Friday, June 8, 2012

Ancient maths = modern learning

There are many issues of concern associated with mathematics teaching today.  One is the emphasis on procedural competence regardless of conceptual understanding. Another is the almost total lack of historical significance of mathematics. This is a pity as one can inform the other.

In our busy classrooms we tend to rush from one mathematical concept to another - often without any explanation given to the students. It is my experience that if we take the time to develop student interest in a topic then sound performance often follows  -  the “diversions” demonstrate that mathematics is more than learning algorithms  and makes the broad topic of mathematics more interesting - which engages the students - and engagement leads to understanding which in turn leads to higher performance.

My views on this are shaped not only by my experience as a teacher - but as a student. My primary years have been largely shrouded in the mist of time but I can still remember being stunned and amazed when I was briefly shown alternate historical methods of calculation. It was a revelation. Maths could be done more than one way? People once did maths differently - and accurately - and it worked … every time?  Ancient peoples were smart enough to do mathematics?  They discovered this stuff?

I’ve covered the same material myself in classrooms and  recognise the same lightbulb moment of understanding in students as they make the same discovery.  It is often the beginning of lasting interest in mathematics.

Today, thanks to web-based video we can show students “historical” mathematics as a way to scaffold and extend their mathematical knowledge.

One of the favourite “ancient” forms of mathematics taught in schools is often known as “Egyptian multiplication”.  It turns out that “Egyptian” part of the label may not be overly accurate - but the technique surely is.  This clip provides a good explanation of the process - together with a brief explanation as to why it works.

Ethiopian multiplication (also known as Egyptian)
From Egypt / Ethiopia / Elsewhere  we can make the short historical journey to the lattice method explained here.
Lattice Multiplication

The use of the diagonal columns will have parallels with the use of vertical columns in our current systems of calculations which will not escape the children and provides a useful opportunity to discuss the use of columns in modern calculations.
From there it is a short step to Napier’s Bones.  
Napier's Bones
 (This is a good clip despite it’s simplicity as it allows the students to study the structure of the “bones” and perhaps identify the pattern in the rods for themselves.) The clip below shows how the bones or rods were used.
The sense of amazement and wonder is enhanced if students actually complete some calculations themselves - in which case the pro forma here will be useful.  Once the students are capable of using the “bones” it is a useful extension to get them to use them to multiply two two digit numbers - and let them discover the technique themselves if they can.

The discussion around why devices such as Napier’s Bones were useful and necessary is a worthy way to spend some time.  Our students find it difficult to understand that it was once the norm for humans to perform all calculations, not silicon chips.  It is also worth raising awareness the universal education is a relatively new social phenomena - and discussing the impact of that.

Another avenue that can be useful is to look at numbers without using numerals at all (well, at first at least). This clip looks at the ancient Greek interest in square numbers and demonstrates the concept effectively using nothing more sophisticated than stones - which is all the Greeks needed to discover the concept.

Counting like an Ancient Greek

I’m not suggesting that the techniques presented here be “taught” in isolation - more that they can be explored and then, once students are familiar with the techniques, compare and contrast them to current methods.  It is often at this stage that students really develop an understanding of the strength of the place value system that we use today - after which their fluency with formal algorithms often improves significantly.  
If nothing else students will have enjoyed interacting with mathematics - and without a worksheet in sight.

If you enjoyed this post you may enjoy my other maths related posts available via the maths page or by clicking here.

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