Michal Yerushalmy

University of Haifa | haifa · Department of Mathematics Education

This study investigated students’ engagement with feedback processes in tasks of comparing fractions using visual representations. The research had two main goals. First, it used automated analysis based on student responses to explore the diversity of work methods deployed by students in completing the tasks. Second, it investigated students’ meta...

This is one of two contributions written to commemorate Professor Uri Leron, who became Associate Editor of what was then the International Journal for Computers and Mathematical Learning, a journal founded by Seymour Papert, and upon which Digital Experiences in Mathematics Education builds on. One contribution addresses Leron’s biographical aspec...

Research on technology and mathematics education has been a longstanding interest of the PME community. In this paper we revisit the interplay between technology and conjecturing within the process of problem-solving with an intention to capture different aspects of the processes in which students make and explore mathematical conjectures, and role...

We report on an innovative design of algorithmic analysis that supports automatic online assessment of students’ exploration of geometry propositions in a dynamic geometry environment. We hypothesized that difficulties with and misuse of terms or logic in conjectures are rooted in the early exploration stages of inquiry. We developed a generic acti...

Much important research on the learning of mathematics with technology-supported inquiry has been devoted to learning with multiple-linked representations (MLR) as a mode of feedback. Like a mirror, MLR feedback helps students see their actions in one representation “reflected” in another. Yet, research has followed learning episodes where MLR feed...

Conjectures are a key component of mathematical inquiry, a process in which the students raise conjectures, refute or dismiss some of them, and formulate additional ones. Taking a design-based research approach, we formulated a design principle for personal feedback in supporting the iterative process of conjecturing. We empirically explored the de...

The article reports on a study of the resources and constraints designed in interactive diagrams to guide students’ development of ways to solve quadratic equations and perform algebraic manipulations. We report qualitatively on an experiment in which four 14- and 15-year-old students were challenged by a task presented as an interactive diagram (I...

We used an assessment platform to study the potential of rich student data obtained online to influence classroom instruction and help teachers respond to students’ needs. The article focuses on technology-based formative assessment in the course of teaching a unit on quadrilaterals. We studied example-eliciting tasks, requiring students to determi...

This article queries how learning analytics systems can support content-specific group formation to develop students’ thinking about a specific mathematical concept. Automated group formation requires identifying personal characteristics, designing tasks to probe students’ perceptions, and grouping them to increase individual learning chances. Desi...

In their enactment of the curriculum, teachers have a substantial role as instructional designers. Accordingly, any evaluation of the progression of students’ learning should first be concerned with the pedagogical intentions of the teacher. In this article we present a method for reconstructing teachers’ implicit and tacit considerations in their...

With the emergence of e-textbooks, along with expectations to integrate technology in instruction, teachers are becoming significant co-designers of the curriculum. Informed selection and sequencing of learning resources requires sensitivity to didactic nuance, and tools to support development and application of such sensitivity. Teachers’ practice...

We argue that examples can do more than serve the purpose of illustrating the truth of an existential statement or disconfirming the truth of a universal statement. Our argument is relevant to the use of technology in classroom assessment. A central challenge of computer-assisted assessment is to develop ways of collecting rich and complex data tha...

Automated online formative assessment of students' work in a rich digital environment has the potential to support and develop students' reasoning process. Previous studies have presented the challenge of designing e-tasks. Here we focus on the challenge of designing a personal online formative assessment that supports the students' reasoning proce...

What are mathematics teachers' considerations in grouping students, and how could automated formative assessment systems help them in doing it? In this study, nine teachers were asked to use data on students' performance in a mathematics task, derived from an automated formative assessment system, to create pairs in which students could contribute...

In the world of print, textbooks were the most important tools for dictating what and how student learn in schools. The introduction of Information and Communication Technology (ICT), however, gave rise to eTextbooks – a multi-modal, hardware mediated, and connectable, curriculum material. Indeed, the emergence of eTextbook creates fascinating oppo...

The paper describes two versions of an inquiry-based activity in geometry, designed as a game between two players. The game is inspired by Hintikka’s semantic game, which is a familiar tool in the field of logic to define truth. The activity is designed in a dynamic geometry environment (DGE). The inquiry is initially guided by the game itself and...

Despite the widespread use of dynamic geometry environments (DGEs) in mathematics classrooms, and despite the emergence of online assessment systems, DGEs still play only a minor role in assessment practices. In this paper, we examined the potential benefits and feasibility of using information gleaned from students' interactions with geometry inte...

With the emergence of e-textbooks, together with expectations to integrate technology in instruction, teachers are coming to take an active role in designing the curriculum they teach. In orderto maintain instructional coherence, teachers need to develop sensitivity to various didactical aspects of the curriculum. We suggest the use of novel tools...

