Tag Archives: ill-structured-problem-based solving

Problem Solving and Ill-Structured Problem Solving

Too often we give our children answers to remember rather than problems to solve.   Roger Lewin

Cadets at BRNC participate in a team problem-s...

Cadets at BRNC participate in a group problem solving exercise. Image via Wikipedia

If school were to prepare children to solve the problems of real life, we would have to consider the nature of real life problems.  So let’s look at two.


At the passport office in Ottawa there is a guard who sits at the door.  His job is to check whether applicants have everything they need to get their passport.  He then directs them in one of three ways: to the right because they have all their paperwork done, to the left because they need to pick up paperwork or sort out a problem or home because a signature or a photograph is missing.  It is very efficient and saves everyone a lot of time.  Whoever thought of this had carefully considered what the bottlenecks are at this point in the bureaucracy and how they could be resolved.

What I think is interesting is the likelihood that a good percentage of those people who don’t go to the right, could have been spared the trip altogether if they had carefully read the instructions and the followed them with equal care.


A single healthy young woman who has been working for a year and lives in a city has been offered a new job.  The new workplace is awkward to get to.  What should she do?

Most peoples’ response would be that we don’t have enough information to answer the question.  In fact this may be all the information the young woman might have.

To solve this problem she and we need to ask questions to get useful information.

What other information do we need to answer the question?

What makes it awkward to get to?

Is it close enough to walk?

Is cycling an option?

What bus route options are available?

Could she take a bus that comes close and walk the rest of the way?

Would using a car help?

Could she afford a car?

Is car-pooling an option?

Is moving an option?

The answers to these questions may create other questions such as costs in time spent traveling, distance from favourite activities, whether the new job is worth the difficulties, are there trade-offs such as walking becoming part of her exercise program?  From our own experiences we know it is rare that immediately we have a problem we will have enough information to solve it. We also know that sometimes there is not one right answer and we are left deciding between two or more equally acceptable but different answers.


This is the essence of what is known as ill-structured problem solving.  Students are given a problem to solve.  In solving it, they discover they need more information and sometimes as they acquire that information, they discover that the problem is not quite what they thought it was.

In the problem above, the young woman may realise that the issue is not how to get to her new job but whether the extra costs and difficulties make it worth taking the job in the first place.  She may be able to negotiate working part time from home or working flexible hours.  She may decide the increase in salary and opportunities for promotion are worth the difficulties and hope that later she can find either an alternate means of transportation or closer accommodation. Some are solutions that are not obvious in the statement of the problem or in the information supplied.

With the same information available to start with and the same information available through research, different people or groups of people may solve the problem differently and may also take different paths in accumulating facts and applying logic to arrive at a solution.


This kind of problem solving as a teaching tool was first used at McMaster medical school* in Hamilton, Ontario.  While it didn’t change the retention of information by much, it did improve diagnostic and other skills in the embryo doctors.  It was so successful it was soon copied by Harvard’s medical school.   In browsing through the Internet I noticed that ISPS seems to be most used in higher education and sometimes in secondary schools.  It is also seen as something appropriate for academically talented students.

This is the kind of problem solving that will be a permanent part of our lives and good decision-making will rely, in part, on our skill in dealing with it.  The question arises, can we teach it earlier? How old do children have to be before they will benefit?


TRIZ process for creative problem solving

One model for solving problems Image via Wikipedia

Take a look at most math books. The word problems often follow the same structure for each concept taught.  If the unit taught were subtraction, most of the word problems would follow a pattern:


Owner        has X  things.    If   it gives away Y things, how many will be left?

Harry              has 7 puppies.     If he gives away 5 puppies, how many will be left?

The teacher   has 25 cookies.    If he gives away 5 cookies, how many will be left?

The merchant has 10 free cars.  If he gives away 7 cars,    how many will be left?

After a couple of questions, the students look for the numbers, plug them into the formula without thinking about the problem: X-Y= right answer, and move on.  To be sure we now require students to write down what the problem is, the method and the answer, but these, too, are formulaic.

Should we throw in a question such as:

Collector has Y whatsits but needs X whatsits, how many more does he need to find?

the child who hasn’t truly grasped the concept of subtraction will be confused.

Adam has 5  flat smooth rocks, but needs 13.  How many more does he need to find?

Should the subtraction problems be mixed with other word problems, such as addition, the child who hasn’t grasped the concepts will be completely stymied.  If she has also not learned her number facts, she will be so slowed and frustrated that arithmetic will become difficult.


