Monday, June 01, 2015

Math Makes Sense with Manipulatives

Mathletes and early computer scientists had recently won World War II, Soviet scientists were threatening to militarize Earth orbit after Sputnik in 1957, and so it was decided in the early 1960's that the United States would aim to quickly boost the quality of mathematics education.

Emphasis was shifted from wrote memorization of “facts and formulas” to discovery-based experiential learning. The goal was for K-16 students to understand deeply the math they were learning, rather than merely repeating facts or algorithms robotically without knowing why. Grade school math homework now involved mathematical theory in addition to traditional drills in basic operations. Parents were left scratching their heads at homework that involved the abstractions of set theory rather than simple, unadorned arithmetic. In answer to the question “Why does 2 plus 2 equal 4?,” it was no longer acceptable to respond with an exasperated “BECAUSE!”

New Math was designed to literally “make sense” to young students by involving their senses and whole brains as they learned mathematics rather than mentally photographing and filing pages of facts “just because.” Objects were counted and distances measured when learning to add and subtract. Rods of equal length were arranged in rectangles to prove multiplication facts.

“No doubt about it: 2+2 does equal 4; five two's are in fact ten."

Students learned not only to memorize facts of arithmetic, but to understand arithmetic processes themselves. And this new comprehension was viscerally anchored, which meant it was deeply understood, and could be more easily and productively connected to other knowledge the student had (or would later have).

Physical objects used to reify numbers and clarify mathematical concepts and processes later came to be called Math Manipulatives. Manipulatives are still are the best way to teach young children basic arithmetic and early mathematics. The result of manipulative-based learning is an indispensable instinctive “feel” for numbers called “number sense.” *Legitimate criticisms of "New Math" notwithstanding, I simply can’t imagine learning arithmetic any other way.

In addition to traditional physical math manipulatives, screen-based "virtual manipulates" are now available for use on computers and other devices to aid in teaching and learning early math. These on-screen versions are superior to physical math manipulatives in many respects. Virtual manipulatives are easier to use, harder to lose, don't need to be picked up and put away after use, they go anywhere a tablet device goes, and can save tons of money and space.

Though virtual manipulatives shouldn't completely replace traditional physical manipulatives at home or school (especially during the first stages of a child's introduction to numbers and mathematical ideas, when sensory information involved in handling physical manipulatives is such an important aspect of developing critical early number sense), virtual manipulatives are well worth considering as an adjunct to traditional manipulatives.

For parents eager to set up a Math Exploration Station at home, I’ve listed below what I believe to be the most generally useful math manipulatives and virtual manipulatives along with selected guidebooks on how to use them in teaching math at home:


Physical Manipulatives

Modeling Clay, Colored Plasticine

Cuisenaire Rods

Cuisenaire Rod Track

Jumbo Cuisenaire Rods

Centimeter Gram Cubes

Centimeter Snap Cubes, Building Base

2cm Snap Cubes

2cm Creative Color Cubes


EAI Master Ruler

Measuring Tapes, Tape Measure: 30m/100ft

Animal Counters, Dinosaur Counters, Family Counters, Transportation Counters

Two-Color Bean Counters

10 Frames

Hundred Numbers Boards

Hundreds Pocket Chart with 100 Number Cards

120 Number Board

Place Value Disks

Play Money

Square Color Tiles

Wooden Pattern Blocks

Fraction Toys, Aids, and Activities

Tangram Toys, Aids, and Activities

Pentomino Toys, Aids, and Activities

Wooden Geometric Solids

Fillable Geometric Solids

AngLegs: 72 Piece Set

Math Dice, Polyhedral Dice: Set of 7 (D&D)

5-Bead Soroban Abacus, Guide Books

XY Coordinate Pegboard

Algebraic XYZ Bosse Tiles Set


Virtual Manipulatives

National Library of Virtual Manipulatives

Matti Math

BrainingCamp Virtual Manipulatives


Guidebooks and Resources

Why Teach Math With Manipulatives?

Manipulative Glossary

Math Tasks Teacher Guides: K-8

Activity Math: Using Manipulatives in the Classroom


* Parents today are experiencing a similar frustration with "Common Core," which could be characterized as "New Math 2." There's good reason for this: For most students, "Discovery-Based Math Instruction" only works well when closely paired with traditional algorithm-based "Direct Instruction." Although currently out of fashion, memorization, algorithms, drills and other traditional approaches to math instruction are at least as important and useful to most students as the "New Math" methods in vogue since the Math Standards movement picked up steam in the 1990's.


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