## What is Number Sense?

September 28, 2014

A person's ability to use and understand numbers:

• knowing their relative values,

• how to use them to make judgments,

• how to use them in flexible ways when adding, subtracting, multiplying or dividing

• how to develop useful strategies when counting, measuring or estimating.

a person to understand numbers and number relationships and to solve mathematical problems that are not bound by traditional algorithms". The National Council of Teachers identified five components that characterize number sense: number meaning, number relationships, number magnitude, operations involving numbers and referents for numbers and quantities. These skills are considered important because they contribute to general intuitions about numbers and lay the foundation for more advanced skills.

Researchers have linked good number sense with skills observed in students proficient in the following mathematical activities:

Usually, when presented with more than five objects, other mental strategies must be utilized. For example, we might see a group of six objects as two groups of three. Each group of three is instantly recognized, then very quickly (virtually unconsciously) combined to make six. In this strategy no actual counting of objects is involved, but rather a part-part-whole relationship and rapid mental addition is used. That is, there is an understanding that a number (in this case six) can be composed of smaller parts, together with the knowledge that 'three plus three makes six'. This type of mathematical thinking has already begun by the time children begin school and should be nurtured because it lays the foundation for understanding operations and in developing valuable mental calculation strategies.

To begin with, early number activities are best done with movable objects such as counters, blocks and small toys. Most children will need the concrete experience of physically manipulating groups of objects into sub-groups and combining small groups to make a larger group. After these essential experiences more static materials such as 'dot cards' become very useful.

Dot cards are simply cards with dot stickers of a single color stuck on one side. (However, any markings can be used. Self-inking stamps are fast when making a lot of cards). The important factors in the design of the cards are the number of dots and the arrangement of these dots. The various combinations of these factors determine the mathematical structure of each card, and hence the types of number relations and mental strategies prompted by them.

Consider each of the following arrangements of dots before reading further. What mental strategies are likely to be prompted by each card? What order would you place them in according to level of difficulty?

Card A is the classic symmetrical dice and playing card arrangement of five and so is often instantly recognized without engaging other mental strategies. It is perhaps the easiest arrangement of five to deal with.

Card B presents clear sub-groups of two and three, each of which can be instantly recognized. With practice, the number fact of 'two and three makes five' can be recalled almost instantly.

Card C: A linear arrangement is the one most likely to prompt counting. However, many people will mentally separate the dots into groups of two and three, as in the previous card. Other strategies such as seeing two then counting '3, 4, 5' might also be used.

Card D could be called a random arrangement, though in reality it has been quite deliberately organized to prompt the mental activity of sub-grouping. There are a variety of ways to form the sub-groups, with no prompt in any particular direction, so this card could be considered to be the most difficult one in the set.

Card E shows another sub-group arrangement that encourages the use (or discovery) of the 'four and one makes five' number relation.

Obviously, using fewer than five dots would develop the most basic number sense skills, and using more than five dots would provide opportunities for more advanced strategies. However, it is probably not useful to use more than ten dots. (See the follow-on article focusing on developing a 'sense of ten' and 'place value readiness'). Cards such as these can be shown briefly to children, then the children asked how many dots they saw. The children should be asked to explain how they perceived the arrangement, and hence what strategies they employed.

Materials: 15 dot cards with a variety of dot patterns representing the numbers from one to five and a plentiful supply of counters or buttons.

Rules: One child deals out one card face up to each other player. Each child then uses the counters to replicate the arrangement of dots on his/her card and says the number aloud. The dealer checks each result, then deals out a new card to each player, placing it on top of the previous card. The children then rearrange their counters to match the new card. This continues until all the cards have been used.

Variations/Extensions

Each child can predict aloud whether the new card has more, less or the same number of dots as the previous card. The prediction is checked by the dealer, by observing whether counters need to be taken away or added. Increase the number of dots on the cards.

Materials: 12 dot cards, consisting of six pairs of cards showing two different arrangements of a particular number of dots, from 1 to 6 dots. (For example, a pair for 5 might be Card A and Card B from the set above).

Rules: Spread all the cards out face down. The first player turns over any two cards. If they are a pair (i.e. have the same number of dots), the player removes the cards and scores a point. If they are not a pair, both cards are turned back down in their places. The second player then turns over two cards and so on. When all the cards have been matched, the player with more pairs wins.

Variations/Extensions

Increase the number of pairs of cards used. Use a greater number of dots on the cards. Pair a dot card with a numeral card.

Materials: A pack of 20 to 30 dot cards (1 to 10 dots in dice and regular patterns), counters.

Rules: Spread out 10 cards face down and place the rest of the cards in a pile face down. The first player turns over the top pile card and places beside the pile. He/she then turns over one of the spread cards. The player works out the difference between the number of dots on each card, and takes that number of counters. (E.g. If one card showed 3 dots and the other 8, the player would take 5 counters.) The spread card is turned face down again in its place and the next player turns the top pile card and so on. Play continues until all the pile cards have been used. The winner is the player with the most counters; therefore the strategy is to remember the value of the spread cards so the one that gives the maximum difference can be chosen.

Variations/Extensions

Try to turn the spread cards that give the minimum difference, so the winner is the player with the fewest counters. Roll a die instead of using pile cards. Start with a set number of counters (say 20), so that when all the counters have been claimed the game ends. Use dot cards with random arrangements of dots.

Number Sense plays into how well order students grasp onto the more difficult concepts such as rounding, place value, and learning the basic math facts. Look for more information to come. Have a great week!

• knowing their relative values,

• how to use them to make judgments,

• how to use them in flexible ways when adding, subtracting, multiplying or dividing

• how to develop useful strategies when counting, measuring or estimating.

### What is number sense?

The term "number sense" is a relatively new one in mathematics education. It is difficult to define precisely, but broadly speaking, it refers to "a well-organized conceptual framework of number information that enablesa person to understand numbers and number relationships and to solve mathematical problems that are not bound by traditional algorithms". The National Council of Teachers identified five components that characterize number sense: number meaning, number relationships, number magnitude, operations involving numbers and referents for numbers and quantities. These skills are considered important because they contribute to general intuitions about numbers and lay the foundation for more advanced skills.

