Mathematical Knowledge for Teaching and the Mathematical Quality of Instruction: An Exploratory Study

This study illuminates claims that teachers' mathematical knowledge plays an important role in their teaching of this subject matter. In particular, we focus on teachers' mathematical knowledge for teaching (MKT), which includes both the mathematical knowledge that is common to individuals working in diverse professions and the mathematical knowledge that is specialized to teaching. We use a series of five case studies and associated quantitative data to detail how MKT is associated with the mathematical quality of instruction. Although there is a significant, strong, and positive association between levels of MKT and the mathematical quality of instruction, we also find that there are a number of important factors that mediate this relationship, either supporting or hindering teachers' use of knowledge in practice.     

An Investigation of the Relationship between Age,Achievement, and Creativity in Mathematics

In this exploratory study, I investigate the relationship between age, knowledge, and creativity in mathematics, by looking at to what extent does grade level, controlled for mathematical achievement, influence mathematical creativity and what characterizes the relationship between grade level, mathematical achievement and mathematical creativity. This was accomplished in two steps. In the first part, 301 students, 184 grade eight students and 117 grade eleven students, were given a creative mathematics test. A 3 × 2 ANOVA indicates that the older students were more creative; however, there was a significant interaction effect between grade level and achievement in mathematics on mathematical creativity. In the second part, an inductive content analysis was performed on the solutions of high achievers in grade eleven and grade eight. The results indicate that high achievers in grade eight are more creative than high achievers in grade eleven, but the nature of the task mediates the relationship between creativity and knowledge.     

Logic of imagination. Echoes of Cartesian epistemology in contemporary philosophy of mathematics and beyond

Descartes’ Rules for the direction of the mind presents us with a theory of knowledge in which imagination, considered as an “aid” for the intellect, plays a key role. This function of schematization, which strongly resembles key features of Proclus’ philosophy of mathematics, is in full accordance with Descartes’ mathematical practice in later works such as La Géométrie from 1637. Although due to its reliance on a form of geometric intuition, it may sound obsolete, I would like to show that this has strong echoes in contemporary philosophy of mathematics, in particular in the trend of the so called “philosophy of mathematical practice”. Indeed Ken Manders’ study on the Euclidean practice, along with Reviel Netz’s historical studies on ancient Greek Geometry, indicate that mathematical imagination can play a central role in mathematical knowledge as bearing specific forms of inference. Moreover, this role can be formalized into sound logical systems. One question of general epistemology is thus to understand this mysterious role of the imagination in reasoning and to assess its relevance for other mathematical practices. Drawing from Edwin Hutchins’ study of “material anchors” in human reasoning, I would like to show that Descartes’ epistemology of mathematics may prove to be a helpful resource in the analysis of mathematical knowledge.     

Platonism and metaphor in the texts of mathematics: Gödel and Frege on mathematical knowledge

In this paper, I challenge those interpretations of Frege that reinforce the view that his talk of grasping thoughts about abstract objects is consistent with Russell's notion of acquaintance with universals and with Gödel's contention that we possess a faculty of mathematical perception capable of perceiving the objects of set theory. Here I argue the case that Frege is not an epistemological Platonist in the sense in which Gödel is one. The contention advanced is that Gödel bases his Platonism on a literal comparison between mathematical intuition and physical perception. He concludes that since we accept sense perception as a source of empirical knowledge, then we similarly should posit a faculty of mathematical intuition to serve as the source of mathematical knowledge. Unlike Gödel, Frege does not posit a faculty of mathematical intuition. Frege talks instead about grasping thoughts about abstract objects. However, despite his hostility to metaphor, he uses the notion of ‘grasping’ as a strategic metaphor to model his notion of thinking, i.e., to underscore that it is only by logically manipulating the cognitive content of mathematical propositions that we can obtain mathematical knowledge. Thus, he construes ‘grasping’ more as theoretical activity than as a kind of inner mental ‘seeing’.

     

Logical structuralism and Benacerraf’s problem

There are two general questions which many views in the philosophy of mathematics can be seen as addressing: what are mathematical objects, and how do we have knowledge of them? Naturally, the answers given to these questions are linked, since whatever account we give of how we have knowledge of mathematical objects surely has to take into account what sorts of things we claim they are; conversely, whatever account we give of the nature of mathematical objects must be accompanied by a corresponding account of how it is that we acquire knowledge of those objects. The connection between these problems results in what is often called “Benacerraf’s Problem”, which is a dilemma that many philosophical views about mathematical objects face. It will be my goal here to present a view, attributed to Richard Dedekind, which approaches the initial questions in a different way than many other philosophical views do, and in doing so, avoids the dilemma given by Benacerraf’s problem.     

