Let's Make It Real!
In this paper, Jim Fey, one of the CMP authors, discusses the dilemmas of writing a problem-centered curriculum using contexts. The paper also offers suggestions for how teachers can make the problem settings more relevant for their students.
Let's Make It Real
February 23, 2016
Two of the most challenging tasks for mathematics teachers are convincing students to persevere in solving non-routine problems and answering the related student question, “When am I ever going to use this stuff?” As if these issues needed emphasis, the latest report from the Program for International Student Assessment (PISA) showed that in mathematics, the U.S. ranked 26th in the world, with American students having particular trouble in geometry, modeling, and real-world applications of mathematical concepts.
The best way to develop skill and confidence in solving non-routine problems is to spend a great deal of time engaged in that activity. So instructional materials for each unit in the Connected Mathematics Project (CMP) curriculum present students with challenging problems on a daily basis. Each problem is introduced with a short story that sets the context and poses the big question to be answered. Then more specific questions provide guidance for solving the problem and identifying the key mathematical understandings embedded in the problem.
In response to the predictable student question about when they will ever use what they are being asked to learn, problems that Connected Mathematics authors develop are frequently embedded in contexts that we believe will intrigue students and engage their attention. Some of those problem contexts are whimsical or fantasy settings like the distortion of Wumps in Stretching and Shrinking and the exponentially increasing ruba rewards by the King of Montarek in Growing, Growing, Growing. Other problems have purely mathematical settings like the Factor Game in Prime Time and the Quadrilateral Game in Shapes and Designs.
In many other problems we have tried to use names, data, and pictures that describe real people, products, and places in the context stories. Unfortunately, a variety of legal concerns make commercial publishers wary about such realism. They fear that a world record holder in some athletic event might, sometime in the future, be shamed by charges of using performance-enhancing drugs or by committing a dastardly crime. So editors replace stories and names of real athletes with generic tales about fictional people. The publishers often need permission to use photos of interesting natural or man-made structures. Since Connected Mathematics is sold overseas, international copyright laws also influence content. Such permissions require time of the publisher’s legal staff and payment of fees to the providers of photos. So potentially engaging graphics are often deleted from the initial text. For the same reasons, data and stories about real name-brand products like bicycles, shoes, clothing, or amusement park rides are usually replaced by fictitious names and data.
Fortunately for students and teachers of Connected Mathematics, only a modest effort makes it possible to replace or embellish the generic contexts of many CMP3 problems with stories about real people, places, and events. With simple Internet searches, teachers and students can locate names, data, pictures, video clips, and stories that provide real problem contexts for developing the mathematical ideas in our curriculum. You might want to use that realism only to help students see the importance of a textbook mathematical investigation set in more generic context. Or you might choose to modify the text presentation to include different realistic data, people, products, and places. In the examples that follow, we give a few of our ideas and concerns about how CMP problem contexts can be modified or enhanced to give a stronger sense of realism to the work we are asking of students. We are sure that creative teachers and students can add to the supply of resources for convincing students that school mathematics is truly preparation for real life.
Comparing Olympians — The first investigation of Decimal Ops in grade six begins with some questions that ask students to think about ways that decimals are used to report various kinds of measurement data. The first example says,
A sixth grader might win a 100-meter race with a time of 13.36 seconds.
Then the first two parts of Problem 1.1 ask students to plan calculations that would compare fictional winning times of fictional students at a fictional middle school.
Comparisons of running speed are of great interest to many people who follow track and field sports, especially in Olympic Games years. The winners of the Olympic 100-meter race are usually celebrated as the world’s fastest men and women. To connect with this real world interest in foot speed, you might use the following replacement for the launching example.
The world record in the men’s 100-meter dash is 9.58 seconds
set by Usain Bolt of Jamaica in 2009.
If you Google ‘Olympic sprint videos’ you’ll quickly be led to some intriguing footage and data about runners that is sure to spark conversation in your class.
To make further connection to real runners and their times, you could replace the problem parts about fictitious middle school runners with questions like the following examples that involve real record holders (their pictures are easy to find on the Internet as well).
For each of the following situations:
- The world record for the women’s 400-meter race is 47.6 seconds, set by Marita Koch of East Germany in 1985. The world record for the men’s 400-meter race is 43.18 seconds, set by Michael Johnson of the United States in 1999. What is the difference between the two record times?
- Suppose that Marita Koch and Michael Johnson were to run in a 800 meter relay at their world record speeds. Marita runs 400-meters and tags Michael who runs the last 400-meters. How long would it take them to run the total distance of 800 meters?
Like our publisher, you might wonder about the consequences if one of the real people whose accomplishments you ask students to analyze is subsequently caught in some embarrassing faux pas. We believe that, instead of causing a disturbance in class, such an occurrence will provide a powerful teachable moment. Drugs, sex, guns, fraud, gangs, politics, and dozens of other illegal acts are part of the world in which kids live and which they must learn how to deal with. They know about athletes and entertainers (and ordinary citizens) using illegal drugs and they have to make personal decisions about the consequences if they follow that path themselves. Discussing the negative consequences for public figures who cheat in one way or another seems more educative than pretending that such activities don’t exist.
Real Games — Not surprisingly, several of the game contexts for CMP problems are only thinly veiled takeoffs on real games. In fact, the text introduction to Problem 1.1 in Accentuate the Negative alludes to such a connection to Jeopardy, and rare is the middle school teacher who has not used that connection in a review lesson at some time. In Connected Mathematics, the Math Fever game is used to connect with students’ prior exposure to negative numbers and provide context clues for operating with them.
