Geometry and the Divine in Proclus

Dominic J. O'Meara , in Mathematics and the Divine, 2005

3. The nature of mathematical science

What precisely are the objects of mathematical science? What is their status in the structure of reality? Arguing against Plato, Aristotle saw mathematical objects as conceptual abstractions which we produce by isolating the quantitative dimension of physical objects. In contrast, Plato, according to Aristotle (Metaphysics I, 6), saw mathematical objects as realities existing independently of physical objects, occupying an intermediate realm between physical objects and the transcendent Forms. In Plato's dialogues, the position is less simple. In the Republic, for example, mathematics concerns objects that relate to the transcendent realm of Forms as images of the Forms. But what is not altogether clear is whether mathematical objects are images in the sense that they exist as image-like realities, or merely in the sense that they are the way in which the mathematician thinks immaterial objects. A further difficulty confronting the ancient interpreter of Plato was that concerning the relation between mathematical objects, as intermediates (in some sense) between Forms and physical objects, and soul, which also mediates in Plato between the Forms and the material world.

It is in connection with these difficulties that ancient Neoplatonists developed the interpretation of mathematical thinking that is presented by Proclus in commenting Euclid's geometry. According to this interpretation, 5 mathematical objects are concepts derived through logical procedures by soul from an innate knowledge common to all souls ('common notions', 'axioms'). This innate knowledge is projected (or constructed) by soul in quantity (arithmetic) and extension (geometry), with the purpose of developing (in the literal sense, 'unrolling') this innate knowledge so as to articulate and comprehend it more easily. The space in which geometrical development takes place is provided by imagination (phantasia): it is an imaginative space in which the contents of innate knowledge emerge as expressed by point, line, figure and solid, whereas arithmetic achieves a more unified, compact, purely numerical articulation of this knowledge. The innate knowledge in question, which mathematics seeks to unroll, might be described as metaphysical: it includes concepts which concern the transcendent principles of reality and the laws governing the progression of these principles from the One.

The following aspects of this theory of mathematical objects might be stressed. (1) Plato's intermediates, mathematical objects and soul, are related on this interpretation in the sense that mathematical objects are elaborations of concepts inherent in the nature of soul. (2) Mathematical objects are images of Forms in the sense that soul projects in a more accessible dimension its concepts of transcendent principles. (3) The theory involves both (Platonist) realism and constructivism in the sense that soul projects mathematical objects, but these objects are concepts expressive of transcendent realities; mathematical objects are not conceptual abstractions from physical objects. (4) Mathematical thinking is eminently discursive in its rigorous logical procedures. But this is also its weakness: it is because the soul has difficulty in grasping metaphysical truths that she has recourse to discursively elaborated expressions of them in mathematics. (5) Mathematics thus promotes perfection in the life of discursive reasoning, but it also prepares the soul for a higher level of reasoning, that of theology or metaphysics, the practice of which prepares the soul in turn for access to yet a higher level of divine life, that of non-discursive, perfect, complete knowledge, i.e. the life of divine Intellect. (6) There is a gradation within the mathematical sciences in the sense that arithmetic is a purely numerical articulation of metaphysical truths, whereas geometry extends these truths further into imaginative space. Among mathematical sciences, geometry is mediational and thus eminently suitable as representing the mediational role of mathematics in general: it is more accessible than arithmetic in its use of imaginative extension and of clearly applied demonstrative procedures. For Proclus, certainly, geometry is a privileged mediational science in the context of the divinisation of human life, a science exemplified perfectly for him in Euclid's Elements.

A passage from Proclus' commentary on Euclid beautifully summarizes his conception of the nature and philosophical importance of geometry:

"So the soul, exercising her capacity to know, projects on the imagination, as on a mirror, the ideas of the figures; and the imagination, receiving in pictorial form these impressions of the ideas within the soul, by their means affords the soul an opportunity to turn inward from the pictures and attend to herself. It is as if a man looking at himself in a mirror and marvelling at the power of nature and at his own appearance should wish to look upon himself directly and possess such a power as would enable him to become at the same time the seer and the object seen. In the same way, when the soul is looking outside herself at the imagination, seeing the figures depicted there and being struck by their beauty and orderedness, she is admiring her own ideas from which they are derived; and though she adores their beauty, she dismisses it as something reflected and seeks her own beauty. She wants to penetrate within herself to see the circle and the triangle there, all things without parts and all in one another, to become one with what she sees and enfold their plurality, to behold the secret and ineffable figures in the inaccessible places and shrines of the gods, to uncover the unadorned divine beauty and see the circle more partless than any centre, the triangle without extension, and every other object of knowledge that has regained unity." 6

For Proclus, the soul can reach self-discovery in geometry. In her geometrical projections, she sees an image of herself and, through this self-knowledge, reaches a knowledge of the presence in her of truths concerning transcendent first principles, the gods. In thinking these truths, soul already lives a life of knowledge that will bring her higher in the scale of divinity. This use of mathematics as an anticipatory imaging forth of the knowledge of the divine Proclus described as characteristic of the Pythagoreans. 7

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Mathematics Learning

L. Verschaffel , ... E. De Corte , in International Encyclopedia of Education (Third Edition), 2010

This article presents a review of important recent themes and developments in research on the learning and teaching of mathematical knowledge and thinking. As a framework, we use a model for designing a powerful environment for learning and teaching mathematics; this model is structured according to four interrelated components, namely competence, learning, intervention, and assessment (CLIA-model) (De Corte et al., 2004).

We argue and illustrate that our empirically based knowledge of each of these four interconnected components has substantially advanced over the past decades, enabling a progressively better understanding of not only the components that constitute a mathematical disposition, but also the nature of the learning and developmental processes that should be induced in students to facilitate the acquisition of competence, the characteristics of learning environments that are powerful in initiating and evoking those processes, and finally, the kind of assessment instruments that are appropriate to help monitor and support learning and teaching.

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Assessment for Math Talent and Disability: A Developmental Model

JULIE BUSSE , ... DENISE HILDEBRAND , in Handbook of Psychoeducational Assessment, 2001

FUNCTIONAL MATH SYSTEM

The functional math system is the set of component processes that must be coordinated for an individual student to engage in optimal, developmentally appropriate mathematical thinking and problem solving. Figure 8.1 graphically portrays the theoretical model of the domain-specific functional math system guiding the development of the Process Assessment of the Learner (PAL) Test Battery for Math (Berninger, in press). The functional math system is organized around the central problem-solving space, which has connections from modules for (a) reasoning, (b) arithmetic computation, and (c) conceptual knowledge. These modules may also have connections with each other. Three kinds of reasoning ability may contribute to mathematical problem solving: verbal reasoning, quantitative reasoning, and visual-spatial reasoning (see Robinson, Abbott, Berninger, & Busse, 1996, for evidence that these three kinds of reasoning are unique and not redundant). At least seven kinds of conceptual knowledge may contribute to mathematical problem-solving ability in elementary and middle school students (concepts in Figure 8.1 are representative, not exhaustive). Counting, which is initially rote and then reflects one-to-one correspondence with objects counted and understanding of the number line, emerges first and may also contribute to learning math facts in the arithmetic module. The place value concept emerges next and contributes to computational algorithms as well as problem solving. Part-whole relationship knowledge then emerges and contributes to conceptual knowledge of fractions, time, measurement, money, decimals, and probability, all of which may contribute to mathematical problem solving. The arithmetic module is organized hierarchically, with math facts feeding into computational algorithms that feed into problem solving. Math facts are the "data" that are entered into the four basic calculating operations—addition, subtraction, multiplication, and division. Not represented in Figure 8.1, but important for the functional math system, is knowledge of the terms used in math problem solving (e.g., first, more than, and so on) and knowledge of how to use math tools such as graphs and charts with data or calculators.

FIGURE 8.1. Functional math system.

Figure 8.2 graphically portrays the theoretical model of domain-general processes that affect the domain-specific components of the functional math system. These domain-general processes also guided the development of the PAL Test Battery for Math. When skills are first learned or novel problems are encountered, students often approach them strategically. However, with practice and experience, most students automatize certain component math skills, such as retrieval of math facts and application of calculation algorithms, and begin to apply strategies to higher-level tasks in the problem-solving space. Strategies also undergo development, with simple strategies evolving into more-complex ones, and managed by a metacognitive system that controls a variety of executive functions such as self-monitoring and self-regulating. In some cases, students with learning disabilities use strategies effectively to compensate for problems in automaticity, but sometimes they use strategies ineffectively.

FIGURE 8.2. Domain-general system dimensions that interact with the domain-specific functional math system.

Three memory mechanisms play a critical role in math problem solving. Short-term memory encodes orally or visually presented numbers in transient stores. Long-term memory contains math facts, procedures for computational algorithms, and conceptual knowledge in long-term stores. Working memory (e.g., Case, 1985; Swanson, 1996) is a processing space where incoming stimulus information is held and where representations from long-term memory are retrieved and stored until processing is completed — that is, the math problem is solved. Short-term and long-term memory are storage mechanisms that differ in duration of the store, whereas working memory is a mechanism that has both storage and processing components. Yates's (1996) multiple-regression analyses showed that for 60 first and second graders, a short-term auditory memory task and a short-term visual memory task uniquely predicted calculation, whereas measures of working memory and rapid automatic naming of switching stimuli (letters, numbers, and colors) were the best predictors of math problem solving. Thus, short-term and working memory are especially important in learning math. The incoming data and retrieved information from long-term memory is often declarative knowledge (factual knowledge that), whereas the processing that occurs in working memory is often procedural knowledge (knowledge how).

