Does Practice Really Make Perfect?

While practice makes perfect – so the saying goes – it is no longer always true in today’s Math class, especially when solving word problems. Word problems today require more than mere computation. You need to consider the context of the problem and give thought to the meaning of the mathematical processes before you can arrive at the solution.

Article highlights

When mathematical problems have more than one solution
Word problems as a change agent

Thinking Time

Try solving this:

Alvin has 2 brothers.
Brian has 2 brothers.
Chris has 2 brothers.
Alvin, Brian, Chris and their brothers went into a van.
How many boys are there in the van?

If you answered 6, you are correct. If you answered 8, you are also correct.

Confused? Could there be more than one correct answer to this problem? Yes – depending on the context:

If Brian and Alvin are brothers and Chris is a boy not related to them, then the answer is 6.
If Chris is a girl and Alvin, Brian and Chris are not siblings, then the answer is 8.
Of course, we are assuming it is an empty van the children are getting into. If the van already has some boys, then the answers are again different.

That was an example of a non-standard word problem, as opposed to a standard word problem like this:

Mary has 10 boxes of apples.
There are 5 apples in each box.
How many apples does Mary have?

You can get the answer by a simple computation:
10 x 5 = 50

In this case, there is no need to consider the context of the problem to obtain this answer. The result of the computation is also the solution to the problem.

In standard word problems, the solution is often obtained by identifying the correct operations and carrying out the corresponding computations. There is no need to consider how to use the answer from the computation to obtain the problem solution.

Background of T3 Project

The Think-Things-Through (T³) project aimed to transform the teaching and learning of mathematics through:

developing teacher’s pedagogical skills
engaging students in sense-making when solving word problems

Teachers were trained to use a set of teaching materials, including student worksheets with standard and non-standard problems and manipulatives. They were given different levels of support, with the majority being given teaching notes and encouraged to discuss the lesson plans with their colleagues.

Changing the Way We Learn

Word problems were the focus of a project to study its use as a change agent in Mathematics classrooms.

Although the intervention didn’t increase student’s success in solving non-standard problems, it did enable more students to think more deeply about the problems. The proportion of students who gave illogical answers using mere computation decreased over time.

Three factors were found to have an impact on how well students were able to handle non-standard word problems:

Teachers who are given more guidance in using non-standard problems in their teaching.
Teachers who have the knowledge and understanding of the mathematics behind the non-standard problems.
Teachers who believe in and are comfortable with a more student-centred approach.

Solutions for Math Teachers

Give your students more opportunities to discuss non-standard problems.
Use more student-centred teaching approaches that focus on engaging students rather than on completing a task.
Internalize the principles of the Math curriculum and adapt them in your teaching.
Seek guidance on how to incorporate non-standard problems in your lessons.

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