Diocese of Covington - Education at PO Box 15550, Covington, KY 41015-0550 US - Mathematics Part 1

The following document contains the Curriculum Guidelines for the Diocese of Covington. It was created by a team of teachers from different schools and grade levels throughout the Diocese. In accordance with Diocesan Policy 4265 concerning curriculum, it was created using the last version of the Diocesan Math Curriculum along with the Program of Studies for the state of Kentucky. It should be closely aligned with both the Terra Nova test and the textbooks that are currently being used.

As with all Curriculum Guidelines it is to be used by the schools as a guide in forming local curriculum. It represents the minimum that should be done by all schools within the Diocese. It is important that all teachers follow the scope and sequence so that we have consistency throughout the Diocese especially when students are entering high school. Grade level objectives are based on national and state standards and can be adjusted by the local schools if necessary.

Mathematics plays an integral role in Catholic Schools since it reflects the order and unity in God’s universe. Mathematics contributes to the formation of Christians who can respond wisely and effectively to a changing world. Contemporary society demands mathematical knowledge which requires students to develop their ability to reason and think logically and to discover creative ways of problem solving. Because of its nature, mathematics can contribute to the development of the whole person by enriching one’s life and providing one with a practical tool for learning.

Toward these ends, students should:

• Learn to value mathematics
• Learn to reason mathematically
• Learn to communicate mathematically
• Learn to use technology to investigate and solve problems
• Become confident of their mathematical abilities
• Become creative mathematical problem solvers

Beliefs:

• All students can learn.
• Students learn by experience and doing.
• Students often have negative feelings regarding mathematics.
• Students can be motivated to achieve maximally their God-given potential.

Learning Expectations:

Students should be able to:

• see a reason for doing mathematics
• know that one can learn from mistakes.
• see mathematics as relevant to daily living
• be conceptually as well as computationally sound.
• use current technology
• communicate their knowledge to others.
• see the connection with other mathematics topics and other subject areas
• know that there is more than one strategy that can be used to solve a problem
• work in groups

1. Equity. Excellence in mathematics education requires equity—high expectations and strong support for all students.

• All students, regardless of their personal characteristics, backgrounds, or physical challenges, must have opportunities to study---and support to learn---mathematics.

2. Curriculum. A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades.

• Mathematics is a highly interconnected and cumulative subject. The mathematics curriculum therefore needs to introduce ideas in such a way that they build on one another.

3. Teaching. Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well.

• Students learn mathematics through the experiences that teachers provide. There is no one "right way" to teach.

4. Learning. Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.

• Research has solidly established the importance of conceptual understanding in becoming proficient in a subject. When students understand mathematics, they are able to use their knowledge flexibly. They combine factual knowledge, procedural facility, and conceptual understanding in powerful ways.

5. Assessment.Assessment should support the learning of important mathematics and furnish useful information to both teachers and students.

• Assessment should be more than merely a test at the end of instruction to gage learning. It should be an integral part of instruction that guides teachers and enhances students’ learning.

6.Technology.Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning.

• Calculators and computers are reshaping the mathematical landscape, and school mathematics should reflect those changes. Students can learn more mathematics more deeply with the appropriate and reasonable use of technology. Technology can not replace the mathematic teacher.

Instructional programs from pre-kindergarten through grade 12 should enable all students to—

1. Number and Operations

• understand numbers, ways of representing numbers, relationships among numbers, and number systems;
• understand meanings of operations and how they relate to one another;
• compute fluently and make reasonable estimate

2. Algebra

• understand patterns, relations, and functions;
• represent and analyze mathematical situations and structures using algebraic symbols;
• use mathematical models to represent and understand quantitative relationships;
• analyze change in various contexts.

3. Geometry

• analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships;
• specify locations and describe spatial relationships using coordinate geometry and other representational systems;
• apply transformations and use symmetry to analyze mathematical situations;
• use visualization, spatial reasoning, and geometric modeling to solve problems.

4. Measurement

• understand measurable attributes of objects and the units, systems, and processes of measurement;
• apply appropriate techniques, tools, and formulas to determine measurements.

