Mathematics teachers need to teach students more than just a laundry list of mathematical skills. Many of us also want to train our students to use mathematical habits of mind. However, traditional grading schemes usually do not allow us to reward intellectual courage and modesty. In this paper, I report on my implementation of specifications grading in a mathematics content course for preservice elementary teachers. This scheme supports my students in revising their mathematical thinking, taking mathematical risks, and effectively communicating their ideas through writing.

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Research in mathematics education show that students learn more mathematics when it is connected to what they hear, see, and do outside of school; students' cultural knowledge and experiences can serve as familiar, relatable contexts for learning mathematics. When teachers adopt a culturally responsive stance to teaching mathematics, they integrate the knowledge and experiences of students as situated in their cultures, homes, and communities. The following paper presents Talavera tiles as one example of a culturally responsive context for teaching geometry. The authors first briefly present the experiences of seven elementary teachers who participated in a professional development workshop, Te'ALaMo (Teachers, Art from Latin@ cultures, and Mathematical modelling) where they used Talavera tiles to explore aspects of symmetry. In the second half of the paper, the authors extend and unpack the mathematics of Talavera tiles in the context of teaching geometry for children and adolescents.

In this chapter, the authors consider the purposeful design of two mathematics content courses (Content 1 and Content II) and one methods course (Methods) as a means of helping teacher candidates (TCs) learn about divergent formative assessment (DFA), which seeks to explore what students understand rather than only if they understand a concept or skill. The authors leverage the research of groupworthy tasks and the Rights of the Learner to describe three tasks they use to help TCs learn mathematics through problem-solving and to learn to teach through problem-solving. The chapter outlines three commonalities across the courses: 1) Shifting from implicit to explicit and informal to formal practices of DFA that reflects teaching through problem-solving; 2) Using DFAs to transition TCs' identities from learners to teacher-learners; and 3) Supporting TCs' self-assessment through DFAs in multiple ways.

Dependency models are the basis of several important products for testability analysis, diagnosability analysis, and generation of optimal fault trees, including generation of dynamic test strategies based on current parameters and available resources. The AI-ESTATE Committee of the IEEE SCC20 is working on a standard representation of dependency models (IEEE 1232.1) and once that is formalized, the resulting increased portability and re-usability of models will make dependency models even more important.The problem with dependency models is that they are difficult to generate. Dependency models generated from netlist information alone are close to useless. Dependency models can also be generated via a circuit simulator, but these run very slowly and for complex circuits, this is not feasible.Intelligent Automation, Inc. (IAI), together with the US. Army Communications and Electronics Command, is developing a new approach to automatic generation of dependency models. The approach is based on use of "intelligent agents." Intelligent agents may also be useful for other aspects of the problem of testing and diagnosing complex electronic circuits, and our success in the automatic generation of dependency models has led us to begin studying other applications of agents in this domain.There are two key benefits of implementing software using the paradigm of autonomous agents. First, agents are independent and asynchronous, which means they can run on any number of computers which might be available as long as the computers are networked together.The second benefit is ease of implementation. The work reported here was done as an experiment in the use of autonomous agents and not with the goal of implementing a usable system. Many problems still need to be addressed, some of which are discussed in the paper and other of which are beyond the scope of this paper.