[MathEd] Proposed New Framework for Mathematics Education

Figure 1 – My Proposed Framework

     In this article, I propose a framework for mathematics education that can be used in curriculum (re-)design, lesson design, evaluation of the attained curriculum, as well as analysing mathematics education or educational technology initiatives.  As a citizen of Singapore, I do hope that at least some of my ideas (in their original spirit) gets considered and adopted in my own country, but I want to share this with everybody in the world –  whoever cares to listen and engage.  I hope to spark a global conversation among students, parents, teachers, industry leaders and educational leaders regarding the future of mathematics education, according to needs and challenges that are currently felt in the 21^st Century world and as well as unforeseen needs.

     When we talk about mathematics curricula, we need to distinguish between the intended curriculum (what we think should be taught), implemented curriculum (what teachers actually teach) and the attained curriculum (what students actually end up learning).  These are different things.  My framework attempts to build upon the strengths of the present Singapore (intended) mathematics curriculum, and is influenced by my post-graduate studies of the academic literature in mathematics education, as well as

my observations and reflections of my personal experiences in my career as a teacher, tutor, instructional designer, educational software developer, researcher and consultant.  To explain my framework fully, it would probably take many pages and chapters.  Here, I shall give an introduction to my main ideas.

Crisis in Mathematics Education – “Iceberg” Metaphor

Figure 2 – What most schools (try to) focus on for maths

     Mathematics is an enterprise of gaining knowledge about the regularities and patterns

in our universe that has been practiced by people from different cultures in history throughout the world.  Most schools around the world try to teach only certain mathematical facts and procedures, which are only a very small part of what mathematics is really about (hence “the tip of the iceberg”).  And even with this, they are already struggling.  Politicians and curriculum developers tend to focus on tests and examinations that assess concepts, skills and processes (algorithms or methods of calculation).  Parents naturally want their children to “do well” in mathematics.  Many people think that the best and “objective” way to indicate this is via paper-and-pen examination and test grades.  For the sake of “accountability”, most teachers around the world seem to be pressured to take an exam-oriented approach to teaching mathematics (and much else).  What tends to get ignored are things like the development of students’ ability to solve real-world problems, the ability to learn on their own, the willingness to engage in life-long learning, the ability to “figure out things” on their own, a love and thirst for knowledge and sense-making of the world, the cultivation of values (e.g. appreciation of beauty, connection with other disciplines, precision, rigour) and dispositions (e.g. creativity, patience, meticulousness, succinctness, critical thinking, a questioning mind … etc).  The “mathematics” most students get at the end of their school career is probably some spotty recollection of a few concepts and a few tricks and this fades in their adult life.

Figure 3 – What most students achieve in maths

     Furthermore, educators around the world generally fail to connect with students’ identities: a sense of who they are as human beings in this world, their roles, goals, wishes, aspiration, ambitions, decisions and life-stories, and how mathematics is relevant to the development of these matters.  This is true even of the best students, what more of the rest of the students.  Students who are “good” in mathematics (in our current education systems) may not like mathematics or see its relevance to their lives.  They may just be able to grudgingly attain good “performances” in mathematical tasks.  The bulk of the students are disengaged.  They think “You know … maths is just one of those things we have to get through in order to get to the university course of my choice”.  The “worst” students hate mathematics and the system, and they become disruptive and anti-social – they literally “deconstruct” the system and yet they do not have anything constructive to show for in its place.

Figure 4 – What “mathematics” computers can already do now

     We now live in the Information Age where information technologies (e.g. graphing calculators, Computer Algebra Systems, Wolfram Alpha, … etc) are getting more and more advanced by the day and can now do many of the numerical and algebraic/symbolic calculations that schools are trying so hard to teach to human beings.  [2015 Update: Recently, mobile apps like PhotoMath appeared on the market.  These apps allow students to snap photos of mathematics homework problems and the apps will solve the maths problems for them, including the working. ]  Actually, we do not need human beings to do the procedures of algebra and calculus anymore – machines can do them much faster and with less hassle.  There is no need to have them sit for mathematics classes to be taught with boring lectures and colourful textbooks, occasionally spiced up by some “math apps”, and then fail to learn perfectly.  Dear reader, if you have not realised it by now, this spells
                                              D I S A S T E R. 
The human beings who graduate from our mathematics education (if ever they do) are redundant!  It is a false comfort that today we have technology that can even do mathematics homework for students.  In fact, this is the very reason that these students are irrelevant in the current and future job market.  Furthermore, they do not acquire a wholistic mathematical education for their adult living.

