Beyond mere knowledge of mathematics: The importance of knowing-to act in the moment

Knowing-to is active knowledge which is present in the moment when it is required. To try to produce knowing-to, formal education focuses on forms of knowing which are easier to teach and to test: knowing-that (factual), knowing-how (technique and skills), and knowing-why (having a story in order to structure actions and from which to reconstruct actions). Together these constitute knowing-about the subject. Expertise is demonstrated by being able to respond to assessments: to write essays and to solve routine problems. The central problem of education is that knowing-about does not in itself guarantee knowing-to, as teachers have attested throughout the ages. For example, Edward Fitzgerald (Harrison, 1937) captures it beautifully in one stanza of his purported translation of the Rubaiyat of Omar Khayyam:Myself when young did eagerly frequent,Doctor and Saint and heard great argument,About it and about: but ever more Came out by the same door as in I went (p. 341).Instead of trying to reach definitions, we illustrate distinctions amongst kinds of knowing as used by various authors in the past. Then we turn to our own experience, for it is in one's own experience that one can locate and enliven sources of metaphoric resonances and metonymic triggers which constitute understanding. Drawing on our experience we distinguish knowing-to from other forms of knowing, and explore implications of that distinction for teaching and learning mathematics. We propose that knowing-to act in the moment depends on the structure of attention in the moment, depends on what one is aware of. Educating this awareness is most effectively done by labelling experiences in which powers have been exhibited, and developing a rich network of connections and triggers so that actions come to mind. No-one can act if they are unaware of a possibility to act; no-one can act unless they have an act to perform.This revised version was published online in September 2005 with corrections to the Cover Date.     

Double-Loop Diversity: Applying Adult Learning Theory to the Cultivation of Diverse Educational Climates in Higher Education

Most U.S. higher education institutions have implemented initiatives to increase campus diversity. The intended outcome is to reduce educational inequities among underrepresented populations. Such initiatives have come under fire, however, for serving as a superficial solution to educational inequities. Here I propose the application of an action-knowledge conceptual framework to explore why money and policy change are not enough to create campus environments where multi-culturalism enhances student learning. Argyris' (1985) notion of double-loop learning recognizes the necessity for actionable knowledge to be instilled in practitioners in order for diversity initiatives to truly achieve desired outcomes and have lasting effects.     

Phonological awareness,verbal working memory and the acquisition of literacy

This paper describes a 2-year longitudinal study of 76 initially prereading children. The study examined the relationships between phonological awareness (measured by tests of onset and rime, phonemic segmentation and phoneme deletion), verbal working memory and the development of reading and spelling. Factor analyses showed that the verbal working memory tests which were administered loaded on two distinct but highly related factors, the first of which,simple repetition, involved the repetition of verbal items exactly as spoken by the experimenter, whereas the second,backwards repetition, involved repetition of items in reverse order. Factor analyses also showed that, whist the phonological awareness variables consistently loaded on the backwards repetition factor at the beginning and end of Grade 1, by Grade 2 the phonological awareness variables loaded on a separate factor which also included sentence repetition. Results of multiple regression analyses, with reading and spelling as a compound criterion variable, indicated that phonological awareness consistently predicted later reading and spelling even when both simple and backwards repetition were controlled. In contrast, verbal working memory did not consistently predict reading and spelling across testing times. Whilst there was some indication that verbal working memory, especially backwards repetition, measured during Grade 1 did predict reading and spelling in Grade 2, these effects were no longer evident when all three phonological variables were controlled. Nevertheless, with 4 individual reading and 2 individual spelling measures as the criterion variables, it was shown that phonological awareness was not quite such a consistent predictor of reading and spelling: it was most highly related to reading pseudowords and spelling real words; but it was not so highly related to spelling pseudowords, apparently because the processing demands of the task for the young children in the study were extremely high. Given the importance of verbal working memory for the completion of phonological awareness, reading and spelling tasks, in particular for spelling pseudowords, the findings are interpreted as providing some support for a theoretical position which posits that both phonological awareness and verbal working memory contribute to the early stages of literacy acquisition. Whilst the findings suggest some support for a general underlying phonological ability, there is also evidence that, as children learn to read and write, verbal working memory and phonological awareness become more differentiated.     

Electronic Roads in Historical Documents: A Student Oriented Approach

The new educational approaches, as far as teaching of history in secondary education is concerned, are characterized by a shift away from sterile memorization and towards a critical approach of historical facts and phenomena; the aim is to contribute to both the development of students' historical concept and conscience and the promotion of critical thought. These teaching goals are pursued by promoting initiative of the students through self study assignments in the form of projects; students can be greatly supported in such tasks by computer-based applications, which can offer access to vast amounts of historical texts and data to be used next to the main scholar textbook and be analyzed by students. Still, existing applications seem to be quite inadequate for this purpose, as they require that the student be already informed on a matter, before the initiation of a quest for data. In this paper, we describe an intelligent information system that is designed to facilitate browsing of educational material and historical sources, thus allowing students to efficiently retrieve information on topics that are not yet known to them and expand in this way their historical knowledge. This can help in fulfilling the aforementioned teaching goals. Our system relies on the notion of the Electronic Road.     

Impact of a Mentoring Experience on College Students’ Beliefs about Early Childhood Development

With the increasing demands placed on working families and the push downward of academics from the K-12 system, there is a critical need to provide high-quality early childhood programming for our nation’s children, particularly those considered at risk for academic failure. This study attempts to (a) understand the beliefs of college students about early childhood development and developmentally appropriate practices, and (b) determine if these beliefs change after their yearlong involvement in Jumpstart, a national intensive mentoring program for academically at-risk preschoolers. Surveys from this national program investigated college students’ beliefs about early childhood development and developmentally appropriate practices. Results indicate that a yearlong mentoring program positively impacts college students’ beliefs. These beliefs become less skills-based more and child-centered and constructivist.     

The effects of using graphing calculators to enhance college students' performance in precalculus

This study extended through three academic semesters and involved 710 subjects. It compared the performance of college students taught precalculus using a graphing calculator and a textbook written to be used with a graphing utility, to the performance of students using the traditional approach, a regular textbook, and a scientific calculator. On a comprehensive common final exam, students who were taught precalculus using the graphing calculator had significantly higher scores than those taught by traditional methods.PosthumousPreliminary partial results of this experiment were presented at the Fourth Annual International Conference on Technology in Collegiate Mathematics, Portland, Oregon, November 15–17,1991.     

Illuminating growth and struggles using mixed methods: Practice-based professional development and coaching for differentiating SRSD instruction in writing

In this mixed methods study, qualitative, quantitative, and single-case methods were combined to provide a comprehensive investigation of teacher and student outcomes following practice-based professional development (PBPD) for self-regulated strategy development (SRSD) in writing. Qualitative observations were used to determine outcomes among the three-fourth grade teachers involved, a nested quantitative model was used to analyze classwide student writing outcomes across 53 students in the three classes, and single case design was used to determine differential outcomes among randomly selected struggling and average writers in each class. PBPD was followed by coaching for differentiation based on student performance and fidelity as teachers taught their students strategies for writing timed imaginary stories (with self as main character), as required by the state writing test. Qualitative results indicated two teachers did not differentiate writing instruction without coaching; one teacher was unresponsive to coaching and did not differentiate instruction. Classwide analysis demonstrated significant growth in writing at the class level for students from baseline to posttesting. Single case design results indicated mixed outcomes among struggling and average writers and instances where instruction was not effective. Teachers indicated high social validity for PBPD and for SRSD; students indicated high social validity for SRSD. Results across the multiple methods illuminate growth and struggles often disguised in group data and are addressed to aid in understanding and enhancing PBPD as well as instruction in SRSD and other evidence-based practices. Limitations and future research directions are discussed.