Both temporal processing (e.g., ordering steps of a multistep procedure) and spatial processing (e.g., moving in vertical, horizontal, or diagonal coordinates in space) are needed to carry out multistep calculations. Students' listening or reading skills can affect their ability to represent a problem accurately, while their speaking and/or writing abilities can affect their ability to communicate their approach to solving a problem or to the final solution. Also, finger function skills (i.e., planning sequential fine-motor acts and sensori-symbol integration), can affect paper-and-pencil calculation. Finally, level of cognitive development affects the functional math system. When students are still in the concrete operational stage, manipulatives are useful in representing and solving problems. However, when students have developed formal operational thought, they may solve problems more efficiently by mentally manipulating abstract symbols represented in their minds.

To date, little research has examined how the domain-general processes affect the functioning of the domain-specific math system. The research that does exist has primarily focused on the transition from strategies to automaticity. For example, a major task of beginning mathematics is acquiring basic number facts in the arithmetic module. Initially, children rely primarily on procedural strategies, such as counting in the conceptual knowledge base, to obtain these math facts (Goldman, Pellegrino, & Mertz, 1988; Groen & Parkman, 1972; Nesher, 1986). With practice, children become more accurate and efficient in their strategy use. To illustrate, initially, students often use a "count-all" strategy—to compute 2 + 3, the student counts "1, 2, [pause] 3, 4, 5," using 2 fingers to represent the quantity 2 and 3 fingers to represent the quantity 3—but later progresses to using the "count-on" or "min" strategy that involves starting to count with the larger addend ("3") and only using the smaller addend ("4, 5") to count higher. Eventually, after much successful practice, students make the transition to direct, automatic retrieval of math facts from memory and no longer need to count (Cooney, Swanson, & Ladd, 1988; Siegler, 1988). Automaticity refers to the ability to make the transition from using effortful strategies when performing low-level basic skills to quick, efficient performance. Resnick and Ford (1981) suggested that automaticity, or the ability to get skills on "automatic pilot," may function in the learning of mathematics in a way similar to reading. This transition from procedural strategies to automatic retrieval is conceptually analogous to the transition in reading from phonological decoding to automatic word recognition. In both reading and arithmetic, automatizing a low-level process, such as word recognition, math fact retrieval, or computational algorithms, frees up limited capacity in working memory for the higher-level cognitive processes of reading comprehension (e.g., LaBerge & Samuels, 1974) or math problem solving. Rapid automatic naming of digits (e.g., Wolf, 1986; Wolf, Bally, & Morris, 1986) may index the degree to which access to numerals is automatized. In some cases, faulty strategies may be automatized, rendering it difficult to unlearn them. Thus, automaticity may have advantages as well as disadvantages.

Clearly, the interactions of the domain-specific functional math system and the domain-general systems are complex and require further research. Fortunately, however, standardized tests are now or will soon be available that permit psychologists to assess the level to which each of the components of these systems is developed relative to age or grade peers. Table 8.1 lists test instruments that might be used to assess each of the components in Figures 8.1 and 8.2. These tests are described more fully later in this chapter.

TABLE 8.1. Assessing Domain-Specific and Domain-General Features of the Functional Math System *

Reasoning Ability Test Instrument
Verbal reasoning WISC-III g Verbal Comprehension Factor
SB-IV d Verbal Reasoning
CogAT a Verbal
Quantitative reasoning SB-IV Quantitative Reasoning
CogAT Quantitative
Visual-spatial reasoning WISC-III Performance IQ
SB-IV Visual-Spatial Reasoning
Arithmetic Module
Fact retrieval (+, −, ×, ÷) PAL c Fact Retrieval (accuracy and automaticity for 4 input- output combinations: oral-spoken, oral- written, written-oral, written-written)
Computational algorithms WIAT e and WIAT II f Numerical Operations
WRAT-3 i Arithmetic
WJ-R h Calculation
KM-R b Operations
(separate subtests for written addition, subtraction, multiplication, division, and mental computation)
PAL Written Computations (verbal think aloud—analyze steps in process and not just final product)
Conceptual Knowledge
Counting and number line Selected items on KM-R Numeration
Selected items on WIAT or WIAT II Mathematical
Reasoning
Selected items of WRAT-3 Arithmetic (depending on student's age)
PAL Counting (automaticity)
Place value Selected items on KM-R Numeration
PAL Place Value
Part-whole relationships KM-R Rational Numbers
Selected items on WIAT or WIAT II
Selected items on WJ-R Quantitative Concepts
Time KM-R Time and Money
Selected items on WIAT or WIAT II
Selected Items on WJ-R Quantitative Concepts
Measurement KM-R Measurement
Selected items on WIAT or WIAT II
Selected items on WJ-R Quantitative Concepts
Money KM-R Time and Money
Selected items on WIAT or WIAT II
Selected items on WJ-R Quantitative Concepts
Geometry KM-R Geometry
Selected items on WIAT or WIAT II
Selected items on WJ-R Quantitative Concepts
Problem-solving space KM-R Estimation and Interpreting Data and
Problem Solving
WIAT or WIAT II Math Reasoning
WJ-R Applied Problems
WISC III Arithmetic subtest
SB-IV Number Series and Matrices subtest
PAL Processes Related to Solving Multi-Step
Word Problems
Strategies Think-alouds while student solves any domain-
specific type problem
Student interviews
Automaticity PAL Counting Automaticity
PAL Calculations
PAL RAN for single-digit and double-digit
numbers
Short-Term Memory WISC-III Digit Span (Digits Forward)
Working Memory PAL Quantitative Working Memory
PAL Visual-Spatial Working Memory
PAL Verbal Working Memory
WISC-III Digit Span (Digits Backwards)
Long-term memory PAL Math Facts
Declarative knowledge versus procedural knowledge Verbalization of knowledge versus application of knowledge
Temporal versus visual-spatial processing Observe student performing written calculations to evaluate whether steps of calculation are out of order or errors are made in moving right to left, left to right, up to down, down to up, or diagonally during computational algorithms
Input-output combinations PAL Calculations
Concrete versus formal operations Testing the limits with concrete manipulatives like cuisenaire rods
Piagetian tasks (see Berninger & Yates, 1993)
*
This table is meant to be a suggestive rather than exhaustive listing of tests for assessing the functional math system.
a
CogAt = Cognitive Abilities Test (Thorndike & Hagen, 1993)
b
KM-R = Key Math—Revised (Connolly, 1988)
c
PAL = Process Assessment of the Learner (PAL) Test Battery for Math (Berninger, in press)
d
SB-IV = Stanford Binet Intelligence Scale, 4th Ed. (Thorndike, Hagen, & Sattler, 1986)
e
WIAT = Wechsler Individual Achievement Test (The Psychological Corporation, 1992)
f
WIAT II = Wechsler Individual Achievement Test—Second Edition (The Psychological Corporation, in press)
g
WISC-III = Wechsler Intelligence Scale for Children—Third Edition (The Psychological Corporation, 1991)
h
WJ-R = Woodcock-Johnson Psycho-Educational Battery—Revised (Woodcock & Johnson, 1990)
i
WRAT-3 = Wide Range Achievement Test—Third Edition (Wilkinson, 1993)

The tests in Table 8.1 can be used to screen primary grade children for early intervention for math talent or math disability. Such tests might also be used to monitor progress during early intervention or in response to curriculum modifications. More-comprehensive, in-depth assessment of all components may be warranted at a later time, if the student does not respond to intervention and is suspected to have a math disability.

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The Construct of Mathematical Resilience

Clare Lee , Sue Johnston-Wilder , in Understanding Emotions in Mathematical Thinking and Learning, 2017

Knowing How to Gain Support

Resilient learners of mathematics know that mathematics requires struggle but they also know how to access help and support in that struggle. Some learners may need to be shown successful models of mathematical thinking and reasoning. All need to be listened to because in formulating thinking to communicate ideas to others, that thinking becomes clarified ( Lee, 2006). Furthermore when the "sticking point" becomes obvious, a more capable peer will be able to question the reasoning or offer targeted and appropriate support. Collaborative discourse may enable students to learn to ask themselves the right questions (Lee, 2006).

In order to work collaboratively, learners need to build up sufficient language or vocabulary so that they can use enough of the mathematics register to be able to express and explore mathematical ideas (Lee, 2006). This is not to say that they must use the "right" words but rather that they see a need to express mathematical ideas themselves and thus seek a way to do so effectively. Effective mathematics communicators can explore their own understandings and connections and support others. Mathematical resilience is based within a social constructivist domain (Vygotsky, 1978); expressing mathematical ideas and talking about mathematical learning within a mathematical community are both vital aspects of developing the resilience that allows for learning mathematics.

Digital technologies have a role to play here. Search engines furnish a process or ready-made procedure that may yield solutions to a problem, and software such as Grid Algebra or dynamic geometry programs provide a safe place to express ideas, experiment, think, and learn. The potential to make mistakes and to learn from those mistakes is high when using IT because feedback tends to be immediate and having "another go" is straightforward. The transitory nature of attempts on screen appeals to many people who suffer acute anxiety, while they build their resilience.

