Michael Battista

• The Ohio State University

This study explored 10 prospective teachers’ (PTs’) understanding of the area of a rectangular region using square and non-square rectangular area-units. In an hour-long interview, each PT was first asked to explain how they would find the area of a given rectangular region in terms of a non-square notecard. For several PTs, this task prompted disc...

We report on a collaborative research study that coordinated two prominent theoretical perspectives (units coordination and spatial-numerical structuring) to build second-order models of students' geometric enumerations and measurement. We focus on one prospective elementary teacher, Jake, and his solution to a geometric enumeration task in two dim...

This report contributes novel insights into how undergraduate students think and reason about the number of tiles within a given tiling of the plane. We juxtapose two theoretical perspectives-units coordination and spatial-temporal-enactive (or S*-) structuring-to provide an analytic account of how two undergraduate students reasoned about the tili...

Permutations are fundamental to combinatorics and other areas of mathematics, and it is important that students develop efficient and conceptually supported ways of mentally constructing, listing, and enumerating them. To date, there is still much to learn about how students reason about enumerating permutations, and how instruction can support stu...

Over half a century has passed since Bruner suggested his three-stage enactive-iconic-symbolic model of instruction. In more recent research, predominantly in educational psychology, Bruner’s model has been reformulated into the theory of instruction known as concreteness fading (CF). In a recent constructivist teaching experiment investigating two...

Lockwood has argued that taking a set-oriented perspective is critical for successful combinatorial enumeration. To date, however, the research literature has not yet captured the cognitive processes involved in taking such a perspective. In this theoretical paper, we elaborate the constructs of spatial structuring and spatial-numerical linked stru...

In prior research, we proposed an initial learning trajectory resulting from analyses of two undergraduate students' schemes for enumerating permutations. We explained the levels of this learning trajectory using an elaborated theory of levels of abstraction for both operations on combinatorial composites and, in more advanced levels, symbolic repr...

Prior research has identified spatial structuring-the mental process of constructing an organization or form for an object or set of objects-as critical to students' development of spatial-geometric reasoning and understanding. We propose an alteration to this construct to include aspects of structuring that are especially salient in combinatorial...

We report on findings from two one-on-one teaching experiments with prospective middle school teachers (PTs). The focus of each teaching experiment was on identifying and explicating the mental processes and types of intermediate, supporting reasoning that each PT used in their development of combinatorial reasoning. The teaching experiments were d...

In recent years, there has been increased attention on teaching transformational geometry. There is also increased recognition of the importance of spatial reasoning in mathematics and science. As a way of integrating research on these interconnected topics, we investigated middle school students’ developing understanding of geometric rotations in...

To highlight mutually beneficial intersections between research in psychology and mathematics education, in this commentary, we connect our measurement research to that of Congdon et al. We illustrate how our qualitative investigation of measurement reasoning can elaborate, deepen, and introduce additional perspectives and insights into the researc...

Numerous studies have found that spatial ability and mathematical ability are positively correlated. But specifying the exact nature of the relation between these types of reasoning has been elusive, with much research focused on understanding correlations between mathematical performance and specific spatial skills as measured by spatial tests. We...

This unique volume surveys recent research on spatial visualization in mathematics in the fields of cognitive psychology and mathematics education. The general topic of spatial skill and mathematics has a long research tradition, but has been gaining attention in recent years, although much of this research happens in disconnected subfields. This v...

The driving forces behind mathematics learning trajectories is the need to understand how children actually learn and make sense of mathematics—how they progress from prior knowledge, through intermediate understandings, to the mathematics target understandings—and how to use these insights to improve instruction and student learning. In this book,...

Learning Trajectories in Mathematics: A Foundation for Standards, Curriculum, Assessment, and Instruction aims to provide: A useful introduction to current work and thinking about learning trajectories for mathematics education An explanation for why we should care about these questions A strategy for how to think about what is being attempted in t...

I do not think any thoughtful researcher today believes that experiments or randomized field trials are the "gold standard" for addressing all the important questions in educa- tional research. Yet, because these designs are now required by the 2001 No Child Left Behind Act (NCLB) and are being strongly encouraged in other federal legislation and f...

Describes assessment tasks and a conceptual framework for understanding elementary students' thinking about the concept of length. Teachers will learn about student difficulties with length and how to differentiate instruction to reach these learners.

