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Full-Text Articles in Science and Mathematics Education

Is Teaching Parallel Algorithmic Thinking To High School Students Possible? One Teacher’S Experience, Shane Torbert, Uzi Vishkin, Ron Tzur, David Ellison Jan 2016

Is Teaching Parallel Algorithmic Thinking To High School Students Possible? One Teacher’S Experience, Shane Torbert, Uzi Vishkin, Ron Tzur, David Ellison

Ron Tzur

All students at our high school are required to take at least one course in Computer Science prior to their junior year. They are also required to complete a year-long senior project associated with a specific in-house laboratory, one of which is the Computer Systems Lab. To prepare students for this experience the lab offers elective courses at the post-AP Computer Science level. Since the early 1990s one of these electives has focused on parallel computing. The course enrolls approximately 40 students each year for two semesters of instruction. The lead programming language is C and topics include a wide ...


Nature And Utility Of Teacher Questioning: A Case Of Constructivist-Oriented Intervention, Jessica Hunt, Ron Tzur, Arla Westenskow Jan 2016

Nature And Utility Of Teacher Questioning: A Case Of Constructivist-Oriented Intervention, Jessica Hunt, Ron Tzur, Arla Westenskow

Ron Tzur

No abstract provided.


Interweaving Tasks And Conceptions To Promote Multiplicative Reasoning In Students With Learning Disabilities In Mathematics, Yan Xin, Ron Tzur, Luo Si, Dake Zhang, Casey Hord, Wei Luo, Suleyman Centintas Jan 2016

Interweaving Tasks And Conceptions To Promote Multiplicative Reasoning In Students With Learning Disabilities In Mathematics, Yan Xin, Ron Tzur, Luo Si, Dake Zhang, Casey Hord, Wei Luo, Suleyman Centintas

Ron Tzur

This case study examined the efficacy of tasks designed for promoting multiplicative reasoning in students with learning disabilities. Chad’s (grade 4) construction of a mixed-unit coordination scheme was nurtured in the context of a teaching experiment with 14 students in two USA Midwest elementary schools. The analysis focuses on how a sequence of tasks, tailored to Chad’s available conceptions, brought forth his transfer of a crucial mathematical idea to novel, realistic problem situations. We argue for the use of such conceptually tailored tasks as a means for promoting students’ progress from what they know to a transfer-enabling stage ...


The Effect Of An Intelligent Tutor On Math Problem-Solving Of Students With Learning Disabilities, Xiaojun Ma, Yan Xin, Ron Tzur, Luo Si, Xuan Yang, Joo Park, Jia Liu, Rui Ding Jan 2016

The Effect Of An Intelligent Tutor On Math Problem-Solving Of Students With Learning Disabilities, Xiaojun Ma, Yan Xin, Ron Tzur, Luo Si, Xuan Yang, Joo Park, Jia Liu, Rui Ding

Ron Tzur

Reform-based math instruction calls for students’ construction of conceptual understanding, solving challenging problems and explanation of reasoning. However, existing literature shows that students with learning disabilities (LD) easily get lost in reform-based instruction. As an outcome of collaborative work between math education and special education in instructing students with D, we’ve developed an intelligent tutor (PGBM-COMPS) to nurture multiplicative reasoning of students with LD. The intelligent tutor dynamically models individual student's evolving conceptions and recommends tasks to promote her/his advancement to a higher level in the learning trajectory and solve complex word problems using mathematical model equations ...


Jake’S Conceptual Operations In Multiplicative Tasks: Focus On Number Choice, Rachael Risley, Nicola Hodkowski, Ron Tzur Jan 2016

Jake’S Conceptual Operations In Multiplicative Tasks: Focus On Number Choice, Rachael Risley, Nicola Hodkowski, Ron Tzur

Ron Tzur

No abstract provided.


