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The Most Powerful Tool You’ve Probably Never Heard Of…. Christina Tondevold & Lynn Rule. Conference for the Advancement of Mathematics Teaching Texas-2011. A mathematician, like a painter or a poet, is a maker of patterns.

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The Most Powerful Tool You’ve Probably Never Heard Of…

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The Most Powerful Tool You’ve Probably Never Heard Of…

Christina Tondevold & Lynn Rule

Conference for the Advancement of Mathematics Teaching


A mathematician, like a painter or a poet, is a maker of patterns.

If his patterns are more permanent than theirs, it is because they are made with ideas.

Godfrey Harold Hardy

A Mathematician’s Apology

Urgency in the teaching of mathematics

  • “The United States suffers from innumeracy in its general population, ‘math avoidance’ among high school students, and 50% failure among college calculus students (Reuben Hersh ) Too many children choose their college major and their career paths based upon how many math courses they need to take. (Boaler, 2008)

Urgency in the classroom

  • Teachers need to see themselves as mathematicians. If we foster environments in which teachers can begin to see mathematics as mathematizing-as constructing mathematical meaning in their lived world-they will be better able to facilitate the journey for the young mathematicians with whom they work.” (Fosnot)

Brain research effecting teaching and learning (Sousa)

  • Creating and using conceptual subitizing patterns help young students develop the abstract number and arithmetic strategies they will need to master counting.

  • Information is most likely to be stored if it makes sense and has meaning

Brain Research cont’d

  • Too often, mathematics instruction focuses on skills, knowledge and performance but spends little time on reasoning and deep understanding

  • Just as phonemic awareness is a prerequisite to learning phonics and becoming a successful reader, developing number sense is a prerequisite for succeeding in mathematics

What is Number Sense?

“ …good intuition about numbers and their relationships. It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms.” (Howden)

How We Learn Best

  • Memorize this eleven digit number:


    Now look for a connection (relationship) within numbers

    2 5 8 11 14 17 20

How many dots are there?

How many dots are there?

The rekenrek

  • is a tool developed at the Freudenthal Institute in the Netherlands by Adrian Treffers to support the natural mathematical development of children

  • in Dutch means “calculating frame” or “arithmetic rack.”

  • looks like a counting frame but is designed to move children away from counting each bead.

  • looks like an abacus but it is not based on place value.

Features of the rekenrek

  • The beads are red and white.

  • There are two rows of beads.

  • There are five red beads and five white beads on the top row, and the same on the bottom.

  • There are ten beads total on the top row, and ten beads on the bottom row.

  • There are ten red beads and ten white beads on the rack.

  • There are twenty beads altogether.

  • “In the United States, the manipulatives most commonly used with young children are single objects that can be counted-Unifix cubes, bottle caps, chips, or buttons, while these manipulatives have great benefits in the very early stages of counting and modeling problems, they do little to support the development of the important strategies needed for automaticity.” (Fosnot)


  • The MathRack has a built-in structure that encourages children to use their knowledge about numbers instead of counting one to one.

  • The built-in structure allows children the flexibility to develop more advanced strategies as well.

Using the MathRack to build Early Numeracy

  • What do you notice?– let the children explore the tool and learn the built in structure before you have them use the tool.

    Builds counting, enumerating, and cardinality

  • Show Me on one row ‘show me___”. Have them show a certain number. Some may count one-by-one to show the number but the structure of the tool allows for more advanced strategies.

    Builds counting, enumerating, cardinality and subitizing

  • Flash forward – once children become more confident with the tool, show the MathRack of a certain number (1-10) for a few seconds and have them determine which number was flashed. When first starting allow enough time that children who need to can still count one-by-one, gradually shorten the time so that it encourages children to see groupings.

    Builds subitizing, but some kids may still be working on enumerating and cardinality

Quick Images

How many beads are there?

How do you know?

How many beads?

Read this side

How many beads?

How many beads? How do you know?

How many beads? How do you know?

How many beads? How do you know?

How many beads? How do you know?

Turn and talk

What are all the possible ways

children will figure out how many?

Developing the landmark strategies

  • Subitizing

  • Using the 5-structure

  • Using the 10-structure

  • Counting on

  • Doubles and near-doubles

  • Compensation

  • Skip counting

  • Part/whole

Contexts for the MathRack

…”mathematical meaning in their lived world”

  • Attendance chart

  • Bunk beds

  • Double-decker bus

  • Bookshelves

Taking attendance

How many children are here today? How did you figure it out?

The Double-Decker Bus

Games with the MathRack

How many empty seats on top?

