# Mathematics

### Mathematics possesses not only truth, but also supreme beauty.

### —Bertrand Russell

### The goals of the mathematics program are for students to:

acquire a deep understanding of mathematical concepts

develop the habits of mind of mathematicians

use mathematics as a tool to understand their world

### More about Math at NCCL:

Concepts and problems are presented in a context in a developmentally appropriate way and time is allowed for students to explore, discuss, revisit and reflect. Models and manipulatives are used in all grades to represent problems, help clarify mathematical thinking, and to extend questioning. Fluency with numbers and operations is developed from an understanding of the procedures used.

The math program from K through Grade 8 is based on recommendations made in the principles and standards of the National Council of Teacher of Mathematics (NCTM), on-going research conducted by Trends in International Math and Science Study (TIMSS), research and materials from the National Science Foundation (NSF), and the National Common Core Standards for Mathematics of 2010 and adopted by Delaware in 2011.

## Below is a general guide to the concepts and skills development

## through the grades at NCCL.

There is definite overlap between sections and you will notice that Grade 1 is covered in two sections, reflecting the developmental nature of mathematical concepts.

### Grades K and 1

Children in grades K and 1 are working on beginning number concepts which include one-to-one correspondence, conservation of number, inclusion (understanding that 5 includes 4,3,2 & 1), and number relationships. These are complex concepts for young children and they need to work with real objects and have many experiences with numbers to develop true understanding.

**Mathematical concepts in the early grades include**: one to one correspondence, conservation of number (understanding that number does not change,) counting, number relationships, more and less, estimation, number combinations, beginning addition and subtraction, recognizing and interpreting patterns, non-standard measurement, spatial concepts in geometry, and an introduction to grouping in tens.

**Mathematical skills include:** writing and interpreting symbols and signs for whole number concepts (children are starting to associate the numerals with the quantities they represent and are beginning to record their math work), gathering and representing data by taking meaningful surveys and making graphs about information children are interested in, and learning to identify and describe basic attributes of shapes.

### Grades 1, 2

In grades 1 and 2, children are extending their understanding of number relationships with a focus on the parts of numbers and number combinations for addition and subtraction. They are starting to see the patterns that repeat in place value representations of two-digit numbers and thinking of groups of tens and ones. This is when most children develop a beginning understanding of place value and regrouping. Children continue to need repeated experiences with objects to further develop their mathematical understanding.

**Mathematical concepts and skills include:** "More and less" number relationships, comparing larger groups and estimating with larger numbers. Number combinations and beginning strategies for addition and subtraction become more automatic. Multi-digit addition and subtraction are introduced (usually 2nd grade). Beginning understanding of place value allows children to make sense of this. Children are recording math work using both natural language and symbols. We do not focus on traditional algorithms. Children learn best when they problem solve in a way that makes sense to them. As children get older their strategies need to become more efficient and fact memorization and some traditional algorithms are incorporated into the curriculum.

### Grades 3, 4

In grades 3 and 4 students are consolidating and expanding their understanding of number relationships for whole number operations. They are more reliably able to work and think in groups, which expands their ability to interpret and work with multiplication and division concepts and to develop a more powerful understanding of place value. They begin to move from thinking additively to thinking multiplicatively. Students begin to develop a sense of the fraction relationships.

**Mathematical concepts and skills include: **Developing efficient strategies for whole number operations, extending an understanding of multiplication and division, beginning fraction concepts of "part to whole" and division, comparing fractions, categorizing geometric shapes by attributes, symmetry, data collection and interpretation, ideas of probability, extension of the understanding of mathematical symbols.

### Grades 5, 6

The major conceptual breakthrough in grades 5 and 6 is the beginning awareness of proportional relationships. This is an extension of multiplicative reasoning that began in earlier grades. Proportional relationships are fundamental to rational number understanding and efficiency; many experiences over time are required to build the foundation necessary for success in algebra. Students in these grades are able to think abstractly and make generalizations from specific examples which moves them toward an algebraic interpretation of math situations.

**Mathematical concepts and skills include: **The development of strategies for standard fraction, decimal and percent operations, fraction interpretations of ratios, rate and scaling, extension of place value understanding and bases other than ten, exponential notation, statistics and data collection, use of different graphing systems, patterns and functions including linear and exponential expressions, interpreting problems algebraically, plane and solid geometry: deriving formulas for area and perimeter of polygons and circles, discovering the Pythagorean Theorem, measuring angles, calculating volume and surface area of prisms, and metric measurement.

### Grades 7, 8

In grades 7 and 8 students use their strong understanding of rational numbers to move into the abstract world of algebra and the power of generalized formulas. The goal is to give students a flexible and intuitive understanding of mathematical relationships and how to use the effectiveness of algebraic and geometric techniques to solve real-world and scientific problems.

**Mathematical concepts and skills include:** Understanding algebraic expressions, use andinterpretation of variables, x-y graphing, generating equations from both graphs and data, and three dimensional graphing. Functions, including linear and non-linear equations, logarithmic and trigonometric functions, Boolean algebra, polynomials and quadratics, extended use of powers and roots, and scientific notation. Computer programming, use of spreadsheets, and financial analysis. Geometry topics are extended with a focus on terms and constructs, finding surface area and volume of combined and truncated shapes.

### Why do our NCCL students love Math?

Why do we teach math differently here at NCCL? Researchers from around the world are sharing work that demonstrates the importance of teaching math using our philosophy and methods. These videos truly exemplify the how and why of our teaching methods!

### Number Talks part 1.

Step inside the world of number talks. This video by Jo Boaler, inspired by Ruth Parker and Kathy Richardson, is a very informative video of showing how number talks can provide a successful way of teaching kids number sense. While learning about what a number talk is, you will get to experience the journey yourself. She explains how a number talk begins with mentally thinking of ways to solve a problem without pencil and paper. Having someone, like a teacher, notate what’s being discussed provides a visual representation to observe and will give children a way to see multiple strategies used for solving the same problem. Looking at a problem and finding many ways of breaking down and building numbers, provides children the opportunity and ability to look at Math through a creative discovery process.

### Number Talks part 2.

Number Talks are a significant part of our math program at NCCL. Through number talks, students develop flexibility and fluency with an understanding of the properties of operations. In this interactive video, Jo Boaler will give you the opportunity to participate in a number talk, share the work of her graduate students, and connect the solutions to the development of algebraic reasoning. The activities you will observe are a regular part of our program in Group 1 through Group 4.