Figure 5 – What humans need but do not learn in most  schools

     My proposed new framework attempts to address these challenges by putting emphasis on the deeper things.  This is not just about economic survival, but it is about what is most important for us as human beings trying to make sense of this universe as we live in it.

The Context

     In my framework, the learning of mathematics takes place in the context of real-life (symbolised by the land and sky) and a community (represented by the ocean).

     The “Iceberg” points to real-life: that means students link mathematics to contextualised real-life applications and authentic problems.  Students see how mathematics is relevant to their own lives and how mathematics is being applied.  [This does not mean sacrificing generalization and abstraction, but being able to see how the processes of generalisation and abstraction, when done properly, can be transferred to other contexts or new contexts.  This also means connection with other disciplines or subjects.]

     The community refers to fellow learners (not necessarily form the same age or country) and teachers and experts.  Instead of competitive individual learning, collaboration and connection with the world beyond classroom walls, contribution to society is encouraged.

The First Five Layers

     The First Five Layers of my “Iceberg” framework cover the following:-

     (1)  Concepts

            § Numerical     § Algebraic       § Geometrical

            § Statistical      § Probabilistic   § Analytical

     (2)  Skills

            § Numerical calculation   § Algebraic manipulation   § Spatial visualization
            § Data analysis   § Measurement   § Use of mathematical tools   § Estimation

     (3)  Processes

            § Reasoning   § Communication and connections
            § Thinking skills and heuristics   § Application and modelling

     (4)  Metacognition
            § Monitoring of one’s own thinking   § Self-regulation of learning

     (5)  Attitudes

            § Beliefs   § Interest   § Appreciation   § Confidence   § Perseverance

The Deep Layer

     Below these five layers, we have the following:-

     (1) Problem Solving
           § Understanding   § Planning   § Executing   § Evaluating   § Reflecting

     (2) Dispositions

           § Habits of Mind   § Transfer of Learning

     (3) Values

           § Purpose of Learning   § Utility   § Aesthetics

     (4) Epistemology
          § Ways of knowing  § Logical reasoning  § Plausibility and number sense
          § Life-long Learning   § Self-Directed Learning   § Critical Thinking

     The above are very important, but these are just aspects surrounding identity

     (§ Character-building   § Roles   § Life-story   § Being and becoming)

What these all mean

     What these all means is in my conception of an ideal student who graduated under this mathematical education framework, this person is someone who has a strong sense of who he/she is in this world and what he/she wants to do with his/her life (identity).  As part of this core identity, he/she is able to solve problems, has desirable mathematical dispositions, values, is a life-long self-learner who knows how to figure things out on his/her own.  The mathematical concepts, skills (including the appropriate use of technology), thinking processes, metacognition and attitudes are built upon this core.  This person is able to collaborate with other people in real-life (including the ability to connect with other subject disciplines).

Connection with other disciplines/subjects

     I have alluded to connection with other disciplines/subjects.  What I have said regarding the crisis in mathematics education is probably largely true in disciplines/subjects.  One can imagine that other disciplines/subjects (e.g. biology, physics, literature, history, geography … etc) all have their similar “icebergs”.  Actually all these other icebergs are connected at the deep level, with “identity” as the common core.  Human knowledge has traditionally been dissected and put into silos for different disciplines for ease of handling, but in reality, all aspects of knowledge and learning are interconnected and there are no artificial boundaries.

Questions to Ponder

1.  Do you think my framework is practical?  Do you know of any places and/or
     schools already implementing all aspects of my framework (without necessarily
     putting them in the format that I have described)?  Which school?
     Which district / province / country?

2.  How would you redesign your province’s mathematics curriculum?

3.  If you are a current school teacher, would you want to redesign your next mathematics
     lesson after reading this article?

