Isn’t it time we replaced  the word  “Mathematics” with another word?  

Mathematicians who created the earliest curricula for dispensing it among young minds, stuffed it with exclusive and inclusive definitions fortified by formal language. The language often sounds like Latinised legal jargon: incomprehensible,  intent on mystifying things instead of clarifying them. Over the centuries “Mathematics” has increasingly taken on a  forbidding image, like a fortress in a spooky forest, standing ominously against a dark and stormy sky. The knowledge that lurks within it feels inaccessible, a preserve of  those who wear cloaks and mantles, and speak in riddles. It evokes fear of a sort associated with authority and power, which in turn  is somehow  associated with deep, penetrating minds. Not kind, gentle and generous ones but the unfriendly kind, the kinds that speak in stern voices. 

Mathematics’  earliest association goes back to  those who performed curious calculations to predict  impending doom, warn of catastrophes, wars and plagues, and help the powerful ward off evil with the magical power of  certain numbers and patterns. So the ancient priests and Mathematicians injected into Mathematics,  for generations to follow, a fearful dread among ordinary folks who have always been in awe of   their other-worldly omnipotence. 

That power, centuries old, can still be felt  in the haloed halls of Mathematicans’ culture. It has been transmitted  to the school curriculum centuries ago and now contaminates it with its dark academic spirit  of formalism and linguistic mystifications of the simplest Mathematical notions. Its adult, male-dominated precepts and perceptions define the context from which it has drawn its metaphors, terms, examples…..and even teachers. 

The opaque, disconnected  world of the past, anchored in the permanence and certainty of its conjectures and truths, has changed considerably since. Today, in startling contrast, the world is  an increasingly connected one, made more complex and uncertain by a continuous flood of changing information. Soon it will be filled with gadgets and gizmos that will speak to each other within an “Internet of Things”. Already, today’s technology can let you  open your front door from 20 feet away with a breezy wave of your hand. Three seconds after you enter your house, the water starts to boil to your desired temperature, a robot takes off your shoes and a foot-rest rises from the floor before your chair. The armrest has an embedded device that can be removed  to scan messages sent and received,  check out home security, select your favourite TV show, check if your coffee is ready, make a call, see where you left your glasses, obtain a brief health report, read your favourite book, or catch up on all international developments that have a direct bearing on your personal or professional life. In short it can  keep you informed of every aspect of your life over which you exercise your personal executive power.  

All the how-to for the above is already within reach. It conjures up a fragmented, divided world at one level, but a seriously connected one at another.  A complex, sometimes paradoxical,  mesh (or mess?) of choices and decisions confront critical  decision-making on travel options, work, jobs, relationships, enterprise, international economy, and family issues. Technology does not just connect people. It also connects their problems, their rights and wrongs. 

What does the word “Mathematics”, with its pre-medieval connotations and its medieval associations  have to do with this world? The word “Mathematics” in education ought to be re-contextualized as a complex body of knowledge for transmission to our children throughout their school lives. Such a task should be trusted to  teachers and educators who can give it a fresh new start. Perhaps that fresh start should begin with a new name.  

Like Neuristics, for example.  It fuses into a single word :  (a) number patterns (b)  numerical reasoning   (c) heuristics (or stratagems for developing  problem-solving approaches) and (d)  developmental neurology that indicates substrates of Mathematical learning cues in the evolving human brain. 

Neuristics sounds a lot more contemporary, derived from the body and spirit of today’s (and tomorrow’s) world. It is  sharp, brief, neutral sounding, and embedded in our reality like micro-chips  in our environment. It is evocative of something new and exciting e.g. the study of a discipline that teaches you how to think in ways that integrate (a) Connection-making (b) Pattern matching and correspondence (c) Quantification and Measurement (d) Functional Relationships (e) Solving for Unknown Quantities  (f) Analysis and Synthesis (g) Numerical reasoning  and so on . One could easily come up with a list that can serve as  the foundation for defining a new curriculum . It would have something in common with “Soft Psychology” as much as with the “Hard Sciences” in so far as Neuristics  guides our understanding of reality from the point of view of  devising innovative  interventions (not always inventions) to produce desirable changes. (Is it any wonder that “hard” and “difficult” are semantic siblings?) 

