Cabinetmakers’ Workplace Mathematics and Problem Solving

This study explored what kind of mathematics is needed in cabinetmakers’ everyday work and how problem solving is intertwined in it. The informants of the study were four Finnish cabinetmakers and the data consisted of workshop observations, interviews, photos, pictures and sketches made by the participants during the interviews. The data was analysed using different qualitative techniques. Even though the participants identified many areas of mathematics that could be used in their daily work, they used mathematics only if they were able to. The cabinetmakers’ different mathematical skills and knowledge were utilized to their skill limit. Cabinetmakers were found to constantly face problem solving situations along with the creative processes. Being able to use more advanced mathematics helped them to solve those problems more efficiently, without wasting time and materials. Based on the findings, the paper discusses the similarities and differences between problem solving and creative processes. It is suggested that the combination of craftsmanship, creativity, and efficient problem solving skills together with more than basic mathematical knowledge will help cabinetmakers in adapting and surviving in future unstable labour markets.     

Using Problem Solving to Assess Young Children’s Mathematics Knowledge

Mathematics problem solving provides a means for obtaining a view of young children’s understanding of mathematics as they move through the early childhood concept development sequence. Assessment information can be obtained through observations and interviews as children develop problem solutions. Examples of preschool, kindergarten, and primary grade children’s approaches to problem solving are provided in the article. Prekindergarten and kindergarten age children discover problems during play. For example, they figure out how to use informal measurement to use construction materials such as unit blocks and Lego to build a desired building or make a desired object. Moldable materials such as clay and play dough provided shape experiences. The daily sequence of activities builds on their concept of time. Primary grade children solve adult- and child-generated problems. They may use manipulatives and/or drawings to generate problem solutions prior to using symbols and notation. Teacher and/or student devised rubrics can be used to guide evaluation.     

Reform of Mathematics Curriculum and Open-ended Problem Solving

The basic assumption of reform mathematics is that ＂no one can teach mathematics＂. Therefore, the important teachers＇ role is to stimulate students to learn mathematics and support their development. Open-ended problem solving is an instructional strategy that creates interest and stimulates creative mathematical activity in the classroom through students＇ collaborative work. Lessons using openended problem solving problem solving activities result. emphasize the process of rather than focusing on the     

Opportunities to learn: Mathematics textbooks and students’ achievements

This study explores how textbooks function in education. It asked whether opportunities provided in math textbooks to engage in tasks demanding different levels of understanding correlate with students’ achievements on tasks demanding equivalent levels of understanding on a standardized exam. The textbooks evaluated were two 8^th grade mathematics textbooks used by students in the Arab community in Israel, showing that Textbook A makes more cognitive demands than Textbook B. The study correlated textbooks’ cognitive demand with the scores of all 8^th grade students in the Arab community who completed the national math test in 2015 and studied in schools using either Textbook A or B (N = 4040), while attending to mediating variables. The findings show that if a textbook provides the opportunity to engage in tasks demanding higher levels of understanding, students using this book will have higher scores. The study shows that gender and SES play an important role in how opportunities provided in textbooks interact with students’ scores. Many factors influence variations in mathematics achievements within and between nations. The findings illuminate textbooks’ ability to provide opportunities to learn mathematics. As a result, they raise new questions about how teachers use textbooks and about the role of textbooks in promoting access and equity in mathematics education. Although the work explored specific textbooks, its findings shed light on how learning opportunities relate to achievements more generally.     

Relations Between Students’ Mathematics Anxiety and Motivation to Learn Mathematics: a Meta-Analysis

Educational Psychology Review - The current meta-analysis examined the association between K-12 students’ motivation to learn mathematics and mathematics anxiety, and explored the effects of...     

