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Explaining and Exploring Mathematics

Christian Puritz
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

This is a series of Socratic dialogs about a variety of math topics in middle and high school, aimed at teachers. It has a strong discovery or inquiry-based learning flavor, which is done deliberately to combat the rote learning method with is commonly used at these levels. It’s not enrichment material in the sense of covering things students wouldn’t normally see (although there are a few of these, such as Farey sequences). It does attempt to teach the students why mathematical rules are true. It starts at age 11 and so assumes they already know arithmetic, and focuses on sizes of numbers, algebra, trigonometry, geometry, and calculus. It’s not comprehensive on any of these, but does give a good idea of the kind of thinking needed for them.

The book is aimed at the British curriculum, which seems to be similar to the American one although some topics may be introduced earlier in Britain. The book is divided into three age ranges (11–14, 14–16, and 16–18 years old), although there’s no need to stick to this division. There are a few terms that may be confusing to an American audience, such as a problem involving muesli bars on p. 14 and the order of operations being summarized as BIDMAS rather than PEMDAS.

I thought the dialog approach was awkward, although it may be useful in making the teachers more comfortable with the roundabout path that discovery learning takes. The players are the teacher and about five students in each age range. All the students seem very smart and articulate and probably indistinguishable. I think the fictional students did much better at figuring out the answers than real students would be. These dialogs are more upbeat than the original Socratic dialogs: there’s no attempt to show the listeners how ignorant they are before leading them to the truth.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.


Part I: 11-14 years old

1. Decimals and multiplication by 10 etc.
2. Multiplying and dividing by decimals
3. Adding fractions
4. Multiplying and dividing by fractions; and by 0?
5. Using patterns with negative numbers
6. Use hundreds and thousands, not apples and bananas!
7. Angles and polygons
8. Special quadrilaterals
9. Basic areas
10. Circles and pi
11. Starting trigonometry
12. Square of a sum and sum of squares
13. The difference of two squares
14. Another look at (a-b)(a+b)
15. Number museum: how many factors?

Part II: 14-16 years old

1. The difference of two squares revisited
2. The m,d method: an alternative approach to quadratics
3. Negative and fractional indices
4. A way to calculate pi
5. Pyramids and cones
6. Volume and area of a sphere
7. Straight line graphs
8. Percentage changes
9. Combining small percentage changes
10. Trigonometry with general triangles
11. Irrational numbers
12. Minimising via reflection
13. Maximum area for given perimeter
14. Farey sequences
15. Touching circles & Farey sequences again

Part III: 16-18 years old

1. Remainder theorem…
2. Adding arithmetic series
3. D why? by dx; or What is differentiation for?
4. Integration without calculus
5. Integration using calculus
6. Summing series: using differencing instead of induction
7. Geometric series, perfect numbers and repaying a loan
8. Binomial expansion and counting
9. How to make your own logarithms
10. The mysterious integral of 1/x
11. Differentiating exponential functions
12. Why do the trig ratios have those names?
13. Compound angle formulae
14. Differentiating trig ratios
15. Fermat centre of a triangle