Neuristics evokes nothing of the dread and horror associated with the word “Mathematics”.  The feeling associated with the M-word  is pervasively negative in the minds of learners. The anxiety it generates is generic and global.  All the glitzy, gimmicky, animated  depictions of it to project user-friendliness on the computer cannot disguise its essential, intimidating  complexity. If anything, such transparent attempts to sweeten what is essentially understood, accepted and acknowledged to be a bitter pill, only serves to reinforce old fears. Why after all would the Disneysian power of high-tech  be harnessed to dilute Math(s)’ unsavoury taste if  it didn’t really nauseate the brain? From inside these fancy new techie-bottles, the sour old wine of numbers and definitions, the same old traditional templates, compressed  with oppressive abstractions, (cleverly disguised) and the antiquated language that defines them…..all these inky dollops from the past keep spilling out, staining one’s contemporary consciousness with Neanderthal fears. Only  those exceptional Math(s)-smart kids are spared this primordial sense of doom and defeat that the M-word invokes in most minds.  

A new-generation program of Neuristics  will need to break away from past traditions and develop novel and new  ways of communicating  Mathematical concepts. I suspect that it will initially have to go the CLSO way in which the  program-design’s interface with the user (not teachers) gently guides much of the learning process.  The Neuristics teacher will (a) supervise the learning process, (b) remove impediments to learning by designing absent communicative materials (c) diagnose the reasons why a learner is experiencing difficulties based on a deeper knowledge of his/her personality and patterns of learning . All new-generation programs like CLSO should bring to convergence three aspects essential to neuristic learning: (a) re-organizing  the hierarchical structure of the neuristic content in new, flexible ways (b) employing highly communicative educational  designs for displaying neuristic concepts and operations and (c) making the design and content of Neuristics congruent with the mind’s learning patterns. 

 Mathematicians please note: we need you increasingly,for consultancy !  J 

Okay. Snap! End of bubble! You can return to your dreary old world of Math(s) now!

Stay keyed-in-and-connected!Until next week.

Shad

THINKING AND REASONING TOOLS  FOR ALL 

Once upon a time….or was it really such a long time ago ?…..learning Mathematics in school was mostly  about finding right answers to set problems. Indeed, so much so that the problems were treated as vehicles for taking you to the right answers.  This message seemed to descend  from an invisible authority who lived somewhere up in the clouds, beyond and above all the schools of the world. The message was given in a universal voice, one that had traveled through time and through many lands.  It went something like this:

“One day when you are older, you may want to be a scientist, an engineer, a mechanic, a doctor, a pharmacist or an architect , etc. All these professions require a knowledge of science. And Mathematics is the language of science. You have to learn this language so you can solve problems. That is why you need to be trained to find the correct answers to problems. If you come up with wrong answers, then Ouch!! Wrong answers lead to expensive consequences and  will  not be tolerated. Only stupid, lazy, inattentive  or neglectful people make mistakes and come up with wrong answers. If you want to be successful, focus on getting the right answers. If you are committed  enough to this task, you will always find a way, even if it means going over your calculations many times, until you find your error”.

This may sound familiar to many even today. Indeed, today’s classrooms and curriculii still function under the shadow of history, and the Math class is no exception. There is still greater emphasis on correct computation than on correct understanding, despite the acceptance of scientific calculators as  an integral part of our existential repertoire, like e-mail. Until only 25 years ago, the  traditional class teacher emphasized mathematical operations first and foremost. By contrast, mathematical concepts were regarded as a lot of fanciful talk. After all, concepts  don’t even sound mathematical. They feel so obtuse and  difficult to articulate, rarely measurable or quantifiable  What’s worse, concepts can branch out on their own, in different directions like the tributaries of a river, and merge into other concepts! They also offer conditional solutions to specific problems! How can answers in Math not be unconditionally wrong, or right? How can students be allowed to approach Math problems in uniquely different ways and cause confusion all around? And how in heavens does one practice Math concepts? 

