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How maths should be taught in high school

This is a follow up to our no more algebra article, where we criticize the way maths are currently taught. Here, we provide an example of a great math lesson and how it can spark interest in maths in mainstream kids (including girls), not just geeks.

The Pythagoras Theorem Revisited

In traditional US high school teaching, it's presented as a theorem totally out of context, and the focus is on doing repetitive, boring-to-death drill exercises. Here's how I would teach this subject:

  1. Lets start with the fundamental triangle with sides of length 3, 4 and 5 (whatever the unit is). How many right triangles have integer numbers for their length? Can you identify all of them (is there a finite of infinite numbers of such triangles)? Let's discuss units.
  2. Now let's focus on the right triangle with sides of length 1, 1 and SQRT(2): is SQRT(2) a rational number? How to disprove this fact? What is a non-constructive proof? How to approximate SQRT(2) by a simple iterative algorithm? How to boost convergence of this iterative algorithm? Obviously, a quantity such SQRT(2) can be represented using a compass and a ruler. But can the number Pi be?
  3. About 2,000 years ago, a Greek mathematician discovered that SQRT(2) can not be represented as a fraction of two integers. The mathematician in question was eventually murdered for revealing this secret.
  4. How to best approximate irrational numbers by fractions? Why is (1+SQRT(5))/2 - the Fibonnacci gold ratio - is the number most difficult to approximate by a fraction of integers?
  5. What is the nature of the animal {SQRT(2)}^{SQRT(2)}? Prove that it is irrational. Is it transcendent? (nobody knows)
  6. Are decimals of SQRT(2) random? More random than Pi? How do you determine randomness? (as of today, nobody knows if SQRT(2) or Pi decimals are randomly distributed)
  7. Pythagoras theorem generalized to 3, 4, 5 and any dimensions. What's the theorem in 3 dimensions? How do you project 3 (or 4) dimensions onto a 2 (or 3) dimensional space? Make the connection with data visualization (use Tableau or some other software for illustration) Introduction to the concept known as the curse of dimensionality.
  8. What's the shape of the surface with minimal perimeter (a square, a circle, a triangle, something else)? And in 3 dimensions?
  9. Can you create tessellations with right triangles? What about in 3 dimensions? 
  10. Introduction to the simplex problem (the triangle is the most basic simplex)
  11. Why is the Pythagoras Theorem known as "Pont aux Anes" ("Donkey Bridge") in French? What does it mean?

The way I would teach such a lesson is by focusing on history, fun facts, and have kids discover the proofs of a few simple facts, by themselves. Then allowing the kids to think about generalizations (3-D, other irrational numbers etc.)

Related article: Is Algebra Necessary? | New York Times

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Comment by Bill Cormier on August 10, 2012 at 9:37am

We have a critical shortage of problem solvers in the US and likely worldwide. A big part of the reason is that education from K to 12 or perhaps K to 16 is focused on ONLY information (2 + 2 = 4).  My son at age 6 did his first math proof (prove there is no biggest number). A simple proof yes, but it was a great thinking process.  We spent about a year (10 or 20 mins. every month or so) working on is there a biggest prime and he was able to prove there is no biggest prime.  I let it run as long as it needed to go.  

 

This thinking part was only 2% to 5% of his math work, the other 95% to 98% was basic foundation information, facts and process.  So, no do not get rid of the foundations, just add some thinking and problem solving to the mix.  The real problem is that well over 99% of teachers have no clue if there is a largest prime or not. I would not want to speculate on the % who know if there is a biggest number or not. This will add some light to the teacher problem http://econfaculty.gmu.edu/wew/articles/04/ineptitude2.html

 

I live in a county with 100K students.  The Math Counts program would only get 8 or 9 teams. Typically 2 public schools (the math magnet schools), 1 home school team and 5 or 6 from private schools with less than 100 students total attending.   

 

The bottom line is that the quality of US K-12 academics is sinking compared with other nations. We spend near the top in funds and are near the bottom in results.  There are existing efforts to take out more basics such as 2+2=4. The student needs to know the basic skills in order to use them to solve a problem AND they need to work on solving problems.

Comment by Vincent Granville on August 9, 2012 at 3:53pm

I think what is being called math, geometry, calculus or algebra in high school or college is actually... not math. It's something that could be called drill or memory exercises, but has nothing to do with real maths. 

Comment by Vincent Granville on August 7, 2012 at 7:24pm

My 9 years old daughter is very interested in discussing maths with me: she comes back from school, I asked her what she learned, and then I tell her about interesting generalizations. She's proud to be the only kid in the classroom to know about square roots, irrational numbers, tessellations, math illusions and the regular polyhedrons, thanks to me. And she's proud that her dad makes a living (working from home) using highly applied analytic wizardry. I don't help her with her homework though. Also, she does not look like a geek.

The lack of interest in maths in US is probably caused both by teachers and school system in general, other kids and the parents - not just by high school teachers alone. Interest has to start in primary school, even before. In the future, I could see the situation improving as parents buy math/stats training apps for kids to play on their iPads (rather than watching TV or doing sports). My 5-year old boy  is already learning how to count when he wants money to buy stuff (game upgrades, unlock a new level, get fake money etc.) on his iPad! He already managed the "hack" the password a few times to buy stuff with real money :-)

Comment by Bill Cormier on August 7, 2012 at 9:14am

http://www.siam.org/news/news.php?id=240  SIAM looked at the fluf in math text books, it is a good read.

 

In the US most math text books have far too much fun things and not enough solid math.  We homeschooled our kids and were delighted to find Saxon Math text books (it has since changed ownership and I do not know about the quality od its current books).  It was basic solid math and had plenty of real world and practical problems, it was very applied.

 

Comment by Razvan on August 6, 2012 at 10:30am

Re:"What is the nature of the animal {SQRT(2)}^{SQRT(2)}? Prove that it is irrational. Is it transcendent? (nobody knows)".

I think the animal above is transcendental. By Gelfand-Schneider's theorem, 2^{sqrt(2)} is transcendental, and this is the square of the animal. If the animal would be algebraic, its square would also be algebraic, by a simple algebraic construction. So the animal is transcendental.

After having taught at the University level for more than 10 years, I would like to submit the opinion that the quest for "fun facts" too many times obliterates the need for a solid foundation. The books become more and more colorful, with more and more photos that are less and less related directly to the subject to be presented. And oh yes, they cost a fortune, and are changing every year. Seriously? In more than 3 centuries, there is no good exposition of (Pre)Calculus (Algebra included) that can become a de-facto standard for undergraduates? Do we really get 15 different new and improved ways to teach Calculus every year? I doubt it.

In the end, it is not the amount of "fun facts" that stays with a student, but the amount of solid, useful knowledge. The difficult part is balancing the personality and learning curve/type of each student, with the need to transmit a set body of knowledge. The students who have the drive and focus to learn on their own, regardless of the resources (online/books/class notes etc.) are few. For the 99.9%, following a clear laid out plan of study works best - and sometimes better than for the other 0.1%.

While hoping to avoid a flame war, I submit the opinion that the "example of a great math lesson" above is great only in regards to a specific target student population. Which may not be representative of the whole student population.

It would be really interesting though to collect, from the professionals in this forum, impressions and suggestions - keeping in mind the very personal nature of education - on how to improve the teaching of the new generation. Which will build the bridges (analytic or not) on which all of us will walk one day...

Respectfully,

Razvan V.

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