Many people perceive
as being something different than
Yet, when asked “what operation connects the “3” to the “x”, most will think a second and respond “multiplication”. So, they can figure out what it stands for – but they do not perceive it that way initially.
This mis-perception contributes to a number of algebra errors. Most often, people do not seem to see that
or that
I recommend three steps to resolve this situation:
– rewrite terms with coefficients explicitly as products (either by putting a dot between coefficient and variable, or putting each in parentheses)
– expand the product into a sum of like terms (a version of repeated addition)
– put a coefficient of “1” in front of any variables without coefficients. It is usually worth revisiting the notion that multiplying by one does not change anything, so students are always welcome to do this if it helps.
Faced with
people often fail to notice the common factor. However, if it is rewritten as
people often spot that three is also a factor of six more quickly and are confident rewriting the expression as
before finally factoring to produce
Faced with
people are often uncertain about what they can, or cannot do. The lack of a coefficient in front on the 
or perhaps even
this will often resolve uncertainty, as they might see that it can be factored
and then simplified
However if a student consistently does not recognize 
then expanded into
At this point people are usually confident in stating that the answer is
It is also worth noting that expanding multiplication into a sum of terms also works with decimal coefficients, but the terms will not all be identical and thus is not the usual “repeated addition”. For example,
Once a person become skilled at visualizing
as either
and
as either
without conscious effort, mistakes in these situations usually tend to begin happening less frequently.
MathMaine 