The very notion of communicating math concepts made teachers nervous. This was territory that was both unfamiliar and uncharted in terms of well-articulated pedagogy. Nevertheless, within a decade or so, the critical significance of teaching Math concepts  spread among the teaching community. Evidence supporting the need to teach these descended from the ivory towers (think-tanks and research labs) and  took almost an entire decade to penetrate the walls of Elementary classrooms, and another  decade to impact upon classroom practices. Times, after all, have changed.  So has society. The new culture of tech-knowledgy demands a very different mindset. The citizen of the future is not going to be obsessing about getting right answers any more. S/he would be more concerned about getting the right ideas: new ideas, innovative and creative ideas, ideas that lead to products that transform human habits.

As traditional roles of production shift their axis towards the developing world, the many uses and applications  of tech-knowledgy occupies center-stage as  the dynamo for growth, global change and wealth-creation. A post-industrial world such as ours, belongs to men and women who understand complexity. It needs people  who know how to live with problems, who like  to tinker with problems, not people who feel insecure without the immediate self-gratification that comes from  finding “ the right solution”. We will need people who can micro-manage processes within processes,  people who can learn to enjoy creating new answers to old questions, or multi-layered answers to single questions. They will need a different mindset than that of their fore-fathers: one that  is able to scrutinize complex process from very close, and one that is capable of identifying  those changes in the procedural aspects of problem-solving  that might produce  desired results . The desired result could be a product to be created e.g. an e-pod, a device that offers educational programs for use everywhere we go.  So in a sense, the answer (output = an e-pod) is already provided. The presenting problem is: what procedures, strategies, approaches, planning and calculations are needed to arrive at a desired answer?   People  would have to ask  out-of-the box questions like: “What if we tweaked this procedure here or that procedure there?” to see what the effect would be in approximating the “output”.

Math learning should not just be about cranking a machine and checking if the correctly computed answer comes out the other end. Checking if the computation is correct by tracing it backwards, will still be required, but that’s  a pretty clerical task . For those aspiring to be in the drivers’ seats, the more important question would be: “Does this make any sense??!! What are we doing here? Let’s step back and think about it” . Math mechanics can worry about: “Where is the calculation error?”.

Here is a very simple example of the “correct solution”-oriented student. The following Math question  is posed in a test: “Bill’s Mom likes to knit sweaters. She needs 12 pounds of wool to knit him a sweater. She  had  18 pounds of wool.  How many sweaters could she knit for Bill?”  

Here is the student’s solution:

                   1 . 6         Answer: 2 sweaters

             ________

             ]   1  8 . 0

     12    ]

     ____]  -1 2

              ________

                  6  0

             -    6  0

                _______

                      0

When told to obtain the “correct” solution, he promptly rounds off the 1.6  to 2 sweaters, (since he just learned “rounding off” when using decimals).  When the teacher insists that his answer is still wrong, he  retraces his steps to look primarily for computation errors. Sure enough he finds one, and his correction now produces the following solution:

                   1 . 5       Answer: 1.5 sweaters

             ________

             ]    1  8 . 0

     12    ]

     ____]  -1 2

              ________

                  6  0

             -    6  0

                _______

                      0

He now plays safe and keeps his answer as it is, since the “rounding off” feels tricky because he forgot the rule with the decimal 5:  does one round it off upwards or downwards? By contrast, the more savvy student will not even bother to compute. She will reflect on the problem, examine the data until it takes on significance and meaning. Next, by a process of simple numerical reasoning she will conclude that the mother can knit only 1 sweater for Bill, since 1.5 sweaters makes no sense.

What if she had twice the wool than before (i.e. 36 pounds). Does that mean she can now knit twice the number of sweaters, too  (i.e. 2 sweaters)?  If yes, then how? If not, then,  how many? And why not double the amount = double the amount of sweaters?

Such What-if questions become important because they help students tinker with the problem to see how the solution changes and why. It opens up new areas of numerical reasoning which  in turn leads to new mathematical concepts (ratios, in the above example). The original problem becomes a vehicle for exploring more advanced mathematical ideas. It should not just be a means to arrive at “correct solutions”. 