Paper tasks are often redesigned to function as digital tasks. The research and design literature (Pead, 2010; Burkhardt & Pead, 2003) has reported on the challenges of such a transformation. We report on a study exploring the design principles of an e-task, originally designed as a paper-and-pencil task and converted into an interactive diagram. W...

Teachers face challenges when conducting lessons built on student-designed solu tion strategies. The work we describe here is part of an effort to explore how tech nological support for classroom formative assessment may play a role in attempts to reform mathematics instruction. We focus on the challenges involved in teach ers' attempts to design a...

Our research focuses on the e-assessment of challenging ‘construction’ e-tasks designed to function as a dynamic interactive environment of multiple linked representations (MLR); we explore the effect of constraints on the variation in the students’ response space. Students are asked to determine whether an existential statement is correct. If they...

Our research focuses on the e-assessment of challenging ‘construction’ e-tasks designed to function as a dynamic interactive environment of multiple linked representations (MLR); we explore the effect of constraints on the variation in the students’ response space. Students are asked to determine whether an existential statement is correct. If they...

Interactive textbooks appear to be the tools of choice in mathematics instruction in the foreseeable future. It is important, therefore, to establish the theoretical foundations of design that define student-textbook-teacher interactions. In our long-term research, we suggested, tested, and refined a semiotic framework that offers a set of terms he...

In the present study, we ask whether and how online assessment can inform teaching about students’ understanding of advanced concepts. Our main goal is to illustrate how we study design of tasks that support reliable online formative assessment by automatically analyzing the objects and relations that characterize the students’ submissions. We aim...

Rich technological environments present many opportunities for guided inquiry in the mathematics classroom. In this paper we focus on the role of the teacher supporting the forming and proving of conjectures by the students, during a whole class discussion. We examine the practices of an expert teacher that conducts a classroom discussion based on...

Multiple choice (MC) items are the natural choice for automated online assessment. Ideally, making a choice should be based on knowledge and reasoning. Nevertheless, studies demonstrate that often various techniques (e. g. guessing) are the common practices. In the last decade technology is recruited to support real-time feedback as formative asses...

This paper describes the design process and the underlying principles of an innovative mathematical gaming activity. This game, somewhat similar in the playing principles to ”Guess who?” , is based on interaction between two players, one questioning the other in order to deduce which mathematical object the opponent had chosen. This interaction req...

In light of the recent growing interest in conceptual learning and teaching of calculus, and especially with the focus on using technological environments, our study was designed to explore the learning processes and the role played by multiple-linked representations and by interactive technological environment in objectifying the accumulation func...

Over the past decade, a wide variety of PME papers have focused on the use of digital technologies in mathematics education. The 2006 PME Handbook separated research on the use of digital technologies across three main chapters (geometry, arithmetic and algebra, and proof); however, this 2016 PME Handbook unites the topic areas together.

In most educational systems, the textbook remains the core external authority. However, as textbooks rapidly change from print to digital formats, it is assumed that the ways in which they will be used will also change. In this situation, the main challenge is to rethink the sets of concepts and images used to guide us in thinking about the structu...

Our research focuses on the e-assessment of challenging calculus construction e-tasks designed to function as a dynamic interactive environment of multiple linked representations (MLR) that provide feedback to the learner. A construction e-task requires students to use technological affordances to construct examples that satisfy specific conditions...

The history of visualization within mathematics education is a long one. Since the beginning of the 1980s mathematics educators are interested in the practical challenges of teaching visualization, in visualization of mathematics as exhibits in school or aligned with educational psychology and are looking for theoretical frameworks.

The present study was designed to identify the objectification processes involved in making sense of the concept of an indefinite integral when studied graphically in a dynamic technological environment. The study focuses on 11 pairs of 17-year-old students familiar with the concept of differentiation but not of integration. The students were asked...

This case study is one in a series of studies regarding the learning of the integral concept. Its focus is on the processes involved in the convergence of Riemann accumulation function (RAF) to the accumulation function (AF). The study is guided by the objectification theory which considers learning to be a process of becoming aware of the knowledg...

As part of a long-term research project about the design of guided inquiry in mathematics classrooms, we studied the design principles and the functions of a specific class of interactive digital diagrams called narrating interactive diagrams. These diagrams are designed to provide a structure for the available options in explorations. They provide...

The present study focuses on the accumulation process involved in the integration of a single-variable function. Observing the work of two high-school calculus students who had not yet learned any other integral-related ideas, we analyze the emergence of the semiotic relationship between personal and mathematical meanings, as expressed through the...

In an attempt to study the emerging questions about design of mathematical tasks that could support the solving of challenging problems, we designed two settings of interactive diagrams that share an example represented as an animation of multi-process motion but differ in their organizational functions. The interactive settings, each comprising of...

