[AM_20161105ITFF] False Friends in Integration (Calculus)

Question

Introduction

          False friends are words in two languages that look/sound alike, but differ significantly in meaning.  Do you know that there are also false friends in mathematics?  Can you distinguish and explain the difference between the two integrals?

Solution

          The integrand on the left has the variable  x  as the base and the constant  e  as the index.  So we integrate it using the Power Law.

          By contrast, for the integrand on the right, the base  e  is a constant whereas the index is the variable  x.  Integrating  e  to the power of  x  is the eeeeeeeeeeeeeeeasiest.  You just get back the same thing, plus the arbitrary constant of course.

Remark

          Many students make the mistake of trying to apply the Power Law for the exponential.  As a learner of mathematics, one needs to cultivate the habit of being observant and paying attention to detail.  This is part of developing one’s identity and character which is important in life.

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* GCE ‘A’ Levels (revision)

* revision for IB Mathematics HL & SL (revision)

* Advanced Placement (AP) Calculus AB & BC

* University / College Calculus

* other syllabuses that involve integral calculus

* whoever is interested

[EM_20160204PBIE] “Hillarious” Mathematics of Politics?

Problem

Six coins are tossed to decide a result for either “C” or “S”.  Assuming that the coins are fair, and that the results of the tosses are independent, calculate the probability that all the tosses are in favour of “C”.

Introduction

     Hot in recent news is the story of the purported six coin tosses that were needed to determine certain county delegates in the race between Hillary Clinton and Bernie Sandersin the state of Iowa.  All six coin tosses were in favour of Hillary Clinton, and the result is so improbable that some people said it was “hillarious”.
     How is the probability calculated?  This is an example of mathematics in real life events that has the potential to affect the United States of America, and the whole world (including Singapore).

Solution

Discussion

     In order to make the calculation, we make two assumptions: 

          (1) that the coins were fair, and

          (2) that the coin toss results are independent.
     So, what does it mean that the coins are “fair”?  It means that the probability of getting a “heads” is the same as the probability of getting a “tails”, which means ½ for each.
     Coin toss results can be “heads” or “tails”.  These are examples of events.  An event is something that can happen or not happen, and we associate a probability with it.  The probability is a number that indicates how likely the event happens.  It is between  0  and  1  inclusive.  Zero probability means a practically impossible event.  A probability of  1  means a practically certain event.  [The reason for me using the word “practically” is technical, which I shall not discuss.]  If the events do not affect one another (i.e. in our case, the coin tosses are not affected by the other coin tosses) then the events are said to be independent.  If the events are independent, then we can simply multiply the individual probabilites together, as above.  If the events are dependent, the calculation would be more complicated.

     So are the coin toss results valid?  I do not know.  All I can say is: improbable does not mean impossible.  Mathematics cannot tell whether the above assumptions (1) and (2) are correct.  But at least I “lay all the cards on the table”, so that astute students of probability know the basis of these calculations.  It is up to you to decide, but at least you would have made a mathematically-informed decision.  There could be other twists to the story, which is beyond the scope of this article.  This is one of the reasons why you need to learn mathematics carefully and think critically, whether or not you would become a mathematician,engineer, teacher or have a mathematics-intensive career.

Heuristics Used

H08. Make suppositions

H10. Simplify the problem

H11. Solve part of the problem

Suitable Levels

* GCE ‘O’ Level “Elementary” Mathematics (Number patterns, with algebra)

* other syllabuses that involve probability

* anybody in the whole wide world!

[Maths_Education] Is Learning Mathematics a "Performance"?

Article
The Math-Class Paradox


My Comments
     This is an excellent article addressing the issue of mathematics education in schools today.  School "mathematics" barely touches even the tip of my iceberg.  Is the learning of mathematics merely a matter of performing under time pressure on tests and exams (and that puts many students off)?  What about exploration, creativity, asking questions and discovering connections ... etc?
     The thought provoking article is written by Jo Boaler, one of my favorite professors of mathematics education.  It is definitely worth your while reading it!

[MathEd] Critique: Problematising Mathematics Education

In Response to Article
The Politics of Math Education


My Comments

1. It is good to problematise mathematics education.  Certainly politics is involved in the choice of mathematics curricula.

2. However, is it good to argue and debate so much that nothing gets done?  Are you chasing down false dichotomies?  Are you assuming that you cannot have it all?

3. In Singapore, we do not argue so much and students go on to perform well in "mathematics".  Unfortunely, I feel, they get a very narrow and distorted view of what mathematics really is.