It can be productive to make connections to other television game shows in other CMP units. For instance, on the syndicated version of Who Wants To Be A Millionaire, contestants often face the dilemma of whether to make a guess on a question, with the risk of losing almost all of their money earned to that point, or to quit with a lesser guaranteed amount. From a mathematical perspective, this problem calls for calculation of expected value for a random variable—probability of success or failure combined with potential win or loss associated with each possible outcome. So the real-life Millionaire game can be used to motivate fundamental questions in the CMP What Do You Expect? unit.
The Millionaire program and other television game shows are also the basis for several questions in the CMP Function Junction investigation of arithmetic and geometric series. In many shows the prize values increase in various kinds of progressions—often exponential doubling (as in 100, 200, 400, 800, 1600, …). So asking students about their favorite game shows can be a real-life connection to launch mathematical investigations. It turns out that the prize values on Millionaire as of 2016 increase in a sequence that is neither arithmetic nor geometric (500; 1,000; 2,000; 3,000; 5,000; 7,000; 10,000; 20,000; 30,000; 50,000; 100,000; 250,000; 500,000; 1,000,000). So one real television game show provides many interesting openings for mathematical analysis and learning experiences.
Comparing Products — One of the classic CMP problems—used to launch study of ratios, proportions, and percents—presents data from a soft drink taste test comparing Bolda Cola and Cola Nola. The obvious real life comparison suggesting this particular context is competitive comparison of preferences for Coke or Pepsi. Once again, it seems quite possible that asking students in your class for their preferences among various available soft drinks would engage them in the underlying question that structures Problem 1.1 of Comparing and Scaling. But mentioning those real products in the commercial textbook has permission complications.
The cola taste test problem also raises another issue that could stimulate discussion about the context—whether so-called ‘junk foods’ should or should not be available in schools. To focus on this nutrition issue you might alter the problem as presented to compare preferences for some popular soft drink and a more healthful alternative. This question makes connections across disciplines and it has the potential to raise the important point that few problems can be solved solely by mathematical analysis—expressions of preference might well conflict with health policy advice!
Comparing Real People—The Common Core State Standards for grade eight statistics call for developing student understanding of correlation and variation of data. One context we use for developing those understandings and skills is a claim by the first-century Roman scientist Vitruvius that for most humans their height is approximately the same as their arm span.
When we looked for an engaging launch to the question of how height and arm span are related, we thought naturally about basketball players for whom both height and arm span are important traits. We proposed launching Investigation 4 of Thinking with Mathematical Models by showing an action photo of NBA star Kevin Durant and providing his height and arm span data (both readily available on the Internet). Unfortunately, because of the ‘no real people allowed’ rule in commercial publishing, we were constrained to a staged photo of two nameless young men in generic gym clothes. But you can easily find relevant data, video, and still photos about well-known (male and female) athletes. You might even use such data to inject some doubt into the minds of students who claim no need to study because they will be rich NBA basketball players someday!
Cutting Both Ways—One of the Common Core State Standards geometry objectives for grade seven calls for students to develop visualization skills that allow them to predict the shapes formed when solid figures are cut by planes. In a near final draft of Filling and Wrapping, this CCSS objective was addressed by a problem that we introduced with a story about carving ice sculptures at the Saint Paul, Minnesota Winter Carnival and a picture of a sculptor using a chain saw to carve an ice dinosaur. It might not surprise you that the publisher expressed concern about that photo, worrying that such material might entice students into dangerous chain saw experiments and make the publisher liable for damages. You might have the same fears. But since chain saw sculpture is a fairly well known artistic medium, it seems fair (and intriguing) game for introducing mathematics that ends up being far less artistically interesting!
This particular problem suggests that some hands on experiments involving cutting materials like blocks of cheese or Styrofoam or Play-Doh would be helpful in developing abstract visualization skills. But teachers always worry about sharp instruments in the hands of students. The CMP textbook probably ends up being far more cautious than necessary, and teachers can certainly supplement that cautious presentation with at least some demonstrations in their own hands.
Real Data—If there is any set of CMP units where it is easiest to make the mathematics seem real, it would be those in the data analysis and statistics strand. Local and national newspapers display graphs of interesting data sets everyday. Internet searches will readily reveal data about almost any question students can imagine. Private and government institutions collect and report all kinds of data about people, places, organizations, and life activities. Using those data makes learning the basic concepts of statistics very real.
One of the key goals in the CMP statistics and data analysis strand is developing student skill in analysis of data distributions using the concepts of mean, median, range, and standard deviation. The usefulness of such analytic tools can be made very real by considering questions such as the distribution of earnings and wealth among American individuals and families. A simple Internet query will lead quickly to the U. S. Census Bureau web site and tables showing such data, with breakdown by race, age, and education level as well. In addition to the distribution of family income levels, the Census tables give summary statistics, so one can ask why the median income of about $50,000 is so much below the mean income of about $60,000. Not only does this sort of real data analysis underscore the meanings of key statistical concepts, it raises questions about social structure and equity that are almost certain to engage student interest.
Local Connections—The examples described above are all suggestions of ways that CMP3 problems might be adapted to make stronger connections to people, places, products, and activities in the real world of all students. There are, of course many ways that you can make similar connections by modifying the problems to fit people and situations in your individual classrooms and communities. Having students generate data from surveys of themselves or asking them to look for examples of mathematical ideas in their local environments can also be very engaging. For instance, the data CMP3 units offer many invitations to local surveys. The geometry units invite students to find examples of rigid structures and symmetries in local buildings, bridges, towers, and construction projects in progress.
In addition to addressing the question about usefulness of mathematics, urging students to see the mathematics around them can have the important further payoff of developing a broader disposition to reason mathematically in the wide range of situations where such behavior is appropriate. No amount of knowledge is meaningful unless it comes with a disposition to use that learning.