Researchers have linked good number sense with skills observed in students proficient in the following mathematical activities:

- mental calculation
- computational estimation
- judging the relative magnitude of numbers
- recognizing part-whole relationships and place value concepts and;
- problem solving

#### How does number sense begin?

An intuitive sense of number begins at a very early age. Children as young as two years of age can confidently identify one, two or three objects before they can actually count with understanding. Piaget called this ability to instantaneously recognize the number of objects in a small group 'subitizing'. As mental powers develop, usually by about the age of four, groups of four can be recognized without counting. It is thought that the maximum number for subitizing, even for most adults, is five. This skill appears to be based on the mind's ability to form stable mental images of patterns and associate them with a number. Therefore, it may be possible to recognize more than five objects if they are arranged in a particular way or practice and memorization takes place. A simple example of this is six dots arranged in two rows of three, as on dice or playing cards. Because this image is familiar, six can be instantly recognized when presented this way.Usually, when presented with more than five objects, other mental strategies must be utilized. For example, we might see a group of six objects as two groups of three. Each group of three is instantly recognized, then very quickly (virtually unconsciously) combined to make six. In this strategy no actual counting of objects is involved, but rather a part-part-whole relationship and rapid mental addition is used. That is, there is an understanding that a number (in this case six) can be composed of smaller parts, together with the knowledge that 'three plus three makes six'. This type of mathematical thinking has already begun by the time children begin school and should be nurtured because it lays the foundation for understanding operations and in developing valuable mental calculation strategies.

#### What teaching strategies promote early number sense?

Learning to count with understanding is a crucial number skill, but other skills, such as perceiving subgroups, need to develop alongside counting to provide a firm foundation for number sense. By simply presenting objects (such as stamps on a flashcard) in various arrangements, different mental strategies can be prompted. For example, showing six stamps in a cluster of four and a pair prompts the combination of 'four and two makes six'. If the four is not subitised, it may be seen as 'two and two and two makes six'. This arrangement is obviously a little more complex than two groups of three. So different arrangements will prompt different strategies, and these strategies will vary from person to person.
If mental strategies such as these are to be encouraged (and just counting discouraged) then an element of speed is necessary. Seeing the objects for only a few seconds challenges the mind to find strategies other than counting. It is also important to have children reflect on and share their strategies. This is helpful in three ways:

- verbalizing a strategy brings the strategy to a conscious level and allows the person to learn about their own thinking;
- it provides other children with the opportunity to pick up new strategies;
- the teacher can assess the type of thinking being used and adjust the type of arrangement, level of difficulty or speed of presentation accordingly.

To begin with, early number activities are best done with movable objects such as counters, blocks and small toys. Most children will need the concrete experience of physically manipulating groups of objects into sub-groups and combining small groups to make a larger group. After these essential experiences more static materials such as 'dot cards' become very useful.

Dot cards are simply cards with dot stickers of a single color stuck on one side. (However, any markings can be used. Self-inking stamps are fast when making a lot of cards). The important factors in the design of the cards are the number of dots and the arrangement of these dots. The various combinations of these factors determine the mathematical structure of each card, and hence the types of number relations and mental strategies prompted by them.

Consider each of the following arrangements of dots before reading further. What mental strategies are likely to be prompted by each card? What order would you place them in according to level of difficulty?

Card A is the classic symmetrical dice and playing card arrangement of five and so is often instantly recognized without engaging other mental strategies. It is perhaps the easiest arrangement of five to deal with.

Card B presents clear sub-groups of two and three, each of which can be instantly recognized. With practice, the number fact of 'two and three makes five' can be recalled almost instantly.

Card C: A linear arrangement is the one most likely to prompt counting. However, many people will mentally separate the dots into groups of two and three, as in the previous card. Other strategies such as seeing two then counting '3, 4, 5' might also be used.

Card D could be called a random arrangement, though in reality it has been quite deliberately organized to prompt the mental activity of sub-grouping. There are a variety of ways to form the sub-groups, with no prompt in any particular direction, so this card could be considered to be the most difficult one in the set.

Card E shows another sub-group arrangement that encourages the use (or discovery) of the 'four and one makes five' number relation.

Obviously, using fewer than five dots would develop the most basic number sense skills, and using more than five dots would provide opportunities for more advanced strategies. However, it is probably not useful to use more than ten dots. (See the follow-on article focusing on developing a 'sense of ten' and 'place value readiness'). Cards such as these can be shown briefly to children, then the children asked how many dots they saw. The children should be asked to explain how they perceived the arrangement, and hence what strategies they employed.

#### What games can assist development of early number sense?

Games can be very useful for reinforcing and developing ideas and procedures previously introduced to children. Although a suggested age group is given for each of the following games, it is the children's level of experience that should determine the suitability of the game. Several demonstration games should be played, until the children become comfortable with the rules and procedures of the games.__Deal and Copy (4-5 years) 3-4 players__Materials: 15 dot cards with a variety of dot patterns representing the numbers from one to five and a plentiful supply of counters or buttons.

Rules: One child deals out one card face up to each other player. Each child then uses the counters to replicate the arrangement of dots on his/her card and says the number aloud. The dealer checks each result, then deals out a new card to each player, placing it on top of the previous card. The children then rearrange their counters to match the new card. This continues until all the cards have been used.

Variations/Extensions

Each child can predict aloud whether the new card has more, less or the same number of dots as the previous card. The prediction is checked by the dealer, by observing whether counters need to be taken away or added. Increase the number of dots on the cards.

__Memory Match (5-7 years) 2 players__Materials: 12 dot cards, consisting of six pairs of cards showing two different arrangements of a particular number of dots, from 1 to 6 dots. (For example, a pair for 5 might be Card A and Card B from the set above).

Rules: Spread all the cards out face down. The first player turns over any two cards. If they are a pair (i.e. have the same number of dots), the player removes the cards and scores a point. If they are not a pair, both cards are turned back down in their places. The second player then turns over two cards and so on. When all the cards have been matched, the player with more pairs wins.

Variations/Extensions

Increase the number of pairs of cards used. Use a greater number of dots on the cards. Pair a dot card with a numeral card.

__What's the Difference? (7-8 years) 2-4 players__Materials: A pack of 20 to 30 dot cards (1 to 10 dots in dice and regular patterns), counters.

Rules: Spread out 10 cards face down and place the rest of the cards in a pile face down. The first player turns over the top pile card and places beside the pile. He/she then turns over one of the spread cards. The player works out the difference between the number of dots on each card, and takes that number of counters. (E.g. If one card showed 3 dots and the other 8, the player would take 5 counters.) The spread card is turned face down again in its place and the next player turns the top pile card and so on. Play continues until all the pile cards have been used. The winner is the player with the most counters; therefore the strategy is to remember the value of the spread cards so the one that gives the maximum difference can be chosen.

Variations/Extensions

Try to turn the spread cards that give the minimum difference, so the winner is the player with the fewest counters. Roll a die instead of using pile cards. Start with a set number of counters (say 20), so that when all the counters have been claimed the game ends. Use dot cards with random arrangements of dots.

Number Sense plays into how well order students grasp onto the more difficult concepts such as rounding, place value, and learning the basic math facts. Look for more information to come. Have a great week!

