Wait! What are we counting? On Ambiguity and Units

  • Wait! What are we counting? On Ambiguity and Units

    By Christopher Danielson, posted October 07, 2016 —

    In my last post, I argued that ambiguity is an important source of mathematical activity and that it is too often overlooked in favor of certainty in math classrooms. Although certainty and right answers have an important and unique role in mathematics, we would do well to offer ambiguity to students as well. 

    My examples in the earlier post were of geometry. In this post, I’ll introduce you to the possibilities afforded by ambiguous number and counting tasks. Ambiguity in counting usually stems from unstated units. Many situations allow for use of multiple units; a simple example is eggs. At my local grocery store, I can buy eggs in a container labeled 2 1/2 dozen. Why are 30 eggs labeled as 2 1/2 dozen, but 18 eggs are labeled just as 18 eggs? Who knows? But it makes clear that counting eggs has two important units: (1) eggs, and (2) dozens of eggs.

    There is no ambiguity when buying eggs at the grocery store because the units are clearly labeled. But I recently had lunch with my son at a Japanese restaurant that had its menu displayed on wooden slats on the wall. One of the menu items caught my eye.

    2016_10_07_Danielson_2_Fig1

    In this case, there are two units—pot stickers and dollars. Which number refers to which unit? How do you know? The ambiguity of missing units gives us a chance to discuss. Before you read further, stop the nearest person and ask him or her which number refers to which unit. I’ll wait.

    Now contrast that conversation you just had with what likely follows from this unambiguous textbook-like treatment of the pot sticker situation:

    A Japanese restaurant sells pot stickers 4 for $4, 8 for $7 or 12 for $9. What is the price per pot sticker for each of these servings?

    The pot sticker situation is one with a right answer—one of those numbers really does refer to dollars, and the other really does refer to pot stickers. Next, let’s consider a couple of situations that have multiple right answers. Look at the picture below and consider the question, How many?

    Did you say two? Then you were probably counting shoes. Did you say one? Maybe you were counting boxes or pairs of shoes. Other common answers are four and twenty. What are the people with those answers counting? “How many?” is an ambiguous question because it doesn’t specify what you should count. 

    In most elementary school classrooms that I have visited, asking this question quickly develops into a wonderful game in which students find new things to count, challenge themselves to count more difficult items, and even assign informal probabilities to their counts. (“There are probably about fifty yellow stitches, but there could be more—or less—because I can’t see them all.”)

    2016_10_07_Danielson_2_Fig2

    Another fun “How Many?” prompt is about the following photograph.

    2016_10_07_Danielson_2_Fig3

    One, seven, eight, fifteen, and seven-and-a-half are all common answers here. Each of them is correct as long as it is attached to the right unit; and, of course, there are probably more possibilities that your students will see that I haven’t anticipated. 

    The ambiguity of the question, “How many?” with unspecified units can lead to discussions of whether a half-avocado counts as a unit (answer: yes) and to the question of whether there is one-to-one correspondence of pits to avocados (answer: not in this photo). These are deep, important, and, above all, mathematical questions that come to the surface because the task allows children the space to think and to be creative.

    Your turn

    I hope you’ll share these tasks with your students and colleagues, and I especially hope you’ll share the results of doing so—either here in the comments or on Twitter (I’m @trianglemancsd). What new possibilities do you see opening up as you and your students wrestle with unstated units and ambiguous counting in math class?


    2016_09_26_Danielson_1auPic

    Christopher Danielson is on the teaching faculty at Desmos, which offers a set of free digital math tools along with a growing library of activities developed by the community of users. He is the author of two books—Common Core Math for Parents For Dummies, and Which One Doesn't Belong? A Shapes Book. You can find more of his writing at his website: Talking Math with Your Kids. For full-size, classroom-ready versions of the images used in this post, go to http://talkingmathwithkids.com/2016/10/10/images-for-counting-and-units/.

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