In the September 2014 edition of Mathematics Teacher, reader Thomas Bannon reports that his research group has found that the applications of algebra haven’t changed much throughout history.
Demochares has lived a fourth of his life as a boy; a fifth as a youth; a third as a man; and has spent 13 years in his dotage; how old is he?
A man bought a horse and carriage for $500, paying three times as much for the carriage as for the horse. How much did each cost?
The Panama Canal will be 46 miles long. Of this distance the lower land parts on the Atlantic and Pacific sides will together be 9 times the length of the Culebra Cut, or hill part. How many miles long will the Culebra Cut be? Prove answer.
Shandra’s age is four more than three times Sherita’s age. Write an equation for Shandra’s age. Solve if Sherita is 3 years old.
I’m grateful for Bannon’s research but his conclusion is, in my opinion, overly sunny:
Looking through these century-old mathematics book can be a lot of fun. Challenging students to find and solve what they consider the most interesting problem can be a great contest or project.
My alternate reading here is that the primary application of school algebra throughout history has been to solve contrived questions. Instead of challenging students to answer the most interesting of those contrived questions, we should ask questions that aren’t contrived and that actually do justice to the power of algebra. Or skip the whole algebra thing altogether.
If you told me there existed a book of arithmetic problems that didn’t include any numbers, I’d wonder which progressive post-CCSS author wrote it. Imagine my surprise to find Problems Without Figures, a book of 360 such problems, published in 1909.
For example, imagine the interesting possible responses to #39:
What would be a convenient way to find the combined weight of what you eat and drink at a meal?
That’s great question development. Now here’s an alternative where we rush students along to the answer:
Sam weighs 185.3 pounds after lunch. He weighed 184.2 before lunch. What was the weight of his lunch?
So much less interesting! As the author explains in the powerful foreword:
Adding, subtracting, multiplying and dividing do not train the power to reason, but deciding in a given set of conditions which of these operations to use and why, is the feature of arithmetic which requires reasoning.
Add the numbers back into the problem later. Two minutes later, I don’t care. But subtracting them for just two minutes allows for that many more interesting answers to that many more interesting questions.
This is a series about “developing the question” in math class.