scores as well as significant gains on five of the eight science domains and on three of the four mathematics domains. The males scored significantly higher than the females on the science domain, phase—Sun/Earth/Moon positions, and on the mathematics domain geometric spatial visualisation. GSA results found both male and female groups achieving a significant increase in their test scores on the overall GSA. Females made significant gains on the GSA mathematics domains, periodic patterns and cardinal directions, while males made significant gains on only the periodic patterns domain. Findings suggest that both scientific and mathematical understandings can be significantly improved for both sexes through the use of spatially focused, inquiry‐oriented curriculum such as REAL.     

Discourses of time and maturity structuring participation in mathematics and further mathematics

Abstract

This paper examines how young people account for choosing mathematical subjects, and how these processes sustain, or not, their continued participation. It draws on a 2-year qualitative study of 24 young people’s accounts of following advanced mathematical pathways within a widening participation programme. Working within a post-structural framework, I combine two arguments: firstly, that local discourses of time, age and maturity position contemporary adolescence as a time of ‘becoming’ that aligns personal aspirations with mathematical progress, and secondly that students’ accounts of choice and aspiration require multiple imaginings of present and future selves. I identify distinct discourses –moving/improving and getting ahead - that structure the intelligibility of participation in mathematics and further mathematics respectively. I argue that tracing the alignments between students’ accounts of themselves and/ in mathematics offers potential to understand emergent practices in mathematics participation but also how exclusions are re-inscribed along classed and gendered lines.     

Interactive anatomical and surgical live stream lectures improve students' academic performance in applied clinical anatomy

Tuebingen's Sectio Chirurgica (TSC) is an innovative, interactive, multimedia, and transdisciplinary teaching method designed to complement dissection courses. The Tuebingen's Sectio Chirurgica (TSC) allows clinical anatomy to be taught via interactive live stream surgeries moderated by an anatomist. This method aims to provide an application‐oriented approach to teaching anatomy that offers students a deeper learning experience. A cohort study was devised to determine whether students who participated in the TSC were better able to solve clinical application questions than students who did not participate. A total of 365 students participated in the dissection course during the winter term of the 2012/2013 academic year. The final examination contained 40 standard multiple‐choice (S‐MC) and 20 clinically‐applied multiple‐choice (CA‐MC) items. The CA‐MC items referred to clinical cases but could be answered solely using anatomical knowledge. Students who regularly participated in the TSC answered the CA‐MC questions significantly better than the control group (75% and 65%, respectively; P < 0.05, Mann‐Whitney U test). The groups exhibited no differences on the S‐MC questions (85% and 82.5%, respectively; P > 0.05). The CA‐MC questions had a slightly higher level of difficulty than the S‐MC questions (0.725 and 0.801, respectively; P = 0.083). The discriminatory power of the items was comparable (S‐MC median Pearson correlations: 0.321; CA‐MC: 0.283). The TSC successfully teaches the clinical application of anatomical knowledge. Students who attended the TSC in addition to the dissection course were able to answer CA‐MC questions significantly better than students who did not attend the TSC. Thus, attending the TSC in addition to the dissection course supported students' clinical learning goals. Anat Sci Educ 10: 46–52. © 2016 American Association of Anatomists.     

Relationship between affect and achievement in science and mathematics in Malaysia and Singapore

Background : The Trends in International Mathematics and Science Study (TIMSS) assesses the quality of the teaching and learning of science and mathematics among Grades 4 and 8 students across participating countries.

Purpose : This study explored the relationship between positive affect towards science and mathematics and achievement in science and mathematics among Malaysian and Singaporean Grade 8 students.

Sample : In total, 4466 Malaysia students and 4599 Singaporean students from Grade 8 who participated in TIMSS 2007 were involved in this study.

Design and method : Students’ achievement scores on eight items in the survey instrument that were reported in TIMSS 2007 were used as the dependent variable in the analysis. Students’ scores on four items in the TIMSS 2007 survey instrument pertaining to students’ affect towards science and mathematics together with students’ gender, language spoken at home and parental education were used as the independent variables.

Results : Positive affect towards science and mathematics indicated statistically significant predictive effects on achievement in the two subjects for both Malaysian and Singaporean Grade 8 students. There were statistically significant predictive effects on mathematics achievement for the students’ gender, language spoken at home and parental education for both Malaysian and Singaporean students, with R ² = 0.18 and 0.21, respectively. However, only parental education showed statistically significant predictive effects on science achievement for both countries. For Singapore, language spoken at home also demonstrated statistically significant predictive effects on science achievement, whereas gender did not. For Malaysia, neither gender nor language spoken at home had statistically significant predictive effects on science achievement.

