Connecting Research to Teaching: Visual and Analytic Thinking in Calculus

2009 ◽  
Vol 103 (2) ◽  
pp. 140-145
Author(s):  
Erhan Selcuk Haciomeroglu ◽  
Leslie Aspinwall ◽  
Norma C. Presmeg
Keyword(s):  
Mathematics Education ◽  
Multiple Representations ◽  
Reform Movement ◽  
National Council ◽  
Analytic Thinking ◽  
Mathematical Concepts ◽  
Mathematical Objects ◽  
Ways Of Thinking ◽  
Calculus Reform

A frequent message in mathematics education focuses on the benefits of multiple representations of mathematical concepts (Aspinwall and Shaw 2002). The National Council of Teachers of Mathematics, for instance, claims that “different representations support different ways of thinking about and manipulating mathematical objects” (NCTM 2000, p. 360). A recommendation conveyed in the ongoing calculus reform movement is that students should use multiple representations and make connections among them so that they can develop deeper and more robust understanding of the concepts.

Connecting Research to Teaching: When Visualization Is a Barrier to Mathematical Understanding

2002 ◽  
Vol 95 (9) ◽  
pp. 714-717
Author(s):  
Leslie Aspinwall ◽  
Kenneth L. Shaw
Keyword(s):  
Mathematics Teachers ◽  
Mathematics Learning ◽  
Mathematical Understanding ◽  
Mathematical Representation ◽  
National Council ◽  
Mathematical Task ◽  
Mathematical Concepts ◽  
Mathematical Objects ◽  
Graphic Representations ◽  
Ways Of Thinking

AS MATHEMATICS TEACHERS, OUR INTUITION TELLS us that students benefit from being able to understand a variety of representations for mathematical concepts and being able to select and apply a representation that is suited to a particular mathematical task. Students develop mathematical power as they learn to operate on mathematical objects and to translate from one mathematical representation to another when necessary. For example, graphic representations convey mathematical information visually, whereas expressions represented symbolically may be easier to manipulate, analyze, or transform. The National Council of Teachers of Mathematics (NCTM) reinforces our intuition: “Different representations support different ways of thinking about and manipulating mathematical objects. An object can be better understood when viewed through multiple lenses” (NCTM 2000, p. 360). Notwithstanding our intuition and experience with students' representations, we teachers tend to think that students' graphic and visual abilities are always an advantage in mathematics learning. Indeed, we accept on pedagogical faith that students' conceptual understandings are enhanced whenever they use visualization.

Elementary school mathematics in the 1970s

1971 ◽  
Vol 18 (6) ◽  
pp. 385
Author(s):  
John R. Clark
Keyword(s):  
Elementary School ◽  
Mathematics Education ◽  
National Science Foundation ◽  
School Mathematics ◽  
Reform Movement ◽  
National Council ◽  
Elementary School Mathematics ◽  
National Science ◽  
Mathematics And Science

Following the successful launching of Sputnik, Congress created the National Science Foundation with instructions and funds to upgrade the scholarship of teachers of mathematics and science. Prestigious professors of mathematics, in cooperation with committees of the Mathematics Association of America and the National Council of Teachers of Mathematics, set out to produce a modern program of instruction in school mathematics. The then-existing programs were analyzed and found to be seriously inadequate in structure, in definitions and assumptions, in development of properties of operation with their appropriate symbolism, and in precision of vocabulary. During the early 1960s institutes and writing teams were engaged in producing and promoting the so-called new mathematics. The resulting reform movement in mathematics education eclipsed any previous one, both in scope and in speed of implementation.

