How much of symbolic manipulation is just symbol pushing?

This paper deals with the complex relationship between the meanings and symbols of algebraic expressions. The study reported here uses a theoretical model to interpret some basic dynamics of algebraic thinking in the analysis of students' protocols. The strong difference noticed in protocols by students who have been given very different instructional approaches, suggests further investigation on the sources, mainly in view of educational implications.

2010

The thesis addresses questions of relating diagrams and mental representations in diagrammatic reasoning through eye movement research. It proposes model-based representations of attention for improved human-computer cooperation. In particular, the thesis proposes in theory and details in practice a computational framework for live capture and analysis of eye movement data in diagrammatic reasoning scenarios. The analysis includes an on-line generation of hypotheses about spatial mental representations currently held by human reasoners. The generated hypotheses may be employed to guide reactive and proactive behavior in semiautomated reasoning systems more (cognitively) adequately than previously possible, for example, in live human-computer collaboration or tutoring settings. Among other fields of application, the framework may be used to externally influence mental visuo-spatial reasoning in selective ways through administering specific patterns of sensory cues. Additionally, the ...

Memory & Cognition, 2001

Mathematical Thinking and Learning, 2014

This paper analyses the use of the software Grid Algebra 1 with a mixed ability class of twenty-one 9-10 year old students who worked with complex formal notation involving all four arithmetic operations. Unlike many other models to support learning, Grid Algebra has formal notation ever present and allows students to 'look through' that notation and interpret it either in terms of physical journeys on a grid, or in terms of mathematical operations. A dynamic fluidity was found between the formal notation, imagery of movements on a grid, and the process of mathematical operations. This fluidity is interpreted as a 'dance' between these three. The significant way in which this dynamic took place reflects the scaffolding and fading offered by the software which was crucial to the students' fluency with formal notation well beyond what has been reported from students of that age. This paper analyses a number of episodes from a teaching sequence of three lessons where a class of mixed ability students, of age 9-10 years old, learnt to interpret and use formal notation in tasks such as substituting and solving linear equations. The number of lessons was a constraint from the school but the focus was on what students might be able to achieve within this given timeframe. As such this paper cannot say anything about the students' longer term understanding of algebra but indicates what is possible for students of that age to achieve within just three lessons. The confidence with which students of that age worked with complex formal notation was noticeable, especially given the limited timeframe. By formal notation I mean the conventional way in which a series of arithmetic operations is expressed. In particular this includes:

Pragmatics & Cognition, 2015

Many types of everyday and specialized reasoning depend on diagrams: we use maps to find our way, we draw graphs and sketches to communicate concepts and prove geometrical theorems, and we manipulate diagrams to explore new creative solutions to problems. The active involvement and manipulation of representational artifacts for purposes of thinking and communicating is discussed in relation to C.S. Peirce’s notion ofdiagrammatical reasoning. We propose to extend Peirce’s original ideas and sketch a conceptual framework that delineates different kinds of diagram manipulation: Sometimes diagrams are manipulated in order to profile known information in an optimal fashion. At other times diagrams are explored in order to gain new insights, solve problems or discover hidden meaning potentials. The latter cases often entail manipulations that either generate additional information or extract information by means of abstraction. Ideas are substantiated by reference to ethnographic, experim...

