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Monday, May 9, 2011

Do 4.92 hours really exist?

  A parent recently wrote requesting advise on how to help her student  round her answers for a POW problem. It reminded me of how many times I write comments to students asking them to reconsider the reasonableness of their answers. In fact, many problems based on the real world are answered more effectively with rounded numbers, yet I continually see numerical answers carried deeply into the decimal place values, such as 13.2948372 centimeters,  2.3333333338 degrees  or even 284.34950 people! 

While it is great to be able to calculate so completely, it should not be  considered “more accurate”. A crucial component of working with numbers in the age of cheap and ubiquitous calculators is determining the relevance of the answers you find. In science there are rules governing what is known as “significant figures”. These rules are based on the fact that human beings are imperfect observers of the world, so we have to cushion our measurements with a healthy dose of humility. When scientists work with numbers (such as measurements from a ruler), the numbers are not exact, but carry some amount of inaccuracy with them (because, for example, no ruler is absolutely, perfectly straight).  As a result, they round their calculations to a decimal place value that is mutually agreed upon according to the measuring tool they are using.                      

Many of the POW problems are based on a real world context. The answers do not easily follow a set of rules for rounding such as I just described in science. Nevertheless, your answers should be rounded in a way that makes sense with the situation described in the problem. What I am looking for is the development of “number sense”. Number sense refers to "an intuitive understanding of numbers, their magnitude, relationships, significance and how they are affected by operations." In many ways, the definition of number sense succinctly describes the goals of the entire 7th grade math curriculum I have designed. 

People with a better developed sense of the meaning and relationships of numbers often find math more relevant, interesting and (dare I say) enjoyable.         I consider the development of number sense to be similar to developing our own physical fitness. While it is true that some people find exercise either easier or more rewarding that others, there is little doubt that every person’s health and well being is enhanced from exercise and a healthy diet.
One way to develop number sense is to take risks in the way we interpret our answers. Take the question I pose at the top of this article:  Is there such a thing as “4.92 hours”?  Of course there is. But what is the context of needing to know time down to the hundredths place of an hour? In the first place, few of us ever measure time as decimals of one hour. Secondly, time has a distinctly subjective feel to it. We naturally “bunch” up remaining minutes in our mind. In effect, we round our description of time to fit our perception of it. Finally, many problems involving time, for example, do not imply absolutely constant change. Things often speed up and slow down in the real world, so even if our calculations feel “exact” to the hundredths place, they are not. 

I was thinking of a situation involving “4.92” hours to use as an example. My family has a cabin in the Sierra Nevada mountains between Lake Tahoe and Yosemite.          I know that it takes us approximately 5 hours to drive there from my house in SF. I can imagine that over the years I could have potentially arrived in slightly less than 5 hours, say 4.92 hours, but I would not have known or even cared. In my mind, anything between 4.5 and 5.5 hours falls within the range of “about 5 hours”. In terms of planning my next trip to my cabin, that is the number I use to calculate by departure.

So, while 4.92 hours does exist, it has limited practical use. Estimation is an important part of mathematics and a very handy tool for everyday life. Rounding off is a kind of estimation. Get in the habit of rounding amounts of money, lengths of time, and distances as way of making better sense of your answers.


David said...

I think that some students may neglect to round their answers because they are risk averse, and they do not wish to be penalized for excessive or incorrect rounding. The disconnect with reality occurs because students aren't necessarily trying to find the most correct answer, but rather the answer that will earn the teacher's approval.

ERKO said...

So true, David, so true.