The Turtle Deserves a Star

1986 ◽  
Vol 33 (7) ◽  
pp. 14-16
Author(s):  
Rick Billstein ◽  
Johnny W. Lott
Keyword(s):  
Mathematics Education ◽  
Problem Solving ◽  
Major Influence ◽  
Computer Language ◽  
National Council ◽  
Computing Technology ◽  
Mathematical Concepts ◽  
Problem Solving Skills ◽  
Excellent Opportunity ◽  
Mathematics Curricula

The National Council of Teachers of Mathematics recently published “The lmpact of Computing Technology on School Mathematics: Report of an NCTM Conference” (NCTM 1985). This report addresses the need for mathematics curricula and instructional methods to respond to the influence of computing technology. This report states that “the major influence of technology on mathematics education is its potential to shift the focus of instruction from an emphasis on manipulative skills to an emphasis on developing concepts, relationships, structures, and problem-solving skills.” The use of the computer language Logo offers an excellent opportunity to use technology to help develop the problem-solving skills advocated in mathematics. This article gives examples not only of how Logo might be used to teach some mathematical concepts but also of how it can be used as a problem-solving tool.

Parental Involvement in the Reform of Mathematics Education

1997 ◽  
Vol 90 (6) ◽  
pp. 421-427
Author(s):  
Dominic Peressini
Keyword(s):  
Mathematics Education ◽  
Parental Involvement ◽  
Educational Reform ◽  
Reform Movement ◽  
National Council ◽  
Content Areas ◽  
Parent Teacher Association ◽  
Recent Reform ◽  
Year 2000 ◽  
The U.S

High Schools across the nation have been, and are, engaged in efforts to implement the recommendations in the three Standards documents of the National Council of Teachers of Mathematics (1989, 1991, 1995). These efforts continue in the context of a larger educational-reform movement that spans all content areas (Fullan 1991; U.S. DOE 1994a). Many of the more recent reform documents, which point the direction for this movement, call for increaes in parental involvement and the promotion of partnerships between schools and communities (National Parent-Teacher Association 1994; U.S. DOE 1994a, 1994b). These aims are apparent in the U.S. DOE's national goals for the year 2000 (U.S. DOE 1994a):

A Role for Technology in Mathematics Education

1996 ◽  
Vol 178 (2) ◽  
pp. 15-32 ◽  
Author(s):  
Albert A. Cuoco ◽  
E. Paul Goldenberg
Keyword(s):  
Mathematics Education ◽  
Curriculum Change ◽  
Mathematics Curriculum ◽  
New Technology ◽  
Habits Of Mind ◽  
Mathematical Research ◽  
Mathematical Concepts ◽  
Mathematical Ideas ◽  
Mathematics Educators

New technology poses challenges to mathematics educators. How should the mathematics curriculum change to best make use of this new technology? Often computers are used badly, as a sort of electronic flash card, which does not make good use of the capabilities of either the computer or the learner. However, computers can be used to help students develop mathematical habits of mind and construct mathematical ides. The mathematics curriculum must be restructured to include activities that allow students to experiment and build models to help explain mathematical ideas and concepts. Technology can be used most effectively to help students gather data, and test, modify, and reject or accept conjectures as they think about these mathematical concepts and experience mathematical research.

scholarly journals You can count on your fingers: The role of fingers in early mathematical development

2018 ◽  
Vol 4 (1) ◽  
pp. 107-135 ◽  
Author(s):  
Firat Soylu ◽  
Frank K. Lester ◽  
Sharlene D. Newman
Keyword(s):  
Mathematics Education ◽  
Human Cognition ◽  
Diagnostic Tools ◽  
Mathematical Concepts ◽  
Simple Arithmetic ◽  
Road Map ◽  
Mathematics Ability ◽  
Finger Counting ◽  
Bodily Activity ◽  
And Mathematics

Even though mathematics is considered one of the most abstract domains of human cognition, recent work on embodiment of mathematics has shown that we make sense of mathematical concepts by using insights and skills acquired through bodily activity. Fingers play a significant role in many of these bodily interactions. Finger-based interactions provide the preliminary access to foundational mathematical constructs, such as one-to-one correspondence and whole-part relations in early development. In addition, children across cultures use their fingers to count and do simple arithmetic. There is also some evidence for an association between children’s ability to individuate fingers (finger gnosis) and mathematics ability. Paralleling these behavioral findings, there is accumulating evidence for overlapping neural correlates and functional associations between fingers and number processing. In this paper, we synthesize mathematics education and neurocognitive research on the relevance of fingers for early mathematics development. We delve into issues such as how the early multimodal (tactile, motor, visuospatial) experiences with fingers might be the gateway for later numerical skills, how finger gnosis, finger counting habits, and numerical abilities are associated at the behavioral and neural levels, and implications for mathematics education. We argue that, taken together, the two bodies of research can better inform how different finger skills support the development of numerical competencies, and we provide a road map for future interdisciplinary research that can yield to development of diagnostic tools and interventions for preschool and primary grade classrooms.