Labels:common core,lesson plan,math | 0
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## Math Big Ideas

September 19, 2014

Concrete - Representational - Abstract: Sequence of Instruction

The purpose of teaching through a concrete-to-representational-to-abstract sequence of instruction is to ensure students truly have a thorough understanding of the math concepts/skills they are learning. When students who have math learning problems are allowed to first develop a concrete understanding of the math concept/skill, then they are much more likely to perform that math skill and truly understand math concepts at the abstract level.

What is it?

What are the critical elements of this strategy?

When initially teaching a math concept/skill, describe & model it using concrete objects (concrete level of understanding).

How Does This Instructional Strategy Positively Impact Students Who Have Learning Problems?

The concrete level of understanding is the most basic level of mathematical understanding. It is also the most crucial level for developing conceptual understanding of math concepts/skills. Concrete learning occurs when students have ample opportunities to manipulate concrete objects to problem-solve. For students who have math learning problems, explicit teacher modeling of the use of specific concrete objects to solve specific math problems is needed.

Understanding manipulatives (concrete objects)

To use math manipulatives effectively, it is important that you understand several basic characteristics of different types of math manipulatives and how these specific characteristics impact students who have learning problems. As you read about the different types of manipulatives, click on the numbers beside each description to view pictures of these different types of manipulatives.

General types of math manipulatives:

Discrete - those materials that can be counted (e.g. cookies, children, counting blocks, toy cars, etc.).

Continuous - materials that are not used for counting but are used for measurement (e.g. ruler, measuring cup, weight scale, trundle wheel). See example - 1

Suggestions for using Discrete & Continuous materials with students who have learning problems:

Students who have learning problems need to have abundant experiences using discrete materials before they will benefit from the use of continuous materials. This is because discrete materials have defining characteristics that students can easily discriminate through sight and touch. As students master an understanding of specific readiness concepts for specific measurement concepts/skills through the use of discrete materials (e.g. counting skills), then continuous materials can be used.

Types of manipulatives used to teach the Base-10 System/place-value (Smith, 1997):

Proportional - show relationships by size (e.g. ten counting blocks grouped together is ten times the size of one counting block; a beanstick with ten beans glued to a popsicle stick is ten times bigger than one bean).

Non-linked proportional - single units are independent of each other, but can be "bundled together (e.g. popsicle sticks can be "bundled together in groups of 'tens' with rubber bands; individual unifix cubes can be attached in rows of ten unifix cubes each).

Linked proportional - comes in single units as well as "already bundled" tens units, hundreds units, & thousands units (e.g. base ten cubes/blocks; beans & beansticks).

Non-proportional - use units where size is not indicative of value while other characteristics indicate value (e.g. money, where one dime is worth ten times the value of one penny; poker chips where color indicates value of chip; an abacus where location of the row indicates value). A specified number of units representing one value are exchanged for one unit of greater value (e.g. ten pennies for one dime; ten white poker chips for one blue poker chip, ten beads in the first row of an abacus for one bead in the second row). See example - 1

Suggestions for using proportional and non-proportional manipulatives with students who have learning problems:

Students who have learning problems are more likely to learn place value when using proportional manipulatives because differences between ones units, tens units, & hundreds units are easy to see and feel. Due to the very nature of non-proportional manipulatives, students who have learning problems have more difficulty seeing and feeling the differences in unit values.

Examples of manipulatives (concrete objects)

Suggested manipulatives are listed according to math concept/skill area. Descriptions of manipulatives are provided as appropriate. A brief description of how each set of manipulatives may be used to teach the math concept/skill is provided at the bottom of the list for each math concept area. Picture examples of some of the manipulatives for each math concept area can be accessed by clicking on the numbers found underneath the title of each math concept area. This is not meant to be an exhaustive list, but this list does include a variety of common manipulatives. The list includes examples of "teacher-made" manipulatives as well "commercially-made" ones.

Colored chips

Beans

Unifix cubes

Golf tees

Skittles or other candy pieces

Packaging popcorn

Popsicle sticks/tongue depressors

Description of use: Students can use these concrete materials to count, to add, and to subtract. Students can count by pointing to objects and counting aloud. Students can add by counting objects, putting them in one group and then counting the total. Students can subtract by removing objects from a group and then counting how many are left.

Base 10 cubes/blocks

Beans and bean sticks

Popsicle sticks & rubber bands for bundling

Unifix cubes (individual cubes can be combined to represent "tens")

Place value mat (a piece of tag board or other surface that has columns representing the "ones," "tens," and "hundreds" place values)

Description of use: Students are first taught to represent 1-9 objects in the "ones" column. They are then taught to represent "10" by trading in ten single counting objects for one object that contains the ten counting objects on it (e.g. ten separate beans are traded in for one "beanstick" - a popsicle stick with ten beans glued on one side. Students then begin representing different values 1-99. At this point, students repeat the same trading process for "hundreds."

Containers & counting objects (paper dessert plates & beans, paper or plastic cups and candy pieces, playing cards & chips, cutout tag board circles & golf tees, etc.). Containers represent the "groups" and counting objects represent the number of objects in each group. (e.g. 2 x 4 = 8: two containers with four counting objects on each container)

Counting objects arranged in arrays (arranged in rows and columns). Color-code the "outside" vertical column and horizontal row helps emphasize the multipliers

.

Counting objects, one set light colored and one set dark colored (e.g. light & dark colored beans; yellow & blue counting chips; circles cut out of tag board with one side colored, etc.).

Description of use: Light colored objects represent positive integers and dark colored objects represent negative integers. When adding positive and negative integers, the student matches pairs of dark and light colored objects. The color and number of objects remaining represent the solution.

Fraction pieces (circles, half-circles, quarter-circles, etc.)

Fraction strips (strips of tag board one foot in length and one inch wide, divided into wholes, ½'s, 1/3's, ¼'s, etc.

Fraction blocks or stacks. Blocks/cubes that represent fractional parts by proportion (e.g. a "1/2" block is twice the height as a "1/4" block).

Description of use: Teacher models how to compare fractional parts using one type of manipulative. Students then compare fractional parts. As students gain understanding of fractional parts and their relationships with a variety of manipulatives, teacher models and then students begin to add, subtract, multiply, and divide using fraction pieces.

Geoboards (square platforms that have raised notches or rods that are formed in a array). Rubber bands or string can be used to form various shapes around the raised notches or rods.

Description of use: Concepts such as area and perimeter can be demonstrated by counting the number of notch or rod "units" inside the shape or around the perimeter of the shape.

Containers (representing the variable of "unknown") and counting objects (representing integers) -e.g. paper dessert plates & beans, small clear plastic beverage cups 7 counting chips, playing cards & candy pieces, etc.

Description of use: The algebraic expression, "4x = 8," can be represented with four plates ("4x"). Eight beans can be distributed evenly among the four plates. The number of beans on one plate represent the solution ("x" = 2).

