# On mentalities of arithmetic

May 23, 2016 1 Comment

It is a challenge for math teachers to understand the different ways students make sense (or not) of arithmetic.In Chile the currency is the peso. 680 pesos are equal to the US dollar. 1000 pesos—written 1.000 not 1,000—is close to $1.47. To get the dollar equivalent of a Chilean bill, some friends divide by 700 (rounding up 680 to the nearest hundred). I like to multiply by 1 1/2, ignoring what comes after the dot. To do this I divide the number by two and added to the original. 3.000 pesos is $4.50. My friends say that they can’t understand this method. Now I wonder how they divide by 700. Do they ignore the zeros, divide by seven to get 4.30, 43 or 430, and then adjust the answer so it is in the right ballpark, namely, $4.50?

I haven’t yet asked them to explain their method(s), but I did pose this problem: “How much do you tip in the US?” 15% was one answer. “How do you calculate the tip?” Answer: “I divide by 10 then I take a half of that and add it on.” I asked” “Why not divide the bill by seven? That comes to almost the same thing.” “Oh, don’t be ridiculous.”

Now, I appreciate that the latter tipping method is unusual – I don’t think I’ve met anyone else who calculates tips that way. (Or *calculated*: these days, after my son worked a lot in restaurants, I divide by five to give a 20% tip, or double then divide by 10.) The point, however, is that my friends understand the “divide by two and add that to it” method. It must be something else about how the method get visualized. At the same time, I know how to divide by seven, so I why do I prefer my method of converting Chilean currency?

Thinking now like a teacher: How do I probe to find the different visualizing? How much will that vary among any group of students? How much do I as teacher have to bridge between the current visualization and the one I want to move the students to – or, at least, in this case, want to move them to consider as a feasible alternative?

Perhaps it would be possible to show my friends the methods in writing(see below) and in that form they would get it. Then the teacher could duck the question about the mentalities of arithmetic – at least for the time being.

Currency conversion
30.000 pesos = 30,000 ÷ 700 = $42.80 ≈$43
30.000 pesos 30 One half of this = 15 Add $45 |
Restaurant tipping
$30
÷ 7 $4.28 ≈$4.50
$30 Divide by 10 $3 One half of this = $1.50 Add $4.50 |

Or everybody just learns Vedic Math 😉