These thoughts come from an excellent article by professor of Mathematics Jerome Dancis. You can read the full article here.

Comprehending word problems correctly and then translating them into organized mathematical expressions and equations, is a crucial part of doing math and science. Reading includes paraphrasing a word problem, following directions, understanding vocabulary, realizing that the order of words matters, as well as the reading of tables, charts and graphs.

Herb Gross, math instructor at a community college, has written: “The ability to paraphrase is one of the greatest aids I have when I solve problems. I find that most of my students are very weak in this regard. As important as mathematics is, it is a distant second to the need for good reading comprehension. The shortage of training in translating word problems into mathematical expressions is why we teachers so often hear students summarize a course by saying ‘I could do everything except the word problems’. Sadly, in the textbook of life, there are only word problems.”

In solving word problems, children have to comprehend and translate into mathematics a multitude of words indicating actions and relations between objects, such as put, give, take, bring, fill, drain, move, meet, overtake, more, less, later, earlier, before, after, from, to, between, against, away etc.

Teaching the connections between the roots of words is useful. For example, it is useful to note that:

~ A quart is a quarter of a gallon. This is better than defining a quart as a (seemingly random word chosen for) measure of capacity.

~ 100 percent makes the whole, just like 100 cents makes exactly one whole dollar and 100 centimeters make exactly one whole meter. Relatedly, 17 cents is 17 percent of a dollar and 17 centimeters is 17 percent of a meter. Similarly 236 centimeters make 2.36 meters, just like 236 cents make $2.36.

Here are some problems to ponder. (Solutions and more problems can be found in the full article)

Problem 1. The Geese Problem – SAT Level 5

“A flock of geese on a pond were being observed continuously.

At 1:00 P.M., 1/5 of the geese flew away.

At 2:00 P.M., 1/8 of the geese that remained flew away.

At 3:00 P.M., 3 times as many geese as had flown away at 1:00 P.M. flew away, leaving 28 geese on the pond.

At no other time did any geese arrive or fly away or die. How many geese were in the original flock?”

 Problem 2.  From a Singapore Math textbook

“Mrs. Chen made some tarts. She sold 3/5 of them in the morning and 1/4 of the remainder in the afternoon. If she sold 200 more tarts in the morning than in the afternoon, how many tarts did she make?”

Problem 3.

For the set of numbers {1, 2 and 3},

(a) Find the sum of the squares of these three numbers.

(b) Find the square of the sum of these three numbers.

(c) Find the average of the squares of these three numbers.

(d) Find the square of the average of these three numbers.

Problem 4.

The store sale sign reads: “Buy one tie, get a second tie at half price, when the second is of equal or lesser value (price)”. John buys two ties, one priced at 4 dollars, the other at 6 dollars.

(a) How much is the sale price on these two ties?

(b) What percentage saving does John obtain on his purchase of the two ties?

How will you help your students understand the vocabulary, actions, and relationships in word problems and model them with manipulatives, pictures, diagrams, and, eventually, arithmetic or algebraic expressions and equations?

Post your thoughts under the Questions/Comments/Solutions link at the top of the page.

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