As children begin to engage with, and enjoy maths games and puzzles, they realise that reaching the correct answer is not the greatest source of satisfaction, or indeed of paramount importance. What really matters is the speed with which the problem is solved and the elegance of the solution. People who aren't mathematicians sometimes struggle with the concept that some answers to maths problems are elegant and others ugly, when both result in the correct answer. More perplexing still is the idea that sometimes a child who produces a wrong answer using a mathematically correct approach should be awarded higher marks than a child who arrives at the right answer by guesswork or, dare one suggest, cheating.
Revisiting the Lily Pond Problem
In the opening article we considered a maths problem or brain teaser involving a lily and a lily pond. Stripped of its accompanying narrative, the mathematical bones of the problems are a pond 15 feet by 8 feet, in which has been planted a lily which doubles in area each day. After seven days the plant covers half the area of the pond. How long does it take before the pond is completely covered? The problem can be solved logically, mathematically or practically.
The Mathematical Approach
To approach the problem using conventional mathematical techniques, one would develop a formula using the variables to first work out the area of the lily when planted, and then to determine the number of days needed for it to cover the pond. The initial formula would read
Area after n days = Initial Area x 2^n (two to the power n)
By substituting the area covered after a known number of days, which is 60 square feet when n equals 7, one can calculate the initial area of the pond as being 0.46875 square feet. Having substituted the final area that is covered by the lily, 120 square feet, it is calculated that it takes 8 days for the lily to reach an area of 120 square feet. All very mathematically correct and all very long winded.
The Logical Approach
Children who grasp that both the size of the lily pond and the initial area covered by the lily are irrelevant to the problem will quickly arrive at the solution, which has an elegant simplicity. If the lily doubles in size every day and covers half the area of the pond in n days, the time taken to cover the whole are is n plus one days. A child who immediately sees the logical path may answer the question in only a few seconds but be unable to explain the problem's solution; 'well it's obvious' and 'it just is' are typical responses.
The Practical Approach
The practical approach to the problem might involve drawing a scale model of the lily pond on squared paper and then dividing the area successively in half to show the area covered after 7, 6, 5 days etc. For some children, the process of visualising the problem will provide a short cut to the logical solution, but nearly all will arrive at the correct solution eventually if they persist with the practical approach.
The Dangers of Cheating
Earlier, I mentioned cheating. Perhaps cheating is a strong word to use, but if you set maths games as homework you must be prepared for the fact that a proportion of children will seek the solution to the problems online, rather than using their own mental capabilities. For this reason I would advocate using maths games and puzzles as fun classroom activities, rather than as homework exercises.
The Educational Value of Maths Games
There is no doubt that some children find this type of maths game or puzzle enthralling, whilst others find them extremely frustrating. We've also discovered in our detailed analysed of the solution to the 'the lily pond problem' that some puzzles which are presented as maths problems are best solved using a non-mathematical approach. This leaves us to consider, in the final article in the series, how best to use maths games in an education context and how to integrate them effectively with other classroom activities.
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