## Old School vs New School

Some time ago, I was hired as a mathematics tutor in Connecticut at the Learning Center of a national tutoring organization. It was a very interesting experience for two reasons: It was tutoring in a classroom environment and the method of tutoring focused mainly on memorization. These concepts seemed odd to me because I had always considered tutoring as a one-on-one event and I had always emphasized understanding the material rather than just memorizing it. This organization evidently had a different perspective. My job was to sit at a desk in front of a classroom full of kids ranging from 4 to 16 years old, and answer any questions that they might have regarding mathematics. There was no chance that all of these kids, studying different levels of mathematics, with only one tutor for help, could benefit as effectively as one-on-one tutoring, since I could only allot 10 minutes of focused attention to each student within the three hour session.

Even though my job title was "math tutor", there was very little actual tutoring. These kids arrived at the Center directly after school, and spent 4 hours per day filling out worksheets which illustrated the organization's system of new math. These worksheets were lists of numerical equations. The equations varied from "4 + 4 = ?" to "16 X 13 = ?" to "If X + Y = 12 and 4X + 3Y = 46, solve for X and Y". Each worksheet contained 40 or 50 questions and, on the simpler worksheets, there were easily 100 questions. The Center's concept was to have the students memorize combinations of numbers. If the students could memorize 6 + 6 = 12, then when they came across that specific combination (perhaps as part of a more difficult problem), they would not have to spend extra time calculating 6 + 6 = 12. They would instantly recognize the combination, thus saving time. As the student progressed, then they would begin to memorize harder combinations such as 14 X 14 = 196.

Fundamentally, I took issue with this teaching method. When it comes to learning methods in mathematics, I use memorization as a last resort. For example, one formula that is used frequently in mathematics is "y = mx + b". This is called the "Slope Intercept Formula" and it is used to determine the equation of a line, when given the slope and where the line crosses the y-axis. The odds that someone can determine the equation of a line, given the slope and where the line crosses the y-axis, without knowing "y = mx + b", are slim to none. And if they happened to be brilliant enough to figure it out without the "Slope Intercept Formula", it would still take days, weeks, or maybe years to solve. With the formula, it would take seconds. This is an example where memorization in mathematics is applicable and necessary. Mathematicians have spent their lives deriving formulas such as "Slope Intercept" and the "Quadratic Formula" and "Sine, Cosine & Tangent", so that we can apply them without the hassle or futility of deriving them ourselves.

Personally, I don't see a good reason to memorize 4 + 4 = 8 or 14 X 14 = 196. Of course, most adults know these combinations without having to calculate them, but it's not because as kids, we sat down with flash cards and memorized them. We just know them because we've seen them so many times during the courses of our lives. When you also consider the technology that kids have access to nowadays (calculators on their watches, cell phones, etc.), it makes even less sense that they should memorize combinations of numbers. If anything, they should be taught mathematical concepts, and then let them use their calculators for everything else.

This organization's reason for using this technique is to increase the speed with which students solve problems. That's a valid point but for two reasons: My generation and those prior, never memorized these combinations (except for the times tables through 12 X 12) yet we still managed, and how much time are they really saving? If I didn't know that 14 X 14 = 196, it would take me 5 seconds to calculate with a pencil and paper. Saving 5 seconds in order to enforce a teaching method that is manifested through the concept of taking shortcuts, is not an equitable trade for our kids. Memorization can be a useful tool for history, spelling, music, etc. But using this particular teaching method for basic number combinations is really not necessary and might even be hazardous. Memorization can only take you so far. It you use too much memorization to get through school, then once you graduate, you'll be shocked when you realize how much you don't know.

--Editor, Tutor Pros

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