 # Dr. Cotter on Multiplication

Multiplication has been the mathematical downfall of many children, not so much because of the algorithms, but memorizing the 100 facts. Before expecting the child to learn the facts, we need to teach the meaning of multiplication. Describing it as repeated addition is a limited view, which doesn’t work well for multiplying fractions or decimals.

Since the primary application of multiplication is finding areas, an array, an arrangement of objects in rows and columns, makes a better model. A row with six objects repeated three times is 6 multiplied by 3, or as Montessorians say “Six taken three times.” This array produces a product of 18.

Note how this interpretation also corresponds to the Cartesian coordinate system. Representing the array arrangement of 6 across in 3 rows with the expression 6 × 3 is similar to finding a point (6, 3) on a grid. The first number, 6, indicates the horizontal number and the 3, the vertical number.

Some Missteps in Teaching the Facts

All too often the multiplication facts are taught by rote, an overwhelming task, especially when 6 and 3 was previously learned as 9, but now it’s 18. Some programs increase the burden by extending the number of facts to be learned from 10 × 10 to 12 × 12. The 11s and 12s are not basic facts and increase the amount to be memorized by 44%. The 11s and 12s can easily be figured out as a sum of 10 times the factor times plus 1 or 2 times the factor. For example, 12 × 3 is 10 × 3 plus 2 × 3, or 30 + 6 = 36.

Sometimes children learn songs or rhymes for the facts. One drawback is that the child needs to sing the song until the desired fact is reached. A second drawback is the additional time the brain needs to transfer the information from the language section of the brain to the mathematics section.

Another faulty approach to learning the facts incorporates pictures, one for each fact. For instance, to remember 4 × 4, one image shows a 4-wheel drive truck with the caption that the driver needs to be 16 to drive it. When I first saw this, the legal age to drive in North Dakota was 14. Does that mean 4 × 4 might be equal to 14? Seriously, these types of pictures do cause a delay in fact retrieval caused by translating unrelated pictures into mathematical concepts.

The belief that the product obtained by multiplication is always greater than either factor is a common misconception. Think about 7 × 1: the product, 7, is equal to, not greater than the factor, 7. Again, look at what happens when multiplying by zero: 257 × 0 is 0, certainly not greater than 257. And multiplying proper fractions always results in a product less than either fraction.

What about using skip counting for teaching the facts? It seems to make sense to teach the facts through skip counting. However, children often resort to counting on their fingers to find the desired fact. I saw this in a school in England where the children were becoming fast counters, but were not mastering the facts. This simply becomes another rote procedure.

Learning the Facts through Visualization

For learning the multiplication facts, there is nothing like the commutative property to simplify the task. I still think it’s amazing that 6 × 4 is equal to 4 × 6. That alone reduces the number of facts to be learned in a 10 by 10 table from 100 to 55.

Learning the 1s facts is easy: 1 × 8 means 1 repeated 8 times which is 8 and 8 × 1 means 8 taken 1 time, which also is 8. The 2s are already known from the addition facts and the 10s are known from place-value work. Now we’re down to 28 facts to learn.

The AL Abacus provides great visualizable strategies. For example, enter 6 × 4, shown below in a rough sketch.

o o o o o x
o o o o o x
o o o o o x
o o o o o x

See the two groups of 10 and the four ones, which makes 6 × 4 = 10 × 2 + 4 = 24.

The fact 9 × 3 can be seen as 10 × 3 – 3 to give 27.

One last example is 7 × 7. See below.

o o o o o x x
o o o o o x x
o o o o o x x
o o o o o x x
o o o o o x x
x x x x x o o
x x x x x o o

Here you can see the five rows of five o’s that make 25. Also see a 10 along the right two columns and another 10 in the bottom two rows. The lower right corner has the leftover 4. It takes less than 2 or 3 seconds to summon the visual image and find the product by thinking 25 + 10 + 10 + 4 = 49. Remember a child knows a fact when they can answer in 2 or 3 seconds.

Each of the multiplication facts can be found on the AL Abacus by looking for similar patterns.

Multiplication Algorithms

Except possibly for historical purposes, I do not see any reason to teach lattice multiplication. The lattice procedure is time-consuming, does not aid in understanding, and does not naturally lead to the traditional algorithm. The standard multiplication algorithm is fairly easy to teach with understanding.

1. Dirceu Aparecido Borges says
2. Dirceu Aparecido Borges says