Concept Development of Decimals in Chinese Elementary Students: A Conceptual Change Approach

The aim of this study was to examine the concept development of decimal numbers in 244 Chinese elementary students in grades 4–6. Three grades of students differed in their intuitive sense of decimals and conceptual understanding of decimals, with more strategic approaches used by older students. Misconceptions regarding the density nature of decimals indicated the progress in an ascending spiral trend (i.e., fourth graders performed the worst; fifth graders performed the best; and sixth graders regressed slightly), not in a linear trend. Misconceptions regarding decimal computation (i.e., multiplication makes bigger) generally decreased across grades. However, children's misconceptions regarding the density and infinity features of decimals appeared to be more persistent than misconceptions regarding decimal computation. Some students in higher grades continued to use the discreteness feature of whole numbers to explain the distance between two decimal numbers, indicating an intermediate level of understanding decimals. The findings revealed the effect of symbolic representation of interval end points and students' responses were contingent on the actual representations of interval end points. Students in all three grades demonstrated narrowed application of decimal values (e.g., merchandise), and their application of decimals was largely limited by their learning experiences.     

The use of cross multiplication and other mal–rules in fraction operations by pre-service teachers

Many studies show that prospective teachers often have misconceptions about fractions. In this case study, we report on some of the mal–rules used by a group of 60 prospective South African primary school teachers. The students’ written responses to two items focusing on addition and multiplication of fractions which formed part of an assessment, were analyzed. Semi-structured interviews were also used to elicit the reasoning used in the students’ calculations. Less than half of the participants completed both items correctly, and many of the other students displayed various mal–rules. To interpret the pre–service teachers’ misconceptions, we studied the rules used by the participants, and expressed them as theorems–in–action. An interesting mal–rule governing the multiplication of fractions was the widespread ‘cross multiplication’ rule which after some mutations led to other mal–rules, illustrating how students’ misconceptions can persist many years after their initial learning.     

Helping Students Come to Grips With the Meaning of Division

Many years ago, Arons pointed out the incomprehension science students exhibit of the basic mathematical operations multiplication and division and the need to address the problem in physics classes to assure student understanding of the physical world. McDermott et al.'s Physics by Inquiry program does address this need directly and in detail (by defining two meanings for division). However, in the author's classes many students had relatively low scores (ranging from 60–80%) when trying to explain simple operations. Reported in this paper are ways to supplement the text that force students to address the actual meaning of division by stressing the relation between a “whole” and a “package,” and connect that meaning with previously learned operational definitions for area and volume.     

Facilitating an Elementary Engineering Design Process Module

STEM education in elementary school is guided by the understanding that engineering represents the application of science and math concepts to make life better for people. The Engineering Design Process (EDP) guides the application of creative solutions to problems. Helping teachers understand how to apply the EDP to create lessons develops a classroom where students are engaged in solving real world problems by applying the concepts they learn about science and mathematics. This article outlines a framework for developing such lessons and units, and discusses the underlying theory of systems thinking. A model lesson that uses this framework is discussed. Misconceptions regarding the EDP that children have displayed through this lesson and other design challenge lessons are highlighted. Through understanding these misconceptions, teachers can do a better job of helping students understand the system of ideas that helps engineers attack problems in the real world. Getting children ready for the 21st century requires a different outlook. Children need to tackle problems with a plan and not shrivel when at first, they fail. Seeing themselves as engineers will help more underrepresented students see engineering and other STEM fields as viable career options, which is our ultimate goal.     

An exploration of the role natural language and idiosyncratic representations in teaching how to convert among fractions, decimals, and percents

Using qualitative data collection and analyses techniques, we examined mathematical representations used by sixteen (N = 16) teachers while teaching the concepts of converting among fractions, decimals, and percents. We also studied representational choices by their students (N = 581).In addition to using geometric figures and manipulatives, teachers used natural language such as the words nanny and house to characterize mathematical procedures or algorithms. Some teachers used the words or phrases bigger, smaller, doubling, and building-up in the context of equivalent fractions. There was widespread use of idiosyncratic representations by teachers and students, specifically equations with missing equals signs and not multiply/dividing by one to find equivalent fractions. No evidence though of a relationship between representational forms and degree of correctness of solutions was found on student work. However, when students exhibited misconceptions, those misconceptions were linked to teachers’ use of idiosyncratic representations.     

