Do "Motivating Problems" Really Motivate Students?
Summary
Many "motivating problems" in math are practical but are unlikely to garner investment from students. Although such problems tend to be realistic and accessible, they aren't very compelling. Typical motivating problems are boring and don't facilitate context-independent learning. In this post, I instead propose that motivating problems should be stories written with minimal technical language and including tangible scenarios. These sorts of problems have potential to improve student engagement, investment, persistence, and understanding.
The problem is in the problems
Mary needs to order pizza for 18 students. Each student should get ¼ of a pizza. How many pizzas should Mary order? How much pizza will be left over? (From Fraction Multi-Step Word Problems)
You are building an armadillo exhibit at the zoo. You have 18 armadillos, and each one needs a quarter of a bag of cotton for its bed. How many bags of cotton should you buy to make sure every armadillo is comfortable? Will you have any cotton left over?
Which of these would you prefer to solve if you were not yet comfortable multiplying fractions by integers? Which would a student prefer?
These are both examples of motivating problems: questions which provide justification for learning a mathematical concept or technique. Teachers commonly use motivating problems to introduce new content. If students need the new content to solve the motivating problem, the thinking goes, they will care more about what they learn in class. A motivating problem could also appear shortly after useful tools are introduced as justification for those tools. However, just because a problem justifies a tool it does not necessarily make the tool compelling. Motivating problems often do nothing to actually motivate students. The issue is, real-world utility is not the most relevant metric of whether a problem invests a student in class content. While it is true that we often use math in the "real world," students will not care about math just because it is useful. Furthermore, our goal is to enable students to use math where they may need it, not necessarily to force them to actually use it in that context.
Motivating problems are foundational to the problem-centered approach advocated by Kahan & Wyberg (2003). Under this paradigm, "students’ engagement in making sense of the problem is the mechanism by which they come to understand the underlying mathematics." A good problem should therefore connect naturally to the topic being taught, allowing students to discover mathematics for themselves. In so doing, students access mathematical tools through discovery, thus understanding and remembering the tools better than they would under directive (lecture-like) teaching. Pursuing engaging problems for a similar approach, Crespo & Sinclair (2008) describe three aesthetic characteristics that can make a problem "mathematically interesting": surprise (unexpected outcomes), novelty (divergence from familiar problems), or fruitfulness (access to abundant further learning). Naturally, we would like to write motivating problems to possess some or all of these characteristics.
Typical characteristics of motivating problems
In my experience, familiar motivating problems tend to share some traits in common:
- They are realistic and often practical.
- The full conditions of the problem are explicitly presented so that they are easy to translate into symbolic mathematics.
- Students already have most or all of the tools the problem is meant to justify.
- The connection between content and problem is arbitrary.
Consider this example from a private tutoring website:
The perimeter of a rectangular pool is 56 meters. If the length of the pool is 16 meters, then find its width. (Word Problems, 2022)
The problem is realistic: a pool is a fairly mundane object, and it could reasonably have the dimensions in this problem. The conditions are explicit: the words “perimeter,” “rectangular,” “length,” and “width” all appear, cuing students as to the relevant mathematics. The students already know most of the underlying content: the question is posed using vocabulary that a student would only know from prior experience with the underlying mathematical ideas. In short, the problem could replace the phrase “rectangular pool” with “rectangle” and be functionally identical, so there is little point in using a word problem at all.
This problem presents a scaffolded example of why someone might want to know how to work with perimeters; it does not, however, justify why a geometry student would care in the slightest. It perhaps fails most dramatically at getting initial interest from a student. The problem could theoretically be interesting in the surprising-novel-fruitful sense of Crespo & Sinclair, but if so, a student has no way to see that before actually attempting the problem. A mathematically interesting problem is not the same as one that engages students enough to get that far. Unfortunately, most real-world applications of grade-school math are not fun. In fact, if a student first sees a technique applied in a math class, they may view it with the same attitude as the class as a whole. Since so many students already dislike math classes, a length-perimeter pool problem is more likely to get them to dislike measuring pools than to like math. What geometry student wants pool time to feel like math class?