Preservice Middle and High School Mathematics Teachers’ Strategies when Solving Proportion Problems

The purpose of this study was to investigate eight preservice middle and high school mathematics teachers’ solution strategies when solving single and multiple proportion problems. Real-world missing-value word problems were used in an interview setting to collect information about preservice teachers’ (PSTs) reasoning about proportional relationships. An explanatory case study methodology with multiple cases was used to make comparisons within and across cases. Analysis of the semi-structured interviews with each PST revealed that using practical problems, in which plastic gears and a mini balance system were provided, and multiple proportion problems facilitated the PSTs’ recognition of the proportional relationships in their solutions. Therefore, they avoided using cross-multiplication and erroneous strategies in those problems. Among the strategies that the PSTs used in solving single and multiple proportion problems, the ratio table strategy was the most frequent and effective strategy. The ratio table strategy enabled the PSTs to recognize the constant ratio and product relationships more than the other strategies. The results of this study illuminate how PSTs reason about proportional relationships when they cannot rely on computation methods like cross-multiplication.     

Constructivism in Mathematics Classrooms: Listening to Ghanaian Teachers’ and Students’ Views

One of the challenges of implementing a new curriculum is how to bridge the gap between the underlining principles of the curriculum and the cultural and social orientations of the society which includes teachers and students. This article reports on a study that explored how the cultural and social orientations of teachers and students can influence the implementation of a constructivist curriculum in mathematics classrooms. The data for the study came from 250 students and 41 mathematics teachers, using questionnaires, observations, and interviews. The results showed that inasmuch as mathematics teachers and their students acknowledge the importance of student’s active participation and teamwork, these practices have not been fully conceptualised into the Ghanaian mathematics classroom due to some cultural factors. Two main cultural factors were discovered from the analyses of the results. Firstly, the culture of acknowledging only correct answers in class has a negative impact on individual students’ confidence and participation during mathematics lessons. Also, the culture of teamwork is not fully accepted within Ghanaian classrooms as most students find it difficult working in groups and accepting and appreciating each other’s view. It was evident in all lessons that students were ridiculed by their peers when they provide a wrong answer to a question and this affected individual students’ participation in the classroom. Therefore, we suggested that teachers should be pro-active in promoting a classroom environment which is free from fear and intimidation to motivate students to be actively involved in the classroom discourse.     

Understanding and Teaching Problem‐Solving in Physics

Summaries

English

We describe a systematic study of skills for solving problems in basic physics, a domain of practical significance for instruction, but not of prohibitive complexity. Our studies show that an inexperienced student tends to solve a problem by assembling individual equations. By contrast, an expert solves a problem by a process of successive refinements, first describing the main problem features by seemingly vague words or pictures, and only later considering the problem in greater detail in more mathematical language. We have formulated explicit theoretical models with such features and have supported them by some detailed observations of individuals. In addition, experimental instruction incorporating such features seems to improve problem‐solving performance significantly. These investigations yield thus some basic insights into thinking processes effective for problem‐solving. Furthermore, they offer the prospect that these insights can be used to teach students improved problem‐solving skills and to modify common teaching practices which inhibit the development of such skills.     

Knowledge and Technique in Statistical Problem‐Solving

Summaries

English

Subjects’ knowledge and technique in statistical problem‐solving, as well as their ability to detect their own problem‐solving errors, were studied. Twenty subjects were asked to solve two statistical problems, and then to explain their solutions to the experimenter. Finally, each subject was asked successively more detailed questions about his/her solution, particularly in reference to its erroneous parts. The results showed that subjects tended to describe rather than explain their solutions. Subjects eliminated three errors (4%) when accounting for their solutions and another 20 errors (24%) in the following interview. About one‐third of all errors were analysed as being due to deficiencies in subjects’ problem‐solving techniques. For 62% of the errors where subjects showed knowledge deficiencies, there were deficiencies in propositional knowledge, i.e. in subjects’ understanding of statistical concepts and their interrelations. The results showed that misunderstandings of concepts and methods of solution were much more common than lack of conceptions as such. In addition, the results suggest that deficiencies in subjects’ problem‐solving are related to motivational factors, such as a tendency to minimize cognitive strain. Pedagogical implications of the results are discussed.     

