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Achievement Results for Second and Third Graders Using the Standards-Based Curriculum Everyday Mathematics
Abstract
Students using Everyday Mathematics (EM), developed to incorporate ideas from the NCTM Standards, were at normative U.S. levels on multidigit addition and subtraction symbolic computation on traditional, reform-based, and EM-specific test items. Heterogeneous EM 2nd graders scored higher than middle- to upper-middle-class U.S. traditional students on 2 number sense items, matched them on others, and were equivalent to a middle-class Japanese group. On a computation test, the EM 2nd graders outperformed the U.S. traditional students on 3 items involving 3-digit numbers and were outperformed on the 6 most difficult test items by the Japanese children. EM 3rd graders outscored traditional U.S. students on place value and numeration, reasoning, geometry, data, and number-story items.
... The support for standards-based reform in the classroom has been documented by other research studies (Bay, Beem, Reys, Papick, & Barnes, 1999;Fuson, Carroll, & Drueck, 2000;Manouchehri & Goodman, 1998;Riordan & Noyce, 2001). Manouchehri and Goodman studied and evaluated the implementation of four standards-based curricular materials of over 60 middle school mathematics teachers from 12 rural, suburban, and urban school districts in the state of Missouri. ...
... Their scores were used as the focus of this study, and the analysis of the data showed positive results supporting a standards-based approach to teaching mathematics when compared to the NAEP comparison group. Both studies showed students can perform better with standards-based curriculum than with traditional curriculum approaches (Fuson et al., 2000). Riordan and Noyce (2001) conducted a study with approximately 9,000 fourth-grade and eighth-grade students. ...
... indicated the attitudes of the students are important to the learning process. Students who are actively engaged in their learning have an increased opportunity to master mathematical concepts and problem solving abilities that are promoted by the PSSM (Fuson et al., 2000;Riordan & Noyce, 2001). ...
... The recommendations made in Principles and Standards for School Mathematics (2000) were made only after years of analyzing an extensive research base on children"s thinking and learning. Evidence is now available demonstrating that instructional techniques that use the standards recommended by the NCTM have a positive effect on student achievement (Fuson et al., 2000;Hiebert, 1999;McGinley et al., 2007;Riordan & Noyce, 2001). When McGinley, Bynum, Rose and Kokkinis (2007) began to study the characteristics of a few great teachers (based on students" test scores) in a Charleston, South Carolina school district, they discovered a few patterns. ...
... On tradition addition and subtraction, EM students performed as well as students using traditional approaches. On a wide range of other mathematically and conceptually demanding tasks, EM students outperformed other groups (Fuson et al., 2000). In addition, the EM students had opportunities to study a wider range of curriculum topics, e.g. ...
... The research literature indicates that direct instruction continues to dominate as the primary method of teaching mathematics (Allen & Johnston -Wilder, 2004;Boaler, 2008;Rosenstein, 2007; (Fuson et al., 2000;Hiebert, 1999;Johnson, 2000;McCaffrey et al., 2001;NCTM, 2000;Riordan & Noyce, 2001). Math stations that incorporate these instructional practices, may therefore, have a decided advantage over traditional math instruction. ...
... This section investigates the response to the Everyday Mathematics curriculum that was developed in Chicago. Carroll (1997) and Fuson, Carroll and Drueck (2000) investigated the Everyday Mathematics Curriculum and found that students achieved higher test results than with traditional mathematics curriculum. Sood and Jitendra (2007) The goal of this curriculum is for students to work in small groups and to investigate real-world context with the use of manipulative, calculators and other tools to solve problems. ...
... While the previous study was conducted in 1997, Fuson, Carroll and Drueck (2000) conducted another study on the achievement results of third grade mathematics in relation to the Everyday Mathematics Curriculum (EM). The researchers combined the results of the achievement of second and third graders into two different studies. ...
... The participants of that test consisted of 29 U.S. Second graders in San Francisco from an upper to middle class families and 33 Japanese students attending a middle class public school in Japan. Fuson et al. (2000), stated that while the two tests given to the San Francisco and Japanese students as a comparison, that they were biased against the EM curriculum. The test results comprised of different analysis areas: Numbers-Sense test, Mathematics Achievement Test and Additional items. ...
... En una tercera alternativa más eficaz, los niños tratan los conceptos en profundidad y en una secuencia lógica. Tal profundidad y coherencia permiten a los niños desarrollar, construir, probar y reflexionar sus conocimientos matemáticos [10,23,59,60]. Esta alternativa también aumenta las posibilidades de que los maestros detecten lagunas en el conocimiento de los niños y dediquen tiempo a tratarlas. ...
... Al tomar estas opciones, los buenos maestros de educación infantil se basan en el conocimiento matemático informal de los niños y en sus experiencias previas, siempre considerando el bagaje cultural y lingüístico de los niños [23]. 10. Apoyar el aprendizaje de los niños mediante la evaluación continua y reflexiva del conocimiento, destrezas y estrategias de todos los niños. ...
Declaración conjunta de posición de la National Association for the Education of Young Children (Asociación Nacional para la Educación Infantil, NAEYC) y el National Council of Teachers of Mathematics (Consejo Nacional de Profesores de Matemáticas, NCTM) sobre Matemáticas en la Educación Infantil. Adoptada en 2002. Actualizada en 2010.
... In addition to these, working in groups that consisted of 3 or 4 students, group and class discussions, sharing ideas with the peers in the group and whole class, and the researcher were other prominent properties of learning environment. We preferred this kind of intervention because many studies showed that such learning environments led to greater mathematical understanding and problem solving achievement, especially for middle and high achieving students (Boaler, 1998;Fuson et al., 2000). During the instruction, the researcher took roles such as presenting the problem situations, encouraging children to express their solutions without worrying about making mistakes, guiding and giving hints to students when needed, and leading discussions. ...
... First of all, when we evaluated influence of the designed learning environment regardless the domain of fractions, we found that classroom interaction and working in group enhanced pupils' comprehension at almost each achievement levels. This finding matched up with the results of studies by Boaler (1998) and Fuson, Carroll, and Drueck (2000). ...
The purpose of this experimental study is to investigate whether low, middle and high achieving students could benefit at the same extent from a fraction instruction which was prepared according to basic principles of Socio-constructivism and Realistic Mathematics Education. To this end, an instruction starting with sharing situations, and focusing on group and class discussions was carried out with experimental group. Meanwhile, the students in the control group attended their regular lessons. Both groups were consisted of 27 fourth and 28 fifth grade students. Three tests were administered to the participants: General Mathematical Achievement Test, Fractional Understanding Pre Test and Fractional Understanding Post Test. According to t-test and ANCOVA results, the positive effect of the designed learning environment on fractional understanding of high, middle and low achieving students was substantially similar. Likewise, high, middle and low achievers in the control group also did not show any difference with regard to effect of the traditional learning environment on their fractional understanding.
... Thoroughly, all the indicators include both orders of thinking skills, namely Lower-Order Thinking skills (LOTs) and Higher-Order Thinking skills (HOTs), according to the age-based cognitive development of the sampling frame in the respective articles. de León, Jiménez, and Hernández-Cabrera (2020), Fuson, Caroll, and Drueck (2000), Keijzer, and Terwel (2003), Sarama, Clements, Swaminathan, McMillen, and Gomez (2003), Somasundram, Akmar, and Eu (2019) Finally, throughout all the codes, the core themes emerged were labeled as the analogous numerical cognition. As the research in mathematics education has seemed to categorize the indicator of number sense complexity according to age, the analogous numerical cognition was a broad term referred as the biological or natural human perception of numerical conception acquired in primary school years. ...
