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| Authors (date) | Primary learning objective (math topic) | Secondary learning objective | Delivery characteristics | Manipulatives |
|
| Bouck et al. [31] | More advanced mathematics (equivalent fractions) | / | 15 weeks, 1-2 lessons per week, 10–15 minutes, individual | VRA sequence teaching (virtual manipulatives: app Fraction Tiles) |
| Bouck et al. [45] | Precursor skills (place-value) and arithmetic computation (whole numbers; subtraction) | / | 3 phases (concrete manipulatives, virtual manipulatives, no manipulatives) and 1 phase with best condition, 5 lessons by phase, individual | Manipulatives (base 10 blocks) and virtual manipulatives (app Base 10 Blocks Manipulative) |
| Shin and Bryant [46] | More advanced mathematics (fractions; problem solving) | / | 13 weeks, twice per week, 9–10 lessons for each child, 20 minutes, individual | Virtual manipulatives: Fun Fraction program |
| Satsangi et al. [47] | More advanced mathematics (algebra: linear equations) | / | 3 phases (concrete manipulatives, virtual manipulatives, no manipulatives) and 1 phase with best condition, 10 sessions per phase, 10–15 minutes, individual | Manipulatives (an Ohaus® school balance, chips, and canisters) and virtual manipulatives (Algebra Balance Scales, NLVM website) |
| Bouck et al. [48] | More advanced mathematics (geometry: area and perimeter) | / | 3 days, 3 lessons | Virtual manipulatives: polynominoes (virtual tiles), NLVM website |
| Satsangi and Bouck [49] | More advanced mathematics (geometry: area and perimeter) | / | 5–9 lessons, 40 minutes, individual | Virtual manipulatives: polynominoes (virtual tiles), NLVM website |
| Flores et al. [81] | Arithmetic computation (whole numbers; subtraction, multiplication) | / | 3 months, 4 days per week, 25 minutes, until 40 lessons, individual | CRA sequence teaching (concrete manipulatives: base ten blocks) |
| Fuchs et al. [50] | More advanced mathematics (fractions; conceptual understanding (measurement and part-whole) and procedural skills (addition and subtraction)) | / | 12 weeks, three times per week, 36 lessons, 30 minutes, in small group | Concrete materials: objects with shaded regions |
| Yang et al. [51] | Precursor skills, arithmetic computation, word problem solving, and more advanced mathematics (general math achievement) | Mathematics interest and confidence | 8 weeks, twice per week, 16 lessons, 40 minutes, small group | Concrete materials |
| Fuchs et al. [52] | More advanced mathematics (fractions; conceptual understanding (measurement and part-whole) and procedural skills (addition and subtraction)) | / | 12 weeks, three times per week, 36 lessons, 30 minutes, in small group | Concrete materials: objects with shaded regions |
| Watt [53] | Arithmetic computation (whole numbers and fractions; algebra: evaluating equations and simplifying fractions) | / | 2 weeks, 5 times per week, 10 lessons, 30 minutes, in whole class | CRA sequence teaching: (concrete manipulatives: some examples are indicated, such as popsicle sticks, string, tongue depressors, and cups) |
| Mancl et al. [54] | Arithmetic computation and word problem solving (whole numbers; computation fluency and problem solving) | / | 11 times, 30 minutes, individual | CRA sequence teaching (concrete manipulatives: 3D plastic base ten blocks) |
| Sealander et al. [55] | Arithmetic computation (whole numbers; subtraction) | / | 9 lessons, 15 minutes, individual | CRA sequence teaching (concrete manipulatives: concrete materials that are not described) |
| Strickland and Maccini [56] | Word problem solving and more advanced mathematics (whole numbers; algebra: equation) | / | 3 lessons, 30–40 minutes, individual | CRA sequence teaching (concrete manipulatives: algebra blocks) |
| Miller and Kaffar [57] | Arithmetic computation and word problem solving (whole numbers, addition computation and problem solving) | / | 4 weeks, 16 lessons, 60 minutes, in whole class | CRA sequence teaching (concrete manipulatives: base ten blocks) |
| Flores [58] | Arithmetic computation (whole numbers; addition and subtraction with regrouping) | / | 3 times per week, 10 lessons, 30 minutes, individual | CRA sequence teaching (concrete manipulatives: base ten blocks) |
| Powell and Fuchs [59] | Arithmetic computation and word problem solving (whole numbers; prealgebra, equal-sign, and equations) | / | 5 weeks, 3 times per week, 15 lessons, 25–30 minutes, individual | Manipulatives (bears and blocks) |
| Flores [60] | Arithmetic computation (whole numbers; subtraction with regrouping) | / | 10 lessons, individual | CRA sequence teaching (concrete manipulatives: base ten blocks) |
