Mathematics - Essential Characteristics
Essential Characteristics of a World-Class Curriculum in Mathematics
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
- Deep Conceptual and Procedural Knowledge
- Rigor and Relevance
- Effective Use of Technology
- Integrated 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 teaching paradigm of "memorize and practice" to one of "understand and apply."
Teaching for understanding means:
- Posing problem-based instructional tasks
(See description of problem-based instructional tasks 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 situations
- 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 problem-based 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
Meaningful Purposeful Distributed Practice
Practice is essential to learn mathematics. However, to be effective in raising student achievement, practice must be distributed, purposeful, and meaningful.
Meaningful Purposeful Distributed Practice:
- Meaningful: Builds on and extends understanding
- Purposeful: Is linked 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
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
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As a brief example of mathematical modeling, consider the following diagram, which shows the use of quadratic functions to model the problem in business of finding the break even point in a manufacturing process.
Example of Mathematical Modeling: Break Even
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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.
The prevalent view is that conceptual knowledge is deep knowledge and procedural knowledge is superficial. However, recent research (e.g., Star, 2005) suggests that this view conflates type of knowledge with quality of knowledge. Separating out these two dimensions yields the following table.
Type and Quality of Knowledge
|
|
Knowledge of Concepts |
Knowledge of Procedures |
|
Superficial Knowledge |
|
|
|
Deep Knowledge |
XX |
XX |
Deep understanding includes both deep conceptual knowledge and deep procedural knowledge. 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.
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 next table.
Prevalent Fragmented View of the Mathematics Curriculum
|
What to teach: |
How to teach: |
|
Procedures, Skills, Facts |
Memorize and Practice |
|
Concepts |
Understand and Apply |
|
Problem Solving |
Heuristics and Solving Problems |
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. Instead:
|
World-Class Mathematics Curriculum What to teach:
How to teach:
Result:
|
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 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 toenhance their students' learning opportunities by selecting or creating mathematical tasks that take advantage of what technology can do efficiently and well--graphing, 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. 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 high school mathematics curriculum should be rigorous and relevant. These terms, while open to a variety of interpretations, are used in this document with reference to their meaning as given by Daggett (2005). According to the Iowa Department of Education's document on "Improving Rigor and Relevance in the High School Curriculum" (Iowa DE, September 2005):
Daggett asserts that secondary 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. (p. 4)
Integrated Content
The United States is virtually the only country in the world in which the high school mathematics curriculum is generally not integrated. 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 an integrated high school mathematics curriculum.
What is an integrated mathematics curriculum? One can consider the content to be integrated and the method of integration. Concerning the content to be integrated, one might integrate across the strands of mathematics (such as algebra, geometry, and statistics) or integrate across different disciplines (such as mathematics, science, and history). Concerning methods of integration, one might integrate through the use of "thematic units," whereby a particular theme or application is the organizing principle for the unit and you bring in all the mathematics necessary to pursue that theme or application. Alternatively, you might integrate through use of "big-idea units," whereby a big idea is the main organizing principle for the unit and you bring in a variety of contexts in which the big idea is developed and applied.
The content integration prevalent throughout the world, and recommended here, is integration across the strands of mathematics. Thus, mathematics courses are taught, not separate courses in algebra, geometry, advanced algebra, trigonometry, statistics, 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 integrated 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" (Burkhardt, 2001). A world-class high school mathematics curriculum should be an integrated curriculum.