 hidden symbols

Hidden symbols, or defaults, abound in text and education beginning at very young ages.  Possibly it is the fear of confusion that could easily occur if a symbol is revealed or mentioned as it requires prerequisite knowledge to understand.  This fear also could be indicative of a hole in either the teacher’s knowledge, the curriculum, or both.  The concept of hidden symbols is currently not being addressed properly in education.

Many math symbols are hidden for economies of time to write them, economies of space to write them, or relevance.  Take for instance the equation:

y = x + 4

A rational exponential form of the same equation reveals: This exploded view of the same equation reveals common assumptions that are poorly understood by too many high school students.  A few examples of hidden symbols rules are:

1. -If a coefficient is a positive one then it can be hidden if it is at the start of an equation or immediately follows the equal sign.

2. -If the exponent is a positive one then it can be hidden when exponents are not germane to the topic.

3. -If the denominator is a one then the number does not need to be expressed in rational form.

Mathematically the equation x + y = 3 is read as “x plus y equals three” rather than “one x plus one y equals three.”  While mathematically equivalent, the use of ‘one’ is considered unnecessary for clarification.  However, a common problem is observable in many classrooms when students are asked to solve, x + 3x = 12, for x.  As the only number associated with the x’s is three, the solution must be four.  This error among some students completely disappears when asked to solve, 1x + 3x = 12.  There are clearly four x’s and, therefore, the solution is three.