Students With Learning Disability In Math Are Left Behind In Multiplicative Reasoning? Number As Abstract Composite Unit Is A Likely ‘Culprit’, Ron Tzur, Yan Xin, Rachael Kenney, Adam Guebert Jan 2016

Students With Learning Disability In Math Are Left Behind In Multiplicative Reasoning? Number As Abstract Composite Unit Is A Likely ‘Culprit’, Ron Tzur, Yan Xin, Rachael Kenney, Adam Guebert

Ron Tzur

This study addressed the problem of why students with learning disabilities in mathematics too often fail to develop multiplicative and divisional concepts/operations. We conducted a constructivist teaching experiment with 12 students (nine 5th and three 4th graders). This report focuses on three students’ conceptual progress, particularly on Sandy’s (most pronounced). Our analysis indicates that, at the outset, those three could only reason additively because they lacked a robust concept of number as an abstract composite unit and were bound to rely on strategies of counting units of one. Once teaching engendered a concept of number in them, they ...


Non-Traditional Use Of ‘Math-As-Human-Endeavour’: Two Chinese Middle School Teachers’ Attempts To Inspire Students’ Curiosity, Xianyan Jin, Ron Tzur Jan 2016

Non-Traditional Use Of ‘Math-As-Human-Endeavour’: Two Chinese Middle School Teachers’ Attempts To Inspire Students’ Curiosity, Xianyan Jin, Ron Tzur

Ron Tzur

No abstract provided.


The Power Of Natural Thinking: Applications Of Cognitive Psychology To Mathematics Education, Ron Tzur Jan 2016

The Power Of Natural Thinking: Applications Of Cognitive Psychology To Mathematics Education, Ron Tzur

Ron Tzur

Concurring with Uri Leron’s cross-disciplinary approach to distinct modes of mathematical thinking, intuitive and analytic, I discuss his elaboration and adaptation to our field of the cognitive psychology dual-processing theory (DPT). I reflect on (a) the problem significance, (b) aspects of the theory he adapts, and (c) elegance of presentation. Then, I further discuss DPT in light of a constructivist stance on the inseparability of thinking and learning. I link DPT to accounts of (i) brain-based conceptual learning and (ii) how mathematics teaching may promote such learning—and discuss advantages of those accounts.


Want Teaching To Matter? Theorize It With Learning..., Ron Tzur Jan 2016

Want Teaching To Matter? Theorize It With Learning..., Ron Tzur

Ron Tzur

No abstract provided.


Explicating A Mechanism For Conceptual Learning: Elaborating The Construct Of Reflective Abstraction, Marty Simon, Ron Tzur, Karen Heinz, Margaret Kinzel Jan 2016

Explicating A Mechanism For Conceptual Learning: Elaborating The Construct Of Reflective Abstraction, Marty Simon, Ron Tzur, Karen Heinz, Margaret Kinzel

Ron Tzur

We articulate and explicate a mechanism for mathematics conceptual learning that can serve as a basis for the design of mathematics lessons. The mechanism, reflection on activity-effect relationships, addresses the learning paradox (Pascual-Leone, 1976), a paradox that derives from careful attention to the construct of assimilation (Piaget, 1970). The mechanism is an elaboration of Piaget's (2001) reflective abstraction and is potentially useful for addressing some of the more intractable problems in teaching mathematics. Implications of the mechanism for lesson design are discussed and exemplified.


Children's Development Of Multiplicative Reasoning: A Schemes And Tasks Framework, Ron Tzur, Heather Johnson, Evan Mcclintock, Rachael Kenney, Yan Xin, Luo Si, Jerry Woodward, Casey Hord, Xianyan Jin Jan 2016

Children's Development Of Multiplicative Reasoning: A Schemes And Tasks Framework, Ron Tzur, Heather Johnson, Evan Mcclintock, Rachael Kenney, Yan Xin, Luo Si, Jerry Woodward, Casey Hord, Xianyan Jin

