A world-class mathematics curriculum should be built around and focused on:

• Teaching for Understanding
• Problem-Based Instructional Tasks
• Distributed Practice that is Meaningful and Purposeful
• Mathematical Modeling (secondary school emphasis)
• Deep Conceptual and Procedural Knowledge
• Rigor and Relevance
• Effective Use of Technology
• Connected and Coherent Content

Teaching for Understanding

First and foremost, teaching mathematics for understanding is the basis of the world-class core curriculum in mathematics that all Iowa students deserve. We must shift from a paradigm of "memorize and practice" to one of "understand and apply."

Teaching for understanding involves:

• Developing deep conceptual and procedural knowledge of mathematics
  (See description below.)
• Posing problem-based instructional tasks
  (See description below.)
• Engaging students in the tasks and providing guidance and support as they develop their own representations and solution strategies
• Promoting discourse among students to share their solution strategies and justify their reasoning
• Summarizing the mathematics and highlighting effective representations and strategies
• Extending students' thinking by challenging them to apply their knowledge in new situations, especially in real-world settings
• Listening to students and basing instructional decisions on their understanding

Problem-Based Instructional Tasks

Problem-based instructional tasks are at the heart of teaching for understanding. A world-class mathematics curriculum should be built around rich instructional tasks focused on important mathematics.

Problem-based instructional tasks:

• Help students develop a deep understanding of important mathematics
• Emphasize connections, especially to the real world
• Are accessible yet challenging to all
• Can be solved in several ways
• Encourage student engagement and communication
• Encourage the use of connected multiple representations
• Encourage appropriate use of intellectual, physical, and technological tools

Distributed Practice that is Meaningful and Purposeful

Practice is essential to learn mathematics. However, to be effective in raising student achievement, practice must be meaningful, purposeful, and distributed.

Meaningful Purposeful Distributed Practice:

• Meaningful: Builds on and extends understanding
• Purposeful: Links to curriculum goals and targets an identified need based on multiple data sources
• Distributed: Consists of short periods of systematic practice distributed over a long period of time

Mathematical Modeling (secondary school emphasis)

Mathematical modeling is the process of applying mathematics to solve real-world problems. As such, it is an essential characteristic of a world-class mathematics curriculum. The diagram below summarizes the process of mathematical modeling.

Process of Mathematical Modeling
 

Deep Conceptual and Procedural Knowledge

The goal of a world-class curriculum in mathematics is for all students to develop a deep understanding of important mathematics, which can be applied flexibly and powerfully to solve problems. An ongoing debate in mathematics education revolves around conceptual knowledge (knowledge of mathematical concepts such as function and rate of change) versus procedural knowledge (knowledge of mathematical procedures such as factoring and equation solving). In particular, questions persist about how to teach procedures, when to teach them, how much time to spend teaching them, and the relation of procedural knowledge to conceptual knowledge.

A common view is that conceptual knowledge is deep knowledge and procedural knowledge is superficial. However, recent research (e.g., Star, 2005) suggests that this view confusingly combines type of knowledge with quality of knowledge. Separating out these two dimensions yields the following table, where XX indicates the goal of deep knowledge for both procedures and concepts.

Deep-level knowledge is characterized by comprehension, abstraction, flexibility, critical judgment, and evaluation. It is structured in memory so that it is maximally useful for performance of tasks. This is in contrast to superficial knowledge, which is rote or at best inflexible knowledge.

The debates about conceptual knowledge versus procedural knowledge and about deep versus superficial knowledge are in fact based on false dichotomies. Students must develop deep knowledge of both concepts and procedures. Furthermore, concepts and procedures should be connected.

"As students develop a view of mathematics as a connected and integrated whole, they will have less of a tendency to view mathematical skills and concepts separately. If conceptual understandings are linked to procedures, students will not perceive mathematics as an arbitrary set of rules. This integration of procedures and concepts should be central in school mathematics" (NCTM, 2000, p. 65).

In addition to procedures and concepts, a typical third goal of mathematics instruction is problem solving. One often sees mathematics curricula and assessments discussed and organized in terms of skills, concepts, and problem solving. The prevalent view is that each of these three tends to be taught in a specific way, as summarized in the left-hand column of the following table. However, in a world-class mathematics curriculum, practice is not just for skills, understanding is not just for concepts, and problem solving is not just for developing the ability of solving problems, as shown in the right-hand column of the table.

Effective Use of Technology

Technology is an integral part of contemporary life, and as such should be an integral part of mathematics education. Technological tools, such as graphing calculators, computers, and the Internet, should be used to enhance teaching and learning. As stated in NCTM's Principles and Standards:

