COMPLEX, Collaborative, Real-World Problem Solving: Open-Ended Problems

COMPLEX, Collaborative, Real-World Problem Solving: Open-Ended Problems

Purpose

Students are going to try to attack a complex, real-world, open-ended problem the way professional engineers and scientists do, which is in small working groups over an extended period of time.

Overview

If you think about the types of problems scientists, doctors, engineers, computer scientists, city planners, politicians, business leaders, lawyers, and others try to figure out on a daily basis, it will likely become apparent very quickly those problems have an extra layer of complexity on them compared to the problems we try to solve for class on a daily basis.

While we focus on problems that try to develop knowledge and applications of one or two physics concepts in the context of ‘physics land’ problems, professionals are forced to think of real-world problems and all the additional complications and complexity reality poses and our textbook or AP problems ignore! Plus, real-world problems rarely have an absolute final answer – rather, the real world tends to offer complex, ‘open-ended’ problems that have many different approaches to try and solve, and we often do not know what the final answer is, or even if there is a ‘correct’ answer.

In this project, students will be given a real-world problem that is complex and open-ended. The will need to work in teams to develop a possible solution to the problem, using reasonable assumptions and approximations, and a mathematical model developed from relevant data they find from their research. They will write up their proposed solution, and present it to the class.

Student Outcomes

Because this is the type of problem students may get on occasion in college, and certainly when out in the world working, they will learn to:

·  Develop high-level thinking and problem solving skills

·  Be able to research and use large data sets, and figure out what to do with an overload of data

·  Improve on your already good analysis skills, and possibly learn some new techniques for crunching numbers

·  Develop and use computational thinking techniques, such as technology to assist in developing some plausible solution to a complex problem

·  Learn about the creation and use of mathematical models;

·  Develop approximations, assumptions and estimates that are reasonable, and justify why they are reasonable within the context of the data and math model;

·  Learn how to think about and identify limitations to the model and solution(s) you develop, and think about how to improve upon your model and solution given more time;

·  Gain an appreciation for life’s complexities;’

·  Get better at communicating technical solutions to the world;

·  Develop ‘21st century’ skills colleges and future employers will be looking for;

·  Gain experience doing complex problem solving collaboratively;

·  Offering creative solutions to tough problems;

·  Improve your technical writing and presentation skills;

·  Hopefully have some fun with some tough problems!! These are relevant.

This activity addresses the following skills from the Computational Thinking (CT) STEM Taxonomy chart:

-  1a Collecting Data

-  1d Manipulating Data

-  1e Analyzing Data

-  1f Visualizing Data

-  3a Using Computational Models to Understand a Concept

-  3c Assessing Computational Models

Next Generation Science Standards:

·  ETS1.C Optimizing the design solution (break a problem into simpler ones and approach systematically to see relevance/importance)

·  Science and Engineering Practices: Planning and Carrying Out Investigations

·  Science and Engineering Practices: Developing (mathematical) Models

·  Science and Engineering Practices: Using Math and Comp. Thinking

·  Science and Engineering Practices: Constructing Explanations and Designing Solutions (this will

Time

This is a longer-term, team collaborative project. Typical length of time is one month.

Level

This could be done in some form with any high school science or math class.

Materials and Tools

There are no real materials needed in class for this. Students will be working in teams outside of class time. There are no lab materials or supplies necessary. Students will need to find time to meet with their teams outside of class, and will need consistent access to the Internet. They will need some type of analysis software such as Excel or Graphical Analysis, so they can organize and analyze large amounts of data, and also to do fits to those data for the development of a mathematical model.

Teachers may provide hard copies of exemplar papers from national contests, which can be found on the Moody’s site:

http://m3challenge.siam.org/about/archives/.

Judge’s comments can be useful as well, to discuss with students what makes for a good solution paper:

http://m3challenge.siam.org/about/archives/2013/perspective/.

Judge’s perspectives are provided for each year for the Moody’s contest.

Preparation

Teachers need to help students understand, first and foremost, what exactly math modeling is and what the process they are about to use looks like. A helpful graphic to use with students to prepare them is below:

This is found at http://www.indiana.edu/~hmathmod/modelmodel.html.

Prerequisites and Background for Students

Students do need to be comfortable with using the Internet for researching topics, finding good, reliable data relevant to a problem, data analysis of what could be large datasets or databases using software such as Excel (this includes finding best-fit functions to the data to develop a mathematical model for the problem at hand). Students also need to be able to work collaboratively and be comfortable brainstorming with each other. All of this can and should be consistently and regularly modeled and practiced in class. These are some of the 21st century skills teachers need to work on with students, and students need to master.

Students will need to be able to judge and setup a time management schedule to complete this, or any, longer-term project and paper.

Students should be able to use a program such as Google Documents, so they the entire team may be able to work on the paper simultaneously if that is what they choose to do, or to access the most recent version at any time from anywhere. This can be demonstrated and practiced in any and all classes in school.

Students should be familiar with the typical structure and style/format of science research or lab reports. Technical writing is different from creative writing or the reports done in history or language arts classes.

This project will encourage students to use all of these individual skills at the same time!

Teaching Notes

Trying to develop mathematical models as solutions to complex, open-ended problems is something that can be intimidating at first, whether for teachers with little experience or with much experience, and it will certainly be intimidating and challenging to students the first time they try to do this. What makes it so intimidating is that we are all taken out of our comfort zone of knowing the answer! There is no clear-cut, single answer to a good open-ended modeling problem. The reason for this is because there are multiple ways to approach the problem, and, depending so much on the assumptions being made for a particular model, different models and solutions by different groups, even though they are given the same problem statement, can look entirely different from one another.

