During a recent visit to a sixthgrade classroom, students were observed estimating the sums of columns of multidigit numbers by first painstakingly computing the exact sum of each column of numbers on paper, then rounding the answers to obtain estimates. Apparently the students were not bothered by the fact that it didn't make much sense to round the numbers after already finding an exact sum. Furthermore, they seemed unaware that the power of estimming lies in the speed with which a person can obtain an amount that is close enough for the purpose at hand. Instead. estimation had been turned into a cumbersome process intended, at most, to satisfy the direction on a worksheet.

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## Making Connections with Estimation

### Joanne E. Lobato

## Cross-Cutting Themes From International Research on Early Algebra

### Joanne Lobato and Jaime Marie Diamond

A review of Early Algebraization: A Global Dialogue From Multiple Perspectives, edited by Jinfa Cai and Eric Knuth.

## Making Sense of Algebraic Expressions in Context

### Isabel White, Michael Foster, and Joanne Lobato

Explore three challenges that students faced and how they made progress.

## Students' Mathematical Noticing

### Joanne Lobato, Charles Hohensee, and Bohdan Rhodehamel

Even in simple mathematical situations, there is an array of different mathematical features that students can attend to or notice. What students notice mathematically has consequences for their subsequent reasoning. By adapting work from both cognitive science and applied linguistics anthropology, we present a focusing framework, which treats noticing as a complex phenomenon that is distributed across individual cognition, social interactions, material resources, and normed practices. Specifically, this research demonstrates that different centers of focus emerged in two middle grades mathematics classes addressing the same content goals, which, in turn, were related conceptually to differences in student reasoning on subsequent interview tasks. Furthermore, differences in the discourse practices, features of the mathematical tasks, and the nature of the mathematical activity in the two classrooms were related to the different mathematical features that students appeared to notice.

## Initiating and Eliciting in Teaching: A Reformulation of Telling

### Joanne Lobato, David Clarke, and Amy Burns Ellis

We address the telling/not-telling dilemma in mathematics education. Telling is instructionally important, but has been downplayed because of (a) perceived inconsistencies between telling and constructivism, (b) increased awareness of the negative consequences of relying too heavily on telling, and (c) a focus on “non-telling” actions as pedagogical implications of constructivism. In response, we advance a theoretical reformulation of telling as the set of teaching actions that serve the function of stimulating students' mathematical thoughts via the introduction of new ideas into a classroom conversation. We reformulate telling in three ways: (a) in terms of the *function* (which involves attention to the teacher's *intention*, the nature of the teaching *action*, and the students' *interpretations* of the action) rather than the *form* of teachers' communicative acts; (b) in terms of the conceptual rather than procedural content of the new information; and (c) in terms of its relationship to other actions rather than as an isolated action. This reformulation resolves some of the concerns with teaching as telling and helps establish the legitimacy of providing new information within a constructivist perspective on learning.