Supporting teachers' interaction with instructional materials, while developing curricular, subject matter and pedagogical knowledge are major goals of professional development programs world-wide. Mobile Gurukul project focuses on supporting teachers' professional development in the rural areas, especially in the rural of areas India. We address t...

Engaging mathematics students in active exploration of real-life scenarios and supporting inquiry processes are major challenges for teachers. It requires a shift in the teacher’s role from lecturing and telling to listening, observing, facilitating, and guiding. In our exploratory work we found that mobile devices can offer a challenging setting f...

We investigated how students use the representation of data in a given example appearing in an interactive diagram (ID) and how they create additional examples with the ID. Students who worked with the ID that offered limited representations and tools (illustrating ID) looked for ways to bypass the designed constraints: they changed the representat...

This chapter explores the learning experienced by teachers engaged in algebra reform inquiry. The chapter describes two ways of using video episodes in the analysis of learning practices acquired through teaching. First, we perform a comparative analysis of two episodes that involve guided inquiry during whole-group discussion by the same students,...

Software design is a crucial dimension in the educational use of technology and a key for transformative practices. The choice for this chapter, issued of a plenary panel at the study conference, has been to ask creators and designers of well established and widely used environments to contribute from his/her own unique expertise. After introductor...

We investigate whether and how printed diagrams vs. interactive diagrams, video clips vs. interactive animations, create different contexts for mathematics learning.The present paper analyzes an experiment in which two algebra activities are presented to students in a task-based interview. One activity, describing a motion situation, is presented i...

he first. The learning environment designed for this study includes activi ties focusing on the use of computer applications that support direct manipulat ions of graphs and parameters related to the visual representations of accumulati on and integrals. The case study focuses on two 17 year old students who have studie d differentiation but not in...

This interview study takes place in the context of a single small district in the United States. In the algebra curriculum of this district, there was a shift in the conception of equation, from a statement about unknown numbers to a question about the comparison of two functions over the domain of the real numbers. Using two of Shulman’s [Shulman,...

This paper examines the relation between bodily actions, artifact-mediated activities, and semiotic processes that students experience while producing and interpreting graphs of two-dimensional motion in the plane. We designed a technology-based setting that enabled students to engage in embodied semiotic activities and experience two modes of inte...

The domain of digital interactive mathematics textbooks is new and largely unexplored as yet. In our study we seek to identify practices associated with the design of type of textbook and to focus on the functions of interactive visual representations, which we call interactive diagrams (IDs). An ID is a relatively small and simple software applica...

The paper presents a pilot case study intended to examine how socio-cultural and situated learning aspects are reflected in learning experiences within a novel mobile learning environment, Math4Mobile, a cellular application for mathematics learning. The case study focused on four students in a mathematics methods course who were engaged in a mathe...

As is suggested by the title, this chapter presents research on algebra learning and teaching that has been carried out in various technological environments, more specifically those where the focus has been either multiple representations in computer and graphics calculator environments, or dynamic control, or structured symbolic calculation. Disc...

This paper describes a study of the cognitive complexity of young students, in the pre-formal stage, experiencing the dragging tool. Our goal was to study how various conditions of geometric knowledge and various mental models of dragging interact and influence the learning of central concepts of quadrilaterals. We present three situations that ref...

The article discusses the ways that less successful mathematics students used graphing software with capabilities similar to a basic graphing calculator to solve algebra problems in context. The study is based on interviewing students who learned algebra for 3 years in an environment where software tools were always present. We found differences be...

This paper focuses on the cognitive processes that occur while students are exploring motion graphs. In a classroom experiment, we examine how high-school students (aged 17), with backgrounds in calculus and physics, interpret the graphs they create through drawing the path of the movement of their hand with a computer mouse. Based on recent, and e...

The paper explores changes in technology that have implications for the teaching and learning of school mathematics. To this end, it examines aspects of interactive mathematical textbooks; specifically it analyzes functions authors may intend to be carried out by embedded interactive diagrams. The paper analyzes theoretical as well as practical les...

In this study, we focus on the first steps of qualitative modeling of temporal phenomena. In particular, we analyze subtleties involved in the analysis of images of "hand-made motion." We start with the description of the rationale of the environment on which this study is based. We then describe two pairs of seventh graders who are involved in mod...

The sequential organization of actions necessary to construct a figure in a dynamic geometry environment (DGE) introduces an explicit order of construction. Such sequential organization produces what is, in effect, a hierarchy of dependences, as elements of the construction depend on something created earlier. This hierarchy of dependences is one o...

The article focuses on a specific method of constructing the concept of function. The core of this method is a didactic model that plays two roles together- on the one hand a role of a model of the concept of function and on the other hand a role of a model of physical phenomena that functions can represent. This synergy of modeling situations and...