Understanding Math problems depends on a student’s grasp of fundamental mathematical concepts and operations. If poorly understood,  most solution-oriented students flounder on math concepts built on sand. They jump nervously into the problem, manipulate  the numbers wildly while groping for the “right answer”. In the process  they   are guided by their teacher’s facial expression (A for Assent,  D  for Dissent) as they mentally discard one answer for another until the teacher’s face reads an unmistakable  A.

There are some solution-oriented students who  rely on smart heuristic approaches  to perform well-reasoned mathematical operations. Often they succeed  in producing correct or near-correct results. These students, invariably gifted, or otherwise trained, devise clever approaches to problem-solving. If the problem is new and unfamiliar, they rely upon heuristics: a technique by which they (a) think of similar problems in simplified form or (b) try out some well-known approaches to see if they work before unleashing a different  strategy (c) look for a pattern that is identifiable and can lead to some clues (d) employ a process of elimination, gaining insight as they proceed, and so on. It is true that heuristics can, to an extent, be taught. Teachers are, for instance teaching children to look out for cues and clues when reading a problem: identify the key words, circle the numerical information provided, underline what is being asked in the problem, etc. Knowing such techniques helps bolster students’ confidence. Certainly, it makes them more able and willing to tackle word problems. However, heuristics, do not function like guaranteed algorithms. The value of the simplest heuristic is optimized when a student is stuck with a problem and can’t think of a way around it, and is therefore asked to:  “Try imagining those fractions as whole numbers, and see whether it makes sense when you read it”…..or “ Try doing what you did in a similar problem. Do you remember?” etc. 

We can understand all the elementary and most secondary mathematical concepts, their formulations, their connections, their derivatives, etc.  through plain common-sense. Or raw intelligence. However, Math empowers common sense with  booster-powers for higher-level  reasoning and thinking. The same raw intelligence grows more sophisticated  and advances swiftly into the realm of exploratory and innovative thinking.

Such a trained mind is an asset not just to those who study the hard sciences, but to everyone who pursues knowledge for making this a better and more enjoyable world to live in. The fundamental activity of Math learning is to make humans think. All disciplines, be it the arts, the sciences, the humanities and social sciences, etc.   harness this power to think, reason, fathom and derive meaning. Which is why it is preposterous to conceive of the world today and tomorrow as one where Math continues to be reserved for a few privileged  minds. It should be flung open to everybody, for everybody, and offered in the same  spirit as one offers music to all i.e. not just those who are “musically gifted”.

The new-generation Math program, like CLSO-MATH has been designed to do just that.

More next week, so stay tuned!

Shad

Here is another scenario: imagine sipping coffee and having a conversation around a 3-legged coffee table with one leg short. Any weight  placed  on the table makes it wobble and spill coffee. Assuming  there is no remedy for it, we would compensate for the design flaw by modifying our behavior e.g. we would have to watch it! before leaning an elbow, or anything else for that matter,  on the table.  Even worse, we would be on anxious alert for inattentiveness in others. If such  flawed coffee-tables were universally endured because they cannot be cured then coffee-drinking would be elevated to a skill requiring partcular  attention to “Spillage Avoidance”. 

When the slightest spillage of attention can cause a major spillage of coffee, we’re looking at a pretty wobbly environment for conducting intelligent conversation.  A small area of  our brain would attend directly to the problem of exchanging ideas and opinions while large parts of it  manage the stress and anxiety inherent in such a situation. Once again skilled coffee-drinkers would be in the minority and this would be quite acceptable since  smart and minority  go together on everything.
Limitations  in design influence social cultures and create sub-cultures all their own. Within them certain home-truths mushroom spontaneously and naturally, and are accepted as being organic to such an environment.  So,  good drivers would  naturally be highly paid because they are smarter and fewer and perform a task essential to  civilization. There would naturally be fewer cars and more buses. Bus-fares would be naturally quite high. There would naturally be supplementary, private driving schools  teaching sub-skills essential to qualify for entrance into prestigious driving schools. It would be natural to expect fewer people wanting to be drivers, so naturally there would be state-sponsored non-profit programs lauding the financial advantages, the social  benefits of choosing driving as a profession, etc.  Good drivers who also excel in the science of coffee-drinking, would belong to a class all their own…… naturally. So would those considered inept at both.

