[AM_20150616LGDFCA] Logarithms and Knowing that You are Correct

Question

Introduction

     This is a relatively straightforward question once the student has learned the rules of logarithms.  When I was first learning logarithms it took me quite some time to get used to the idea of “logs”.  Are they fallen trees?  So what are “logs”?  They are just the exponents or indices.  For example  2³ = 2 ´ 2 ´ 2 = 8  and we can write  log₂8 = 3.  Logarithm to base  2  of  8  is 3, because  3  is the index  i.e. to get  8  you need to multiply  2  by itself  3-fold.

     In general,  log_ba = x   Û   a = b^x.  Why?  Because that is exactly what logarithm means!  One way to remember this definition is to imagine: if you transport the log to the other side of the equation, the log drops off and you get the base  b  propping up the  x.  You can also do it the other way round.  If the base  b  of a power moves to the other side, it becomes a  “log”  with base  b.  [active mnemonics]

     What about the “common logarithm” lg?  It is the logarithm with base  b = 10.  In the days before pocket calculators were prevalent, students used books and slide-rules with logarithms of base  10  for multiplying and dividing large numbers.  Base 10 logarithms are still commonly used in today for the Richter Scale (in seismology, to measure earthquakes), for decibels (to compare the loudness of sounds or gain / loss in amplifiers), for pH (measurement of acidity / alkalinity in Chemistry) .... etc.  The aforementioned rule works exactly the same way, with  b = 10.

                      lg a = x   Û   a = 10^x                 (lg means log₁₀)

Note that in many calculators, their “log” button is for  lg  or logarithm of base 10.

Solution

Checking Your Answer

     The person who posted this question on Facebook got  33 333 333.3  as his answer, but did not realise that his answer is the same as the “model” answer, which is given to three significant figures in standard scientific notation.  Many students have the habit of checking their answers against the “model” answer usually given at the back of the book or worksheet, which may sometimes be wrong!  Anyway, in tests and examinations, you do not have the luxury of checking your answers like this.  In real life, if an engineer makes a calculation mistake, buildings may collapse and people die.  It is better to make it a habit to check your answers on your own and to know and be sure that you are correct.  One way to do this is to substitute the value of  x  back into the original equation to see if it works.  Nowadays, many models of calculators have a “store” function indicated by a button labelled with “STO” or an arrow “®” or something like that.  You can store the value into a variable (or memory location) like  X  and then key in something like  “log(3X) ”  and see whether you get  9  or something close.  Be aware that calculators can have rounding errors. 

Notations for “log”

     School students are taught to use “lg” to mean “log₁₀”  and  “ln”  to mean the natural logarithm “log_e”  where the special number  e = 2.7182818284 ...  discovered by the visually impaired but brilliant mathematician Euler.  Many calculators take “log” to mean “lg”  or  “log₁₀”.  For adult working professionals, “log” (without indication of the base) usually depends on what field they are in, or on the topic being discussed.  As mentioned before, base 10 is used for Richter scale, decibels and pH.  Computer scientists tend to use base  2  because of the binary system.  For rate of reaction (chemistry) or radioactive decay (chemistry / physics), the natural logarithm “ln” is often used.  In school, for the purposes of learning, we make the logarithm bases explicit.  Do not simply write “log”.  Write “lg”, “ln” or “log₂” or “log₇” or “log_b” (for whatever  b  is).  Note also that the letter “l” in all these notations is not the letter “i” or “I”, but it is the smaller case “L” (for logarithms).

Heuristics Used

H05. Work backwards

H09. Restate the problem in another way

H13* Use Equation / write a Mathematical Sentence

Suitable Levels

* GCE ‘O’ Levels Additional Mathematics

* International Baccalaureate (IB) Mathematics (revision)

* other syllabuses that involve logarithms and exponentials

[OlymUSec_20150412BDP] Guessing Cheryl’s Birthday

Question

Introduction

     This “Primary 5 mathematics” (actually an upper secondary Olympiad) logic puzzle has gone viral.  It has been making its rounds in various forums in Singapore and overseas, stumping adults and children alike.  It is actually a parody of an old puzzle.  Can it even be solved?  It seems that there is no information given by each parties that we can exploit.  Actually there is!  In a subtle way ...

Solution

     In the beginning, everybody knows that Albert knows only the month and Bernard knows only the numerical day of the month.

     When Albert tells us “I don’t know when Cheryl’s birthday is, but I know that Bernard does not know too.” he is leaking out information (from his knowledge of the month) that the day of the month appears more than once and cannot be (June 18 or May 19).  Actually, the original phrasing is more like “If I don’t know when Cheryl’s birthday is, then Bernard does not know too.”.  The person who set this question merely changed the names of the people and the dates, without appreciating the subtle but crucial difference between a statement of fact and an implication (an “if ... then ... ” statement). 

     Ruling out June 18 and May 19, we also know that Albert knows that the birthday month is neither June nor May.  Otherwise, how would he have been so confident in saying that he knows Bernard would not know Cheryl’s exact birthday?  So we can eliminate those months.

