Middle Grades Mathematics Research Articles

Questioning Channels of Mathematics Teachers in Technology and Non-Technology Supported Mathematics Classrooms

The objective of the present study was to gain a deeper understanding of how middle-grade mathematics teachers implement questions in both technology supported and non-technology supported classrooms. The investigation demonstrated that in-service mathematics teachers teaching in a technology supported classroom and in a non-technology supported classroom employed seven questioning channels: teacher-created static nature of drawings, teacher-created dynamic nature of drawings, student-created nature of drawings, student verbal discourse, a ready-to-use educational animation and textbook definition, analogies, and real-life examples. Commonly, both teachers used student-created static nature of drawings on worked examples, teacher-created static nature of drawings on worked examples, and mathematical procedures or concepts in their questioning, as well as students’ verbal discourse. The integration of technology introduced three additional questioning channels in technology supported classrooms: the teacher-created dynamic nature of drawings, a ready-to-use educational animation and textbook definition, and a real-life example. In conclusion, this study indicates that teacher questions can be influenced by the questioning channels employed, and technology might, to some extent, affect teachers’ variety of questioning channels. The results are discussed in the light of related literature.

Why ask why? An analysis of teachers’ why-questions in elementary and middle grade mathematics classrooms

Why ask why? An analysis of teachers’ why-questions in elementary and middle grade mathematics classrooms

Reasoning with Two Types of Multiplicative Units Structures in Solving Middle-Grade Mathematics Problems

ABSTRACT In the present study, we illuminate students’ multiplicative reasoning in the context of their units-coordinating activity. Of particular interest is to investigate students’ use of three levels ofunits as given material for problem-solving activity, which we regard as supporting a more advanced level of multiplicative reasoning. Among 13 middle school students with whom we have conducted clinical interviews, this study focuses on eight students who conceived fractions with three levels of units and reports their units-coordinating activities in solving diverse middle-grade level problems. The result of the study indicates that the ability to coordinate two multiplicative units structures was an evident characteristic by which we classified them into two distinctive groups. Specifically, two students outperformed the other students in solving advanced equal sharing and bar transformation problems by the coordination of two explicitly multiplicative units structures and in representing a multiplicative relationship between two unknown quantities by the coordination of two implicitly multiplicative units structures. The findings suggest that the ability to use the coordinated result of two multiplicative units structures needs to be attended to as a sophisticated form of multiplicative reasoning, which we believe potentially undergirds students’ advanced mathematical learning beyond elementary school.

Middle Grades Math with Ice Cream Sundaes: Connecting Math to the Real World

This qualitative case study presents findings related to the experiences of one preservice teacher as they attempted to create student-centered math experiences in their middle school mathematics classroom. The participant was enrolled in a graduate secondary mathematics teacher education program and was student teaching at a middle school. In this research study, I asked, “How does one preservice middle grades mathematics teacher understand ways to connect math to the real world?” The data collected included interviews, classroom artifacts, and researcher memos. The data were analyzed using thematic analysis. The findings suggested that, although the preservice teacher wanted to infuse real world experiences in their lessons, they were unsure of ways to do so; they struggled to infuse a real world context in math at the middle school level. This study draws attention to the importance of ensuring that middle grades mathematics educators receive instruction on ways to help their students make connections to the real world and thus create student-centered mathematics classrooms.

Connections Between Prospective Middle-Grades Mathematics Teachers’ Technology-Enhanced Specialized Content Knowledge and Beliefs

The authors explore the influence of instructional technologies on teachers’ subject-matter knowledge as well as how this knowledge is mediated by context, experience, and beliefs about teaching, mathematics, and technology. This study examined how four pre-service middle grades mathematics teachers’ (PSTs) beliefs about teaching, mathematics, and technology influenced their subject matter knowledge specific to the teaching profession (i.e, specialized content knowledge [SCK]). Embedded-unit single case study analysis revealed that PSTs’ different beliefs about teaching, mathematics, and technology indicated various impacts of the use of dynamic geometry software on their SCK. PSTs who viewed teaching mathematics with technology as an effective pedagogical decision to support students’ understanding and self-discovery of their subject matter knowledge demonstrated more dynamic SCK with dynamic geometry software. The authors discuss interconnections between PSTs’ beliefs and knowledge and share implications for technology-enhanced teacher education of middle grades mathematics teachers.