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An Overview of the Growth and Trends of Current Research on Emotions and Mathematics

Ulises Xolocotzin Eligio , in Understanding Emotions in Mathematical Thinking and Learning, 2017

First Half of the 20th Century: Early Explorations

Nowadays, researchers study a range of emotional phenomena in relation to mathematics. It is worth recalling that emotions were widely considered worth studying until after the second half of the 20th century. Before that, emotions were largely neglected by the dominant approaches to mathematical thinking and learning.

The views of mathematical activity that dominated the first half of the 20th century did not include the belief that emotions could be related to mathematics. For example, Gillette (1901) argued that the abstract nature of mathematical truths makes them distant from mundane experiences. Therefore, mathematics does not provoke any emotional reactions in the layman. Following this rationale, it is logical to conclude that emotions and mathematics are fundamentally separated dimensions of human experience. Views of this kind, however, were about to change soon.

Polya (1945) proposed the integration of emotions in the analysis of mathematical activity, specifically in relation to problem solving. He argued that "teaching to solve problems is education of the will" (p. 94), suggesting that students need to become familiar with the emotional struggle that is required to find a solution. At about the same time, the study of mathematics anxiety appeared and became the first research line that connected mathematics with emotional phenomena.

The construct Mathematics Anxiety is rooted in accounts of what was described as "mathemaphobia" (Gough, 1954), or "number anxiety" (Dreger & Aiken, 1957). Richardson and Suinn (1972) defined mathematics anxiety as the feelings of tension and anxiety that disrupt the manipulation of numbers and the solving of mathematical problems. A wealth of research has been produced in an attempt to understand the origins of math anxiety and alleviate its consequences.

Thorough reviews by Ashcraft (2002), Hembree (1990), and Ma (1999) outline the conclusions of 30 years of research on mathematics anxiety. First, it is clear that mathematics anxiety dampens mathematical performance. However, there is not enough evidence to conclude that poor performance causes anxiety. The effects of mathematics anxiety seem to be more pronounced in males than in females. The negative relationship between anxiety and performance, however, is stable across grade levels and ethnic groups. Individuals who suffer from mathematics anxiety tend to avoid mathematics and learn less when they are exposed to the topic. Nevertheless, it has been proved that anxiety does not relate to overall intelligence. Research on mathematics anxiety is constantly expanding. Extensive reviews of the state of the art can be consulted in Suárez-Pellicioni, Núñez-Peña, and Colomé (2016) and Chang and Beilock (2016).

This chapter concentrates on literature outside the mathematics anxiety domain. This research line played a crucial role in emotions being recognized as an inherent component of mathematical activity. Moreover, scholars working on the topic produce large amounts of literature—so much that it can be considered to be a standalone research domain that requires a complete volume for its review. However, mathematics anxiety is only one way in which emotions and mathematics can be related.

Anxiety mainly refers to the experience of fear. There are many more emotions that can be experienced in relation to mathematics. Also, performance is the main concern in mathematics anxiety research. Although this is important, emotions are likely to influence a wider range of issues involved in the process and outcome of mathematical activity.

A variety of emotional issues other than mathematics anxiety have attracted the attention of researchers. Below we will see that the amount of literature that stems from these interests is growing steadily. Before we continue, it is worth remembering where the current trends in emotion research originate. The following section describes the historical context in which contemporary research lines emerged.

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The Emotions Experienced While Learning Mathematics at Home

Janet Goodall , ... Rosemary Russell , in Understanding Emotions in Mathematical Thinking and Learning, 2017

Education

Ideally, education is about "leading out" the growth potential in all of us. At all times this process should be guided by salutogenesis, meaning what keeps us well, and developing a personal "sense of coherence" (Antonovsky, 1993) that involves comprehensibility, manageability, and meaningfulness for the learners, rather than focusing on risks which must be avoided. In relation to mathematics per se, learners encounter math when exploring pattern, change, size, quantity, shape, and uncertainty. Mathematical activity can include pattern seeking, experimenting, describing, tinkering, inventing, visualizing, conjecturing, or guessing (Cuoco, Goldenberg, & Mark, 1996). Math education is concerned with increasing awareness of these processes (Gattegno, 1970) and their role in modern life.

Developing mathematical thinking is about developing habits of mind: defining, systematizing, abstracting, making connections, developing new ways to describe situations and make predictions, creating, inventing, conjecturing, and experimenting ( Cuoco et al., 1996). Developing mathematical thinking may also be understood to include imagining and expressing, specializing and generalizing, conjecturing and convincing, focusing and defocusing, sense-making, looking back, getting unstuck (Mason et al., 1982), and articulating (Lee, 2006). This focus on describing mathematical thinking in terms of processes rather than content leads to the concept of mathematical literacy.