Traditional Math(s) learning has influenced our culture in similar ways. For a number of historic reasons, success in Math(s) is naturally associated with  a clutch of secondary and tertiary skills and abilities like high attention levels, extreme alertness and exceptional memory. We are expected to possess them, develop them and hone them in order to be “good” at Math(s). The counter-pedagogical design  in Math(s) teaching-and-learning materials remain unscrutable, un-discussable, even inconceivable as the dominant reason for mass under-achievement. It is easier to  see and hear  the  frustrated attempts of learners struggling to grasp this understandably “complex”  subject. The solution to this problem is seen not in a radical and much-needed transformation of  the design and delivery of Math(s) but in another area altogether: e.g. teacher-training and student attitudes. Thus, (a) teachers should be trained  to apply creative and imaginative approaches when teaching Math(s), and (b) students should deploy more intellectual energy learning it. Typical demands of (b) are: 

(1) a high concentration and intelligence for figuring out implicit connections in Math(s). The connections rarely display explicit logic.

(2) a curiosity that collapses into trial-and-error thinking to grasp  simple concepts masquerading as complex.

(3)  extremely good recall of rules and algorithms that are often vaguely understood.

(4) a rote-learning memory for mathematical  facts to compensate for an unreliable grasp of mathematical concepts and operations.

(5) a blinkered focus on learning the Mathematical language (formal terminologies and definitions) first and foremost on the assumption that learning such a language will facilitate the understanding of Math(s) concepts. Usually, it is the other way around: if done right, a well understood Math(s) concept seamlessly assimilates the Math(s) language corresponding to it. We know that language learning in babies is preceded by pre-linguistic concept formation of the immediate environment. As many concepts become  familiar via incidental learning, they  are smoothly integrated  into a child’s linguistic comprehension.
High concentration.  Curiosity. Good recall. Mathematical language. One could argue: They all sound good. What’s wrong with developing these?? They are wonderful abilities to develop and use! No doubt. But why should they be learned at the expense of making Mathematics easier to learn? Why should they be fostered primarily to compensate for limitations in the design of Math(s) education materials? 

If a movie-camera cannot be rotated, raised, rolled or turned in any direction, then one would have to be pretty inventive in order to make good movies. Even worse, if film was very expensive to use, then mistakes in filming would have to be reduced to a minimum to avoid expensive retakes. The camera’s limitations, along with the budget’s,  would ignite the proverbial Necessity that is the Mother of all Inventions. Therefore, movie sets  would have to be designed carefully, very methodically. Each shot would have to be pre-conceived, possibly even pencil-sketched from various perspectives. The immobile cameras would need to be placed in precise positions to capture the intended effect. To prevent retakes, actors would need to practice their lines and roles almost to perfection, with no room for mistakes and retakes.  The point being made here is that mental abilities galvanized   to circumvent limitations, do impel the creative and the perfectionist in us. Ultimately, such movie-sets during earlier years produced giants in the movie-world like Eisenstein, Kurosawa, Fellini and Ray. But those limitations also spurred innovations and improvements. Our creative and imaginative abilities harnessed to  today’s digitized movie camera in special-effects techie environments  reach  higher, more complex levels. They render possible what was once unthinkable. Which is why more people  produce more excellent movies today compared to the few memorable ones from earlier  decades.
Let’s face it: cars without serious design limitations are simply easier to drive.  They enable the driver’s swift mastery of all the operational features  e.g. steering, gears, brakes, accelerator pedals, lights, indicators, wipers etc.  And mastery lends automaticity to our physical act of driving. The resulting “auto-pilot” mode liberates a wide spectrum of attention for attending to traffic signals and road signs,   motorists, pedestrians,  road-turns, one-way signals, numbered blocks (when seeking an address), etc. Freedom from the mechanics of driving also helps us relax and attend to more complex mental tasks such as thinking about work-related problems, reflecting upon domestic issues,  engaging in a lively conversation with a companion, or toying with abstract ideas. In short, driving a well-designed car, or drinking coffee on a level table, should not demand an excessively  high (or peak) level of  alertness or attentiveness. As learners attain  mastery in the basic skills needed to drive  (and drink coffee??),  large portions of the brain get disengaged to attend to the more superior function of problem-solving and thinking purposefully and creatively.  
        