The definitions serve the purpose of communication and preservation of knowledge in scientific inquiry. However, it is quite often to perceive number sense concept without well-accepted definitions in the field of mathematics education research. Despite the mentioned issue, the current literature on children's number sense provide a glean for introducing, implementing, and even measuring the number sense using the specific and contextualized indicators in early mathematics. Consequently, this phenomenon offers to bridge a gap in the literature concerning definitions of number sense and its indicators. This article systematically reviews on the indicators in measuring children's number sense based on the past research guided by the PRISMA statements. The metadata were analysed using open-coding and were further re-coded through axial coding and selective coding to form a definition of number sense. This article discusses on limitations, implications, and the future research directions for studying children's number sense in the primary schools' mathematics.
... For one, the modeling and coteaching coaching cycles differed with respect to the coaches and teachers involved. The cycles also differed in the curriculum that was used during the lessons: the modeling coaching cycles focused on Calendar Math, whereas the coteaching coaching cycles primarily used Everyday Mathematics 4. That is important to consider given that some scholars have criticized Calendar Math for a variety of reasons (Beneke et al., 2008;Ethridge & King, 2005), including that it may not offer the types of learning opportunities envisioned by Common Core (Parks, 2008), whereas Everyday Mathematics is thought to promote greater conceptual understanding for students (Fuson et al., 2000). Given that the two curricula differed in the extent to which they aligned with the types of mathematical learning opportunities we looked for, we do not seek to make general claims about differences in learning opportunities that modeling and coteaching can offer from the data reported here. ...
Coaching is a popular, yet costly, professional development structure. Therefore, understanding the learning opportunities coaching provides is essential. Following a framework by Campbell and Griffin (2017), we explore five elementary school teachers’ learning opportunities during 15 meetings and 23 lessons with two instructional coaches in two schools. Using Greeno’s (2005) situative perspective, we focus on coach–teacher dyads, examining the substance and depth (Coburn, 2003) of their conversations. Results indicate that most coach–teacher talk centered on logistics, whereas mathematics conversations were rare. We consider institutional contexts that shaped the dyads’ discussions, ultimately viewing the coaches and teachers as “sensible beings" (Leatham, 2006) with valid reasons for focusing discussions as they did. Coaching implications and directions for research are discussed.
... It was found that the extent of the social turn was not dominant after this period (Gates & Joegensen, 2015;Jablonka & Bergsten, 2010) and socio-cultural perspective started to place in the studies (Lerman, 2006). By the end of the decade, a number of studies were found helpful for reform-based mathematics methods and curricula (Cohen & Hill, 2001;Fuson, Carroll, & Drueck, 2000;McCaffrey et al., 2001;Schoenfeld, 2002). According to the changes in mathematical content, the interest in the addition and subtraction, problem solving has showed a peak during the 1980s. ...
The purpose of this study is to establish the evolution and expose the trends of research in mathematics education between 1980 and 2019. The bibliometric analysis of the articles in Web of Science database indicated four-clustered structure. The first cluster covers the items related to the theoretical framework of mathematics education whereas the second cluster has the terms defining the methods for effective mathematics instruction. The third cluster includes the concepts interrelated to mathematics education while the fourth cluster encloses the studies about international mathematics assessments. The earlier studies look mathematics education mostly in students’ perspective and investigates generalization, restructuring, interiorization and representation. Between 1995 and 2010, curriculum and teacher-related factors were dominant in mathematics education studies. After 2010, the articles investigated specific topics and carried the traces from all stakeholders in mathematics education. The investigation on the trends of mathematics education would provide gain insight about the areas that need more research, contribute to the researchers, teachers, students and policy makers in this field and light the way for further studies.
... Studies involving elementary students showed that students taught through the PSA had higher levels of mathematical understanding and problem solving skills on a computation test than those taught with the conventional methods (Fuson et al., 2000). Other studies involving middle school students (Romberg & Shafer, 2002) revealed that students taught with the Problem Based Instruction had higher levels of mathematical understanding than the students taught by the traditional instruction. ...
... Early studies stated that students' performance in problem-solving within the standards-based curriculum is better (Bell, 1998;Boaler, 1998;Fuson, Carroll and Drueck, 2000;Hiebert, 2003;Reys, Robinsoni Sconiers, and Mark, 1999;Riordin and Noyce, 2001;Stein, Grover, and Hennigsen, 1996;Stein and Lane, 1996;Wood and Sellers, 1996, 1997cited in van de Walle, Karp, Bay-Williams, 2007. When evaluated with the findings of this research, it can be said the improvement of student performance will be accomplished through improving the areas that have stayed behind in meeting the process standards. ...
Bu araştırma ilkokul matematik dersi öğretim programının Ulusal Matematik Öğretmenleri Konseyi-National Council of Mathematics Teachers-(NTCM) tarafından önerilen beş süreç standardı temelinde gösterdiği dağılımı belirlemek amacıyla gerçekleştirilmiştir. Çalışma bir nitel araştırma olup araştırma verileri doküman analizi ile analiz edilmiştir. NTCM' in önerdiği süreç standartlarına kod ve sayılar verilerek İlkokul Matematik Dersi Öğretim Programında (1-4) belirtilen kazanımların karşılık geldiği standartlar belirlenmiştir. Araştırma sürecinde beş temel süreç standardının içinde yer alan toplam on sekiz standart, kazanımların karşılık gelme durumlarına göre detaylı olarak değerlendirilmiştir. Araştırma sonucunda programda yer alan kazanımların sırasıyla en çok; ilişkilendirme ve temsil standartlarına karşılık geldiği, karşılanmakta en geride kalınan standardın ise akıl yürütme ve ispat olduğu görülmüştür.
Abstract This study aims to determine the distribution of learning outcomes, which the Turkish primary school mathematics curriculum includes, based on five process standards proposed by the National Council of Mathematics Teachers (NTCM). The qualitative research design was used in the study. The data were analyzed by the document analysis technique. Codes and numbers were given to the process standards to determine which learning outcomes (Primary School Mathematics Curriculum, 1-4) matching up to the standards. In the research process a total of eighteen standards, among the five process standards, were evaluated in detail according to the being equipoise to the learning outcomes. Results showed the learning outcomes in the curriculum be in mostly equipoise to the connection and representation standards. Also, it is another result of the research that the learning outcomes have stayed behind in meeting reasoning and proof standards.
... This approach, proponents argue, results in students who deeply understand the mathematical ideas they have generated and refined through their problem solving and are prepared to apply the mathematics they have learned in unfamiliar situations (Cai 2003;Schroeder and Lester 1989). Empirical studies of elementary (Carpenter et al. 1998;Cobb et al. 1991;Fuson et al. 2000;Hiebert and Wearne 1993;Wood and Sellers 1997) and middle school students (Ridgway et al. 2003;Romberg and Shafer 2003) support the notion that students who learn through solving problems (including CPs) demonstrate higher degrees of conceptual understanding and problem solving competencies than those who learn through a more traditional approach. ...
