concrete cues

Concrete cues

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1. Become a Sought-After Opportunity Maker

More than title, talent, hard work, money or even charisma, the most sought-after individuals in our increasingly connected yet complex world are those who can recruit the right team to seize opportunities and solve problems faster and better than others. They share three “mutuality minded” traits: capacity to connect with unexpected allies, recognize ways to leverage value for all participants and be frequently quoted. Kare literally demonstrates the concrete cues that enable you to hone these skills. Nothing optimizes collective performance and personal meaning more than enabling people to use their best talents together around strong sweet spots of mutual interest. This approach clearly resonates as it has attracted over 1.9 million views of Kare Anderson’s TED talk on this topic. Ready to become the glue that holds groups together?

2. Hidden Behavioral Cues That Boost or Bust Credibility

From the angle at which you face others to the shape of the room and the sound of your voice, you emanate a presence that is never neutral. Discover how to turn more situations into opportunities to bring out others better side so they are more likely to see and support yours. Kare has spent the past decade translating research into ways to become more likable, trusted and deeply connected

3. Make Your Public-Serving Place More Memorable by Storyboarding the Sequence of Scenes They Experience

From the opening scene to the satisfying ending that movie directors create to pull audiences into their story, you can design the sequence of multi-sensory and interactive times your customers feel when stepping into your place. Recognize the three key moments that are often not optimized. Learn how to conduct an “Exposures Audit” to then craft the moment-by-moment, one-of-a-kind experiences that others will rave about.

4. Make Your Conference More Meaningful by Storyboarding the Sequence of Scenes They Experience

Just as a movie director storyboards the sequence of scenes to pull audiences into the story, you can increase the number of positive multi-sensory and human interaction moments your “audience” feels when stepping into your event.  From the first sight to the last touch, increase the multi-sensory cues that nudge people to participate more fully.  Those who design conferences or other events can evoke a more memorable and meaningful experience, using Kare’s behavioral research-based storyboarding method.  Discover how to conduct an “Exposures Audit”, to then craft the moment-by-moment, one-of-a-kind experiences that attendees and exhibitors rave about and return to see again.


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Why Cue - "Neva Heard" (Official Music Video)

Last UpdatedMonday, July 26th, 2010Created

Myths and misconceptions about concrete and water

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More information about concrete cues

Sequence of Instruction

  • 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.
  • As a teacher moves through a concrete-to-representational-to-abstract sequence of instruction, the abstract numbers and/or symbols should be used in conjunction with the concrete materials and representational drawings (promotes association of abstract symbols with concrete & representational understanding)
  • 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 teaching at each level of understanding, use explicit teaching methods (see the instruction strategy Explicit Teacher Modeling).
  1. When initially teaching a math concept/skill, describe & model it using concrete objects (concrete level of understanding).
  2. Provide students many practice opportunities using concrete objects.
  3. 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).
  4. Provide many practice opportunities where students draw their solutions or use pictures to problem-solve.
  5. When students demonstrate mastery drawing solutions, describe and model how to perform the skill using only numbers and math symbols (abstract level of understanding).
  6. Provide many opportunities for students to practice performing the skill using only numbers and symbols.
  7. After students master performing the skill at the abstract level of understanding, ensure students maintain their skill level by providing periodic practice opportunities for the math skills.
  • 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

Research Support For The Instructional Features Of This Instructional Strategy: Allsopp (1999); Baroody (1987); Butler, Miller, Crehan, Babbit, & Pierce (2003); Harris, Miller, & Mercer (1993);  Kennedy and Tips (1998); Mercer, Jordan, & Miller (1996); Mercer and Mercer (2005); Miller, Butler, & Lee (1998); Miller and Mercer, 1995; Miller, Mercer, & Dillon (1992); Peterson, Mercer, & O'Shea. (1988); Van De Walle (2005); Witzel, Mercer, & Miller (2003).

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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:

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):

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

Multiplication/Division Pictures

See examples - 1 | 2 |

  • 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

See examples - 1

  • 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:

Fractions Pictures

See examples - 1 | 2 | 3 |

  • 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

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

Beginning Algebra Pictures

See examples - 1 | 2 |

  • 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

Suggestions for using manipulatives (Burns, 1996)

  • 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


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.



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:

  • 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.
  • 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

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.

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Posted on: November 7, 2016

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