Conclusions : It is important for educators to consider implementing self-concept enhancement intervention programmes by incorporating ‘affect’ components of academic self-concept in order to develop students’ talents and promote academic excellence in science and mathematics.     

Attainment in Coeducational and Segregated Schools

Kyöstiö, O. K. (1970). Attainment in Coeducational and Segregated Schools. An inquiry into the prediction of secondary school final success. Scand. J. Educ. Res. 14,113‐170. The problems of this study concerned on one hand the scholastic results of mixed versus segregated education, and on the other the predictiveness of the selection system in Finnish secondary schools. The study was a follow‐up type. The age, social status, and ability of the students were held constant by matching the different groups. The results show that an assumed null‐hypothesis, which concerned the attainment of students in the examined subjects of the matriculation examination in different types of schools, could not be approved. Girls in segregated schools showed themselves to be significantly better than the other groups in total point value. They were also best in most subjects. Only boys from mixed schools were equal to them in mathematics and in general subjects. The correlation coefficient between the sum variable of the entrance selection and the total point value of the matriculation exam variable was significant, showing that the selection had positive predictive value, as was assumed. A regression analysis showed that only the total sum variable of the selection procedure had a significantly predictive value on all success variables, except mathematics. Another regression analysis without sum variables indicated the significant influence of the single determinants too, but in both analyses the variances they could explain were very small. The influence of the continuous and strong selection (high dropout rate) may weaken the generalization of the results.     

The relationship between young children’s enjoyment of learning to read,reading attitudes,confidence and attainment

Background: While there is a considerable body of research exploring the relationship between older primary school children’s reading attitudes, confidence and attainment, there is a noticeable lack of research with younger children. Furthermore, there is relatively little research exploring the extent to which children’s reading attitudes, confidence and attainment are related to their enjoyment of learning to read.

Purpose: To understand the relationship between young children’s reading attitudes, reading confidence, enjoyment of learning to read and reading attainment.

Sample: Two hundred and three children (103 boys) with an average age of 6 years and 9 months (.32 years standard deviation) participated in this study. Children were from 11 primary schools in England, located within a wide range of socio-economic neighbourhoods.

Design and methods: All children completed questionnaires examining their attitudes to reading, reading confidence (using Progress in International Reading Literacy Study items) and enjoyment of learning to read. Children also completed an assessment of word reading. All assessments were administered individually.

Results: Children’s attitudes to reading, reading confidence and enjoyment of learning to read correlated with their word reading skill; however, only reading attitudes and reading confidence predicted variance in reading success. With regard to gender differences, girls reported more positive reading attitudes than boys, even after controlling for differences in reading skill.

Conclusions: Results highlight that the relationship between children’s reading skills and the affective aspects of learning to read develops from a young age. Therefore, methods of reading instruction need to foster affective aspects of reading, in addition to developing cognitive skills to best support young children’s reading development.     

Quantitative skills as a graduate learning outcome: exploring students’ evaluative expertise

In the biosciences, quantitative skills are an essential graduate learning outcome. Efforts to evidence student attainment at the whole of degree programme level are rare and making sense of such data is complex. We draw on assessment theories from Sadler (evaluative expertise) and Boud (sustainable assessment) to interpret final-year bioscience students’ responses to an assessment task comprised of quantitative reasoning questions across 10 mathematical and statistical topics. The question guiding the study was: do final year science students graduate knowing the quantitative skills that they have, and knowing the quantitative skills that they do not have? Confidence indicators for the 10 topics gathered students’ perceptions of their quantitative skills. Students were assigned to one of four categories: high performance-high confidence; low performance-low confidence; high performance-low confidence; or low performance-high confidence – with those in the first two categories demonstrating evaluative expertise. Results showed the majority of students effectively evaluated their quantitative skills as low performance-low confidence. We argue that the application of evaluative expertise to make sense of this graduate learning outcome can further the debate on how assuring graduate learning outcomes can enhance student learning.     

