Same & Different As It Ever Was


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.

[via @lucyefreitas]

This is a series about “developing the question” in math class.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. I’m enjoying this series a bunch. After reading this post I found myself trying to pin down exactly what it means to “develop the question.” Does this sound right?

    There’s a question that you think is interesting. Feh, you think it’s interesting because you’re you. In order to get lots of kids as excited as you are about the question, you’ll need to develop the question with some prior teaching.

    Reading through this series, it’s seemed to me like you’re further arguing that a particularly effective way to develop a question is to start with questions that require less precision en route to a question that requires more precision.

    Do I have this right?

  2. This is fascinating! I wonder what would happen if we spent as much time in math class focusing on representing relationships as we do on calculations. I can’t recall it right now, but seems like I read some research about how less successful problem solvers pay attention to types of numbers and key words in problems vs. successful solvers who pay attention to the situation described in the problem. That says to me that problem solving ability might be inhibited by the types of numbers and particular operations we’re comfortable with. Now that calculating is trivial (which is different than number sense), maybe we can do problems like it’s 1909.

  3. @Michael, so far so good. I have taken Willingham’s quote, meant to apply broadly to all disciplines, and adapted it for math.

    “Rush to the answer” has become (for me) “rush to the most precise, most abstract, most formal answer as possible.”

    I envision a dial on the math class wall. It specifies the level of formality and precision of the task at hand. In many classes it’s set to “OMG GET THE CORRECT ANSWER NOW!” all the time. And no one knows the dial even exists because it’s setting is constant. It becomes part of the background anxiety.

    So lately I’m enjoying looking at tasks and trying to determine the earlier settings on the dial. The process of turning up the dial goes by many names, I think. Around here: “mathematization,” “ascending the ladder of abstraction,” “teaching in three acts,” and “creating intellectual need.”

    It kind of ties the place together.

  4. …“mathematization,” “ascending the ladder of abstraction,” “teaching in three acts,” and “creating intellectual need.”

    Are all of these equivalent ways of developing the question, or are they different ways of developing the question?

  5. They’re more alike than they’re different. But they aren’t really equivalent. Some pertain more to modeling (“real world math”). Others are more general. Call them cousins, maybe.

  6. “152. If you know how much a man paid for an automobile, how much it cost to run it two months, what the current rate of interest is, the amount of the doctor’s bill for setting the man’s leg and for a month’s attendance, and how much the machine then sold for, how can you compute the cost of the fun the man had?”

  7. Just curious (not highly ciritical–you never know how a printed message might sound…).

    Mr. Meyer, I read in a yahoo article that you taught math. For how long did you teach before moving on to advanced studies?

  8. Oops. Just checked out the “About” page. Found my answer. 6 years. More later, gotta grade papers!

  9. I find this to be quite true on so many levels. If there are no numbers in the problems my students don’t think they are even doing math. Our new CCSS curriculum has a lot of scenarios like this and I love that it is challenging not only their math abilities but also their critical thinking skills. If they understand the reasoning behind problems, then most times the numbers (and/or variables) become a moot point.

  10. It goes back further than Bannon has taken it.

    In my Algebra 1 and Geometry classes I like to pull out and show some of the word problems (with worked-out solutions) found on clay tablets in Old Babylonian excavations. Apart from the base-60 number system, these are pretty much identical to what we might see in a textbook today.

    To give one favorite example (I think this one comes from a Sippar in the 9th century BCE but I’d have to check):

    “A textile. The length is 48 [rods]. In 1 day she wove 0:20 [rods]. In how many days will she cut [it] off [the loom]?
    You solve the reciprocal of 0:20. You will see 3. Multiply 48 by 3. You will see 2:24. She will cut [it] off after 4 months, 24 days.”

    (I especially like to show a picture of this tablet because the scribe ran out of room at the bottom of the tablet and the second half of this particular solution is marked along the bottom edge. Students can usually sympathize with this if they have ever run out of room on a page while solving an equation).

  11. Very interesting book. Great premise and some great problems. A lot of figures in it though, just in word form. I had hoped it was more pure. Anyway, here’s some more that help flesh out the possible a bit:

    Describe the configuration of scales on a snake using 3D transformations.

    Given a closed two-dimensional shape, invent a statistic that describes how “round” the shape is. When your statistic is applied to a circle, it should yield the highest possible score.

    A tree’s primary branches are those branches that sprout directly from the trunk. Secondary branches are those that sprout from a primary branch. Tertiary branches sprout from secondary ones, etc. What is the ratio of a tree’s trunk circumference to the circumference of it’s thickest primary branch? What about the ratio of the circumference of a tree’s thickest primary branch to the thickest secondary branch? Thickest secondary to tertiary, etc?

    How many kisses will happen today?

    How close can you get the tip of your right index finger to the tip of your right elbow? What is the farthest distance you can make between that finger and elbow? Answer the same two questions for your right index finger and left elbow.

    Imagine two line segments that are connected at one end. What could this be a model of?

    How much do the raw materials used to make a single wooden pencil cost?

    A lattice point is a point in the coordinate plane with integer coordinates. Is it possible for a line to go through exactly one lattice point?

    Think of as many words as you can that are related to geometry and contain the letters c and t.

    Make a serious effort to find the best word or phrase to complete the following analogy and explain your choice. Math is to beauty as ___________ is to emotions.

  12. @Michael, let me try one more out:

    Developing the question means attending to processes that interest normal people before attending to the processes that interest mathematicians.

    Too glib perhaps.