数学真理的问题

在当前数学实践中,数学知识（如果有这样的知识的话）是通过在定义和公理的基础上证明定理来获得的。问题在于该怎样理解证明中所得到的东西是如何构成知识的,具体而言,即是要给出一个关于数学真理和数学知识的统一的解释,该解释能够揭示两者的内在联系。此处的困难是,根据贝纳塞拉夫的为人熟知的论证,由于塔斯基语义学认为真与对象的联系（通过单称词项或通过量词）是不可消去的,因此在数学中无法将塔斯基语义学与完整的认识论相结合：数学知识要么是通过证明得到的,这种情况下数学知识与数学对象是无关的,因此我们就无法解释数学真理;要么数学对象是数学真理的构件,从而数学知识不是通过证明得到的,这种情况下我们就无从理解数学知识。接着,本文通过一系列阶段,将这些困难一直追溯到最基本的逻辑观念,即将之看作形式的和纯粹解释性的：如果数学是从概念出发仅仅使用逻辑的推理实践,依照康德,那么数学应该是分析的,也即,仅仅是解释性的,根本就不是通常意义上的知识。我认为,这对数学真理是真正困难的问题。本文概括了四种回应,其中仅有一个有希望解决我们的困难,也即皮尔斯和弗雷格的回应。根据他们的方案,逻辑是科学,因此是实验性的和可错的;符号语言是有内容的,尽管并不涉及与任何对象的关联;证明是构成性的,因此是富于产出的过程。通过充分发展这些观点,我们将有可能最终解决数学真理的问题。     

Defining and measuring conceptual knowledge in mathematics

A long tradition of research on mathematical thinking has focused on procedural knowledge, or knowledge of how to solve problems and enact procedures. In recent years, however, there has been a shift toward focusing, not only on solving problems, but also on conceptual knowledge. In the current work, we reviewed (1) how conceptual knowledge is defined in the mathematical thinking literature, and (2) how conceptual knowledge is defined, operationalized, and measured in three mathematical domains: equivalence, cardinality, and inversion. We uncovered three general issues. First, few investigators provide explicit definitions of conceptual knowledge. Second, the definitions that are provided are often vague or poorly operationalized. Finally, the tasks used to measure conceptual knowledge do not always align with theoretical claims about mathematical understanding. Together, these three issues make it challenging to understand the development of conceptual knowledge, its relationship to procedural knowledge, and how it can best be taught to students. In light of these issues, we propose a general framework that divides conceptual knowledge into two facets: knowledge of general principles and knowledge of the principles underlying procedures.     

The Innateness Hypothesis and Mathematical Concepts

In historical claims for nativism, mathematics is a paradigmatic example of innate knowledge. Claims by contemporary developmental psychologists of elementary mathematical skills in human infants are a legacy of this. However, the connection between these skills and more formal mathematical concepts and methods remains unclear. This paper assesses the current debates surrounding nativism and mathematical knowledge by teasing them apart into two distinct claims. First, in what way does the experimental evidence from infants, nonhuman animals and neuropsychology support the nativist hypothesis? Second, granting that infants have some elementary mathematical skills, does this mean that such skills play an important role in the development of mathematical knowledge?     

Preschool children's mathematical knowledge: The effect of teacher "math talk."

This study examined the relation between the amount of mathematical input in the speech of preschool or day-care teachers and the growth of children's conventional mathematical knowledge over the school year. Three main findings emerged. First, there were marked individual differences in children's conventional mathematical knowledge by 4 years of age that were associated with socioeconomic status. Second, there were dramatic differences in the amount of math-related talk teachers provided. Third, and most important, the amount of teachers' math-related talk was significantly related to the growth of preschoolers' conventional mathematical knowledge over the school year but was unrelated to their math knowledge at the start of the school year.     

Metacognition,Executive Functions and Aging: The Effect of Training in the Use of Metacognitive Skills to Solve Mathematical Word Problems

The aim of our research was to study the effects of metacognition training on the performance of older adults solving mathematical word problems. A further aim was to study the links between metacognition and executive function. Thirty-two subjects aged over 60, divided into an experimental and a control group, took part in this study which involved five training sessions. Results show that metacognition training enhanced the two metacognition components (knowledge and skills) and the mathematical problem-solving capacities of the participants. They also suggest that the use of metacognition by older people to solve mathematical word problems is supported by executive functions (updating, shifting) and particularly by processing speed.     

The Microevolution of Mathematical Representations in Children's Activity

In this article, I discuss children's design of mathematical representations on paper, asking how material displays are constructed and transformed in activity. I show that (a) the design of displays during problem solving shapes one's mathematical activity and sense making in crucial ways, and (b) knowledge of mathematical representations is not simply recalled and applied to problem solving but also emerges (whether constructed anew or not) out of one's interactions with the social and material settings of activity. A detailed characterization of student-designed tables of values to solve problems about linear functions is also presented.     