Teachers are encouraged to connect mathematic instruction to the real-world by posing tasks that are situated in rich, relevant contexts, but research has found that many teachers integrate contextual problems (CPs) as motivators rather than as supports for conceptual development. To provide insight into how teachers’ conceptions about CPs shift as they teach through contextual problem solving, we interviewed six teachers before and after they taught from a unit designed from principles of realistic mathematics education, an instructional design theory which positions realistic contexts as learning supports. Our findings indicate that teachers initially viewed CPs primarily as affective or motivating enhancements, but after teaching the RME unit with university-based support, the teachers articulated integrated understandings of how CPs can function as supports for conceptual development. The teachers articulated how CPs provide initial access, sites for progressive representational formalization, and references to which students can fall back in order to interpret subsequent tasks. The authors identify connections between these ideas and the support provided by the university team and the teachers’ guides.
... There is a growing body of research examining the effects of different levels of tasks and these studies demonstrate that high-level cognitive demand tasks make a difference in how much and how well students learn [4,17,[24][25][26][27]. Specifically, Stein and Lane [17] assert that the greatest student learning gains occur if teachers are able to maintain the cognitive demand of high-level tasks during instruction. ...
The purpose of this paper is to examine the cognitive demand levels of tasks used by an in-service primary teacher during length measurement and perimeter instruction and to examine a possible link between these tasks and the teacher’s mathematical knowledge in teaching. For this purpose, a case study approach was used and the data was drawn from classroom observations, semi-structured interviews, and field notes. Specific tasks from length measurement and perimeter instruction were presented and analyzed according to the Mathematical Tasks Framework. Then, how these tasks gave information about the teacher’s mathematical knowledge in teaching in the length measurement and perimeter topics was examined according to the Knowledge Quartet model. According to the findings of the study, the tasks used during length measurement and perimeter instruction were mostly categorized as low-level tasks. In addition, teacher’s mathematical knowledge in teaching affected the implementation of the tasks.
... Also, it was identified that the opportunity led to enhance learning. For instance, the EM group considerably well performed in the National Assessment of Educational Progress sample on geometry items (Fuson et al, 2000). ...
The multifarious role play by mathematics instructional materials in teaching and learning is evident and the process of adoption of textbooks for students is critical to the selection of high-quality instructional materials. Hence, the instructional material in the textbooks is one that must be scrutinized through a comprehensive and reliable procedure and must be supported by published research in the field of mathematics education. The selected curriculum materials must align with and support the need for implementation of state or country level curriculum standards. The purpose of this paper is to provide a comprehensive research-based framework that analyzes salient features of instructional materials which build mathematical proficiency among students of middles grades. The framework will assist state-level textbooks evaluation teams, school administrators, and teachers in selecting mathematics curriculum materials that support the implementation of mathematics curriculum standards. The proposed framework may also provide significant information that can be useful for curriculum developers and schools while they are making decisions regarding modifications in available mathematics materials and monitoring the quality of published materials for better attainment of students" learning.
... Precisamente desde ese enfoque de acompañamiento progresivo, y dado el propósito investigativo de favorecer la comprensión del valor posicional y dar cuenta del avance de los estudiantes, se consideraron en la estructuración del marco conceptual de este estudio, aquellos trabajos teóricos o empíricos, que asumen la comprensión del valor posicional y del SDN en términos del desarrollo gradual de habilidades, y de la construcción progresiva de unidades (simples o compuestas) para incrementar la flexibilidad en el conteo, en la realización de particiones, en la agrupación y en la comparación de cantidades; constructos que desde la investigación en educación matemática (Alba y Quintero;2016;Angulo y Herrera, 2009;Baturo, 2002;Bruno y Noda, 2014;Chandler y Kamii, 2009;Fischer, 1990;Fuson, Carroll y Drueck, 2000;Fuson et al, 1997;Kamii y Joseph, 1988;Salazar y Vivas, 2013;Zúñiga, 2015) han mostrado ser esenciales para analizar la enseñanza y el aprendizaje del valor posicional. ...
RESUMEN Se reporta un estudio en el cual se propone una estrategia de enseñanza para hacer frente y, eventualmente, superar las dificultades que presentan estudiantes de 2º de educación básica primaria (5-7 años) de dos colegios públicos de Bogotá (Colombia), en relación con el concepto de valor posicional. A partir de un análisis del contexto de aula de ambos grupos y de la revisión de literatura relacionada, se diseñó, implementó y evaluó una unidad didáctica bajo el enfoque de la Enseñanza para la Comprensión, con el propósito de que los estudiantes avanzaran en la comprensión del concepto de valor posicional. El estudio se realizó desde un enfoque cualitativo de alcance descriptivo, utilizando el diseño metodológico de investigación-acción. Los hallazgos evidencian que los estudiantes avanzaron en la comprensión del concepto de valor posicional, a través de los desempeños de comprensión propuestos en la unidad didáctica. Palabras clave: Educación matemática infantil, valor posicional, estrategia de enseñanza, Enseñanza para la Comprensión.
ABSTRACT
This paper reports a study that proposes an alternative teaching strategy to cope with and eventually overcome some difficulties presented by second graders (ages 5-7) at two public Elementary schools in Bogota (Colombia). Based on the analysis of the classroom contexts in both groups and on a literary review on the field, a teaching unit, founded on the Teaching for Understanding framework, was designed, implemented and evaluated. This strategy intended to enhance students' understanding of the concept of place value. This was a qualitative-descriptive study framed within the Action-Research design. The findings show that students made progress in understanding the concept of place value through the activities proposed in the teaching unit.
... A number of such curricula have been studied in well-designed comparative studies (i.e., the curricula emphasizing understanding have been compared with more conventional curricula, which involve more direct teaching of formula and routines). In general, such curricula have fared very well in such comparisons, with student mathematical achievement generally higher when understanding, reflection, and teacher-assisted discovery of strategies is emphasized (e.g., Boalar, 1998;Carroll, 1997;Cramer, Post, & delMas, 2002;Fuson, Carroll, & Drueck, 2000;Hollar & Norwood, 1999;Huntley, 2000;McCaffrey, Hamilton, Stecher, Klein, & Robyn, 2001;Reys, Reys, Lapan, Holliday, & Wasman, 2003;Riordan & Noyce, 2001;Thompson & Senk, 2001). ...
... Significantly, scholars have advocated a shift from the teaching of standard computational algorithms ( Ebby, 2005), not only because students who use standard algorithms rarely understand what they are doing ( McNeal, 1995), but because algorithms unteach place value and hinder children's development of number sense ( Kamii & Dominick, 1997). Moreover, children who are encouraged to develop informal methods perform better on computational tasks than children taught algorithms ( Fuson, Carroll, & Drueck, 2000). Such matters are of considerable interest, particularly in light of evidence from countries like Taiwan, one of the world's more successful nations on tests of mathematical competencewhere research shows that students' algorithmic competence exceed that of their number sense ( Reys & Yang, 1998). ...
Many large-scale tests include tasks for which students are expected to write only an answer. With no expectation that students should show their written working such tasks have limited diagnostic value in that student thinking has to be inferred. However, by means of a pair-wise comparison of students' answers to a several such tasks, we show how the thinking processes involved in the production of students' incorrect answers can be meaningfully inferred. Data derive from tasks that appeared repeatedly in year five Swedish national tests. In particular, the results of one task, 399+4, are compared with those of three differently structured subtraction tasks and one differently structured addition. Analyses are framed hypothetically against situations in which students exploited algorithmic or number sense reasoning to explain how students who appear successful on one type of question may fail on another, typically structurally simpler, type of question.
