## How to Build Number Sense

February 02, 2014

WHAT IS NUMBER SENSE?

Number sense involves understanding numbers; knowing how to write and represent numbers in different ways; recognizing the quantity represented by numerals and other number forms; and discovering how a number relates to another number or group of numbers. Number sense develops gradually and varies as a result of exploring numbers, visualizing them in a variety of contexts, and relating to them in different ways.

In the primary and intermediate grades, number sense includes skills such as counting; representing numbers with manipulatives and models; understanding place value in the context of our base 10 number system; writing and recognizing numbers in different forms such as expanded, word, and standard; and expressing a number different ways—5 is "4 + 1" as well as "7 - 2," and 100 is 10 tens as well as 1 hundred. Number sense also includes the ability to compare and order numbers—whole numbers, fractions, decimals, and integers—and the ability to identify a number by an attribute—such as odd or even, prime or composite-or as a multiple or factor of another number. As students work with numbers, they gradually develop flexibility in thinking about numbers, which is a distinguishing characteristic of number sense.

WHY IS IT IMPORTANT?

Number sense enables students to understand and express quantities in their world. For example, whole numbers describe the number of students in a class or the number of days until a special event. Decimal quantities relate to money or metric measures, fractional amounts describing ingredient measures or time increments, negative quantities conveying temperatures below zero or depths below sea level, or percent amounts describing test scores or sale prices. Number sense is also the basis for understanding any mathematical operation and being able to estimate and make a meaningful interpretation of its result.

HOW CAN I MAKE IT HAPPEN?

In teaching number sense, using manipulatives and models (e.g., place-value blocks, fraction strips, decimal squares, number lines, and place-value and hundreds charts) helps students understand what numbers represent, different ways to express numbers, and how numbers relate to one another.

When students trade with place-value blocks they can demonstrate that the number 14 may be represented as 14 ones or as 1 ten and 4 ones. They can also demonstrate that 10 hundreds is the same as 1 thousand. By recording the number of each kind of block in the corresponding column (thousands, hundreds, tens, or ones) on a place-value chart, students practice writing numbers in standard form.

Using fraction strips, students find that 1/4 is less than 1/3 and that it names the same amount as 2/8. Using decimal squares, students see that 8 tenths can be written as 0.8 or 8/10. By pairing up counters to identify even numbers and marking these on a hundreds chart, primary-grade students discover that, beginning with 2, every other number is an even number. Intermediate-grade students can mark multiples of 3 and 6 on a hundreds chart and find that every number that has 6 as a factor also has 3 as a factor. Using a number line, students see how fractions with different denominators relate to the benchmark quantities of 0, 1/2, and 1. From these concrete experiences, students build the foundation for number sense they will bring to computation, estimation, measurement, problem solving, and all other areas of mathematics.

DAILY ROUTINES

Many teachers use the calendar as a source of mathematics activities. Students can work with counting, patterns, number sequence, odd and even numbers, and multiples of a number; they can also create word problems related to the calendar. A hundreds chart can help them count the number of days in school, and the current day’s number can be the "number of the day." Students can suggest various ways to make or describe that number. For example, on the 37th day of school, children may describe that number as 30 plus 7, 40 minus 3, an odd number, 15 plus 15 plus 7, my mother’s age, or 1 more than 3 dozen. The complexity of student’s responses will grow as the year goes on and as they listen to one another think mathematically. This is a great language building time.

THY THESE

When a teacher publicly records a number of different approaches to solving a problem–solicited from the class or by introducing her own—it exposes students to strategies that they may not have considered. As Marilyn Burns explains, “When children think that there is one right way to compute, they focus on learning and applying it, rather than thinking about what makes sense for the numbers at hand.”

Mental math encourages students to build on their knowledge about numbers and numerical relationships. When they cannot rely on memorized procedures or hold large quantities in their heads, students are forced to think more flexibly and efficiently, and to consider alternate problem solving strategies.

Classroom discussions about strategies help students to crystalize their own thinking while providing them the opportunity to critically evaluate their classmates’ approaches. In guiding the discussion, be sure to track ideas on the board to help students make connections between mathematical thinking and symbolic representation.

Most of the math that we do every day—deciding when to leave for school, how much paint to buy, what type of tip to leave in a restaurant, which line to get in at the grocery store relies not only on mental math but estimations. However traditional textbook rounding exercises don’t provide the necessary context for students to understand estimating or build number sense. To do that, estimation must be embedded in problem situations.

Asking students about their reasoning—both when they make mistakes AND when they arrive at the correct answer—communicates to them that you value their ideas, that math is about reasoning, and, most importantly, that math should make sense to them. Exploring reasoning is also extremely important for the teacher as a formative assessment tool. It helps her understand each student’s strengths and weaknesses, content knowledge, reasoning strategies and misconceptions.

Problems with multiple answers provide plenty of opportunities for students to reason numerically. It’s a chance to explore numbers and reasoning perhaps more creatively than if there was “one right answer.”

NUMBER SENSE EVERY DAY

All of these number sense activities contribute to your students’ abilities to solve problems. When children have daily, long-term opportunities to work (and play) with numbers, you will be continually amazed by the growth in their mathematical thinking, confidence, and enthusiasm about mathematics. By helping your children develop number sense, especially in the context of problem solving, you are helping them believe in themselves as mathematicians.

Number sense involves understanding numbers; knowing how to write and represent numbers in different ways; recognizing the quantity represented by numerals and other number forms; and discovering how a number relates to another number or group of numbers. Number sense develops gradually and varies as a result of exploring numbers, visualizing them in a variety of contexts, and relating to them in different ways.