Facilitating Students’ Problem Solving in a Technological Context: Prospective Teachers’ Learning Trajectory

In the past decade, there has been an increased emphasis on the preparation of teachers who can effectively engage students in meaningful mathematics with technology tools. This study presents a closer look at how three prospective teachers interpreted and developed in their role of facilitating students’ mathematical problem solving with a technology tool. A cycle of planning–experience–reflection was repeated twice during an undergraduate course to allow the prospective teachers to change their strategies when working with two different groups of students. Case study methods were used to identify and analyze critical events that occurred throughout the different phases of the study and how these events may have influenced the prospective teachers’ work with students. Looking across the cases, several themes emerged. The prospective teachers (1) used their problem solving approaches to influence their pedagogical decisions; (2) desired to ask questions that would guide students in their solution strategies; (3) recognized their own struggle in facilitating students’ problem solving and focused on improving their interactions with students; (4) assumed the role of an explainer for some portion of their work with students; (5) used technological representations to promote students’ mathematical thinking or focus their attention; and (6) used the technology tools in ways consistent with the nature of their interactions and perceived role with students. The implications inform the development of an expanded learning trajectory for what we might expect as prospective teachers develop an understanding of how to teach mathematics in technology-rich environments.     

Correction to: Kindergarten and First-Grade Students’ Understandings and Representations of Arithmetic Properties

Early Childhood Education Journal - The original version of this article unfortunately contained a typo in co-author name.     

A Mathematics Teachers’ Perspective and its Relationship to Practice

In this paper we try to characterize the pedagogical approaches that mathematics teachers are developing to meet the challenges posed by education reforms. A key aspect is the identification of the perspectives that underlie those pedagogical approaches, using the term perspective to include a broad pedagogical structure composed of multiple conceptions that are related to some aspects of a teacher’s practice. Through the study of the practice of a secondary mathematics teacher, we try to explore how his/her pedagogical approaches on mathematics, mathematics learning, and mathematics teaching are related to the relational architecture that is established in the classroom during the development of an instructional unit of similarity at a secondary school level, and we examine if that relationship can be explained in terms of the underlying perspective. The results of the study have shown the characteristics of that relationship, and the important role that the teacher’s knowledge of the students’ difficulties plays both in making decisions and in developing the teachers’ actions.     

Some Issues in Problem Solving

Asweknow ,theproblemsolvingisanimportanttopicinmathematicseducation .Itcouldbediscussedwithrespecttoclassroompractices,curriculum ,research ,orprofessionaldevelopment.Whatisproblemsolvingtoyou ?Howisproblemsolvingincorporatedintoyourcurriculum ?Whataretheinstructionalissuesrelatedtoproblemsolving ?Howdoweassessproblemsolving ?Whatdoesresearchsayaboutproblemsolving ?Whatarethe politicsofproblemsolving ?Thefollowingaresomeissuesin problemsolving :———Whataredifferentviewsormeaningsofproblems…     

Commitment to Teach in Under-Resourced Schools: Prospective Science and Mathematics Teachers’ Dispositions

In this exploratory study, we sought to gain an understanding of what motivates prospective teachers who are Noyce Scholars at a research-intensive southeastern US university to commit to teaching secondary level science or mathematics in school districts that have a high proportion of students who come from low-socioeconomic households. An interpretive methodology revealed three themes associated with Noyce Scholars’ motivations to teach (1) awareness of educational challenges, (2) sense of belonging to or comfort with diverse communities, and (3) belief that one can serve as a role model and resource. The paper describes and compares the significance of each theme among six prospective teachers who identify with the schooling experiences of students who came from low-income or poor households and nine prospective teachers who identify with the schooling experiences in a middle-income school or district. The implication of this study supports the importance of recruiting prospective science and mathematics teachers who have knowledge of and a disposition to work with learners from low-income or poor households, even if those prospective teachers are not themselves the members of under-served populations.     

