Elementary Mathematics Curriculum Research Articles

Examining Bridges in Mathematics and Differential Effects Among English Language Learners

Determining the effectiveness of core mathematics curricula is foundational to evidence-based practice. Examining effectiveness heterogeneity is also crucial to better understanding mathematics achievement among English language learners (ELLs). In this study, we used a quasi-experimental design (difference-in-differences) to examine the impact of a standards-based elementary mathematics curriculum (Bridges in Mathematics) on fifth graders’ annual gains in mathematics achievement in a large midwestern school district compared to the district’s prior curriculum (Investigations). We also investigated whether the effect of Bridges varied across English language proficiency (ELP) levels of ELLs. Students in schools that implemented Bridges (n = 1,839) showed significantly greater mathematics gains compared to those receiving the prior curriculum (n = 3,354; g = 0.25 in change score standard deviations). This effect did not vary significantly across ELP levels. Limitations of this study as well as implications for research and practice with core curricula are discussed. Impact Statement We used a quasi-experimental design to investigate the effect of the Bridges in Mathematics curriculum on student mathematics achievement gains in fifth grade in a large, urban school district. Students who received the curriculum grew measurably more on mathematics scores from the fall to spring of fifth grade than students who received the previous curriculum. English language learners with heterogenous English language proficiency levels and English-proficient peers benefited similarly from the curriculum.

Features of teaching combinatorial elements in primary school mathematics

This article discusses the methods of solving and explaining problems related to the elements of combinatory that are newly introduced into the elementary mathematics curriculum and the teaching methods need to be studied. Solving such problems allows students their knowledge of the problem itself. The decision-making process requires the search for several different solutions. In addition, training aimed at solving combinatorial issues will help develop the quality of thinking, such as variability. Calculation is always carried out according to some properties of objects and is directly related to the activity of classification of objects.

A study on the plane figures of elementary school in the view of transformation geometry

One of the main purposes of geometry learning is the classification of shapes. Thus the activity of classifying shapes from a geometric symmetry (invariant) point of view has sufficient educational significance from Klein's point of view. The concept of translation, line symmetry, and rotational symmetry is introduced as part of activities to raise spatial sense in the current elementary school mathematics curriculum. The above three transformations are basic activities in the concept of congruence. But the current mathematics curriculum tends to deal with three basic concepts and congruence regardless of whether they are related. In fact, three activities and congruent concepts for developing spatial senses was introduced in different the grade-band for each other. In this paper, we shall classify triangles and quadrilaterals from the viewpoint of a symmetry breaking

Pre-Service Elementary Teachers’ Knowledge of Students: The Case of Subtraction

Abstract Introduction: Although there is ambiguity about the elements of teacher knowledge, all researchers accept that being able to anticipate what errors can be made, the reasons for and the strategies to overcome these errors, in short, the knowledge of students is important for student achievement. In this study, knowledge of students refers to being aware of students’ possible errors and underlying reasons for these errors and knowing how to overcome these errors. Based on this consideration, the purpose of this study is to investigate pre-service elementary teachers’ knowledge of students on the subtraction topic. Methods: Considering the purpose, the data were collected from 118 pre-service elementary teachers who were enrolled in a four-year Elementary Teacher Education program via a task-based questionnaire related to the topic of subtraction and semi-structured interviews following the questionnaire. The task-based questionnaire included three completed incorrect subtraction tasks and was prepared considering the related literature and the elementary school mathematics curriculum of Turkey. Each task in the questionnaire contained a different type of error. The pre-service elementary teachers’ answers to the task-based questionnaire were categorized as correct, partially correct, wrong, or no answer by means of categorical analysis. The pre-service elementary teachers who gave correct and partially correct answers to the tasks were asked to participate in the second part of the study to learn their possible strategies to overcome the errors made in the tasks. Results: As a result of the quantitative and qualitative analysis of the pre-service elementary teachers’ responses, it was found that their knowledge of students for subtraction is limited. Specifically, although the pre-service elementary teachers were partially able to identify the errors in the first and second task, they were not able to identify the error in the third task. Furthermore, they were better able to determine a more commonly occurring subtraction error compared to the uncommon ones. The pre-service elementary teachers could not identify the underlying reasons that led the students to the errors. They could not explain what conceptual knowledge related to the topic of subtraction the student lacked that resulted in the errors. As the pre-service teachers did not attain these reasons, they were not able to provide strategies to overcome these errors different than restating the rules or procedures to overcome students’ errors. Discussion: Only being able to identify the errors is not enough to make the instruction effective. Teachers also need to know and provide a rationale for why the errors happen and how to overcome them (Even & Tirosh, 1995). Contrary to this statement, the pre-service elementary teachers in this study could not attempt to understand the students’ thinking or could not explain the reasons behind students’ errors with the notion of subtraction. Son (2013) emphasizes that pre-service teachers tend to explain the reasons for students’ errors as procedural. Similar to this emphasis, the pre-service teachers identified the errors as resulting from not applying procedures carefully or not enough knowing algorithms to find correct solutions. However, without addressing the reasons or focusing on procedures does not promote students’ understanding of the related topic (An, Kulm, & Wu, 2004). The pre-service teachers’ difficulties in identifying the possible conceptual reasons may result from deficits in knowledge of students (Fennema & Franke, 1992; Ball et al., 2008). Limitations: The findings were limited with the responses of the pre-service elementary teachers participating in this study. Moreover, the pre-service elementary teachers’ knowledge of students was investigated within the scope of subtraction. Conclusions: Although the pre-service elementary teachers could identify students’ errors on subtraction, they had difficulty in identifying the reasons and suggesting strategies to overcome the errors. Therefore, it can be concluded that the pre-service teachers’ knowledge of students for the topic of subtraction is limited. An implication of this is that teacher educators need to include error-analysis tasks to help pre-service teachers be aware of the importance of their knowledge of students.