Some types of parent number talk count more than others: relations between parents' input and children's cardinal-number knowledge

Before they enter preschool, children vary greatly in their numerical and mathematical knowledge, and this knowledge predicts their achievement throughout elementary school (e.g. Duncan et al., 2007; Ginsburg & Russell, 1981). Therefore, it is critical that we look to the home environment for parental inputs that may lead to these early variations. Recent work has shown that the amount of number talk that parents engage in with their children is robustly related to a critical aspect of mathematical development - cardinal-number knowledge (e.g. knowing that the word 'three' refers to sets of three entities; Levine, Suriyakham, Rowe, Huttenlocher & Gunderson, 2010). The present study characterizes the different types of number talk that parents produce and investigates which types are most predictive of children's later cardinal-number knowledge. We find that parents' number talk involving counting or labeling sets of present, visible objects is related to children's later cardinal-number knowledge, whereas other types of parent number talk are not. In addition, number talk that refers to large sets of present objects (i.e. sets of size 4 to 10 that fall outside children's ability to track individual objects) is more robustly predictive of children's later cardinal-number knowledge than talk about smaller sets. The relation between parents' number talk about large sets of present objects and children's cardinal-number knowledge remains significant even when controlling for factors such as parents' socioeconomic status and other measures of parents' number and non-number talk.     

The contribution of logical reasoning to the learning of mathematics in primary school

It has often been claimed that children's mathematical understanding is based on their ability to reason logically, but there is no good evidence for this causal link. We tested the causal hypothesis about logic and mathematical development in two related studies. In a longitudinal study, we showed that (a) 6‐year‐old children's logical abilities and their working memory predict mathematical achievement 16 months later; and (b) logical scores continued to predict mathematical levels after controls for working memory, whereas working memory scores failed to predict the same measure after controls for differences in logical ability. In our second study, we trained a group of children in logical reasoning and found that they made more progress in mathematics than a control group who were not given this training. These studies establish a causal link between logical reasoning and mathematical learning. Much of children's mathematical knowledge is based on their understanding of its underlying logic.     

Cognitive skills in mathematical problem solving in Grade 3

Background. Research on the relationship between cognitive skills and mathematical problem solving is usually conducted on adults or on participants with acquired deficits associated with brain injury (e.g. Cipolotti, 1995 ; Cohen, Dehaene, & Verstichel, 1994 ; McCloskey, 1992 ). Aims. In these studies we wanted to make a contribution to the field of children's mathematical problem solving. The first aim of this study was to investigate whether mathematical problem solving in children is merely determined by semantic elaboration, as hypothesized in some of the models of adult processing (semantic hypothesis). In addition, we aimed to investigate whether there is a continuum from very good to very poor mathematical problem solving among children with mathematical learning disabilities showing immature cognitive skills (maturational lag hypothesis). Sample. The participants were 376 third graders and 107 second graders. Method. The internal structure of the data was analysed with a principal components analysis. In addition, two MANOVA were conducted to compare children with learning disabilities or problems with age‐matched and performance‐matched subjects. Results. Two components, a semantic and a non‐semantic one, were needed to account for an adequate fit of the dataset. In addition, children with mathematical learning disabilities had less‐developed cognitive skills compared with peers without learning disabilities, but they did not differ from younger children on seven of the nine cognitive skills. Conclusions. This study highlighted that children's mathematical problem solving is not determined by one general component. The picture is more complex, since two mathematics components were found. In addition, although our findings point in the direction of the maturational lag hypothesis it may be important to assess the different cognitive skills and especially assess the number system knowledge, since it seems below average in children with mathematical learning disabilities, compared with the knowledge of younger children with comparable skills in mathematics.     

Investigating Links from Teacher Knowledge,to Classroom Practice,to Student Learning in the Instructional System of the Middle-School Mathematics Classroom

Using data collected in 125 seventh-grade and 56 eighth-grade Texas classrooms in the context of the “Scaling Up SimCalc” research project in 2005–07, we examined relationships between teachers’ mathematics knowledge, teachers’ classroom decision making, and student achievement outcomes on topics of rate, proportionality, and linear function—three important and cognitively demanding prealgebra topics. We found that teachers’ mathematical knowledge was correlated with student achievement in only one study out of three. We also found a lack of correlations between teachers’ mathematical knowledge and critical aspects of instructional decision making. Curriculum and other learning resources (e.g., technology, student–student interactions) are clearly important factors for student learning in addition to, and in interaction with, teachers’ mathematical knowledge. Our results suggest that mathematics knowledge for teaching may have a nonlinear relationship with student learning, that those effects may be heavily mediated by other instructional factors, and that short-term content knowledge gains in teacher workshops may not persist in classroom instruction. We discuss a need in the field for richer models of how “mathematical knowledge for teaching” works in the context of complete instructional systems.     

