- What are "curriculum focal points"?
Answer: “Curriculum focal points” are the most important mathematical topics for each grade level. They comprise related ideas, concepts, skills, and procedures that form the foundation for understanding and lasting learning. They are the topics that should be considered as the basis for decisions about curriculum development.
- What exactly is a "focal point"?
Answer: A focal point is a cohesive cluster of related knowledge, skills, and concepts. It is more than a single objective, standard, expectation, or indicator. It is not a discrete topic for teachers to present to students and check off as having been mastered. Rather, a focal point specifies the mathematical content that a student needs to understand deeply and thoroughly for future mathematics learning.
- Why has NCTM developed a set of curriculum focal points?
Answer: With current mathematics curricula, students are expected to become acquainted with a wide range of topics in a short period of time, keeping them from developing deep mathematical understanding and connections. These curricula typically include long lists of concepts and skills at each grade level but never answer the question, “What are the key mathematical ideas or topics on which the others build?” The focal points are an example of how to answer this question. NCTM is offering a focused framework to guide states and school districts as they design and organize the next revisions of their expectations, standards, curriculum, and assessment programs.
- What is the purpose of NCTM's Curriculum Focal Points?
- NCTM’s Curriculum Focal Points present a way to bring focus to the teaching, learning, and assessing of mathematics. They provide a framework for designing and organizing curricular expectations and assessments. Collectively, they describe an approach that can be used in developing a mathematics curriculum for prekindergarten through grade 8.
- Organizing a curriculum around a set of focal points, with a clear emphasis on the processes of mathematics, as outlined in NCTM’s Principles and Standards for Schools Mathematics, can provide students with a connected, coherent, ever expanding body of mathematical knowledge and ways of thinking.
- The Curriculum Focal Points are intended to address curriculum, or what is taught, rather than instruction, or how it is taught. Although the initial impact of the focal points will be on curriculum, ultimately the focal points will affect teaching, learning, and assessment, as well.
- Who should use the Curriculum Focal Points?
Answer: The Curriculum Focal Points are intended for use by a variety of constituencies committed to improving the learning and teaching of mathematics:
- State and district leaders as they take the next steps in the development of future curriculum, instruction, and assessment materials
- Publishers as they reconsider the scope, sequence, and organization of textbooks and assessment tools
- Teacher educators as they organize content and methods courses and help teachers and prospective teachers develop their knowledge of critical mathematics understandings across the grades
- Classroom teachers and instructional leaders as they prioritize existing lists of curriculum objectives, expectations, and textbook materials
- How does NCTM's publication, Curriculum Focal Points, relate to the publication Principles and Standards for School Mathematics?
Answer: Curriculum Focal Points is an extension of Principles and Standards for School Mathematics. Principles and Standards describes the development of mathematics content across the grades; Curriculum Focal Points identifies areas of emphasis within the mathematics curriculum at each grade level, prekindergarten through grade 8. Together, these two publications provide valuable resources to guide mathematics curriculum, instruction, and assessment.
- Will this study affect math curriculum recommendations by NCTM for high school?
Answer: The Council has formed a task force to consider the need for and advisability of developing something similar to focal points for grades 9–12.
- Does the Standard Algorithm have to be mastered by all students?
Subtraction (grade 2) says "including the standard algorithms" which I assume means borrowing or something like the left to right subtraction with "trouble’s coming" approach. The multiplication (grade 4) and division (grade 5) focal points say "including the standard algorithm" as if students must learn carrying with multiplication and the old long division.
Our goal should be that students can use a method to subtract that allows them to get the right answer, to do so reasonably fast, and to be able to do this repeatedly.
Answer: At grade 2, quick recall of addition and subtraction facts and fluency with supporting algorithms is a focus. Note that the focal point suggests efficient procedures, including the standard algorithm—meaning including, not exclusively, access to the standard algorithm. Students need to understand how and why algorithms work. The standard algorithms are historically validated mathematically efficient procedures. But we want children to understand how these procedures work, which means, in this case, having classroom experience with place value, and the important properties (e.g. commutative and associative) which influence how we add and subtract. As you state, “Our goal should be that students can use a method to subtract that allows them to get the right answer, to do so reasonably fast, and to be able to do this repeatedly.”
Similarly, for grade 4 the quick recall of multiplication facts and fluency with efficient procedures, including the standard algorithm, is a focus. Most important is that fluency emerges through understanding of the multiplication process—how multiplication is represented, how properties are used when multiplying, etc. Students become fluent through their understanding of efficient procedures that include the standard algorithm.
Also note that a focal point is an area of emphasis in a given year; earlier work with multiplication would occur in grade three and extend in grade five.
Curriculum focal points are designed to promote discussion and dialogue about important mathematics. It is hoped that the focal points impact curriculum revision at the state and local level. However, that discussion and dialogue certainly may push focal point topics or elements of the focal points to a lower or higher grade level. Additionally, the extent any of the mathematics suggested is actually implemented is determined by the user. Our intent is that access to the standard algorithm come through understanding, but we recognize that use of the standard algorithms may be an issue with some. The key here is the understanding of the algorithm, any algorithm, however it is defined.