Ron Tzur

In this paper we propose a developmental framework that makes distinctions and links among schemes—conceptual structures and operations children construct to reason in multiplicative situations. We provide a set of tasks (problem situations) to promote construction of such schemes. Elaborating on Steffe et al.’s (Steffe & Cobb, 1998) seminal work, this framework synthesizes findings of our teaching experimentsi with over 20 children who have disabilities or difficulties in mathematics. This empirically grounded framework contributes to articulating and promoting multiplicative reasoning—a key developmental understanding (Simon, 2006) that presents a formidable conceptual leap from additive reasoning for students and teachers (Harel & Confrey, 1994; Simon & Blume, 1994). In place of pedagogies that focus primarily on multiplication procedures, our framework can inform teaching for and studying of children’s conceptual understandings. Such understandings provide a basis not only for promoting multiplication and division concepts and procedures but also for reasoning in place-value number systems, and in fractional, proportional, and algebraic situations (Thompson & Saldnha, 2003; Xin, 2008). We contrast our stance on children’s cognitive change and teaching that promotes it with the Cognitively Guided Instruction (CGI) approach (Carpenter, Franke, Jacobs, Fennema, & Empson, 1998)). CGI grew out of research on children’s solutions to addition and subtraction tasks. By asserting that “children’s solution processes directly modeled the action or relationships described in the problem” (Carpenter, Hiebert, & Moser, 1983, p. 55), CGI researchers seemed to equate children’s cognitive processes with tasks. In contrast, we argue for explicitly distinguishing between task features as adults conceive of them and schemes children bring forth for solving tasks. Consider a Join task such as, “We had 7 toys and got 4 more; how many toys we then had in all?”

A child may solve such a task by counting-all 1s (1-2-3-…10-11), by counting-on (7; 8-9-10-11), or ...


Culturally-Mathematically Relevant Pedagogy (Cmrp): Fostering Urban English Language Learners' Multiplicative Reasoning, Ron Tzur, Heather Johnson, Evan Mcclintock, Rachael Risley Jan 2016

Culturally-Mathematically Relevant Pedagogy (Cmrp): Fostering Urban English Language Learners' Multiplicative Reasoning, Ron Tzur, Heather Johnson, Evan Mcclintock, Rachael Risley

Ron Tzur

In this paper we articulate an approach, termed culturally-mathematically relevant pedagogy (CMRP), for fostering urban English language learners’ mathematical progression. CMRP integrates three aspects, the use of (1) adaptive teaching to build on students’ funds of knowledge for mathematics, (2) tasks that make sense to students given their current mathematical conceptions, and (3) manipulatives and representations that, for the students, meaningfully signify quantities linked to numbers and operations used in a task. To situate CMRP, we use a continuum of conceptual transitions in multiplicative reasoning, which are critical for supporting students’ development of algebraic reasoning.


Conceptually Based Task Design: Megan’S Progress To The Anticipatory Stage Of Multiplicative Double Counting, Jerry Woodward, Rachael Kenney, Dake Zhang, Adam Guebert, Suleyman Centintas, Ron Tzur, Yan Xin Jan 2016

Conceptually Based Task Design: Megan’S Progress To The Anticipatory Stage Of Multiplicative Double Counting, Jerry Woodward, Rachael Kenney, Dake Zhang, Adam Guebert, Suleyman Centintas, Ron Tzur, Yan Xin

Ron Tzur

No abstract provided.


An Account Of A Teacher's Perspective On Learning And Teaching Mathematics: Implications For Teacher Development, Ron Tzur, Marty Simon, Karen Heinz, Margaret Kinzel Jan 2016

An Account Of A Teacher's Perspective On Learning And Teaching Mathematics: Implications For Teacher Development, Ron Tzur, Marty Simon, Karen Heinz, Margaret Kinzel

Ron Tzur

This report presents an account of one teacher's mathematics teaching and a perspective that underlies his teaching. Nevil was a fifth grade teacher participating incurrent mathematics education reforms in the United States. Through the account, we make distinctions about teachers' thinking and practice that can inform teacher education efforts. We constructed an account by analyzing four sets of classroom observations and interviews. We observed that Nevil decomposed hisunderstandings of the mathematics into smaller components and connections among those components. He created situations that he believed made those components and connections transparent and attempted to elicit those connections from ...


How Do Changes Happen? Transition From Intuitive To Advanced Strategies In Multiplicative Reasoning For Students With Math Disabilities, Dake Zhang, Yan Xin, Ron Tzur, Casey Hord, Luo Si, Suleyman Centintas Jan 2016

How Do Changes Happen? Transition From Intuitive To Advanced Strategies In Multiplicative Reasoning For Students With Math Disabilities, Dake Zhang, Yan Xin, Ron Tzur, Casey Hord, Luo Si, Suleyman Centintas

Ron Tzur

This study investigated how students with mathematics learning disabilities (MD) or at-risk for MD developed their multiplicative reasoning skills from intuitive strategies to advanced strategies through a teaching experiment. The participants consisted of two fifth graders with MD and one at-risk. A micro-genetic approach with a single subject design was used. Investigators coded and analyzed five strategies children used. Results showed that the participants had fewer strategies than normal-achieving students, but they improved their performance throughout the teaching experiment. The participants increased their use of double counting and direct retrieval, and decreased their use of unitary counting during the intervention.


