# Mistakes I Do and Don’t Correct

Math Mistakes I Will Not Tolerate:  Stream of Consciousness
Some students use the equals sign to mean “…and then…”  As in, I have $12$  apples and then someone takes away $3$ and then I’m left with $9$ and then someone gives me $4$ more apples and then I have $13$ would look like this:

Now we have an illogical statement where on the left we have $12-3$ and on the right we see that it is equal to $13$.  I think this is part of the reason so many students find math frustrating: they see illogical statements like this one and think math is a subject where rules can be made and broken at whim.  I am quick to correct errors like this.

Math Mistakes I Allow:  Inefficient Exploration
My students who are new to algebra will not always choose the most efficient solution.  For example, when solving this equation for x:

an experienced student will first subtract $3$ from both sides and then divide both sides by $5$.  Some of my less experienced students have more creative ways of solving the equation.  They may choose to first divide by sides by $5$, yielding

Or they will begin by subtracting $5x$ from both sides or any number of unusual attempts to isolate $x$.  My mantra is “As long as it’s legal, we can try it.”  The most fundamental thing a student needs to learn about algebra is the idea that when you do the same thing to both sides of an equation, you still have equality of both sides.  This is my main focus, that if they do something to the left hand side of the equation, they must do the same thing to the right side.  If the cost is a little extra time spent in going down some inefficient rabbit holes, so be it.

The other reason I tolerate Inefficient Exploration is that it won’t always be obvious how to solve a problem, until you try to solve it.  Many students when they encounter a problem they don’t know how to solve will freeze.  They may have some ideas, but since they don’t know if it will work, they don’t take the risk of trying.  I want my students to be like the one who decides to divide by $5$ first, just to see if it will work.  It might, it might not, but we’ll never know unless we try.