| Scheuermann et al. [61] | More advanced mathematics (whole numbers; algebra, conceptual understanding of equation) | / | Many lessons, 55 minutes, in small groups | CRA sequence teaching (concrete manipulatives: 3D objects such as buttons, Unifix cubes) |
| Smith and Montani [62] | Arithmetic computation (whole numbers; fluency) | / | 7 months, each week, 11–13 lessons, 40 minutes, in whole class | Multisensory materials: base ten blocks, fractions stacks |
| Tournaki et al. [63] | Arithmetic computation (whole numbers; addition and subtraction and fluency) | / | 3 weeks, 15 lessons, 30 minutes, individual | Rekenrek, frame (5-10-2 × 10), plastic counters |
| Witzel [64] | More advanced mathematics (whole numbers; algebra, and equation) | / | 19 lessons, 50 minutes, in whole class | CRA sequence teaching (concrete manipulatives: algebra material without indication) |
| Butler et al. [65] | More advanced mathematics (fractions; conceptual understanding, computational fluency, and problem solving) | / | 2 weeks, twice per day, 10 lessons, in whole class | CRA/RA sequence teaching (concrete manipulatives: fraction circles, small white dried beans, and fraction squares of construction paper) |
| Cass et al. [66] | More advanced mathematics (whole numbers; geometry (area and perimeter), conceptual understanding, computational fluency, and problem solving) | / | Each day, 7 lessons, 15–20 minutes, individual | Geoboard (rubber bands and tape) |
| Witzel et al. [67] | More advanced mathematics (whole numbers; algebra and equation) | / | 19 lessons, 50 minutes, in whole class | CRA sequence teaching (concrete manipulatives: objects without indication) |
| Wisniewski and Smith [68] | Arithmetic computation (whole numbers; addition, fluency) | / | 14 weeks, each day, 20 minutes, in whole class | Math Addition Mastery Kit |
| Maccini and Hughes [69] | Word problem solving (whole numbers; conceptual understanding (representation), problem solving) | / | 20–30 minutes, individual | CRA sequence teaching (concrete manipulatives: tiles) |
| Maccini and Ruhl [70] | Word problem solving (whole numbers; conceptual understanding (representation), computational fluency, and problem solving) | / | 6 lessons, individual | CRA sequence teaching (concrete manipulatives: tiles) |
| Jordan et al. [71] | More advanced mathematics (fractions; conceptual understanding, computational fluency) | / | In whole class | CRA sequence teaching (concrete manipulatives: paper circles cut into fraction pieces and stripes) |
| Miller et al. [72] | Arithmetic computation (whole numbers; multiplication basic facts and fluency) | / | 21 lessons, in whole class | CRA sequence teaching (concrete manipulatives: paper plates and plastic discs) |
| Marshe and Cooke [73] | Word problem solving (whole numbers) | / | 13 lessons, 20 minutes, in whole class | Cuisenaire rods |
| Harris et al. [74] | Arithmetic computation (whole numbers; multiplication basic facts, fluency) | / | 21 lessons, in whole class | CRA sequence teaching (concrete manipulatives: paper plates and plastic discs) |
| Miller and Mercer [75] | Arithmetic computation (whole numbers; addition, division, and fluency) | / | 5–7 lessons, 20 minutes, individual | CRA sequence teaching (concrete manipulatives: checkers, popsicle sticks, buttons, pennies, dimes, etc.) |
| Mercer and Miller [76] | Precursor skills and computational fluency (whole numbers; place-value and multiplication basic facts) | / | 21 lessons, 11 hours in all (30 minutes per lesson), in whole class | CRA sequence teaching (concrete manipulatives: checkers, cubes, buttons, etc.) |
| Miller et al. [77] | Precursor skills, computational fluency, and word problem solving (whole numbers; place-value and multiplication basic facts; word problems) | / | 9 lessons, 20 minutes per day, individual | CRA sequence teaching (concrete manipulatives: no indication) |
| Peterson et al. [78] | Precursor skills (whole numbers; place-value) | / | 9–15 days, 5 days per week, 15 minutes | CRA sequence teaching (concrete manipulatives: plastic cubes and place-value strips) |
| Hudson et al. [79] | Precursor skills (whole numbers; place-value) | / | Individual | CRA sequence teaching (concrete manipulatives: plastic cubes and place-value square strips) |
| Peterson et al. [80] | Precursor skills (whole numbers; place-value, conceptual understanding) | / | 18 lessons, one lesson per day, 10–15 minutes, in whole class | CRA sequence teaching (concrete manipulatives: plastic Unifix cubes, place-value sticks (popsicle sticks), place-value strips) |
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