## Abstract

Decades ago Hans Freudenthal referred to the school mathematics experienced by most students as the “fossilized remains” of reasoning processes. Indeed, the facts and procedures of school mathematics may seem as frightening to some as the fossilized remains of a tyrannosaurus rex, although they are empty; like dinosaur skeletons, they bear only partial resemblance to the real thing. The challenge is to see the substance behind the structure and to understand how the mathematics fits together. That is a matter of mathematical thinking, reasoning, and problem-solving – the *how* and the *why* beneath the fossilized surface. Opportunities for such understandings are accessible through mathematical sense making, but they are rare in schools. This chapter indicates that there is more to learning and understanding mathematical content and practices than it would appear. Moreover, understanding mathematics is only one component of effective or “ambitious” teaching – better framed as the creation of mathematically rich and equitable learning environments. The challenge is to create robust learning environments that support every student in developing not only the knowledge and practices that underlie effective mathematical thinking, but that help them develop the sense of agency to engage in sense making. This implicates issues of race and equity, which are a challenge not only in classrooms but in society at large; structural and social inequities permeate the schools. Major obstacles to addressing the challenges of powerful mathematics within schools include a general absence of curricular support for rich and meaningful mathematics, instructional practices that do not invite students into mathematics, assessments that fail to focus on thinking, professional development that focuses on what the teacher does rather than the students’ learning opportunities and experiences, and a vastly inequitable cultural context both outside and inside schools. This chapter points to existence proofs that at least some these challenges can be addressed, while documenting the substantial challenges to making progress at scale.

### Keywords

- Ambitious instruction
- Bias
- Formative assessment
- Problem-solving
- Summative assessment
- Teaching for robust understanding
- Sense making
- Systemic inequities
- Thinking mathematically

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## Acknowledgments

This chapter was produced with support from the US National Science Foundation Grant 1503454, “TRUmath and Lesson Study: Supporting Fundamental and Sustainable Improvement in High School Mathematics Teaching,” a partnership between the Oakland Unified School District, Mills College, the SERP Institute, and the University of California at Berkeley. It has profited immensely from comments by Abraham Arcavi, Hugh Burkhardt, Diana Casanova, Gabriel Davis, Heather Fink, Vicki Hand, Nicole Louie, Dragana Martinovic, Sandra Zuñiga Ruiz, Alyssa Sayavedra, Xinyu Wei, and Anna Weltman.

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Schoenfeld, A.H. (2022). Why Are Learning and Teaching Mathematics So Difficult?. In: Danesi, M. (eds) Handbook of Cognitive Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-44982-7_10-1

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