... This study focuses on a school district where both the NCTM and the Renaissance Learning perspectives for math fact acquisition were present. The three elementary schools in the district are blue ribbon schools and administrators within the district have long valued the acquisition of math reasoning skills and selected Everyday Mathematics (The University of Chicago School Mathematics Project, 2007) as the district math curriculum in order to focus on math problem-solving skills over rote memorization of procedures (Fuson, Carrol, & Drueck, 2000). Teachers at these three elementary schools attended the same districtbased professional learning to assure fidelity with the curriculum and commonly engaged in practices to encourage sense-making including number talks, use of manipulatives, and individual/small group mathematics projects to explore mathematics connections and concepts. ...
While conceptual understanding of properties, operations, and the base-ten number system is certainly associated with the ability to access math facts fluently, the role of math fact memorization to promote conceptual understanding remains contested. In order to gain insight into this question, this study looks at the results when one of three elementary schools in a school district implements mandatory automaticity drills for 10 minutes each day while the remaining two elementary schools, with the same curriculum and very similar demographics, do not. This study looks at (a) the impact that schoolwide implementation of automaticity drills has on schoolwide computational math skills as measured by the ITBS and (b) the relationship between automaticity and conceptual understanding as measured by statewide standardized testing. The results suggest that while there may be an association between automaticity and higher performance on standardized tests, caution should be taken before assuming there are benefits to promoting automaticity drills. These results are consistent with those that support a process-driven approach to automaticity based on familiarity with properties and strategies associated with the base-ten number system; they are not consistent with those that support an answer-driven approach to automaticity based on memorization of answers.
... Learning trajectories include several levels of thinking and sequential steps needed to achieve a goal (developmental progression), which children follow in developing understanding of a mathematical idea. Effective teachers use them to design instructional tasks from childrens perspectives and help children to move through levels of understanding (Fuson, Carroll, & Drueck, 2000). Instructional tasks are the activities that should match each level of childrens developmental progression and should promote childrens progression to the next level. ...
... Clarke et al., 2002;Clements & Sarama, 2007;Fennema et al., 1996;Kühne, van den Heuvel-Panhulzen, & Ensor, 2005;Thomas & Ward, 2001;Wright, Martland, Stafford, & Stanger, 2002). Scale-up projects using curricula and professional development based on learning trajectories have also shown strong positive and enduring results (Clements, Sarama, Spitler, Lange, & Wolfe, 2011;Clements, Sarama, Wolfe, & Spitler, 2013Fuson, Carroll, & Drueck, 2000;Thomas & Ward, 2001). ...
PK–3 " has become a rallying cry among many developmental scientists and educators. A central component of this movement is alignment between preschool and the early elementary grades. Many districts have made policy changes designed to promote continuity in children's educational experiences as they progress from preschool through third grade— to provide children with a seamless education that will sustain the gains made in preschool and lead to better developmental and learning outcomes overall. This report proposes a conceptualization of productive continuity in academic instruction, as well as in the social climate and classroom management practices that might affect children's social-emotional development. It also considers ways in which schools might seek to achieve continuity in parents' and children's experiences. Finally, the report proposes specifi c state and district policies and school practices that are likely to promote continuous and meaningful learning experiences.
... The interdependence of teacher action, student action, and the content of studies: How the content that is available to be learned is influenced by instruction. Researchers interested in determining the impact of curriculum (doing what Henderson, 1963, called "curriculum research") sometimes set aside questions of the nature of curricular implementation and represent teaching as simply a conduit for the curriculum, one that leaves the mathematics itself unchanged (e.g., Fuson, Carroll, & Drueck, 2000). By way of contrast, with a focus on curriculum implementation, research on mathematics teaching has increasingly viewed instruction as actively shaping the nature of students' opportunities to learn (e.g., Tarr et al., 2008). ...
... This developmental progression can and should be nurtured in order for meaningful learning to occur in math classrooms. Students need support to learn how to move through a progression or range of solution methods as suggested by Fuson, Carroll and Drueck (2000). Hiebert and Grouws (2007) state that students' effort to make sense of mathematics, to figure something out that is not immediately apparent can advance their thinking and play an important role in deepening their understanding, if supported carefully toward a resolution and given appropriate time. ...
Struggle and its connection to learning are central to improve student learning and understanding of mathematics. A description of what a student’s productive struggle looks like in the setting of classrooms can provide insight into how teaching can support or hinder the student’s learning process. In order for any struggle to be productive, these struggles with mathematics must be documented. However, prior studies on student struggles are limited and have primarily focused on examining whether or not struggle occurred. Thus, the purpose of this study is to describe pre-service middle grade teachers’ (PSTs) struggle types in details as well as to investigate how enagaging in a non-routine high level task (doing mathematics) fosters (or inhibits) productive struggle during instruction.
... On standardized tests measuring computational skills and procedural knowledge, students using problem--solving approaches performed at least well as students using traditional curricula. In addition, students using problem--solving approaches performed better than students using traditional curricula on tests specifically designed to measure conceptual understanding and problem solving Carpenter et al., 1998;Cobb et al., 1991;Fuson et al., 2000;Hiebert & Wearne, 1993). For example, Cobb et al. (1991) examined the performance on a standardized mathematics achievement test of ten classes, whose students had participated in a year--long, problem--centered mathematics project and compared them with eight non--project classes. ...
In this chapter, the authors note that during the past 30 years there have been significant advances in our understanding of the affective, cognitive, and metacognitive aspects of problem solving in mathematics and there also has been considerable research on teaching mathematical problem solving in classrooms. However, the authors point out that there remain far more questions than answers about this complex form of activity. The chapter is organized around six questions: (1) Should problem solving be taught as a separate topic in the mathematics curriculum or should it be integrated throughout the curriculum? (2) Doesn’t teaching mathematics through problem require more time than more traditional approaches? (3) What kinds of instructional activities should be used in teaching through problems? (4) How can teachers orchestrate pedagogically sound, problem solving in the classroom? (5) How can productive beliefs toward mathematical problem solving be nurtured? (6) Will students sacrifice basic skills if they are taught mathematics through problem solving?
... Studies examining the effect of standardsbased textbooks on student learning reported that students studying from standards-based textbooks use a greater variety of solution methods than students studying from traditional US textbooks. [5,16,17] However, given that various important mathematics topics are still underrepresented in the current literature, more studies analyzing standards-based texts and traditional texts and comparing US textbooks with textbooks from other countries are needed to find a better way to promote student understanding. ...
In order to provide insight into cross-national differences in students’ achievement, this study compares the initial treatment of the concept of function sections of Chinese and US textbooks. The number of lessons, contents, and mathematical problems were analyzed. The results show that the US curricula introduce the concept of function one year earlier than the Chinese curriculum and provide strikingly more problems for students to work on. However, the Chinese curriculum emphasizes developing both concepts and procedures and includes more problems that require explanations, visual representations, and problem solving in worked-out examples that may help students formulate multiple solution methods. This result could indicate that instead of the number of problems and early introduction of the concept, the cognitive demands of textbook problems required for student thinking could be one reason for differences in American and Chinese students’ performances in international comparative studies. Implications of these findings for curriculum developers, teachers, and researchers are discussed.
... Results from a growing body of research support the positive influence of high-level tasks on students' learning of mathematics. This research indicates that curricular materials specifically developed to contain high-level tasks (USDE, 1999) are successful at improving students' performance on state and national tests of mathematical achievement (e.g., Fuson, Carroll, & Druek, 2000;Riordan & Noyce, 2001;Schoen, Fey, Hirsch, & Coxford, 1999), at improving students' understanding of important mathematical concepts (e.g., Ben-Chaim, Fey, Fitzgerald, Benedetto, & Miller, 1998;Huntley, Rasmussen, Villarubi, Sangtong, & Fey, 2000;Thompson & Senk, 2001;Reys, Reys, Lappan, & Holliday, 2003), and at improving students abilities to reason, communicate, problem-solve and make mathematical connections (e.g., Ridgeway, Zawojewski, Hoover, & Lambdin, 2003;Schoenfeld, 2002). Engaging students with tasks that elicit high-level cognitive demands appears to have a positive effect on students' development of mathematical understanding. ...