The multiplicative meaning conveyed by visual representations

Research and practitioner articles advocate the use of visual representations in scaffolding elementary students’ learning of multiplication and division. Prior research suggests students use different strategies when provided with different visualized representations of multiplication and division. However, there is relatively little study examining how children’s multiplicative reasoning corresponds with different representations. The present study collected data from 182 elementary students responding to set, area, and length representations of multiplication/division. Rasch modeling was used to estimate item difficulty statistics to measure differences between visual representations. Results suggest that visual representations differed primarily in how unit was represented and quantified, and not regarding the form of representation (set, area, length).     

A Cross‐Cultural Lesson Comparison on Teaching the Connection Between Multiplication and Division

This study compared one lesson across four U.S. “traditional” textbook series, two U.S. reform‐based textbook series, and one Chinese mathematics textbook series in teaching the connection between multiplication and division. The results showed the differences across U.S. and Chinese lessons in both the teaching and the practice parts of the lesson across three dimensions (i.e., problem schemata, response requirement, and algebra readiness). In particular, the Chinese lesson's penetrating analysis or explanation of the topic is reflected in its deliberately constructed examples and wide range of problems (pertaining to problem types and difficulty levels) present in the teaching and practice sections of the lesson. None of analyzed U.S. lessons are comparable with the Chinese lesson with respect to the breadth and depth in teaching the topic. A deliberate emphasis, both arithmetically and algebraically, on problem schema acquisition as found in the Chinese lesson represents a promotion of symbolic or higher order of conceptual understanding. The findings are discussed within the context of teaching big ideas through problem schemata acquisition and the importance of symbolic level of conceptual understanding.     

Exploring high-achieving sixth grade students’ erroneous answers and misconceptions on the angle concept

The aim of this research was to investigate high achievers’ erroneous answers and misconceptions on the angle concept. The participants consisted of 233 grade 6 students drawn from eight classes in two well-established elementary schools of Trabzon, Turkey. All the participants were considered to be current achievers in mathematics, graded 4 or 5 out of 5, and selected via a purposive sampling method. Data were collected through six questions reflecting the learning competencies set out in the grade 6 curriculum in Turkey and the findings of previous studies that aimed to identify students’ misconceptions of the angle concept. This questionnaire was then applied over a 40-minute period in each class. The findings were analysed by two researchers whose inter-rater agreement was computed as 0.97, or almost perfect. Thereafter, coding discrepancies were resolved, and consensus was established. We found that although the participants in this study were high achievers, they still held several misconceptions on the angle concept such as recognizing a straight angle or a right angle in different orientations. We also show how some of these misconceptions could have arisen due to the definitions or representations used in the textbook, and offer suggestions concerning their content in the future.     

Using Form and Function Analogy Object Boxes to Teach Human Body Systems

This study compares the use of form and function analogy object boxes to more traditional lecture and worksheet instruction during a 10th‐grade unit on human body systems. The study was conducted with two classes (N= 32) of mixed ability students at a high‐needs rural high school in central New York State. The study used a pretest/posttest design, in which the two classes alternated between conditions for the four systems (skeletal, digestive, immune, nervous). Both conditions involved students in quality instruction addressing the same concepts for the same amount of time. Additionally, all students participated in hands‐on labs. The experimental condition presented students with a set of objects analogous in form and function to parts of a human body system. Students matched objects with cards describing body system parts, mapped the analogies on a chart, generated alternative objects that could be used for the analogy, and finally, created new analogies for other body system parts. Students made significantly higher posttest and gain scores on material learned in the experimental condition, with a mean gain score average of 12.4 points out of 25, compared to 6.2 points in the control condition. Cohen's Effect Size was large, 1.36.     

Encouraging problem-solving disposition in a Singapore classroom

In this article, we share our learning experience as a Lesson Study team. The Research Lesson was on Figural Patterns taught in Year 7. In addition to helping students learn the skills of the topic, we wanted them to develop a problem-solving disposition. The management of these two objectives was a challenge to us. From the lesson observation and the students’ classwork, it turned out better than we expected.     

