Well-prepared beginning teachers of mathematics recognize the difference between access to and advancement in mathematics learning and work to provide access and advancement for every student. |

Well-prepared beginners know the meanings of *access* and *advancement,* understanding that denial of access or advancement leads to inequities. Access to mathematics is essential for equitable mathematics education. Access includes ensuring that students have opportunities to learn important mathematics taught by qualified teachers. Well-prepared beginners recognize that access involves structures in schools and in classrooms and recognize classroom practices that threaten access. Access becomes particularly important in the placement of students into higher level courses, in which the focus is on doing mathematics rather than practicing procedures. But, *access* also refers to opportunities within a classroom. Well-prepared beginners realize that access is increased when students can approach a problem from multiple routes (e.g., using a method that is familiar to solve the problem) or when they use curriculum materials that include high-quality, meaningful tasks that go beyond the basic skills often tied to standardized testing. Access is threatened when false hurdles are inherent in the system (e.g., denying access to calculators until students master particular skills such as basic facts).

*Advancement *is the opportunity to go beyond grade-level expectations to learn additional content. Advancement includes participating in an advanced group in elementary schools, taking algebra prior to ninth grade, taking advanced or college-level courses in high school, and pursuing additional courses or taking honors sections of courses. Well-prepared beginners are prepared to advocate for equitable practices for identifying students for advanced study, recognizing the inadequacy of defining *success* solely by the teacher or standardized tests but also the goals that students hold for themselves, especially students who are Black, Latinx, American Indian, emergent multilinguals, or students of any ethnicity living in poverty (Gutiérrez, Bay-Williams, & Kanold, 2008).