4.  Using my framework, how would you approach the evaluation of the attained
     curriculum (what students end up learning) in your country / school / district?

     Remember: students do not just learn facts (e.g. “1+3=4” ) and skills (e.g. factorisation)
     They also learn knowingly or unwittingly attitudes (e.g. that “mathematics is boring”,
     “it has nothing to do with my life”, “Oh!  It’s just a bunch of calculations”, “it has got
     nothing to do with logic”, “answers are what matters, not how you got it”, “just learn it
     from the teacher”, “don’t give me that cr** about reasoning, just give me the facts”
     etc.).

5.  Using my framework, how would you evaluate your country / school / district’s
     implementation of your curriculum?  Do you see any gaps in the way teachers actually
     teach your curriculum?  What are you going to do about it?

6.  Do you agree with everything I have said?  Do you have anything to add or take away?

7.  How does your school’s technology use fit into this framework?  How does it, for
     example, support students’ mathematical epistemologies (i.e. they way they learn, and
     the way they critically assess the knowledge that they have acquired via searching,
     experimentation, … etc)?

8.  Consider Apple’s latest initiative to put cheaper-than-paper-textbooks material on the
     market.  If you were to use a red-coloured pencil to shade the areas being covered in
     my framework, what areas would be shaded?

9.  Any other business …

Conclusion

     Actually, there is no conclusion.  We have only just begun.  With my introduction, I hope everybody has a clear idea of the issues we face today and what areas need to be addressed.  You may agree or disagree with me, or you may want to suggest some things.  Let the conversation begin.  Put your comments/feedback below or email me.

About this blog

     This is a blog discussing real **hard-core** Singapore school Mathematics by a true-blue Singaporean educator for struggling students and parents in Singapore and around the world, as well as for fellow educators.

     Note that this is not "Singapore Math". Firstly, mathematics is universal -- it just tastes slightly different in different countries. Secondly, the power of the Singapore Mathematics Curriculum did not originate from Singapore, but rather the curriculum planners (chief among whom was Dr Kho Tek Hong, now retired) integrated the best ideas of Bruner, Dienes, Piaget, Polya, Vygotsky and Skemp into a curriculum and the schools (more or less) implemented it.  Thirdly, in Singapore, we just call it "Maths" (with an 's' at the end).

     So actually, thre is no such thing as Singapore Mathematics as such.  The so called "Singapore Mathematics" is just mathematics as done in Singapore, including the syllabii in schools, homes, tuition/enrichment centres and wherever.  I hope to expose the secrets of the maths whizzes (e.g. slick tricks, clever short-cuts, heuristics and metacognition), and show how these are applied to solve difficult mathematics questions.  I also hope to discuss issues regarding mathematics education, and education in general.

WARNING: This blog may seriously improve your mathematics and/or challenge your cherished ideas regarding education.

AJC 2009/I/14(a)(ii) Application of Integration: Area

figure 0 – problem statement

Introduction

     In this article, we look at question 14 part (a)(ii) taken from the preliminary examination paper of Anderson Junior College H2 Mathematics Paper 1 in year 2009.  In Singapore schools, you either get challenging questions or very challenging questions.  This part of the question is challenging, and yet is worth only worth 3 marks’ credit.  We can apply the same processes of metacognition (self-monitoring, self-awareness, self-questioning) and heuristics (guidelines, rules-of-thumb, tactics) to solve this problem.  These processes are generally applicable in problem-solving, and more important than the mathematical content itself (which you probably will forget anyway after you graduate from school).  One should learn mathematics not just for the sake of clearing examinations, but to get educated.

     “Education is what remains after one has forgotten what one has learned in
     school. ”                                                                              – Albert Einstein

Stage 1 – Understanding the problem

What topic is this under?

     Integration Applications (Area)

What are you trying to find?

     The area of the shaded region.

Stage 2 – Planning

What methods can you use?  Which is easier?

     We can integrate by cutting the area vertically or horizontally.  [Imagine cutting the area into very thin rectangular strips.]  Horizontal slicing looks easier.

What is the correct formula for that?

