by Margaret Taplin
Institute of Sathya Sai Education, Hong Kong
|
| For many reasons, the state of society has
reached a stage where it is more critical than ever to educate people in the traditional
values of their culture. In recent years there has been considerable discussion about
whether it is the responsibility of schools to impart values education. There is growing
pressure for all teachers to become teachers of values, through modelling, discussing and
critiquing values-related issues. |
| There are many opportunities to teach the
principles of values education through existing subjects and topics. The purpose of this
article is to suggest one of the many ways in which values education can be incorporated
into existing mathematics curricula and approaches to teaching mathematics. In particular,
it will focus on ways in which values education can be enhanced by utilising a
problem-solving approach to teaching mathematics. The articles include quotations, printed
in italics, from the Sathya Sai Education in Human Values program, which originated in
India and is now active in more than 40 countries around the world. |
| These quotations are concerned with the following
values: |
- equipping students to meet the challenges of life
- developing general knowledge and common sense
- learning how to be discriminating in use of knowledge, that is to know what knowledge is
appropriate to use for what purposes
- integrating what is learned with the whole being
- arousing attention and interest in the field of knowledge so it will be mastered in a
worthy way
|
|
| Why Can Values be Enhanced by
Teaching Mathematics via Problem Solving? |
| Increasing numbers of individuals need to be able
to think for themselves in a constantly changing environment, particularly as
technology is making larger quantities of information easier to access and to manipulate.
They also need to be able to adapt to unfamiliar or unpredictable situations more
easily than people needed to in the past. Teaching mathematics encompasses skills and
functions which are a part of everyday life. |
| Examples: |
- reading a map to find directions
- understanding weather reports
- understanding economic indicators
- understanding loan repayments
- calculating whether the cheapest item is the best buy
|
|
| Presenting a problem and developing the skills
needed to solve that problem is more motivational than teaching the skills without a
context. It allows the students to see a reason for learning the mathematics, and hence to
become more deeply involved in learning it. Teaching through problem solving can enhance
logical reasoning, helping people to be able to decide what rule, if
any, a situation requires, or if necessary to develop their own rules in
a situation where an existing rule cannot be directly applied. Problem solving can also allow
the whole person to develop by experiencing the full range of emotions associated with
various stages of the solution process. |
| Examples: |
- The problem that we worked on today had us make a hypothesis. Through testing, our
hypothesis was proven incorrect. The problem solving approach allowed our group to find
this out for ourselves, which made the "bitter pill" of our mistake easier to
follow.
- I found this activity to be quite a challenge. I felt intimidated because I could not
see an immediate solution,and wanted to give up. I was gripped by a feeling of panic. I
had to read the question many times before I understood what I had to find. I really had
to "dig down" into the depths of my memory to recall the knowledge I needed to
solve the problem.
- Seeing patterns evelop before my own eyes was a powerful experience: it had a
stimulating effect. I felt that I had to explore further in a quest for an answer, and for
more knowledge.
Extracts from a student teacher's journal after
three separate problem solving sessions |
|
| The student who wrote the extracts above, has
illustrated how interest rooted in the problem encouraged steady interest needed
to master worthy knowledge. Experience with problem solving can develop curiosity,
confidence and open-mindedness. |
|
|
| How To Teach Human Values By
Incorporating Problem Solving Into The Mathematics Program. |
| This section will describe the types of problem
solving which can be used to enhance the values described above, and will give some
suggestions of how it can be used in the mathematics program. |
There are three types of problems to which
students should be exposed:
- word problems, where the concept is embedded in a real-world situation and the student
is required to recognise and apply the appropriate algorithm/rule (preparing pupils for
the challenges of life)
- non-routine problems which require a higher degree of interpretation and organisation of
the information in the problem, rather than just the recognition and application of an
algorithm (encouraging the development of general knowledge and common sense)
- "real" problems, concerned with investigating a problem which is real to the
students, does not necessarily have a fixed solution, and uses mathematics as a tool to
find a solution (engaging pupils in service to society).
Each of these problem types will be described in more detail below. |
|
| Problems which require the direct use of
a mathematics rule or concept. |
| By solving these types of problems, students are
learning to discriminate what knowledge is required for certain situations, and
developing their common sense. The following examples have been adapted from the HBJ
Mathematics Series, Book 6, to show how values such as sharing, helping and conserving
energy can be included in the wording of the problems. They increase in difficulty as
they require more steps: |
| Examples: |
- 7 children went mushrooming and agreed to share. They picked 245 mushrooms. How will
they find out how many they will get each?