DEVELOPING SECONDARY MATHEMATICS TEACHERS'KNOWLEDGE OF AND CAPACITY TO IMPLEMENT INSTRUCTIONAL TASKS WITH HIGH LEVEL COGNITIVE DEMANDSMelissa D. Boston, EdD University of Pittsburgh, 2006This study analyzed mathematics teachers' selection and implementation of instructional tasks in their own classrooms before, during, and after their participation in a professional development workshop focused on the cognitive demands of mathematical tasks. Eighteen secondary mathematics teachers participated in a six-session professional development workshop under the auspices of the Enhancing Secondary Mathematics Teacher Preparation (ESP) Project throughout the 2004-2005 school year. Data collected from the ESP workshop included written artifacts created during the professional development sessions and videotapes of each session. Data collected from teachers included a pre/post measure of teachers' knowledge of the cognitive demands of mathematical tasks, collections of tasks and student work from teachers' classrooms, lesson observations, and interviews. Ten secondary mathematics teachers who did not participate in the ESP workshop served as the contrast group, completed the pre/post measure, and participated in one lesson observation.Analysis of the data indicated that the ESP workshop provided learning experiences for teachers that transformed their previous knowledge and instructional practices. ESP teachers enhanced their knowledge of the cognitive demands of mathematical tasks; specifically, they improved their ability to identify and describe the characteristics of tasks that influence students' opportunities for learning. Following their participation in ESP, teachers were more frequently selecting high-level tasks as the main instructional tasks in their own classrooms. ESP teachers also improved their ability to maintain high-level cognitive demands during implementation. Student work implementation significantly improved from Fall to Spring, and comparisons of the implementation of high-level student work tasks indicated that high-level demands were less likely to decline in Spring than in Fall. Lesson observations did not yield statistically significant results from Fall to Spring; however, significant differences existed between ESP teachers and the contrast group in task selection and implementation during lesson observations. ESP teachers also outperformed the contrast group on the post-measure of the knowledge of cognitive demands of mathematical tasks. None of the significant differences were influenced by the use of a reform vs. traditional curricula in teachers' classrooms. Teachers who exhibited greater improvements more frequent contributions and more comments on issues of implementation than teachers who exhibited less improvement.
... Results indicated that standards-based curriculum materials such as graphing calculators assisted in solving algebraic problems within applied contexts, increasing student achievement. Similar findings have been reported while investigating the impact of elementary standards-based units by Fuson et al. (2000), in a quantitative study. They used two sets of studies, the first study with 392 second graders in 22 classes and the second with 620 third graders in 29 classes. ...
... The approach encourages students to invent their own procedure for the arithmetical operations (Kamii et al, 1993). Instruction in "…which students construct meaning for the mathematical concepts and procedures they are investigating and engage in meaningful problem-solving activities" are considered ideal (Fuson, Carroll, & Drueck, 2000). According to Kamii (1991), the only way students can learn mathematics is by making their own decisions and evaluating the results of their decisions. ...
... Whilst many teacher education programs reflected a constructivist approach during the 1990s (and continue to do so), accumulating evidence indicates that it has not translated into classroom practice as expected (Foss & Kleinsasser, 1996;Fuson, Carroll, & Drueck, 2000). Furthermore, studies by Foss and Kleinsasser (1996) and Klein (1999) concluded that not only do preservice primary teachers' mathematical beliefs remain unchanged as a result of their teacher education but that their conceptualisations of mathematics and themselves as teachers of mathematics do not change either. ...
This paper reports on the initial findings of a longitudinal study aimed at investigating the impact on preservice primary teachers' mathematical beliefs, knowledge and practices of a mathematics education course utilising a situated learning perspective. Other research in this area has found that preservice primary teacher's beliefs have remained unchanged as a result of their teacher education programs. Preliminary findings from this study contradict these results. It is suggested that the utilisation of a multiplicity of learning contexts may explain this discrepancy. There is a growing number of studies confirming the influence teachers' mathematical beliefs play on their classroom practice (Buzeika, 1996; Ernest, 2000). In regard to preservice teachers, it is widely accepted that they enter their teacher education studies holding 'traditional' views on how to teach mathematics and what mathematics should be taught (Benbow, 1993). These views, in part, are acquired from the preservice teachers' personal experiences based on literally hundreds of hours of listening and observing mathematics school teachers. Unless challenged, such views ensure that the traditional model of teaching mathematics is perpetuated. During the past 15 years the notion of constructivism has challenged the traditional model of teaching and learning (Klein, 1999). In relation to the discipline of mathematics, it has challenged both the nature of mathematics (dynamic versus static) and how knowledge is acquired (actively versus passively). In response to growing support for teaching mathematics from a constructivist perspective (Australian Education Council, 1990; DEET, 1989), a number of teacher education programs adopted a constructivist approach in the way their courses were delivered. According to constructivist principles, this translated into establishing a learning environment in which students construct their own knowledge by linking prior experiences (including knowledge and beliefs) to new knowledge (Jones & Vesilind, 1996), creating 'learning communities' in which students engage in rich discourse about important ideas (Putman & Borko, 2000) and using reflection as a powerful vehicle for reconceptualising knowledge and beliefs (Beattie, 1997). It was anticipated that such modelling of a constructivist approach within preservice education programs would translate into classroom practice (DEET, 1989). Whilst many teacher education programs reflected a constructivist approach during the 1990s (and continue to do so), accumulating evidence indicates that it has not translated into classroom practice as expected (Foss & Kleinsasser, 1996; Fuson, Carroll, & Drueck, 2000). Furthermore, studies by Foss and Kleinsasser (1996) and Klein (1999) concluded that not only do preservice primary teachers' mathematical beliefs remain unchanged as a result of their teacher education but that their conceptualisations of mathematics and themselves as teachers of mathematics do not change either. Such findings directly contrast the initial findings of the current investigation.
... Boaler (2000,) found that girls in a school that used an open problem-solving teaching developed increased confidence and enjoyment of mathematics, and attained statistically significant higher grades on the GCSE examination than girls in a school with a similar population using a textbook-based traditional approach. Other studies have reported similar equitable gains for boys and girls in their confidence, problem solving and conceptual understanding in mathematics as a result of open approaches to mathematics teaching (Carpenter et al, 1998;Fuson et al, 2000). ...
The paper reports on an ongoing Malawian case study that is part of the Forum for African Women Educationalists (FAWE) initiatives to improve the learning of girls in Science Mathematics and Technology (SMT) which is being implemented in a number of FAWE chapters in Africa. The study has so far run for one school calendar year. The data for this study was collected from three co-educational pilot schools in Malawi. The current report will focus on the students' attitude towards mathematics. Data collection for the study was carried out through an administration of an attitude questionnaire to all Form one (year 9 of formal schooling) students in the three pilot schools. The questionnaire was administered at the beginning and at the end of the 2007 academic year. The initial data was used as baseline line data. Several intervention activities were implemented in the schools during the year. Such interventions included training of teachers on Gender Responsive Pedagogies, visits to industry to see mathematics at work, remedial classes, team teaching amongst teachers, reflective meetings amongst teachers to discuss issues concerning girls' learning of mathematics and holding of Science, Mathematics and Technology (SMT) camps during school term holidays. Analysis of data from the questionnaire responses revealed that girls had started with a relatively low attitude towards mathematics compared to boys at the start of the study. However, the girls' attitude had improved more that the boys' by the end of the first year. This indicates that the intervention of the project did not impede the boys' attitude but had more positive impact on the girls than the boys. Implications for the findings are discussed in the paper.