Suggestions for using manipulatives:

At the representational level of understanding, students learn to problem-solve by drawing pictures. The pictures students draw represent the concrete objects students manipulated when problem-solving at the concrete level. It is appropriate for students to begin drawing solutions to problems as soon as they demonstrate they have mastered a particular math concept/skill at the concrete level. While not all students need to draw solutions to problems before moving from a concrete level of understanding to an abstract level of understanding, students who have learning problems in particular typically need practice solving problems through drawing. When they learn to draw solutions, students are provided an intermediate step where they begin transferring their concrete understanding toward an abstract level of understanding. When students learn to draw solutions, they gain the ability to solve problems independently. Through multiple independent problem-solving practice opportunities, students gain confidence as they experience success. Multiple practice opportunities also assist students to begin to "internalize" the particular problem-solving process. Additionally, students' concrete understanding of the concept/skill is reinforced because of the similarity of their drawings to the manipulatives they used previously at the concrete level.

Drawing is not a "crutch" for students that they will use forever. It simply provides students an effective way to practice problem solving independently until they develop fluency at the abstract level.

Examples of drawing solutions by math concept level

The following drawing examples are categorized by the type of drawings ("Lines, Tallies, & Circles," or "Circles/Boxes"). In each category there are a variety of examples demonstrating how to use these drawings to solve different types of computation problems. Click on the numbers below to view these examples.

A student who problem-solves at the abstract level, does so without the use of concrete objects or without drawing pictures. Understanding math concepts and performing math skills at the abstract level requires students to do this with numbers and math symbols only. Abstract understanding is often referred to as, "doing math in your head." Completing math problems where math problems are written and students solve these problems using paper and pencil is a common example of abstract level problem solving.

Potential barriers to abstract understanding for students who have learning problems and how to manage these barriers

Students who are not successful solving problems at the abstract level may:

Re-teach the concept/skill at the concrete level using appropriate concrete objects (see Concrete Level of Understanding).

Re-teach concept/skill at representational level and provide opportunities for student to practice concept/skill by drawing solutions (see Representational Level of Understanding).

Provide opportunities for students to use language to explain their solutions and how they got them (see instructional strategy Structured Language Experiences).

- Have difficulty with basic facts/memory problems

Suggestions:

The purpose of teaching through a concrete-to-representational-to-abstract sequence of instruction is to ensure students truly have a thorough understanding of the math concepts/skills they are learning. When students who have math learning problems are allowed to first develop a concrete understanding of the math concept/skill, then they are much more likely to perform that math skill and truly understand math concepts at the abstract level.

What is it?

- Each math concept/skill is first modeled with concrete materials (e.g. chips, unifix cubes, base ten blocks, beans and bean sticks, pattern blocks).
- Students are provided many opportunities to practice and demonstrate mastery using concrete materials
- The math concept/skill is next modeled at the representational (semi-concrete) level which involves drawing pictures that represent the concrete objects previously used (e.g. tallies, dots, circles, stamps that imprint pictures for counting)
- Students are provided many opportunities to practice and demonstrate mastery by drawing solutions
- The math concept/skill is finally modeled at the abstract level (using only numbers and mathematical symbols)
- Students are provided many opportunities to practice and demonstrate mastery at the abstract level before moving to a new math concept/skill.

What are the critical elements of this strategy?

- Use appropriate concrete objects to teach particular math concept/skill (see Concrete Level of Understanding/Understanding Manipulatives-Examples of manipulatives by math concept area). Teach concrete understanding first.
- Use appropriate drawing techniques or appropriate picture representations of concrete objects (see Representational Level of Understanding/Examples of drawing solutions by math concept area). Teach representational understanding second.
- Use appropriate strategies for assisting students to move to the abstract level of understanding for a particular math concept/skill (see Abstract Level of Understanding/Potential barriers to abstract understanding for students who have learning problems and how to manage these barriers).

When initially teaching a math concept/skill, describe & model it using concrete objects (concrete level of understanding).

- Provide students many practice opportunities using concrete objects.
- When students demonstrate mastery of skill by using concrete objects, describe & model how to perform the skill by drawing or with pictures that represent concrete objects (representational level of understanding).
- Provide many practice opportunities where students draw their solutions or use pictures to problem-solve.
- When students demonstrate mastery drawing solutions, describe and model how to perform the skill using only numbers and math symbols (abstract level of understanding).
- Provide many opportunities for students to practice performing the skill using only numbers and symbols.

How Does This Instructional Strategy Positively Impact Students Who Have Learning Problems?

- Helps passive learner to make meaningful connections
- Teaches conceptual understanding by connecting concrete understanding to abstract math process
- By linking learning experiences from concrete-to-representational-to-abstract levels of understanding, the teacher provides a graduated framework for students to make meaningful connections.
- Blends conceptual and procedural understanding in structured way

#### Concrete

What is it?The concrete level of understanding is the most basic level of mathematical understanding. It is also the most crucial level for developing conceptual understanding of math concepts/skills. Concrete learning occurs when students have ample opportunities to manipulate concrete objects to problem-solve. For students who have math learning problems, explicit teacher modeling of the use of specific concrete objects to solve specific math problems is needed.

Understanding manipulatives (concrete objects)

To use math manipulatives effectively, it is important that you understand several basic characteristics of different types of math manipulatives and how these specific characteristics impact students who have learning problems. As you read about the different types of manipulatives, click on the numbers beside each description to view pictures of these different types of manipulatives.

General types of math manipulatives:

Discrete - those materials that can be counted (e.g. cookies, children, counting blocks, toy cars, etc.).

Continuous - materials that are not used for counting but are used for measurement (e.g. ruler, measuring cup, weight scale, trundle wheel). See example - 1

Suggestions for using Discrete & Continuous materials with students who have learning problems:

Students who have learning problems need to have abundant experiences using discrete materials before they will benefit from the use of continuous materials. This is because discrete materials have defining characteristics that students can easily discriminate through sight and touch. As students master an understanding of specific readiness concepts for specific measurement concepts/skills through the use of discrete materials (e.g. counting skills), then continuous materials can be used.

Types of manipulatives used to teach the Base-10 System/place-value (Smith, 1997):

Proportional - show relationships by size (e.g. ten counting blocks grouped together is ten times the size of one counting block; a beanstick with ten beans glued to a popsicle stick is ten times bigger than one bean).

Non-linked proportional - single units are independent of each other, but can be "bundled together (e.g. popsicle sticks can be "bundled together in groups of 'tens' with rubber bands; individual unifix cubes can be attached in rows of ten unifix cubes each).

Linked proportional - comes in single units as well as "already bundled" tens units, hundreds units, & thousands units (e.g. base ten cubes/blocks; beans & beansticks).