4. What is the answer the dilemma?  I think it's a question of identity.  I think students should make well-informed negotiated decision about the kinds of people they want to be and how a wholistic mathematics education serves to develop them not just in terms of skills and content, but also in terms of values, habits / dispositions, problem solving ability and critical thinking, ... etc.  I would be interested to learn of and even work with curriculum planners heading in this direction.

[AM_20151117] An “Unorthodox” Technique for Trigonometric Proof

Question

Introduction

     When proving trigonometrical identities, one usually starts from the “more difficult” side and try to work towards the other side.  The above identity looks like a tough nut to crack.  Both sides look equally complicated.  Where do we even begin?

     Here is one way to “cheat”.  Starting from, say, the LHS, we multiply the RHS expression and also multiply by its reciprocal i.e. dividing by the same amount.  It is like using a magic wand to create something out of nothing (无中生有 in Chinese), but doing so does not change the value of the LHS expression.  Now we do not touch the part that is equal to the RHS (highlighted in yellow below), but we try to find a way to cancel away the other stuff, as shown here.

Solution

Remarks

     In the third step, I had replaced  cos²A  with  1 – sin²A  and  sin²B  with  1 – cos²B.  This leads to the required cancellation and we are left with the yellow patch, which was never touched since the first step and it is the RHS.  This completes the proof.

     Just as in martial arts where deadly opponents require deadly strokes to counter them, evil questions require “unorthodox” techniques.  Even though most people would not have thought of it, all the steps presented above are actually legitimate.  This is because at every step, the equality is preserved.  All the “=” are really equal, and it’s legit (either work hard or you might as well quit), although my “can’t touch this” tactic seems a bit clairvoyant.

. 

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H08. Make suppositions

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘O’ Level Additional Mathematics

* other syllabuses that involve proof of trigonometric identities

* anyone who loves mathematical challenges

[U_Complex20151117] Lagrange Identity for Complex Numbers

Question

Introduction

     I was revising my complex analysis just for fun (I had graduated almost 3 decades ago) and I came across this problem which is question 5 from page 9 of the book Complex Analysis by Ahlfors.  This book is pretty hard core for a first course in complex analysis, but this is not surprising since Lars Ahlfors was no less than a Field’s medallist.

     Intuitively, I knew that this identity looks like the vector identity

                                        |a · b|² = |a|² |b|² – |a ´ b|²

and if one strips away the |a|² |b|², it boils down to the Pythagorean Trigonometric Identity

                                         cos² q = 1 – sin² q

I thought if I could just define the appropriate dot and cross products (actually this can be done), I could solve it easily.  However, this itself requires proof.  I was begging the question.  In fact, it is precisely because of this Lagrange Identity (and the related Cauchy-Schwarz Inequality) that allows  cos q  and   sin q  to be meaningfully defined.

     OK, it looks like I have to do it the hard way.  The tricky part in the manipulation of those sums in sigma notation is to ensure, at each step, that I did not introduce any spurious terms, nor miss out any terms.  To simplify the notation, in what follows I shall assume that  i  and  j  are indices in the range of whole numbers  [n] = {1, ... , n}.

Solution

Remarks

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in another way

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* University / College level Complex Analysis

* other syllabuses that involve Complex Analysis

* any independent learner game for a challenge

[S2_20151027ACS] Completing the Square

Question

Introduction

     Many students in Singapore are taught to the method of “completing the square” in a rote fashion.  For example, to handle a quadratic expression like  x² – 6x + 8,  take half of the coefficient of the  x term, put it with  x, square that resulting binomial, and then also subtract the square of that same number, like this: 

                                        x² – 6x + 8 = (x – 3)² – (3)² + 8 = (x – 3)² –1

Note that the sign inside the squared binomial always follows the sign of the  x  term.  Just follow the procedure!  It works!

     Does the student understand why this works?  And, by the way, does this always work?  What if the coefficient of  x²  is not  1?  Well, we need to pull out that coefficient.  Many teachers teach pulling out that coefficient from all three terms  e.g. 

     3x² + 12x + 5 = 3[x² + 4x + ⁵/₃] = 3[(x + 2)² – (2)² + ⁵/₃] = 3[(x + 2)² – ⁷/₃]  = 3(x + 2)² – 7

A better way to do this is to just pull it out from the first two terms: 

     3x² + 12x + 5 = 3[x² + 4x] + 5  = 3[(x + 2)² – (2)²] + 5  = 3(x + 2)² – 7

Now it seems that as the questions get harder and harder, students need to memorise more and more procedures by rote.  And what if they encounter a question such as the one featured above?