... 457). Fuson, Carroll, and Drueck (2000) used two studies to test the achievement of second and third graders using a Standards-based curriculum. ...
... There is evidence that superior teachers use a conceptual structure similar to an LP (Clements & Sarama, 2004b). For example, in one study of a reform-based curriculum, the teachers who had the most valuable in-class discussions saw themselves not as moving through a curriculum but as helping students move through a progression or range of solution methods (Fuson, Carroll, & Drueck, 2000); that is, they were simultaneously using and modifying a type of learning trajectory (Clements & Sarama, 2004b). Simon (1995) discussed the knowledge of a hypothetical learning trajectory (HLT) as being essential to developing pedagogical thinking. ...
Prior work on the CBAL™ mathematics competency model resulted in an initial competency model for middle school grades with several learning progressions (LPs) that elaborate central ideas in the competency model and provide a basis for connecting summative and formative assessment. In the current project, we created a competency model for Grades 3–5 that is based on both the middle school competency model and the Common Core State Standards (CCSS). We also developed an LP for rational numbers based on an extensive literature review, consultations with members of the CBAL mathematics team and other related research staff at Educational Testing Service, input from an advisory panel of external experts in mathematics education and cognitive psychology, and the use of small-scale cognitive interviews with students and teachers. Elementary mathematical understanding, specifically that of rational numbers, is viewed as fundamental and critical to developing future knowledge and skill in middle and high school mathematics and therefore essential for success in the 21st century world. The competency model and the rational number LP serve as the conceptual basis for developing and connecting summative and formative assessment as well as professional support materials for Grades 3–5. We report here on the development process of these models and future implications for task development.
Curriculum materials are instructional materials produced to be used for teaching and learning. Successful teaching requires new materials and innovative approaches. In the present study, I shall present DGS Cui-rods an instructional material for effective teaching and learning of mathematical concepts. I created them in the Geometer's Sketchpad environment. Dynamic geometry environments allow students to discover and reinvent a wide range of mathematical concepts and their relationships. During problem-solving situations, students are able to construct meanings. They are led to create their personal representations of mathematical concepts and transform them. The design of activities in the learning environment as a part of the instruction thus has a crucial role to play. In the next sections, I shall describe how learning through DGS Cui-Ros affects students' cognitive structure's transformations and consequently their cognitive growth.
In this study we aimed to understand teaching mathematics through problem posing based on an analysis of 22 teaching cases. Teaching mathematics through problem posing starts with problem-posing tasks. This study provides not only specific examples of problem-posing tasks used in classrooms but also related task variables to consider when developing problem-posing tasks. This study also contributes to our understanding of how teachers can deal with student-posed problems in the classroom. In these 22 teaching cases, there was a typical pattern to how teachers dealt with the students’ posed problems in the classroom according to the instructional goals. For future research, we need to accumulate additional teaching cases and explore possible discourse patterns concerning how teachers handle students’ posed problems, as well as identify the most effective discourse patterns when teaching mathematics through problem posing.
Η παρούσα μονογραφία αποτελεί επιλογή και σύνθεση αποσπασμάτων (α) της ενδελεχούς μελέτης μου στην έννοια της αναπαράστασης (β) του θεωρητικού πλαισίου της διδακτορικής μου διατριβής (Πατσιομίτου, 2012α), (γ) της μονογραφίας για τη διδακτική των μαθηματικών «A trajectory for the teaching and learning of the didactics of mathematics: linking visual active representation” (Patsiomitou, 2019c), που είναι ανοικτής πρόσβασης στο διαδίκτυο, (δ) άρθρων μου που έχουν δημοσιευθεί σε ελληνικά ή διεθνή έγκριτα συνέδρια και περιοδικά, (ε) άλλου υλικού που προέκυψε από διερεύνηση στο διαδίκτυο, όπως περιγράφεται στη συνέχεια:
απάνθισμα υλικού που συγκέντρωσα, σταχυολόγησα, και επέλεξα να αποτελέσει μία πρώτη προσέγγιση σε σημαντικές έννοιες της διδακτικής και ψυχολογίας των μαθηματικών, κάνοντας χρήση υπολογιστικών περιβαλλόντων για τη διδασκαλία και μάθηση των μαθηματικών εννοιών.
Το βιβλίο απευθύνεται σε φοιτητές και εκπαιδευτικούς που βρίσκονται στη φάση διερεύνησης της διασύνδεσης θεωρίας–πράξης, όσον αφορά τη διδακτική και διδασκαλία των μαθηματικών, ως αλληλεπίδραση της επιστήμης και της τέχνης των μαθησιακών μεθόδων.
Purpose
Students with attention-deficit/hyperactivity disorder (ADHD) often demonstrate language deficits requiring speech-language pathologist (SLP) interventions. With the number of students diagnosed with ADHD on the rise, SLPs are being called upon increasingly to address the needs of these students. Ensuring SLPs are comfortable treating students with ADHD is of growing importance. To date, there is limited evidence examining what factors may predict how comfortable SLPs are working with students with ADHD. This study investigated the relationships between variables to better predict comfort levels of school SLPs in their management of students with ADHD.
Method
A total of 86 school SLPs from the American Speech-Language-Hearing Association's Special Interest Groups 1 (Language, Learning and Education) and 16 (School-Based Issues) whose clinical caseloads included students diagnosed with ADHD participated in an anonymous, online survey to determine the relationship between various factors and SLPs' comfort level when working with students with ADHD.
Results
Correlations revealed feeling adequately trained predicted SLPs' comfort level in working with students with ADHD. SLPs who either received coursework that included content regarding ADHD in school or sought out professional development reported feeling adequately trained. Regression analyses revealed feeling adequately trained and having a reasonable caseload as significant predictors of SLPs' comfort level.
Conclusions
Not all SLPs were comfortable working with students with ADHD. Many sought courses to increase their level of comfort, and many did not feel adequately trained. Results suggest the need for SLPs to have more preparation about managing ADHD prior to beginning their professional clinical careers.
In many countries, in contemporary math curricula, particularly in early childhood and childhood classrooms, it is considered crucial to encourage children to show their work, to concretely represent their thinking. This article takes up an historiography of this trend (from an American perspective), and draws on case studies of elementary classrooms to show that sometimes, encouraging children to represent their thought betrays sensible epistemological and psychological understandings of the nature of thought and knowledge. The author draws on psychoanalytic theory in conversation with educational philosopher Gert Biesta to argue for a rehumanizing pedagogy, one that enables teachers to see children and themselves as whole and complex even in the context of a mandated and predetermined curriculum. Though the article focuses on math classrooms, its implications for childhood education are broader and trans-curricular.