- Nick helps his elderly neighbour for 1/4 of an hour every week night and for 1/2 an hour
at the weekend. How much time does he spend helping her in 1 week?
- Recently it was discovered that a clean engine uses less fuel. An aeroplane used 4700
litres of fuel. After it was cleaned it was found to use 4630 litres for the same trip. If
fuel cost 59 cents a litre, how much more economical is the clean plane?
|
| Sometimes it is important to give problems which
contain too much information, so the pupils need to select what is appropriate and
relevant: |
| Example: |
| Last week I travelled on a train for a distance
of 1093 kilometres. I left at 8 a.m. and averaged 86 km/hour for the first four hours of
the journey. The train stopped at a station for 1 1/2 hours and then travelled for another
three hours at an average speed of 78 km/hour before stopping at another station. How far
had I travelled? |
|
| To be able to solve these problems, the pupils
cannot just use the bookish knowledge which they have been taught. They also need
to apply general knowledge and common sense. |
| Another type of problem, which will encourage
pupils to be resourceful, is that which does not give enough
information. These problems are often called Fermi problems, named after the mathematician
who made them popular. When people first see a Fermi problem they immediately think they
need more information to solve it. Basically though, common sense and experience
can allow for reasonable solutions. The solution of these problems relies totally on
knowledge and experience which the students already have. They are problems which are
non-threatening, and can be solved in a co-operative environment. These problems
can be related to social issues, for example: |
| Examples: |
- How many liters of petrol are consumed in your town in a day?
- How much money would the average person in your town save in a year by walking instead
of driving or taking public transport?
- How much food is wasted by an average family in a week?
|
|
| Using a Fermi Problem to
Promote Human Values |
| Ms. Lam wanted to teach her class of
ten-year-olds about the value of money, and to appreciate what their parents were doing
for them: |
| "I believe that students should be aware of
this important issue and thus can be more considerate when a money issue raised in their
own family, such as failure to persuade their parents to buy an expensive present. In
solving the problems, I think that students can have a better understanding of the concept
of money, not simply as a tool of buying and selling things. |
| "First I told the class a story about
Peter's argument with his family. Peter failed to persuade his parents to buy expensive
sportshoes as his birthday present and thought that his parents did not treat him well.
The parents also felt upset as they regarded this son as an inconsiderate child. They
thought that he should understand that the economy is not so good. They asked Peter if he
knew about how much money was being spent on him throughout the whole year. Unfortunately,
Peter could not produce the answer immediately. So I asked the class if they could help
Peter. I asked them to find answers to the following problems: |
- How much money do your parents spend on you in a year?
- How much money have your parents spent on you up till now?
- How much money will your parents have spent on you by the time you finish secondary
school?
- How much money will be spent on raising children in the whole country this year?
|
| "The students were formed into groups of 4
to find out the possible data that they need to know. Later, the groups were asked to
present their data and the way of finding out the answer. Finally, I concluded that this
is an open question as each person may have different expenditure along with some common
human basic needs such as food, clothes and travelling fares. Anyway, the answer should be
regarded as a large sum of money and thus give them a better understanding of their
parents' burden." |
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|
| Sometimes pupils can be asked to make up their
own problems, which can help to enhance their understanding. This can encourage them to be
flexible, and to realise that there can be more than one way of looking at a
problem. Further, the teacher can set a theme for the problems that the pupils make
up, such as giving help to others or concern for the environment, which can
help them to focus on the underlying values as well as the mathematics. |
|
| Non-Routine Problems |
| Non-routine problems can be used to encourage
logical thinking, reinforce or extend pupils' understanding of concepts, and to develop
problem-solving strategies which can be applied to other situations. The following is an
example of a non-routine problem: |
| What is my mystery number? |
- If I divide it by 3 the remainder is 1.
- If I divide it by 4 the remainder is 2.
- If I divide it by 5 the remainder is 3.
