Mathematics education uncovered and recovered

Mystery continues


A new edition of a Pearson A level textbook has been published recently, and although there have been some changes made to the text, the mystery surrounding square roots has not been fully resolved. The good news is that the ambiguity about the meaning of the square root sign is gone as the book clearly says

  \(\sqrt{9}\) is the positive square root of 9.

The bad news is that the case of \(\sqrt{0}\) has been overlooked, but at least we can hope that weird statements like \(\sqrt{49}=\pm 7\) from the previous edition are expelled from this textbook.

Unfortunately, when it comes to using square roots for solving simple quadratic equations, there is still no clarity. To solve the equation \( (2x-3) ^{2} = 25\)  , the instruction is

Take square root of both sides.

Remember \( 5 ^{2} = ( -5 )^{2} = 25\)

which leads to \( 2x-3 = \pm 5 \) and then to \( x =-1\) and \( x =4\) .

Although the equation is solved, and the answer is correct, the process does not look convincing despite the line-by-line comments in the margins.

For example, what happens to the left-hand side when the square root is taken? Does it become \( 2x-3 \) as one may conclude from the suggested solution? Or is it  \( \pm (2x-3) \) similar to \(  \pm 5 \)  on the right-hand side?

It’s neither of those. Taking square roots of both sides of the equation leads to \( \left | 2x-3 \right | = 5\), which then produces two options: \( 2x-3 =  5 \) and \( 2x-3 = – 5 \).

Alternatively, to solve the original equation one can rearrange it as

\( (2x-3) ^{2} – 25 = 0\) and then factorise using the difference of squares formula.

Both methods are crystal clear and rigorous, while the solution suggested in the textbook is in fact wholly based on remembering that both 5 and −5, when squared, produce 25. If we choose this approach to solving the equation, then taking square root of both sides is not needed at all, contrary to what the textbook comment suggests. At the same time, apart from “remembering” about 5 and −5, one have to realise that no other number squared gives 25, but there is nothing on that in the margins.

To sum it up, the comments given lack some crucial detail, instead offering  irrelevant “explanations”. And sadly, the whole book is written in the same style, going against everything mathematics should teach: logic, reasoning, rigour. I feel really sorry for the students who will have to rely on this textbook for their A level course.

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