"Mathematical literacy is an individual's capacity to identify and understand the role that mathematical thinking plays in the world, to make well-founded judgements and to use and engage with mathematics in ways that meet the needs of that individual's life as a constructive, concerned and reflective citizen" (OECD, 2002). Further, an educated individual is always willing and able to learn more knowledge and skills as needed in life. They are resilient. As learning progresses and the individual becomes increasingly independent, educators may assume more of a coach role: listening, exploring options, supporting goal setting, and reviewing (Egan, 2013).

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Mathematics Teacher Education

K. Krainer , S. Llinares , in International Encyclopedia of Education (Third Edition), 2010

Teachers' Reflections

Partially connected with the social shift mentioned above, we can see a stronger emphasis on the reflection of teachers as a second trend in MTE. Reflection is increasingly considered a key element in the development of processes required for ongoing learning, since it is a means by which teachers continue learning about teaching and about themselves as professionals. Often teachers are involved in teacher-education activities where they reflect on the mathematical thinking processes of students or on the reflections of their own learning (e.g., in contexts where they do mathematical activities themselves or in discussions about their practice and beliefs). Here, it is assumed that the attitudes, skills, and knowledge (student) teachers need are generated when they treat the knowledge and theory produced by others as generative material (points of reference) for interpretation of mathematics teaching, students' mathematical thinking, and the contextual conditions of teaching.

In addition, teachers' reflection can be used as a key element in investigating the growth of the teachers. For example, a focus in this type of research examines what teachers notice when observing their classes, the interpretations they give of events and the changes they propose in their practice. The findings from these studies reveal that differences in the beginning teachers' reflection were linked to changes in practice.

Other studies investigate the changing views of teachers on the students' learning processes and outcomes. For example, a researcher studied teachers' expectations on how secondary students verify conjectures in geometry. The mathematics teachers tended to underestimate students' reasoning on a four-level scale. However, when the same teachers were demonstrated typical examples of the reasoning of students (thus getting a better feeling for students' reasoning levels), their assessment improved. This shows that theory-based models (here for levels of students' thinking) can not only be used as research instruments but also as a means to involve teachers in meaningful reflections and discussions.

In general, research reports show that teachers' learning is not only promoted by meaningful activities, but also by reflections on these activities. In many cases, teachers' reflections play a double role: they aim at increasing the teachers' understanding (of mathematics, of students' mathematical thinking, of institutional constraints on teaching, etc.), and at the same time they are used by researchers as means to describe and interpret teachers' learning.

Since about 10   years, increasingly interactive multimedia learning environments were used to support (student) teachers' learning about teaching mathematics and to study their growth when analyzing teaching events. Research instruments influence practice, and practice itself is the place where new instruments are generated. In particular, supporting teachers to write down experiences in a systematic and self-critical way is an intervention toward an investigative attitude which is the entrance card to participation in a research community and culture. Nowadays, the new forms of discourse generated by the currently available communication tools such as video paper, blogs, bulletin boards, and so on, use writing as an instrument for collaborative reflection and as a tool for inquiry. The new type of text generated by these communication tools can be used as an improvable object that favors the generation of a progressive discourse and as one that acts as the focus of collaborative knowledge building. For example, the process of creating a video paper – as a multimedia document that integrates and synchronizes different forms of representation, including text, video, and images, in one single nonlinear cohesive document – supports teacher reflection. The same is true for writing in order to participate in online discussions as a means to notice mathematics teaching. These processes of writing are qualitatively different from simply watching a video from a whole lesson to reflect on one's practice: when mathematics (student) teachers' have to select relevant clips or particular events from a videotaped lesson to discuss, their reflection is fostered by initiating a more analytical process that might not necessarily happen if they had just watched a whole lesson without needing to produce a written text. In this sense, for example, the multimodal character of video papers (combining video and text) enriches reflection and thus also the process of meaning-making and knowledge construction.

In combination with the assumption that teachers are key persons of educational change, it makes sense to regard them as reflective practitioners, researchers, and experts. Recently papers on teacher educators' growth have been written indicating that mathematics teacher educators' learning and their self-reflection is an important issue as well.

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Evaluation of Mathematics Education Programs

J. Cai , in International Encyclopedia of Education (Third Edition), 2010

Assessment tasks

Even though various methods can be used to measure students' learning, the heart of measuring mathematical performance is the set of tasks on which students' learning is to be evaluated (NRC, 2001 ). It is desirable to use various types of assessment tasks, thereby measuring different facets of mathematical thinking. For example, different formats of assessment tasks (such as multiple-choice and open-ended tasks) may be used to measure students' learning. Multiple-choice tasks have many advantages. For example, more items can be administered within a given time period, and scoring responses can be done quickly and reliably. However, it is difficult to infer students' cognitive processes from their responses to multiple-choice items. Thus, in addition to multiple-choice tasks, open-ended tasks may be used. For open-ended tasks, students are asked to produce answers as well as show their solution processes and provide justifications for their answers. In this way, the open-ended tasks provide a better window into the thinking and reasoning processes involved in their   learning of mathematics. Of course, a disadvantage of open-ended tasks is that only a small number of these tasks can be administered within a given period of time. Also, grading students' responses is labor-intensive. To help overcome the disadvantages of using open-ended tasks, a matrix sampling design of administering open-ended tasks to students is recommended. In this way, we can reduce both testing time and grading time but still obtain a good overall estimate of students' learning of mathematics.