Non-proportional - use units where size is not indicative of value while other characteristics indicate value (e.g. money, where one dime is worth ten times the value of one penny; poker chips where color indicates value of chip; an abacus where location of the row indicates value). A specified number of units representing one value are exchanged for one unit of greater value (e.g. ten pennies for one dime; ten white poker chips for one blue poker chip, ten beads in the first row of an abacus for one bead in the second row). See example - 1

Suggestions for using proportional and non-proportional manipulatives with students who have learning problems:

Students who have learning problems are more likely to learn place value when using proportional manipulatives because differences between ones units, tens units, & hundreds units are easy to see and feel. Due to the very nature of non-proportional manipulatives, students who have learning problems have more difficulty seeing and feeling the differences in unit values.

Examples of manipulatives (concrete objects)

Suggested manipulatives are listed according to math concept/skill area. Descriptions of manipulatives are provided as appropriate. A brief description of how each set of manipulatives may be used to teach the math concept/skill is provided at the bottom of the list for each math concept area. Picture examples of some of the manipulatives for each math concept area can be accessed by clicking on the numbers found underneath the title of each math concept area. This is not meant to be an exhaustive list, but this list does include a variety of common manipulatives. The list includes examples of "teacher-made" manipulatives as well "commercially-made" ones.

__Counting/Basic Addition & Subtraction Pictures__Colored chips

Beans

Unifix cubes

Golf tees

Skittles or other candy pieces

Packaging popcorn

Popsicle sticks/tongue depressors

Description of use: Students can use these concrete materials to count, to add, and to subtract. Students can count by pointing to objects and counting aloud. Students can add by counting objects, putting them in one group and then counting the total. Students can subtract by removing objects from a group and then counting how many are left.

__Place Value Pictures__Base 10 cubes/blocks

Beans and bean sticks

Popsicle sticks & rubber bands for bundling

Unifix cubes (individual cubes can be combined to represent "tens")

Place value mat (a piece of tag board or other surface that has columns representing the "ones," "tens," and "hundreds" place values)

Description of use: Students are first taught to represent 1-9 objects in the "ones" column. They are then taught to represent "10" by trading in ten single counting objects for one object that contains the ten counting objects on it (e.g. ten separate beans are traded in for one "beanstick" - a popsicle stick with ten beans glued on one side. Students then begin representing different values 1-99. At this point, students repeat the same trading process for "hundreds."

__Multiplication/Division Pictures__Containers & counting objects (paper dessert plates & beans, paper or plastic cups and candy pieces, playing cards & chips, cutout tag board circles & golf tees, etc.). Containers represent the "groups" and counting objects represent the number of objects in each group. (e.g. 2 x 4 = 8: two containers with four counting objects on each container)

Counting objects arranged in arrays (arranged in rows and columns). Color-code the "outside" vertical column and horizontal row helps emphasize the multipliers

.

__Positive & Negative Integers Pictures__Counting objects, one set light colored and one set dark colored (e.g. light & dark colored beans; yellow & blue counting chips; circles cut out of tag board with one side colored, etc.).

Description of use: Light colored objects represent positive integers and dark colored objects represent negative integers. When adding positive and negative integers, the student matches pairs of dark and light colored objects. The color and number of objects remaining represent the solution.

__Fractions Pictures__Fraction pieces (circles, half-circles, quarter-circles, etc.)

Fraction strips (strips of tag board one foot in length and one inch wide, divided into wholes, ½'s, 1/3's, ¼'s, etc.

Fraction blocks or stacks. Blocks/cubes that represent fractional parts by proportion (e.g. a "1/2" block is twice the height as a "1/4" block).

Description of use: Teacher models how to compare fractional parts using one type of manipulative. Students then compare fractional parts. As students gain understanding of fractional parts and their relationships with a variety of manipulatives, teacher models and then students begin to add, subtract, multiply, and divide using fraction pieces.

__Geometry Pictures__Geoboards (square platforms that have raised notches or rods that are formed in a array). Rubber bands or string can be used to form various shapes around the raised notches or rods.

Description of use: Concepts such as area and perimeter can be demonstrated by counting the number of notch or rod "units" inside the shape or around the perimeter of the shape.

__Beginning Algebra Pictures__Containers (representing the variable of "unknown") and counting objects (representing integers) -e.g. paper dessert plates & beans, small clear plastic beverage cups 7 counting chips, playing cards & candy pieces, etc.

Description of use: The algebraic expression, "4x = 8," can be represented with four plates ("4x"). Eight beans can be distributed evenly among the four plates. The number of beans on one plate represent the solution ("x" = 2).

Suggestions for using manipulatives:

- Talk with your students about how manipulatives help to learn math.
- Set ground rules for using manipulatives.
- Develop a system for storing manipulatives.
- Allow time for your students to explore manipulatives before beginning instruction.
- Encourage students to learn names of the manipulatives they use.
- Provide students time to describe the manipulatives they use orally or in writing. Model this as appropriate.
- Introduce manipulatives to parents

#### Representational

What is it?At the representational level of understanding, students learn to problem-solve by drawing pictures. The pictures students draw represent the concrete objects students manipulated when problem-solving at the concrete level. It is appropriate for students to begin drawing solutions to problems as soon as they demonstrate they have mastered a particular math concept/skill at the concrete level. While not all students need to draw solutions to problems before moving from a concrete level of understanding to an abstract level of understanding, students who have learning problems in particular typically need practice solving problems through drawing. When they learn to draw solutions, students are provided an intermediate step where they begin transferring their concrete understanding toward an abstract level of understanding. When students learn to draw solutions, they gain the ability to solve problems independently. Through multiple independent problem-solving practice opportunities, students gain confidence as they experience success. Multiple practice opportunities also assist students to begin to "internalize" the particular problem-solving process. Additionally, students' concrete understanding of the concept/skill is reinforced because of the similarity of their drawings to the manipulatives they used previously at the concrete level.

Drawing is not a "crutch" for students that they will use forever. It simply provides students an effective way to practice problem solving independently until they develop fluency at the abstract level.

Examples of drawing solutions by math concept level

The following drawing examples are categorized by the type of drawings ("Lines, Tallies, & Circles," or "Circles/Boxes"). In each category there are a variety of examples demonstrating how to use these drawings to solve different types of computation problems. Click on the numbers below to view these examples.

#### Abstract

What is it?A student who problem-solves at the abstract level, does so without the use of concrete objects or without drawing pictures. Understanding math concepts and performing math skills at the abstract level requires students to do this with numbers and math symbols only. Abstract understanding is often referred to as, "doing math in your head." Completing math problems where math problems are written and students solve these problems using paper and pencil is a common example of abstract level problem solving.

Potential barriers to abstract understanding for students who have learning problems and how to manage these barriers

Students who are not successful solving problems at the abstract level may:

- Not understand the concept behind the skill

Re-teach the concept/skill at the concrete level using appropriate concrete objects (see Concrete Level of Understanding).

