Spatializing math “facts” builds executive functioning skills….

Math facts are much easier to learn and remember and more rapidly recalled from memory when we embed them within physical, visual investigative tasks. The intent of working with blocks, tiles, grid paper and the like, in math class, is to ENGAGE the senses in the learning. The mistake teachers often make is to turn using the manipulative into just another set of rules. The materials should be used flexibly. COMPARE!




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These are all ways to see 7. What do they have in common?

Students should never be afraid to test out a turn, or a flip, or a slide…  What changed, what stayed the same.

The COMMUTATIVE PROPERTY emerges when we encourage students to move materials and images:

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I store blocks in fives of one colour. I want students to trust the five from mid Grade one on…  I can move the image, I can move the symbols…


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Understanding is grounded in the physical experiences. Language is a symbol system for communicating understanding. When we start in the real world of objects and materials, every student can begin.


The imagery continues with models into 2 and 3 digit numbers. The blocks become cumbersome so we gently but intentionally move to hundred grids and numberlines.

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B.E.R.C.S. cards and tasks are designed with spatial reasoning in mind. But any tools for thinking can be proceduralized into meaningless rules if we do not learn how to use them. Teachers have no choice. We must continually learn……

do do birdEVOLVE or GO EXTINCT…








The imagery continues into multiplication…


Students build multiplication models with grid paper or tiles, with cusienaire rods & diagrams. Flips (reflections), slides (translations) turns (rotations) do not change the product.

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Students physically experience the commutative property. Link the images to the equations and practice mentally transforming both imagery and symbolism.




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The Commutative Property for multiplication is spatial.


So what’s up with xy or yx?  x + 4 or 4 + x?

Algebra becomes a puzzle not a chore!