Future teachers’ use of co-occurrence and same extent perspectives when generating linear equations

We used video recorded interviews to investigate how 6 future middle grades mathematics teachers generated equations to express relationships between 2 quantities that co-vary in a proportional relationship. We found that well established results, including those having to do with meanings for letters and for the equal sign, did not explain what the future teachers found challenging. In particular, we found that future teachers were more facile generating inscriptions to express 2 quantities that co-occur than to express 2 quantities that have the same extent. We introduce the co-occurrence and same extent constructs, present 2 detailed cases in which we use the constructs to locate what was challenging about modeling problem situations with equations, discuss possible pathways for overcoming those challenges, and identify directions for future research.

Characterizing whole class discussions about data and statistics with conversation profile analysis

Whole class discussions (WCDs) are an important pedagogical tool for mathematics classes but are challenging to characterize across large numbers of observations because of their dynamic and complex nature. In this paper, we report on an exploratory method to characterize WCDs in mathematics classes across large numbers of observations that we refer to as Conversation Profile Analysis (CPA). CPA uses Latent Class Modeling (LCM) with live observation data to generate profiles of WCDs in middle-grade mathematics classes. We report on our exploratory use of CPA to analyze observation data from 259 WCDs about data and statistics in middle school classes making use of an innovative approach to instruction called Data Modeling. We identified 4 profiles of WCDs and found that these profiles varied in likelihood across time and were associated with different ways students talked about key mathematical ideas. We also discuss broader implications of the CPA approach to studying WCDs in math classes.

A Curriculum-Based Approach to Learning Trajectories in Middle School Algebra

Our aim is to contribute to the body of research on learning trajectories (LTs) in mathematics by making visible a process for articulating a hypothetical learning trajectory implicit in a widely adopted, reform-based, middle-grades mathematics curriculum. In doing so, we highlight considerations, decisions, and challenges we faced as part of this work. By describing our LT articulation process, our aim is to highlight ways in which curriculum-specific LTs can be articulated to serve as a more proximal and instrumental tool for teachers’ instructional practice. Furthermore, to illustrate we describe how the products of the work were used in practice-based professional learning experiences with middle-grades mathematics teachers.

Investigation of a Middle School Preservice Teacher’s Knowledge of Content and Students

The purpose of this study was to explicate one preservice middle grades mathematics teacher’s Knowledge of Content and Students (KCS) in the context of multiple solution strategies. This study’s purpose is to underline the importance of preservice teachers’ KCS and provide possible investigative methods for evaluating preservice teachers’ KCS. Specifically, the research inquiry guiding this study focused on how a middle school preservice mathematics teacher displays KCS when engaging with tasks about pattern recognition and linear functions in the context of multiple solution strategies. The data consisted of three videotaped semi-structured interviews with the preservice mathematics teacher as well as the written work she produced during the interviews. This study explicated one preservice mathematic teacher’s performance regarding two important themes of KCS: generating multiple possible solution strategies of middle school students and explaining multiple student solution strategies. In terms of generating multiple solution strategies of middle school students, the study found that the preservice mathematics teacher provided the same solution strategies that she employed when she solved the problems by herself. Regarding explaining multiple student solution strategies, this study revealed that the preservice teacher did not explicate how typical middle school students reason. The preservice teacher had limitations when explaining the possible procedures that students might have used to solve problems when given the final student solutions. With regard to the teacher’s abilities to recognise and understand students’ typical understandings and misunderstandings, the study demonstrated that the preservice teacher was capable of explaining some solution strategies but not all of them.

