Problem Solving Structures

As a note of caution, Vanderbilt University Centre for Teaching notes that:

"Experts (teachers) in a particular field are often so fluent in solving problems from that field that they can find it difficult to articulate the problem solving principles and strategies they use to novices (students) in their field because these principles and strategies are second nature to the expert. To teach students problem solving skills, a teacher should be aware of principles and strategies of good problem solving in his or her discipline."

Activity 1.1

Consider how problems are solved in your discipline?

Read the following website from Vanderbilt University Centre for Teaching and discuss the following questions with your colleagues;

Teaching Problem Solving | Center for Teaching | Vanderbilt University

How feasible are the tips provided for your classroom?
Does the "two Column solution" provide a potential format for summative assessment or should lit be reserved for formative assessment?
Should we fear group-based problem solving work?

Activity 1.2

Consider Polya's model below.

Does it have application beyond Mathematics?
Could this be developed into a set of generic problem solving steps in your discipline for students to follow?

"99 Problems", Snorgtees.com

Polya's Model for "How to Solve it?"

G. Polya, How to Solve It

Summary taken from G. Polya, "How to Solve It", 2nd ed., Princeton University Press, 1957.

UNDERSTANDING THE PROBLEM
- First. You have to understand the problem.
- What is the unknown? What are the data? What is the condition?
- Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?
- Draw a figure. Introduce suitable notation.
- Separate the various parts of the condition. Can you write them down?
DEVISING A PLAN
- Second. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.
- Have you seen it before? Or have you seen the same problem in a slightly different form?
- Do you know a related problem? Do you know a theorem that could be useful?
- Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.
- Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?
- Could you restate the problem? Could you restate it still differently? Go back to definitions.
- If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other?
- Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?
CARRYING OUT THE PLAN
- Third. Carry out your plan.
- Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?
Looking Back
- Fourth. Examine the solution obtained.
- Can you check the result? Can you check the argument?
- Can you derive the solution differently? Can you see it at a glance?
- Can you use the result, or the method, for some other problem?

Activity 1.3

Assess the model by Pretz, Naples and Sternbergy below.
Could students in your discipline use it to solve problems?

Caspar David Friedrich, "Wanderer above the Sea of Fog" 1818

These skills, among others, target the following problem-solving steps (Pretz, Naples, & Sternbergy, 2003):

Recognize or identify a problem
Define and represent the problem mentally
Develop a solution strategy
Organize your knowledge about the problem
Allocate mental and physical resources for solving the problem
Monitor your progress toward the goal
Evaluate the solution for accuracy

How Problem Solving Interacts with Other Key Skills

Minnette de Silva with Pablo Picasso (left) at the World Congress of Intellectuals in Defense of Peace, 1948, Wikipedia.com

Communication

A key skill for problem solving is knowing how to communicate their understanding of a problem, that is how to define the problem, explain their chosen methodology for solving the problem, and clearly argue for their solutions. In attempting to communicate their understanding or a problem, they will refine their understanding of it themselves.

They also need to understand which problems are amenable to being solved through the practices of their discipline. This is true for all students, regardless of discipline.

If students read in their discipline, they will learn some of the language, rhetorical practices and ways that problems are typically phrased.

Headscratchers.com

Critical and Creative thinking

As Brown University Centre of teaching and learning suggests, "critical thinking is the “ability to assess your assumptions, beliefs, and actions” (Merriam & Bierema, 2014, p. 222) with the intent to change your actions in the future and is necessary when solving problems".

In addition, creative thinking is also necessary as students create and consider alternate possibilities. Critical thinking can help identify problems to be solved and creative thinking allows us to propose possible solutions.

Collaboration/Teamwork

Problems solving contexts can also be used to build teamwork and collaboration skills, which are valuable and desirable skills in themselves. Further, as many problems in the world are solved collaboratively, it also initiates students into the habits of work and further study. However, team and collaborative processes need to be taught as much as any skills. Some of the problem solving students will learn in understanding a poem, conducting an experiment, or preparing a performance, will be about working with other people.

Reflection

As The Brown University Centre for Teaching and Learning argues, "expert researchers, practitioners, and educators incorporate reflection and iteration as part of their practice. Key steps of the problem-solving process include being reflective about the process and what is working or not working towards a goal". Reflective activities suggested by Brown include, short minute papers, to semester-long reflective journals. Think-alouds, or having a student verbally solve a problem with another student, can also help students develop reflective problem-solving skills because it “provides a structure for students to observe both their own and another’s process of learning” (Barkley, 2010, p. 259).

Reflective practice in learning requires students to consider ways to improve.

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