Posted August 21, 2017
(By Dean Ballard, CORE’s Director of Math)
Students are more concerned that a math problem be engaging than that it be real-world. I have had my share of students asking, “When are we ever going to use this?” However, I have never had students ask this question when they found the math interesting. Problems that students find interesting to solve are the ones that are relevant to the students. Hattie, Fisher, and Frey (2017) state that “relevancy is a major condition for transfer learning” (p. 183). Students need problems that are different and that provide meaningful applications of the mathematics they are learning in order to maintain their interest and attention (Sousa, 2008). Woodward et al. (2012) call these problems that provide meaningful applications “non-routine” problems because they “force students to apply what they have learned in new ways” (p. 12).
Word problems in math text books may be real-world but are rarely interesting problems. For example, here is a word problem that is real-world:
Johnny dove (with scuba gear) down to 60 feet below the ocean’s surface. When it was time to return to the surface he rose at the rate of 1.25 feet per second. How long did it take Johnny to reach the surface?
The scenario is realistic and it requires the use of specific math knowledge to solve. In the text book, there may even be an appealing illustration next to the word problem of a diver swimming in the ocean. It is important to recognize that problems like this provide important practice for students to apply facts and procedures previously learned. Unfortunately, few students find a problem like this engaging because it does not spark their imaginations or interest. Along with routine problems used for practice, we need non-routine problems that not only require application of previously learned skills and concepts, but maintain interest and attention.
Non-routine problems come in many forms. Real-world problems can be introduced with a little kick. For example, suppose the diver problem above was introduced with a video from a movie showing a diver ascending because he or she is trying to get away from a shark. Then we have an interesting situation. I could ask students to figure out the rate at which the diver ascended, determine if the diver was likely to suffer from decompression sickness (the bends) by ascending too quickly, and decide which was worse, the bends or the shark (may also discuss what would be a better strategy to keep from becoming shark bait). With this scenario, a diver rising from the ocean depths goes from mundane to interesting.
Many hands-on problems are very engaging for students and when done well lead to extending or deepening important mathematical knowledge. For example, students can choose and use linear and exponential models to predict bounce heights for tennis balls based on data they collect. Students can also create, compare and use linear models for rope lengths based on the number of knots they untie from the ropes. Another example is for students to determine a rule for a relationship among the lengths of the sides of a triangle based on an analysis of triangles students create using various lengths of sticks. Although these and many other activities I have used are contrived situations, students find these types of activities interesting and therefore relevant to them. I call these “unreal” world problems with real math connections. For other engaging real-world and unreal-world math tasks see my list of resources above.
Many great math tasks are simply math challenges. These are interesting to solve and mathematically beneficial for students. Examples of math challenges include KenKen puzzles (KenKenPuzzles.com) and problems in Spend Some Time with 1 to 9 (Ballard, 2014). Students see these as puzzles rather than as math problems.
Fifteen years ago, a wise math teacher told me students are only problem-solving if the problem they are solving is one for which they do not already know the route to the answer and if students are interested in solving the problem. Without these conditions, it isn’t problem-solving for that student because either it is just practice or it is unimportant. Students are most engaged when they find math interesting. Real-world problems are no guarantee of interest or perceived relevance to students. However, real-world problems with a kick, unreal-world problems, and math challenges are usually interesting and engage students in solving non-routine problems. Interest generated by these problems makes the math relevant to the student at that moment and that relevance leads to greater learning.
Ballard, D. (2014). Spend Some Time with 1 to 9: Building Number Sense and Fluency Through Problem Solving for K-8. Oakland, CA: Consortium on Reaching Excellence in Education
Hattie, J., Fisher, D., & Frey, N. (2017). Visible Learning for Mathematics. Thousand Oaks, CA: Corwin.
Sousa, D. A. (2008). How the Brain Learns Mathematics (p. 138). Thousand Oaks, CA: Corwin Press.
Woodward, J., Beckmann, S., Driscoll, M., Franke, M., Herzig, P., Jitendra, A., & Koedinger, K. R. (2012). Improving Mathematical Problem Solving in Grades 4 Through 8 (NCEE 2012-4055 ed.). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from https://ies.ed.gov/ncee/wwc/PracticeGuide/16
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