The CLSO-MATH Program demands average, not peak concentration. CLSO exercise templates are designed  to guide learning in a series of smooth incremental steps. The visual templates arouse curiosity and dispel anxiety because they lure the mind into learning and understanding what once seemed so baffling. Even anxious students feel relieved and  become curious  to exhibit what they finally understand. Once mastery is achieved in (a) understanding Mathematical concepts (b) reasoning numerically and quantitatively,  the learner’s mind is   liberated to perform higher level problem-solving tasks requiring peak-level concentration. Summoning such concentration  for higher-level thinking (opposed to exhausting it in low-level  tasks) makes better economic use of the students’ intellectual energies. 
More next week!

Shad         

Copyright © Shad Moarif, April 2007, Vancouver, Canada.  All rights reserved. No part of  “Shad’s Blog” may be reproduced or copied in any form, electronic or otherwise, without the author’s permission.

There is now convincing research to suggest that the highly anxious person who exerts himself/herself excessively over a problem, uses smaller areas of the brain. By contrast, the relaxed un-rushed  person who approaches a problem calmly employs larger areas of her/his brain. High-strung, distraught minds, rushing to complete their tasks battle emotional demons like worry, fear, anxiety and guilt, leaving less brain-power to attend to the problem at hand. Which is why tense, anxious people who feel very pressured take longer to solve a problem and often make more errors. Their more relaxed peers with their calm and collected minds have larger segments of the brain available to turn a cerebral problem over in their heads, enabling them to reach correct solutions quickly.  

Math anxiety is a well documented phenomena. Much of the literature on the subject silently acknowledges Math to be intrinsically “difficult” and therefore naturally prone to induce anxiety. Symptoms of anxiety can be traced by children to their own parents first e.g. Johnny notices that Mom refers only Math problems to Dad,  who in turn refers it to Johnny’s teacher. Later, Johnny notices that his own Math anxiety  is reflected in a  majority of students in his class. As a result, an unimpeachable truth gets permanently entrenched in his mind:  Math is mysteriously difficult. Even adults  like Mom and Dad feel jittery around it.

The difficulty does not, however,  lie in the intrinsic nature of Math as a discipline. Far from it. More often, it is  creative design (or the lack of it)  in conventional Math teaching and  materials that determines the levels of difficulty experienced by Mom, Dad and Johnny.  Students, even teachers, who use traditional teaching approaches and materials have to  struggle to make un-explained connections. Often, students do it entirely on their own, sometimes several grades later, via hindsight.   When consulting a Math text-book, or listening to a class-teacher’s explanation, they have to switch to  peak levels of concentration to grasp some of the mathematical ideas being explained.  This is because  Math concepts are illustrated minimally, using implicit, highly formal language. 

Learners and teachers (quite unknowingly)  contend  with basic limitations in design in the delivery of Math concepts, operations and applications. Admirably, though, a few teachers  compensate for these limitations by developing highly creative and imaginative activities to unravel central ideas or connections in Math learning. Doing this demands among other things,  a display of incandescent passion, intense motivation and a curious absorption with Math on the part of the teacher. These attributes are sometimes borrowed from untapped reserves within the teacher, or role-modeled or simply acted out as best as one can. 

Often, exceptional high-energy integrated-learning engagements that promote team member’s interactions among each other (to assimilate Math ideas properly), can create an intimidating learning environment.  Both teacher and “bright” students  charge such group sessions with  a  kind of competitive energy that can be anxiety-inducing (or simply off-putting) for those barely managing to  keep up. On the other hand, students may be exposed to an equally dreaded alternative: a teacher standing by the board blissfully disconnected from his/her students, pointing and explaining, making poorly designed doodles  and expecting tuned-out students to exert  maximum attention throughout their sterile, disinterested silence.  

