Tag Archives: mathematics

Ideas, lessons and units currently available on for Teaching Outside the Box http://teachingideasoutsidethebox.wordpress.com/


6. TEN BOOK REPORTS IN A YEAR: THE PACKAGE

This unit is aimed at getting twelve year olds to read in quantity and quality.  It could be adapted for other grades and might need to be adapted for other marking systems.  The list in number 5 was originally written to go with this unit.

5.         Have You Read?

A list of books aimed at academically talented grade seven and eight students with the intention of broadening their usual tastes in reading and pushing them to try something new or a bit more challenging.

4.  Finding the Poetry

A lesson aimed at teaching the important parts of writing poetry: words and feelings.

3.  Learning to Love Teaching Poetry

It’s tough teaching poetry well.  This is a suggestion for a unit used for grades two, three and four using The Walrus and the Carpenter and The Tyger.

2.  Lessons in Perspective (Art, Empathy, Math, Literature)

A unit that combines lessons in perspective in art, empathy and mathematics.  Can be expanded to include literature and writing. Can be adapted K-12

1. Using the Internet to teach and teaching students how to use the Internet

Ideas on teaching research skills to all grade levels, including appropriate use of Internet, identification of bias, Boolean logic, using indexes, encyclopaedias and other resources.

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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

Related articles

Guess Who Is Stronger in Math?


Is he better at math?

Is she better at math?

Two articles on a subject dear to my heart: the first is a summary by a journalist, the second is the original academic article.  Before you get excited, remember that the plural of anecdote is not evidence.  I think that applies to academic articles, too. Click on the links for the articles

Written by a Journalist

From io9

There really is no difference between men and women’s math abilities

By Alasdair Wilkins

http://io9.com/5867401/there-really-is-no-difference-between-men-and-womens-math-abilities

Written by two academics

From the pages of Notices of the American Mathematical Society

JANUARY 2012 NOTICES OF THE AMS            10         VOLUME 59, NUMBER 1

 Debunking Myths about Gender and Mathematics Performance

Jonathan M. Kane and Janet E. Mertz

http://www.ams.org/notices/201201/rtx120100010p.pdf

The Notices is the world’s most widely read magazine aimed at professional mathematicians.

As the membership journal of the American Mathematical Society, the Notices is sent to the approximately 30,000 AMS members worldwide, one-third of whom reside outside the United States. It appears monthly except for a combined June/July issue in the summer.

By publishing high-level exposition, the Notices provides opportunities for mathematicians and students of mathematics to find out what is going on in the field. Each issue contains one or two such expository articles that describe current developments in mathematical research, written by professional mathematicians. The Notices also carries articles on the history of mathematics, mathematics education, and professional issues facing mathematicians, as well as reviews of books, plays, movies, and other artistic and cultural works involving mathematics. Members keep abreast of official AMS reports, activities, and actions, and the news of the mathematical world, through articles the Notices.