Embracing Math Manipulatives

Math seems hard for so many. I've always thought it's becuase we teach it wrong... focusing on pattern recognition rather than conceptual understanding. If we understand what the numbers are, what they really are, most math becomes quite easy.

At its most basic essence, math is just adding or taking away ones (1+1 = 2; 2+1 = 3; 3-1 = 2; 2-1 = 1). The rest of math is just learning faster way to do it. While you could do 17 x 17 by adding 1+1, seventeen times in a row, seventeen times, learning to multiply makes this much faster. And with areas like calculus, with its infinite sets, it would take eternity (literally) to find the answer. But at the end of the day, it's still just all adding 1+1 or subtracting 1-1. And this is where I love to use manipulatives. As a reminder, we use the Math in Focus program as our guide to make sure we don't miss anything.

Counting Cubes and Subatizing

When Charlie was two, we first started with counting cubes. Connecitng them together, counting them, having competitions for who could build the highest tower, and of course, making swords and other such things toward which he gravitates. By grouping always grouping them in sets no more than five, subatizing (the art of recognizing how many are in a small group without counting) came naturally! As a bonus - snapping these blocks together also helps with finger strength and dexterity.

Here is what we use: Math Cubes.

Dice and Subatizing

We like to play a dice game where we roll and see who has the highest number and wins! We started with one dice and then moved up to two, three, and four. Dice are another awesome way to help little ones with subatizing as long as you get the dice wiht dots on them rather than numbers.

Here is what we use: six-sided dice and 10-sided dice.

Base 10 Blocks

As the math progresses and we start dealing with place value, a base 10 set has proven invaluable for visualizing what numbers are and how they interact. Make sure you get the one with metric cubes that are all 1cm on each side as this helps also helps to start understanding volume and mass. I also use these blocks to help understand time!

Time: As we study history, I've asked our son to imagine the smallest block as one year and then we can see how long 100 or 1000 years is with the largest cube. It really helps to visualize how long ago things like the Battle Hastings were! Imagining each small cube as 1000 years, we can start an early understanding of millions when we speak about space, distance, speed, and time!

Volume, the largest block is exactly 1 liter, containing 1000 of the smallest blocks (a cubic centimeter). That makes the smallest block 1 cubic centimeter which happens to be the original definition of 1 gram if that cubic centimeter is pure water at 4 degrees celcius (we now use the kilogram and Planck's constant but that's a subject for future grades!).

Mass: Having an understanding that mass is the stuff inside of a volume and that weight is how much that stuff in that volume is pulled down by gravity starts our understanding of forces (gravity), space (in discussing different weights on different planets because of different gravity), and physics (energy and momentum - why a larger mass will hurt more if it falls on your foot than a smaller mass and why, even in space, it's easier to get a feather moving than a bowling ball). I deeply believe that describing the world, the definitions, the whys and the hows from the start will lead to a more fluid and natural understanding in the future (but will still save Plank's constant for later)!

Here is what we use: Base Ten Set.

The Scales

We use both a digital kitchen scale with grams and ounces and a balance scale. Once again, we can visualize how numbers work and it leads to understaning of force, weight, and mass per the above. We can show that 2 cubes + 3 cubes = 5 cubes with the balance. We can prove that 5 cubes minus 3 cubes = 1 cube.

Here is what we use: Balance Scale and Kitchen Scale.

The Soroban

The first time I took the abacus seriously was when we started learning math with bigger number. The soroban (aka Japanese abacus) is what I like the best as it lends itself well to understanding the finger abacus. While there are much more advanced things one may do with a soroban, in its most basic form, it's awesome for place values, addition, subtaction, and visualizing how math works and how the numbers interact. It's also a pretty awesome way to learn subatizing.

As a quick overview: the top bead is worth 5, the bottom beads are each worth 1. After you reach 9, it gets reset to zero and you add one to the next row. It's that easy! The same goes for the finger abacus. The thumb is worth 5, the fingers are woth 1. Once you reach nine on the right hand, just go to the left hand for 10!

Here is what we use: Soroban. (I'll cover 3d printers soon).

Making it Real

We try to involve our son in cooking and building things. He is the one who measures the teaspoons and cups. He is the one who marks how many centimeters or inches we need to cut a board to make the lemonade stand. When we recently studies Alexander the Great and his excursions into India, Chalrie is the one who measured out the length of the spears Alexander ordered to combat the war elephants they encountered. Making math real is what brings it all together instead of a useless, abstract exercise in pure boredom! But we all still need to get through some boring parts!

Happy adding!