Mathematics Curriculum Implementing a Course of Study Local Course of Study Process of Writing a Course of Study Goals vs. Objectives, Standards and Benchmarks Selecting Textbooks Alternative Sources of Materials, Resource Files Mathematics Curriculum C2 Local Course of Study There is no national curriculum. The constitution places the responsibility of education on the states. The Standards does not constitute a curriculum; it is a guideline. Textbooks and other curricular material do not make a curriculum; they are aids. The same is also true for the Nebraska standards. At present the responsibility for developing a curriculum falls on the shoulders of the local school districts. There are examples in the state where the curriculum is actually determined by individual buildings in the district; especially at the lower grade levels. Using Standards as a Framework Standards should be nationally determined so that curricular materials have a common focus. That appears true in mathematics. Standards are the ―what‖ of what you will teach. A nationally recognized curriculum specialist, Robin Fogarty, advises teachers to use standards to create kid-friendly activities that will entice students to learn. She goes on to say that you need to group objectives and outcomes together ―so that content, process, and performance standards are combined and clustered, textured and tiered, and infused and integrated‖. This will create a very dynamic curriculum. Process of Writing a Course of Study There are differently opinions about what the different elements of a curriculum are called. With that said, the typical basic elements are: Mission/Philosophy Goals General Objectives Specific Objectives & Outcomes In a standards-based curriculum, the basic elements are: Mission/Philosophy – What are the global purposes of the curriculum? Principles – Principles are infused throughout the entire curriculum. Standards – Standards are the ―what‖ of what you teach. They are broad strokes. Performance Standards – Performance standards narrow the vision and are more specific. What ―exactly‖ will students do? Benchmarks – Benchmarks are how you know when a performance standard has been met. You should always think about assessment when you write a curriculum. As seen in the Standards both the content and process standards are consistent across grade bands. Performance standards and benchmarks should change from one grade level to another so Mathematics Curriculum C3 as to reflect student growth and sophistication. Goals vs. Objectives, Standards and Benchmarks Objectives-Based Curriculum Goals are broad statements about what is to be accomplished by a curriculum and should be directly linked back to the mission and/or philosophy of the curriculum. Goals often address mathematical disposition. There are two basic types of objectives, (1) general and (2) specific. General objectives address what the student ―will‖ be able to accomplish in a course and/or unit of study. General objectives should be linked to the goals. Specific objectives address what the student will be able to accomplish in a lesson. Specific objectives should be linked to the general objectives. Students should be assessed on their achievement of specific objectives. A common school of thought is to only include specific objectives if you plan to test over them. Standards-Based Curriculum As stated previously the basic elements of a standards-based curriculum are: Mission/Philosophy – What are the global purposes of the curriculum? Principles – Principles are infused throughout the entire curriculum. Standards – Standards are the ―what‖ of what you teach. They are broad strokes. Performance Standards – Performance standards narrow the vision and are more specific. What ―exactly‖ will students do? Benchmarks – Benchmarks are how you know when a performance standard has been met. The mission/philosophy statement is the same as in an objectives-based curriculum. A philosophy statement could refer to the principles. Principles are ideas and practices that are infused throughout the curriculum. The NCTM’s six underlying principles of school mathematics are: Equity. Excellence in mathematics education requires equity—high expectations and strong support for all students. Curriculum. A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades. Teaching. Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well. Mathematics Curriculum C4 Learning. Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge. Assessment. Assessment should support the learning of important mathematics and furnish useful information to both teachers and students. Technology. Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning. All six principles should be kept in mind when developing a curriculum of study, but are not necessarily addressed in the curriculum itself. Students are not assessed over principles. Standards are the ―what‖ of what you teach. They are written in broad language. The NCTM uses category headings for standards. Five Content Standards of School Mathematics Number and Operation Patterns, Functions, and Algebra Geometry and Spatial Sense Measurement Data Analysis, Statistics, and Probability Five Process Standards of School Mathematics Problem-Solving Reasoning (and Proof) Communication Connections Representation As stated previously, all standards should be incorporated at each grade level. They are not necessarily included in each lesson. Performance standards narrow the vision and are written in more specific terms. What will students be able to do? They are similar to objectives in the fact that they are written as affective (attitudes or feelings) or cognitive (skills and concepts). Benchmarks are how you know when a performance standard has been met. They should be tied to a rubric. Cognitive benchmarks should measure: Knowledge and Skill Concept Application A standards-based curriculum should include real-life applications. Mathematics Curriculum C5 Determine the number of board feet that can be taken from a circular log that is 18 inches in diameter and 20 feet long. A board foot measures 12 in x 12 in and is 1 inch thick. Final Note A mathematics curriculum should include mathematical history and historical problems. A Hindu arithmetic problem from Mahāvīra (ca. 850): A powerful, unvanquished, excellent black snake which is 80 angulas in length enters into a hole at the rate of 7-1/2 angulas in 5/14 of a day, and in the course of ¼ of a day its tail grows 11/4 of an angula. O ornament of arithmeticians, tell me by what time this serpent enters fully into the hole? Alcuin of York (ca. 775) may have compiled the Latin collection, Problems for the Quickening of the Mind. A problem from the collection: A dying man wills that if his wife, being with child, gives birth to a son, the son shall inherit ¾ and the widow ¼ of the property, but if a daughter is born, she shall inherit 7/12 and the widow 5/12 of the property. How is the property to be divided if both a son and a daughter are born? (The solution given in Alcuin’s collection is not acceptable.) Another problem taken from Problems for the Quickening of the Mind: In his Geometry, Gerbert expressed the area of an equilateral triangle of side a as (a/2)(a – a/7). Show that this is equivalent to taking 3 = 1.714. Selecting Textbooks TEXTBOOKS ARE NOT THE CURRICULUM!!! Ideally textbooks should be chosen after the curriculum is written. Some questions to ask appear on page 96 of Brahier’s Teaching Secondary and Middle School Mathematics. Some additional questions to consider: How does the text fit with the curriculum? What portion of the curriculum is covered by the book? How much additional material will you need to provide? Do you need a textbook? Alternative Sources of Materials, Resource Files Additional sources of materials?