Mathematics education uncovered and recovered

The root mystery



What is $$\sqrt{25}$$? You probably know but let’s see what textbooks say. One Year 7 textbook explains:

Key point: The inverse of square is square root.

$$3^{2}=9 \enspace$$ so the square root of  $$9=\sqrt{9}=3$$

However, just next to this explanation there is a copy-and-complete exercise that reminds students that both $$4^{2}$$ and $$(-4)^{2}$$ give $$16 \enspace$$ and suggests that they write $$\sqrt{16}=4 \enspace$$ or $$-4$$.

Write down both answers to each of these: $$\sqrt{25}$$, $$\sqrt{81}$$, $$\sqrt{9}$$

So it looks like $$\sqrt{9}$$ does mean two numbers at once, despite what the Key point note above says. Moving on to the next page though, we see a worked example like this:

$$\sqrt{49}=7 \enspace$$ and  $$\sqrt{64}=8$$

$$\sqrt{55}$$ lies between $$7$$ and $$8$$

Well, should students now forget that $$\sqrt{49} = 7 \enspace$$ or $$-7 \enspace$$ as they were told on the previous page? Perhaps so.

I am not discussing the validity of the reasoning behind the estimation example, but it certainly deserves some explanation. Otherwise, by the same logic we can conclude that $$1^{2}=1 \enspace$$ is between $$(-2)^{2}=4$$   and $$3^{2}=9$$

Thus the explanation from the textbook “based on academic research into what improves learning in mathematics” leaves us none the wiser.

Maybe it’s because the proper explanation is beyond understanding of young Year 7 students? Let’s have a look at what happens at A level then.

Starting with the statement that $$\frac{\sqrt{5}+1}{2} \approx$$ 1.618, the first page of a popular Core 1 textbook promises clear answers. The hope doesn’t last long though. A few pages later we see that $$49^{\frac{3}{2}}=\pm343 \enspace$$ probably implying both $$7$$ and $$-7$$ to be the answers for $$\sqrt{49}$$. Two pages later the textbook firmly claims that $$\sqrt{4} = 2 \enspace$$ but then after a few more pages we see the warning:

Remember that $$\sqrt{25} = 5 \enspace$$ or $$-5$$

And, of course, when the quadratic formula is presented, the square root is dutifully preceded by $$\pm \enspace$$, which would not be required if taking the square root produced both positive and negative answers.

So what is $$\sqrt{25}$$? The textbooks cannot solve this mystery and the only hope for the students is that their teachers can.

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