"Bear in mind that the wonderful things you learn in your schools are the work of many generations, produced by enthusiastic effort and infinite labor in every country of the world. All this is put into your hands as your inheritance in order that you may receive it, honor it, add to it, and one day faithfully hand it to your children. Thus do we mortals achieve immortality in the permanent things which we create in common." - Albert Einstein

Wednesday, July 1, 2015

How We Test and How We Teach

Serving in the GRE chemistry committee for the Educational Testing Service for six years taught me how much careful thought is required to write fair questions. A question cannot be testing too many things. A test haphazardly prepared often includes extraneous or even distracting material. During a test, a student is asked to demonstrate how much he or she has learned. A test that does not target what needs to be measured cannot really paint an accurate picture of where a student currently stands.

Copied from Austintown Local Schools
Here are two examples of how a problem in mathematics might begin:
  • “Ms. Williamson woke up one morning to find her basement flooded with water. She called two different plumbers to get their rates. The first plumber charges $75 just to walk in the door plus $25 an hour. The second plumber charges a flat $40 an hour....” 
  •  “You have just become CEO (chief executive officer) of a company that is heavily in debt. The company’s balance sheet currently shows a balance of $525,000. The company is paying the debt off at the rate of $12,500 per month.”
The above are two examples given in a paper published in the Journal of Educational Psychology, "How Readability and Topic Incidence Relate to Performance on Mathematics Story Problems in Computer-Based Curricula". The first example clearly includes extraneous information and uses a proper noun, while the second example contains three sentences that are cohesive and uses a pronoun. It turns out that students perform better with problems that are similar to the second example. The authors conclude, "In summary, problems with three or fewer sentences, third-person singular pronouns, and consistent sentence overlap are associated with higher performance levels." Apparently, the topic also matters as students perform better on math questions that have a social context than those that pertain to finance and health.

Obviously, there is a need for students to learn how to navigate math problems that are given in the form of a story. Real problems in life do not usually present themselves in a cohesive clear-cut manner. However, from the point of view of teaching students mathematics, it is important that extraneous or distracting materials are avoided. The authors correctly point out that what students demonstrate in these tests is how they cognitively process a math problem. Teaching a math concept or procedure for the first time must therefore work with text that is easily comprehensible. First examples must cater to topics that are either interesting or familiar to students. Only after students have gained some mastery of the underlying math concepts should a teacher begin introducing complexity in text, topic and situation.

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