Conservation refers to an ability in logical thinking according to the psychologist Jean Piaget who developed four stages in cognitive development. During the third stage, the Concrete operational stage, the child of age 7-11 masters this ability, to logically determine that a certain quantity will remain the same despite adjustment of the container, shape, or apparent size.

Conservation tasks test a child’s ability to see that some properties are conserved or invariant after an object undergoes physical transformation. Conservation itself is defined as the ability to keep in mind what stays the same and what changes in an object after it has changed aesthetically. One who can conserve is able to reverse the transformation mentally and understand compensation.

Piaget’s most famous task (there are many others e.g. conservation of substance, weight, number etc) involved showing a child two beakers, both of which were identical and which contained the same amount of liquid. The child was asked whether the two beakers had the same amount of liquid in both. Then liquid from one of the glasses was poured into a taller, thinner glass. The child was then asked whether there was still the same amount of liquid in both glasses. A child who cannot conserve would answer "No, there is more in the tall thin glass".

He furthered the conclusion to suggest that this confusion was born from a pre-operational child’s inability to understand the notion of reversibility; the ability to see the reversal of a physical transformation as well as the transformation itself. These ideas were used to create the ‘Principle of Invariance’.

The ages at which children are able to complete conservation tasks has been questioned by subsequent research. Research has suggested that asking the same question twice leads young children to change their answer as they assume that they are being asked again because they got it wrong first time around ^[1]. The importance of context was also emphasised by researchers who altered the task so that a 'naughty teddy' changed the array rather than an experimenter themselves. This seemed to give children a clear reason for the second question being asked, and reduced the age at which children passed the tests ^[2].

Newman's prompts

Finding out why students make mistakes

The Australian educator Anne Newman (1977) suggested five significant prompts to help determine where errors may occur in students attempts to solve written problems. She asked students the following questions as they attempted problems.

1.       Please read the question to me. If you don't know a word, leave it out.

2.       Tell me what the question is asking you to do.

3.       Tell me how you are going to find the answer.

4.       Show me what to do to get the answer. "Talk aloud" as you do it, so that I can understand how you are thinking.

5.       Now, write down your answer to the question.

These five questions can be used to determine why students make mistakes with written mathematics questions.

A student wishing to solve a written mathematics problem typically has to work through five basic steps:

1.Reading the problem	Reading
2. Comprehending what is read	Comprehension
3. Carrying out a transformation from the words of the problem to the selection of an appropriate mathematical strategy	Transformation
4. Applying the process skills demanded by the selected strategy	Process skills
5. Encoding the answer in an acceptable written form	Encoding

The five questions the teacher asks clearly link to the five processes involved in solving a written mathematics problem.

If when reworking a question using the Newman analysis the student is able to correctly answer the question, the original error is classified as a careless error.

Research using Newman's error analysis has shown that over 50% of errors occur before students get to use their process skills. Yet many attempts at remediation in mathematics have in the past over-emphasised the revision of standard algorithms and basic facts.

George Pólya (December 13, 1887 – September 7, 1985, in Hungarian Pólya György) was a Hungarian mathematician.

Pólya's four principles

First principle: Understand the problem

This seems so obvious that it is often not even mentioned, yet students are often stymied in their efforts to solve problems simply because they don't understand it fully, or even in part. Pólya taught teachers to ask students questions such as:

• Do you understand all the words used in stating the problem?


• What are you asked to find or show?


• Can you restate the problem in your own words?


• Can you think of a picture or a diagram that might help you understand the problem?


• Is there enough information to enable you to find a solution?


• Do you need to ask a question to get the answer?

Second principle: Devise a plan

Pólya mentions (1957) that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:

• Guess and check


• Make an orderly list


• Eliminate possibilities


• Use symmetry


• Consider special cases


• Use direct reasoning


• Solve an equation

Also suggested:

• Look for a pattern


• Draw a picture


• Solve a simpler problem


• Use a model


• Work backward


• Use a formula


• Be creative


• Use your head/noggen

Third principle: Carry out the plan

This step is usually easier than devising the plan. In general (1957), all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don't be misled, this is how mathematics is done, even by professionals.

Fourth principle: Review/extend

Pólya mentions (1957) that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn't. Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem.