2014 CRUME Plenary Speakers

Prof. Andrea diSessa , University of California, Berkeley

Title
Knowledge in Pieces: How to Analyze the Process of Learning at High Resolution

Abstract
I aim to give an overview of how the epistemological perspective of “Knowledge in Pieces” (KiP) has allowed the creation of high-resolution analyses of learning in process. “High resolution” entails analysis of real-time data, so that one can actually see learning steps at the grain-size at which learners experience it, and at which teachers and curriculum developers try to manage it. As such, this very rare kind of analysis might be extraordinarily helpful in designing instruction and learning materials.
I will first try to characterize the overall KiP program of studies and contrast it with other programs of studying learning. Then, I will use data from two recent studies to illustrate the principles in action. (1) I will show a case of a small class of students developing, on their own, some normative physics (Newton’s law of thermal equilibration: Temperature difference drives rate of change of temperature). Here, we can see, element by element, what incoming knowledge was invoked, and how it changed and combined to result in the normative idea. (2) The other study involves micro-analysis of student learning from a well-studied instructional sequence (Brown and Clement’s “bridging analogies”). In this, we track differences in incoming student knowledge well enough to see why some students succeeded and others failed to achieve the instructional goal.

 

Prof. Anna Sfard , University of Haifa

Title
Mathematics learning: Does language make a difference?

Abstract
Mathematics and its learning are generally believed to be relatively independent of the language in which they are practiced. This assumption tacitly underlies the nowadays popular idea of international comparisons such as TIMSS or PISA, in which young people from all over the world are being tested with the help of a single mathematical questionnaire. The fact that the questions appear in different languages does not diminish examiners’ conviction that, wherever they go, they are testing “the same mathematics,” thus assessing fully comparable types of learning.
And yet, in the view of recent theoretical developments and some new empirical findings, the assumption about the language-proof nature of mathematics and its learning may be questioned. This issue is of particular importance to those who teach mathematics in schools and universities. Indeed, if it turns out that the way people learn is shaped by their main language, there may be significant differences in the needs of learners gathered in the same multilingual classroom.
The question of the impact of language on mathematics learning is the focus of this talk. I will begin with a brief historical survey of research guided by the famous Sapir-Whorf Hypothesis, according to which all human thinking is shaped by language. I will follow with a theoretical reflection on the relation between thinking and communication, undertaken in an attempt to reconceptualize the topic. I will then use the resulting conceptual apparatus while summarizing and interpreting results of two studies, one on learning limits and infinity and the other on learning fractions and probability, both of them launched in the quest for dissimilarities in mathematical discourses of learners coming from different linguistic backgrounds.

 

Prof. Ron Tzur , University of Colorado

Title
Promoting Teachers’ and Students’ Learning to Reason Multiplicatively: A Units-and-Operations Developmental Approach

Abstract
This paper presents an approach to the learning and teaching of multiplicative reasoning that focuses on units and operations students may construct, and use, when solving and posing mathematical problems.  To explain learning, this approach includes a 6-scheme developmental framework rooted in studies on children’s construction of multiplicative and divisional schemes.  This framework (a) distinguishes between two types of units—singletons (1s) and composite—each possibly comprised of concrete, figural, or abstract items, and (b) articulates advances in students ways of coordinating operations on either or both unit types. To promote students’ learning and teacher development, this approach foregrounds a student adaptive mathematical pedagogy (STAMP; shorthand – adaptive teaching). Adaptive teaching stresses the need to tailor goals for student learning (“what should we teach next?”) and activities for accomplishing these goals (“how should we teach this?”) to students’ available ways of operating on various units.  Specifically, teachers learn to design and implement tasks for reactivating, and transforming, schemes that are both available to the students and instigate conceptual pathways to the intended mathematics.  Data collected and analyzed in studies that employed this approach will be presented to elucidate and substantiate how it can contribute to teacher change and to student learning and outcomes (both those with learning disabilities and their normal-achieving peers). Implications of this approach to teaching undergraduate students entering university mathematics courses as well as prospective elementary teachers will be discussed.