Embedded teaching procedures have become a recommended practice and a commonly implemented teaching strategy in early childhood classrooms. Embedded teaching is characterized by instructions and feedback regarding target skills being delivered within child-initiated activities during varied and typical routines. Although embedded teaching strategies have been widely adopted, little research has been conducted on the critical elements of embedded teaching. The purpose of this project is to determine the relative efficacy of and child preference for different embedded teaching conditions that vary in the rate of embedded teacher prompts. Three preschool aged children experienced three teaching conditions 1) Dense Condition (four embedded prompts per min were programmed), 2) Lean Condition (one embedded prompt per min was programmed) and 3) Control Condition (no embedded prompts). The relative efficacy of the teaching conditions were evaluated within a multielement design while a concurrent chains procedure was implemented to directly assess each child's preference for the teaching conditions. The dense and lean embedded teaching strategies were equally efficacious for teaching preschool children food name relations and the participant's preference assessment results were variable across the three participants. One participant showed a preference for the control condition, and the other two participant's preference was variable between the three conditions. The importance of choice making and using evidence based teaching strategies in preschool environments is discussed.

In school algebra, using different methods including factorization to solve quadratic equations is one common teaching and learning topic at upper secondary school level. This study is about analyzing the algebra content related to solving quadratic equations and the method of factorization as presented in Swedish mathematics textbooks with subject matter content knowledge (CK) and pedagogical content knowledge (PCK) as analytical tools. Mathematics textbooks as educational resources and artefacts are widely used in classroom teaching and learning. What is presented in a textbook is often taught by teachers in the classroom. Similarly, what is missing from the textbook may not be presented by the teacher. The study is based on an assumption that pedagogical content knowledge is embedded in the subject content presented in textbooks. Textbooks contain both subject content knowledge and pedagogical content knowledge. The primary aim of the study is to explore what pedagogical content knowledge regarding solving quadratic equations that is embedded in mathematics textbooks. The secondary aim is to analyze the algebra content related to solving quadratic equations from the perspective of mathematics as a discipline in relation to algebra history. It is about what one can find in the textbook rather than how the textbook is used in the classroom. The study concerns a teaching perspective and is intended to contribute to the understanding of the conditions of teaching solving quadratic equations. The theoretical framework is based on Shulman’s concept pedagogical content knowledge and Mishra and Koehler’s concept content knowledge. The general theoretical perspective is based on Wartofsky’s artifact theory. The empirical material used in this study includes twelve mathematics textbooks in the mathematics B course at Swedish upper secondary schools. The study contains four rounds of analyses. The results of the first three rounds have set up a basis for a deep analysis of one selected textbook. The results show that the analyzed Swedish mathematics textbooks reflect the Swedish mathematics syllabus of algebra. It is found that the algebra content related to solving quadratic equations is similar in every investigated textbook. There is an accumulative relationship among all the algebra content with a final goal of presenting how to solve quadratic equations by quadratic formula, which implies that classroom teaching may focus on quadratic formula. Factorization method is presented for solving simple quadratic equations but not the general-formed quadratic equations. The study finds that the presentation of the algebra content related to quadratic equations in the selected textbook is organized by four geometrical models that can be traced back to the history of algebra. These four geometrical models are applied for illustrating algebra rules and construct an overall embedded teaching trajectory with five sub-trajectories. The historically related pedagogy and application of mathematics in both real world and pure mathematics contexts are the pedagogical content knowledge related to quadratic equations.