As part of a discussion of cognition-based assessment (CBA) for elementary school mathematics, I describe assessment tasks for area and volume measurement and a research-based conceptual framework for interpreting students' reasoning on these tasks. At the core of this conceptual framework is the notion of levels of sophistication. I provide detail...

Because cognition is the core substance of understanding and sense making, cognition-based assessment is essential for understanding and monitoring students' development of powerful mathematical thinking. The Cognition-Based Assessment System (CBAS) project is applying the results, theories, and methods of modern research in mathematics education t...

NCTM's Principles and Standards for School Mathematics suggests that interactive geometry software can be used to enhance student learning (2000). This article shows how using such software can foster the development of students' understanding and reasoning about two-dimensional shapes. The article first describes basic principles that underlie hig...

Professional standards for school mathematics recommend that dynamic geometry programs such as the Geometer's Sketchpad can and should be used to enhance student learning of geometry. This article illustrates how a geometry computer microworld containing screen manipulable, dynamically transformable shape-making objects can promote the development...

Measurement is one of the principal real-world applications of mathematics. It bridges two critical realms of mathematics: geometry or spatial relations and real numbers. Done well, education in measurement can connect these two realms, each providing conceptual support to the other. Indications are, however, that this potential is usually not real...

This chapter discusses the way that mathematics curricula are developed in the United States. It suggests that curriculum development in mathematics education makes little progress because it fails to adhere to scientific methodology. Weaknesses in the current development process, how such development can become scientific, how curriculum developme...

Second-grader Katy was shown that a plastic inch square was the same size as one of the indicated squares on the seven-by-three-inch rectangle displayed in figure 1a. She was then asked to predict how many plastic squares she would need to cover the rectangle completely.

In this study I utilize psychological and sociocultural components of a constructivist paradigm to provide a detailed analysis of how the cognitive constructions students make as they enumerate 3D arrays of cubes develop and change in an inquiry-based problem-centered mathematics classroom. I describe the classroom work of 3 pairs of 5th graders on...

Presents an overview of the geometry findings of the Third International Mathematics and Science Study (TIMSS) and ways to use problems from the study to assess students' understanding of geometry. (ASK)

To perform a reasonable analysis of the quality of mathematics teaching requires an understanding not only of the essence of mathematics but also of current research about how students learn mathematical ideas, Mr. Battista points out. Without extensive knowledge of both, judgments made about what mathematics should be taught to schoolchildren and...

We define spatial structuring as the mental operation of constructing an organization or form for an object or set of objects. It is an essential mental process underlying students' quantitative dealings with spatial situations. In this article, we examine in detail students' structuring and enumeration of 2-dimensional (2D) rectangular arrays of s...

We define spatial structuring as the mental operation of constructing an organization or form for an object or set of objects. It is an essential mental process underlying students' quantitative dealings with spatial situations. In this article, we examine in detail students' structuring and enumeration of 2-dimensional (2D) rectangular arrays of s...

For students to find algebra conceptually meaningful, as well as useful in modeling and analyzing real-world problems, they must be able to reflect on, make sense of, and communicate about general numerical procedures (Kieran 1992). Such procedures consist of set sequences of arithmetic operations performed on numbers. Examples include computing an...

Fifth graders ben and Jessica were trying to determine the amount of space contained in their classroom. They chose as their unit of measure the 9-inch-by-9-inch-by-18-inch concrete block that had been used to construct the classroom walls. Because other students in the class had mentioned this idea, the teacher had previously brought an actual blo...

Investigates the development of shape concepts within an instructional unit on geometric paths, including the role of noncomputer and computer interactions in that development. Concludes that the computer environment and the activities facilitated students' development of their conceptualizations of the properties of geometric shapes and their conn...

Elementary school students have considerable difficulty determining the number of cubes that are contained in three-dimensional rectangular buildings like the one shown in figure 1 (Battista and Clements 1996). The reasoning required to complete such tasks is important because it builds the cognitive frame-work for understanding the measurement of...

SEE MY WORD DICTIONARY ON ALL THIS! This from Keono Grav’s chapter: Everyday experiences, and the informal knowledge that children construct from them, can and should serve as the foundation of the mathematics of space. Children can mathematize (redescribe and elaborate mathematically; reinvention and abstraction) what they first understand intuit...

