Tag Archives: primary

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.

REAL LIFE: THE PASSPORT OFFICE

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.

REAL LIFE IMAGINED: THE NEW JOB

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.

ILL-STRUCTURED PROBLEM SOLVING

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.

ILL STRUCTURED PROBLEM SOLVING AS A TEACHING TOOL

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?

CHILDREN AND PROBLEM SOLVING

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.

CHILDREN AND ILL-STRUCTURED PROBLEMS

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.

WHY IS THIS IMPORTANT?

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)

BUT YOU ARE TALKING ABOUT UNIVERSITY STUDENTS

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

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Once More Into the Blog, Dear Readers


The Remains of the Dock

Events have rendered me unwilling to think about education and despairing of teachers in Ontario ever being treated as more than technicians in the near future.  I have not written about education for many months now, but the little girl next door is leaving her Montessori school to start grade one at our local school.   My niece is half way through high school and two young cousins are returning to the francophone primary school.  We talk about school:  what their parents like; what works for the children; the inequities; the little miracles wrought by their teachers; the rules, ridiculous and important; the children’s biases and prejudices.  For the most part both the parents and children recognise that compromises have to be made in what is essentially still an industrial model of education but sometimes I hear frustration in their voices.

I am surrounded by teachers, too.  Primary, Montessori, kindergarten to grade 8 French, high school language, supply, university psychology, adult ESL teacher trainers, graduate supervisors.  They all have their delights and concerns as they return to the classroom.  Sometimes one or two will honour me by turning over a problem with me or asking my advice and I get some insight into their particular corners of the teaching profession.

All of us see things which don’t make sense, which hinder teachers in their teaching, students in their learning and parents in their support of both.  So inevitably I want to write again to point out the illogical, the wasteful and the effective events taking place in our schools.  I want to talk about what does work, especially the simple easy techniques.

I have been looking over the many thoughtful comments I have received from readers.  Please keep them coming; even when I disagree with you, they provoke me to think and consider other possibilities.  Writing in a vacuum is a dangerous thing as the writer may begin to believe everything she writes.

Watch this space for more about equity for the learning disabled, sense in teaching second languages and reflections on morality.  I hope to eventually have some comments to make on university teaching, too.  One might say that teaching is a new discovery in all university faculties, except, perhaps the education faculties.  And I am not too sure about them!

Rebuilding the Dock

“Education for All” and the Myth of Universal Design


Part 2 of Education for All

Universal Design in Architecture

Universal design is a concept that has come to us from architecture.  Architects noticed that as the handicapped were increasingly accommodated not only by retrofitting buildings, but by designing buildings from scratch to meet the needs of the handicapped, a curious thing was happening: the general public was also benefiting from the designs.

The wheelchair ramps worked well for cyclists and parents with children in strollers.  The lower wash basins and levered handles made life easier for small children.  Even the subtitles on television have proven a boon in places such as noisy hospital waiting rooms.  And who hasn’t hit the big blue button with their elbow or their foot when trying to wrangle too many packages through a heavy door?  In fact, the accommodations that many of us thought would be expensive extras for the few have proven to be welcome improvements in the lives of many.

Now it sometimes happens that the lessons or ideas of one discipline cross over successfully to another.  A well-made woodworking tool becomes a surgical instrument and leads to an unexpected partnership between the toolmaker and the surgeon.   The sciences of psychology and neurology, the art of religion and the discipline of philosophy discover that more and more often they are saying similar things until a psychiatrist finds himself writing a guide to meditation, a neurologist writes a book on Zen and the Brain and another neurologist explores the importance of emotions and the meaning of consciousness.  The Ontario Institute for Studies in Education is offering collaborative graduate work in areas such as neuroscience.  These developments come from a willingness to explore an unordinary take on things using a combination of imagination, openness to new ideas and critical thinking.

Universal Design and Education

At first blush, the Ontario Ministry of Education’s embrace of universal design might seem to be one of these happy crossovers.  Their idea is that if the curriculum is taught in a way that students with learning disabilities can learn it, then the same method will work for all the students in the class.  The corollary is that with rare exceptions all students should be in the same class because they all can be taught well at the same time.

