The researcher conducted a six-month ethnographic study as an apprentice carpenter in Cape Town, South Africa, to document the valid mathematical ideas that are embedded in the everyday woodworking activities of a group of carpenters. A secondary objective was to examine and to give a firsthand account of the teaching and learning of mathematical ideas in the context of the researcher's apprenticeship. Finally, the study offers methodological techniques for identifying mathematics in thought and action and for differentiating the mathematics from routine applications of procedures. The results showed that many conventional mathematical concepts are embedded in the practices of the carpenters. They made extensive use of such concepts as congruence, symmetry, proportion, and straight and parallel lines in their everyday work. Furthermore, the carpenters' problem solving was enhanced by their strength in spatial visualization. Their explanations, in the form of convincing arguments, showed the sequential, logical reasoning that is related to the need in mathematics for proof and substantiation. The results also showed that the carpenters' mathematics has several unique characteristics: there was tacit mathematical knowledge in their actions, and reflection on actions led them to articulate their tacit knowledge; decontextualized questions posed were revised into concrete, contextualized problematics; and their ideas were framed by the context of the workshop and carpentry tools. Comparison, using the senses of touch and sight, was preferred to measuring and usually resulted in optimal solutions. To solve a problem such as "How many table legs can be cut from this plank?" spatial visualization practices were used to construct functional units, producing an optimal result that could not be obtained with formal procedures. The results are presented as a series of 20 narrative episodes, followed by an analysis. The epistemological, educational, and methodological implications of these results are discussed. (Contains over 100 references.) (Author)