Examination of a middle School mathematics preservice teacher’s interpretation skills

This study aims to explore one preservice teacher’s interpretation of students’ mathematical thinking using pattern recognition and linear function tasks. Specifically, the research question for this study was as follows: How does a preservice middle grade mathematics teacher interpret students’ mathematical thinking using tasks designed to test students’ pattern recognition and linear function abilities? The preservice teacher, Amy, was a middle school preservice teacher from the Mathematics and Science Education Department of a major land-grant university in the southeastern Unites States. The data consists of Amy’s responses during a face-to-face, semi-structured interview, written work she produced during the interview, and a careful analysis of a transcript of the interview. With regards to interpreting important mathematical ideas in students’ work in the National Council of Teachers of Mathematics [NCTM] videos, this study found that the preservice teacher provided explanations of specific important mathematical ideas and their meaning. When the preservice teacher interpreted students’ mistakes or misconceptions, she explained the misconceptions mathematically rather than providing specific examples from the video. In terms of interpreting the similarities between students’ reasoning, the preservice teacher’s description of the similarities between students’ reasoning was broad. Depending on the task and how many times she watched the videos, either one or two times, the extent to which she provided evidence varied when she interpreted the differences between students’ reasoning. This study revealed that the strength of the preservice teacher’s interpretation strategies varied depending on the tasks. The preservice teacher did not provide detailed explanations of students’ reasonings when watching NCTM videos even though she solved all the tasks in multiple ways and explained the key concepts while solving the problems.

Investigation of Middle Grade Mathematics Teachers’ Suggestions for Eliminating Student Misconceptions According to Different Variables

Bu araştırmanın amacı, ortaokul matematik öğretmenlerinin matematik derslerinde karşılaştıkları kavram yanılgılarının gidermeye yönelik sundukları çözüm önerilerini ve bu çözüm önerilerini etkileyen faktörleri incelenmektir. Bu çalışmada tarama modeli kullanılmıştır. Araştırma 2015- 2016 eğitim-öğretim yılında Ankara il Milli Eğitim Müdürlüğü’ne bağlı ortaokullarda, görev yapan 129 matematik öğretmeni ile gerçekleştirilmiştir. Araştırmanın verileri 6 sorudan oluşan Kavram Yanılgıları Çözüm Anketi (KYÇA)’nden elde edilmiştir. Öğretmenlerin KYÇA’ da sundukları çözüm önerileri puanlandırılarak elde edilen nicel verileri analiz etmek için Mann- Whitney U ve Kruskal Wallis Testleri kullanılmıştır. Elde edilen bulgulara göre ortaokul matematik öğretmenlerinin kavram yanılgılarını gidermeye yönelik ürettikleri çözüm önerilerinin sayısında cinsiyet, öğrenim durumu, lisans akademik başarı düzeyi, mesleki kıdem, meslektaşlarla iletişim, eğitim yaklaşımı ve sınıf mevcudu değişkenlerine göre anlamlı bir fark gözlenmemiştir (p>0.05). Nitel verilerin analizinde ise içerik analizi yöntemi kullanılmıştır. Analiz sonuçlarına göre öğretmenlerin kavram yanılgılarının giderilmesine yönelik sundukları çözüm önerileri öğretim bilgisi, öğretim için teknoloji bilgisi, öğrenci bilgisi ve diğer olmak üzere dört temada ele alınmıştır. Öğretmenlerin bu temalar altında kritik noktaları vurgulama, matematiksel şekil ve modelleme, konu tekrarı, somut öğretim materyali kullanma, kavram yanılgılarına hitap etme vb. gibi çeşitli çözüm önerileri sundukları görülmüştür.