We define and describe how subitizing activity develops and relates to early quantifiers in mathematics. Subitizing is the direct perceptual apprehension and identification of the numerosity of a small group of items. Although subitizing is too often a neglected quantifier in educational practice, it has been extensively studied as a critical cognitive process. We believe that subitizing also helps explain early cognitive processes that relate to early number development and thus deserves more instructional attention. We also contend that integrating developmental/cognitive psychology and mathematics education research affords opportunities to develop learning trajectories for subitizing. A complete learning trajectory includes three components: goal, developmental progression, or learning path through which children move through levels of thinking, and instruction. Such a learning trajectory thus helps establish goals for educational purposes and frames instructional tasks and/or teaching practices. Through this chapter, it is our hope that early childhood educators and researchers begin to understand how to develop critical educational tools for early childhood mathematics instruction. Through this instruction, we believe that children will be able to use subitizing to discover critical properties of number and build on subitizing to develop capabilities such as unitizing, cardinality, and arithmetic capabilities.
Purpose:
The purpose of this descriptive study was to investigate why some elementary children have difficulties mastering addition and subtraction calculation tasks.
Design/methodology/approach:
The researchers examined error types in addition and subtraction calculation made by 697 Portuguese students in elementary grades. Each student completed a written assessment of mathematical knowledge. A system code (e.g., FR = failure to regroup) was used to grade the tests. A reliability check was performed on 65% randomly selected exams.
Findings:
Data frequency analyses revealed that the most common type of error was miscalculation for both addition (n =164; 38.6%) and subtraction (n = 180; 21.7%). The second most common error type related to failure to regroup in addition (n = 74; 17.5%) and subtraction (n = 139; 16.3%). Frequency of error types by grade level is provided. Findings from the hierarchical regression analyses indicated that students’ performance differences emerged as a function of error types which indicated students’ type of difficulties.
Research limitations/implications:
There are several limitations of this study: (1) the use of a convenient sample; (2) all schools were located in the northern region of Portugal; (3) the limited number of problems; and (4) time of the year of assessment.
Practical implications:
Students’ errors suggested that their performance in calculation tasks is related to conceptual and procedural knowledge and skills. Error analysis allows teachers to better understand the individual performance of a diverse group and to tailor instruction to ensure that all students have an opportunity to succeed in mathematics.
Originality/value:
This paper adds to the international literature on error analysis and reinforces its value in diagnosing students’ type and severity of math difficulties.
As a science, knowledge created during curriculum development should be both generated and placed within a scientific research corpus, peer reviewed, and published. In the context of science, the knowledge generated during the process of developing curriculum should be generated and placed within the public domain in a scientific manner. This chapter will describe a framework for curriculum development, study and evaluation of research based curricula. It will also provide a description of the framework, which will include three categories of activities and 10 phases that are embedded within those categories. It will propose that curriculum research should provide an ideal context for building a scientific knowledge base for education curriculum development.
This study is considered to be significant in objectively analyzing the negative-positive effects of the program during the process of learning, teacher's acquaintance to the new program in a closer way and acquiring a positive perspective of the program. This study has been conducted in order to discover the effects of 2005-2006 academic year primary education program on cognitive-emotional-psychomotor acquisitions, and expectations of the students in learning process. The sampling of the research consisted of 836 teachers (Women: 438, Men: 397) of state and private primary education institutions selected by random sampling method in Istanbul in 2008-2009 academic year. Research data were collected by "the Scale of Teachers' Attitude towards New Primary Education Program" consisted of 27 questions developed by the researcher. This study aims objectively at designating the negative-positive effects of the program during the process of learning. Moreover, perceptions and attitudes of primary education teachers towards the 2005 academic year primary education program are studied; the relationship between the previous and the new program is tried to be designated; and the differences and limitations of the new program are also discovered. Hence, the study aims at contributing to the concerned officials and researchers. The attitudes of the teachers towards the new educational program do not show significant differences in respect to gender, experiences, educational levels and state of graduation. However, there is a significant difference of their perspectives about new educational program in respect to their marital status and the institutions they work at. The results of the study were compared and discussed with several former program evaluation examples.
This article reports on the first 4 years of an effort to develop comprehensive and sustainable mathematics education reforms in high poverty middle schools. In four related analyses, we examine the levels of implementation achieved and impact of the reforms on various measures of achievement in the first 3 schools to implement the Talent Development (TD) Middle School Model's mathematics program that combines coherent research-based instructional materials from the University of Chicago School Mathematics Project with a multi-tiered teacher support system of sustained professional development and in-class coaching. A moderate level of implementation was achieved. TD students outperformed students from control schools on multiple measures of achievement. The average effect size, Δ, by the end of middle school was.24.
The purpose of this research is to examine the impact of a Standards-based elementary mathematics curriculum on third grade students' mathematics performance. A total of 707 students participated in this study. Of this total, 368 students were from eight schools located within the same school district in a racially and ethnically diverse large city in the Midwest. Another 339 of the students were from four schools located in two different school districts in a middle- to high-SES, largely white, suburban area in the Northeast. Third grade students using Investigations curriculum outperformed in most cases and performed the same in the other cases matched comparison groups who were using a range of conventional curricula on the mathematics assessment. The Investigations groups did the same or better on the decontextualized, contextualized, and algebraic-reasoning constellations compared with their counterparts. The revised Investigations curriculum was not as effective with the low SES African-American students as it was for middle to high SES, white students. The curriculum fidelity measures did not differentiate achievement differences.
Research on the impact of Standards-based, K-12 mathematics programs (i.e., written curricula and associated teaching practices) and of reform calculus programs has focused primarily on student achievement and secondarily, and rather ineffectively, on student attitudes. This research has shown that reform programs have competed well with traditional programs in terms of student achievement. Results for attitude change have been much less conclusive because of conceptual and methodological problems. We critically review this literature to argue for broader conceptions of impact that target new dimensions of program effect and examine interactions between dimensions. We also briefly present the conceptualization, design, and broad results of one study, the Mathematical Transitions Project (MTP), which expanded the range of impact along those lines. The MTP results reveal substantial diversity in students' experience within and between research sites, different patterns of experience between high school and university students, and surprising relationships between achievement and attitude for some students.
This book puts a spotlight on the practices of teachers across the nation who have implemented effective mathematics instruction for students of different ethnicities. Among the ethnic groups represented are African Americans, Latinos, Native Americans, Haitians, Arab Americans, and Euro–Americans.
This study investigates how and to what extent preservice and inservice early childhood educators are being prepared to develop mathematics skills and processes in young children, age birth to five, in one southeastern state in the United States. Utilizing a multi-phase mixed methods study and a follow-up survey central to the research question, findings from this study indicate that professional development in mathematics is inadequate. Few of the professional development opportunities focus on specific mathematical content areas or address the Common Core State Standards in Mathematics. In addition, the majority of the professional development opportunities are short, hourly sessions with little to no follow-up. Implications of the results and suggestions for future research are discussed.
The present study aims to investigate Turkish kindergarten teachers' beliefs toward learning and teaching and their current instructional practices in kindergarten classrooms. To this end, teacher interviews and classroom observations were conducted with 28 kindergarten teachers. The interview results showed that teachers did not realize the critical role of hands-on materials, peers, colleagues, teacher education programs, and professional development programs on young children's learning. In addition, the observation results showed that students spent almost one third of their time for waiting in line, moving from one activity to another, and so on. Also, limited time is spent on mathematics, science, and social studies. Thus, the study has critical implications for future research in terms of teacher education programs as well as professional development programs for in-service kindergarten teachers.
In this study, we examined how curricular resources supported three expert teachers in their enactment of progressions. Using a video-stimulated interview process, we documented the multiple types of progressions identified, described, and enacted by the teachers. Results indicate that the teachers used four different types of progressions—mathematical concepts, instructional activities, number choices, and student solutions—and that the progressions were embedded within each other. This study contributes to the field’s understanding of the ways in which expert teachers make use of curriculum- and research-based progressions in their teaching practices.