     Bernard acknowledges the above state of affairs and the embedded hint.  With the choice narrowed down and with his knowledge of the numerical date, he now knows Cheryl’s birthday.  Since we know that Bernard knows Cheryl’s birthday, we know that it cannot be a numerical date that appears more than once (otherwise he would not have been able to know).  So we can cross out July 14 and August 14.

     Now Albert would telepathically thank Bernard for this helpful hint.  Because now he is able to deduce Cheryl’s birthday with his knowledge of the month.  That would mean that this cannot be a month with two candidate dates.  We blot out the August dates and see for ourselves the only remaining possibility.

Conclusion: Cheryl’s birthday is  July 16.

Remarks

     This puzzle was solved using the process of elimination and analysing our knowledge of what each party knows and can know.  Thus we successively narrow down the possibilities until the answer becomes obvious.  Here we learn that

     knowledge of other people’s knowledge can itself give us knowledge. 

This principle is actually employed in cryptology (the use of secret codes) which finds applications in fields like banking, the military (cf. interesting story of how the German Enigma code was broken in WorldWar II) and communications.  As an example, radio communication can tell the enemy of troop positions and warn of an impending attack, and that is why radio silence is imployed as a precaution.  Sensitive information in certain organisations is restricted on a “need to know” basis.

Suitable Levels
* Upper Secondary Olympiad
* other syllabuses that involve knowledge or epistemology
* application of mathematical principles in real life
* for all people interested in logic puzzles

[OlymPri20150412LGC] Your #logic a House of Cards?

Question

Introduction

     This is an olympiad type of question that tests one’s use of logic.  Sadly, logic is seldom taught in schools, except maybe in courses like knowledge inquiry, theory of knowledge, or in selected topics like geometry.  Mathematics is actually very much connected with logic, but it seems to be regarded as difficult to teach.  Being able to use logic and think critically is very important in our lives in various aspects.

     Let us study how we can analyse conditional statements using logic.

Analysing Conditional Statement with Logic

     A conditional statement is a statement of the form “If  H  then  C.”  An example would be: “If the blue litmus paper turns red, then the liquid is an acid”.  The front clause H (the part about the litmus paper turning red) is called the hypothesis (or premise).  The latter clause  C  (the part about the acid) is called the conclusion.  Every conditional statement has three other related forms: the contrapositive, the converse, and the inverse.

     For example, let say there is a NC16-rated movie and you can watch it only if you are at least 16 years old.  So  H := “watch NC16”  and  C := “16 years old and above”.  The conditional statement means “If you want to watch the movie, then you must be 16 years old and above”.  The contrapositive is “If you below 16 years old, then you cannot watch the movie”.  The contrapositive always has exactly the same meaning as the original conditional statement.  If one is true, so is the other.  If one is false, so is the other.  The contrapositive is merely phrased in a different way.

     The converse says “If you are 16 and above, then you will watch the movie.”  The inverse would say “If you don’t want to watch the movie, then you are not 16 and above.”  Note that the inverse is the contrapositive of the converse.  The inverse always has exactly the same meaning as the converse.  If one is true, so is the other.  If one is false, so is the other.

     The converse (and the inverse) has a different meaning from the original statement.  If the original statment is true, the converse may or may not be true.  In our example, “If you want to watch the movie, then you must be 16 years old and above” is true, but the converse “If you are 16 years old and above, then you want to watch the movie.”  may not be true, as some people who are 16 years old and above may want to do something else.  Like reading a book.  Or playing golf.  So you see, there are really two camps: the original conditional statement and its contrapositive in one camp, and the converse and the inverse in the other.

     Let us analyse the given problem using logic.

Analysis

     The statement  ‘any card with a letter A on one side always has the number “1” on the other side’ can be rephrased as ‘If letter A appears, then the other side is “1”’.  Let us analyse each position in order.  For (a), since the letter A appears, we definitely need to know what is behind the card to test whether the claim is true.  For (b), note that the claim does not say ‘If the letter is not A, then the other side is not “1”’ (inverse)  nor  ‘If the number is “1”, then the other side must be an A.’ (converse).  It is possible that A is on the other side and we are fine.  But it is also possible to have a non-A with the number  “1”.  The given condition does not prohibit that.  So, if the other side is not an A, it is still OK.  So the other side can be anything, and it does not matter.  For (c) we definitely have to flip to check the other side in case the other side is an A, then the claim would be proven false.  For (d), since the card is not an A, it is irrelevant.  The claim is talking about  A.  It is not talking about B.  We can summarise the analysis in the diagram below:-

Ans:  We need to flip (a)  and  (c)

Suitable Levels
* Primary School Olympiad

* syllabuses that involve logic and epistemology

[IB-HL_FTPR20150402] Periodicity cos a fortiori

     In this article, I give a rigorous proof of the period of cosine functions, using an a fortiori argument, signaled by the use of the words “in particular”.  When something is generally true for, say, all real numbers, it is “all the more” true for a particular case of a real number like zero.  I use this logic to nail down the answer.

Question

Solution

Suitable Levels

* revision for IB Mathematics HL / SL

* other syllabuses that teach trigonometry

[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.