Students’ images of two-variable functions and their graphs

This paper presents a conceptual analysis for students’ images of graphs and their extension to graphs of two-variable functions. We use the conceptual analysis, based on quantitative and covariational reasoning, to construct a hypothetical learning trajectory (HLT) for how students might generalize their understanding of graphs of one-variable functions to graphs of two-variable functions. To evaluate the viability of this learning trajectory, we use data from two teaching experiments based on tasks intended to support development of the schemes in the HLT. We focus on the schemes that two students developed in these teaching experiments and discuss their relationship to the original HLT. We close by considering the role of covariational reasoning in generalization, consider other ways in which students might come to conceptualize graphs of two-variable functions, and discuss implications for instruction.     

Students’ images of mathematics

Students’ judgments about “what counts” as mathematics in and out of school have important consequences for problem solving and transfer, yet our understanding of the source and nature of these judgments remains incomplete. Thirty-five sixth grade students participated in a study focused on what activities students judge as mathematical, and how they make their judgments. Students completed a photo sorting activity; took, viewed, and captioned their own photos of mathematics; viewed and commented on classmates’ photos; and participated in a small group discussion. Across multiple sources of data, findings showed that students attended to two major features of photos and activities when making judgments: surface cues present in the photos, such as numbers and money, and the possibility for mathematical action. Some students looked for the possibility of mathematics, while others asked if mathematics was necessary. Students also gave higher ratings to activities with which they had personal experience. The article concludes with possible implications for practice.     

Students’ understanding of the structure of deductive proof

While proof is central to mathematics, difficulties in the teaching and learning of proof are well-recognised internationally. Within the research literature, a number of theoretical frameworks relating to the teaching of different aspects of proof and proving are evident. In our work, we are focusing on secondary school students learning the structure of deductive proofs and, in this paper, we propose a theoretical framework based on this aspect of proof education. In our framework, we capture students’ understanding of the structure of deductive proofs in terms of three levels of increasing sophistication: Pre-structural, Partial-structural, and Holistic-structural, with the Partial-structural level further divided into two sub-levels: Elemental and Relational. In this paper, we apply the framework to data from our classroom research in which secondary school students (aged 14) tackled a series of lessons that provided an introduction to proof problems involving congruent triangles. Using data from the transcribed lessons, we focus in particular on students who displayed the tendency to accept a proof that contained logical circularity. From the perspective of our framework, we illustrate what we argue are two independent aspects of Relational understanding of the Partial-structural level, those of universal instantiation and hypothetical syllogism, and contend that accepting logical circularity can be an indicator of lack of understanding of syllogism. These findings can inform how teaching approaches might be improved so that students develop a more secure understanding of deductive proofs and proving in geometry.     

Three concepts or one? Students’ understanding of basic limit concepts

In many mathematics curricula, the notion of limit is introduced three times: the limit of a sequence, the limit of a function at a point and the limit of a function at infinity. Despite the use of very similar symbols, few connections between these notions are made explicitly and few papers in the large literature on student understanding of limit connect them. This paper examines the nature of connections made by students exposed to this fragmented curriculum. The study adopted a phenomenographic approach and used card sorting and comparison tasks to expose students to symbols representing these different types of limit. The findings suggest that, while some students treat limit cases as separate, some can draw connections, but often do so in ways which are at odds with the formal mathematics. In particular, while there are occasional, implicit uses of neighbourhood notions, no student in the study appeared to possess a unifying organisational framework for all three basic uses of limit.     

Students’ understanding of a groundwater system and attitudes towards groundwater use and conservation

The purpose of this study was to investigate students’ understanding of groundwater as a system, and to further examine whether and how their understanding is related to attitudes towards groundwater use and conservation. A total of 676 junior high school students completed a groundwater survey including a drawing activity and a questionnaire. An additional group of university students with environmental science majors were recruited for the survey for the purpose of determining reasonable expectations and validating interpretations of the results. Analysis of student drawings indicated that students generally held an incomplete and unconnected understanding of the groundwater system. Correlation analysis showed that student levels of understanding were associated with their perceptions of environmental impacts as well as environmental concern regarding groundwater problems. These findings re-emphasised the importance of promoting students’ understanding of environmental systems, as well as their ability to think systemically.     

