Purpose of Word Problems

Objectives of Word Problems

Word Problems are universally considered as a difficult aspect of mathematics. Before considering the difficulties, let us see the objectives or reasons why word problems are an important part of the math curriculum.

Some of the objectives are easily apparent to most teachers.

1. Introduction to Mathematical Modelling of real-life situations

   1. Word problems describe situations in life where math can be used to analyse the situation, find alternatives and find a solution. This enables students to understand the relevance of math to their lives and possibly reinforces their motivation to learn the subject. It also strengthens conceptual thinking.

   2. Word problems give opportunities to describe situations from other subjects like science, sports, social studies etc. AI tools like ChatGpt can help in constructing word problems which connect varous disciplines.

   3. Students realise, over a period of time, that they at not just solving word problems in math but that they are solving problems which they may come across in life.

   4. But teachers usually have a simplistic understanding of life situations. Teaching of "operation metaphors" are not explicitly a part of the math curriculum.

      1. Teachers do not even realise that these are one of the sources of the difficulties students face while doing word problems. We will deal with this issue in detail in the next chapter.

2. Fluency with computations & procedures

   1. Word problems present a variety of mathematical computations and procedures, in contexts relevant to life. Hence, they enable students to practice the computations & procedures with an understanding of their relevance.

      1. This strengthens logical & structured thinking and enables learning at a deeper level.

3. Precise use math-related vocabulary & syntax

   1. Understanding a word problem, writing its solution in logical steps and interpreting the solution in terms of the rea-life context, enhance appreciation of the precise nature of math vocabulary & syntax.

   2. A math student has to be as careful with language as a lawyer!

Mathematical Thinking

The most important objective of solving word problems is to develop mathematical thinking. Most teachers may not be aware of this aspect. We will see this in detail in Chapter 29.7 "Types of Thinking".

The ultimate purpose of learning is problem solving. Word problems are the first steps to learning problem solving.

Let us now look at several other objectives which may also not be apparent to teachers.

1. Introducing a new topic
   Using a familiar life situation or a story reveals its relevance. It kindles interest of the students and increases motivation to learn the topic. History of math has many such stories of how new ideas were thought of.
   For example, before introducing the idea of exponents, introduce and discuss with students a problem which needs the use of exponents. The famous chess story where a king is requested to place 1 grain of rice in the 1stsquare and keep on doubling the number of grains in the subsequent squares could be used. Let the students arrive at the idea of “repeated” multiplication before introducing the notation as a short cut to represent such operations.

2. Connecting a topic to previous topics
   Using a story from history of math can also reveal its connection with the previous topics. When doing decimal numbers (which are usually done after fractions) use a problem which uses both representations thus emphasising their equivalence and interchangeability. Here the social conventions of using these representations in certain fields can be brought out. For example teachers give fractional marks to questions in answer papers in rational number format, whereas all prices are denoted in decimal notations.

Developing Logical & Structured Thinking

Teachers also need to realise that most applications of math in real life, which are understandable to students would be from arithmetic and elementary geometry. Most other applications in science & engineering use more complex topics like calculus, trigonometry & differential equations. It will be difficult for both the teachers & students to provide easy to understand examples.

But what needs to be emphasized at all levels of schooling is that the basic purpose of learning math is to “develop logical & structured thinking”. At every level of school from K to 12, math provides ample opportunities for developing these skills. The tragedy is that instead of doing the above, math has been reduced to memorizing computations & procedures. This has been a major reason for the development of math anxiety & phobia among students.

We give a few explorations for developing logical thinking in K-5 level.

1. Given any number, we can always provide a bigger number. Hence there is no “biggest” number

2. Visualise the difference between odd, even, composite & prime numbers

3. Visualise number 324. Write it in Base 8 number system.

4. Why is ODD + ODD = EVEN whereas ODD X ODD = ODD?

5. What exactly is “Carry over” in addition?

6. Give an example in life where it makes no sense to “put together” 2 numbers stated in the same unit of measurement.

7. Why do we take LCM while adding 2 fractions?

In the next chapter, we will deal with some of the difficulties students face in understanding & solving word problems.

Mechanical Approach to Word Problems

Often students are taught mechanical means for solving word problems without the need for thinking deeply. One such strategy is called CUBES (Circle, Understand, Box, Eliminate, Solve) which is to focus on "key words." Such approaches lead to many misunderstandings.

One of them is that every number which occurs in a word problem has to be used. A famous problem is the "age of the shepherd" problem - "There are 125 sheep and 5 dogs in a flock. How old is the shepherd?"

Researchers have documented that a majority of students tried to us the numbers given in the problem to arrive at an answer! Some critical thinking would have revealed that the question is an absurd one.

It is important to remind ourselves that developing mathematical thinking is far more important that just getting the answer to a problem.