Conceptual And Brain Processing Of Unit Fraction Comparisons: A Cogneuro-Mathed Study, Ron Tzur, Brenden Depue Jan 2016

Conceptual And Brain Processing Of Unit Fraction Comparisons: A Cogneuro-Mathed Study, Ron Tzur, Brenden Depue

Ron Tzur

This mixed-method, qualitative/quantitative study examined (a) how a constructivist- based intervention (CBI) effected adults’ learning of unit fractions and performance on whole-number (WN) or unit fraction (FR) comparisons and (b) brain circuitry implicated (fMRI) when processing these comparisons. The CBI used unit-iteration based activities to foster a shift in participants’ understanding of FR, from the prevalent, limiting “one-out-of-so-many-equal-parts” idea to a multiplicative relation conception and thus inverse magnitude relation among FR (1/n>1/m though m>n). Pre- and two post-intervention tests indicated CBI impact on decreased reaction time in comparing not just FR but also WN and ...


Intermediate Participatory Stages Of The Concept Of Unit Fraction: Two Students With Learning Disability, Jessica Hunt, Ron Tzur, Arla Westenskow Jan 2016

Intermediate Participatory Stages Of The Concept Of Unit Fraction: Two Students With Learning Disability, Jessica Hunt, Ron Tzur, Arla Westenskow

Ron Tzur

No abstract provided.


Engendering Multiplicative Reasoning In Students With Learning Disabilities In Mathematics: Sam's Computer-Assisted Transition To Anticipatory Unit Differentiation-And-Selection, Evan Mcclintock, Ron Tzur, Yan Xin, Luo Si Jan 2016

Engendering Multiplicative Reasoning In Students With Learning Disabilities In Mathematics: Sam's Computer-Assisted Transition To Anticipatory Unit Differentiation-And-Selection, Evan Mcclintock, Ron Tzur, Yan Xin, Luo Si

Ron Tzur

We examined how a student with learning disabilities (SLD) in mathematics constructed a scheme for differentiating, selecting, and properly operating on/with units that constitute a multiplicative situation, namely, singletons (‘1s’) and composite units (abbreviated UDS). Conducted as part of a larger teaching experiment in a learning environment that synergizes human and computer-assisted teaching, this study included 12 videotaped teaching episodes with a 5th grader (pseudonym-Sam), analyzed qualitatively. Our data provide a window onto the conceptual transformation involved in advancing from absence, through a participatory, to an anticipatory stage of a UDS scheme—a cognitive root for the distributive property ...


A Comparison Of Instructional Sequence In Intelligent Tutor-Assisted Math Problem-Solving Intervention Program, Joo Park, Yan Xin, Ron Tzur, Luo Si, Casey Hord Jan 2016

A Comparison Of Instructional Sequence In Intelligent Tutor-Assisted Math Problem-Solving Intervention Program, Joo Park, Yan Xin, Ron Tzur, Luo Si, Casey Hord

Ron Tzur

No abstract provided.


Curricular Change Agenda For Failure-Experienced Mathematics Students: Can Success-Promoting Assessment Make A Difference?, Ron Tzur, Nitsa Movshovitz-Hadar Jan 2016

Curricular Change Agenda For Failure-Experienced Mathematics Students: Can Success-Promoting Assessment Make A Difference?, Ron Tzur, Nitsa Movshovitz-Hadar

Ron Tzur

The study reported in this paper addressed the question: Can a success promoting assessment schema (SPAS) be designed so as to have a positive impact on mathematics learning of failure-experienced students? Addressing the problem of the study is important because assessment of students' mathematics learning greatly impacts the way mathematics is taught and learned in schools (National Council of Teachers of Mathematics, 1995). Learning mathematics is essential for one's functioning in today's society, and it is considered desirable that all students know and use mathematics (For Good Measure, 1992; National Council of Teachers of Mathematics, 1989). However, these ...