This study is an attempt to analyze students' construction of function based problem solving methods in introductory algebra. It claims that for functions to be a main concept for learning school algebra, a complex process that has to be developed during a long period of learning must take place. The article describes a longitudinal observation of...

This study attempts to investigate how teaching function’s based approach to algebra with graphing technology challenges thinking about traditional introductory algebra. It enlightens the rational for studying teachers who reform their algebra teaching with technology. Analyzing the mathematical knowledge and didactical approaches as reflected in d...

The goal of the workshop "On Tools and Classrooms: New approaches to algebra and calculus using innovative software." is to explore the mathematical and educational rational for the design of the software and the activities that support the approach of the "Visual Mathematics" algebra and Calculus curriculum1. In the workshop we will concentrate on...

Many recent projects have lately explored function-based approaches to algebra. For the last five years we developed and tested such an approach to algebra, the 'Visual Mathematics'1 curriculum, in grades seven to nine. While teaching this curriculum we learned from our students' work that while functions form an important basis for solving various...

There is plenty of software in use in the secondary school mathematics curriculum. It runs on computers or on advanced calculators. Introducing software for exploration in the secondary mathematics class does not require a radical change in the curriculum agenda because it allows and promotes new organization of both content and learning styles. Th...

This work is part of a larger attempt to explore the nature of symbolic understanding involving graphic technology. This study describes learning advanced mathematics that occurs through constructing qualitative reasoning methods using graphic technology. Data was gathered from a precalculus class who, for a few weeks, investigated and explored asy...

One major part of the effort to reform secondary school mathematics is the project of changing the goal of studying school algebra from the mastery of symbolic manipulations to the ability to reason algebraically. Another major component of these reform efforts is the creation of opportunities for students to communicate within and about mathematic...

One major part of the effort to reform secondary school mathematics is the project of changing the goal of studying school algebra from the mastery of symbolic manipulations to the ability to reason algebraically. Another major component of these reform efforts is the creation of opportunities for students to communicate within and about mathematic...

In this article, I concentrate on the linguistic aspects of modeling, primarily investigating the uses of notations over tasks that consist of multievent temporal phenomena. Specifically, I describe attempts by a group of 7th graders who were algebra beginners to reformulate narratives using two lexical sets: verbal and iconic. The 2 sets are paral...

What is it that a rich society owes to its citizens? Not, I think, warehousing under the pretext that they are in “ninth-grade algebra or learning Macbeth,” but something far deeper, more valuable, more personal, more meaningful. When a student is bored, I do not think it is necessarily a failure of the student, nor of the teacher, but more often a...

The rationale for teaching geometry as part of the standard high school curriculum is twofold: 1) to teach students about the measurement, properties, and relationships of points, lines, angles, surfaces, and solids; and 2) to teach students deductive reasoning by exposing them to classical Euclidian geometry, the archetype deductive system. Most g...

This study attempted to enhance 8th grade students' acquisition of basic geometrical concepts through an instructional intervention which focused on constructing meaningful definitions by incorporating the use of the GEOMETRIC SUPPOSER software. Results of a test administered subsequent to the learning indicated that students in the experimental gr...

The development of computer programs with multiple representation capabilities allows us to manipulate the numeric, symbolic, and graphic aspects of functions and to observe the effects and results of these manipulations in each of these aspects, either separately or simultaneously. The combined use of symbolic and visual representations is expecte...

This study examined the effect of graphic representation of algebraic functions on performance of tasks involving expressions' transformations. Twenty eight teachers, who received instruction and practice using multiple representation software, were tested on their ability to debug faulty transformed expressions, either with or without graphic feed...

This article presents research about the effects of computerized feedback on the way students carry out algebraic transformations and how they debug their own working processes. Twenty-five seventh graders met for two-hour weekly sessions in a computer lab, over the course of twelve weeks. They were divided into four groups, each group working with...

Multiple representation software was used here as the core of the algebra curriculum of eighth graders (age 14 years) learning algebraic functions and graphs. The connection between the symbolic representation of functions and their graphs received special emphasis. Thirty-five students met 20 times over a 3-month period in computer-facilitated les...

This paper presents research about the effect of intensive work with diagrams on high school students' use of diagrams in geometry. It identifies three obstacles, culled from previous research, which students must overcome when examining and interpreting diagrams: Diagrams are particular; common usage confuses certain standard diagrams with the cla...

From 1984 through 1988, the authors worked with teachers using an inquiry approach to teach high school geometry courses with the aid of the GEOMETRIC SUPPOSERS. Problems are a critical component of the approach, as they are of any instructional process, because they focus attention and energy and guide students in the application, integration, and...

The Geometric Supposer computer software is described. It allows users to make any geometric construction that Euclid knew how to make. Some explorations with the Supposer are presented. (MNS)