Knowing students as mathematics learners and teaching numbers 10–100: A case study of four 1st grade teachers from Romania

Researchers have increasingly linked teacher effectiveness with teacher knowledge of subject matter, curriculum, and teaching. Moreover, teacher knowledge of students has been regarded as another very significant component of teacher knowledge, influencing the classroom practice and student performance. Knowing students as mathematics learners means being aware of the ways students learn certain topics. This study examined the knowledge of students as mathematics learners displayed by four 1st grade teachers from Romania when designing and implementing a lesson on numbers 10–100. Findings show that knowledge of students as mathematics learners influenced the ways teachers planned and implemented their lesson. Teachers learned about students as mathematics learners from one series to another, and they tailored their use of manipulatives and classroom activities to meet the needs of their current students.     

Improving Algebra Preparation: Implications From Research on Student Misconceptions and Difficulties

Through historical and contemporary research, educators have identified widespread misconceptions and difficulties faced by students in learning algebra. Many of these universal issues stem from content addressed long before students take their first algebra course. Yet elementary and middle school teachers may not understand how the subtleties of the arithmetic content they teach can dramatically, and sometimes negatively, impact their students' ability to transition to algebra. The purpose of this article is to bring awareness of some common algebra misconceptions, and suggestions on how they can be averted, to those who are teaching students the early mathematical concepts they will build upon when learning formal algebra. Published literature discussing misconceptions will be presented for four prerequisite concepts, related to symbolic representation: bracket usage, equality, operational symbols, and letter usage. Each section will conclude with research‐based practical applications and suggestions for preventing such misconceptions. The literature discussed in this article makes a case for elementary and middle school teachers to have a deeper and more flexible understanding of the mathematics they teach, so they can recognize how the structure of algebra can and should be exposed while teaching arithmetic.     

Toeplitz Operators and Division Theorems in Anisotropic Spaces of Holomorphic Functions in the Polydisc

This work is an introduction to anisotropic spaces, which have an ω-weight of analytic functions and are generalizations of Lipshitz classes in the polydisc. We prove that these classes form an algebra and are invariant with respect to monomial multiplication. These classes are described in terms of derivatives. It is established that Toeplitz operators are bounded in these (Lipschitz and Djrbashian) spaces. As an application, a theorem about the division by good-inner functions in the mentioned classes is proved.     

Using Culturally Responsive Stories in Mathematics: Responses From the Target Audience

This study examined how Black students responded to the utilization of culturally responsive stories in their mathematics class. All students in the two classes participated in mathematics lessons that began with an African American story (culturally responsive to this population), followed by mathematical discussion and concluded with solving problems that correlated to the story. The researcher observed and recorded responses by students during each part of these lessons with protocols. Students independently reflected weekly by answering five questions to share their perspective on the African American stories. The teacher reflected on each lesson as well, describing thoughts on how these students responded to the story in each lesson. This paper examines the analyzed data from the target audience: Black students. Results revealed that Black students responded to the use of African American stories with high self‐rated levels of engagement and enjoyment and that the stories helped them think about mathematics to varying degrees. Since students who are engaged and are thinking about mathematics are more likely to achieve mathematical understanding, the researcher concludes that this strategy should continue to be tested in diverse classrooms with an emphasis on student reflection to determine if the outcomes are transferable and generalizable.     

Exploring a structure for mathematics lessons that initiate learning by activating cognition on challenging tasks

While there is widespread agreement on the importance of incorporating problem solving and reasoning into mathematics classrooms, there is limited specific advice on how this can best happen. This is a report of an aspect of a project that is examining the opportunities and constraints in initiating learning by posing challenging mathematics tasks intended to prompt problem solving and reasoning to students, not only to activate their thinking but also to develop an orientation to persistence. Data were sought from teachers and students in middle primary classes (students aged 8–10 years) via online surveys. One lesson focusing on the concept of equivalence is described in detail although mention is made of other lessons. The research questions focused on the teachers’ reactions to the lesson structure and the specifics of the implementation in a particular school. The results indicate that student learning is facilitated by the particular lesson structure. This article reports on the implementation of this lesson structure and also on the finding that students’ responses to the lessons can be used to inform subsequent learning experiences.     

