Math CentralQuandaries & Queries


Pandora, a parent:

Is there a easier way to explain to my child regarding "pages" problem - like when to add one? when to sub tract one?
John read 46 pages in his new book. If he began reading at the top of page 33, on what page did he finish?
and John needs to do 45 pages of problems in his workbook. He finished this assignment at the the bottom of page 99. On what gate did John begain?

Thank you.

With problems like this, I do not begin with a rule (though I can create one after analysis) but with a picture, really a small scale simulation.

What if the numbers were smaller?
- In mind I imagine a small book.

If I start on page 3, and end on page 5, how many pages I have read? I imagine the pages

Part of the difficulty is that we are not quite reading the whole page (and turning each page).
So - if I start at the top of page 3, and turn page 5, I will be looking at the top of page 6. That is where the bookmark will go anywhere.
Now subtraction of page number does work!

Similarly, if I need to end at the bottom of page 4, then I might start turning from the bottom of page 2 (top of page 3) in order to cover two pages.

Alternatively, I might turn page 4 (looking at the top of page 5) so 2 pages starts at the top of page 3.

You are actually trying to count 'whole pages' - but the words suggest an imaginary set up with is not quite complete! Therefore the numbers 'slip' and subtraction/ addition do not quite fit.

You will see I am using visual thinking - but in a way that only some features matter, and I could imagine the picture with pages 99, 100,
54, 55, ... without all the pages in between. This type of reasoning works well in these 'counting problems' - but is often hidden in the books - which quietly assume you will make these 'pictures in your mind' without drawing them (and without them drawing them).

Walter Whiteley
York University

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