These problems are not ill structured because all the information necessary to solve the problem is available, but the issues I have described are part of the skills involved in being able to solve an ill-structured problem.  The child needs to understand what kind of problem is in front of her, whether she has all the information she needs to solve it and what tools she could use to solve it.  She needs to have the confidence to examine the problem to see if she can extrapolate or calculate the information she needs and especially the confidence to declare that there is not enough information.

If one of the problems read:

 Justin needs 13 smooth white stones.  He found some beside the river and 6 in the schoolyard.  How many more does he need?

the child should recognise what she needs to do solve the problem and that she cannot do it without a certain piece of information.

Depending on her age, it might not be essential that she can voice the necessary operation; it would be sufficient to demonstrate the difficulty using drawings or beans.  She might say:

He has 6 stones and some of the 7 he needs to make 13.  That means he must have at least 1 stone.  The best estimate I can make is that he needs between 6 stones and none to make up the 13.  

Or she might say:

I know that 13 – 6 = 7 so the stones he got in the school yard are between 1 and 7.  If the number of stones he found in the schoolyard is subtracted from 7, the answer is the number needed. 

There are lots of ways for a child in grade two or three to talk about a problem like this.  The point is that she is considering the problem itself, rather than plugging in a formula.  I am not knocking learning formulae or number facts; I believe they are worth the effort, but without learning to play with ideas to solve problems, a student is only being trained to be a calculator.

It also really doesn’t matter if she is using mathematical terms.  In fact it is probably too much to ask her to use what is new vocabulary for her.  What matters is that she is solving the problem to the point where she can see her way through to an answer or why she can’t reach an answer.

TN2020: Problem solving through storyboarding

TN2020: Problem solving through storyboarding (Photo credit: Zadi Diaz) There are many processes that are useful in solving a problem.


In many grade 11 and 12 academic math and physics classes today, students complain that the teacher is unfair if she gives problem sets on tests or exams that are not more or less identical to the ones they studied in class.  In other words, they expect not to have to figure out a problem, but simply recognise it, match it with the correct formula and plug in the numbers.   They want this in order to get the highest possible marks to aid their applications to universities. This is neither math nor thinking.

This story astonished me when I first heard it, as I naively assume that the last two years in an academic stream should be used to hone students’ analytical abilities.  I wondered how these students would cope if they were given and ill-structured problem in science or in math.  How would they cope if it were their summative?

These students do not see variety in their problem sets, much less ill-structured problems.  They arrive at universities unprepared to think, expecting to memorise facts and formulas.  Professors who expect them to think are resented and courses they expect to be bird courses are unpleasant surprises when the professors demand thought.

The professors are distressed, too.  They expect to teach concepts that the students will take away and make an effort to understand.  They expect to have embryo scientists and mathematicians in front of them, eager to learn and understand; they do not expect clever calculators waiting for more formulae and numbers.

Math and the sciences aren’t the only subjects where students are allowed to slip through using formulae.  It is not uncommon for students to leave high school for university never having progressed beyond the five-paragraph essay.  For those of you who are not familiar with the concept, the five-paragraph essay is another formula.  I won’t go into it as you can find it on the Internet.  Suffice it to say that no student starting first year in the Humanities should be stuck knowing only how to write a five-paragraph essay.  For a start, their ideas should be too complex and too subtle to be expressed in such a crude instrument.

Problem Solving PDCA

Problem Solving PDCA (Photo credit: Luigi Mengato)


It isn’t only true for academics.  What kind of job is a plumber or electrician or cleaner going to do if their only thinking is formulaic?  How will parents deal with their children and the school or medical system if they can’t think things through to ask the questions that will help their children or themselves?  Just because students are not going on to university is no reason to condemn them to simplistic thinking.

Going back to our grade two student: if every year she is in school she is taught and expected to think and apply the facts she has also learned, consider how she will be empowered to make good decisions for her own life.

If she has the talents to go on to university, imagine how little time she will waste as she engages with new ideas.  The same applies no matter what post-secondary education she chooses because she will have learned to look beyond the obvious. In a world, we are told, where she can be expected to change jobs and learn new skills with some regularity, isn’t that what her education system should do for her?

English: Mimi & Eunice, “Problems”. Categories...

Image via Wikipedia

* McMaster Medical School:  the Little School that Could and Did  http://www.scribd.com/doc/20150938/McMaster-University-Medical-School

Harvard Dean Gives McMaster an A   http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1491910/pdf/cmaj00138-0091.pdf

McMaster’s Innovations in Medical Innovation Honoured in NewsWeek   http://fhs.mcmaster.ca/main/news/news_archives/newsweek.htm

Related articles

What’s in “Education for All” Besides the Concept of Universal Design?