5. Data Analysis and Probability

• formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them;
• select and use appropriate statistical methods to analyze data;
• develop and evaluate inferences and predictions that are based on data;
• understand and apply basic concepts of probability

6. Problem Solving

• build new mathematical knowledge through problem solving;
• solve problems that arise in mathematics and in other contexts;
• apply and adapt a variety of appropriate strategies to solve problems;
• monitor and reflect on the process of mathematical problem solving.

7. Reasoning and Proof

• recognize reasoning and proof as fundamental aspects of mathematics;
• make and investigate mathematical conjectures;
• develop and evaluate mathematical arguments and proofs;
• select and use various types of reasoning and methods of proof.

8. Communication

• organize and consolidate their mathematical thinking through communication;
• communicate their mathematical thinking coherently and clearly to peers, teachers, and others;
• analyze and evaluate the mathematical thinking and strategies of others;
• use the language of mathematics to express mathematical ideas precisely.

9. Connections

• recognize and use connections among mathematical ideas;
• understand how mathematical ideas interconnect and build on one another to produce a coherent whole;
• recognize and apply mathematics in contexts outside of mathematics.

10. Representation

• create and use representations to organize, record, and communicate mathematical ideas;
• select, apply, and translate among mathematical representations to solve problems;
• use representations to model and interpret physical, social, and mathematical phenomena.

Estimation activities should be a step used in all computational activities, providing students with a problem-solving aid and as a means of judging reasonableness of solutions.

Deliberate and thorough development of the ability to estimate and do mental arithmetic must be a regular part of instruction.

These skills should be used to enhance number sense and spatial sense to help children develop insights into concepts and procedures, flexibility in working with numbers and measurements, and awareness of reasonable results. Estimation skills and understanding enhance the abilities of children to deal with everyday quantitative situations.

It is important that students learn a variety of methods of estimating, and develop reasoning and judgment in using estimation.

Mental arithmetic enables students to arrive at exact solutions without the use of paper-and-pencil algorithms. Students should be encouraged to develop non-standard techniques for performing calculations involving number properties and operations, including compensation and using the distributive property.

Increased use of calculator increases the need for both mental computations and estimation.

The learner will:

1. explore estimation strategies
2. recognize when an estimate is appropriate
3. use estimation to determine the reasonableness of results and as an aid in selecting a method for exact calculation
4. apply estimating in working with quantities, measurement, computation, and problem solving
5. use mental arithmetic for all simple operations and for manipulations

The mathematics curriculum identifies skills needed to be mathematically literate in a world that increasingly relies on calculators and computers to carry out computational procedures.

Because technology is changing mathematics and its uses, appropriate calculators should be available to all students, and every student should have access to computers for individual and group work to be used as tools for processing information and performing calculations to investigate and solve problems. However, access to technology is no guarantee that any student will become mathematically literate. Calculating tools simplify, but do not accomplish the work at hand. Thus this curriculum is based on the fundamental mathematics students will need.

At the same time, they should learn that for some simple computations, use of the calculator is either cumbersome or, worse, can obscure the understanding of the calculation being performed. As they gain experience, students should be expected to judge whether use of the calculator will be effective and efficient. Calculators do not replace the need to learn basic facts, to compute mentally, or to do reasonable paper and pencil computation. On the contrary, proper use of the calculator requires a knowledge of basic facts and strengthens number skills. It also requires the development of students’ understanding of the meaning of arithmetic operations and when to apply each, and highlight the importance estimation skills and the ability to recognize whether computed results are reasonable.

In mathematics education, computers have special importance. Their value in creating geometric display, organizing and graphing data, simulating real-life situations, and generating numerical sequences and patterns is already recognized. In general, computers should be used as an integral part of instruction, enabling teachers to demonstrate concepts, and students to explore and experiment with mathematical ideas.

The mathematics classroom envisioned in the Standards is one in which calculators, computers, courseware, and manipulative materials are readily available and regularly used in instruction. Although no rigid criteria exist for judging what constitutes adequate resources and equipment, every program should provide as many opportunities for learning, using these tools, as resources will allow.