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Digging Beneath Dual Systems Theory and the Bicameral Brain

John Mason , Martina Metz , in Understanding Emotions in Mathematical Thinking and Learning, 2017

Introduction

In this chapter, we use a phenomenological approach to explore the behavior of students exploring mathematics. At the core of our approach is an ancient psychological description of the human psyche, consisting of enaction, affect, cognition, attention, will, and witness. The notion of attention as being multiply structured (Mason, 1998; van Hiele, 1986) is part of the proposed elaboration. The ideas draw upon and develop the notion of frames of mind (Hudson, 1968; Minsky, 1986), which align with the notion of polyphrenia, otherwise known as multiple selves (Bennett, 1964), and microidentities (Varela, 1999). The stance taken develops the notion of adherences and coordinations among enaction, affect, cognition, attention, and will, observed by the witness. This, it is proposed, can be used to make useful and informative sense of experience of mathematical thinking and of working with others to help them to develop their mathematical thinking (see also Mason, 1998; Metz & Simmt, 2015).

The milieu within which mathematical thinking is initiated, sustained, and promoted is of course highly significant, and much discussed currently in a vast literature that promotes discussion. Often, however, this is done without attending to the practices and ethos that make that social immersion effective. For example, on the one hand there is the notion of apprenticeship, illustrated by Lave (1988) observing Liberian tailors; and Seeley-Brown, Collins, and Duguid (1989) promoting a modern day version of apprenticeship. On the other hand there is the notion of structured group work as promoted by Boaler (2010) among others. We suggest that the practices identified in these various approaches work most effectively when they take into account the full human psyche as well as drawing upon didactic specifics of learning and doing mathematics (Mason, 2002a).

Our whole approach is informed by the use of the discipline of noticing (Mason, 2002a), which provides philosophically well-founded methods for enriching the noticing of opportunities to act freshly in the moment, or in other words, to learn from experience.

The chapter will show how automaticities (habitual actions that require no explicit in-the-moment conscious cognitive input) often have affective as well as enactive and cognitive components, how they are structured by the structure of attention, how they enable or are enabled by will, and how this is all informed by inner witnessing. It will suggest ways in which emotional reactions can be worked at and around, through the ways in which students begin to trust both their own mathematical powers and the teacher. Habits of mind, emotion, action, and attention will be contrasted with intuition and with experiencing the ineffable in mathematics, including how the ineffable can be brought to expression through entry into the implicit (Gendlin, 1978, 2009).

Trying to appreciate students' attempts at learning how to solve mathematical problems, by which we mean sophisticated exploration rather than routine exercises, we return to the theme of McLeod and Adams (1989), who in their ground-breaking book drew attention to the central role of emotions during mathematical problem solving. Our purpose is not simply to re-emphasize the important role that emotions play in mathematical thinking, but to integrate this into a more sophisticated view of the human psyche.

The thrust of our study concerns the combining of dual process theory as adumbrated by Kahneman (2012) and Kahneman and Frederick (2002) with a more comprehensive view of the human psyche involving six interacting components: enaction, affect, and cognition, together with attention, will, and witness. We include along the way the bicamerality of the human brain (Jaynes, 1976; McGilchrist, 2009) and the notion of micro-identities (Varela, 1999) or coordinated adherences, which some describe using the term multiple selves (Bennett, 1964; Minsky, 1986). We consider it vital not to try to isolate emotions or affect, but rather to integrate them into a complex comprehensive view of the psyche. As with the Sufi story of the blind men and the elephant (Shah, 1970; see also Wikipedia 1, n.d.), dual process theory, emotions, and cognition are but components of a complex phenomenon known as human beings.

We are interested in lived experience, and so our approach to research is to begin phenomenologically. Because we wish to act consistently with our principles, our approach to reporting research is again phenomenological. The data we offer is what comes to you as you read brief-but-vivid descriptions of incidents and accounts-of phenomena, rather than transcripts of incidents observed by researchers accounting for what is observed. This is part of the discipline of noticing (Mason, 2002a, 2002b). Thus the chapter begins with descriptions of some phenomena that we hope readers will recognize. The intention is that that experience goes beyond mere recognition by summoning up images, triggering recollection of past experience, and perhaps even then resonating with, or triggering access to, more recent experiences with similar qualities. In our accounts we use the personal pronoun as the events recounted are personal.

