The Common Core Math Standards stresses modeling with mathematics. Modeling is a K-12 Practice Standard as well as a High School Conceptual Category. Developing a clear understanding of what is meant by modeling is the first step in being able to implement it in math classrooms. The next step is knowing which types of tasks elicit modeling practice in students.
Although most resources on modeling align it with real world tasks, I argue that a modeling problem does not have to be related to the real world. It can be any well-designed task in which a student has to create a mathematical model (equation, graph, table, flow chart) to solve. The connection to the real world does not by itself make a problem a modeling problem. The depth of creative thought needed to solve the problem is, in my opinion, the marker of a good modeling problem.
The idea for this post was stirred up last week as I was working with a group of high school geometry teachers. They had chosen tasks from the textbook bank of problems to include in a common assessment that would be administered to all geometry students in the school. Here are two of the tasks:
The door on a spacecraft is formed with 6 straight panels that overlap to form a regular hexagon. What is the measure of angle YXZ?
A pillow is the shape of a kite. Heath wants to create a design connecting opposite corners from point B to point D, and from point A to point C. Show your work to find the amount of cording needing. Round your answer to the nearest tenth.
A spacecraft door? A pillow in the shape of a kite? What is meant by cording? Where does it go? On the perimeter or on the diagonals? Or on both? Yikes! Both of these problems were a lame attempt to dress up a routine task in a real world context. It’s not that these tasks are trivial. They require students to apply theorems about shapes that we would like a geometry student to know. In the first one, the students need to know either (1) the sum of the interior angles in a hexagon, or (2) the sum of the exterior angles in any polygon is 360 degrees, (3) know that the angles in a regular polygon are congruent, and (4) know that each exterior angle and interior angle form an ordered pair. In the second problem, they need to know (1) that the diagonals of a kite are perpendicular and (2) the Pythagorean Theorem, and it helps if they know the shortcut of creating a 3-4-5 right riangle . In both of these problems, the diagrams give clues that help to scaffold the problem. Both problems attempt to engage students by masquerading as problems that real people might solve. However, it would be a stretch to call these problems that would help students learn to model mathematically. Real world does not necessarily imply engagement.
Some good resources around modeling can be found in the California Mathematics Project website. Most of the resources found refer to modeling being around real world phenomenon. Both 100 ideas for modeling and four modeling problems revolve around real life scenarios. In the CCSSM definition of modeling, the examples given are again focused on real life scenarios.The “How to Solve It” by G. Polya definition closely resembles the Modeling Cycle described in the CCSSM definition of modeling. The Modeling Definition by Henry Pollak introduces some examples of modeling that are not real world, but the final discussion focuses on modeling real world situations.
Here is one rather verbose, and perhaps too academic, definition of modeling found there. But the last line (bolding is mine) provided me with a great phrase to describe modeling. It is one of the few definitions that does not specifically require modeling to apply to to the real world. However, the use of the word “phenomenon” indicates that the author considers that the conceptual representation (mathematical model) describes an observable fact or event. Can a fact or event be non-real world? I believe it can be.
Modeling refers to the process of generating a model as a conceptual representation of some phenomenon. Typically a model will refer only to some aspects of the phenomenon in question, and two models of the same phenomenon may be essentially different, that is in which the difference is more than just a simple renaming. This may be due to differing requirements of the model’s end users or to conceptual or esthetic differences by the modellers and decisions made during the modelling process. Esthetic considerations that may influence the structure of a model might be the modeller’s preference for a reduced ontology, preferences regarding probabilistic models vis-a-vis deterministic ones, discrete vs continuous time etc. For this reason users of a model need to understand the model’s original purpose and the assumptions of its validity. Models are basically known to generate creativity from chaos.
source: Elwin Savelsbergh
I love that closing sentence. Our goal in “school math” is to give students plenty of opportunities to practice generating creativity from chaos. The ultimate goal is that students will transfer that knowledge of how to generate creative solutions out of chaos and be able to use it in their real life. That doesn’t mean that every opportunity we give students to practice this skill has to relate to the real world. There are plenty of interesting problems that build students modeling skills that have no immediate connection to a practical application in real life.
The paper Exploring Mathematics in Imaginative Places: Rethinking What Counts as Meaningful Contexts for Learning Mathematics by Cynthia Nicol (University of British Columbia) and Sandra Crespo (Michigan State University) published in School Science and Mathematics, Volume 105(5), May 2005 gives examples of modeling problems that are in imaginative non-real world settings. Here is an excerpt from the introduction:
This paper explores what happens when students engage with mathematical tasks that make no attempt to be connected with students’ everyday life experiences. The investigation draws on the work of educators who call for a broader view of what might count as real and relevant contexts for studying mathematics….
Findings suggest that students can indeed engage productively with mathematics when it is explored in imaginative settings and that such contexts can help students support and sustain their engagement with the mathematics in the task.
The authors quote Sierpinka (1995) to point out the challenges of forcing real life contexts into mathematics:
“If one wants to make students’ activity more authentic and take children out of school into supermarkets and other kind of environment…their cognitive activity will become more authentically every day, but much less authentically mathematical.”
They quote Egan (1992) who suggested that “the more distant and different something is from students’ everyday experience and environments the more imaginatively engaging it is”. Egan also “challenged the assumption that to engage students in learning subject matter requires that they make connections with everyday experiences.”
