Monday, January 16, 2012

Metacognition in Real Life – an Example with MRT

Figure 0.  Problem statement


     In London, they call it The Tube.
     In New York, they call it The Subway.
     In Hong Kong, they call it the MTR (Mass Transit Railway).
     In Singapore, we call it the MRT (Mass Rapid Transit).

     Singapore’s local railway system may not be as complicated as those in some other cities, but it does pose some navigation challenge for certain people, especially if they are easily confused or if they are absent-minded.  In this article, I shall show how we can apply metacognition and heuristics to solve a real-life problem of getting to one’s destination on the MRT.

     I wrote this in response to “intmath” in Twitter, who is an ex-colleague of mine, saying that he liked my solutions to the problems in the previous articles, but those problems only appear in textbooks.  Actually, I do agree with him that school mathematics is rather theoretical and I feel that the mathematics curriculum needs to be changed.  The good news is that metacognition and heuristics are general problem solving skills that can be applied in real life problems and in any situation, or indeed any new curriculum.  Let us begin our journey …

Stage 1:  Understanding the Problem

     I want to go from Tampines Mall to Nee Soon South Community Children’s Library, via the MRT.  That means I want to go from Tampines MRT station  (the nearest station to the shopping mall) to Khatib MRT station (the nearest station to the library).

Stage 2:  Planning the Method of Attack

What navigation aids can I use to help me?

     Maybe I could use Google Maps on my Android mobile phone.  Oh no!  My battery is running low.  So I’d better not take up so much bandwidth and energy.

     Ah!  Here is a schematic map of the MRT System.  I can use that.  Let’s see … I want to go from here (Tampines) … to here (Khatib).
Figure 1.  System Map

Can I go straight to my destination on a single train?
     No, the stations are on different lines (indicated by different colours on the map).  So I need to switch trains at one or more interchange stations.

What routes are available?

     I could take the train to Jurong East, change over to the North-South line, and go up north and back down south to Khatib ...  No.  That’s waaaaaaay too long.
Figure 2.  It’s a long way to Khatib station, it’s a long way to go …
     How about this: go to Outram Park, switch to the North-West line and go to Serangoon, then switch to the Circle line and get to Bishan, then switch to the North-South line and travel to Khatib.  Hmmmm … this looks like a snake.  Naaaah!  Too long and too many switches.  No can do.
Figure 3.  Snake Manoeuvre?
What other options do I have?

     OK!  This one: go to Paya Lebar interchange, switch to the Circle Line and go to Bishan, then use the North-South line to reach Khatib.  Two switches.  Not bad, we can consider that.
Figure 4.  Taking the Circle Line

What else?

     We could go downtown to City Hall, switch to the North-South line and go up north to Khatib.  Only one switch.  Not bad.  I can consider that.
Figure 5.  Going down-town first?

So what are the real options you have now?  Which is better?
     The last two.  The first two were basically cr**.  If I go downtown to City Hall, it will be more crowded there.  By the time I reach that place, it would be the evening rush-hour.  Even though the journey requires only one switch, there are more stations and the journey is longer.  The Circle Line route seems to be the best.  Go to Paya Lebar, switch line and go to Bishan, then switch line and go on to Khatib.  Two stations, shorter route, avoiding the city centre.  That’s the one I take.  Let’s go!

Stage 3: Execution
     So I hop on a train at Tampines.  It goes to Simei … Kembangan …

Stage 4:  Evaluation

Figure 7.  Tanah Merah Station – time to get off?

     I am at Tanah Merah.

Is it time to switch?

     No.  This station forks to Changi Airport, which is not where I want to go.

Where should you alight then?

     Paya Lebar station, four stops away..  OK, so I stay on.

Back to stage 3: Execution

     I remain on the train.  It goes to Bedok … Kembangan … Eunos … Paya Lebar!
Time to hop off and switch to the Circle Line!
Figure 6.  Paya Lebar – time to switch lines

     I walk over to the platforms for the Circle Line.  Now let’s see … er … which is the correct platform …

Stage 4: Evaluation

Is this the correct platform?

