Singapore Math has gained in popularity in the United States in recent years.  Numerous schools, both private and public, have adopted it with varying degrees of intensity and success. But where did Singapore Math come from, and what makes it different?

Ironically, Singapore Math has its roots in the United States. Born October 1, 1915, American psychologist Jerome Bruner has spent a lifetime dedicated to understanding how humans perceive and learn. He graduated from Duke University in 1937, and obtained a PhD in pyschology from Harvard in 1941.

A pioneer of modern psychology, his work on human perception and sensation was important in establishing the concepts of cognitive psychology. Following his work on perception and sensation, he turned his attention to learning and developmental psychology. His work there led to a framework of how learning takes place that can be applied to various situations. The framework is broken down into 3 stages, also known as the 3-step learning process.

Enactive – Sometimes called the ‘concrete’ stage, the enactive stage involves interactions with the physical world and objects — how they fit together or come apart, how they can be grouped, etc.

Iconic – The iconic or pictorial stage is when learning occurs by looking at pictures or models.

Symbolic – The symbolic or abstract stage is where learning can take place in abstract terms.

Singapore Math aims to utilize these three stages, progressing deliberately through each. In the enactive stage, students are taught by using physical objects in the classroom — paper clips, crayons, or other objects that they can pick up, pile up, and shift around.  Subtraction, then, would be taking a few crayons from a pile, and figuring out how many are left.  This isn’t described as subtraction, so much as it is a demonstration of play.

In the iconic stage, Singapore Math has students drawing pictures — usually bar models — to describe ratios and simple equations.  The emphasis is on visualizing what happens

during the mathematical exercise. At this point students still aren’t thinking in terms of algorithms, equations, or even numbers.

In the symbolic stage, students start using numbers and equations. With concrete and iconic bases, the idea is that the symbolic stage will be more easily understood.

The application of Singapore Math in the United States has mad mixed results. Unless schools utilize the curriculum completely, it can fail. Also, since most Americans learned math a different way, parents may have difficulty helping their children with Singapore Math homework, instinctively teaching algorithms (stage 3) when the children are counting paperclips (stage 1).  Still, the concrete applications like this of cognitive psychology to learning are exciting steps toward improving education.

Hu, Winnie. “Making math as easy as 1, pause, 2, pause, 3.” Sept 2010, The New York Times, retrieved from http://www.nytimes.com/2010/10/01/education/01math.html?_r=0

Macleod, Saul. “Bruner.” 2008, retrieved from http://www.simplypsychology.org/bruner.html

As a software developer, one of my heroes for many years has been the English scientist Alan Turing.  He was one of the first people to talk about the concepts of machine intelligence and machine learning.  His insight was brilliant, and, beyond his contributions to computer science, he was a hero who came to a tragic end.

I first heard of the Turing Test when I was 13.  I’d already been programming in BASIC, and was fascinated by the idea of artificial intelligence.  The Turing Test seemed simple enough; a computer program had to convince a human that it wasn’t a computer program — it was also a human.  Essentially, a person would sit down at a computer and start a natural language conversation with two entities — one a computer and one another human.  The tester would need to determine which responses were coming from a computer, and which were coming from a human. While not a sign of actual intelligence, per se, it is still a valuable exercise in understanding communication and thought.

As a teenager, I thought this wouldn’t be that difficult, with enough lines of code.  So I started making up a Q&A Turing Test, just to see how it would work.  Let’s look at an example of some input, computer ‘deliberation’, and responses:

Human: “Hello, I’m Lee”

(simple introduction… respond in kind.)

Computer: “Hi Lee, I’m Hal.  How are you?”

Human: I’m well.  Looking forward to the Superbowl.  How about those Seahawks?

(… assuming the software knows what a Superbowl is… and the Seahawks… and can put the Seahawks into context, and not determine the human is talking about a type of bird — three pretty huge assumptions already)

Computer: Yes, it will be exciting.  The Seahawks are great this year.

Lee: Do you have any plans?

(… assuming the computer is still understanding that Lee is talking about the superbowl… and that people ‘do’ things for the superbowl – going to parties or sports bars and such…)

It quickly becomes clear just what a huge undertaking this kind of thing is. Conversation, it turned out, is incredibly complex.  So, even with a million if-then-else statements, and a computer fast enough to comb through them all to come up with an appropriate response, tricking a human that a computer program isn’t a computer program can be quite difficult. It can be done, but not consistently, and in many instances not convincingly.  In fact, there’s an annual prize that’s been around since the 1990s — the Loebner Prize — that aims to challenge programmers to write software that does what the Turing Test was getting at. The ‘winners’ so far have been not completely convincing ‘chatterbots’ that don’t demonstrate intelligence so much as good trickery.

It is impressive that Turing came up with his idea when computers were in their infancy. In fact, one of the most complicated machine Turing had encountered was not a computer at all. It was critically important to the future of the world, though. During World War II, Turing was the leader of a team of engineers and scientists that unlocked the secrets of Germany’s Enigma machine – a complex device the Nazis used during the war to send and receive coded messages.  The code generated by the machine was seemingly unbreakable.  Turing managed to essentially reverse engineer the Enigma machine and break the code, though, so the Allies were able to intercept and understand German transmissions.  Some historians estimate that Turing’s efforts shortened the war in Europe by 1-2 years (Fitzsimmons, 2013).

For all of his successes, Turing’s life ended sadly.  In the early 1950s, the British government convicted him of being a homosexual, which was a crime in Britain then.  He was forcibly chemically castrated.  Two years later, before he turned 42, Turing committed suicide.  Thus ended the life of one of the greatest contributors to computer science – and tangentially to cognitive psychology – as well as a great hero of World War II. Ironically, he was pardoned for his ‘crime’ by Queen Elizabeth II, just a little over a month ago.

Fitzsimmons, Emma.  “Alan Turing, Enigma Code-Breaker and Computer Pioneer, Wins Royal Pardon”, NYTimes.com, December 24, 2013.  Last accessed February 2, 2014. Original Link: http://www.nytimes.com/2013/12/24/world/europe/alan-turing-enigma-code-breaker-and-computer-pioneer-wins-royal-pardon.html?_r=0