scholarly journals Digital Representations of Mathematical Objects in the Context of Various Forms of Representation of Mathematical Knowledge

2020 ◽  
pp. 58-86
Author(s):  
Semjon F. Adlaj ◽  
◽  
Sergey N. Pozdniakov ◽  
Keyword(s):  
Comparative Analysis ◽  
Digital Media ◽  
Mathematical Knowledge ◽  
Information Environment ◽  
Teaching Mathematics ◽  
Mathematical Concepts ◽  
Mathematical Objects ◽  
Computer Tools ◽  
Digital Representations ◽  
The Impact

This article is devoted to a comparative analysis of the results of the ReMath project (Representing Mathematics with digital media), devoted to the study of digital representations of mathematical concepts. The theoretical provisions and conclusions of this project will be analyzed based on the theory of the information environment [1], developed with the participation of one of the authors of this article. The analysis performed in this work partially coincides with the conclusions of the ReMath project, but uses a different research basis, based mainly on the work of Russian scientists. It is of interest to analyze the work of the ReMath project from the conceptual positions set forth in this monograph and to establish links between concepts and differences in understanding the impact of computer tools (artifacts) on the process of teaching mathematics. At the same time, the authors dispute the interpretation of some issues in Vygotsky’s works by foreign researchers and give their views on the types and functions of digital artifacts in teaching mathematics.

scholarly journals How Does Language Impact the Learning of Mathematics? Let Me Count the Ways

2007 ◽  
Vol 5 (1) ◽  
Author(s):  
Genevieve Boulet
Keyword(s):  
Mathematics Education ◽  
Current Literature ◽  
Teaching And Learning ◽  
Mathematics Classroom ◽  
Mathematical Discourse ◽  
Mathematical Concepts ◽  
Teachers And Students ◽  
Key Factor

The role that language plays in the teaching and learning of mathematics is at the forefront of current literature in mathematics education. In this paper, I give particular attention to the manner in which teachers and students engage in the exploration of mathematical concepts and procedures with the goal of revealing how language impacts students’ learning. Through a series of examples of language commonly used in the mathematics classroom, I address specific issues pertaining to language used to describe mathematical processes, to read and interpret notation, and to define mathematical terms. Considering that communication is a key factor in the building of understanding, it is hoped that these examples will motivate teachers to examine and to adapt their own practices in order to cultivate productive and meaningful mathematical discourse in their classrooms.

scholarly journals Students’ Cognitive Barrier in Problem Solving: Picture-based Problem-solving

2020 ◽  
Vol 11 (1) ◽  
pp. 73-82
Author(s):  
A.Wilda Indra Nanna ◽  
Enditiyas Pratiwi
Keyword(s):  
Problem Solving ◽  
Primary Education ◽  
Qualitative Method ◽  
Thought Processes ◽  
Structured Interviews ◽  
Mathematical Objects ◽  
New Findings ◽  
Mathematical Problems ◽  
Ways Of Thinking ◽  
Descriptive Qualitative

Pre-service teachers in primary education often have difficulty in solving mathematical problems, specifically fractions that are presented with a picture. In solving problems, some thought processes are needed by the teacher to reduce students' cognitive barriers. Therefore, this study aimed to reveal the cognitive barriers experienced by students in solving fraction problems. The cognitive barriers referred to in this study are ways of thinking about structures or mathematical objects that are appropriate in one situation and not appropriate in another situation. This study employed a descriptive-qualitative method. Furthermore, participants were followed up with in-depth semi-structured interviews to find out the cognitive barriers that occurred in solving fraction problems. This study discovers that the participants, in solving fraction problems, experienced all indicators of cognitive barrier and two cognitive obstacles are found as new findings that tend to involve mathematical calculations and violates the rules in dividing images into equal parts in the problem-solving procedure. 

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