Marveling At The Historical

Math Oldies But Goodies

  • About This Blog

    This blog is mostly about math procedures in textbooks dated from about 1825-1900. I’m writing about them because some of the procedures are exquisite and much more powerful, and simpler, than some of the procedures in current text books. Really!

    I update this blog as frequently as possible ... every 2-3 days. And, if you are a lover of old texts and unique procedures, you might want to talk to me about them, at I’m not an antiquarian; the books I have are dusty, musty, brown-paged scribbled-in texts written by authors with insights into how math works. Unfortunately, most of their procedures have vanished. They’ve been overcome by more traditional perspectives, but you have to realize that at that time, they were teaching the traditional methods.

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Rephrase That Impossible Application Problem

Posted by mark schwartz on July 19, 2016


As I was presenting a topic one day, a student said that what I was saying didn’t make any sense and could I please say it differently. My first reaction was to ask the class if it was true for them too; some agreed. It wasn’t a rude statement and I took the comment seriously and did rephrase what I said and asked if that made more sense and apparently I got it right. Then I got to thinking not only about that moment but other moments where what I was saying may not have made sense, but nobody bothered to stop me. As far as I can tell, it wasn’t the math content but the language I used to describe the content that bothered them. Thus the story that follows.

The Story

Question: Let’s say that the first city 4th of July fireworks I attended was in 2005. Since then, I attended the city fireworks every year including this year, 2016. How many fireworks have I attended?

Before considering the answer, consider if that question is the same as: how many years have I attended the city fireworks display on the fourth of July?

If your answer to the first question is 11, you’re wrong and as vague as the second question is, the answer is 11.

In both cases, which seems to be the same case, I suspect you got your answer by simple subtraction, 2016 – 2005 = 11. The thing to consider though is what exactly is being subtracted? Let’s bring this into a more manageable range, like 5 – 1. If you were to do this operation on a number line, you could put your finger on the five and move to the one, counting as you go and thus you would get 4. That four represents the number of movements from point 5 to point 1 on the number line. When you move from 5 to 4, you say “1”, in essence, “scaling” the distance between 5 and 4 as 1 unit, regardless of the actual distance. Given that it’s a number line, the distance between the points on the number line will all be the same. So, when we say 5 – 1, we are asking how many distances are there between 5 and 1. By the way, this distance analogy is similar to the idea of having 5 kittens and giving 1 away – how many kittens have you? In this case, it’s not distance, it’s kittens but conceptually it’s the same. We need not bother with scaling the number of kittens, because it’s a quantity not a distance, although some consider distance a quantity. As far as “how many years have I attended the fireworks?”, what’s being counted here is the number of years – an amount of time scaled rather than a distance. So, 2016 – 2015 is one, etc. as far as counting.

What’s the point? Remember the first question? To repeat: “Let’s say that the first city 4th of July fireworks I attended was in 2005. Since then, I attended the city fireworks every year including this year, 2016. How many fireworks have I attended?”

What is being counted here? Again, consider the number line. We’re not counting the distance between points on the number line rather were counting the number of points. The first question then has to be a subtraction plus one, which really is asking for the inclusive count.

You might say “so what?” to the difference between the first and second questions but looking at them as I did points out that there is a difference. The real issue here is the nature of asking questions in a math class. If we, as instructors, ask ambiguous questions, or questions which require students to reflect on the context of the information as well as the information in the question (and students don’t see the need to reflect on these issues) then we are, in a sense, misleading them and adding to their confusion about math. The context in this case is the words we instructors use.

I’ve seen this in questions in texts. We glibly accept the questions and answers at the end of the chapter and if some of those questions are questionable, we simply don’t assign them. But it’s not just the questions in texts. It’s how we state information, it’s how we use the language to structure questions and present concepts. The difference between the first and second question demonstrates this.

We should be attempting to be better at some precision in our questions and presentations because, like it or not, instructors are math role models for students. If we expect precision and accuracy from students, we should also expect that they can phrase good questions and it’s the instructor that establishes the idea of a well-phrased statement. It also seems it’s a critical component of being able to arrive at a correct answer to a problem. The caution to read “word” problems until you understand it is reasonable, but what if you never “understand” the problem? My thought is that students have to have license to and practice in rephrasing problems, without changing any of the relationships in the problem.

For example, when teaching percent using the percent proportion model (you can see how this is presented in the Percent Proportion posting in this blog), I point out to students that most percent problems can be rephrased. An example: A farmer sold 180 sheep, which represented 16% of all the sheep he had. How many sheep had he after the sale?

There are a lot of extra words in this problem, but only two numbers. I asked the class to rephrase this problem focusing on the relationship between the numbers. The students wrote all their attempts on the board, so that we could discuss the thinking that brought about their answer. What they produced, based on the percent proportion model that I presented to them, were two rephrased problem: First, “180 is 16% of what number of sheep?” to get the total number of sheep and then subtract the 180 he sold. The second was to see that if 180 sheep were subtracted from the total (written as x – 180), this would represent the number of sheep left and the percent left would be 84 (100 – 16). The rephrasing then would be “The number of sheep remaining is 84% of what number of sheep?” The result of this rephrased problem still is the total but again, simply subtract 180 sheep. If you want to play with it, the answer is 945. Even if the percent proportion model isn’t used, this rephrased problem is much more manageable.

But here’s how to set up both results using the percent proportion model.

180  16
 x   100      … solving, x = 1125 … total after sale = 945
x - 180  84
  x      100   … solving, x = 1125 … total after sale = 945

We practiced this rephrasing idea some more and I reminded them that they don’t have to rephrase every problem, but if the problem seems “impossible”, rephrase it.


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