by Don S. Balka and Ruth Harbin Miles

A new manipulative has emerged that appears to provide many students with a better understanding of early number concepts. The Rekenrek, arithmetic rack, or counting rack was developed at the Freudenthal Institute in the Netherlands by Adrian Treffers. Resembling an abacus, the Rekenrek typically consists of two rows of ten beads, with each row having five red beads and five white beads. The Rekenrek takes its place in the primary grades classroom alongside other popular models for developing early mathematics concepts, such as Unifix Cubes, base ten blocks, ten-frames and counters.

The focus in using a Rekenrek is on fives and tens. Rekenreks are also available with four rows of beads or ten rows of beads to deal with numbers 21 through 100.

## State Mathematics Standards and the Rekenrek

Schools across the country are busy implementing the Common Core State Standards for Mathematics (CCSS-M) or other comparable state standards such as the Texas Essential Knowledge and Skills, the Indiana Academic Standards, and the Virginia Standards of Learning.

The Rekenrek is a useful adjunct in meeting the Grades K-1 CCSS-M standards in the domains of Counting and Cardinality, Operations and Algebraic Thinking, and Number and Operations in Base Ten. Rekenrek activities also incorporate most of the CCSS Mathematical Practice Standards:

• Make sense of problems and persevere in solving them. (MP1)
• Reason abstractly and quantitatively. (MP2)
• Model with mathematics. (MP4)
• Use appropriate tools strategically. (MP5)
• Attend to precision. (MP6)
• Look for and make use of structure. (MP7)
• Look for and express regularity in repeated reasoning. (MP8)

For Texas, the content standard areas are (1) Number and Operations and (2) Algebraic Reasoning for all three grades; for Indiana, the areas are (1) Number Sense and (2) Computation and Algebraic Thinking; and for Virginia, (1) Numbers and Number Sense and (2) Computation and Estimation.

## Subitizing and the Rekenrek

Subitizing is an important early number sense skill. It is the ability to perceive at a glance the number of items presented in a group without counting. This ability allows students to make quick mental images of arrangements of objects and then write the corresponding number symbol or reproduce a representation of the quantity. Subitizing, for example, allows game-players to recognize the pattern of dots on regular six-sided dice or double-six or double-nine dominoes.

For students using the Rekenrek, the numbers 5, 10, 15, and 20, plus or minus 1, 2, 3, or 4 become key to the subitizing. The initial position for all beads on either row is to the far right. Thus, there are 0 beads on the left. When all five red beads are moved to the left, students "know" that the beads represent the number 5 without counting:

Likewise, when all ten beads are moved to the left, students understand without counting that the number represented is 10. Students also see that 5 + 5 = 10.

Simple subitizing activities with the Rekenrek strengthen initial counting skills and beginning addition and subtraction for students. They are able to observe 1, 2, 3, 4, or 5 red beads moved to the left. They then see 6, 7, 8, 9, and 10 as 1, 2, 3, 4, and 5 white beads more than the respective numbers.

## Counting, Cardinality, and the Rekenrek

Counting is the typical way young children learn to construct number relationships. At home and in preschool, they are taught to count orally with little understanding of the meaning of the number words. The idea of the cardinality of a set of objects is missing. Later, they begin to attach meaning to the numbers by physically pointing to objects and saying the number names. Children are also shown number symbols and asked to draw the corresponding number of objects. Activities with the Rekenrek help children to attach meaning to counting and cardinality, which is critical to the development of more advanced number concepts.

## Composition and Decomposition

Fundamental components of number sense for young children to master are the tasks of composing and decomposing numbers. For example, the number 7 may be composed (made up) or decomposed (broken down) into a set of 6 objects and 1, 5 objects and 2, 4 objects and 3. Children visualize these relationships by sliding the beads to the left or right on the Rekenrek. Being able to make quick mental calculations aids students as they move on to place value concepts.

## Place Value

Children in the primary grades need to be able to make groups of 10 and visualize them as a single item. For example, 16 means "1 group of ten and 6 ones." With its emphasis on multiples of 5, the Rekenrek helps children visualize this relationship: All 10 beads on the first row are moved to the left; 5 red beads and 1 white bead are moved to the left on the second row. In writing the number 16, children now understand that the 1 in the tens place represents the 10 beads that were moved to the left on the top row of the Rekenrek, while the number 6 represents the beads moved to the left on the bottom row. Use of the Rekenrek assists in developing place value understanding.

Counting Conventions on the Rekenrek

For certain number concepts, sometimes both rows of the Rekenrek are used instead of just one row. When students are first learning addition, we might show one addend on the top row of beads and the second addend on the bottom row of beads. Here is an example showing 5 + 2.

Students count the beads on the left to find the sum, 7. Later, one row of beads is used to do the addition.

Basic addition fact strategies are another area where two rows of beads are used. Doubles or doubles plus one can be illustrated easily with two rows. For example, 4 + 4 would look like this:

and the doubles plus one, 5 + 6, could be shown like this:

In the last example, students visualize (5 + 5) + 1 = 11.

## Summary

Well-chosen activities using the Rekenrek can be a valuable aid to teachers as they work with young children to develop understandings of numbers, operations, and place value. Students are attaching meaning to numbers based on relationships with 5 and 10, and eventually 15 and 20. Learning experiences with the Rekenrek engage students and help promote early success with mathematics.