     This is the Circle Line alright.  But it goes downtown to Dhoby Ghaut. Let’s check the route …
Figure 8.  Platform B goes to Dhoby Ghaut

     None of the stations look correct.  So no, it’s not the correct platform.

Back to Stage 2: Planning

You have completed part of the journey.  What are you trying to do now?

     I want to go towards Bishan, and I want to find the correct platform to go to … *looking around* … ah-hah!

Figure 9.  The sign says go to Platform A

What does the sign say?

     It says go to platform A.

Stage 3: Execution

     So I go to platform A, and I hop on the next train on the Circle Line, towards Bishan.
Figure 10.  On the Circle Line

     The train goes to Mac Pherson … Tai Seng … Bartley … Serangoon … Lorong Chuan … Bishan!  OK, time to hop off and switch lines again.  So I go to the platforms on the North-South Line.  And I go north:  Ang Mo Kio … Yio Chu Kang … OK time to get off!

Stage 4: Evaluation

Figure 11.  Oops!  Wrong station!  I overshot!

     Yishun!  Oh dear!  I missed my station!  How dumb of me!  Silly stupid nincompoop!
Oh dear!  What am I going to do?  What am I going to do?  What am I going to do? Oh dear!  Oh dear!  Arrrrrrrrrrrrrrrrrrrrgh!

Come on!  Cool it man!  Don’t be too hard on yourself.  Let’s do some re-planning …

Back to Stage 2: (Re-)Planning

Where are you going to?

Where are you now?

Let’s look at the map again.  How many stops away is that?
     Er … one.  Hey!  That’s easy.  So I just go back south one stop, and I will reach Khatib.

Stage 3: Execution

     So I do as planned.  Take the next train south and there’s Khatib!

Stage 4: Evaluation
Figure 12.  Finally, at Khatib

Have you reached your destination?
     Yes!  Yes!  Yes!  Ha!  Ha!  Am I smart or am I smart?

Stage 5: Reflection

So what have you learned from this experience?
     I learned to apply the 5-stage problem solving process in a real-life setting, to get me from Tampines to Khatib on the MRT.  I use heuristics (e.g. reading a schematic map) and metacognition (asking myself questions and being aware of what I am doing) in doing so.  I learned to understand, plan, execute, evaluate and (now I am) reflecting.  Sometimes I need to loop back and forth between planning, executing and evaluating.

What else did you learn?
     I must remain alert at all times, so as not to miss my station.
     If I do miss my station, I must learn to keep calm and backtrack.  I “go back to the drawing board” (re-plan) and move on.


1)  Problem solving is not a straight-forward process.  Even expert go back and forth between planning, executing and evaluating e.g. Andrew Wiles cracking the infamous “Fermat’s Last Theorem”.

2)  Human problem solvers have emotions.  It is natural to feel bad when one stuck or when one is wrong.  However, the difference between good problem solvers and bad problem solvers is this:   Bad problem solvers allow their emotions to dominate them.
When wrong, bad problem solvers either do not realize that they are wrong, or they keep doing the same thing over and over again, getting the same ineffective results.  Does that sound like a lot of people?  Maybe, but do not take it personally.  Learn from the good problem solvers.  So what do good problem solvers do?  Well, they manage their emotions.  What they catch themselves on the wrong track or when stuck, they backtrack and/or try something else.  They do not give up, but keep trying until they succeed.

3)  Schematic diagrams are related to branches of mathematics called topology and graph theory.  These branches of mathematics can be used to analyse electrical circuits, internet/computer networks, social networks, traffic, scheduling, logistics … etc.

4)  The Circle Line is not really a complete loop, although it is a "large arc".

I hope you enjoyed reading this article, as much as I have enjoyed authoring it.  Cheers!

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