Equality – Awakening to a Problem Part I

Share Cognitive Science Implications for Teaching Mathematics
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  • The following essay describes how I became aware of the fact that some students misunderstand the ubiquitous = symbol. I will simple present the paper as I originally wrote it in 2015.
    The other day a student wrote the following on his quiz paper.
    (5 + 3i)(2 – 4i)
    10 – 20i + 6i – (12i2)
    10 – 20i + 6i + 12
    = 22 – 14i
    In an effort to correct his writing I corrected his work as follows
    (5 + 3i)(2 – 4i)
    = 10 – 20i + 6i – (12i2)
    = 10 – 20i + 6i + 12
    = 22 – 14i
    I added these notes:
    1.   If two expressions are equal, indicate that fact by using of the = symbol.
    2.  Just because two expressions are written beneath one another does not indicate any relationship between the two expressions except in the case of solving an equation or an inequality.
    I also circled the one equal symbol he had provided, and I asked, “Where did this come from?”
    The basis for my actions were.
    •  I assumed the missing equality symbols was nothing but “lazy writing”
    •  I was truly baffled by the fact that the student felt no compulsion to indicate equality of any two expressions until the very last line.
    Since that time, I have read a number of papers about some of these errors with the = symbol and have concluded that my feedback to the student really did not address his problems.
    It turns out that many (if not most) beginning students use and read the = symbol to mean “the answer is”. Their view of the = symbol is operational rather than the correct relational understanding of the symbol. Thus, this student was simply telling me “the answer is 22 – 14i”.
    What I read was that 10 – 20i + 6i + 12 and 22 – 14i represent the same complex number and that meant they had the same real components as well as the same complex components. Not at all what the student thought he had said.
    The missing = symbols are a consequence of our instructional methods which are more directed at “getting the answer” than understanding.
    We permit students (and everyone else) to write something like the following accepted convention when solving an equation.
    5x – 7 = 3x + 5
    2x – 7 = 5
    2x = 12
    x = 6
    Almost every College Algebra student will claim incorrectly that the first and second equations are equal, the second and third equations are equal, and so on to the end. The student quite naturally develops the idea that if things are written beneath one another they are presumed to be equal. Consequently, it makes good sense to write the following.
    (5 + 3i)(2 – 4i)
    10 – 20i + 6i – (12i2)
    10 – 20i + 6i + 12
    = 22 – 14i
    These and other misconceptions exist because we as teachers permit them to exist. In fact, what we do in our classrooms fosters and encourages such misconceptions. We teach students to write using certain conventions without insisting that they know what is meant by the convention.

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