A content analysis of over 28,000 pages from 141 elementary school mathematics textbooks published between 1900 and 2000 shows that widely used mathematics textbooks have changed substantially. Textbooks from the early part of the century were typically narrow in content but presented substantial amounts of advanced arithmetic and also asked students simultaneously to engage with material in effortful and conceptual ways. A period of change marked the middle of the century, when less advanced topics were presented and problem-solving tasks were simplified. From the mid-1960s onward, however, the trend reversed, and 3 major changes occurred in primary school mathematics curricula over the next 4 decades: (a) expansion of topics and the number of pages devoted to each topic; (b) a shift of traditionally more advanced topics from higher to lower grades; and, (c) within arithmetic, an increase in the number, abstraction, and cognitive demand of problem-solving strategies. Implications of these findings are discussed in terms of the historical study of mathematics and curriculum in U.S. schools.
In this study we discuss convergence between instructional practices suggested by research on achievement motivation and practices promoted in the mathematics instruction reform literature, and we assess associations among instructional practices, motivation, and learning of fractions. Participants included 624 fourth- through sixth-grade students and their 24 teachers. Results indicated that the instructional practices suggested in literature in both research areas positively affected students' motivation (e.g., focus on learning and understanding; positive emotions, such as pride in accomplishments; enjoyment) and conceptual learning related to fractions. Positive student motivation was associated with increased skills related to fractions.
Constructivist theory has been prominent in recent research on mathematics learning and has provided a basis for recent mathematics education reform efforts. Although constructivism has the potential to inform changes in mathematics teaching, it offers no particular vision of how mathematics should be taught; models of teaching based on constructivism are needed. Data are presented from a whole-class, constructivist teaching experiment in which problems of teaching practice required the teacher/researcher to explore the pedagogical implications of his theoretical (constructivist) perspectives. The analysis of the data led to the development of a model of teacher decision making with respect to mathematical tasks. Central to this model is the creative tension between the teacher's goals with regard to student learning and his responsibility to be sensitive and responsive to the mathematical thinking of the students.
Five second-grade classes in two schools participated in a project that was generally compatible with a constructivist theory of knowing. At the end of the school year, the students in these classes and their peers in six non-project classes in the same schools were assigned to ten textbook-based third-grade classes on the basis of reading scores. The two groups of students were compared at the end of the third-grade year on a standardized achievement test and on instruments designed to assess their conceptual development in arithmetic, their personal goals in mathematics, and their beliefs about reasons for success in mathematics. The levels of computation performance on familiar textbook tasks were comparable, but former project students had attained more advanced levels of conceptual understanding. In addition, they held stronger beliefs about the importance of working hard and being interested in mathematics, and about understanding and collaborating. Further, they attributed less importance to conforming to the solution methods of others.
Longitudinal analyses of the mathematical achievement and beliefs of 3 groups of elementary pupils are presented The groups consist of those students who had received 2 years of problem-centered mathematics instruction, those who had received 1 year, and those who had received textbook instruction. Comparisons are made for the groups using a standardized norm-referenced achievement test from first through fourth grade. Next, student comparisons are made using instruments developed to measure conceptual understanding of arithmetic and beliefs and motivation for learning mathematics. The results of the analyses indicate that after 2 years in problem-centered classes, students have significantly higher achievement on standardized achievement measures, better conceptual understanding, and more task-oriented beliefs for learning mathematics than do those in textbook instruction. In addition these differences remain after problem-centered students return to classes using textbook instruction. Comparisons of pupils in problem-centered classes for only 1 year reveal that after returning to textbook instruction, these students' mathematical achievement and beliefs are more similar to the textbook group. Also included are exploratory analyses of the pedagogical beliefs held by teachers before and after teaching in problem-centered classes, and those held by teachers in textbook classes. The results of the student and teacher analyses are interpreted in light of research on children's construction of nonstandard algorithms and the nature of classroom environments.
This 3-year longitudinal study investigated the development of 82 children's understanding of multidigit number concepts and operations in Grades 1-3. Students were individually interviewed 5 times on a variety of tasks involving base-ten number concepts and addition and subtraction problems. The study provides an existence proof that children can invent strategies for adding and subtracting and illustrates both what that invention affords and the role that different concepts may play in that invention. About 90% of the students used invented strategies. Students who used invented strategies before they learned standard algorithms demonstrated better knowledge of base-ten number concepts and were more successful in extending their knowledge to new situations than were students who initially learned standard algorithms.
Year-long classroom teaching experiments in two predominantly Latino low-socioeconomic-status (SES) urban classrooms (one English-speaking and one Spanish-speaking) sought to support first graders' thinking of 2-digit quantities as tens and ones. A model of a developmental sequence of conceptual structures for 2-digit numbers (the UDSSI triad model) is presented to describe children's thinking. By the end of the year, most of the children could accurately add and subtract 2-digit numbers that require trading (regrouping) by using drawings or objects and gave answers by using tens and ones on various tasks. Their performance was substantially above that reported in other studies for U. S. first graders of higher SES and for older U. S. children. Their responses looked more like those of East Asian children than of U. S. children in other studies.
In this article we present and describe a pedagogical framework that supports children's development of conceptual understanding of mathematics. The framework for Advancing Children's Thinking (ACT) was synthesized from an in-depth analysis of observed and reported data from 1 skillful 1st-grade teacher using the Everyday Mathematics (EM) curriculum. The ACT framework comprises 3 components: Eliciting Children's Solution Methods, Supporting Children's Conceptual Understanding, and Extending Children's Mathematical Thinking. The framework guided a cross-teacher analysis over 5 additional EM 1st-grade teachers. This comparison indicated that teachers often supported children's mathematical thinking but less often elicited or extended children's thinking. The ACT framework can contribute to educational research, teacher education, and the design of mathematics curricula.
Researchers from 4 projects with a problem-solving approach to teaching and learning multidigit number concepts and operations describe (a) a common framework of conceptual structures children construct for multidigit numbers and (b) categories of methods children devise for multidigit addition and subtraction. For each of the quantitative conceptual structures for 2-digit numbers, a somewhat different triad of relations is established between the number words, written 2-digit marks, and quantities. The conceptions are unitary, decade and ones, sequence-tens and ones, separate-tens and ones, and integrated sequence-separate conceptions. Conceptual supports used within each of the 4 projects are described and linked to multidigit addition and subtraction methods used by project children. Typical errors that may arise with each method are identified. We identify as crucial across all projects sustained opportunities for children to (a) construct triad conceptual structures that relate ten-structured quantities to number words and written 2-digit numerals and (b) use these triads in solving multidigit addition and subtraction situations.
To investigate relationships between teaching and learning mathematics, the six second-grade classrooms in one school were observed regularly during the 12 weeks of instruction on place value and multidigit addition and subtraction. Two classrooms implemented an alternative to the more conventional textbook approach. The alternative approach emphasized constructing relationships between place value and computation strategies rather than practicing prescribed procedures. Students were assessed at the beginning and the end of the year on place value understanding, routine computation, and novel computation. Students in the alternative classrooms, compared with their more traditionally taught peers, received fewer problems and spent more time with each problem, were asked more questions requesting them to describe and explain alternative strategies, talked more using longer responses, and showed higher levels of performance or gained more by the end of the year on most types of items. The results suggest that relationships between teaching and learning are a function of the instructional environment; different relationships emerged in the alternative classrooms than those that have been reported for more traditional classrooms.