When technological tools are available, students can focus on decision making, reflection, reasoning, and problem solving. Students can learn more mathematics more deeply with the appropriate use of technology (Dunham and Dick 1994; Sheets 1993; Boers-van Oosterum 1990; Rojano 1996; Groves 1994).
Technology enhances mathematics learning: — Students' engagement with, and ownership of, abstract mathematical ideas can be fostered through technology. Students can examine more examples or representational forms than are feasible by hand, so they can make and explore conjectures easily thus allowing more time for conceptualizing and modeling.
Technology supports effective mathematics teaching: — The effective use of technology in the mathematics classroom depends on the teacher. Technology is not a panacea. As with any teaching tool, it can be used well or poorly. Teachers should use technology to enhance their students' learning opportunities by selecting or creating mathematical tasks that take advantage of what technology can do efficiently and wellùgraphing, visualizing, and computing.
Technology influences what mathematics is taught: — Technology not only influences how mathematics is taught and learned but also affects what is taught and when a topic appears in the curriculum. With technology at hand, young children can explore and solve problems involving large numbers, or they can investigate characteristics of shapes using dynamic geometry software. Elementary school students can organize and analyze large sets of data. Middle-grades students can study linear relationships and the ideas of slope and uniform change with computer representations and by performing physical experiments with calculator-based-laboratory systems. High school students can use simulations to study sample distributions, and they can work with computer algebra systems that efficiently perform most of the symbolic manipulation that was the focus of traditional high school mathematics programs. The study of algebra need not be limited to simple situations in which symbolic manipulation is relatively straightforward. Using technological tools, students can reason about more-general issues, such as parameter changes, and they can model and solve complex problems that were heretofore inaccessible to them. Technology also blurs some of the artificial separations among topics in algebra, geometry, and data analysis by allowing students to use ideas from one area of mathematics to better understand another area of mathematics. Technology can help teachers connect the development of skills and procedures to the more general development of mathematical understanding. As some skills that were once considered essential are rendered less necessary by technological tools, students can be asked to work at higher levels of generalization or abstraction. (NCTM, 2000, pp. 24-26)

Rigor and Relevance

A world-class school mathematics curriculum should be rigorous and relevant. These terms, while open to a variety of interpretations, are used in the Iowa Core Curriculum with reference to their meaning as given by Daggett (2005).

"Daggett asserts that schools can no longer afford to teach only a discrete set of facts, but instead must teach students how to think. It is insufficient to teach students how to do things by rote; now schools must teach people how to do things with deeper levels of understanding. He recommends levels of cognitive knowledge [rigor] applied to real-world situations [relevance], that is, academic rigor applied in open-ended and unpredictable ways. Daggett advises educators to use the Rigor/Relevance Framework to move beyond the what of curriculum to the how of instruction" (Iowa Department of Education 2005, p. 4).

Connected and Coherent Content

"Mathematics comprises different topical strands, such as algebra and geometry, but the strands are highly interconnected. The interconnections should be displayed prominently in the curriculum . A coherent curriculum effectively organizes and integrates important mathematical ideas so that students can see how the ideas build on, or connect with other ideas thus enabling them to develop new understandings and skills" (NCTM, 2000, Curriculum Principle, p. 15). The school mathematics curriculum, in kindergarten through grade 12, should be connected and coherent.

The United States is virtually the only country in the world in which the high school mathematics curriculum is generally not connected across strands. In particular, the countries that consistently outperform the U.S. on international mathematics achievement tests, including those countries often looked to for solutions such as Singapore and Japan, have a connected high school mathematics curriculum.

What is a connected mathematics curriculum? One can consider the content to be connected and the method of connecting the content. With respect to the content that is connected, the strands of mathematics (such as algebra, geometry, and statistics) might be connected or different disciplines (such as mathematics, science, and social studies) might be connected. Concerning methods of connecting the content, connections might be made through use of thematic units, whereby a particular theme or application is the organizing principle for the unit and targeted mathematics is developed to pursue that theme or application; or connections could be made through use of big-idea strand-dominant units, whereby a big mathematical idea, typically from a specific strand, is the main organizing principle for the unit and a variety of contexts and mathematical connections are utilized to help develop that big idea.

The curriculum content connection prevalent throughout the world is across the strands of mathematics, with courses typically consisting of several connected blocks each focused on a particular mathematical strand. Thus, mathematics courses are taught, not separate courses in algebra, geometry, advanced algebra, trigonometry, etc. According to Burkhardt (2001), "Nowhere else in the world would people contemplate the idea of a year of algebra, a year of geometry, another year of algebra, and so on." The advantages of connected mathematics courses are that "they build essential connections, help make mathematics more usable, avoid long gaps in learning, allow a balanced curriculum, and support equity. I know of no comparable disadvantages, provided that the 'chunks' of learning are substantial and coherent."

As stated in the NCTM Connections Standard, "Mathematics is not a collection of separate strands or standards, even though it is often partitioned and presented in this manner. Rather, mathematics is an integrated field of study. When students can connect mathematical ideas, their understanding is deeper and more lasting. " (NCTM, 2000, p. 64).

"In a coherent curriculum, mathematical ideas are linked to and build on one another so that students' understanding and knowledge deepens and their ability to apply mathematics expands" (NCTM, 2000, Curriculum Principle, p. 15).

Thus, the value of a connected and coherent curriculum is that students gain deeper understanding of mathematics and greater ability to apply mathematics. The essential content and skills specified in the Iowa Core Curriculum can be taught in integrated or non-integrated courses, and there is no requirement to restructure schools or adopt any specific materials. It is essential that, whatever courses or materials are chosen, the mathematics content should be connected and coherent.