What now?

     Let us go back to basics.  The square-of-sum identity is                                        

and the square-of-difference identity is                                       

Instead of memorising procedures (not that these are wrong in themselves), why not use the above identities and think backwards (which is a type of heuristic)?  You can even make it a game of “filling in the blanks”, as shown below.

Solution

     According to the identities, the two green patches must be the same and likewise the orange patches must be equal.  Working from the right end, we figure out that the orange patch must be  2  since  2² = 4  or  Ö4 = 2 .  Once we know this, we try to figure out the green space by matching the middle  y  terms:  2( ? y)(2)  =  32y.  So the unknown number (?)  must be  32 ¸ 4 = 8.  So the green spaces must be filled with  8y.  Since  (8y)² = 64y², we deduce that  k = 64.  Bingo!

Learning Points

     · You definitely need to know formulas and procedures, but ...

     · memorising procedures is not the holy grail of mathematics.

     · There is no holy grail of mathematics, but ...

     · heuristics (e.g. thinking backwards, pattern matching) are powerful problem solving tactics.

Heuristics Used

H04. Look for pattern(s)

H05. Work backwards

H09. Restate the problem in another way

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

· Lower Secondary Mathematics

· other syllabuses that involve algebraic identities and completing the square

· anyone who loves a challenge!

[S2_20150501AXC] 2011: Chunking and Substitution with Algebra

Question

Introduction

     This lower secondary algebra question seems complicated, doesn’t it?  Can you spot any chunk that is repeated, or almost the same?  This is one of the keys to solving the problem.  Another key that you need is the relevant algebraic identities and tricks.

Reminders

     First, let us review some of these useful formulas.                                        

This is the square-of-difference identity, illustrated here.

This swapping trick is illustrated here. 

     Looking back at the question, do you notice anything that is repeated?  Can you see any chunks that are the same or almost the same?  (n – 2011)  is almost the same as  (2012 – n)  isn’t it?  Whenever you see a repeated chunk, it is a good idea to substitute that chunk with another variable that you invent.  To name this new variable, you can use any letter that is not used before, so as not to conflict with existing letter(s).

Solution

Summary

     To solve the given problem, we have used the following:-

     (1)  square-of-difference identity

     (2)  swapping technique

     (3)  observation of repeated chunks

     (4)  using substitution with the chunks

Heuristics Used

H04. Look for pattern(s)  e.g. chunking, observation

H09. Restate the problem in another way  e.g. swapping, identities

H10. Simplify the problem e.g. substitution for chunks

H11. Solve part of the problem

H12* Think of a related problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

· Lower Secondary Mathematics

· other syllabuses that involve whole numbers and ratios

[Pri20150302PBW] Pebbles on a Road Backwards

Question

A jar contained some pebbles. Nigel took out half of them plus 2 more for display. Jenny took out half of the remaining pebbles plus 1 more. Finally, Alice took out half of what remaining plus 5 more for her project. In the end, there were only 8 pebbles left. How many pebbles were in the jar at first?

Solution

We can trace what happens to the remaining pebbles at each stage, and represent the information given in a sequence as above.  Note that when “Nigel took out half of them plus 2 more for display”, we would have half and then two less the number of pebbles.  Similary for the Jenny’s and Alice’s stages.

                            

Now we just do the opposite of all the operations.  The inverse operation of ‘–5’ is ‘+5’, the inverse operation of ‘´ ½’ is ‘´ 2’ and so on.  Working our way backwards, we arrive at 112.

Ans: There were 112 pebbles at first. 

Commentary

     Remember: In mathematics, there is no such thing as "the only way" to solve problems.  There are many ways to skin the cat, as they say.  Other methods to solve this question include: using algebra, and “the branching method” (taught by some tutors/teachers) which tracks both the remaining pebbles as well as the ones taken away.

     If you do use algebra (of some sort), remember to use different letters e.g. v, w, ... etc to refer to different things.  Likewise, do not just write  ‘½’  to represent ½  a unit, otherwise it could be misinterpreted as literally the number ½.  I suggest you can surround the ‘½’ with different shapes (e.g. circle, square, triangle) .

Precision is very important in maths, as well as in life. You don't want an imprecise person to be your aircraft designer, accountant or doctor, do you?

Heuristics Used

H05. Work backwards

Suitable Levels

* Primary School Mathematics

* other syllabuses that involve whole numbers and fractions

* any learner who is up for a challenge

[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.