     Observing that the area is the area between two curves/lines, the formula is

Which means … ?

     Obviously, y_upper = ⁸/₃  and  y_lower = ¹/₆.  These integration limits correspond with the variable of integration ‘y’.  If the integration is ‘dy’ (with respect to y), then the upper and lower values must be  y  values.  If the integration is ‘dx’ (with respect to x), then the definite integral’s limits must be  x  values.

     x_right is the equation of the (straight) oblique line on the right boundary of the region, with  x  expressed in terms of  y.

     x_left is the equation of the elliptical curve on the left boundary of the region, with  x  expressed in terms of  y.

What heuristics can you use?

     1.  I can split this task into smaller sub-tasks.

     2.  Try to use the result in the previous part, part(a)(i).

What do you need to do?

     I need to find the x_right and x_left and then do the calculation.

Stage 3 – Execution

     Let us find the  x-formulas for the right and left lines.

figure 1 – determining the formulas for right and left parts

Remark: Most of this is straightforward for a JC student.  You are expected to be very familiar with secondary school algebra by now.  Regarding the last two lines: there are two choices for the equation of the curve.  Since we are using the left side of the ellipse (x < 1), we choose the one with the ‘–’ square root instead of the one with the ‘+’.  This is a common trick that schools like to catch students with.  Make sure you don’t stumble on this point.

     With these formulas, we can now work out our solution.

     Applying the formula (line #2) we have line #3 and taking out the brackets leads to line #4.  We do a bit of algebra, and then split the integration into two parts (#line 5).  For lines #6 and #7, the left integral is a straight-forward calculation and the right integral is the answer from part (a)(i).  In line #8, we consolidate our answer by pulling out 25 as common factor.

Stage 4 – Evaluation

Is your answer correct?

     The upper and lower limits match the variable of integration and they make sense.

     Although an exact answer (i.e. non-decimal) is required, we can use the Graphing Calculator to check the calculation numerically.  Here is one way (there are other ways too) to do it:-

We are correct.  The slight differences in the 7^th decimal place is because the Graphing Calculator itself uses an approximation to numerically calculate this definite integral.

Stage 5 – Reflection

What lessons did you learn by solving this question?

     ·  decide whether do to the area by slicing “horizontally” or by slicing “vertically”.  Whichever is easier.

     ·  if doing the integral by slicing “horizontally” always take the right curve
         minus the left one.

     ·  split the task into smaller sub-tasks:-

         1.  determine the upper and lower limits of the definite integral

         2.  if you integrate by slicing “horizontally”, determine the equations of the right
              and left curves, with  x  as the subject.

     ·  for the equation of the elliptical curve with  x  as the subject: choose ‘–Ö’ for the left

         half and ‘+Ö’ for the right half.  [But usually Singapore schools like to set ‘–Ö’ to
         catch unwary students off-guard, and that becomes so predictable: If you do not
         know which to choose, just choose the ‘–’ and you’d probably be right!  LOL! J ]

What if we took the left minus the right?

What will happen if we did the integration by slicing vertically (i.e. with respect to x)?

     Type your comments below.

JCCDQBHWHSS076 Telescoping Sum and Inequalities

[original source unknown]

Introduction

     In keeping the spirit of discussing genuinely “hard core” Singapore school mathematics (not “Singapore Math”, the Americanised parody) in this blog, I discuss a particularly “pernicious” problem on summation, taken from a book whose authors did not bother to credit the questions’ source.  Aside: Do you wonder why these guys never get caught for copyright infringement, and why they can get away with selling these books blatantly at a popular book chain?  My theory is that the police officers or lawyers themselves have children who are also struggling with maths … and they “need” these examination-paper compilations … so … .

     Anyway, do not be overwhelmed when you encounter a question like this that seems out of your reach.  Good problem solving involves not just regurgitating formulas and performing set procedures, but knowing how to react when one encounters unfamiliar problems.  There are also some examination-paper compilations (illegally) sold at road-side stalls at various places in Singapore.  They often provide full solutions copied straight from teachers’ marking schemes.  However, even if you have the full solutions, they may not explain how these solutions were obtained.  In this article (as in others), I will reveal how we can tackle hard questions like this using metacognition and heuristics. 