Principled Improvement in Science: Forces and proportional relations in early secondary-school teaching

In response to continuing concerns about student attainment and participation in science and mathematics, the epiSTEMe project took a novel approach to pedagogy in these two disciplines. Using principles identified as effective in the research literature (and combining these in a fashion not previously attempted), the project developed topic modules for early secondary-school teaching in the UK, arranged for their implementation in classrooms, and evaluated the results. This paper reports the development, implementation, and evaluation of one of the epiSTEMe science modules. Entitled Forces and Proportional Relations, the module covers standard curricular material in the domain of forces, while paying particular attention to the proportional nature of many key constructs. It was developed in collaboration with a small group of teachers; implemented subsequently in 16 classrooms, in all cases involving students from the first year of secondary school; and evaluated through comparison with first-year students in 13 control classrooms who were studying the topic using established methods. Evaluation addressed topic mastery and opinions about the topic and the manner in which it was taught. While further research is required before definite conclusions are warranted, results relating to topic mastery provide grounds for optimism about the epiSTEMe approach. Furthermore, student opinions about the module were positive.     

K-8 Teachers’ Overall and Gender-Specific Beliefs About Mathematical Aptitude

Teachers’ beliefs play a significant role in students’ academic attainment and career choices. Despite comparable attainment levels between genders, persistent stereotypes and beliefs that certain disciplines require innate ability and that men and women have different ability levels impede students’ academic career paths. In this study, we examined the prevalence of US mathematics teachers’ explicit general and gender-specific beliefs about mathematical ability and identified which teacher characteristics were associated with these beliefs. An analysis of data from 382 K-8 teachers in the USA indicated that overall, teachers disagreed with the idea that general and gender-specific mathematical ability is innate and agreed with the idea that hard work and dedication are required for success in mathematics. However, our findings indicate that those who believed mathematics requires brilliance also tended to think girls lacked this ability. We also found that teachers who were teaching mathematics to 11- to 14-year-old students seemed to believe that mathematics requires innate ability compared with teachers who were teaching mathematics to 5- to 10-year-old students. In addition, more experienced teachers and teachers who worked with special education students seemed to believe less in the role of hard work in success in mathematics, which could have serious consequences for shaping their students’ beliefs about their academic self-concept and future career-related decisions.

     

Student engagement with feedback and attainment: the role of academic self-efficacy

Abstract

Academic self-efficacy, the belief that one can achieve desired academic goals plays an important role in learning. This study aimed to determine the extent to which academic self-efficacy mediates relationships between students’ perceptions of feedback and their academic attainment. An opportunity sample of 232 students (123 female) in their first year of higher education reported their academic self-efficacy and evaluated their assessment experience, including the perceived quantity and quality of feedback and the extent to which this feedback elicited an active response. Positive associations were observed between academic attainment and students’ confidence that they could achieve their desired grades and adopt appropriate study behaviours. A negative association was identified between attainment and confidence to talk about their studies. Attainment was not related to the perceived quantity or quality of feedback, but did bear a significant association with the reported use to which feedback was put. Positive associations were generally identified between academic self-efficacy and perceptions of feedback. Path models revealed that inter-relationships were best represented by a model wherein academic self-efficacy mediated links between students’ perceptions of feedback and academic attainment. The findings highlight the need to incorporate characteristics of the individual into an understanding of student engagement with feedback.     

To teach or not to teach the conceptual structure of mathematics? Teachers undervalue the potential of Principle-Oriented explanations

Principle-oriented explanations have demonstrated to foster students’ mathematical understanding, as they integrate conceptual and procedural information to make the solution process tangible to novice students. Teachers, however, often omit conceptual information when explaining procedures. In two experimental studies, we tested the hypothesis that teachers’ tendency to omit conceptual information may have occurred, as teachers generally devalued the potential of principle-oriented explanations. In Study 1, we randomly provided two cohorts of secondary students (N = 129) with principle-oriented versus procedure-oriented explanations on four mathematical topics. Afterwards, students answered a knowledge test. We replicated previous findings that students with principle-oriented explanations outperformed students with procedure-oriented explanations on the knowledge test (application test, transfer test). In Study 2, we gave mathematics teachers (N = 69) these explanations as judgment materials. Teachers randomly rated a balanced set of four explanations of Study 1 which varied in their procedure- versus principle-orientation. We found no significant differences between teachers’ judgments of principle- versus procedure-oriented explanations. Content analyses of the justifications revealed that teachers were more concerned about students’ overload when judging principle-oriented explanations than procedure-oriented explanations. Our findings replicated the beneficial effects of principle-oriented explanations for novice students’ understanding. Furthermore, they highlight the need to sensitize teachers for potential effects of providing principle-oriented explanations.     

