Friday, November 11, 2016

Why do High Schools teach (mostly) Medieval Mathematics? A Guest Post

A guest post from my father:

When I was a high school teacher I taught mathematics and sciences.  And while most scientific discoveries were presented along with the scientists and (with any luck) a historical context, the way mathematics was taught gave the students the impression that mathematics has always been there.  Perhaps it was first inscribed on the back of the Ten Commandments, or perhaps it was found in the cave paintings of southern France.  People don’t create mathematics; it just is.
But like any human endeavor, mathematics has a history, and I decided to find out what it was.  During my studies I realized most of what we teach in High School Mathematics was known during the Middle Ages.
Arithmetic goes way back into the mists of history.  Based on the evidence of the Ishango Bone, people may have been doing arithmetic by 18,000 BC.  It is also possible that ancient Sumerians invented ways to write numbers before they invented letters, just so they could count livestock and other saleable commodities.
Geometry was also known to the ancients.  Possibly the most famous textbook on geometry is Euclid’s Elements and it was written around 300BC.  Scholars believe it was based on earlier work by mathematicians such as Pythagoras.
Our knowledge (and word for) algebra comes from al-Khwarizmi who lived in the 8th century CE.   But algebra has roots that go back to Babylonian mathematics.  Even quadratic equations, which fascinated the Renaissance mathematicians, were known to Indian scholars such as Brahmagupta (598-c.670 CE).   Thus, most of what High School students learn would have been known in the Middle-Ages.
There are some exceptions of course. Analytic geometry started in the 17th century CE, as did probability.  Complex numbers started during the Renaissance.  Logarithms as we know them were developed during the 17th and 18th centuries CE, but go back to Babylonian times.
Even with the exceptions, most mathematics that high school students learn is at least 300 years old.  In fact, it was during one of my 4th Year Analysis classes in university (I have a B.Sc. in mathematics) that the Professor proudly announced.  “And now we are going to prove a theorem that comes from the 20th century.”
So why is this?  Partly it comes concentrating on the “how” of mathematics and not on the “why.”  It is important for students to understand arithmetic for business purposes and it is important they understand geometry so they can build things.  Some of the other stuff gets a bit dicey: “You need to learn in case you go to university.”  As an aside, one topic I wish was better covered in high school is statistics.  How many high school students really know what the phrase “This election poll is estimated to be accurate to within 3.1 percentage points, 19 times out of 20?”  How many people with a higher education even know what this means? 
I am not saying we should scrap the curriculum.  But it would be beneficial if we spent some time on the “why” of mathematics.  It could be taught as a brief history which could emphasize certain themes:
People invented things as they needed them.  Arithmetic came about because of needing to count things.  The Egyptians developed geometry because they had to contend with the Nile flooding each year.  They needed to replace their surveyor’s marks each year and geometry helped them do that.  They also needed geometry to build the pyramids.
People were/are constantly looking for refinements and easier ways to do things:  The Babylonians had a system for dealing with fractions that worked with parts of 60.  It was extremely accurate and allowed many kinds of divisions (e.g. 1/2, 1/3, ¼, 1/5, 1/6, 1/10, 1/12, 1/15, 1/20, and 1/30 are all easily represented as parts of 60).  The Indians realized they could simplify their number system by inventing a number for nothing, i.e. 0.  Europeans realized they could simplify their number system by borrowing the Indian one.
People were/are constantly looking to solve harder and harder problems.  Arab scholars like al-Khwarizmi realized that arithmetic expressions with unknown numbers could be manipulated as long as certain rules were followed and thus we got Algebra.
People realized that mathematical things could be combined.  For example, geometry and arithmetic could be combined to make analytical geometry.  Infinitesimals which Archimedes had used to solve geometric problems could be combined with algebraic concepts to create calculus.
People noticed patterns among certain things.  For example, rational numbers, real numbers, matrices, complex numbers, and polynomials all behave in similar ways. I.e. they could be added and multiplied.  Sometimes order mattered and sometimes it didn’t.  This gave rise to group theory and eventually to modern abstract algebra.
Mathematics could be used to predict things as well as solve for them.  This notion is imbedded in arithmetic, (e.g. If I work 9 hours at $10 per hour how much money will I have?) but it could also predict things that seemed unpredictable.  For example, if we played a game where you paid $1 for me to roll a die and I paid you $5 every time I rolled a one, probability can predict that if we play this game long enough I will clean you out, even if the dice is fair.
And finally, mathematics is constantly evolving.  It is not carved in stone in a cave or on the back of some tablets.  It grows as people’s needs grow.  And maybe if students understood the “why” better they would be more interested in the “how.”


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