The Haylock Bookmark

When I am in the middle of reading a book I need a bookmark that not only helps me to find the page I am on, but also exactly where on the page I should start to continue my reading! I offer readers my design for an effective bookmark for this purpose and a simple example of mathematics applied to another area of the curriculum!

First measure the height of the page of the book (h mm) and the height of the smaller of the top and bottom margins for the text on the page (x mm). Cut a rectangular piece of firm card so that it has a length that is equal to h/2 + x. The width can be whatever you wish, say, about a third of the width of the book.


For example, for a standard paperback novel I find that h = 198 and x = 19. So we need the bookmark to be 99 + 19 = 118 mm long.

Then on one side of the card draw a double-headed arrow x mm from the top (i.e. 19 mm for our standard paperback). The diagram shows a bookmark made for this standard paperback.

That's it! Now, when you have finished your reading session place the bookmark in the fold of the book on the opposite page to the one you are on, with the arrow showing and pointing to your place on the page. This might involve rotating the bookmark, of course, depending whether you are on the top half or the bottom half of the page. Then just close the book. When you open it next time just make sure you have the side of the card with the arrow visible, and remember that the arrow tells you where on the opposite page you should start reading again!

With this system any starting point on either the even-numbered page (verso) or the odd-numbered page (recto) can be marked, without the bookmark sticking out of the closed book (which I don't like).

The diagrams below show the bookmark placed inside the book to mark (a) a point half-way down the left-hand page; and (b) a point near the bottom of the right-hand page.

 (a)

(b)

Well, it works for me. And it is a good application of simple measurement and spatial reasoning!

Measuring scales

One of the important principles to understand about measurement is that there are different kinds of measuring scales. And what you can do with a particular set of measurements will depend on what kind of measurement scale is involved.

For example, you can compare two lengths (say, a = 6 metres and b = 2 metres) by their ratio (a is 3 times longer than b). So, 'length' is what is called a ratio scale. One of the features of a ratio scale is that 0 represents 'nothing'. So, '0 metres' is no length at all.

Temperature measured in degrees celsius is not therefore a ratio scale, because '0 degrees' does not represent 'no temperature at all'. So, you could not compare, say, a temperature of 12 degrees with one of 4 degrees, by saying that the the first one is three times hotter! To compare these you could only use the difference in temperature: the first is 8 degrees hotter than the second. This is an example of an interval scale.

A more primitive type of measurement, often met in everyday life, is an ordinal scale. Essentially all this does is to put various entities into order, according to some criteria.

An interesting example of the misuse of an ordinal scale is provided by the Amazon customer reviews for a book. To give a book 3 stars means simply that it is judged better than 2 but not as good as 3, by whatever criteria the reviewer is using.

The thing to note is that you cannot do arithmetic with ordinal scales. You cannot say that a 4-star rating is twice as good as a 2-star rating; or even that the difference between a 2-star rating and a 3-star rating is in some sense 'the same size' as the difference between a 3-star rating and a 4-star rating! So, you definitely cannot add up all the customer ratings and divide by the number of them to calculate an average rating (i.e. the arithmetic mean). That could only be done if the interval between various ratings represented the same amount of something! If there was something that could be measured that we could call, say, a 'unit of worth', so that 1 star meant 1 unit of worth, and 2 stars meant 2 units of worth, and so on, and 0 stars meant 'absolutely worthless', then we would have a ratio scale and we could merrily calculate averages and so on. But customer ratings are not like that!

I mention this because I noticed that my book, Mathematics Explained for Primary Teachers, is given an average rating of what looks like four and a half stars. This is the result of doing some arithmetic with 21 five-star reviews, 3 four-star, 1 three-star and 1 one-star. [This 1-star reviewer wrote 'not fore parents' (sic).] If you were to treat these as measurements on a ratio scale, the arithmetic mean would be 4.65, which appears in an icon as 4.5 stars.

But, as explained above, you cannot do arithmetic with an ordinal scale, so the use of the arithmetic mean is unacceptable to find an average. What you could use for an average ranking is either the median or the mode, since these require nothing more than putting items in order or putting them into a set number of categories.

If you use the median as an average you line up all the reviews from the lowest to the highest and choose the one in the middle to represent the whole set. In this case the median value is 5 stars!

If you use the mode then you just select the most commonly occurring grade: which again is 5 stars.

So, clearly, if the maths is done appropriately, using either of the legitimate averages for ordinal measurements (median or mode), then my book should get an average ranking of 5 stars! I rest my case.