Re-teach concept/skill at representational level and provide opportunities for student to practice concept/skill by drawing solutions (see Representational Level of Understanding).

Provide opportunities for students to use language to explain their solutions and how they got them (see instructional strategy Structured Language Experiences).

- Have difficulty with basic facts/memory problems

Suggestions:

- Regularly provide student with a variety of practice activities focusing on basic facts. Facilitate independent practice by encouraging students to draw solutions when needed (see the student practice strategies Instructional Games, Self-correcting Materials, Structured Cooperative Learning Groups, and Structured Peer Tutoring).
- Conduct regular one-minute timings and chart student performance. Set goals with student and frequently review chart with student to emphasize progress. Focus on particular fact families that are most problematic first, then slowly incorporate a variety of facts as the student demonstrates competence (see evaluation strategy Continuous Monitoring & Charting of Student Performance).
- Teach student regular patterns that occur throughout addition, subtraction, multiplication, & division facts (e.g. "doubles" in multiplication, 9's rule - add 10 & subtract one, etc.)
- Provide student a calculator or table when they are solving multiple-step problems.

__Repeat procedural mistakes____Suggestions:__

- Provide fewer #'s of problems per page.
- Provide fewer numbers of problems when assigning paper & pencil practice/homework.
- Provide ample space for student writing, cueing, & drawing.
- Provide problems that are already written on learning sheets rather than requiring students to copy problems from board or textbook.
- Provide structure: turn lined paper sideways to create straight columns; allow student to use dry-erase boards/lap chalkboards that allow mistakes to be wiped away cleanly; color cue symbols; for multi-step problems, draw color-cued lines that signal students where to write and what operation to use; provide boxes that represent where numerals should be placed; provide visual directional cues in a sample problem; provide a sample problem, completed step by step at top of learning sheet.
- Provide strategy cue cards that student can use to recall the correct procedure for solving problem.
- Provide a variety of practice activities that require modes of expression other than only writing

#### The Big Idea

Student learning & mastery greatly depends on the number of opportunities a student has to respond!! The more opportunities for successful practice that you provide (i.e. practice that doesn't negatively impact student learning characteristics), the more likely it is that your student will develop mastery of that skill.
Labels:math,teaching | 0
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## Building A School Culture

September 03, 2014

It's funny as teachers we build culture in our classroom and it becomes "the way things are done." We hard to build it so that everyone soon becomes a family. We never give a second thought to what our School Building Culture is. We may think we know what it is such as warm and welcoming; everyone shares ideas; everyone has an open door. But is it really.? I think about School Culture the same way I do in my class.

I would like to offer here an approach to learning where the locus of classroom standards is centered neither in the teacher nor the individual student, but rather in a

Culture is a powerful concept. It goes deeper than what is spoken, deeper, even, than what is consciously understood Students in my classroom probably couldn't articulate exactly why they try so hard at their work and probably haven't even stopped to fully analyze it. Similarly, as members of a larger culture, we rarely stop to think about how many of our personal attitudes and actions are simply reflections of cultural norms. Culture defines how people function, and to some extent, even how they think. If a notion of high standards is not simply included in classroom culture but is actually at the core of this culture, then high standards become the norm.

This is all fine in the abstract, but a picture and understanding of how this culture looks in practice is needed. Painting this picture is, for me, a difficult task. A visit to my classroom in action, where I can often leave the room for twenty minutes and students don't even notice, as they are gripped in project work, would be much more compelling and persuasive than my words here could hope to be. Even this, however, would be inadequate, as it would fail to disclose how the environment was built, and what unspoken norms define its social structure.

Schools can sometimes take on the feel of a production shop, students cranking out an endless flow of final products without much personal investment or care. The emphasis is on keeping up with production, not falling behind in classwork or homework, rather than in producing something of lasting value. Like a fast food restaurant, the products are neither creative nor memorable. Teachers create and fuel this situation, despite the fact that we grow tired of repetitive, trivial assignments, and dread correcting piles of such work.

Turning in final drafts of work every day, often many times in one day, even the most ambitious of students must compromise standards continually simply to keep up with the pace. Internalized high standards are no defense against a system which demands final draft work at this rate. If an adult writer, scientist, historian, visual artist, was asked to turn in a finished piece of work every day, two or three finished pieces on some days, how much care could he or she put into each?

An alternative is a project-centered approach. In this structure, students still work hard every day, but their work is instead a small part of a long-range, significant project. Daily work entails the creation or revision of early drafts of a piece, the continued research of a topic or management of an experiment, or the perfection of one component of a large piece of work. Final drafts or presentations of completed projects are no longer trivial events occurring every day, but are special events, moments of individual and class pride and celebration.

Universal success does not mean uniformity. Though the structure which braces and guides student progress is common to every project, each student's project is unique. The structure provides a frame for common learning and critique, as well as appraisal of progress, but it also has room for significant creative expression and direction by individual students. If every student in a classroom prepares a guidebook to a different local building, the steps and skills involved may be somewhat prescribed: conducting interviews, researching local history, consulting city records, trying to obtain blueprints, doing sketches, taking photographs, preparing diagrams, writing and proofreading drafts of text, preparing illustrations, composing book layout, and learning book binding. Within this frame though, individual students have substantial latitude for artistic choices: the choice of building, the choice of whom to interview, the use of research and interviews, the nature of text and illustrations, the balance of text and illustrations, the use of photographs or diagrams, the tone of presentation, and the layout of the finished book.

My comments thus far may have raised a lot of questions. What about "untalented" students? Or teachers, for that matter. People who feel incompetent in particular academic or artistic areas. What about students with pronounced disabilities in perception, memory, or motor skills? If high standards are applied to the work of everyone, won't it just emphasize the painful weaknesses of some?

What about assessment? If so much of project work is creative and individualized, and even deadlines and requirements often differ from student to student, how can fair and constructive assessment take place?

What about a sense of community? If you work to dissolve the boundaries between school and outside life, don't you destroy the precious asylum of school which provides safety and security to children?

What about the cultivation of positive personal qualities in students: politeness, thoughtfulness, cooperation, initiative, self confidence, equanimity. Environments of "high standards" are often high-stress; a high standards classroom often means fighting for teacher recognition.

I'd like to say that I, and my school, have found the definitive answers to all these questions. Of course, we haven't. We are forever confused with trying to fashion a schoolwide assessment system which fits our teaching approach and documents student progress for the community or district in a real way. We are just now undertaking an effort to begin using student portfolios as archives of individual projects and achievement, as many schools have already done. And we are continually trying to restructure our school resources and community traditions to make the school more supportive for students with disabilities, insecurities, or other factors which make them "marginal".