A Brief Review of Learning Disabilities

To clarify (and those of you who know all about LD should skip the next couple of paragraphs): by definition, a student with a learning disability has average intelligence or above and has a deficit in one or more areas that affect learning.  The student may have difficulty reading but no difficulty with mathematical concepts or may have problems using a pen or pencil to write neatly at a reasonable speed but can write wonderful stories using a computer.   Each person who has a learning disability is different from any other one; the only thing they all have in common is they are all at least as smart as the normal person.

Many students with learning disabilities (or as one student I know puts it: learning differences) only need some simple accommodations to be very successful.  One student may need to have someone else’s notes photocopied for him (because he can’t write fast enough and, at the same time, pay attention to the material being taught), more time on tests, to take fewer courses and have textbooks recorded (because he can’t read fast enough even though his comprehension is excellent).  Another student might need to use a laptop in class and answer some questions in note form.  A third might use a non-programmable calculator for a math test because she can do mathematics but not arithmetic.

As an aside, I must note that these so-called disabilities would hardly have been noticed in another time and place.  In a society where minimal literacy and numeracy was needed, what counted would have been how hard people worked, how well they did their jobs and character.  Few people a hundred years ago would have been expected to do much in school after grade eight and many would not have gone even that far.  I think we should remember that an LD is a very subtle insult to the brain and in the larger scheme of things should not have an impact on our view of the student as a whole.

To return to universal design and education, it would be a great savings in time and money if universal design worked in the classroom.  However, for most LD students it is helpful to have highly structured assignments whereas the intellectually talented profit from open-ended assignments.  Combining high structure and open-endedness in an assignment can be done and I have created assignments that worked for everyone, however they generally took a considerable amount of planning and the most successful one took three teachers to deliver it.  The teacher librarian and the computer teacher participated in the delivery of the project.

While teachers of primary students are guaranteed a maximum of 23 students in their classrooms, teachers of grade seven and eight may have 34 students in theirs.  By the intermediate years, the gap in learning has grown considerably between the weak students and the strong students.  Throw in students who are learning disabled and some who have behavioural problems and a teacher will have her hands full trying to successfully meet everyone’s needs.  Should she try to teach using a single design the equivalent of an architectural ramp or lowered sink, she will find herself with a large group of bored, restless students.

An Example of Universal Design or Differentiation?

The example of Universal Design (P. 12) in Education for All is more like the architectural equivalent of a sign saying UP! at the base of a ramp, ladder, elevator and escalator; a health professional beside the sign would choose the most appropriate way for each individual to ascend and still get their heart rate to an aerobic level.  However, it is a wonderful example of differentiation in teaching a lesson in literature; what it does not explain clearly is how a teacher might handle the differentiation in a class of thirty if some of the students will need the teacher’s guidance for periods of time.  It especially does not address how to deal with students with behavioural challenges in this kind of situation; many of them need direct supervision.

The lay person reading this may assume that an educational assistant would be available but EAs are becoming more and more restricted to students with physical handicaps and are rarely assigned to classrooms anymore.  Administrators might point out that special education teachers are now coming into the classroom to support the regular classroom teachers.  This is true, but they are not available full time to any class so they focus on the three Rs, Reading, Writing and Arithmetic.

I have a sneaking suspicion that as time goes by, money for special education teachers will be cut until only the very severely affected students will receive support.  The argument will be that since many teachers are now receiving the equivalent of the old Special Education Certificate Part 1 as part of their B. Ed., they will soon be equipped to deal with behavioural, LD and other special education needs.  What the Ministry, administrators, academics and the public will forget is that this kind of teaching not only requires knowledge but also experience and time.

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:  EDUCATION FOR ALL: an analysis of the content.  August 14th

FOURTH POST:  Giftedness and EDUCATION FOR ALL: August 21st

FIFTH POST: Charts for Teachers derived from EDUCATION FOR ALL:  August 26st