We then elaborate on the theories that we are combining. This provides a complex framework for considering in detail people's response to mathematical tasks in order to highlight but not overstress the emotional component of experiences of thinking mathematically. The complexity of the human psyche is not clarified or appreciated by isolating different components and trying to study them in isolation. Our claim is that by sensitizing oneself to the complexity of the human psyche, and by striving to notice various components, it is possible to be more in tune with and more aware of the lived experience, and this in turn enriches the ability to notice what may be happening for students, both in terms of appreciating potential stumbling blocks they may experience in a particular mathematical context and in terms of noticing (and helping them to notice) the way their attention is structured. In this way, we may better choose effective pedagogic actions rather than reacting in habitual ways to students who need help.

For example, when a student gives an incorrect answer to a question, describing the student as "unable" to answer the question correctly is a habituated reaction that ignores or overlooks the many reasons why a student might not think to say what the teacher wants to hear; or if a student gives a muddled or inarticulate response to a question that is not what the teacher had in mind, thinking that the student does not understand and turning to another student in search of a clearer response can become a habituated action, rather than, for example, treating the response as an attempt to articulate an almost inchoate sense and working with that student in order to clarify and streamline their narrative.

The chapter concludes with an extended description of work on a particular mathematical problem concerning goldfish, making use of our elaborated theories to make sense of and to draw lessons from that experience.

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The Divined Proportion

Albert van der Schoot , in Mathematics and the Divine, 2005

3. Potentia mirabilis

Can we find anything golden in Cusanus' writings? We can, but not the golden section. In 1459, Cusanus wrote down a theorem which he referred to as the aurea propositio in mathematicis. The theorem claims that three straight lines, originating in the same point and forming two equal angles of 45° or less, are always in the same ratio to the boundary line, whether this boundary line is a chord or an arc. In Fig. 3: ab + ah + ac is to bhc as ab + ad + ac is to the arc bdc, etc.

Cusanus reflects on this theorem not only as a mathematician, but also as a theologian. Mathematical contemplation opens the gate to loftier speculations beyond the sphere of empirical perception. In his reflections on the squaring of the circle (as well as in other texts, such as the Idiota de mente ), Cusanus explains his belief in the importance of mathematics for theology: mathematical thinking resides in the highest regions of the mind, as it regards the figures in their true Forms, that is, not contaminated by the tricks of ever changeable matter. Mathematics, therefore, assists the theologian in coming closer to the understanding of the Form of Forms, the primary Form in which all the other forms coincide.

Fig. 3. Cusanus' aurea propositio.

The (in)commensurability of the linear and the curved is a recurring theme in Cusanus' mathematical writings, by which he illustrates the incomparability between the objects of the sublunary world and the austere power which has neither beginning nor end. This infinite power of eternity is as incommensurable to any finite power as the surface of the circle is to any other surface. 13 Yet, the golden proposition leads him to think of an infinitely large circle, with the implication that the difference between chord and arc would then be dissolved. Referring to the common origin of the three lines, he ends his explanation of the golden proposition by the statement that the highest speculation of the wise man will be directed to the Trinitarian origin from which all things emanate.

Although it is clear that this golden proposition bears no relation to the golden proportion, Cusanus' comment is still helpful in understanding the vein of the exalted exclamation which we find added to one of the theorems concerning the divine proportion in another manuscript: the thirteenth century translation of Euclid's Elements by Johannes Campanus, another mathematically gifted clergyman. This translation was later, in 1482, to become the first Euclid in print.

Admirable therefore is the power of a line divided according to the ratio with a mean and two extremes; since very many things worthy of the admiration of philosophers are in harmony with it, this principle or maxim proceeds from the invariable nature of superior principles, so that it can rationally unite solids that are so diverse, first in magnitude, then in the number of bases, then too in shape, in a certain irrational symphony. 14

This is the comment as we find it added to book XIV, prop. 10—a theorem concerning the proportional relationship between the volumes and the sides of a dodecahedron and an icosahedron inscribed in the same sphere. As previously mentioned, the XIVth book is not originally Euclidean, but it was considered as such during Campanus' days (and long after). And Campanus might just as well have written these words as a comment on the authentic book XIII, with its many propositions concerning the properties of the division in extreme and mean ratio.

One may consider Campanus' statement a sign of aesthetic admiration for the golden section but it certainly has nothing to do with art or with human creativity. It is aesthetic in the sense that Campanus is impressed by the formal properties or, if you wish, the beauty of this proportion, in much the same way as one may be aesthetically impressed by the Pythagorean proposition or by one of its many proofs. Yet, it is a metaphysical rather than an aesthetic statement, in the same vein as Cusanus' reflections on the aurea propositio.

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