Design flaws or limitations, have consequences that are strikingly similar in any situation. Just imagine for a minute that all cars have steering wheels that loosen and tighten unexpectedly whenever one turns the steering wheel. Such a “normally accepted” feature (not flaw!)  creates a unique set of attention demands for drivers. With practice, the astute driver might discern a curious, (but unreliable), pattern in the whimsical behavior of the steering wheel e.g. on smooth roads it tightens when rotated clockwise. On slightly bumpy roads it tightens when rotated anti-clockwise. 

Paying attention to such supplementary learning becomes part of the driver’s repertoire of good driving skills.   When roads alternate unpredictably between smooth and not-so-smooth, all drivers would have to summon peak attention levels just to exercise normal caution. Not surprisingly, learning to drive a vehicle that behaves so inconsistently would be very stressful.   Even after mastering the skill, drivers would continue feeling  terribly stressed out while chatting  with a companion seated alongside. Navigating on auto-pilot would court certain disaster.  Furthermore, avoiding accidents (a problem-solving activity) would require a separate, considerably higher level of  training. This is now harder because  a smaller area of the brain would be activated towards avoiding accidents while a large portion of the brain would be managing anxiety, worry and fear due to the inherently stressful nature of driving itself. A chilling thought.

And yet no one would be surprised to see a majority of student learners perform poorly on  driving tests. They would explain such large-scale failure using the principle demonstrated by the statistical bell-curve: i.e. in any given population anywhere, smart people will always be  in the minority; the majority are always average and not-so-smart.  Therefore only a small minority can be good drivers. The rest will have to (a) work harder  or (b) seek special assistance.  Part II to be continued next week.

Stay tuned!

Shad 

If I started to think like a conventional Mathematician in order to teach Math(s), my students would suffer from a common ailment known as VTL or Very Twisted Learning. VTL is associated as you know, with VTP (Very Twisted People).

Imagine explaining the practical application of Number Properties in these terms to Grade 6 students:

“The general idea of numbers comes from the properties we ascribe to the way they develop naturally, in a known sequence, like 1, 2, 3. okay. All mathematical reasoning is derived from our implicit faith in these properties. Here is the first practical application of these properties. Observe closely. You will notice that 7 + 2  =  2 + 7. Ah-ha! What is the general rule here? Right! That the sum does not depend on the order of  its terms. Which means that addition is a commutative operation…..etc.  etc”

“What…..are …..you……thinking??!!” is how  10-12 year olds will likely respond to such raving. Sheer blinking   bewilderment.I can just see them roll their eyes inside their heads and wonder:  “Why is s/he  making such a big deal about 7 + 2 = 2 + 7, and what is so practical about these properties?”.

The teacher attempting to sound like a mathematician might end up taking  his/her students anywhere but to the gates of Math(s) learning. Before long,  s/he may start filtering out the “smarter” ones from among  the students, while punishing (psychologically, of course) the “dumb ones”. The latter may appear too stupid to understand the essential Math(s) principles, the lofty  theories on which they are based, the underlying philosophy that guides mathematics learning… blah-blah-blah. God alone knows how hard s/he tries to simplify them, to reduce those ideas  as much as possible to help  these kids  get it!

The mathematician and the Math(s) teacher are two different  beings, both on the outside and from the inside. They live in two different universes, come from very different sources of knowledge and  have very different goals. They work in  contrasting environments  and are guided by different motives, which in turn impacts on their  reasoning processes.  There is yet another difference that often separates them. The proverbial absent-minded mathematicians are invariably  authentic versions of their true selves. Not so  those obsessive, my-way-or-the-highway minded  Math(s) teachers with correction-impulse genes grafted into their  DNA’s.  Many of them, in all seriousness and good faith,  wear  the mathematician’s mantle in order to portray themselves as true believers in the faith they espouse. 