중학교 사회 교과서 분석을 통한 초등학교 통계교육 개선 방안 탐색: 그래프 지도를 중심으로

In this study, in order to explore ways that statistical graphs teaching in elementary mathematics education can be improved, first, the types of statistical graphs that appear in middle school social textbooks were examined, and second, the characteristics that differentiated these from statistical graphs that appeared in elementary school mathematics were investigated. This analysis indicated that 88.2% of the graphs presented in the middle school social textbooks were addressed in the elementary mathematics curriculum. The frequency of use (excluding mixed graphs) was in the order of bar graphs, line graphs, ratio graphs, and picture graphs. However, most of the statistical graphs did not follow the standardized framework suggested in elementary school mathematics textbooks, as they were instead transformed in various ways. The features that differentiated middle school mathematics graphs from graphs experienced in elementary school mathematics differed for each type of graph. For example, in the case of bar graphs, there were the omission of axes, deformation of bars, and expression of multiple variables, while in the case of line graphs, there were expressions of multiple variables and accurate representations of variables.

초등 수학에서의 미지수 표현 방식에 대한 다차원 교육과정적 관점에서의 논의

This study discussed the representation method of unknowns from the perspective of multi-dimensional curriculum. For this, The elementary mathematics curriculum revised in 2015 as the intended curriculum was compared and analyzed from the 7th curriculum to the curriculum revised in 2009. The mathematics textbooks of 5th and 6th grade of curriculum revised in 2015 as the written curriculum were compared and analyzed with the 5th and 6th grade textbooks for each curriculum. For elementary school teachers, a survey study was conducted regarding the representation method of teaching unknown concepts as an implemented curriculum. A survey was conducted on elementary students' understanding of the unknown concepts as an accomplished curriculum. As a result, first, it is necessary to actively discuss the representation method of unknowns so that early algebra education can be effectively performed in the upper grades of elementary school. Second, when introducing unknown concepts, it is necessary to consider a step-by-step instructional method of using character symbols in elementary school, in order to help understand the concept using shape symbols and ultimately to connect with secondary mathematics. Third, in terms of strengthening the connection between elementary mathematics and secondary mathematics, early algebra education in elementary schools linked to full-scale algebra education in secondary schools can be pursued through the introducing the unknown representation method using character symbols in the 5th grade and 6th grade of elementary school mathematics.

다각형 개념 지도 시 오목다각형 도입에 대한 다차원 교육과정적 관점에서의 고찰

This study discusses including concave polygons in teaching polygon concepts from the perspective of a series of multi-dimensional curriculum. The elementary mathematics curriculum revised in 2015 as the intended curriculum was compared and analyzed from the primary curriculum to the curriculum revised in 2009. The elementary mathematics textbooks of the curriculum revised in 2015 as written curriculum were compared and analyzed. For elementary school teachers, a survey study was conducted regarding the use and method of concave polygons when teaching polygon concepts as an implemented curriculum. A survey was conducted on elementary students' understanding of polygonal concepts as an accomplished curriculum. As a result, first, in consideration of the developmental characteristics of elementary school students, it is necessary to suggest clear limitations related to the polygon concept map in textbooks which is the intended curriculum and written curriculum. Second, since the teachers' PCK has significant impact on the students' mathematical concept formation, teacher training programs and pre-service teacher education should be strengthened to enable teachers to form accurate PCK on polygonal concepts. Third, teaching polygonal concepts by presenting concave polygons in accordance with the definition of polygons in the written curriculum, which can facilitate students' learning and understanding precise polygonal concepts, prevents the formation of misconceptions.

