Investigating Quadrilaterals as an Ongoing Task

In this article we discuss an open-ended problem involving quadrilaterals that we continually offer each semester. The task has been posed to undergraduate and graduate students in methods and problem solving classes. The task involves drawing all possible four sided figures with corners at the dots. A four by four array of dots is included in the instructions and students are asked to develop a system for knowing when they have identified all the quadrilaterals. Students are also encouraged to classify them in as many ways as they can and to look at the perimeters and angle measures. The focus of the discussion is on the potential richness of a task and how students engage in non-routine explorations

Understanding assessment while developing equitable teaching practices

Our research focuses on a growth model of teachers’ ability to assess student learning as a result of creating equitable instruction for students in informal school settings. We describe data collected as part of a study examining the mathematical reasoning of Grades 3–5 students. Our research context took place in six elementary schools from rural and urban settings. Here, we focus on one of the schools by describing how a teacher began her instruction and over time, how she developed her assessment strategies to ensure that students obtained access to and support for algebraic reasoning, mathematical content, and discourse

At the intersections of the embodiment and emergence for a mathematics teacher educator

Research in mathematics education and curriculum theory currently has a very limited set of intersections. Few education researchers claim to work in both fields. I draw on the work of those few researchers for my own understandings as a mathematics teacher educator. Now as a part of this small community, I continue to struggle with what it means to be a mathematics educator from a curriculum theorist‘s perspective. In my journey and in my research, I have come to realize that mathematics is often perceived as an external truth, a fixed set of ideas, and based on that perception, mathematics pedagogy is proffered as basics-as-breakdown (Jardine, Clifford, & Friesen, 2003). As an alternative, I propose that a different way to consider mathematics education is to imagine how one can experience being in the world with mathematics. This being with idea emerged by reviewing two topics in particular: curriculum and the history of mathematics, which are central to my understandings of teacher education, specifically mathematics teacher education. Coupled with this investigation is an autobiographical reflection of how I have experienced being in the world with mathematics and how this investigation allows for a more meaningful engagement in the teaching and learning of mathematics. The intertwining of the personal with the contextual displays how the idea of being with is an interconnected and dynamic notion

Understanding children’s reasoning in multiplication problem-solving

This article investigates two children’s intuitive thinking in solving multiplication problems from different educational backgrounds. One of the children is in a southern elementary school in the US. He was given the same problems both in first and second grades. The other child was a first grader in a southwest region of China, and she was given the same problems. The findings reveal a variety of intuitive thinking in solving the multiplication problems through addition beyond direct modeling and counting strategies. The authors also discussed how different educational backgrounds in early elementary mathematics education may affect children’s intuitive ideas and reasoning in solving multiplication problems. The study implies the importance of understanding children’s intuitive ideas of multiplication and highlights potential opportunities for developing children’s understanding of multiplicative thinking and algebraic thinking in earlier stages of arithmetic learning

Exploring Long Division Through Division Quilts

As a collective, all of the authors agree that at some point in our teaching careers we recognized that there were minimal ways to demonstrate division when teaching the algorithm in isolation; furthermore, there are rare opportunities to adhere to the expectations in mathematics education (Common Core Standards Initiative, 2011; NCTM, 2000) that students should be able to identify and use relationships between operations. Imaging is an important activity (Richardson, Pratt & Kurtts, 2010; Wheatley, 1998) that allows for such opportunities. In this article, we outline a series of activities that provide students different representations of division to help them achieve understanding and proficiency of the algorithm due to their understandings of the visual aide. It is important to state here that we believe students should be able to use the division algorithm, but only as it is attached to meaning making and images. We argue that by using Division Quilts, students are better able to demonstrate their comprehension of division, through student work as well as standardized assessments

The nature of feedback given to elementary student teachers from university supervisors after observations of mathematics lessons

This research explores the frequency and nature of mathematics-specific feedback given to elementary student teachers by university supervisors across a collection of post-lesson observation forms. Approximately one-third of the forms (n=250) analysed from five large universities had no comments related to mathematics. Forms that did have mathematics-specific feedback varied in terms of the number of summary, strength, and suggestion (i.e., type) comments and in the pedagogical focus (e.g., tasks, discourse) of those comments. Chi-square tests of independence indicated the frequency of forms with mathematics-specific feedback differed significantly by university. Results of additional Chi-square tests showed significant interactions between the type of comments and university and between the pedagogical foci of the comments and comment type. Contributing factors and implications, including connectedness of the university supervisors to the programs, professional development provided to university supervisors, and the organization of the forms, are discussed

Signs and tools of algebraic reasoning: A study of models among fifth grade students.

This study focuses on the types of models created by students during algebraic pattern finding tasks. Attention is also given to the change in models over time. This is an important area of study because a closer look is needed to better understand the models created during mathematical activity, especially in the elementary classroom. It is reported here how fifth grade students used given concrete models and created new representations of models to reason algebraically about pattern finding tasks. Twenty-five fifth grade students participated in the three-day teaching experiment. Results indicate that students' recursive models were abandoned and then transformed to explicit models, and finally adopted from others during whole class discussions. These adopted models in most cases were enduring over a six-week period

Imagery and Utilization of an Area Model as a Way of Teaching Long Division: Meeting Diverse Student Needs

The teaching and learning of long division at the elementary level and beyond has presented a longstanding challenge for teachers and students alike. As mathematics teacher educators and as a specialized educator, we address the issue by analyzing some of the challenges involved in the teaching and learning of long division – particularly focusing on students who struggle in mathematics. Our inspiration comes from two shared experiences. First, a lesson taught by one of our graduate level, in-service special education teachers inspired us to consider how other teachers could consider teaching division by using an area model. The lesson that began our initial conversations will be shared later in this article to exhibit one teacher’s use of area in teaching division in an interactive manner. Second, these conversations led to our collaborative work on a book chapter that centered on specialized mathematics education (Pratt, Richardson, & Kurtts, in press) In our chapter we focused on the significance of epistemological perspectives and how imagery relates to effective mathematics teaching and learning

Traveling Representations in a Fifth Grade Classroom: An Exploration of Algebraic Reasoning

In this three-day teaching experiment along with follow up interviews, algebraic concepts related to pattern-finding tasks were examined with 25 fifth grade students. The specific focus centered on representations from a realistic mathematics education perspective, meaning a model “of” a situation toward a model “for” a situation. Within this context, certain situational models were found that seemed to travel and permeate throughout the entire class. Students were able to generalize and justify based on the models developed during whole class discussions. Several weeks after the teaching experiment, follow up interviews indicated that the representations generated were still prevalent in students’ descriptions of the activities. Findings, analysis of findings, and implications of the study will be discussed

Connected Tasks: The Building Blocks of Reasoning and Proof

Do you find it challenging to find mathematical tasks that promote reasoning in your classroom? What type of tasks do you feel are the most important for children to investigate? Finding patterns, and making and justifying conjectures are considered the building blocks of mathematical reasoning and proof. Curriculum revisions in the United States and Australia place increased emphasis on problem solving and reasoning in the primary school curriculum (National Council of Teachers of Mathematics [NCTM], 2000; Australian Curriculum, Assessment and Reporting Authority [ACARA], 2010). A number of curriculum resources for teachers are available (e.g., NCTM, 1993; Sullivan & Lilburn, 1997) but under current reform efforts, primary teachers require additional ideas to extend problem solving and reasoning in their classrooms