The term “real” is extremely ambiguous. A trip to an art museum brings out one interpretation. For example, the French Impressionist Edgar Degas (1834–1917) painted ballet dancers practicing in mirrored rooms. Some dancers in Degas’ paintings show dancers in front of, and reflected in a mirror; some show mirror images of dancers off to the side. Which, if any, is a reflection of a real person? Many other artists’ paintings challenge viewers to ponder reality. Sometimes the work shows reflections in windows, outside open windows, or even reproductions within a painting inside the painting.

Our research provides a rather simple way to determine whether an artist assumed that the reflections or reproductions in their paintings were “real” or not. This is to ask observers to count the number of separate objects in a particular painting. If a mirror captures the reflection of an object in front of it, it cannot be counted, but an object shown outside an open window, can. Why? Our implicit counting knowledge prohibits the double counting of an item. So, if you do not count the reflection in the mirror, you do not think it is a real item. To explain, I need to cover a bit about your counting knowledge.

When we count, we must honor the one–one principle and, therefore, uniquely tag every object once and only once. If we double tag an item, we will use too many tags and get the wrong answer to a “how many?” question. All normal adults and many young children are good counters. Even four-year-olds will correct a puppet that counts an item twice. If one counts an exact mirror copy of a foregrounded item, one commits a principled violation. In our experiments with adults we took advantage of this fact.

Dana Chesney and I showed college students line drawings of frames that we dubbed a mirror, picture frame, or an open window. Within each there was a reproduction of the objects drawn in the foreground (e.g., two chairs). So, two chairs were shown in the mirror, two were outside the open window, and two smaller ones were in a painting on the wall. People simply were asked how many were present in each kind.

Introduction

Our efforts to study the acquisition of new knowledge, especially in the areas of mathematics and science, take advantage of a combination of methods. These include experiments, controlled training studies, and qualitative analyses of videotaped classroom learning. In addition, we have been developing ways to treat the classroom as an in situ learning laboratory. Here we focus on a particular example, a science-into-ESL (English as a Second Language) classroom that functioned as a concept and language learning laboratory.

A key research goal was to gather “on-line” data on whether students moved on to and along relevant learning paths during the course of a semester. Given the classroom setting, we wanted these to be inconspicuous “probes” rather than more traditional learning test trials. Our solution shares important aspects of the microgenetic method, the topic of this book. We requested that students keep notebooks and make regular entries that could function as learning experiences for them and probes as to whether learning was happening over time. Notebook entries are a common feature of how scientists work, especially when it comes to trying to keep track of experiments and evolving ideas. They also have served historians of science as documents for analyzing the conditions of conceptual change. As we shall see, notebooks have the potential to serve researchers' efforts to obtain on-line information about students' progress in a non-intrusive manner.

There is a growing body of evidence that infants attend selectively to some fundamental aspects of number and music. Such findings suggest that attention to and learning about number and music are perhaps due to the presence of innate, skeletal principles in each domain. In our chapter, we develop this position while showing how it is consistent with the different ways that cultures support learning and development in specific knowledge areas. Pairing considerations of number and music enables us to show that domain specificity and cultural variation need not be treated as antithetical.

Whereas it is still common for scholars in some fields to assume that “primitive” peoples lack a concept of number (see the following discussion), there is wide acceptance of the idea that all peoples develop musical competence, mainly because the latter can happen without benefit of formal instruction, use of special symbol systems, or the need to represent abstract, relevant dimensions like pitch, key, harmony, rhythm, and so forth. Indeed, accounts of the evolutionary function of the human music capacity often include the idea that, in preliterate societies, music serves to efficiently organize information that cannot be written down. For example, Gardner (1983) describes a possible role for music in organizing religious rites and work groups in the Stone Age, and Sloboda (1985) hypothesizes that music provides a mnemonic framework within which the structure of cultural knowledge and societal relations is stored and communicated. Whatever the role of music in preliterate groups, Donald (1991) points out that these societies always have complex rituals based on some form of music.