Number Rack App¶

Strategy · All strategies
Evidence · 2 claims (1 for, 1 against) · 9 studies (6 causal, 2 review, 1 quant-synthesis), q2–q3 · 1 of 9 report an effect size

Description¶

Number Rack is a free virtual manipulative from The Math Learning Center that replicates the rekenrek — a two-row bead frame with five red and five white beads per row. Learners slide beads to model quantities, compose and decompose numbers, and act out addition and subtraction. The app includes a text tool for writing equations alongside bead models, and supports multiple racks for work with larger numbers. It is the digital counterpart of the physical rekenrek used in early elementary classrooms.

Design Implications¶

The rekenrek's 5-and-10 structure exploits subitizing and the fives-structure of the base-ten system, letting children see quantities as groups (e.g., 8 as 5+3) rather than counting individual beads [~M]. Virtual manipulatives with linked visual-symbolic representation produce learning gains comparable to or better than physical manipulatives when the visual is directly connected to the mathematical concept [~M]. The app enacts Dual Coding by pairing bead configurations with written equations, and supports Chunking by making the five-group structure perceptually salient Chunking reduces working memory load. [+S].

Context¶

Requirements¶

• A device (iOS, web, or Chromebook) running the free Number Rack app
• Teacher tasks or prompts that direct bead manipulation toward a specific number goal — the tool itself provides no instruction
• Occasional access to physical rekenreks or counters so learners connect screen models to concrete objects [~M]

Constraints¶

• The app is a representational tool, not a curriculum: without teacher-posed problems and discussion, bead sliding becomes unstructured play with little learning [~M]
• Effectiveness of manipulatives drops when learners perceive them as objects to play with rather than representations of mathematical ideas [~M]
• Perceptually rich or decorative features can distract from the target structure and impose extraneous load Decorative illustrations do not improve learning. [-M]
• Limited to early number, addition, and subtraction; not suitable for multiplicative or fractional reasoning

Implementation Variability¶

• Quick images: briefly show a bead configuration, hide it, and ask learners what they saw and how — building subitizing and decomposition fluency
• "Show me" tasks: learners represent a given number, then a number one more/less, one ten more
• Comparison tasks: two racks side by side to model and compare quantities or find the difference
• Equation linking: use the text tool to record the equation matching each bead action, connecting representation to symbol

Target Learners¶

• Kindergarten through Grade 2 learners building number sense and early arithmetic [+M]
• Learners who still count one-by-one and need to develop structured, non-counting strategies for quantities within 20
• Less valuable for learners already fluent in basic facts, for whom the beads add no new information [~M]

Target Learning Goals¶

• Subitizing and structured quantity recognition (seeing 7 as 5+2)
• Composing and decomposing numbers within 20
• Modeling and solving addition and subtraction problems, including missing-addend and comparison structures
• Bridging from concrete counting to symbolic equations

Instructions¶

1. Pose a number or problem (e.g., "Show 9. Now show one more. What equation matches what you did?") before touching the beads — Activation of the goal first.
2. Have learners model the quantity by sliding beads in one motion, grouping fives first, rather than counting bead-by-bead.
3. Record the matching equation with the text tool so the bead action and the symbol are linked (Practice).
4. Use quick-image flashes and ask "How did you see it?" to elicit and compare decomposition strategies (Class Discussion).
5. Assess Performance by whether learners use the five-structure (e.g., sliding 8 as 5+3 in one move) rather than counting.

Related Strategies¶

• Cognitively Guided Instruction (CGI for Math) — the rack is an ideal modeling tool for CGI problem types
• Concrete-Representational-Abstract — the app occupies the representational tier between counters and symbols

Related Elements¶

• Practice — bead tasks must be frequent and varied to build fluency
• Analogies — the rack models the base-ten structure children will later meet in place-value work
• Act It Out — bead sliding is a physical enactment of the arithmetic operation

Examples¶

• The Math Learning Center — Number Rack — the free app (iOS, web) with sharing and equation tools; part of the Bridges in Mathematics curriculum toolkit.
• Bridges in Mathematics (K–2) — classroom lessons using physical and digital rekenreks for quick images, ten-frame equivalents, and story-problem modeling.

Key Sources¶

• Carbonneau, K. J., Marley, S. C., & Selig, J. P. (2013). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. Journal of Educational Psychology, 105(2), 380-400. doi:10.1037/a0031084
• Clements, D. H., & Sarama, J. (2020). Learning and teaching early math: The learning trajectories approach (3rd ed.). Routledge. doi:10.4324/9781003083528
• Moyer-Packenham, P. S., & Westenskow, A. (2013). Effects of virtual manipulatives on student achievement and mathematics learning. International Journal of Virtual and Personal Learning Environments, 4(3), 35–50.
• Gravemeijer, K., & van Galen, F. (2003). Facts and algorithms as products and goals of mathematics learning. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to Principles and Standards for School Mathematics (pp. 114–122). NCTM.