Part 3 of “Education for All”

Education for All was written by a collection of administrators and professors lightly seasoned with a teacher.  It is a collection of good ideas for clarifying the strengths and needs of students and working out ways to teach them well.  Making the best use of it will require taking time to read it carefully and make note of the most useful ideas.  Some teachers may find the ideas too time consuming.  Not every idea is practical in every classroom, especially if there is only one teacher, but in principle, the ideas are promising.  Here is a brief guide.

As I noted in a previous post, the writers failed to make their point about Universal Design but they did a nice job of demonstrating differentiation in the second chapter.  They go on to make lots of useful suggestions for basic approaches to teaching.  Unfortunately, as you will find throughout the book, the suggestions are general; to get something specific and especially concrete that can go from book to classroom, you will have to turn to other books.  Some are mentioned in the bibliography, but they are not distinguished from the purely academic.  This may seem a trivial point but for a teacher hoarding every minute, the time involved in sieving a bibliography for nuggets of usefulness can be too much.

The writers do refer to a great teaching method, but one I wouldn’t normally recommend for K to 6 students.  The Expert Panel calls it Problem Based Approach (p. 17).  They haven’t quite grasped the concept of Ill-Structured-Problem-Based Solving they are actually trying to describe.  It is effective but needs careful design, firmness and support in order to work.  There is a reason it started with Harvard and the innovative McMaster medical school and not in someone’s grade 4 class.

The chapter on Assessment and Evaluation of students with Special Education Needs is useful for both the new teacher who is unfamiliar with the steps required in getting help outside the classroom for students and any teacher who has been confused by the red tape involved.  I have tried to create some charts to help teachers track steps and required information, but referring to this section would also prove useful.  For the charts, please see the post EDUCATION FOR ALL:  Stuff to Help Teachers in the next post.

Developing Learning Profiles: Know your Students

There are lots more charts in the next chapter to help teachers develop learning profiles of their students and the classroom.  I have adapted them somewhat to make them easier to download and use.  In the next post I will explain and offer the charts.  The one thing the ministry does not provide but strongly suggests is questionnaires for parents and children.  These are things teachers have not been trained to create.  I am working on some and will make them available but I strongly suggest teachers sit down with their own divisions and create their own.  My designs might be useful as a starting or discussion point.  The parent questionnaires might stay the same for all grades but student questionnaires will need modification for most grades.  Students may need some explanations about the questionnaires; doing it as part of a unit on data gathering as an authentic activity to demonstrate real life applications might be an effective way of killing two birds with one stone..

And Then We Get to the Part about Teaching in the Inclusive Classroom

Education for All spends three chapters outlining potential challenges for students in acquiring literacy and numeracy skills and ways of anticipating those difficulties.  This is probably the most useful part of the book.    A chart identifies these challenges and strategies for meeting the challenges.  The suggestions are excellent but abstract; busy teachers need concrete suggestions that can be applied immediately.  On the other hand, it is a good starting point for thinking about meeting these challenges.

Teachers would undoubtedly have been grateful for a list of books which had concrete methods, blackline masters and even lesson plans relevant to specific grades and subjects which met Education for All’s criteria.  For example, I have found the following three books invaluable: When Kids Can’t Read: What Teachers Can Do: A Guide for Teachers 6-12 by Kylene Beers; I Read It, But I Don’t Get It by Cris Tovani and Teaching Gifted Kids in the Regular Classroom by Susan Winebrenner. In today’s inclusive classroom, I would add Teaching Kids with Learning Difficulties in the Regular Classroom, also by Susan Winebrenner.  I don’t know what books I would add for numeracy, but the teachers guides to many recent textbooks do offer excellent suggestions for accommodations.  These books, however, are favourites of this intermediate elementary teacher.  Primary teachers would have others.

Accommodations and Modifications: What’s the Difference?

A good explanation of the differences between accommodations and modifications and a detailed chart of many accommodations a teacher might make is provided in Chapter Nine.  A teacher might find it worthwhile to photocopy the chart and put it with the materials used to create lesson plans; it is a good general source of ideas.  A word of caution to teachers: if you are accommodating four students and modifying lessons for three more in addition to being a good (not a walk on water) teacher you may find yourself putting in extra hours every day.

You Say Syllabus, I Say Curriculum; Either Way There is a Lot of Work to Do!