Brown (2001), has a similar take as Egan:
“If we can speak of what is ‘real’ in a more vibrant sense than what ‘exists’ or what we can ‘touch’ and ‘see’ then we not only legitimize more interesting connections between mathematics and the real world but we also suppress the need to seek real world connections as a slave against an otherwise ‘unreal’ world of mathematics.”
The authors summarize Brown’s and Egan’s ideas by saying:
“Both reject the notion that making subject matter interesting and meaningful to students requires the need for it to be placed in real-life contexts. Instead they challenge mathematics educators to imagine alternatives that engage students’ mathematical imaginations.”
The study focused on students’ emotional and intellectual engagement with two mathematical contexts that they consider having no connection to the real world. They call the contexts “mathematically imaginative”. They studied students working problems in two contexts. The first group, who worked with a place value task, was comprised of 36 pre-service elementary teachers, so one could argue if the findings can be generalized to K – 12 students. One could also argue that there is a connection in this activity to the real world. The second group was a class of 50 Grade 6/7 students who focused on ideas about space and shape in the book Flatland.
The place value activity was, interestingly, an activity that was included in an online history of mathematics grad course that I taught for a few years at Drexel University. The activity involves decoding an archeological document in which Mayan symbols, in base 20, are converted to base 10. The activity is from a book by Bazin and Tamez (2002) Math and science across cultures: Activities and investigations from the Exploratorium. 6_MayanCode. In the Mayan Code activity, “the exploration of Mayan symbols inspired student by presenting the context of human emotions and intentions (both the Mayan and archeologists) that helps to make the exploration of different numeration systems intriguing.” In the Flatland activity, “students were invited to imagine and visualize life on an infinite Euclidean plane – a space quite different from their own.”
The authors found that in both of these activities students generated questions not because the teacher asked them, but “out of genuine curiosity and wonder”. In their conclusion, they do not suggest that all mathematical activities need to be “embedded in contexts that are less than real. Our point here is that imaginative contexts, stemming from real or more imagined situations, can be important resources for mathematical engagement and thinking.”
I have recently stumbled upon a new blog Median by Don Steward. He has a wealth of rich tasks in which the student is asked to generate creativity from chaos. Here is an example of one such task, which is simple, but can be modeled in different ways.
An equilateral triangle and three isosceles triangles together make a rhombus. What must the angles in the rhombus be?
When I gave this task to a group of six high school teachers, every one of them approached it a little differently. Some of them created two equations with two unknowns. I think one person created three equations with three unknowns, and at least one person was able to find the answer with one unknown. The task itself is meaningless in the real world, but the task starts out as chaos and the person solving it slowly makes sense of the issue, thinks back to what they know that can help them solve it, and creates a way to approach the problem. A discussion following the solution could determine which solution was the most efficient, the least efficient, the most creative. The purpose of doing the task and then discussing it is to help students build up their creative muscles. In the next problem they are faced with, an insight they received by working and discussing this problem may help them to be more creative in their next solution.
Too often, we give students a lot of small problems that require no creativity [p. 135 (1 – 31 odd)]. Some of the problems masquerade as real world modeling problems. (Usually those are # 27, #29, and #31.) There is a place for drill in math class, just as there is in sports a place for practicing skills. However, unless we are asking students to regularly generate creativity from chaos…real world chaos, imaginative chaos, or abstract chaos…we are not preparing them for the chaotic world they will encounter outside of school.
If you think about it, any “real world” problem that we put into a textbook has already been partly digested before we ask students to chew on it. All of the needed information is given, or a diagram gives away a lot of information, or a table is set up that organizes the chaos for the students. Dan Meyer has become famous in mathematics education circles for challenging the types of problems typically found in textbooks. If you are one of the few reading this that has not watched his TED talk: Math Class Needs a Makeover, stop and watch it now. Dan, Andrew Stadel, and others have developed rich modeling problems in 3-Acts. Act I presents the chaotic situation which students need to make sense of, where they generate their own question and determine what information they need to know in order to create order out of the chaos. Act II is where the teacher provides just enough information to solve the problem, and only when the student asks for it, but does not prescribe a specific way to solve it. Act III provides the answer. I have found that students working a 3-Act problem not only build their modeling muscles, but also practice the other 7 CCSSM Practice Standards: make sense of the problem, persevere, reason abstractly and quantitatively, construct viable arguments, use appropriate tools strategically, attend to precision, look for and make use of structure, look for and express regularity in repeated reasoning. While the 3-Act tasks are based on real world scenarios, they are not pre-digested for the students.
Fawn Nguyen, a middle school teacher in California who hosts a great blog, has created a site, Visual Patterns, where she invites people to post growing visual patterns. The goal is to find an explicit rule to for the pattern that will allow the “value” of any term to be found without knowing the value of the previous term. “Value” is determined by the number of units in the pattern. Some of the patterns have a single unit, while others that have multiple units. Although these are not “real world problems”, the rule that is created is a mathematical model that makes sense out of chaos.
I will close with a quote from Georg Polya, guru of mathematical problem solving, found in the preface of his book How to Solve It.
Modeling is problem solving. It is generating creativity out of chaos. It is a life skill that reaches across all educational disciplines. Problems that we choose for students to build this skill do not have to directly relate to the real world, although they can. Most importantly, the tasks we choose need to engage students and create just enough disequilibrium or chaos to be challenging so the student can, in Polya’s words, “enjoy the triumph of their discovery.”