First Part of the Question

Stage 1 – Understanding

What are you required to do?

     I am required to show that the first expression is equal to the second one.

Stage 2 – Planning

What can you do?

     This looks like a partial fractions problem.  However, it looks pretty nasty.  We have two quadratic denominators …  Are they factorisable?  No, at least not with “nice numbers” (integer coefficients) … which means we might need something like  An+B  over  n² – 3n + 1 and then  Cn+D  over  n² + n – 1.  Then we solve for four unknowns  A, B, C, D.  Yulk!  This does not look like fun.

Is there another way or a better way? 

     Hmmmm … *thinking hard* … We do not really need to solve for A, B, C and D.  Actually we can think of this “show/ prove” question as something in which the answer is already given (viz. the second expression).  We need to show that the first expression is equal to this.  But we can do it by doing it the other way round.  If we can show that the second expression is equal to the first expression, then of course the first expression is equal to the second expression.  Bingo!

Stage 3 – Execution

Figure 1 – 1^st part of the question

Comment: Most of this is just secondary school algebra, which JC students are expected to be adept at already.  Mentioning the “Symmetric Law of Equality” (not in any Singapore school syllabus, but it is just “common sense” made to look more official) is meant to impress teachers and convince the die-hard skeptics that this “unorthodox” method is indeed a valid method.  But then, most likely, the “unorthodox” teacher who set this “unorthodox” exam question probably expected you to do it by this “unorthodox” method anyway.

Stage 4 – Evaluation

Have you done it correctly? 

     Yeah!  Got it, as required!

Second Part of the Question

Stage 1 – Understanding

What are you required to do?

     To find (i.e. evaluate) the expression given in sigma (S) notation.

What will the answer look like?  Will it be a number?

     No.  It will be an expression ...  In terms of?  … capital ‘N’.  What about the small ‘n’?  This is just the summation index, which is a dummy variable i.e. it is a temporary “use-and-then-throw-away” variable for the sigma notation, but it will not appear in the final answer.

What concept is this part of the question testing you on?  How do you know?

     This part of the question is testing me on “the Method of Differences” technique (also known as “the Telescoping Sum” technique).  I know this because it is a favorite technique of the teachers and ‘A’ level examiners, as a huge variety of questions can be set based on this technique.  Actually the major clue is in the first part of the question, where a difference between two expressions is involved.  The minus ‘–’ sign in the second expression is the dead giveaway, the “smoking gun”.

Stage 2 – Planning

What are you going to do?

     Once I have diagnosed this problem as a “method of differences” problem, it is just a matter of following the SOP (Standard Operating Procedure):  Expand the sigma notation by writing out explicitly the first few terms and the last few terms.  Then look for a cancellation pattern.  After cancelling, there will be some terms from the front bit and some terms from the end bit remaining.

Stage 3 – Execution

     Some rough working seems necessary:-

When  n = 3:   n² – 3n + 1  = … =  1,   n² + n – 1   = … = 11

When  n = 4:   n² – 3n + 1  = … =  5,   n² + n – 1   = … = 19

When  n = 5:   n² – 3n + 1  = … = 11,   n² + n – 1   = … = 29

When  n = 6:   n² – 3n + 1  = … = 19,   n² + n – 1   = … = 41

When  n = N – 1: 

   n² – 3n + 1  =  (N – 1)² – 3(N – 1) + 1  =  N ² – 5N + 5

   n² + n – 1   =  (N – 1)² + (N – 1) – 1   =  N ² – N – 1

When  n = N: 

   n² – 3n + 1  =  N ² – 3N + 1

   n² + n – 1   =  N ² + N – 1

Figure 2 – Telescoping Sum Method (a.k.a. Method of Differences) 

From the given expression (line #1), we replace the summand by the difference expression (line #2) found in the earlier part of the question.  We expand the sigma notation by writing out the first four differences  (by substituting n = 3, 4, 5, 6)

and the last two differences  (by substituting n = N–1, N).  From experience, I know that I can see the pattern more clearly if I write the terms neatly, devoting one row per value of n I substitute.  Indeed once I do that, the cancellation pattern becomes obvious.  In the last two lines, I collect the remaining (uncancelled) terms and simplify the resulting expression.

