Category Archives: arithmetic

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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Learning Chinese And What It Has Taught Me About Teaching


I'm learning more than Chinese!

I am back studying Chinese and my grades are beginning to reflect my assessment of my real comprehension.  I’m studying harder, longer and using a wider variety of methods; I am working for recall and not merely recognition but I still flounder.  I am very frustrated and am too tired at the end of the day to pursue my first love, thinking and writing about education.  Chinese was supposed to be a hobby.

We have the good fortune to have two professors of psychology as neighbours.  I asked Dr. F why I was working so hard with so little effect.  She pointed out that memory starts declining at the age of 25, “… and I bet the class is full of kids in their twenties.”  I am probably 30 years older than the oldest of them.  Learning a language requires memory work and Chinese requires more than usual as there is no connection between a character and its pronunciation.  Not only that, but Chinese has no connection to English or French so there are no connections to use.

In fact, it is all memory work.  I have tried hard to use my analytical skills (which improve with age) by tracking homophones, noting components of characters and using mnemonics of all kinds.  Still I lag.

It did start me thinking about the implications for education.  We, and I among us, have turned up our noses at memory work or “rote learning”.  I am beginning to reconsider its value.  Before I go on, I must assert that I strongly believe in teaching children to think and analyse from their first day in school; however, in order to think, there must be something to think about: facts, ideas, issues.  Architects and engineers are necessary to design large buildings, but if bricks and mortar and construction workers don’t turn up at the site, the design wouldn’t come to fruition.  Nor would it come to fruition if the planners didn’t know the physics and mathematics of solid construction.

While children are at their strongest ability for memorisation, the education system should be taking advantage of that strength to teach them.  Recently I observed a primary class where numeration (arithmetic) was taught as a concept, not to mastery.  It isn’t difficult to teach arithmetic to mastery; I have found five minutes a day throughout the year is very effective.  Some children will go further, some will lag, but mastery will allow them to focus on other math skills that can’t be memorised.

Imagine the panic a student feels as she tries to understand area and volume when she can’t remember the times table.  She looks at l x w x h and instead of thinking about how cool it is that length, width and height are used to create volume, she is looking at the two x’s and worrying about dealing with them.  Instead of thinking that this is just area multiplied by one more number, she is unhappy about multiplying three numbers together.

She may understand the models with blocks and other manipulatives, but she won’t want to translate that into numbers.  Without being able to easily retrieve the product of length, width and depth from memory,  she must resort to her fingers, a multiplication table or a calculator, all of which are slower and more distracting than knowing her times table.

Why is this a problem?  If the student doesn’t have an easy relationship with arithmetic, she won’t be able to comfortably estimate the answer to mathematical problems and see approximately what the answer should be.  It means she will not know if an answer is probably right without looking it up.  She may not get past arithmetic into the fun of mathematics.  If you had to build a desk and chair every time you needed it, wouldn’t you get fed up?

More practically, checking the total of grocery bills, calculating tips, estimating change, doing her own taxes and other mundane applications will become too bothersome to be worthwhile.  She will end up trusting others instead herself to take charge of her money.

History educates our children to become informed citizens.  The big patterns of history repeat and understanding reasons and results is important to understanding the current play of events.

While I am not keen on memorising dates in history, it is important that the sequence of key dates be understood.  One way to aid memorisation is a timeline running through the halls so that all the history that is studied in elementary school is displayed with only the important-to-learn dates on them.  Below are some of the dates I would include.  For the events linked with the dates, see the next post.  As students learned about these events, posters and pictures could be put up below them so students could learn the sequence of events and eventually, the dates.  It is important, for example, that students learn that the Quebec Act preceded the American Revolution and the French Revolution was roughly coincident with the American.  Why was the Quebec Act necessary and what role did it play in the American Revolution?  Is it a coincidence that there were two major revolutions going on at roughly the same time?