Here are some major characteristics of this type of problem teachers need to understand themselves, and then get students to understand. These are shortened comments from Profs. Joanna Leathers and Maynard Thompson : http://www.indiana.edu/~iucme/modules/docs/Comments.doc.

1)  There is motivational value in studying situations of intrinsic interest to students. A good modeling activity should have a purpose, and this purpose should be of interest to the person studying the situation. Clearly this comment involves both the situation and the intended audience, and a situation that is highly interesting for one group may be relatively uninteresting to another group. This can be a challenge.

2)  Assumptions are a basic part of model building. In order to fulfill its potential as a modeling task, a situation should require the student (the term we use to refer to the investigator doing the modeling) to think carefully about the situation, to identify the need for assumptions, and to make appropriate assumptions. Of course, the nature of the assumptions — their features, complexity, and importance — will vary depending on the situation and on the background and experience of the student.

For the beginner, many (or most) assumptions may be provided as part of the description of the situation to be studied. For beginning students, it may be necessary to remind them to identify and list their assumptions before and while studying the situation. To help students avoid making an unnecessary assumption, it is useful to ask students to explain the purpose of making an assumption and to ask that they note how it is used in developing the model.

For the more advanced student, listing and (when appropriate) modifying assumptions should be a standard part of the modeling process. Students should be encouraged to revisit their assumptions several times during the model-building process and to understand precisely how the assumptions are affecting their results.

Also, it may also be interesting to discuss with advanced students how the answer to the question would change if a particular assumption were absent or altered. The model builder should use this approach to help in determining the importance of each assumption. Not only should assumptions be identified (and listed in some way), but only those assumptions that are critical to the solution should be retained.

3)  Organizational skills are required. A good modeling problem should provide an opportunity for developing (or using) organizational skills while carrying out the study and then in describing the approach and presenting the conclusions of the study. After all, the main purpose of mathematical modeling is to solve real problems. Since building and studying a mathematical model is usually a relatively complex multistep activity, a thorough understanding of how the various steps are organized is essential. Students need to keep in mind the relations and connections between the step being considered now and what came before and what will come next in their study.

There is still great value in developing a high-level presentation. Regardless of the method (papers, oral reports, power point, etc.), there should be a clear and reasonable organization and flow to the presentation.

4)  Good communication is essential. Mathematical modeling has more features where communication is essential than most other mathematical activities. Two critical areas for communication are: First, most mathematical modeling involves teams and the individuals in these teams must develop productive interactions. Second, as noted in item 3), the team must communicate the results to an interested external audience.

Group activity on modeling problems requires a fair amount of productive communication among group members. Discussing ideas in groups helps confirm the validity of assumptions and conclusions, and may identify ideas that were previously unnoticed. These discussions involve students explaining their ideas to others and listening to the ideas of others. Both explaining and listening are valuable parts of the learning process.

5)  There may be more than one answer to the question. Mathematics and science at the high school level are comforting subjects to some people because there is traditionally one correct solution to each problem. It is often easy to determine whether the answer is right or wrong. An incorrect answer can usually be corrected by finding an error in logic or technique. With modeling problems, this is not true, as modeling problems may have several solutions. Primarily, the usefulness and completeness of the solution depends on the assumptions and the model used. Changing one assumption may change whether the solution is acceptable. Changing the definition of a single word, such as “best,” may change the nature of the solution. When a modeling problem is solved by three different groups, there may be three different solutions, each of which is correct in the sense that it follows from the assumptions and model used. Keep in mind there can be and are wrong solutions to these types of open-ended problems – this happens if incorrect data are used, assumptions not relevant to the real-world problem are made, and so on.

To determine the correctness of a solution, the entire process must be investigated. What assumptions were made? What model was selected? What calculations were made and why? In some cases, it may take an experienced mathematician to determine whether a solution is acceptable, and she may never say that a solution is “correct.” Instead, solutions may be adequate and solve the problem, or they may not. However, the word “correct” falsely gives the misleading impression that there is a right and a wrong solution. In general, modeling problem solutions are unusual in the way that they may leave the solver with a feeling of incompleteness – this is not unusual, even for experts in this sort of problem solving.

6)  There may be several approaches to finding an answer. Modeling problems frequently have several methods that can be used to solve them. By a method we mean a mathematical structure and a study of the structure. Having several methods is helpful because it allows the solver to decide what sort of mathematical ideas and techniques to use. In real world modeling situations, the modeler is usually not told what methods to use. Instead, he or she is faced with a problem and decision about which method is appropriate for the given situation, assumptions, and goal. It is important for problem solvers to be given the opportunity to decide what methods to use, and to recognize that there are alternatives.

For beginning students, it may be beneficial to brainstorm ideas of how to approach the problem with someone with more advanced modeling skills. Then the students can decide which approach to use. More advanced modeling students should be able to decide which process to use without involving others.

One benefit of having several approaches is that the modeling team can pick the solution path with which they are most comfortable. In general, having several paths to answering a question brings the modeling activity closer to what occurs in the real world. As students become more advanced, the number of ways they recognize to solve a problem may increase. Students should be allowed to use any method as long as the choice leads to a solution that is supported by assumptions, attains an appropriate level of difficulty, and is complete.