Teaching Mathematics via Problem Solving

                                                                                                                                                                                                                         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 utilizing a problem-solving approach to teaching mathematics. 

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.
  • 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.

  • 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 envelop 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.
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:

  • 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:
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:
  • 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.”

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 toothers 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”.
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.




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