Students’ conceptions about a fair assessment of their learning

The objective of this study is to know the conceptions of primary and secondary school students about what constitutes a fair assessment. To achieve this aim, a phenomenographic study was carried out with 32 students. In a broad sense, the results showed that there were two types of conceptions. On one hand, a conception of fair assessment related to equality, transparency, objectivity, and evaluation of class content; that is, an egalitarian conception of fair assessment. On the other hand, a conception associated with ideas such as adaptation, diversification of tests, and qualitative assessment, even taking into account students' effort and attitudes, which define a conception of fair assessment linked to equity. These conceptions are associated with the concepts of legal justice and social justice, respectively and they contrast with the nature of external evaluations currently carried out.     

Students’ conceptions and the learning of science

This introductory article to the Special Issue of the International Journal of Science Education attempts to review the theoretical contexts for research into children's conceptions in science and to identify future directions for research programmes in this field.     

Students’ conceptions of work and the understanding of the economic value of labor: a developmental study of unemployment and job precarity in times of an economic crisis

In the aftermath of the financial and economic recession of 2008, 130 Spanish students of five age groups (8 to 17 years) and two socioeconomic backgrounds were individually interviewed about unemployment and lower wages. The participants were presented with two hypothetical situations, and their responses were qualitatively and quantitatively analyzed. The results indicated that children bring their conceptions of work when attempting to explain those phenomena. Specifically, it was found that the explanations were closely associated with the comprehension of the notion of surplus value (i.e., the idea that workers create more value than the cost of their salaries), and that the development of this notion proceeds in three levels. Younger children were simply unaware of the capacity of workers to create value (Level 1), which gave rise to some alternative conceptions (e.g., the idea that money for salaries comes from external sources). The children within the age range of 12–13 began to understand that workers create some value (Level 2). At this level, however, the notion of surplus value was not fully understood, and the idea that money for salaries comes from external sources persisted in a variety of ways. From the ages of 13–14, the children begin to understand that a wage is the price of labor and that the employer earns a profit because workers create more value than the cost of their wages (Level 3). The article presents a discussion of the possible reasons behind the slow progress observed and highlights some suggestions for educational practice.

     

The relationship between biology teachers’ understanding of the nature of science and the understanding and acceptance of the theory of evolution

ABSTRACT

Despite the importance of the theory of evolution (TE) to scientific knowledge, a number of misconceptions continue to be found among biology teachers. In this context, the first objective of this study was to identify the impact of professional development programme (PDP) on teachers’ understanding of nature of science (NOS) and evolution and on the acceptance of this theory. Its second objective was to study the relationship among these variables. Three instruments were used to quantify these variables: the Views of the Nature of Science Version D (VNOS D+), the Assessing Contextual Reasoning about Natural Selection (ACORN), and the Measure of Acceptance of Theory of Evolution (MATE). The results indicate that the PDP had a positive impact on teachers, significantly improving their understanding of the NOS and natural selection, as well as their acceptance of the TE. Furthermore, a positive correlation between the understanding of the NOS obtained by teachers in the first part of the PDP and the understanding and acceptance of evolution that these teachers showed at the end of the programme was determined. However, no relationship between an understanding of the NOS and gains in the understanding and acceptance of evolution was found.     

Students’ perceptions of parental bonding styles and their academic burnout

This study investigated how parental bonding style affects academic burnout in Korean adolescents. Participants were 447 middle school students, who completed the Parental Bonding Instrument and the Maslach Burnout Inventory-Student Survey. MANCOVA results confirmed that adolescents reporting the optimal bonding parental style, for both mother and father, have lower scores on three subscales of academic burnout (Exhaustion, Cynicism, and Inefficacy) than do their counterparts who report the affectionless control parental style. We discuss the possible implications for reducing students?? academic burnout rates.     