- If I divide it by 6 the remainder is 4.
|
|
| Real Problem Solving |
| Bohan, Irby and Vogel (1995) suggest a seven-step
model for doing research in the classroom, to enable students to become "producers of
knowledge rather than merely consumers" (p.256). |
|
| Step 1: What are some questions you would
like answered. |
| The students brainstorm to think of things they
would like to know, questions they would like to answer, or problems that they have
observed in the school or community. Establish a rule that no one is to judge the thoughts
of another. If someone repeats an idea already on the chalkboard, write it up again. Never
say, "We already said that," as this type of response stifles creative thinking. |
| Step 2: Choose a problem or a research
question. |
| The students were concerned with the amount of
garbage produced in the school cafeteria and its impact on the environment. The research
question was, "What part of the garbage in our school cafeteria is recyclable?" |
| Step 3: Predict what the outcome will be.
|
| Step 4: Develop a plan to test your
hypothesis |
| The following need to be considered: |
- Who will need to give permission to collect the data?
- Courtesy - when can we conveniently discuss this project with the cafeteria manager?
- Time - how long will it take to collect the data?
- Cost - will it cost anything?
- Safety - what measures must we take to ensure safety?
|
| Step 5: Carry out the plan: |
| Collect the data and discuss ways in which the
students might report the findings (e.g. graphs) |
| Step 6: Analyse the data: did the test
support our hypothesis? |
| What mathematical tools will be needed to analyse
the data: recognising the most suitable type of graph; mean; mode; median? |
| Step 7: Reflection |
| What did we learn? Will our findings contribute
to our school, our community, or our world? How can we share our findings with others? If
we repeated this experiment at another time, or in another school, could we expect the
same results? Why or why not? Who might be interested in our results? |
| "The final thought to leave with students is
that they can be researchers and producers of new information and that new knowledge can
be produced and communicated through mathematics. Their findings may contribute to the
knowledge base of the class, the school, the community, or society as a whole. Their
findings may affect their school or their world in a very positive way" (Bohan et
al., 1995, p.260). |
|
|
|
| Mathematical Investigations |
| Mathematical investigations can fit into any of
the above three categories. These are problems, or questions, which often start in
response to the pupils' questions, or questions posed by the teacher such as, "Could
we have done the same thing with 3 other numbers?", or, "What would happen
if...." (Bird, 1983). At the beginning of an investigation, the pupils do not know if
there will be a suitable answer, or more than one answer. Furthermore, the teacher either
does not know the outcome, or pretends not to know. Bird suggests that an investigation
approach is suitable for many topics in the curriculum and encourages communication,
confidence, motivation and understanding as well as mathematical thinking. The use of
this approach makes it difficult for pupils to just carry out routine tasks without
thinking about what they are doing. |
| Bird believes that investigational problem
solving can be enhanced if students are encouraged to ask their own questions. She
suggested that the teacher can introduce a "starter" to the whole class, ask the
students to work at it for a short time, ask them to jot down any questions which occurred
to them while doing it, and pool ideas. Initially it will be necessary for the teacher to
provide some examples of "pooled" questions, for example: |
- Does it always work?
- Is there a reason for this happening?
- How many are there?
- Is there any connection between this and.....?
|
| The pupils can be invited to look at each other's
work and, especially if they have different answers, to discuss "who is right".
|
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| Conclusion |
| This article has suggested some reasons why
problem solving is an important vehicle for educating students for life by promoting
interest, developing common sense and the power to discriminate. In particular, it is
an approach which encourages flexibility, the ability to respond to unexpected
situations or situations that do not have an immediate solution, and helps to develop perseverance
in the face of failure. A problem-solving approach can provide a vehicle for students
to construct their own ideas about mathematics and to take responsibility for their own
learning. While these are all important mathematics skills, they are also important
life skills and help to expose pupils to a values education that is essential to their
holistic development. |
|
| References and Useful Reading |
| Bird, M. (1983). Generating Mathematical
Activity in the Classroom. West Sussex, U.K.: West Sussex Institute of Higher
Education. ISBN 0 9508587 0 6. |
| Bohan, H., Irby, B. & Vogel, D. (1995).
'Problem solving: dealing with data in the elementary school'. Teaching Children
Mathematics 1(5), pp.256-260. |
|
| The ideas presented in this article suggest some
ways in which teachers can explore the integration of values education into the existing
mathematics program without needing to add anything extra. Further ideas have been
presented in a book written by the author (Taplin, 1988). As well as giving teaching
ideas, the book summarises recent research and suggests some questions for action research
or discussion that teachers can use in their own classrooms. For further information about
the article or the book, please contact the author at mtaplin@ouhk.edu.hk.
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| Some Questions
For Discussion With Colleagues, or Action Research In Your Classroom |
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