3 on top

7 on top

2 on top

8 on top

6 on top

Day 5

Games with the MathRack

  • Passenger Pairs matching game:

    Moving from the bus story to a model of the context

Games with the MathRack

  • Rack Pairs matching game:

    Moving away from the context

Games with the MathRack

  • Bus Stops game

    How many on the bus as it pulls away from the bus stop?

    How do you know?



Games using the MathRack

  • Bus Stops game

    How many on the bus as it pulls away from the bus stop?

    How do you know?

- 4


Children’s progression to make sense of the formal symbolism we use in mathematics. (Bruner)

  • Enactive-using tangible items to model the problem; the MathRack, cubes, acting it out, etc

  • Iconic-representing what they did in the enactive phase with an icon (tally marks, circles, etc. on paper

  • Symbolic-writing the formal signs and symbols

Mini-lessons with the MathRack

  • Last day of unit

    • 5 + 5

    • 5 + 6

    • 7 + 3

    • 7 + 8

    • 8 + 5

    • 8 + 6

    • 9 + 7

Modeling Story Problems

  • There are 7 people on the double-decker bus. At the next stop, 8 people got on the bus. How many people are on the bus now?

  • At Kylie’s slumber party, some of the girls were playing on her bunk bed. There were 6 girls sitting on the top bunk and 8 girls on the bottom bunk. How many girls were on the bunk bed?

  • There are fifteen people on the double-decker bus. At the first stop, 7 people got off the bus. How many people are still on the bus?

  • Kylie was having a slumber party. There were 13 girls total at her party and all of them are piled on her bunk bed. Eight girls are on the top bunk, how many are on the bottom bunk?

Fluency and Flexibility

  • Fluency- efficient and correct

  • Flexibility- multiple solution strategies determined by the problem

    Fluency is the by-product of flexibility. Assessing fluency by occasionally using timed tests is acceptable. Using timed tests as an instructional tool to build fluency is ineffective, inefficient, and damaging to student learning.

Focus on Relationships

  • When we focus on relationships, it helps give children flexibility when dealing with their basic facts and extending their knowledge to new task. When we build a child’s number sense it promotes thinking instead of just computing.

Cognitively Guided Instruction-characteristics of each stage

Direct Modeler:

  • Follow time sequence, one to one correspondence, no quotity


  • Independent of time sequence, simultaneous counting, quotity

    Derived Facts:

  • Using what you know to solve what you don’t know


  • Known Fact

Cognitively Guided Instruction Progression

  • 7 + 8

    • Direct Modeler – counts out 7 things, counts out 8, pushes them all together and counts the total.

    • Counter – holds 7 in their head and counts on 8 more.

    • Derived Fact – uses a fact they know to help them

Derived Facts

6 + 7

  • What are some derived facts kids might use to solve this problem?

Using Number Sense to Help

6 + 7

  • What are some derived facts kids might use to solve this problem?



  • What relationships (spatial, one/two more and less, benchmarks of 5 & 10, and part-whole) would students need to have before they can use the derived facts for this problem?

Sample of the Connection to Common Core Standards

  • Kindergarten

    • Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

    • For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

    • Fluently add and subtract within 5

  • 1st Grade

    • Add and subtract within 20

    • Understand that the two digits of a two-digit number represent amounts of tens and ones.

    • Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10…

Core Standards cont’d

  • 2nd Grade

    • Fluently add and subtract within 20 using mental strategies

    • Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction

    • Explain why addition and subtraction strategies work, using place value and the properties of operations.

Like mathematicians, we want kids to…

  • Look to the numbers

  • Make numbers friendly

  • Use landmark numbers

  • Play with relationships

Mathematics can be defined simply as the science of patterns…

For students to become mathematicians they need to organize and interpret their world through a mathematical lens. (Fosnot) It is the teacher’s job to keep the lens in focus…the actions of learning and teaching are inseparable.


  • Contexts for Learning Mathematics, by Catherine Twomey Fosnot and colleagues, is a series of two-week units published by Heinemann.

    • &

    • The Double-Decker Bus: Early Addition and Subtraction

    • Bunk Beds and Apple Boxes: Early Number Sense

  • How the Brain Learns Mathematics, David Sousa

  • MathRack ( ) makes arithmetic rack products, many of which are magnetic.

  • Mastering the MathRack-to Build Mathematical Minds, Christina Tondevold,

  • Teaching Student Centered Mathematics K-3, John a. Van De Walle

  • Young Mathematicians at Work series by Catherine Twomey Fosnot and Maarten Dolk (Heinemann)

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