Some teachers might argue that putting up the same timeline year after year would be boring.  It could, but  the approach to the events might be varied, the children will certainly have grown and have learned and even the dates could be varied.  Within each classroom there would be a more detailed sequence of the period that class is studying.  My hope is that within the classroom there would be more emphasis on the social history of the time.

Presumably as the students get older and have a better understanding of history, they will see the events in different lights. That and remembering the key dates will start them on the road to understanding the sequence of world events.  With those tools, they can begin to look at cause and effect and use history as a lens to look at the events unfolding in the daily news.  At least they can if their teachers help them to do so.

Memory work in geography is even more vital.  How often do we laugh when our comedians go south to ask Americans simple questions about Canada?  I suspect that our ignorance is only less by comparison.  We should be using the children’s strong memories to ingrain the continents, the countries, the rivers, mountains, oceans and major cities. Will they remember it all twenty years down the road?  No, but  literature , the news and the study of other subjects will reinforce some of it.  Their memory for others can be jogged by references to what they do remember; “Tunisia, oh is that on the Mediterranean or the Atlantic – no near Libya somewhere – ahhh and Libya IS on the Mediterranean so Tunisia probably is, too.”  And some they will indeed remember.

They will have a rough idea of the climate, vegetation and the economy of each region.  They will know what the key indicators are of health and welfare in each country.  This kind of information along with the well learned and understood sequence and pattern of events of history will help them make sense of the current politics and business.

The facts are no substitute for understanding.  At the beginning of my Grade Seven geography class, I used to hold up one of those three page plasticised summaries of physical geography and say.  “Everything you need to know for this course is in here but if you were to memorise it all, you still would not pass this course.  What you also need to know is what the facts mean, what you can do with them, how to use them.  In short you are going to start to learn to think like a geographer.”  Then I pinned the summary to the back wall and we went to work.

I would argue that there is much to be gained in memorising poetry.  The first is that by memorising poetry a child goes over and over the words.  Given the right poem, the child may begin to see things in the poem that a teacher could not have pointed out.  Secondly, poems are meant to be spoken.  A child who has memorised a poem and then learned to speak it, has learned viscerally the effects of rhythm and rhyme in poetry, or if there is no rhyme, then the effects of the literary devices.  Finally, the poem has become a gift; the child carries it in his head as something he can turn on and turn over.  If you still hear a poem or song in your head that you learned many years ago, then you know what I am talking about.  Since that is the case, choose carefully the poem you ask a child to memorise.  Make it worthy of the effort.

At one point in my teaching, I looked for an alternative to detentions so I asked students to copy a chapter (about the equivalent of a poem) from the Tao de Ching, an early work in the Chinese philosophy of Taoism.  The writing is not easy to understand but I felt it gave the students something to think about.  Many students began to memorise some chapters by heart and occasionally quote them in class. One lad who was a persistent talker often got Chapter 24 which starts, “To be always talking is against nature…” and goes on to reflect on the power of stillness.  One day he came to me and said that he was beginning to get the passage beyond the message of the first few words.  The repetition had engraved it in his memory so it was always with him to contemplate.  This is one of the gifts of memorisation.

So to go back to Chinese and the dull tools I must use to learn it; will I stop?  No, because I CAN learn it, just much more slowly than the university students in my course.  Instead I remind myself that learning Chinese is a hobby and ask myself if I can pare my goal down to what I can reasonably achieve.  When April comes, I will write the final exam and then hire a tutor or swap English for Chinese conversational lessons.  Since I want to speak and understand, that is what I will focus on.  I am exploring spending time in China in the fall.  In the meantime, I will put reading about education and the brain and writing about education first.

Some dates I think are important, especially to Canadians.  This is not intended to be definitive!

DATE EVENT
~1700
~486
~440
~5 BCE to 650 CE
930
~1000
1095
1190
1215
1244
1265
1497
1534
1605
1607
1608
1707
1759
1774
1777
1783
1787
1867
1869
1914 – 1918
1928