Hume on the social construction of mathematical knowledge

Mathematics for Hume is the exemplary field of demonstrative knowledge. Ideally, this knowledge is a priori as it arises only from the comparison of ideas without any further empirical input; it is certain because demonstration consist of steps that are intuitively evident and infallible; and it is also necessary because the possibility of its falsity is inconceivable as it would imply a contradiction. But this is only the ideal, because demonstrative sciences are human enterprises and as such they are just as fallible as their human practitioners. According to the reading suggested here, Hume develops a radical sceptical challenge for mathematics, and thereby he undermines the knowledge claims associated with demonstrative reasoning. But Hume does not stop there: he also offers resources for a sceptical solution to this challenge, one that appeals crucially to social practices, and sketches the social genealogy of a community-wide mathematical certainty. While explaining this process, he relies on the conceptual resources of his faculty psychology that helps him to distinguish between the metaphysics and practices of mathematical knowledge. His account explains why we have reasons to be dubious about our reasoning capacities, and also how human nature and sociability offers some remedy from these epistemic adversities.

     

Platonism and metaphor in the texts of mathematics: G?del and Frege on mathematical knowledge

In this paper, I challenge those interpretations of Frege that reinforce the view that his talk of grasping thoughts about abstract objects is consistent with Russell's notion of acquaintance with universals and with Gödel's contention that we possess a faculty of mathematical perception capable of perceiving the objects of set theory. Here I argue the case that Frege is not an epistemological Platonist in the sense in which Gödel is one. The contention advanced is that Gödel bases his Platonism on a literal comparison between mathematical intuition and physical perception. He concludes that since we accept sense perception as a source of empirical knowledge, then we similarly should posit a faculty of mathematical intuition to serve as the source of mathematical knowledge. Unlike Gödel, Frege does not posit a faculty of mathematical intuition. Frege talks instead about grasping thoughts about abstract objects. However, despite his hostility to metaphor, he uses the notion of grasping as a strategic metaphor to model his notion of thinking, i.e., to underscore that it is only by logically manipulating the cognitive content of mathematical propositions that we can obtain mathematical knowledge. Thus, he construes grasping more as theoretical activity than as a kind of inner mental seeing.     

The developmental change of young pupils’ metacognitive ability in mathematics in relation to their cognitive abilities

Metacognition is a multidimensional construct with two main dimensions: knowledge about cognition and regulation of cognition. The present study aimed to model the development of young pupils’ metacognitive abilities in mathematics in relation to processing efficiency, working memory and mathematical performance. We developed instruments measuring pupils’ metacognitive ability, mathematical ability, working memory capacity, and processing efficiency, and administered them to 126 pupils (8–11 years old) three times, with breaks of 3–4 months between them. Dynamic modeling indicated that growth in each of the abilities is affected by the state of the others, especially the state of processing efficiency. Growth modeling was used to specify the nature of change in the main aspects of mind and the possible interrelations in the patterns of change in these aspects. The initial mathematical self-image was found to depend on the corresponding processing efficiency and its advancement to rely on the development of mathematical performance and the previous working memory ability.     

Interpretation of Scientific or Mathematical Concepts: Cognitive Issues and Instructional Implications

Scientific and mathematical concepts are significantly different from everyday concepts and are notoriously difficult to learn. It is shown that particular instances of such concepts can be identified or generated by different possible modes of concept interpretation. Some of these modes use formally explicit knowledge and thought processes; others rely on less formal case-based knowledge and more automatic recognition processes. The various modes differ in attainable precision, likely errors, and ease of use. A combination of such modes can be used to formulate an “ideal” model for interpreting scientific concepts both reliably and efficiently. Comparisons are made with the actual concept interpretations of expert scientists and novice students. The discussion elucidates some cognitive and metacognitive reasons why the learning of scientific or mathematical concepts is difficult. It also suggests instructional guidelines for teaching such concepts more effectively.     

Conclusive reasons that we perceive sets

Penelope Maddy has defended a modified version of mathematical platonism that involves the perception of some sets. Frederick Suppe has developed a conclusive reasons account of empirical knowledge that, when applied to the sets of interest to Maddy, yields that we have knowledge of these sets. Thus, Benacerraf's challenge to the platonist to account for mathematical knowledge has been met, at least in part. Moreover, it is argued that the modalities involved in Suppe's conclusive reasons account of knowledge can be handled without recourse to either laws of nature or possible worlds, and that this approach is preferable.