함수적 사고를 강조한 ‘규칙과 대응’ 수업에 참여한 초등학생들의 이해 분석

Recently, there has been an active effort to reflect on the perspective of functional thinking in an elementary mathematics curriculum and to explore its effects. Given this, the purpose of this study was to compare students’ understanding of the unit of “Patterns and Correspondence” between two groups: the intervention group who learned it with new materials emphasizing functional thinking and the non-intervention group who learned it with the textbook. For this purpose, the students from three classes participated in this study and they completed pre-post test. The results of this study showed that in the post-test the intervention group performed better than the non-intervention group in exploring for a correspondence relationship using a non-sequential T-table, representing it by an expression with variables, and finding out real-life contexts appropriate to the given expression of a correspondence relationship. This paper closes with some implications on how to teach functional thinking in elementary mathematics.

Seeking Intersections: Math Degrees, Beliefs, and Elementary Teacher Knowledge

Some views of elementary teacher preparation may include the idea that undergraduate mathematics courses are useful or even crucial for prospective teachers to have. This study sought to identify teacher candidates with mathematics degrees, in order to study their mathematical preparation. The identified prospective teachers were studied to investigate initial understanding of the conceptual underpinnings of elementary mathematics, as well as their beliefs about the importance of conceptual learning. Participants were given a written instrument related to elementary school–level conceptual mathematics knowledge, administered at the beginning and end of an elementary (up to grade 8) mathematics curriculum and instruction (“methods”) course. Prospective teachers with university mathematics backgrounds did not appear generally well-prepared in terms of their ability to model or explain the conceptual basis of common elementary school concepts, although they generally claimed to feel such understanding was important. After taking part in the methods course, which focused strongly on experiences learning the mathematical ideas conceptually, they were able to demonstrate increased understanding of the necessary concepts, models, and reasoning related to elementary mathematics teaching.

Integrating Computational Thinking into Elementary Science Curriculum: an Examination of Activities that Support Students\u2019 Computational Thinking in the Service of Disciplinary Learning

Using an example of a grade 3 science unit about population changes during competition for resources, we describe how we integrated computational thinking (CT) into existing curriculum identifying three levels of depth of integration: identifying connections that already exist, enhancing and strengthening connections, and extending units to include activities that more explicitly develop students’ CT. We discuss students’ understanding of the relationship between a simple model of an ecosystem and the actual phenomenon it represents, their engagement with the unit’s data-gathering and data analysis activities, their ability to engage in sense-making regarding data they generated and analyzed, and how collectively the study supports their understanding of the complex system. This example module is part of “Broadening Participation of Elementary School Teachers and Students in Computer Science through STEM Integration and Statewide Collaboration,” a National Science Foundation-funded collaboration among Massachusetts teacher educators, researchers, teachers, and state-level education administrators that developed and implemented a number of elementary grade, CT-integrated science and mathematics curriculum modules. Collectively, these modules are designed to develop practices related to several key CT topics: abstraction, data, modeling and simulation, and algorithms. These CT topics support the development of core skills related to, but not exclusively the domain of, computer science. The strategy of integrating CT into core elementary STEM subject areas was intended to cultivate CT practices in support of science learning.

Enhancing achievement and interest in mathematics learning through Math-Island

Conventional teacher-led instruction remains dominant in most elementary mathematics classrooms in Taiwan. Under such instruction, the teacher can rarely take care of all students. Many students may then continue to fall behind the standard of mathematics achievement and lose their interest in mathematics; they eventually give up on learning mathematics. In fact, students in Taiwan generally have lower interest in learning mathematics compared to many other regions/countries. Thus, how to enhance students’ mathematics achievement and interest are two major problems, especially for those low-achieving students. This paper describes how we designed a game-based learning environment, called Math-Island, by incorporating the mechanisms of a construction management game into the knowledge map of the elementary mathematics curriculum. We also report an experiment conducted with 215 elementary students for 2 years, from grade 2 to grade 3. In this experiment, in addition to teacher-led instruction in the classroom, students were directed to learn with Math-Island by using their own tablets at school and at home. As a result of this experiment, we found that there is an increase in students’ mathematics achievement, especially in the calculation and word problems. Moreover, the achievements of low-achieving students in the experimental school outperformed the low-achieving students in the control school (a control group in another school) in word problems. Moreover, both the low-achieving students and the high-achieving students in the experimental school maintained a rather high level of interest in mathematics and in the system.