In this study we investigated the application and development of spatial thinking in an instructional unit on area and motions, part of a large-scale curriculum development project funded by the National Science Foundation. We also investigated the role of noncomputer and computer interactions in that development. We collected data from paper-and-p...

We investigated the development of linear measure concepts within an instructional unit on paths and lengths of paths, part of a large-scale curriculum development project funded by the National Science Foundation (NSF). We also studied the role of noncomputer and computer interactions in that development. Data from paper-and-pencil assessments, in...

We investigated the development of linear measure concepts within an instructional unit on paths and lengths of paths, part of a large-scale curriculum development project funded by the National Science Foundation (NSF). We also studied the role of noncomputer and computer interactions in that development. Data from paper-and-pencil assessments, in...

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The present study extends previous research in this area by providing a more elaborate and theoretical description of students' solution strategies and errors in dealing with 3-D cube arrays. It describes several cognitive constructions and operations that seem to be required for students to conceptualize and enumerate the cubes in such arrays, exp...

The present study extends previous research in this area by providing a more elaborate and theoretical description of students' solution strategies and errors in dealing with 3-D cube arrays. It describes several cognitive constructions and operations that seem to be required for students to conceptualize and enumerate the cubes in such arrays, exp...

Educators continue to debate the relative emphasis that formal proof should play in high school geometry. Some argue that we should continue the traditional focus on axiomatic systems and proof Some believe that we should abandon proof for a less formal investigation of geometric ideas. Others believe that students should move gradually from an inf...

Focuses on the contribution of the "Journal for Research in Mathematics Education" to the view of learning and teaching elementary school mathematics embodied in current curricular recommendations for school mathematics. (23 references) (MKR)

This study investigated the effects of computer and noncomputer environments on learning of geometric motions. Two treatment groups, one of which used specially designed Logo computer environments, and one of which used manipulatives and paper and pencil, received eight lessons on geometric motions. Interviews revealed that both treatment groups, e...

Given their graphic capabilities, computers may facilitate the construction of geometric concepts. Comparative media research, however, reveals few differences between media; alterations in curricula or teaching strategies might also explain the positive results of many studies that compare computer to noncomputer media. Yet, there remain certain c...

Over the last decade, major advances in technology have brought new and exciting possibilities to mathematics education. The universal availability of calculators—both as stand-alones and as built-ins for such devices as cash registers—is having a profound impact on what should be taught in mathematics curricula. Computers are furnishing increasing...

Greeno (1991) has recently proposed an alternative to the information processing view of conceptual understanding and reasoning in mathematics. In his “environmental/model” metaphor, a conceptual domain is thought of as an “environment with spatial properties” in which reasoning is accomplished by interacting with mental models (p. 211). Because sp...

In the traditional instructional approach to school mathematics, students Jearn and practice a series of separate mathematical skills, then perhaps use them to solve simple, usually unrealistic, application problems. This approach has produced students whose mathematical knowledge is not only unconnected to the real world but unconnected within its...

School geometry is the study of those spatial objects, relationships, and transformations that have been formalized (or mathematized) and the axiomatic mathematical systems that have been constructed to represent them. Spatial reasoning, on the other hand, consists of the set of cognitive processes by which mental representations for spatial object...

Students grades 5 and 7, linear measurement tasks. Many students applied a direct, point counting process to define units. This depended on task situation. Point counting occured with more students when numerals were juxtaposed with points; so, implications for number line use! Notes that when count units on ruler, do attend to points (as indicato...

Two second-grade students are investigating geometry with Logo. They are working in a TEACH environment in which the turtle-movement commands they enter are not only executed but simultaneously recorded in a procedure (Battista and Clements, in press; Clements 1983-84). During earlier lessons, they had successfully maneuvered the turtle to draw squ...

Why is it that students often do not learn what they are taught? On what do they base their thinking? What can we as teachers do to help them construct accurate and robust understandings?

It has been claimed that in Logo programming children learn mathematics by using concepts that aid them in understanding and directing the turtle's movements. The fundamental basis for this claim is that appropriate Logo environments can help children elaborate on, and become cognizant of, the mathematics implicit in certain kinds of intuitive thin...