The Relationship between Students’ Views of the Nature of Science and their Views of the Nature of Scientific Measurement

The present study explores the relationship between students’ views of the nature of science (NOS) and their views of the nature of scientific measurement. A questionnaire with two‐tier diagnostic multiple‐choice items on both the NOS and measurement was administered to 179 first‐year physics students with diverse school experiences. Students’ views on the NOS were classified into four NOS ‘profiles’, and views on measurement were classified according to either the point or set paradigms. The findings show that students with a NOS profile dominated by a belief that the laws of nature are to be discovered by scientists are more likely to have a view of the nature of scientific measurement characterised by a belief in ‘true’ values. On the other hand, students who believe that scientific theories are inventions of scientists, constructed from observations that are then validated through further experimentation, are more likely to have a view of the nature of scientific measurement that is underpinned by the uncertain nature of scientific evidence. The implications for teaching scientific measurement at tertiary level are discussed.     

Preservice Students’ Perceptions and Level of Satisfaction with their Field Experiences

This paper reports on a study of students’ perceptions and level of satisfaction with the teaching‐learning process while participating in the field‐experience component of the four‐year Bachelor of Education course at the Australian Catholic University Signadou Campus in Canberra. It is argued that successful student‐teacher field‐experience programmes provide opportunities for ongoing renewal and growth. Essential to this process is the implementation of appropriate evaluative procedures which involve all the participating members of the programme. The success of the process is evidenced in the ongoing modification of the programme to accurately reflect the interests and needs of the participants. Such a process leads to higher levels of communication, satisfaction and reflection on practice by all interested parties. Through an analysis of students’ questionnaire responses, this paper provides insights into the changing nature of student perceptions about the quality of their field experiences and their level of satisfaction with the various components of the programme. The results indicate that as the focus moves from the university itself to independent student responsibility the levels of satisfaction increase.     

An Investigation into the Relationship between Students’ Conceptions of the Particulate Nature of Matter and their Understanding of Chemical Bonding

A thorough understanding of chemical bonding requires familiarity with the particulate nature of matter. In this study, a two‐tier multiple‐choice diagnostic instrument consisting of ten items (five items involving each of the two concepts) was developed to assess students’ understanding of the particulate nature of matter and chemical bonding so as to identify possible associations between students’ understandings of the two concepts. The instrument was administered to 260 Grades 9 and 10 students (15–16 years old) from a secondary school in Singapore. Analysis of students’ responses revealed several alternative conceptions about the two concepts. In addition, analysis of six pairs of items suggested that students’ limited understanding of the particulate nature of matter influenced their understanding of chemical bonding. The findings provide useful information for challenging students’ alternative conceptions about the particulate nature of matter during classroom instruction in order to enable them to achieve better understanding of chemical bonding.     

Students’ intuitive explanations of the causes of homologies and adaptations

This paper reports data from a study aiming to explore secondary students’ preconceptions and explanations about evolutionary processes. Students may exhibit both alternative and scientifically acceptable conceptions and bring different ones into play in response to different problem contexts. Hence, the examination of their explanations before instruction within different problem contexts is expected to highlight the concepts that instruction should put more emphasis on. To achieve this, an open-ended questionnaire in conjunction with semi-structured interviews was used to allow students to express their own views on issues related to evolution. Students’ explanations highlighted their lack of knowledge of important evolutionary concepts such as common descent and natural selection. In addition, many students explained the origin of traits as the result of evolution through need via purposeful change or as carefully designed adaptations. Rather than evolutionary, final causes formed the basis for the majority of students’ explanations. In many cases students provided different explanations for the same process to tasks with different content. It seems that the structure and the content of the task may have an effect on the explanations that students provide. Implications for evolution education are discussed and a minimal explanatory framework for evolution is suggested.     

Teachers’ understanding of racial inequity predicts their perceptions of students’ behaviors

Teachers’ racialized interpretations of students’ behaviors likely lead to the racial disparities found in discipline across all education levels (Skiba et al., 2011). To explore how four types of teachers’ racial inequity beliefs (i.e., post-racial, American Dream, cultural deficit, schooling inequity) may be associated with teachers’ racialized interpretations of Black and White boys’ behaviors (i.e., hostility, emotionality, behavioral stability, and seriousness), a sample of 228 teachers in the United States filled out the Racism and Inequity Beliefs (RIB) Questionnaire and observed videotapes of six boys potentially misbehaving in school. Teachers’ cultural deficit beliefs uniquely predicted teachers’ negative perceptions of Black boys’ behaviors, with little to no impact on their perceptions of White boys’ behaviors. Specifically, teachers holding more cultural deficit beliefs perceived the Black boys’ behaviors as relatively more hostile, more representative of a behavioral pattern, and more serious compared to teachers who held fewer cultural deficit beliefs, and compared to their judgments of White boys’ behaviors. American Dream beliefs were also uniquely associated with assessments of Black boys’ behavior as more serious, without impact on White boys’ behavior. These results suggest the application of inequitable judgments of boys’ behaviors based on teachers’ racial inequity beliefs and boys’ identifiable race.     