Generating Scenarios of Division as Sharing and Grouping: A Study of Japanese Elementary and University Students

Elementary school students learn two types of division scenarios: partitive and quotitive. Previous researchers have assumed that the partitive scenario is easier because it reflects the everyday notion of sharing, whereas the quotitive scenario, which represents grouping, is more difficult and is understood gradually in the course of mathematics learning. However, this assumption has not been adequately investigated in empirical studies. The present study examines the assumption in a cross-sectional study. Participants were 336 elementary school students (98 in Grade 3, 82 in Grade 4, 88 in Grade 5, and 68 in Grade 6) and 70 university students who performed two tasks. In the preference task, they generated a division scenario of any type consistent with a given numerical equation. In the problem-posing task, they generated a division scenario consistent with both a numerical equation and a picture representing a partitive or quotitive scenario. On the preference task, students at all grade levels preferred the partitive to the quotitive scenario, and this preference increased with students’ grade level. On the problem-posing task, younger students (Grades 3, 4, and 5) had equivalent success in the partitive and quotitive scenarios, but older students (Grade 6 and university) found the partitive scenario to be easier than the quotitive. Implications for mathematics education are discussed.     

The intangible task – a revelatory case of teaching mathematical thinking in Japanese elementary schools

This paper deals with the nature of teaching mathematical thinking and presents a case study of a single Japanese lesson where the characteristics of mathematical thinking and the teaching thereof are identified in relation to multiplication. The raison d’être for this teaching is questioned and investigated by looking at how multiplication is described in the curriculum and representative textbook material. It is seen how Japanese teachers are institutionally conditioned to incorporate mathematical thinking in the context of multiplication, something which may appear in contrast to other countries. The lesson is analysed using the notion of praxeologies and didactic co-determination conceptualised in the Anthropological Theory of the Didactic.     

Students’ experiences with mathematics teaching and learning: listening to unheard voices

This study documents students’ views about the nature of mathematics, the mathematics learning process and factors within the classroom that are perceived to impact upon the learning of mathematics. The participants were senior secondary school students. Qualitative and quantitative methods were used to understand the students’ views about their experiences with mathematics learning and mathematics classroom environment. Interviews of students and mathematics lesson observations were analysed to understand how students view their mathematics classes. A questionnaire was used to solicit students’ views with regards to teaching approaches in mathematics classes. The results suggest that students consider learning and understanding mathematics to mean being successful in getting the correct answers. Students reported that in the majority of cases, the teaching of mathematics was lecture-oriented. Mathematics language was considered a barrier in learning some topics in mathematics. The use of informal language was also evident during mathematics class lessons.     

Relational Reasoning about Numbers and Operations – Foundation for Calculation Strategy Use in Multi-Digit Multiplication and Division

Theoretical analysis of whole number-based calculation strategies and digit-based algorithms for multi-digit multiplication and division reveals that strategy use includes two kinds of reasoning: reasoning about the relations between numbers and reasoning about the relations between operations. In contrast, algorithms aim to reduce the necessary reasoning processes. In a sample of 221 German fourth graders, both kinds of relational reasoning were operationalized, as well as the use of strategies and algorithms in multiplication and division. The multi-dimensionality of the constructs and their discriminant validity were confirmed by a confirmatory factor analysis. The theoretically proposed, unidirectional relations between the constructs were investigated using a structural equation model: Abilities in reasoning about relations between numbers had a significant positive impact on strategy use in multiplication and division. Abilities in reasoning about relations between operations influenced strategy use in multiplication only. The use of algorithms in multiplication and division was exclusively affected by abilities in reasoning about relations between numbers, and not by abilities about relations between operations. Moreover, a negative effect of the use of digit-based algorithms on the use of whole number-based strategies was identified. Finally, the results of the theoretical and empirical analysis were integrated into a synthesis of existing models about calculation strategy use and development.