“How long does it take you to drive to work” is a very common question asked, have you ever noticed that the answers we get to this question is almost always a range and never a precise single number? As in “About 15 – 20 minutes” as opposed to “18 minutes”.
When it comes to answering questions dealing with quantity (like how long it takes to drive to work) we seem to be incapable of giving a precise single number as answer. When asked how “How many people showed up to the Christmas party?” we answer “about 100, 150”. When asked “How many drinks did you have” we answer ” two, three, or five”. We tend to revert to our instincts to deal in approximations when dealing with questions based on quantities.
It would be reasonable to hypothesize that all humans have comparable abilities when it comes to approximating quantities. After all we all do it. In 2008 a group of researchers from John Hopkins University and the Kennedy Krieger institute did a study that investigated whether or not there were significant differences in the approximate number sense of a group of 14 year olds. The experiment they used was simple, the youngsters were shown different numbers of yellow and blue dots at the same time on a screen for 0.2 second, and they were asked if there were more blue or yellow dots. Two interesting results came from this experiment. One is that there was a wide variation in performance amongst the subjects. Some of the teenagers had no trouble observing the difference between nine blue dots and ten yellow dots, while others had difficulty seeing that five yellow dots is more than three blue dots! The second is that when the researchers compared the subjects dot – comparing scores with their math marks since kindergarten they found that there is actually a strong correlation between a persons ability to approximate and success in formal math. It turns out that the better one’s approximate number sense, the more likely they are to get good grades in math.
So what do the results from this study imply about what we should be doing in math class? Well for starters the results suggest it might not be a bad idea to focus less on precision and more on approximation. Traditionally math classes emphasis accuracy as in get the exact answer and your good. After students work out an answer to a question the first thing they want to do is check the answer key to see if they got it right. If they got the right answer they are happy, if not they get discouraged. However there is more to mathematics than just getting the exact right answer.
When solving problems we need an entry point, a way to get started. With math problems a good way to allow students at every level to be able to enter a problem is to simply allow them to estimate what the solution would be. For example if the problem is how many sugar cubes can be packed in a box, you can ask them “Give me a value that is too many, and give me a value that is too little” and students simply write down a couple of numbers. This is effective because all students, regardless of their confidence level, can take a guess and get started with solving the problem. Once they get started students are ready and excited to tackle another task, then another, until the problem is solved.
Mark Twain once said “The secret to getting ahead is to get started”. Makes sense, how can we accomplish anything if we don’t start anything? In the math class asking students to estimate an answer is a way to get them started on solving a problem.