Examining how middle grade mathematics students seize learning opportunities through conflict in small groups

ABSTRACT Understanding how students develop learning opportunities through peer-to-peer interaction is vital for advancing research and practice on collaborative learning environments. This study investigated the discourse practices middle grade students utilized to resolve conflict in ways that promoted or inhibited learning opportunities. Seventy-seven middle grade students were audio/video recorded as they worked in groups of three on cognitively demanding mathematical tasks over three days. Analysis centered on the nature of learners’ discourse practices within instances of small group conflict. The findings revealed 17 specific discourse practices which promote learning opportunities within conflict and six discourse practices which limit learning opportunities. Practitioners might leverage these findings by explicitly teaching and modeling the discourse practices which promote learning opportunities. Researchers might build upon the theoretical and methodological contributions of this study to examine promotive discourse practices under different constraints than were examined in the present analysis.

Helpful and Hindering Features of GeoGebra: Understanding What Affords Conceptual Understandings of Definite Integrals Among Pre-Service Middle Grades Mathematics Teachers

This paper introduces the Affording Understanding Conceptual Framework as a way for mathematics teacher educators to analyze student’s perceptions of a digital math tool’s helpfulness (e.g., GeoGebra) versus the instructor’s perceptions when seeking insights into how a lesson design could be improved. An example of this action research process is provided using GeoGebra to help pre-service middle grades mathematics teachers (MGMTs) develop their conceptual understandings of definite integrals. Findings showed that the pre-service MGMTs noticed four categories of what they considered to be primarily helpful features: “Ease of Use,” “Pictorial Representation,” “Colored,” and “Slider/Dynamic Movement” types. While the pre-service MGMTs all described these as helpful, the content of their comments showed that there was a material difference in how they viewed the “Ease of Use” type from the other three suggesting ways in which the lesson could be revised to be more effective.

Reducing the Achievement Gap: Middle Grades Mathematics Performance and Improvement

Over the past 30 years, numerous policies have focused on closing the achievement gap in the United States. Despite this emphasis on mastery and proficiency, students continue to perform below expectations. We use standardized math test performance to explore whether schools have been able to eliminate the achievement gap across middle-level education (students aged 12 to 14 in school years 6 through 8), and whether it’s possible to maintain consistent improvement within a school. Additionally, we examine how schools work to improve performance of their lowest-performing students. We find that most schools have very little impact in increasing (or decreasing) the proportion of proficient students across the middle grades. Furthermore, schools that do produce consistent gains in proficiency are generally not the same schools that improve the performance of lowest-performing students. These patterns also hold across key student subgroups: African American, Hispanic, and low-income students. Results from this study bring into question the logic behind focusing on proficiency for all students as a measure of school success.

Working at scale to initiate ongoing validation of learning trajectory-based classroom assessments for middle grade mathematics

The paper reports on the design and validation argument for classroom assessments within a digital diagnostic assessment system built on learning trajectories (LTs). It consists of a learning map of nine big ideas, 25 relational learning clusters, and 62 LTs for grades 6−8 mathematics. Students take cluster assessments, and teachers use the data to adapt instruction. An ongoing validation process is presented with data for an algebra cluster. Validation among practitioners, learning scientists, and psychometricians is conceptualized as examining and adjusting inter-level, intra-level, and construct-irrelevant variation in measures of item difficulty and deploying item response theory modeling followed by sequential regressions. Using data from 37,000 assessments collected over three years at 3 middle schools, 167 potentially non-conforming items of the 676 calibrated items (24 %) were identified and revised. The paper discusses how the trajectories and map were refined through a combination of data analysis and feedback from practitioners.

Academic Agency: The Impact of Underlying Dispositions that Affect Teachers' Sense of Responsibility to Educate all Children in a Middle Grades Mathematics Classroom