For each subject, Ontario’s Ministry of Education sets out in broad strokes an outline of expected knowledge and skills outcomes.  They do not provide a detailed syllabus (or curriculum, depending on which side of the Atlantic you reside) with texts to use in order to reach those goals.  As a result, teachers end up reinventing the wheel.  In the primary and junior grades there are usually math texts with good to excellent teacher guides that the inexperienced teacher can follow.

The ministry is also creating Targeted Implementation and Planning Supports for Revised Mathematics (TIPS) that is available to grades 7 to 10 teacher at the moment. It is worth taking the time to figure out. The grade 7 and 8 teachers whose contracts are negotiated by the Elementary Teachers Federation will have to look under the Secondary section of the Education web site.  Nothing similar has been done for the elementary grades.

Aside from math, textbooks are rare in the primary and junior grades.  English textbooks don’t exist in grade 7 or 8.  It used to be that curriculum consultants created a model curriculum that reflected the goals of the ministry.  I still treasure my history binder from the board that I have interleaved with my notes, overheads and photocopied pictures.  However, budget cuts have put an end to curriculum consultants who have the time to do more than workshops on changes to the curriculum.

That being the case, the Ontario teacher is now expected to create the course that will result in the proposed outcomes, and the accommodations and modifications. At the same time she is dealing with more duties and has less help because there are fewer specialists such as teacher librarians.  Teachers in schools where there is another teacher with the same grade would be smart to team up with each other.  With careful co-operation, teachers may reduce their workload to half the planning, half the returning items to the literacy and numeracy resource room, half the time spent photocopying and a better chance of getting home in time for a game of Ultimate Frisbee or to chill with a good book.

Who Does the Work?

The principal is listed under people who will share responsibility with the teacher for inclusive education.  While a teacher may hope for the ideal principal on page 113, the reality is more likely to be a politician who has to watch her back; with tact and diplomacy teachers may get the facilitation needed to implement inclusive education. Principals who come close to that supportive, collegial ideal should be cherished. Teachers such as the Special Education teachers (the name changes regularly) who provide support in the classroom are not likely to be there for times other than language skills or math.  Classroom teachers might consider switching subjects (such as history, geography and science) which make use of literacy or numeracy skills occasionally with language arts or math time slots.  It would be an effective way to support these skills across the curriculum.

Notice that the principal shares responsibility and the special education teacher will provide support but the burden of work and responsibility rests on the shoulders of the classroom teacher.

Computer Assisted Technology

The document addresses computer assisted technology.  While it has useful things to say about it, the most important things are not said.  First, a child will frequently have to wait until the next school year to receive any technology he or she needs.   In the meantime it will be the parents and teacher who will be improvising and trying to keep the child’s spirits up.

Secondly, all technology requires a learning period; initially the student will learn quickly but then start to slow.  Although this is normal, it is discouraging; a student will get frustrated and want to quit.  The teacher should check to make sure everything is working properly and then work with the parents to encourage the student to keep going.  The assistive technology won’t prove really useful until it becomes a tool to do something else, not an end in itself. If you have read Outliers, you will know that mastery of any skill takes many hours of conscious practice.  Parents and teachers will have to resist pressure from the student to let them give up the new aid.  Things Take Time.

Tools for Implementing Inclusive Classrooms

It is fashionable in the educational world to talk about Professional Learning Communities and this document is not any different.  When I picked myself off the floor where I was rolling around laughing at the concept of having the time for a professional learning community, I indulged myself with the fantasy.  I think that the only thing better for a teacher than a professional learning community is personal time.  One of the difficulties (besides time) is that a professional community involving a principal who will eventually assess a teacher might inhibit frank discussions.  That being said, teachers who do have the time and opportunity for a real PLC should go for it.

The writers take five pages to suggest that Professional Development is a Good Thing.  All levels of education, including universities, should be involved and teachers should have plenty of opportunity to learn the skills needed for the inclusive classroom.  Unfortunately, they didn’t mention where the money is coming from.

Finally, the writers make 12 recommendations that should be read and considered by those who have the power to make them happen.  Parents should read them and trustees should read them and administrators should read them.  If inclusive education has any hope of working, these recommendations, practical professional development and, yes, professional learning communities need to happen.  Successful change does not happen by saying, “make it so”.

For More Information:

Education for All: The Report of the Expert Panel on Literacy and Numeracy Instruction for Students With Special Education Needs, Kindergarten to Grade 6, 2005

NEXT POST: Giftedness & Education for All August 21

FINAL POST: Help for Teachers: Charts taken from, modified and created from Education for All August 26