Nevertheless, I think there is much going on in my classroom and school that addresses these questions in powerful ways. Despite having a full range of "academic abilities" in my students, and even students of real special needs, visitors to the classroom are often bemused and skeptical that this is a "regular" group of students. The work on display looks too impressive, the focus, cooperation, investment, and friendly ease of children too good to be true. The answer, I believe, is that students in my classroom are deeply and genuinely supported in countless ways to do their best and act their best, and have been so since Kindergarten. To a great extent, this is due to the ways in which the school culture deals with issues of assistance and assessment for students, and also for teachers.

People often say to me that this learning approach works in my classroom because I am a strong teacher, or works in the school because the school has strong teachers. In fact, they say, it's importance is trivial, because it can only work with strong teachers.

First, it is suspiciously similar to that of teachers who visit my room and tell me: "Oh, but

Second, on a classroom level, saying that this approach needs a strong teacher may be true, but it doesn't begin to reveal why the approach works. The power in the approach it is that it is based in a classroom community which shares a culture. The assessment, encouragement, accountability and teaching which goes on in the classroom is vastly wider than that which emanates from me, as teacher it is a continual and ubiquitous process among students. When I think of a particular student I had last year, a painfully shy girl who was new to town, I would be crazy to take credit for her astonishing emotional and academic growth-- I could hardly get her to speak to me during the year. I take credit only for managing a classroom where countless students took time to tutor her, support her, welcome her, and guide her success.

Again, where do student standards originate? Partly from me as the teacher, but very much from watching each other. Students did quality work, treated each other well, because that is what their friends did, it was the "cool" way to be, it was what the culture supported. How do students decide what is "good enough" as work, what is good enough behavior? Teacher standards only define the minimum standards allowed, they can't define the upper limit of care. Students in my classroom look to each other, help each other, critique each other's work, most of all, push each other to achieve their best. For all of my teacher power, the power in this total culture is infinitely greater.

The culture is rooted in a class community in which the first priority is supporting everyone. The responsibility for each person's emotional well-being and success is shared by everyone; the teacher bears more of this responsibility, but each student is encouraged and expected to help his or her peers. As a teacher, this means setting up a classroom structure which allows time and space for peer collaboration and tutoring; it means complimenting students, rather than complaining, when they abandon their own work at times to assist others. Supporting others means emotional support and care as much as it means academic support and care. In this team concept, the hope is that no student will be left out, left behind, allowed to fail or feel like a failure. It's everybody's job to look out for others.

For me, a culture of high standards means high standards for kindness and cooperation as much as for academic work. In the same way that careful quality in work is stressed over fast production, careful attention to treating all students fairly and thoughtfully is stressed over efficiency and speed in school logistics. "Simple" classroom decisions, such as which students should make a presentation or attend a limited event, often take a long time as they are discussed carefully with students to insure that all feelings are considered. Events and honors which are exclusionary or individualistic, displays of the "best" work or awards for the "best" students or athletes, are avoided in favor of whole-class, whole-school pride. When visitors to my classroom are impressed with student work, it is often due not to specific outstanding examples but rather to the absence of careless work, the uniform commitment to quality. This is a testament to the degree of cultural pride and peer support in the classroom.

My teachers in elementary school often instructed us to try to do your best". This isn't a bad motto; I'd use it with my class today, and most schools would embrace it without a thought. There's a big step between teachers saying this to students, and students actually doing it. Not too many schools seriously look at what aspects of their structure and culture support and compel students to do their best, to act their best, and what aspects undermine this spirit. Rather than searching for individual teachers or principals who they hope can demand high standards, I feel that schools should be looking at how they can create a spirit of high standards, a school culture of high standards.

What's your building culture like? Have a great week! Say cool:)

I would like to offer here an approach to learning where the locus of classroom standards is centered neither in the teacher nor the individual student, but rather in a

*classroom culture*, imbedded in a consonant*school culture*. This classroom culture contains the teacher and each individual student, the peer groups of students, but also transcends them: it is a framework which governs the learning and social interaction of all classroom members, and builds norms for a new peer culture.Culture is a powerful concept. It goes deeper than what is spoken, deeper, even, than what is consciously understood Students in my classroom probably couldn't articulate exactly why they try so hard at their work and probably haven't even stopped to fully analyze it. Similarly, as members of a larger culture, we rarely stop to think about how many of our personal attitudes and actions are simply reflections of cultural norms. Culture defines how people function, and to some extent, even how they think. If a notion of high standards is not simply included in classroom culture but is actually at the core of this culture, then high standards become the norm.

This is all fine in the abstract, but a picture and understanding of how this culture looks in practice is needed. Painting this picture is, for me, a difficult task. A visit to my classroom in action, where I can often leave the room for twenty minutes and students don't even notice, as they are gripped in project work, would be much more compelling and persuasive than my words here could hope to be. Even this, however, would be inadequate, as it would fail to disclose how the environment was built, and what unspoken norms define its social structure.

Schools can sometimes take on the feel of a production shop, students cranking out an endless flow of final products without much personal investment or care. The emphasis is on keeping up with production, not falling behind in classwork or homework, rather than in producing something of lasting value. Like a fast food restaurant, the products are neither creative nor memorable. Teachers create and fuel this situation, despite the fact that we grow tired of repetitive, trivial assignments, and dread correcting piles of such work.

Turning in final drafts of work every day, often many times in one day, even the most ambitious of students must compromise standards continually simply to keep up with the pace. Internalized high standards are no defense against a system which demands final draft work at this rate. If an adult writer, scientist, historian, visual artist, was asked to turn in a finished piece of work every day, two or three finished pieces on some days, how much care could he or she put into each?

An alternative is a project-centered approach. In this structure, students still work hard every day, but their work is instead a small part of a long-range, significant project. Daily work entails the creation or revision of early drafts of a piece, the continued research of a topic or management of an experiment, or the perfection of one component of a large piece of work. Final drafts or presentations of completed projects are no longer trivial events occurring every day, but are special events, moments of individual and class pride and celebration.

Universal success does not mean uniformity. Though the structure which braces and guides student progress is common to every project, each student's project is unique. The structure provides a frame for common learning and critique, as well as appraisal of progress, but it also has room for significant creative expression and direction by individual students. If every student in a classroom prepares a guidebook to a different local building, the steps and skills involved may be somewhat prescribed: conducting interviews, researching local history, consulting city records, trying to obtain blueprints, doing sketches, taking photographs, preparing diagrams, writing and proofreading drafts of text, preparing illustrations, composing book layout, and learning book binding. Within this frame though, individual students have substantial latitude for artistic choices: the choice of building, the choice of whom to interview, the use of research and interviews, the nature of text and illustrations, the balance of text and illustrations, the use of photographs or diagrams, the tone of presentation, and the layout of the finished book.