Explicating The Teacher's Perspective From The Researchers' Perspectives: Generating Accounts Of Mathematics Teachers' Practice, Marty Simon, Ron Tzur Jan 2016

Explicating The Teacher's Perspective From The Researchers' Perspectives: Generating Accounts Of Mathematics Teachers' Practice, Marty Simon, Ron Tzur

Ron Tzur

In this article we articulate a methodology for studying mathematics teacher development in the context of reform. The generation of accounts of teachers'practice, an adaptation of the case study, provides an approach to understanding teachers' current practice and to viewing their current practice in the context of development toward envisioned reforms. The methodology is an alternative both to studies that focus on teachers' deficits and to teachers' own accounts of their practice. Conceptual frameworks developed within the mathematics education research community are applied to the task of investigating the nature of practice developed by teachers in transition. We characterize ...


Characterizing A Perspective Underlying The Practice Of Mathematics Teachers In Transition, Marty Simon, Ron Tzur, Karen Heinz, Margaret Kinzel, Margaret Smith Jan 2016

Characterizing A Perspective Underlying The Practice Of Mathematics Teachers In Transition, Marty Simon, Ron Tzur, Karen Heinz, Margaret Kinzel, Margaret Smith

Ron Tzur

We postulate a construct, perception-based perspective, that we consider to be fundamental to the practices of many teachers currently participating in mathematics education reform in the United States. The postulation of the construct resulted from analyses of data from teaching experiments in teacher education classes with a combined group of prospective and practicing teachers and from case studies with individuals from that group. A perception-based perspective is grounded in a view of mathematics as a connected, logical, and universally accessible part of an ontological reality. From this perspective, learning mathematics with understanding requires learners' direct (firsthand) perception of relevant mathematical ...


Moving Students Through Steps Of Mathematical Knowing: An Account Of The Practice Of An Elementary Mathematics Teacher In Transition, Karen Heinz, Margaret Kinzel, Marty Simon, Ron Tzur Jan 2016

Moving Students Through Steps Of Mathematical Knowing: An Account Of The Practice Of An Elementary Mathematics Teacher In Transition, Karen Heinz, Margaret Kinzel, Marty Simon, Ron Tzur

Ron Tzur

We present an account of a sixth-grade teacher's practice as she responds to the challenges of current reform initiatives. We analyzed classroom observations and interviews to understand how the teacher, Ivy, teaches and thinks about teaching mathematics to her students. For Ivy, mathematical meaning is available in particular experiences. She creates these experiences for her students by leading them through a predetermined sequence of steps of mathematical knowing. This account contributed to our postulation of a perspective on mathematics learning that we refer to as perceptionbased, in which the goal of instruction is to create opportunities for students to ...


An Integrated Research On Children's Construction Of Meaningful, Symbolic, Partitioning-Related Conceptions And The Teacher's Role In Fostering That Learning, Ron Tzur Jan 2016

An Integrated Research On Children's Construction Of Meaningful, Symbolic, Partitioning-Related Conceptions And The Teacher's Role In Fostering That Learning, Ron Tzur

Ron Tzur

A teaching experiment was conducted with two fourth graders to study the co-emergence of teaching and children's construction of fraction knowledge. The children's learning, i.e., modifications in their fraction schemes, was fostered through working on tasks in a computer microworld. The children advanced from thinking about a unit fraction as one of several equal parts in a whole (the equipartitioning scheme) to operating with a unit fraction as a symbolized, iterable part the magnitude of which is based on the numerosity of the partitioned whole (the partitive fraction scheme). The paper interweaves an analysis of children's ...


An Integrated Study Of Children's Construction Of Improper Fractions And The Teacher's Role In Promoting That Learning, Ron Tzur Jan 2016

An Integrated Study Of Children's Construction Of Improper Fractions And The Teacher's Role In Promoting That Learning, Ron Tzur

Ron Tzur

In this constructivist teaching experiment with 2 fourth graders I studied the coemergence of teaching and children's construction of a specific conception that supports the generation of improper fractions. The children's posing and solving tasks in a computer microworld promoted a modification in their fraction schemes. They advanced from thinking about a unit fraction as a part of a whole to thinking about it as standing in a multiplicative relationship with a reference whole (the iterative fraction scheme). In this article I report an intertwined analysis of the children's construction of this multiplicative relationship and an examination ...


