Let's step back a minute and look at the history of the equal sign. It is intolerable to have two different values on each side of an equals sign. I say western children as research from Ma (1999) cited in Darr (2001) says “From the Chinese teachers’ perspective … the semantics of mathematical operations should be represented rigorously. Using the appropriate symbol to distinguish definition statements from equality statements may go a long way, at least in proportion to the effort of using them, towards alleviating confusion in students of math and computer science.When you see an equal sign what do you think it means?Ī lot of research of western educated children shows that many don't see it as meaning 'is the same as' but see it more as the thing that comes before the 'big' number in a sentence. That is, $X := Y$ means “use $X$ as a symbol for $Y$”, which differs from “use $Y$ as a symbol for $X$.” In contrast, the standard equals sign “=” is appropriately symmetric. I prefer this symbol over the popular “$\equiv$” symbol because it emphasizes the assymetry of the statement. I think it’s important to use the symbol “:=” to denote definition. Use “:=” instead of “=” to denote definition Though I have not seen any data on the topic, I wonder whether teaching these two operators from the very beginning of a student’s mathematical education would alleviate this common confusion. In contrast, the statement x = y returns either True or False depending on whether the value referenced by x is equal to the value referenced by y. That is, the statement x = y assigns the value referenced by symbol y to symbol x. That is, that we can create this definition at all! Overloading the equal sign creates confusion in computer programmingĪnyone who has taught introductory computer programming is familiar with the very common confusion between the assignment operator and equality operator in programming languages.įor example, in many programming languages, like C and Python, the assigment operator uses the standard equals sign. The real interesting quality to this definition is that the ratio of the sides of a right triangle are a function of its angles regardless of the lengths of the sides. The statement is defining $\sin \theta$ to be the ratio between the opposite side to the hypotenuse. The mystery was, at least partly, alleviated by the clarification that $\sin \theta$ is not an object that existed before we saw this statement – rather, this statement created the object for the first time. Their confusion arose from the erroneous interpretation of this statement as describing an equality rather than a definition. Their question was more along the lines of, “What is this mysterious thing? And why on earth is it equal to the ratio of the sides of the triangle?” It wasn’t, “Why are the ratios between the sides of a right-triangle functions of the angles between those sides?” Nor, “Why is this definition important?” Rather, their confusion seemed to stem from the very existence of this mysterious object, “$\sin \theta$”. Given some entity denoted with the symbol $Y$, the statement “let $X$ be $Y$”, also often denoted $X = Y$, means that one should use the symbol “$X$” to refer to the entity referred to by “$Y$”.įor example, in introductory math textbooks it is common to define the sine function in reference to a right-triangle:\[\sin \theta = \frac$?” They never explicitly stated so, but it become evident that their confusion was not the good kind of confusion. Said differently, the quantity $c^2$ is the same quantity as the quantity $a^2 + b^2$. The Pythagorean Theorem says that $a^2 + b^2 = c^2$. The statement “$X$ equals $Y$”, denoted $X = Y$, means that $X$ and $Y$ are the same thing.įor example, let’s say we have a right-triangle with edge lengths $a$, $b$ and $c$, where $c$ is the hypotenuse. Let’s say we have two entities, which we will denote using the symbols $X$ and $Y$. To ensure that we’re on the same page, let’s first define these two notions. Early in my learning days, I believe that this overloading of the equal sign led to more confusion than necessary and I have personally witnessed it confuse students. I find it unfortunate that two of the most important relationships in mathematics, namely equality and definition, are often denoted using the exact same symbol – namely, the equal sign: “=”. In this post, I argue for the use of two different symbols for these two fundamentally different operators. The overloading of this symbol confuses students in mathematics and computer programming. Two of the most important relationships in mathematics, namely equality and definition, are both denoted using the same symbol – namely, the equals sign.
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