Physics and Astronomy


Learning How to Multiply

by Ed Meyer

One of the milestones in the education of young students is a mastery of multiplication. It starts with memorizing the multiplication table for the numbers 1 through 9 and then moves to multiplying multiple-digit numbers.

Most children are trained to remember their multiplication facts just as they remember the ABCs, their home address, their phone number and the names of the states. Because of this, the operation of multiplication is stored in the part of the brain that memorizes facts rather than the part of the brain that involves logical problem solving. That is, the process of responding, “42” to the stimulus “What is 6 times 7?” is very similar to the process of responding “Columbus” to the question, “What is the capital of Ohio?”

Often young students are given sheets of multiplication problems to complete and they are usually given a time limit. This is System One thinking. That is, the brain is just responding to a stimulus without careful consideration. Indeed, often the young student is negatively reinforced if the response to the stimulus of “6 times 4” involves any time delay for System Two thinking.

This type of stimulus-response training often results in students having difficulty using multiplication as a tool to solve word problems. For example, consider the problem “A teacher has 17 boxes of pencils and each box contains 24 pencils. How many pencils are there?”

Anyone who understands the concept of multiplication will be able to multiply the 17 and the 24 to get 408. However, when presented with this problem, many young students will add the 17 and the 24 to get 41. This is because they are being trained to use only their System One. That is, they are not thinking logically, they are just responding to a stimulus. In fact, the student may ask, “Am I supposed to add the numbers or multiply them?”

It seems to me that there is not much sense in memorizing multiplication facts before understanding how to use the multiplication facts solve word problems. Students would never mis-apply addition and multiplication if they used their System Two to understand the problem.

I would suggest that a mastery of the multiplication facts be gained by only solving word problems. For example, “How many horseshoes are needed to put horseshoes on both the front and back hooves of five horses?”

This stimulus-response development of System One produces students that are even more averse to engaging their System Two. It produces students that are only interested in the answer to the question rather than the thoughtful reasoning needed to get there. It produces students who hate word problems and would rather memorize facts with flash cards. This does not produce the creative thinkers needed to solve the myriad problems of the human race.

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