Stage 4 – Evaluation / Checking

Are you correct?  How do you check?

     Yes.  I can check by substituting, say, (capital letter) N = 3, 4, 5 and seeing if the expressions agree.

Third (Final) Part of the Question

Stage 1 – Understanding

What are you required to do?

     To show that the given sum to infinity is less than 1.

What type of question is this?

     This is a “show / prove” question involving inequalities and sum to infinity.

Stage 2 – Planning

Do you notice anything?  How is it connected to the earlier part(s) of the question?

     This looks hard.  The connection (if any) is not obvious.

What are you going to do about it?

     There is a pattern.  I’ll solve part of the problem (this is a heuristic) by considering a finite sum first.  Later on, I can let  N®¥  to get the sum to infinity.  I rewrite it in sigma notation to try to see if there is any connection with the previous part.  I need to slowly manipulate this (“massaging the expression”) to make it look like the expression in previous part.  But … hmmm … this looks quite different from the earlier sigma expression …

Figure 3 – Using finite sum and sigma notation

What are the differences?  Can you point them out?

Figure 4 – Doing a comparison (“Spot the differences”)

     (D1) instead of starting from  n = 3, this sum starts from  n = 1

     (D2) there is no ‘2’ in the numerator, unlike the previous summation

     (D3) there is no ‘(2n –1)’ in the numerator, unlike the previous summation

     (D4) in the denominator, the smaller quadratic is  n²  instead of  n² – 3n + 1

     (D5) in the denominator, the larger quadratic is  (n + 1) ²  instead of  n² + n – 1

Hmmmm … it looks like the person who set this question has set up a minefield.  If I get it wrong in any one of the above, the question will blow me off.

Don’t panic.  What heuristic can you use?

     I can split this big problem into smaller problems.  I can handle it a step at a time.

So how can you handle (D1)?

     I can write out the first two terms explicitly and start the summation from  n = 3.

How do you that?  Just substitute n = 1  and then  n = 2  into the summand’s formula to get the first two terms.  For the rest of the terms, I write it in a similar sigma form, but starting from n = 3.

Stage 3 – Execution

Figure 5 – dealing with the big problem in smaller steps

Back to Stage 2 – Planning

Good.  Now, how do you deal with (D2)?

     I can forcefully introduce a ‘2’ in the numerator, and compensate that by putting a factor of ½, which can be written outside the summation.  Further, I can also evaluate ¼ + ¹/₃₆, which is  ¹⁰/₃₆.

Stage 3 – Execution

Figure 6 – dealing with the 2^nd sub-problem

Back to Stage 2 – Planning

Now, how to deal with (D3)?

     I can forcefully introduce a ‘(2n –1)’ in the numerator …

But wouldn’t that be different from the previous equation?

     Yes.  In fact, the new expression would be bigger.  Why?

For  n = 3, 4, 5, …,  each of the  2n –1  will be at least 5 … definitely more than 1.  Multiplying with the positive summands, each term in the summation will be bigger than before.  Hence the resulting summation will be bigger than the previous line’s summation.  That means I need to replace the ‘=’ sign with the ‘<’ sign.

Stage 3 – Execution

Figure 7 – dealing with the 3^rd sub-problem

Back to Stage 2 – Planning

How to deal with (D4) and (D5)?

     Now this is a tough cookie … hmmm …

Can you compare the pairs of denominators?  For (D4), which is bigger one?

     Comparing  n²  with  n² – 3n + 1,  it looks like the latter is smaller because there is a minus  3n.  So  n²  is larger.  By how much?  If  n² – ¿¿¿ = n² – 3n + 1,

what is the ‘¿¿¿’?  By inspection (i.e. fiddling with the algebra) we observe that

     n² – (3n – 1) = n² – 3n + 1

So the  ‘¿¿¿’ is  3n – 1.  Is this positive?  Yes, for n = 3, 4, 5, …, this is at least 8.  Definitely positive.  Which means  n²  is indeed greater than  n² – 3n + 1, and it is bigger by  3n – 1.