The traditional way of posing motivating problems also fails to foster persistence. A successful solution to a boring problem is practical, but it's not satisfying. As teachers, we hope that something about the mathematical insight itself — such as its novelty or surprise — will satisfy. Yet, not all students like math as much as we do. Expecting the math to speak for itself is a lovely but impractical thought.
Traditional motivating problems also do a poor job of facilitating understanding despite that being their ultimate purpose for existing. They “feel like math problems” and so provide highly context-dependent learning. It may appear that if a problem presents realistic context, such as pool measuring, students will more easily recognize an opportunity to apply math in real life. I hold, however, that doing a math problem on a desk never resembles going outside to measure a pool. A student only associates perimeter with the feelings of math class.
One possible remedy is tangible problems with physical manipulatives, as suggested by Moch (2002). Unfortunately, not every math class can be a new physical problem, nor do all topics lend themselves well to a physical project in a classroom. We still need ways to pose compelling motivating problems on paper.
A better approach
We have seen that interesting mathematics alone doesn't always make for compelling problems. Crespo & Sinclair’s criteria are about the content, but they generally neglect the form of the problem — despite their 2008 paper including fantastic examples of immediately attention-grabbing problems. They give the example of tangrams, which resonated personally for me. Even as a first grader, I enjoyed playing with those tiles for their own sake, on a level that had nothing to do with math class. Tangrams are puzzle-like: I experienced many little “aha!” moments while playing with them.
I propose three ways to make motivating problems more compelling, investing students in the math enough to discover the underlying beauty:
- A fun story. Students derive tools for a purpose they might have appreciated outside of the classroom.
- Tangibility. The problem regards something a student can easily picture.
- Non-mathematical language. The problem does not directly hand the expectations to students.
Functionally, these characteristics combine to make a more game-like or riddle-like problem. Here, I have rewritten the rectangular pool problem according to these principles:
You’ve discovered a super-stretchy material, which you want to use to make the best rubber bands in the world. Your rubber band can stretch to a total length of 56 inches around. Your secret laboratory notes form a stack of notebooks 16 inches wide. If you want your rubber band to hold all the papers together, how tall can the stack be?
Is a given student likely to handle a giant rubber band anytime soon? Of course not, but the story is fun. That whimsy makes the problem feel worth thinking about. In addition, the story provides a reason why we might care about a distance: it corresponds to the height of the stack we want to bind. Furthermore, the problem is scaled down from a large pool to something tangible and familiar, a rubber band around some papers. Finally, this formulation uses only one math-like word, "length," leaving students to analyze the situation for themselves.
This paradigm of a tangible story written in plain language has several advantages over more traditional motivating questions. First and foremost, it engages students from the get-go. A student is more likely to give the problem earnest thought and get to the mathematical meat. Once they are invested, students hopefully take interest in the underlying problem. If the problem is interesting by Crespo & Sinclair’s content-focused definition, students get deep enough to find that out.
My approach also fosters persistence in problem solving. By investing students in the story, the rubber band problem transforms an academic goal into a more game-like one. Where solving the pool problem means doing well at school, solving the rubber band problem feels more personal. This kind of success is comparable to video games — and as with video games, students need not literally believe they are in the situation in order for a story to contribute to the sense of satisfaction. Moreover, the removal of technical language makes dead ends in reasoning less discouraging. A student may engage in thinking that does not lead to a solution, but unlike with traditional motivating questions, the student is not left with the feeling of being bad at math. How could they, when the problem hardly feels like math in the first place?
The third advantage is an improved understanding of the underlying mathematics. When students see keywords like "perimeter," they pick a formula from a sheet and use it regardless of how meaningful it is to them. In the rubber band problem, students truly have to derive the appropriate tools because almost no explicit cues exist. One must first understand the situation of the problem and work forward from that understanding. Hence, the only way to choose an appropriate formula is to know why that formula works. Students may also proceed without a formula but unwittingly invent one in the process of solving the problem using more intuition. Because the problem feels less like a traditional math problem, it is also associated less with the formulae and equations of math class. The main experience of solving the problem is one of analyzing the situation, not of grinding through numbers. As a result, the skills transfer better to real-life applications that may look very different from the specific problem. The problem is also memorable in its own right because of the silly story, so it is an easier point of reference than a rectangle which happens to be a pool.