My comments thus far may have raised a lot of questions. What about "untalented" students? Or teachers, for that matter. People who feel incompetent in particular academic or artistic areas. What about students with pronounced disabilities in perception, memory, or motor skills? If high standards are applied to the work of everyone, won't it just emphasize the painful weaknesses of some?

What about assessment? If so much of project work is creative and individualized, and even deadlines and requirements often differ from student to student, how can fair and constructive assessment take place?

What about a sense of community? If you work to dissolve the boundaries between school and outside life, don't you destroy the precious asylum of school which provides safety and security to children?

What about the cultivation of positive personal qualities in students: politeness, thoughtfulness, cooperation, initiative, self confidence, equanimity. Environments of "high standards" are often high-stress; a high standards classroom often means fighting for teacher recognition.

I'd like to say that I, and my school, have found the definitive answers to all these questions. Of course, we haven't. We are forever confused with trying to fashion a schoolwide assessment system which fits our teaching approach and documents student progress for the community or district in a real way. We are just now undertaking an effort to begin using student portfolios as archives of individual projects and achievement, as many schools have already done. And we are continually trying to restructure our school resources and community traditions to make the school more supportive for students with disabilities, insecurities, or other factors which make them "marginal".

Nevertheless, I think there is much going on in my classroom and school that addresses these questions in powerful ways. Despite having a full range of "academic abilities" in my students, and even students of real special needs, visitors to the classroom are often bemused and skeptical that this is a "regular" group of students. The work on display looks too impressive, the focus, cooperation, investment, and friendly ease of children too good to be true. The answer, I believe, is that students in my classroom are deeply and genuinely supported in countless ways to do their best and act their best, and have been so since Kindergarten. To a great extent, this is due to the ways in which the school culture deals with issues of assistance and assessment for students, and also for teachers.

People often say to me that this learning approach works in my classroom because I am a strong teacher, or works in the school because the school has strong teachers. In fact, they say, it's importance is trivial, because it can only work with strong teachers.

*Your*school has strong teachers, but you don't know the teachers at my school. I would never contend that this approach is easy to use, nor that my colleagues are not full of talent.First, it is suspiciously similar to that of teachers who visit my room and tell me: "Oh, but

*your*kids are all bright (or artists, or interested, or well-behaved), you should see*my*kids. This would never work with*my*class." Now the kids in my class are no special group, but somehow they seem to shine. Is it because they're*inherently strong*, as the school's teachers are described, or is it because the structure and culture inspires them to do their best? Granted, children are more malleable and open than adults, particularly schoolteachers (we tend to be a defensive and protective bunch). I would argue, though, that the school culture where I teach inspires teachers to do their best (though this is an argument which might generate some laughter at times in my staff room). In addition to searching for strong teachers, school cultures can be building strong teachers. Perhaps more accurately, school cultures can inspire and reward teachers for displaying their strengths more fully, by taking risks, working together, and assuming substantially more decision-making power. I would guess that children too good to be true. The answer, I believe there are more "strong" teachers than people realize, teachers whose strengths are hidden by isolation, lack of power, and lack of inspiration. The talent and spark in these teachers can be fueled with the same approach used with students in the classroom. If teachers are expected and supported to design curriculum and projects, to take risks, be original, to work together in critique and learning, teachers can often blossom just as "weak" students do. I don't mean to trivialize the difficulty in effecting teacher change, but I think it's crucial to emphasize that standards for teaching are as much a product of school culture as standards for student learning: teachers, like students, tend to settle into surrounding expectations, standards and norms.Second, on a classroom level, saying that this approach needs a strong teacher may be true, but it doesn't begin to reveal why the approach works. The power in the approach it is that it is based in a classroom community which shares a culture. The assessment, encouragement, accountability and teaching which goes on in the classroom is vastly wider than that which emanates from me, as teacher it is a continual and ubiquitous process among students. When I think of a particular student I had last year, a painfully shy girl who was new to town, I would be crazy to take credit for her astonishing emotional and academic growth-- I could hardly get her to speak to me during the year. I take credit only for managing a classroom where countless students took time to tutor her, support her, welcome her, and guide her success.

Again, where do student standards originate? Partly from me as the teacher, but very much from watching each other. Students did quality work, treated each other well, because that is what their friends did, it was the "cool" way to be, it was what the culture supported. How do students decide what is "good enough" as work, what is good enough behavior? Teacher standards only define the minimum standards allowed, they can't define the upper limit of care. Students in my classroom look to each other, help each other, critique each other's work, most of all, push each other to achieve their best. For all of my teacher power, the power in this total culture is infinitely greater.

The culture is rooted in a class community in which the first priority is supporting everyone. The responsibility for each person's emotional well-being and success is shared by everyone; the teacher bears more of this responsibility, but each student is encouraged and expected to help his or her peers. As a teacher, this means setting up a classroom structure which allows time and space for peer collaboration and tutoring; it means complimenting students, rather than complaining, when they abandon their own work at times to assist others. Supporting others means emotional support and care as much as it means academic support and care. In this team concept, the hope is that no student will be left out, left behind, allowed to fail or feel like a failure. It's everybody's job to look out for others.

For me, a culture of high standards means high standards for kindness and cooperation as much as for academic work. In the same way that careful quality in work is stressed over fast production, careful attention to treating all students fairly and thoughtfully is stressed over efficiency and speed in school logistics. "Simple" classroom decisions, such as which students should make a presentation or attend a limited event, often take a long time as they are discussed carefully with students to insure that all feelings are considered. Events and honors which are exclusionary or individualistic, displays of the "best" work or awards for the "best" students or athletes, are avoided in favor of whole-class, whole-school pride. When visitors to my classroom are impressed with student work, it is often due not to specific outstanding examples but rather to the absence of careless work, the uniform commitment to quality. This is a testament to the degree of cultural pride and peer support in the classroom.

My teachers in elementary school often instructed us to try to do your best". This isn't a bad motto; I'd use it with my class today, and most schools would embrace it without a thought. There's a big step between teachers saying this to students, and students actually doing it. Not too many schools seriously look at what aspects of their structure and culture support and compel students to do their best, to act their best, and what aspects undermine this spirit. Rather than searching for individual teachers or principals who they hope can demand high standards, I feel that schools should be looking at how they can create a spirit of high standards, a school culture of high standards.

What's your building culture like? Have a great week! Say cool:)

Labels:leadership,NBPTS,teaching | 2
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## About Me

Welcome to my all thing special education blog. I'm Ms. Whiteley. I teach in the beautiful Mile High state--Colorado. This is my 13th year teaching in an rural K-6 Elementary school as a Exceptional Needs Teachers. As Exceptional Needs National Board Certified Teacher, I believe that ALL students can learn and be successful. When I'm not in school, I love to take my two Italian Greyhounds hiking 14ers and reaching for the stars. Thanks for Hopping By.

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