Curriculum focal points are important mathematical topics for each grade level, pre-K–8. These areas of instructional emphasis can serve as organizing structures for curriculum design and instruction at and across grade levels. The topics are central to mathematics: they convey knowledge and skills that are essential to educated citizens, and they provide the foundations for further mathematical learning. Because the focal points are core structures that lay a conceptual foundation, they can serve to organize content, connecting and bringing coherence to multiple concepts and processes taught at and across grade levels. They are indispensable elements in developing problem solving, reasoning, and critical thinking skills, which are important to all mathematics learning.
When instruction focuses on a small number of key areas of emphasis, students gain extended experience with core concepts and skills. Such experience can facilitate deep understanding, mathematical fluency, and an ability to generalize. The decision to organize instruction around focal points assumes that the learning of mathematics is cumulative, with work in the later grades building on and deepening what students have learned in the earlier grades, without repetitious and inefficient reteaching. A curriculum built on focal points also has the potential to offer opportunities for the diagnosis of difficulties and immediate intervention, thus helping students who are struggling with important mathematics content.
What characteristics qualify a concept or topic to be a curriculum focal point? For inclusion in Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics, a focal point had to pass three rigorous tests:
- Is it mathematically important, both for further study in mathematics and for use in applications in and outside of school?
- Does it “fit” with what is known about learning mathematics?
- Does it connect logically with the mathematics in earlier and later grade levels?
A curriculum focal point may draw on several connected mathematical content topics described in Principles and Standards for School Mathematics (NCTM 2000). It should be addressed by students in the context of the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation. Without facility with these critical processes, a student’s mathematical knowledge is likely to be fragile and limited in its usefulness.
A complete set of curriculum focal points, situated within the processes of mathematics, can provide an outline of an integrated mathematics curriculum that is different from the outline created by a set of grade-level mastery objectives or a list of separated content and process targets. In contrast with grade-level mastery objectives, which can be interpreted as endpoints for learning, curriculum focal points are clearly areas of emphasis, calling for instruction that will help students learn content that gives them a foundation for increasing their understanding as they encounter richer and more challenging mathematics.
Instruction based on focal points would devote the vast majority of attention to the content identified for special emphasis in a grade. A curriculum for pre-K–8 based on a connected set of such focal points could provide a solid mathematical foundation for high school mathematics.