Launching forth: preservice teachers translating elementary mathematics curriculum into lessons

Translating a mathematics lesson from curriculum materials into instruction is not straightforward, and launching, or beginning, a lesson such that students will become productively engaged in the target mathematics is nontrivial. The purpose of the study was to investigate preservice teachers’ curricular noticing as they draw upon curriculum materials to design instruction, with a focus on how to launch that lesson. We engaged preservice elementary licensure students at two universities in a four-part process of analyzing mathematics curriculum, planning a lesson, demonstrating their visualization of enactment through animating their lesson launch, and reflecting on the process. Findings indicate that the focal case study pair modified the curricular materials to model mathematical aspects of fractions, adapted the introduction of key academic vocabulary, and introduced materials not mentioned in the curriculum to draw children into the lesson. We discuss implications for preservice teachers’ planning of the lesson launch and their curricular noticing.

The Effect of Constructivist Mathematics on Achievement in Rural Schools

International assessment data indicate American students are not competing with their counterparts in other countries. The mathematics curriculum and pedagogy are not preparing students to compete in a global economy. This study compared student achievement using sixth grade mathematics results from the Illinois Standards Achievement Test. Specifically, the study compared the results of students in three different rural school districts, all of whom were receiving instruction in three different mathematics curricula. In one district, students received seven years of the K-6 Everyday Mathematics curriculum which was compared with students who received seven years of instruction using a traditional mathematics curriculum in the second district and in the third district scores were compared with students who were taught using a traditional mathematics curriculum supplemented with Mountain Math. The results of this study indicate the constructivist K-6 elementary mathematics curriculum did not lead to higher levels in math achievement when compared with the traditional methods of instruction.

What is the point? Examining how curriculum materials articulate mathematical goals and how teachers steer instruction

This study examined how authors of four different elementary mathematics curriculum materials articulate the mathematical goals of the lesson and orient the activities toward the goals, as well as the extent to which teachers steered instruction toward the goals during enactment. Through document analysis, we identified four ways in which authors support teachers in identifying and orienting the lesson toward the goals. Analysis of the enacted lessons of 8 teachers suggests a correlation between the depth to which the goals were communicated across the four categories and the extent to which teachers steered the instruction towards the goals. The authors conclude with recommendations to curriculum developers for communicating mathematical goals in ergonomic ways.

The Subjects That the Pre-service Classroom Teachers Perceive as Difficult in Elementary Mathematics Curriculum

OECD-PISA which evaluates the education quality of Europe focused on mathematics, and Turkey was ranked 50 among 72 countries. When the mathematics literacy was analyzed in terms of years, it is seen that the performance of the students in Turkey in PISA 2015 was lower compared to PISA 2009 and PISA 2012. This shows that students have difficulty in comprehending the subjects in the mathematics curriculum. Therefore, learning experiences designed for the pupils to have rich mathematics experiences during elementary school is of higher importance. At this point, it is necessary to know the subjects that the pre-service teachers, who will be teachers of the future, are challenged in teaching mathematics. Therefore, the purpose of study is to determine the subjects in which elementary school pre-service teachers perceive teaching in elementary school mathematics as difficult and to examine the causes of these difficulties. The case study method is used in this study and the participants of the study are 120 pre-service classroom teachers who study in the fourth grade in a state university. Participants were administered a 19-item learning difficulty index covering mathematics learning areas of 1-4 grades (n = 120). The distribution of answers given by the pre-service teachers for each topic and the learning difficulty index for each topic were calculated. Interviews were done with the pre-service teachers in response to their responses to the questionnaire items to understand the causes of the difficulties. In interviews conducted one by one, the pre-service teachers were asked general mathematics lecture notes, mathematics fear level and the reasons for the topics they found difficult to teach.