All students should be able to:

• use a calculator correctly and confidently when mental calculation would be difficult or when paper and pencil calculation would be inefficient, and
• use a computer program, as appropriate, to perform extensive or repetitive calculations to simulate real situations, and to perform experiments that aid in understanding of mathematical concepts.

The heart of an effective, well-taught mathematics program is the careful development of concepts and skills through extensive use of concrete materials. Students must be actively involved in constructing, modifying and integrating ideas by interacting with physical materials. They must investigate quantitative and spatial situations by manipulating the materials and then translating the ideas through pictures or diagrams to corresponding abstract symbolic representations. These are essential elements of a program in which mathematics is done with understanding rather than rote.

It is not sufficient to confine the use of concrete materials to teacher demonstration; each student must manipulate his/her own materials. K-8 classrooms especially must be equipped with a wide variety of physical materials and supplies. Manipulatives should continue to be used in grades 9-12 whenever appropriate.

Classrooms should have quantities of materials such as:

• counters
• interlocking cubes
• connecting links
• bean sticks and/or bundling sticks
• base ten, attribute, and pattern blocks
• geometric models
• rulers and other measuring devices
• spinners
• color rods
• geoboards
• balance scales
• fraction pieces
• graph, grid and dot paper
• upper grades should also have compasses and protractors
• geometric puzzles such as tangrams and pentaminoes

The content in the primary level courses is directly aligned with Kentucky's academic expectations. Presented are the topics fundamental to mathematical literacy and mathematical power for all primary level students. The content statements are organized under common topic headings and can be related to other statements. An integral part of the learning process is the systematic review of earlier concepts and procedures in which students use previously learned skills to develop proficiency with more advanced concepts. Furthermore, the primary level mathematics program includes active, hands-on work with concrete materials and appropriate technologies.

Primary problem solving, mathematical communication, and mathematical reasoning should be a part of the mathematics curriculum. The use of these techniques enhances and extends students' arithmetic skills. Accuracy is an integral part of the mathematics program.

Problem solving involves developing and applying strategies to problems from everyday and mathematical situations and evaluating the solutions relative to the original problem situation.

Mathematical communication includes manipulatives (concrete materials), visual representations, and diagrams that relate language to mathematical symbols in speaking, reading, writing, and listening.

Mathematical connectionsinclude

• understanding how one concept relates to other concepts and procedures (e.g., the link between fractions and decimals)
• understanding how one major math topic relates to another (e.g., the link between geometry and measurement)
• understanding how a mathematical topic relates to other disciplines (e.g., the link between statistics and social studies)

Mathematical reasoning includes recognizing patterns and relationships and using models, known facts, and mathematical properties to explain and justify thinking.

In addition to specifying mathematics content, these guidelines provide connections to Kentucky's Learning Goal 5 (Think and Solve Problems) and Goal 6 (Connect and Integrate Knowledge). These connections provide a comprehensive link between essential content and the skills and abilities important to learning.

The learner will:

• match a number of objects with an equal number of objects (1-1 relationships)
• read, copy, count, and model numbers 0-20
• order and compare numbers from 0-20, using physical manipulatives
• recognize positions first through tenth
• match sets of objects with numerals 0-20
• recognize symbols ( + , -, = )
• explore appropriate estimation procedures

The learner will:

• recognize a calculator as a mathematical tool and know its name
• develop meaning of addition and subtraction by using manipulatives
• recognize part-part-whole relationships (e.g. 3+2=5, 1+4=5 and 3-1=2, 4-2=2)

The learner will:

• recognize the spatial relationship of two objects
• identify, describe, and make plane figures (circle, square, triangle, rectangle)
• compare the size and shape of plane figures
• combine plane figures to form other plane figures (e.g. tangram puzzles)

The learner will:

• recognize that a clock (digital and traditional) is used to tell time
• relate time to daily activity
• recognize that a thermometer is used to read temperature
• recognize that a calendar is used to measure days, weeks and months
• identify a penny and nickel and know the value
• compare and order by size, length, and width
• begin to use estimates of measurement in problem-solving situations

The learner will:

• make a graph using manipulatives
• draw conclusions and make predictions based on data displayed on graphs

The learner will:

• explore the concept that a fraction is part of a whole
• use models to explore and verbalize the meaning of whole and half

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