The functions of shared metacognitive regulation and their differential relation with collaborative learners’ understanding of the learning content

By taking into account the functions of socially shared metacognitive regulation (SSMR) (i.e. the role that SSMR plays in the (dis)continuation of ongoing interaction), the present study sheds light on the differential effectiveness of SSMR. The study more particularly investigates how individual students' involvement in SSMR that confirms, changes, activates, or stops interaction, predicts their understanding of the learning content on the short and middle-long term, taking into account students' prior knowledge. Sixty university students were involved. Multilevel models were run to investigate the relation between individual students' engagement in the functions of SSMR and their conceptual understanding. Contributing to SSMR that changes and that activates collaborative learning appeared significantly positively related to students' immediate understanding of the learning content, whereas participating in SSMR that confirms or that stops ongoing interaction was not. Contributing to SSMR (regardless of its function) appeared not significant for predicting students’ conceptual understanding on the middle-long term.     

Examining the Effect of Enactment of a Geospatial Curriculum on Students’ Geospatial Thinking and Reasoning

A potential method for teaching geospatial thinking and reasoning (GTR) is through geospatially enabled learning technologies. We developed an energy resources geospatial curriculum that included learning activities with geographic information systems and virtual globes. This study investigated how 13 urban middle school teachers implemented and varied the enactment of the curriculum with their students and investigated which teacher- and student-level factors accounted for students’ GTR posttest achievement. Data included biweekly implementation surveys from teachers and energy resources content and GTR pre- and posttest achievement measures from 1,049 students. Students significantly increased both their energy resources content knowledge and their GTR skills related to energy resources at the end of the curriculum enactment. Both multiple regression and hierarchical linear modeling found that students’ initial GTR abilities and gain in energy content knowledge were significantly explanatory variables for their geospatial achievement at the end of curriculum enactment, p < .001. Teacher enactment factors, including adherence to implementing the critical components of the curriculum or the number of years the teachers had taught the curriculum, did not have significant effects on students’ geospatial posttest achievement. The findings from this study provide support that learning with geospatially enabled learning technologies can support GTR with urban middle-level learners.     

Students’ multiple state goals as a function of appraisals,trait goals,and their interactions

Although many motivational theories have emphasized that person- and situation-based aspects should be recognized simultaneously when explaining actual learning behavior, not much is known about the occurrence of state goals. The model of adaptive learning (Boekaerts & Niemivierta, 2000) proposes that state goals are influenced by trait goals, perceptions of the situation (i.e., appraisals), and interactions between trait goals and appraisals. Nevertheless, empirical evidence for this model is scarce. The present study aimed to overcome this research deficit by integrating trait goals and appraisals as well as their interactions in explaining state goals in authentic learning episodes. As suggested by Boekaerts and Niemivierta (2000), we adopted a multiple-goal approach and included achievement goals (i.e., mastery, performance-approach, and performance-avoidance goals) as well as well-being goals (i.e., affiliation and work-avoidance goals). They outlined the meaning of perceived relevance and autonomy for learning behavior. Therefore, we included both appraisals in our study. Using longitudinal data with one trait and two state measurements (N = 542), we applied structural equation modeling to test three different effect models with latent variables: an additive effect model, a reactive effect model, and an interaction effect model. Our data did not fit the reactive effect model: Trait goals did not impact appraisals. The joint influence of trait goals and appraisals on state goals was not unitary but multiform: Additive and multiplicative relations between trait goals and appraisals explained the occurrence of state goals. State goals are important predictors of proximal learning behavior. Thus, these results are important for theory development and for practitioners.