Developing New Understandings Of Pds Work: Better Problems, Better Questions, Nancy Dana, Diane Silva, Belinda Gimbert, Jim Nolan, Carla Zembal-Saul, Ron Tzur, Lucy Mule, Lynne Sanders Jan 2016

Developing New Understandings Of Pds Work: Better Problems, Better Questions, Nancy Dana, Diane Silva, Belinda Gimbert, Jim Nolan, Carla Zembal-Saul, Ron Tzur, Lucy Mule, Lynne Sanders

Ron Tzur

Through sharing examples, the authors demonstrate how the analysis of long-term Professional Development School (PDS) problems and their evolution can serve as one indicator of growth in the PDS. Three persistent problem areas are identified: (a) building trust and relationships between university and school personnel, (b) reconceptualizing existing coursework to fit in the PDS context, and (3) making inquiry a central feature of the PDS. The historical evolution of these problem areas is traced through three phases of PDS development over a six-year period, including PDS Planning, PDS Pilot Year, and PDS Institutionalization. The authors conclude that, through careful analysis ...


Riding The Mathematical Merry-Go-Round To Foster Conceptual Understanding Of Angle, Ron Tzur, Matthew Clark Jan 2016

Riding The Mathematical Merry-Go-Round To Foster Conceptual Understanding Of Angle, Ron Tzur, Matthew Clark

Ron Tzur

This article presents playful activities for fostering students' conceptual understanding of angle--a root concept in mathematics--that revolve around the Mathematical Merry-Go-Round game. The authors focus on activities for two reasons. On one hand, NCTM's Principles and Standards for School Mathematics (2000) stresses the central role of student activity in coming to understand mathematics. This emphasis is consistent with a constructivist stance (Piaget 1971) about learning as an active process. On the other hand, typical activities used for teaching angle, in which an introduction of the definition is followed by operations on angles, such as measuring, adding, comparing, and classifying ...


Is Teaching Parallel Algorithmic Thinking To High School Students Possible? One Teacher’S Experience, Shane Torbert, Uzi Vishkin, Ron Tzur, David Ellison Jan 2016

Is Teaching Parallel Algorithmic Thinking To High School Students Possible? One Teacher’S Experience, Shane Torbert, Uzi Vishkin, Ron Tzur, David Ellison

Ron Tzur

All students at our high school are required to take at least one course in Computer Science prior to their junior year. They are also required to complete a year-long senior project associated with a specific in-house laboratory, one of which is the Computer Systems Lab. To prepare students for this experience the lab offers elective courses at the post-AP Computer Science level. Since the early 1990s one of these electives has focused on parallel computing. The course enrolls approximately 40 students each year for two semesters of instruction. The lead programming language is C and topics include a wide ...


Teacher And Students' Joint Production Of A Reversible Fraction Conception, Ron Tzur Jan 2016

Teacher And Students' Joint Production Of A Reversible Fraction Conception, Ron Tzur

Ron Tzur

Within a constructivist perspective, I conducted a teaching experiment with two fourth graders to study how a teacher and students can jointly produce the reversible fraction conception. Ongoing and retrospective analysis of the data revealed the non-trivial process by which students can abstract multiplicative reasoning about fractions. The study articulates a conception in a developmental sequence of iteration-based fraction conceptions and the teacher’s role in fostering such a conception in students.


Explicating The Role Of Mathematical Tasks In Conceptual Learning: An Elaboration Of The Hypothetical Learning Trajectory, Marty Simon, Ron Tzur Jan 2016

Explicating The Role Of Mathematical Tasks In Conceptual Learning: An Elaboration Of The Hypothetical Learning Trajectory, Marty Simon, Ron Tzur

Ron Tzur

Simon's (1995) development of the construct of hypothetical learning trajectory (HLT) offered a description of key aspects of planning mathematics lessons. An HLT consists of the goal for the students' learning, the mathematical tasks that will be used to promote student learning, and hypotheses about the process of the students' learning. However, the construct of HLT provided no framework for thinking about the learning process, the selection of mathematical task, or the role of the mathematical tasks in the learning process. Such a framework could contribute significantly to the generation of useful HLTs. In this article we demonstrate how ...