It has been claimed that in Logo programming children learn mathematics by using concepts that aid them in understanding and directing the turtle's movements. The fundamental basis for this claim is that appropriate Logo environments can help children elaborate on, and become cognizant of, the mathematics implicit in certain kinds of intuitive thin...

In reality, no one can teach mathematics. Effective teachers are those who can stimulate students to learn mathematics. Educational research offers compelling evidence that students learn mathematics well only when they construct their own mathematical understanding (MSEB and National Research Council 1989, 58).

Extending earlier work by D. H. Clements and B. K. Nastasi (1985, 1988), this study investigated whether children working in two educational computing environments—Logo and computer-assisted instruction (CAI) problem-solving—exhibited differing amounts of behaviors indicative of cooperative interaction, conflict and resolution, effectance motivatio...

The balance between visual-spatial and verbal-logical thought may determine "mathematical casts of mind" that influence how an individual processes mathematical information. Thus, to investigate the role that spatial thinking plays in learning, problem solving, and gender differences in high school geometry, spatial thought was examined along with...

The balance between visual-spatial and verbal-logical thought may determine “mathematical casts of mind” that influence how an individual processes mathematical information. Thus, to investigate the role that spatial thinking plays in learning, problem solving, and gender differences in high school geometry, spatial thought was examined along with...

To investigate the effects of computer programming in Logo on specific geometric conceptualizations of primary grade children, 48 third graders were randomly assigned to either a Logo or a control group. The Logo group was given 26 weeks of instruction in a Logo environment. The children were then interviewed to ascertain their conceptualizations o...

To investigate the effects of computer programming in Logo on specific geometric conceptualizations of primary grade children, 48 third graders were randomly assigned to either a Logo or a control group. The Logo group was given 26 weeks of instruction in a Logo environment. The children were then interviewed to ascertain their conceptualizations o...

Geometry is an essential part of mathematics. Unfortunately, according to evaluations of mathematics learning, such as the National Assessment of Educational Progress (NAEP), students are failing to understand basic geometric concepts and are failing to develop adequate geometric problem-solving skills (Carpenter et al. 1980; Fey et al. 1984; Kouba...

The computer screen in which the Logo turtle moves represents a small portion of a mathematical plane. A great deal of the geometry taught in junior and senior high school is essentially the study of such a plane and the objects that exist within it. Thus, creating and investigating shapes with the turtle clearly involves geometric thought. However...

Reports on diagnostic testing of eighth grade students' performance using two different methods. Concludes that the computer version was significantly more difficult than the other. Suggests that students may perform mathematics differently due to the characteristics of the mediums used. (RT)

Much enthusiasm has been generated about students' investigating and exploring the world of Logo's “turtle graphics” to learn geometry. This enthusiasm is easy to understand, since the computer screen in which the Logo turtle moves represents a small portion of a mathematical plane. As it is taught in middle and junior high school, geometry is esse...

The purpose of the present study was to investigate the effects of Logo programming and CAI problem-solving software on problem solving that is dependent on specialized conceptual and procedural knowledge, problem solving that is dependent on specific executive-level cognitive skills, and mathematics achievement. No significant differences were fou...

Lists three guidelines for the sound use of manipulatives in teaching geometry. Also describes activities that use manipulatives to teach topics in geometry and measurement and which illustrate the application of the three guidelines. (JN)

Having students draw and investigate geometric shapes in a Logo environment is advocated as an excellent way to have them constructively learn geometric concepts. This document illustrates the potential of Logo for getting high school students involved in mathematical explorations by presenting a sequence of seven activities. The activities concern...

This study investigated the effects of computer use upon the computer literacy of preservice elementary teachers using two methods of instruction: computer assisted instruction in an earth science course and computer programming in a mathematics education course. Computer literacy was measured by the Minnesota Computer Literacy and Awareness Assess...

This study investigated the effect of microcomputer-assisted instruction on the acquisition of computer literacy by fifth grade students. One group of fifth grade students received mathematics drill and practice instruction using a microcomputer; the other group used an equivalent noncomputer mathematics drill and practice program. Based on the res...

Teaching student the four basic operations on the set of integers in a meaningful way is a difficult task. The task could be made easier, however, if there was a single physical model for the integers in which all four basic operations could be represented. The number-line model, though useful, has serious shortcomings and is incomplete. As an alte...