A story-based approach has an additional benefit of framing the goal of the problem practically rather than around the tools a teacher expects students to use. Hence, these problems make the goal a priority. Schoenfeld (1983: 348) describes an issue in posing certain problems: students, knowing that a teacher is observing, try to demonstrate any "mathematical behavior." This "mathematical behavior" is meant to look fruitful but does not always pursue an effective solution to the problem. However, when the problem does not present itself as so mathematical on its face, problem solving is shielded from this ill.
Potential concerns
I foresee several concerns teachers might raise about posing problems in the way I suggest here. First, stories are harder to read than simple, formulaic problems. Complex sentences may be an especially steep challenge for young students, English language learners, and students with some learning disabilities. That said, teachers in reading-heavy subjects like ELA and history work with these populations frequently. Furthermore, it is not the words of the problem that matter for math, it's the meaning behind the words. We can speak them, we can explain the problem in alternative terms, and we can use diagrams. Multimodal input is a facet of universal design and supports learning for all students, not just those who explicitly need extra supports. In this way, a story-based problem is easier to understand than a traditional one.
Teachers may also worry about the extra time investment in writing problems with more creativity and new language constraints. I posit, however, that writing of any sort is a skill that can be practiced and built. In my own experience, these problems are fun to write, certainly more fun than formulaic problems. Still, I concede that fewer such problems exist among online resources, so there is bound to be a greater time investment.
The issue of time investment also goes for students. When the problem doesn't describe the situation in explicit mathematical terms, students must engage in a lengthier process of interpretation and analysis. I fully acknowledge that students may spend longer on a given story problem, but they also learn more from that problem than they would from a traditional word problem. If students learn more, I have no problem with taking more time to do so.
Finally, we might be worried that without technical language, a problem becomes fuzzier and leads to less rigorous thinking. On the contrary, I expect that a functional rather than procedural understanding of the problem gives students a clearer idea of what constitutes a solution. Stories carry meaning, including the reasons we seek some solution. As a result, stories provide goals more naturally than rigid, technical questions do.
Recap
Traditional motivating problems represent simple, realistic scenarios using technical language that makes appropriate methods easy to infer. In pursuit of greater engagement and understanding, I believe that story-based problems serve students better than other word problems. Mathematics should be a process of discovery. Motivating problems are fantastic — but they ought to foster that discovery. In order for students to experience the joy in rediscovering mathematics, we need motivating problems that get them far enough into math to see the beauty. Students deserve better than procedure. They deserve to love math from the moment they sit down for class.
Sources & further reading
- Crespo, S., & Sinclair, N. (2008). What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems. Journal of Mathematics Teacher Education, 11(5), 395–415. https://doi.org/10.1007/s10857-008-9081-0
- Fraction Multi-Step Word Problems. (n.d.). Closing the Math Achievement Gap. Retrieved August 9, 2022, from http://www.uwosh.edu/coehs/cmagproject/many_word/documents/fraction_multistep.pdf. Now available only through the Internet Archive
- Kahan, J. A., & Wyberg, T. R. (2003). Mathematics as Sense Making. In H. L. Schoenfeld (Ed.), Teaching mathematics through problem solving: Grades 6-12 (pp. 15–25). National Council of Teachers of Mathematics.
- Moch, P. L. (2002). Manipulatives Work! The Educational Forum, 66(1), 81–87. https://doi.org/10.1080/00131720108984802
- Schoenfeld, A. H. (1983). Beyond the Purely Cognitive: Belief Systems, Social Cognitions, and Metacognitions As Driving Forces in Intellectual Performance*. Cognitive Science, 7(4), 329–363. https://doi.org/10.1207/s15516709cog0704_3
- Word Problems: Area and Perimeter of a Rectangle. (2022). Varsity Tutors Hotmath. https://www.varsitytutors.com/hotmath/hotmath_help/topics/word-problems-area-and-perimeter-of-a-rectangle