In the primary and intermediate grades, number sense includes skills such as counting; representing numbers with manipulatives and models; understanding place value in the context of our base 10 number system; writing and recognizing numbers in different forms such as expanded, word, and standard; and expressing a number different ways—5 is "4 + 1" as well as "7 - 2," and 100 is 10 tens as well as 1 hundred. Number sense also includes the ability to compare and order numbers—whole numbers, fractions, decimals, and integers—and the ability to identify a number by an attribute—such as odd or even, prime or composite-or as a multiple or factor of another number. As students work with numbers, they gradually develop flexibility in thinking about numbers, which is a distinguishing characteristic of number sense.

WHY IS IT IMPORTANT?

Number sense enables students to understand and express quantities in their world. For example, whole numbers describe the number of students in a class or the number of days until a special event. Decimal quantities relate to money or metric measures, fractional amounts describing ingredient measures or time increments, negative quantities conveying temperatures below zero or depths below sea level, or percent amounts describing test scores or sale prices. Number sense is also the basis for understanding any mathematical operation and being able to estimate and make a meaningful interpretation of its result.

HOW CAN I MAKE IT HAPPEN?

In teaching number sense, using manipulatives and models (e.g., place-value blocks, fraction strips, decimal squares, number lines, and place-value and hundreds charts) helps students understand what numbers represent, different ways to express numbers, and how numbers relate to one another.

When students trade with place-value blocks they can demonstrate that the number 14 may be represented as 14 ones or as 1 ten and 4 ones. They can also demonstrate that 10 hundreds is the same as 1 thousand. By recording the number of each kind of block in the corresponding column (thousands, hundreds, tens, or ones) on a place-value chart, students practice writing numbers in standard form.

Using fraction strips, students find that 1/4 is less than 1/3 and that it names the same amount as 2/8. Using decimal squares, students see that 8 tenths can be written as 0.8 or 8/10. By pairing up counters to identify even numbers and marking these on a hundreds chart, primary-grade students discover that, beginning with 2, every other number is an even number. Intermediate-grade students can mark multiples of 3 and 6 on a hundreds chart and find that every number that has 6 as a factor also has 3 as a factor. Using a number line, students see how fractions with different denominators relate to the benchmark quantities of 0, 1/2, and 1. From these concrete experiences, students build the foundation for number sense they will bring to computation, estimation, measurement, problem solving, and all other areas of mathematics.

DAILY ROUTINES

Many teachers use the calendar as a source of mathematics activities. Students can work with counting, patterns, number sequence, odd and even numbers, and multiples of a number; they can also create word problems related to the calendar. A hundreds chart can help them count the number of days in school, and the current day’s number can be the "number of the day." Students can suggest various ways to make or describe that number. For example, on the 37th day of school, children may describe that number as 30 plus 7, 40 minus 3, an odd number, 15 plus 15 plus 7, my mother’s age, or 1 more than 3 dozen. The complexity of student’s responses will grow as the year goes on and as they listen to one another think mathematically. This is a great language building time.

THY THESE

- Model different methods for computing:

When a teacher publicly records a number of different approaches to solving a problem–solicited from the class or by introducing her own—it exposes students to strategies that they may not have considered. As Marilyn Burns explains, “When children think that there is one right way to compute, they focus on learning and applying it, rather than thinking about what makes sense for the numbers at hand.”

- Ask students regularly to calculate mentally:

Mental math encourages students to build on their knowledge about numbers and numerical relationships. When they cannot rely on memorized procedures or hold large quantities in their heads, students are forced to think more flexibly and efficiently, and to consider alternate problem solving strategies.

- Have class discussions about strategies for computing:

Classroom discussions about strategies help students to crystalize their own thinking while providing them the opportunity to critically evaluate their classmates’ approaches. In guiding the discussion, be sure to track ideas on the board to help students make connections between mathematical thinking and symbolic representation.

- Make estimation an integral part of computing.

Most of the math that we do every day—deciding when to leave for school, how much paint to buy, what type of tip to leave in a restaurant, which line to get in at the grocery store relies not only on mental math but estimations. However traditional textbook rounding exercises don’t provide the necessary context for students to understand estimating or build number sense. To do that, estimation must be embedded in problem situations.

- Question students about how they reason numerically.

Asking students about their reasoning—both when they make mistakes AND when they arrive at the correct answer—communicates to them that you value their ideas, that math is about reasoning, and, most importantly, that math should make sense to them. Exploring reasoning is also extremely important for the teacher as a formative assessment tool. It helps her understand each student’s strengths and weaknesses, content knowledge, reasoning strategies and misconceptions.

- Pose numerical problems that have more than one possible answer:

Problems with multiple answers provide plenty of opportunities for students to reason numerically. It’s a chance to explore numbers and reasoning perhaps more creatively than if there was “one right answer.”

NUMBER SENSE EVERY DAY

All of these number sense activities contribute to your students’ abilities to solve problems. When children have daily, long-term opportunities to work (and play) with numbers, you will be continually amazed by the growth in their mathematical thinking, confidence, and enthusiasm about mathematics. By helping your children develop number sense, especially in the context of problem solving, you are helping them believe in themselves as mathematicians.

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Welcome to my all thing special education blog. I empower busy elementary special education teachers to use best practice strategies to achieve a data and evidence driven classroom community by sharing easy to use, engaging, unique approaches to small group reading and math. Thanks for Hopping By.

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