What are well-structured and ill-structured problems?

Well-Structured Problem: Yields a correct answer through the application of an appropriate algorithm (SERC)

  • Response: right answer
    • Examples
      • 1+2=3
      • When you let go of an object, why does it fall?

Ill-Structured Problem: Does not yield a particular, certain answer (SERC)

  • Response: claim and justification
    • Examples
      • How did the United States win the revolutionary war?
      • Which economic system allows for the most human flourishing?

How should you teach in a well-structured domain? How should you teach in an ill-structured domain?

My thesis is that you should teach both in largely the same manner because the underlying structures of each domain are the same. In order to successfully answer a question in either domain, students must retrieve content from long-term memory and then successfully apply it to the problem following the correct procedures.

Simon (1973) stated that a problem’s domain (well/ill-structured) depended on the individual learner, “The boundary between well structured and ill structured problem solving is indeed a vague and fluid boundary.” He also said that the processes used in solving ill-structured problems can be successfully applied to well-structured problems.

If the categorization of a well/ill-structured problem depends on the learner, then the categorizations are relative and subject to change.

Academia has not universally agreed with Simon, however. Reitman (1965) argues that ill-structured domains are not easily defined and are often conditional. As a result, students must combine various separate schemas in order to answer an ill-structured problem. This essentially results in the student creating “new” knowledge whereas in answering a well-structured problem, the student would be recalling and applying “old” knowledge. It is argued that solving an ill-structured problem involves content knowledge, structural knowledge, domain-specific strategy, and general searching strategy (Sinnot, 1989). It is my basic argument that these are needed to solve a well structured problem too.

Take the simple problem of 13+9=x.

You obviously cannot solve this problem without content knowledge. You also need to understand the structure of addition. You must know not only how to add individual numbers, but you must carry out the procedure in the correct manner.

Components of solving ill structured problems (Voss, 1988 and copied from Hong, 1998)

  1. Recognizing that there is a problem
  2. Finding out what the problem is
  3. Searching and selecting some information about it
  4. Developing justification by identifying alternative perspectives
  5. Organizing obtained information to fit a new problem situation
  6. Generating some possible solutions
  7. Deciding on the best solution by the solvers perception of problem constraints
  8. Implementing the solution and evaluating it by developing arguments and articulating personal belief or value

In his 1998 dissertation, Hong states, “There are critical differences between the two problems as shown by the literature review. Well-structured problems require cognition, including domain-specific knowledge and structural knowledge, and knowledge of cognition (e.g., general strategies)…In well-structured problems, there is only one correct, guaranteed solution, achieved by using specific pre established rules and procedures. Well-structured problems require finding and applying the correct algorithm for a successful solution rather than decision-making using epistemic cognition and their value or perception about problems (Churchman, 1971). Therefore, solvers do not need to consider alternative arguments, finding new evidence or evaluating the collected information for successful solution of well-structured problems (Kitchener, 1983)….In contrast to well-structured problems, ill-structured problems have multiple potential valid solutions which can be effectively determined by using a particular decision making procedure. Solvers must use their epistemic cognition, values, attitude, belief, motivation and emotion in order to make decisions in novel real life problem situations.”

While Hong would disagree with me, it is my contention that students essentially follow this same process for solving all types of problems. The only significant difference seems to be that well-structured problems can only have one correct answer whereas ill-structured problems could have many justifiable answers. While that may be true, I think that calling the problem “open” or “closed” would be more accurate because you can follow the same steps to solve either, it’s just that one has a clear path to a solution and the other has no clear path.
Take addition as an example that very clearly falls in the well-structured domain.

Addition Word Problem

Susie has thirteen apples and her friend Mark gives her nine more. How many apples does Susie have now?

  1. Recognizing that there is a problem
    1. To solve this problem, the student must recognize it as asking a question. “I have to find the numbers of apples Susie has”.
  2. Finding out what the problem is
    1. The student must recognize it as an addition problem, thus limiting the problem space to the realm of addition. “Mark gave her ‘more’ apples, so I should use addition to solve this problem.”
  3. Searching and selecting some information about it
    1. The student must select the given numbers and add them together, further defining the problem space. “Ok, so thirteen ‘more’ nine…”
    2. This would hold true even if the example was not a word problem. 13+9, the student must still select the numbers and use the proper symbol. This step must happen even in a well-structured problem, even if the problem “gives” this step to the student, the student must accurately select it. A common selection error in this example would be reading ‘more’ to mean subtraction and placing a ‘-’ sign into their equation.
  4. Developing justification by identifying alternative perspectives
    1. For this example, there are no other valid perspectives (hopefully the student will realize this quickly). The student would just check and justify their actions. “Yes, ‘more’ means addition, not a minus or multiply, so I must add the numbers together.”
  5. Organizing obtained information to fit a new problem situation
    1. The student must convert the word problem into a traditional number problem. “Thirteen is 13 and nine is 9. So I have 13+9=Total # of apples.”
  6. Generating some possible solutions
    1. This problem only has one possible solution. However, the student must still generate an accurate answer that takes the problem domain into account. “13+9=22. Susie has 22 apples.”
  7. Deciding on the best solution by the solvers perception of problem constraints
    1. In a “well-structured” problem, this step is combined with step 6 because there is only one reasonable answer.
  8. Implementing the solution and evaluating it by developing arguments and articulating personal belief or value
    1. In this example, simply following the addition algorithm expresses a developed argument and the student’s trust in the algorithm.

My purpose in this example is to show how you can apply the “ill-structured problem” process to well-structured problems with no negative effects. I believe this shows there to be a false dichotomy and, think, like Simon, that “there is no real boundary between well and ill-structured problems.”

It is undoubtedly true that there are problems with more and less structure and that changes the difficulty level of those problems, however, it appears that we can use the same general process to solve both types of problems. Problems with less structure may require more time, background knowledge of the domain and problem structure (conceptual knowledge), along with the knowledge about the problem’s particular context than well-structured problems but, again, the process of solving either type remains the same.

IMPLICATIONS

For teachers, the implications are relatively straightforward. If there are no true boundaries between well and ill-structured problems as Simon said, then…

  1. We should recognize that problems classified as ill-structured often require more scaffolding and pre-teaching because answering them will force students to draw upon a wealth of information and concepts whereas a well-structured problem will often be relatively straightforward.
    1. We can reduce these challenges by explicitly teaching students and modeling and explaining each step with a similar example problem after. (for both well/ill structured problems)
  2. We must take a student’s prior learning into account and realize that the solution to an “ill-structured” problems will often build upon the solutions of multiple “well-structured” problems.
  3. We should teach students how to justify their answers with data.
    1. This is especially important for problems that have been traditionally identified as ill-structured because these problems, by their very nature are more difficult and cognitively demanding, often with multiple possible answers, thus requiring a justification for why they chose their answer. (Well-structured problems also require justification. It is just that their justification is often more straightforward. The justification for the above example is the standard algorithm, 13+9=22.)

Note: Follow this link to the PhD dissertation I primarily used for this article. While I disagree with the author’s finding that well-structured and ill-structured problems utilize different processes there is a wealth of great, applicable information inside (Especially in the Lit Review).

 

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