One curriculum and two textbooks: opportunity to learn in terms of mathematical problem solving

This paper explores how problem solving is approached in two 6th grade mathematics textbooks and national elementary mathematics curriculum in Turkey. The curriculum and textbooks’ problem solving approaches are analysed and compared in the light of three approaches of ‘teaching for, about and through problem solving’. Four learning objectives from different content areas (e.g. algebra, data processing) from the curriculum are determined and their treatments in the textbooks are specifically analysed. The analysis showed that the curriculum did not have a prescriptive stance on any one of problem solving approaches. Similarly, the textbooks also did not follow any approach faithfully. ‘Teaching for problem solving’ approach was not employed while ‘teaching through problem solving’ approach was used for only one learning objective in the textbooks. ‘Teaching about problem solving’ approach was encountered in terms of only one problem solving heuristic (table drawing). We discuss the implications of these findings and note some further research areas.

“It Does Not Exist”: Infinity and Division by Zero in the Ontario Mathematics Curriculum

This study traces curricular support for concepts related to infinity and division by zero, in both elementary and secondary mathematics curriculum documents in our province. It was motivated by results of our previous and ongoing work related to transition to post-secondary studies in mathematics, in which we identified consistent gaps and misconceptions around limits and infinity. Our findings are startling in that, in the school curricula, many expected references to concepts related to division by zero, infinity, and limits are simply not present (expected in the sense of being needed to provide essential support for the material which, according to curriculum documents, must be covered). Recommendations for future curriculum revisions are included.

우리나라와 일본의 초등학교 수학과 교육과정 측정 영역 비교・분석

In this study, the actual state of teaching seven extensive quantities (time, length, capacity, weight, area, angle measure, volume) of measurement domain are analyzed comparatively between the 2015 revised elementary school mathematics curriculum in Korea and the 2017 revised elementary school mathematics curriculum in Japan in terms of comparison in measurement, direct measurement, indirect measurement, and estimation in measurement. From the results of this comparative analysis, some implications for discussion on the development of the next elementary school mathematics textbook and the next elementary mathematics curriculum can be suggested. First, it is necessary to discuss on clarifying the range of handling of comparison, direct measurement, indirect measurement, estimation of seven extensive quantities respectively. Second, it is necessary to discuss on doing direct comparison when intuitive comparison is difficult. Third, it is necessary to discuss on reconsidering indirect comparison of weights. Fourth, it is necessary to discuss on reconsidering measurement using arbitrary units in case of angular measures. Fifth, it is necessary to discuss on dealing with estimating the area of 1 ㎠ and 1 ㎡ and the volume of 1 ㎤ and 1 ㎥ for the purpose to make rough guesses their size respectively.

미군정기의 초등학교 6학년 수학의 지도 내용 분석

In this study the contents of the 6th grade elementary school mathematics in U.S. military government period in Korea were analyzed as part of restoring the history of Korean elementary school mathematics education. The literature to be analyzed is ≪Elementary Sembon 6-1≫ and ≪Elementary Sembon 6-2≫. An analysis was made in terms of three aspects. In 2015 revised elementary mathematics curriculum, what contents in this two textbooks are not dealt with, what contents in this two textbooks are not dealt with explicitly but implicitly, and what contents in this two textbooks are dealt with? Especially, there are three kinds of contents which are not dealt with in 2015 revised elementary mathematics curriculum. Those are the contents which are shifted from elementary school mathematics to middle school mathematics, the contents which are dealt with in elementary school mathematics without shifting to middle school mathematics, and the contents which do not belong to school mathematics by their character. On the contents of the 6th grade elementary school mathematics in U.S. military government period in Korea, the findings based on results of the analysis can be obtained. First, the contents dealt with at that times were very different from the ones currently dealt with. Second, mathematics oriented contents and lifestyle oriented contents were mixed.

Curriculum spaces for connecting to children’s multiple mathematical knowledge bases

Elementary mathematics curriculum materials can serve as a lever for instructional change. In this paper, we promote a particular kind of instructional change: supporting teachers in learning to integrate children’s multiple mathematical knowledge bases (MMKB), including children’s mathematical thinking and children’s home and community-based mathematical funds of knowledge, in instruction. A powerful means of supporting pre-service teachers in integrating children’s MMKB in instruction may be to scaffold teachers’ noticing of potential spaces in elementary mathematics curriculum materials for connecting to children’s MMKB and then developing practices for leveraging these spaces during instruction. We focus on existing and potential spaces in written curriculum materials, or curriculum spaces, so as to better support teachers in enacting curriculum that opens spaces for connecting to children’s MMKB.