A lot of cognition seems to depend on the interaction of procedural and conceptual knowledge. Here we will discuss only procedural knowledge and we will focus mainly on how it is executed or used, as opposed to how it is obtained. More specifically, we will examine how linear equations are solved in a moment-to-moment way and will restrict our comments to how ordinary knowledgeable people solve such equations in an ordinary way, that is, by writing several lines on paper. Our hope is that if this "simple" situation can be understood, then that understanding might offer hints as to how procedural knowledge is used in more complex settings, such as in constructing a proof or solving a non-routine problem. A knowledgeable person is likely to solve a moderately long linear equation (moderately long in that person's view) in several steps, indicated by written equations, the last of which is the solution. How does the person get from one step to the next? We conjecture that the process occurs in roughly three stages, all extremely brief. Suppose the person has completed the n-th step and the associated intermediate equation has just been written on paper. During Stage 1, the person becomes conscious of the equation just written on paper-by perceiving (seeing and understanding) it. Of course, the person has just written the equation so, depending on the equation's length, he or she may remember part of it. However, this remembering may not be sufficient for the person to be conscious of the entire equation because, during the writing process, the person may have been focusing on actions (transformations) needed to construct the current equation from the previous one, rather than focusing on the equation itself. During Stage 2, outside of consciousness, the person "recognizes" various features of the equation and "decides" what would be an appropriate action (transformation) among several possible actions "associated" with those recognized features. For example, the equation might be 5 + 3x = 2x +1, and the person might recognize it as one with several features associated with possible appropriate actions, say, moving the 5, the 2x, or the 1. The person might then decide to move the 5. "Recognize" and "decide" often refer to conscious mental actions, but we are suggesting there are similar mental actions that are not conscious. We do not mean to suggest that the person has a name for the features recognized, but only that he/she has worked with linear equations long enough to have persistent, rememberable mental structures associated with them. ii Also, "associated" often refers to a physical or abstract relationship outside a person's mind, but here we are referring to a mental association specific to a particular individual's mind. The information associate with the recognized features, that is the appropriate actions and how to carry them out, seems to be immediately available and does not require remembering, that is, consulting one's "knowledge base." We mean by an appropriate action something like this: Remove the 5 from the left side of the equations and place a-5 on the right side.

18 Unconventional Essays on the Nature of Mathematics

Robotics, artificial intelligence and, in general, any activity involving computer simulation and engineering relies, in a fundamental way, on mathematics. These fields constitute excellent examples of how mathematics can be applied to some area of investigation with enormous success. T his, of course, includes embodied oriented approaches in these fields, such as Embodied Artificial Intelligence and Cognitive Robotics. In this chapter, while fully endorsing an embodied oriented approach to cognition, I will address the question of the nature of mathematics itself, that is, mathematics not as an application to some area of investigation, but as a human conceptual system with a precise inferential organization that can be investigated in detail in cognitive science. The main goal of this pi ece is to show, using techniques in cognitive science such as cognitive semantics and gestures studies, that concepts and human abstraction in general (as it is exemplified in a sublime form by mathematics) is ult imately embodied in nature.

Although a general sense of the magnitude, quantity, or numerosity of objects is common both in untrained people and in animals, the abilities to deal exactly with large quantities and to reason precisely in complex but well-specified situations—to behave formally, that is—are skills unique to people trained in symbolic notations. These symbolic notations employ typically complex, hierarchically embedded structures, which all extant analyses assume are constructed by concatenative, rule-based processes. The primary goal of this article is to establish, using behavioral measures on naturalistic tasks, that the some of the same cognitive resources involved in representing spatial relations and proximities are also involved in representing symbolic notations: in short, formal notations are a kind of diagram. We examine self-generated productions in the domains of handwritten arithmetic expressions and typewritten statements in a formal logic. In both tasks, we find substantial evidence for spatial representational schemes even in these highly symbolic domains.

Frontiers in Psychology

This article discusses the cognitive process of transforming one representation of mathematical entities into another representation. This process, which has been called mathematical metaphor, allows us to understand and embody a difficult-to-understand mathematical entity in terms of an easy-to-understand entity. When one representation of a mathematical entity is transformed into another representation, more cognitive resources such as the visual and motor systems can come into play to understand the target entity. Because of their nature, some curves, which are one group of visual representations, may have a great motor strength. It is suggested that directedness, straightness, length, and thinness are some possible features that determine degree of motor strength of a curve. Another possible factor that can determine motor strength of a curve is the strength of association between shape of the curve and past experiences of the observer (and her/his prior knowledge). If an indivi...