Preparing Teachers’ to Raise Students’ Mathematics Learning

The purpose of this study was to investigate the effectiveness of the ORIGO Stepping Stones program in a large suburban school district in the Midwestern U.S.. The sample included 11 elementary schools that implemented the program during the 2013–2016 school years. Findings are presented from teacher surveys, classroom observations, and analysis of standardized student mathematics achievement scores on the Northwest Evaluation Association Measurement of Academic Progress (NWEA MAP). Using multilevel models, the program did not demonstrate a detectably larger effect on mathematics achievement than the comparison. Classroom observations and views of teachers that support program implementation were explored. In terms of fidelity of implementation, findings suggested that the program was delivered differently from how it was intended. With respect to researchers and policymakers, the approach used to evaluating mathematics program effectiveness is informative and could be used as part of larger accountability systems.     

Yup’ik Cosmology to School Mathematics: The Power of Symmetry and Proportional Measuring

This article shows how Yup’ik cosmology, epistemology, and everyday practice have implications for the teaching of school mathematics. Math in a Cultural Context (MCC) has a long–term collaborative relationship with Yup’ik elders and experienced Yup’ik teachers. Because of this long–term ethnographically–oriented relationship, the authors – both insiders and an outsider – have been able to understand the mathematical implications of everyday Yup’ik practice. As the article demonstrates, body proportional measuring and symmetry/splitting are two generative solution strategies used by Yup’ik elders in solving everyday problems. We argue that proportional measuring coupled with symmetry/splitting can provide school mathematics with an alternative pathway to the teaching of some aspects of geometry and rational number reasoning.     

A Social-Studies Unit that Developed Pupils’ Powers of Problem Solving

This article reports the author's experiences using graphic novels with pre-service teachers in a young adult literature course. Drawing on critical response papers two students composed after reading Pride of Baghdad, a graphic novel by Brian K. Vaughan and Niko Henrichon, the author argues that when readers possess the background knowledge needed to approximate the role of the implied reader—that is, the imaginary audience for whom authors envision themselves writing—they are capable of engaging with graphic novels in ways that readers who lack experience with the form, or who question its literary merit, are not.     

Correction to: A Critical Review of Students’ and Teachers’ Understandings of Nature of Science

Science & Education - The original article unfortunately contains incorrect presentation of Tables 1, 3, 4, 5 and 6.     

Solution Stories: A Narrative Study of How Teachers Support Children’s Problem Solving

Young children’s self-regulation and problem-solving skills are significant predictors of school success. While early childhood educators shape the development of these skills, providing effective and timely assistance can be challenging. Drawing on complementary theories of Vygotsky, Pekrun, and Lerner, this article chronicles the instructional approaches and strategies employed by one team of teachers to support preschool children’s solutions to complex functional and social problems in the classroom. Findings from this narrative study highlight the focal teachers’ use of modeling, mindful language, and other proactive strategies to develop students’ problem-solving skills and foster independence. In an age of results-focused education, this article argues for the importance of cultivating intentional teacher pedagogies that build young children’s autonomy and efficacy by working through problems, as opposed to seeking resolution only. In so doing, this study elucidates the value of these intuitive and often nuanced aspects of early childhood educators’ classroom practices.     

The Role of Students’ Scientific Epistemic Beliefs in Computer-Simulated Problem Solving

Research on how epistemic beliefs influence students’ learning in different contexts is ambiguous. Given this, we have examined the relationships between students’ scientific epistemic beliefs, their problem solving, and solutions in a constructionist computer-simulation in classical mechanics. The problem-solving process and performance of 19 tenth-grade students, with different scientific epistemic beliefs, were video recorded and inductively coded. Quantitative analysis revealed that different sets of epistemic beliefs were conducive to different aspects of students’ problem-solving process and outcomes. Theoretically sophisticated beliefs were in general associated with logical strategies and high solution complexity. However, authority dependence was associated with high degree of adherence to instructions. Hence, there might not be a universal relationship between the theoretical sophistication of students’ epistemic beliefs and quality of learning outcomes. We suggest that the conduciveness to desired outcomes is a better measure of sophistication than theoretical non-contextualized a priori assumptions.