In this chapter we identify issues pertinent to ‘out of field’ teachers of mathematics. Strategies are offered to assist ‘out of field’ teachers of mathematics while considering the invaluable knowledge a primary-trained teacher can bring to the secondary mathematics classroom. Examples are presented of teachers’ critical pedagogical content knowledge that has a high impact on students’ growth and development across the main strands of the curriculum.

General capabilities and cross-curriculum priorities, literacy, critical and creative thinking, digital literacy, ethical understanding, intercultural understanding, personal and social capability, Aboriginal and Torres Strait Islander histories and cultures, Asia and Australia’s engagement with Asia, Sustainability

A comparison of disciplines is helpful for teaching creativity to identify similarities and differences in the creative process. A challenge for all disciplines is to create a balance between teaching higher-level abilities, such as creativity, and the lower-level technical skills required by the discipline. But there are also differences among disciplines. Scientific training emphasizes avoiding mistakes so it is more risk-aversive than training in the arts in which taking risks is often encouraged. Research on science and mathematics learning includes evaluating the effects of exposing preservice elementary teachers to multiple representations, measuring scientific creativity in elementary school students, identifying competencies for scientific reasoning in junior high school, and designing instruction on complex systems at all levels in the curriculum. TRIZ, an acronym for the Russian phrase ‘theory of inventive problem solving’, has influenced the design and evaluation of curricula for engineering students.

An influential measure of innovation developed by the Drucker Institute provides an annual ranking of companies based on the principle that innovation creates resources for building new and greater wealth. Predictors that underlie the creation of wealth include research and development spending, hiring in cutting-edge fields such as robotics and artificial intelligence, the number and value of patents, customer satisfaction, and employee engagement. Innovation requires overcoming various constraints such as insufficient funding and a lack of time to complete goals. Leaders can become amplifiers by engaging followers, possessing the kind of knowledge that fits corporate needs, and avoiding cognitive entrenchment – knowledge stability that causes experts to be inflexible in their thinking. The dynamic problem-solving model synthesizes a variety of theoretical constructs such as the types of constraints on problem solving, the distinction between a problem-first and an idea-first approach to invention, and the zigzag pathway to creativity.

Chapter 1 begins with the distinction between reasoning from associations and reasoning from rules – a distinction that will resurface in subsequent chapters on creativity and innovation. The associative system is reproductive, automatic, and emphasizes similarity. The rule-based system is productive, deliberative, and emphasizes verification. Daniel Kahneman’s (2011) best-selling book Thinking Fast and Slow introduced readers to how associative and rule-based reasoning influence the speed of responses. The third section on biases in reasoning describes Kahneman’s classic research with Amos Tversky on how the use of heuristics such as availability and representativeness influence frequency estimates. The final section discusses monitoring reasoning in which people use knowledge to improve their thinking skills. Monitoring reasoning is a metacognitive skill that controls the selection, evaluation, revision, and abandonment of cognitive tasks, goals, and strategies.

Successful design depends on overcoming cognitive obstacles such as failures of attention, confirmation bias, fixation, and sunk-cost traps. A countermeasure is to keep multiple ideas alive by developing several prototypes rather than focusing on a single idea. A Design Heuristics tool consisting of cards helps by describing a strategy on one side of a card and an example of applying the strategy on the other side. Sketching offers opportunities for new interpretations and unexpected discoveries as illustrated by successive designs of the Sydney Opera House by Jorn Utzon. The Industrial Design Engineering program at Delft University of Technology in the Netherlands, the Harvard T. H. Chan School of Public Health, and the New School’s Parson School of Design in New York City offer various approaches to design education. Although traditional design education should not be discarded, it requires a broader approach that includes material on cognitive science, anthropology and culture, political science, business, and ethics.

Following the description of cultural-historical activity theory in Chapter 6, this chapter provides a narrative case example of a formative intervention in early childhood education leadership practice using the principles and practices of Change Laboratory. Research data from five time points during the Change Laboratory intervention illustrate the work of two teachers responsible for leading the development of curriculum and pedagogy in the center.

We rely on other people’s ideas because they often know more than we do about many aspects of the world. A negative consequence of shared beliefs occurs when people focus too much on information that originates from people who hold the same opinions. A group is particularly vulnerable to groupthink when its members have similar backgrounds, the group is insulated from outside opinions, and there are no clear rules for decision making. Shared beliefs can nonetheless contribute to group cohesion, coordination of ideas, and shared mental models. The flipside of shared beliefs are unshared beliefs that can cause conflicts. Advice for resolving conflicts includes engaging in persuasive listening, acknowledging common ground, and discussing reasons for a lack of progress. Considering alternative perspectives also broadens views. High levels of task discourse enable team members to resolve ambiguities, refine their ideas, and discuss the potential innovation of those ideas. Training should therefore emphasize a diversity of perspectives, the open exchanges of ideas, the reduction of biases, and an increased motivation for accuracy.