the only difference between different bases is the number that determines a fair trade—not the actual making of the trade—adding and subtracting in Ten Land becomes *meaningful* base 10 arithmetic.

*The Ten Land Blocks Are TRUE Base 10 Blocks *

The Ten Land blocks deserve special mention. They are *true* base 10 blocks, unlike the so-called base 10 blocks that litter America’s elementary school classrooms. Both types of blocks are the . The unit is a cubic centimeter, the long is 10 centimeters long, the flat is 10 centimeters squared, and the cube is 10 centimeters cubed. The difference is in how the flats and cubes are *scored*. The Ten Land flat shows that it is made of ten 10s, whereas the base 10 flat shows that it is made of 100 units*.* (Click to see a comparison.) Even more striking, whereas the shows that it is made of ten 100s, the shows that it is made of *1,000 units* (or 600 according to many children: 100 for each side). *All those units have nothing to do with adding and subtracting and are a huge distraction.*

Base 10 arithmetic is *not* about working just with units. It is about working with units or 1s, then with 10s, then with 100s, then with 1,000s, and so on and keeping track of every time ten the same are traded for one of the next bigger thing or vice versa. In adding, it is about trading ten small things for one of the next bigger thing: *ten* 1s for a 10, *ten* 10s (not 100 units) for a 100, *ten* 100s (not 1,000 units) for a 1,000, and so on. In subtracting, it is about the reverse, trading a big thing for ten of the next smaller thing: a 1,000 for *ten* 100s (not 1,000 units), a 100 for *ten* 10s (not 100 units), and a 10 for *ten* 1s.

*Base 10 Arithmetic is NOT Developmentally Appropriate for Young Children*

Research conducted decades ago by and associates showed that many children are *non-conservers of number*, meaning that their thinking about quantities is not logical because they base their belief about the size of quantities more on how they look than on their actual amount. The classic experiment to determine if a child can conserve number is one you can easily conduct yourself if you have access to a child ranging in age from 5 to 7, the closer to 5 the better. (All kids are non-conservers of number at some point in their arithmetic development and then grow out of it, however at different rates.)

*The Experiment *

Assuming a willing young child, place two arrangements of candy in front of the child, one with ten pieces of candy spread out, the other with ten pieces close together. Assuming the child can count to ten, ask him or her to count the pieces of candy in each arrangement. Assuming he or she counts them correctly and says “ten” for each arrangement, ask which one he or she would like to have and why. If the child picks the one where the pieces are spread out “because it has *more*” (that is, “looks” like more), he or she is a non-conserver of number, at least for *ten* pieces of candy at this point in his or her arithmetic development.

Does being a non-conserver of number matter for school-age children? You bet it does! Numerous studies have shown that non-conservers of number tend to fall behind in math during their early years in school, which is not surprising. For non-conservers of number, much of the math they experience in school is meaningless. For example, if shown 10 beans glued to a stick to illustrate that ten 1s make a 10, they perceive the amount of beans on the stick as *less* than when they were apart. Although they will eventually outgrow their lack of logic in comparing quantities, catching up is hard to do, and many never do. Fortunately, there is a way to avoid their falling behind.

*Try the Experiment Again*

Do the candy experiment again with the *same* *child* except with five pieces of candy in each arrangement. With only five pieces, the child may indicate that their preference for a particular arrangement is unaffected by how it appears because “both are the same.” Most children, if they can count at all, *can* think logically about quantities if they are not too big. The boundary line is about eight. For arrangements with seven or fewer pieces of candy, most children are not confused by the appearance of the arrangements even if they are confused with arrangements of more pieces of candy than that. This is why it is important to begin instruction on addition and subtraction in Two Land and Three Land instead of Ten Land. Since the number of blocks the same in Two Land and Three Land is never allowed to exceed 2 or 3, respectively, *no child will be left behind* because virtually *every* child can think logically (conserve number) about quantities this small.

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**Fair Lands™ Blocks**

Multi-base arithmetic blocks for base 2 and base 3 out to *seven* places. Scored to illustrate the structure of a place value numeration system: that base 2 is based (or built) on 2: each digit to the immediate left of another digit is exactly twice the value of that digit. This relationship is shown with the red or Two Land blocks by having two the same make the next bigger block. Likewise, base 3 is based on 3: each digit to the immediate left of another digit is exactly three times the value of that digit. This relationship is shown with the blue or Three Land blocks by having three the same make the next bigger block.

Available only from this website.

**Ten Land (base 10) Blocks**

Arithmetic blocks for base 10 out to four places—only four because the blocks get too big after that. (A Mega Cube—literally, a million Unit Cubes—would be a cubic meter.) Scored to illustrate the nature of base 10 numeration: that each digit to the immediate left of another digit is exactly ten times the value of that digit (just like on the odometer of a car). This relationship is shown with the Ten Land blocks by having ten the same make the next bigger block.

*True* base 10 blocks showing that base 10 numeration is literally based (or built) on 10: 10 unit cubes make a long (or ten rod), 10 longs make a flat (or hundred flat), and 10 flats make a cube (or thousand block). Different entirely from base 10 *place value* blocks (even though called base 10 blocks) where the flat is made of 100 unit cubes and the cube is made of 1000 unit cubes.

Available only from this website.

Colored Counters

Colored counters are readily available in most pantries. Just about anything that can be sorted by size or color will do, like colored macaroni, different kinds of beans, or certain kinds of cereal. Cereal is risky, though, because of its tastiness.

The Multilink Cubes in the picture are available from ETA/hand2mind.