Episode #125 of the Stack Overflow podcast is here. We talk Tilde Club and mechanical keyboards. Listen now
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A surprisingly large number of students don't know what the equals sign means. Their understanding of the symbol "=" is essentially operational, not relational — they think it means "the next step" or "the answer" or is an instruction to perform some operation. Knuth et al. ("The importance of equals sign understanding in the middle grades", Mathematics ...


15

I can't add much more than what has been stated already. But I come from a programming background, so maybe this example may add a different perspective (or explain some problems interpreting equals for other students). In many programming languages, statements like this are perfectly acceptable: x = x + 1 In this case, the programming language DOES ...


9

I think in your expansion example this is caused by wanting to solve one subpart of the problem first. In essence, they're trying to expand $(x+1)(x-1)$ first as a "subroutine" and then substitute it back into the original formula, but lacking a clear way to delineate this from the solution to the main problem. They chose to mark the substitution with an ...


8

I believe you are mixing up three problems here. One is the mistaken idea that "mathematics is all compact, weird symbols, if I don't use them to abbreviate my text, it's wrong" (your $polynomial = 3$ example), there is the "next step" interpretation, and finally the "assign a value" from programming.


6

You write the following: Obviously, the question is about what fraction of what is shaded. More generally, the question is about unit conversion in disguise. I believe what you have recorded here is the importance of specifying the unit when broaching fractions. This is one of the principal sources of difficulties for students (and some teachers) in ...


6

I'd say that an equation is a statement that two things are equal. This is slightly more general than Amy B's answer, since I don't insist that the statement be mathematical, and it's more general then the second definition in the question, since I don't insist that the two things be quantities. So for example, I consider $$\text{Andreas Blass}=\text{the ...


4

Having taught math in elementary school, I can assure you the second definition is correct. According to Math Open Reference (and many other math sites) "An equation is a mathematical statement that two things are equal" Of course this definition includes equations with variables. It also includes: 1+3 = 4 1 + 3 = 11 - 7 4 = 19 - 15 This question is a ...


4

I would write this as: Earnings over $\$37,000$: $\$78,540-\$37,000 = \$41,540$ Base tax: $\$3,572$ Additional tax: $\$41,540\times 0.325 = \$13,500.50$ Total tax: $\$3,572+\$13,500.50 = \$17 072.50$


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Your question is, in some sense, purely mathematical: What is "the" definition of an equation? You identify two different definitions, so I do not know if any single one of them can be claimed as canonical. Instead, here are some remarks about defining 'equation' from the perspective of the teaching and learning of mathematics. The first name that comes to ...


3

Your findings, as far as I'm concerned, have nothing particular to say about fractions and nothing at all to say about unit conversion. You started with a question that was not well-posed. Since we are generally expected to answer questions even if they are not well-posed (mathematicians not doing so annoys people), people voting chose what they interpreted ...


1

Answer 1, $$78 540-37 000 = 41 540 \times 32.5\% = 13 500.50 + 3572 = $17 072.50$$ could be rewritten as: \begin{align*} 78540-37000 &=41540\\ 41540\times 32.5\% & =13500.50\\ 13500.50+3572 & =17072.50 \end{align*} The use of $\leadsto$ has been suggested in the comments below to indicate each next step in the reasoning. At the level where a ...


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Update. It appears that the intent of this answer was not clear to some readers, so I will elaborate. The most common way to fix the student's computations is simply to break out the subexpressions onto separate lines (as in the other answers). There are however other noteworthy approaches that offer some advantages. Below we briefly present one such ...


1

I believe there are different things we use the term 'equation' to mean over the course of education, and students can get confused because it's usually never explained where the distinctions come in. At a simple level, an equation is any statement that two things are equal. It is a logical statement with a true/false value that uses 'equals' as the ...


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