Based on the novel framework of Academic Agency, this study explored how a teacher's sense of efficacy, commitment, and knowledge influenced their overall sense of responsibility in instructing all children in the science of mathematics. This study employed a two-phase data collection process in which surveys, classroom interviews, and observations were used to examine practicing teacher's beliefs regarding efficacy, commitment, content knowledge, and responsibility. Pearson Correlation and Multiple Linear Regression were employed as analysis tools to identify the direction and strength of the relationships existing between variables. Findings suggest that there is a positive correlation between a teacher's sense of efficacy and their overall sense of responsibility and also a teacher's sense of efficacy and of mathematics teaching efficacy. The second phase of the study, participant interviews, and observations provided a more detailed look at how four purposefully selected educators enacted their articulated beliefs. Hour-long semi-structured interviews were conducted along with classroom observation. The qualitative data collected from the four individuals were analyzed using NVivo software. Based on recorded coding structures, individuals were placed into four predetermined categories ranging from Custodial Emphasis to Focused Academic Agent. Findings suggest that a sense of responsibility may be a factor in educational decisions.

Middle Grades Mathematics Teachers’ Mixed Perceptions of Content-Focused Professional Development

The purpose of this study was to understand how and why middle grades mathematics teachers did or did not fully participate in sustained, content-focused professional development. Qualitative data were collected from middle grades teachers who demonstrated varying levels of participation. The teachers identified opportunities to engage in novel ways with mathematics content as effective in improving their own mathematics understanding and, consequently, their own instruction. Participants who were more critical of the professional development felt the mathematics content was often beyond their specific grade-level. Factors identified as affecting teachers’ participation in the training included administrative support and school-related responsibilities. Offering content-focused professional development designed to reflect the structures and characteristics established in research to most effectively provide teachers with needed knowledge and skills was not enough to ensure teacher participation.

Characterizing reasoning about fraction arithmetic of middle grades teachers in three latent classes

ABSTRACT We analyzed item responses provided by a nationwide sample of 990 in-service U.S. middle grades mathematics teachers to a novel survey focused on fraction arithmetic. The survey targeted four components of reasoning in terms of measured quantities: Referent units, Reversibility, Partitioning and iterating, and Appropriateness. Using the mixture Rasch model, a psychometric model that combines unidimensional scaling with classification, we detected 3 latent classes. To understand what distinguished the performance of teachers in the three classes, we analyzed their responses to each item. One main result is that the performance of teachers across the three classes was more clearly distinguished on items targeting Reversibility and Partitioning and iterating than it was on items targeting Referent units and Appropriateness. A second main result is that the three classes formed a nested structure that aligned with good performance on more and more components of reasoning targeted by the survey items.

Students' gendered perceptions of mathematics in middle grades single‐sex and coeducational classrooms

AbstractWith the introduction of single‐sex classroom settings in coeducational public schools, there is an ongoing debate as to whether single‐sex education may reduce or reinforce traditional stereotypes and gender roles. In this article we present findings from a study that investigated the extent to which mathematics is perceived as a gendered domain among adolescent students enrolled in single‐sex classes and coeducational classes. Further we analyzed the relationships between student characteristics, class‐type, and teacher variables on students' perceptions of gender in mathematics. Findings from this study challenge the traditional view of mathematics as a male domain. Female participants more frequently considered mathematics to be a female domain than the male participants. Male participants, on the other hand, typically did not stereotype the mathematics as a gendered domain. Results from this study do not indicate, for girls at least, that participation in single‐sex classes results in a greater propensity to stereotype mathematics as a gendered domain than would be the case in coeducational classes. This study contributes to the evolving discourse and understanding of adolescents' gendered attitudes and beliefs towards mathematics—especially in light of stereotyped assertions that have a bearing on efforts to promote the learning of mathematics and science.

Investigating Teaching in Conceptually Oriented Mathematics Classrooms Characterized by African American Student Success

This article outlines several forms of instructional practice that distinguished middle-grades mathematics classrooms that were organized around conceptually oriented activity and marked by African American students' success on state assessments. We identified these forms of practice based on a comparative analysis of teaching in (a) classrooms in which there was evidence of conceptually oriented instruction and in which African American students performed better than predicted by their previous state assessment scores and (b) classrooms in which there was evidence of conceptually oriented instruction but in which African American students did not perform better than predicted on previous state assessment scores. The resulting forms of practice can inform professional learning for preservice and in-service teachers.