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