The question asks: “After students have explored the meaning of an operation, such as multiplication, and constructed meaningful ways to do this, they should be encouraged to adopt ever more efficient ways to do this process.” This is our intent.
- Does the NCTM consider Curriculum Focal Points in any sense to be a "reversal" of its position on students learning computational skill, as has been reported in the press?
Answer: Much of the news media got it wrong and letters were written to newspapers. The Curriculum Focal Points are in no way a reversal of the Council's long-standing position on teaching students to learn critical foundational topics (e.g. multiplication) with conceptual understanding, and they are in no way a retreat from Principles and Standards for School Mathematics. The Curriculum Focal Points are the next step in the implementation of the Standards. The focal points fully support the Council's Principles and Standards for School Mathematics. In fact, the appendix in Curriculum Focal Points directly links the focal points to virtually all the expectations in Principles and Standards.
Question: Do the Curriculum Focal Points mean we should rewrite our current standards and replace the curriculum and textbook we're now using?
Answer: Making use of Curriculum Focal Points is not about replacing one set of Standards or one set of curriculum materials with another; it is about beginning a serious discussion of what we want students to learn in grades pre-K through 8 and how that mathematics content can best be organized to support learning and teaching. This effort will require work on several fronts, including review of state and district standards, assessments, and curriculum materials. A coordinated and concerted effort is needed to make these important curricular content decisions.
- Question: How can I determine whether the curriculum I’m using is aligned with the new Curriculum Focal Points?
Answer: No current curriculum materials or textbooks are likely to be “correlated” or “aligned” with the Curriculum Focal Points; the Curriculum Focal Points have just been released, and states, school systems, and curriculum developers will need time to study, consider, refine, and enact curriculum to reflect the focus suggested by the Curriculum Focal Points.
- Question: I don’t see specific mention about when to teach time and money concepts in the focal points, especially for the early grades. Can you clarify this for me?
Answer: Time and money are wonderful and important components of any primary grade mathematics experience; however they are not points of focus in the sense of the Curriculum Focal Points concept. In fact, one could say that time and money could (maybe should) provide the context for number and algebra focal topics at these levels. As for when these are taught, certainly they are introduced early and extended through the grades, with the sophistication of time zones and monetary exchange serving as valuable contexts at the intermediate grade levels.
- Question: How can we use the focal points to create progress-monitoring tools to use in our district?
Answer: Formative assessments could and should be developed to monitor student progress on the elements of the focal points for a particular grade level. This is a good idea and something that is needed.
- Question: How were the focal points created? At a recent meeting someone stated that they were simply a compilation of the most common state standards at that grade level.
Answer: The focal points were developed over more than 2 years. This included an examination of the work that went into a review of state standards and curriculum that was published as "A View from the Nation," based on a conference in July 2004. Work on the Curriculum Focal Points included analysis of all 49 states that have preK–8 curriculum frameworks, as well as curriculum frameworks from Singapore, Japan, China, and Korea. In short, this was an analysis, not a compilation. Multiple drafts of the focal points were shaped, revised, and reviewed before publication.
- Question: To what degree should fifth-grade students be able to divide and multiply? Two digit by two digit multiplication, etc.?
Answer: Multiplication of whole numbers is a focal point at the grade 4 level and division at the grade 5 level. This means that fluency with 2x2 digit multiplication (and perhaps beyond, depending on the learner) is a focus that year (grade 4). Similarly, division of whole numbers is a focal point in grade 5, with fluency expected.
- Question: Upon what research do you base the efficacy of this approach?
Answer: The work of the NCTM Board of Directors and the Association of State Supervisors of Mathematics (see "A View from the Nation," NCTM 2004) and the continuing work of the Center for the Study of Mathematics Curriculum (2006 and more) provide the best resources relative to the sequencing, scope, and emphasis of topics at these grade levels. At best, this research is case-based and survey related. When examining this issue, no experimental research was found on the topic of curriculum alignment at the preK–8 levels.
- Question: I received a promotional letter promoting a software-based program. Its letter seems to imply that they are connected to NCTM by stating that the NCTM's focal points "represents a significant change from the emphasis" on the "1989 recommendations that promoted open-ended problem solving and the use of calculators, rather than adequate practice with pencil and paper. Teaching the basics and solving problems using pencil and paper have been at the heart of our software...." Anyway, I am assuming you are not affiliated with them, but I also want to know is NCTM emphasizing drill and kill as opposed to the open-ended problem solving and use of calculators?
Answer: NCTM has no connection to this program or any other program or product. The promotional materials you received are a misrepresentation of the Curriculum Focal Points, which are the next step in implementing NCTM’s Principles and Standards for School Mathematics by addressing the Curriculum Principle. The Council has not changed its course and continues to advocate learning mathematics with understanding. A full reading of the Curriculum Focal Points reveals the importance of the Process Standards in developing a student’s deep understanding of mathematics.