Examining the implementation of the mathematics strand of the Key Stage 3 Strategy: What are the bases of evaluation?

This article presents findings from a comparative case study examining the implementation of the mathematics strand of the Key Stage 3 Strategy in two contrasting schools—one using setting and whole‐class teaching, the other incorporating mixed‐ability grouping and individualised learning. A number of ‘outcomes’ of implementation are considered, including teachers' and students' experiences, changes in pedagogic practice and students' attainment. Whilst both schools achieved highly in relation to similar schools, higher attainment gains between Key Stage 2 and Key Stage 3 in the school using setting/whole‐class teaching were associated with increasing student disaffection and dependence on teachers. In contrast, the attainment gains at the school using individualised learning were associated with increasing enthusiasm and independence. The mixed picture of outcomes and their complex interconnections suggest that evaluations of implementation resting on attainment‐based outcomes are problematic in terms of longer‐term aims of increasing the proportions of students choosing to study mathematics and developing the skills of independent learning.     

A sociocultural investigation of the efficacy of outdoor education to improve learner engagement

A sociocultural investigation with Year 8 secondary-school students experiencing marginalization and displaying signs of social and emotional difficulties in engaging fully with school life who participated in a mediated outdoor education programme was undertaken for this study. The findings indicate that there was a very statistically significant change in self-concept (p < 0.05) during the course of the programme. In addition, all participants reported positive results in relation to building trust, group cohesion, and emotional regulation with positive results on facing the challenges they experience daily within school. Moreover, 75% reported that they experienced positive gains in their family since beginning participation in the programme, and 100% stated that they had developed a deeper level of trust for the other group members.     

Evaluating the quality of middle school mathematics teachers,using videos rated by college students

Using experts in teaching to evaluate pre-college mathematics teachers is both time consuming and costly. This study examines the potential of letting undergraduate mathematics students perform this task, comparing their ratings of two dimensions, richness of instruction and mathematical correctness, to those previously assigned by an expert. Using 85 undergraduates of two U.S. institutions, who independently watched short videos of teachers, we found that student ratings of teachers’ correctness were a good match to the expert ratings; student ratings of richness of instruction, less so. A "halo effect" was observed in that students did not fully differentiate between richness and correctness in their ratings. Moreover, male students gave harsher ratings than did female students. Whereas undergraduates showed promise in accurately evaluating teachers of younger students, improvements in terms of rating items, attention to bias, and explicit training on the teaching dimensions to be rated should be addressed.     

Social influence or selection? Peer effects on the development of adolescents’ educational expectations in Germany

Abstract

Educational expectations are a key predictor of educational attainment. Throughout adolescence, friends increasingly function as ‘significant others’ and, thus, can affect the development of these expectations. Although scholars often interpret the clustering of students with similar expectations within friendship networks as the outcome of peer influence, a similarity of friends can also be a result of friendship selection processes and preselection due to ability tracking. We apply multilevel social network models to panel data of adolescents from Germany (1,992 ninth-grade students in 91 classes) to disentangle these mechanisms. Beyond selecting similar friends (homophily), we find that adolescents adapt their expectations towards the average expectations of their friends (social influence) but only in secondary-school tracks that support diverse educational paths. We conclude that peer socialization is important for the development of students’ educational expectations in contexts that are sufficiently heterogeneous to allow for the emergence of distinct peer milieus.     

Games Teachers Play

An important social concern in mathematics education is that the educational attainment of pupils may be influenced by teachers' competence in the curricular area they are teaching. This paper provides some insight into the relationship between student primary teachers' mathematics subject knowledge and their reported confidence to teach that subject. Eighty Bachelor of Education first year students completed an attitudes survey as well as an online mathematics competence test which consisted of 28 randomly generated questions from a bank of approximately 300 questions based on the attainment targets of the Scottish curriculum 5–14 document at level F with some E. Students were asked to rank teacher attributes. Though 98% of the students ranked basic numeracy skills as the most important, 65% of the cohort did not possess these skills. Moreover 95% suggested confidence was important, but confidence levels were found to be low even among students with higher than minimum entry requirements to the undergraduate primary teaching programme. It is perhaps not the level of mathematics that needs to be changed but the nature of mathematics taught and learned at that level that needs to be addressed. This in turn has implications for the approaches and the programmes deployed by Initial Teacher Education courses.