What about (D5)?

     (n + 1) ²  is the same as  n² + 2n + 1.  Hence

     n² + n – 1 = (n + 1) ² – n – 2 = (n + 1) ² – (n + 2)

That means  (n + 1) ²  is more than  n² + n – 1, and it is bigger by  (n + 2).

So, are the target denominators  n² – 3n + 1  and  n² + n – 1   bigger or smaller than what we have currently?  These denominators are smaller.

Will the resulting expression be bigger or smaller?

     By the “Monk-Porridge Theorem”, since we are dividing positive quantities by smaller divisors, we will end up with a larger quantity.  So we link to the resulting expression with a ‘<’. 

Stage 3 – Execution

Figure 8 – dealing with the last two sub-problems

Stage 4 – Evaluation

Does this make sense?

     Yes.  This inequality sign ‘<’ is in the same direction as the previous one.

We are using the Law of Transitivity: if  a < b  and  b < c  (conventionally we write this as  a < b < c), then  a < c.  If the inequality signs are in different directions (say,  a < b  and  b > c), then we are in trouble, because we cannot conclude that  a < c.  Here, the inequality signs point the same way, so we are good.  We can go back to stage 3 to complete the rest of the calculations.

Back to stage 3 – Execution

Figure 9 – completing the question

     In line #1, we simplify the previous expression.  Then using the earlier result obtained from the second part of the question, we replace the sigma expression with its equivalent, shown in line #2 between the curly braces.  We have two expressions that are reciprocals of quadratics in ‘N’.  Obviously, these become smaller and smaller as  N becomes larger and larger.  In other words, these terms tend to zero as  N  tends to infinity (line #3).  Hence the infinite sum will tend to something less than ⁷⁹/₉₀ (we can work this out from the line #2 expression).  This is definitely less than 1, which is what we are supposed to demonstrate.

Final Presentation (for the last part of the question)

Figure 10 – putting it all together

Step 5 – Reflection

What did you learn by doing this question?

     I learned that, once again, metacognition and heuristics are useful for the 5-stage problem solving process.  For a complicated problem, one does not have to proceed with the five stages in a strictly linear fashion.  Instead of trying to regurgitate a fixed technique where there is none, I can go back and forth (especially between stages 2 and 4), thinking, doing and re-thinking along the way.  This is the usual way expert mathematics solvers actually solve their mathematics problems, not that they have a method that automatically proceeds from start to finish.

     From the first part of this question, I learned to look out for shortcuts.  We can use the Symmetric Law of Equality (A = B Þ B = A) instead of doing things by the usual way (partial fractions).

     From the second part of this question, I learned to be aware of questions that test the “Method of Differences” (or the “Telescoping Sum” technique).  In this question, the tell-tale giveaway clue is the ‘–’ minus sign.  Once you know it is the “Method of Differences”, the execution of this technique is rather standard.  Write out the terms by substituting the first few values of the summation index (n in this case) and the last few values.  Use one row per value of  n, so that the cancellation pattern can be seen more clearly.  After all the gory cancellations, a few of the first terms and a few of the last terms remain.

     For the last part of the question, I learned to keep calm in the face of difficulties.  I learned to simplify the question and rephrase the question (e.g. by rewriting it into sigma notation).  I look for patterns.  I learned that making comparisons (“what I have” vs “what I want”) is a powerful heuristic that can suggest what steps to take next.  I also learned that a complicated question can be tackled by breaking it down into smaller, more manageable sub-problems.  Then I deal with these sub-problems systematically.

What about you, the reader?  What did you learn from this problem?

Remarks

     The term “Telescoping Sum” comes from the observation that when applying the method of differences, you begin with a long expression like a telescope that is stretched out.  After cancellation of the terms in the middle, the expression is being shorted – just like compressing a telescope.

Metacognition in Real Life – an Example with MRT

Figure 0.  Problem statement

Introduction

     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

Quo Vadis?  Do you know where you’re going to?

     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

Where are you now?

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

Where are you now?

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?

     Khatib.

Where are you now?

     Yishun.

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.

Comments

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!