To mathematicians,  “practical applications” that appear rather petty to the ordinary mortal, carry extremely significant  thought-applications in a mathematically ordered universe.   Here, numbers and their relationships have a well defined  mathematical existence. In this universe, even intuition attains a level of abstraction. The  mind intuits ambiguity like a hunter senses danger in a forest. It also intuits exploratory pathways of logic leading one out of the beguiling uncertainties of ambiguity  into the lush comfort of Mathematical certainty. For the mathematician, mathematics is a language very unlike the language of words that possess content. The purpose of mathematical content is not to convey meaning  (as human words do) but to “construct pure forms of thought”. ( In some ways it does beckon something in us  that strives to reach beyond the senses, beyond the boundaries of human words and their ascribed meanings). The mathematician is  mystified by how the human mind can keep splitting the tiniest of imaginable fragments, repeatedly, into tinier and tinier fragments, forever. Or how we can keep increasing the same given quantity by the same amount over and over again (5, 10, 15, 20…….) until it disappears into  the vague and distant mist of  infinity. Why do we conduct such infinite thought-experiments so naturally?

The world of numbers provide the perfect medium to micro-manage such questions. Most importantly, mathematicians  get a rise out of  a  truth  they stumble upon  over and over again, namely:  (a) certain concrete, tangible phenomena can be  distilled into abstract forms via numerical representations of the relationships that connect them (b) once these are encapsulated  in their abstract forms, the forms grow strikingly vulnerable to human (well, mathematical) manipulation and  (c) certain  manipulations lead to concrete results that alter the real world in breath-taking, mind-boggling ways.

In sharp contrast, the Math(s)-teacher’s world is often a classroom full of noisy kids bursting at their seams with crazy hormones. This world of unpredictable learners can be far more complex  (than the world of numbers!) in terms of how students associate together or singly. How they learn, needs to be considered. As also why. Ideas they find most compelling need to be understood. To add to these headaches, we also need to be informed of their status as  socialized intellectual beings.

Finally, there is, for the Math(s) teacher, the inevitable world of mathematics itself. But what a different world it is from the mathematician’s! As a Math(s) teacher, I can only speak for myself. What is my world of Math(s) like? Well, for one thing, it is an overwhelming  universe all its own that I am compelled to organize into some kind of an architectural structure. I have to visually expose the inherent  composition of its many interconnected parts for learners  to see.  My visual distillations of Math concepts must be simple and  elegant in design. And the design should express both intent and content. This has taken years and years of work  and it ( surfacing as CLSO-MATH ) never ceases.  I have to extract meaning out of Math(s) at a deep personal level, and knowing about the mathematician’s world helps, but never as much as knowing about how the human mind learns number sense, numeracy, and numerical reasoning.  Unlike the mathematician, I have to learn not to speak Math(s) but to explain it, communicate what I understand.   Of course, I  enjoy the  ability  of Math(s) to embellish human  intelligence.  Sometimes a mathematical concept  generates a train of abstract thinking that appears to be converging upon some “essence”, and yet, that essence  eludes definition, like the answer to the question pitched at moody adolescents: What do you want to be when you grow up?. I like to explore my experience of Math(s) as an interested individual  seeking glimpses behind the veil of outward reality not with the eyes of a mathematician, but that of  a teacher.

Finally, I have to hypothesise cautiously about how the learners’ minds react to Math(s): how  they make meaning out of numbers and number patterns; what puzzles them, astonishes them,  excites them or makes them wonder; how they associate Math(s) with brains.  How much of what they think, resonates with me?  How do I invest their gift of  resonance into my teaching practice?

Often I wrestle with questions around  teacher-training:  what does it take to become an effective Math(s) teacher?

If there is a mathematician’s mind lurking  within us somewhere, it could, I think, be transformed into something powerful and compelling as an approach, to  channel a learner’s natural curiosity towards mathematics. But bringing the original cast of the “Mathematician-teacher” into Math(s)-teaching is to regress into medieval times. The medieval-classical approach towards  Math(s) education began a few centuries ago with all Math(s)-teaching materials and approaches  developed and authorised by pure mathematicians. They decided, quite arbitrarily, how Math should be taught to children and teen-agers, without the foggiest notion of how minds develop number sense and learn Math(s).

The shadow of such a tradition falls darkly across many Math(s)-classrooms throughout the contemporary world. We see plenty of evidence of that  in today’s frustrated  majority of Math(s) underachievers.

And so… teacher-training…? Well, that could be  a topic for another blog. 

Shad   

Something is seriously wrong with Math education.

You know it. I know it.

I could be a parent, teacher, an employer or an adult who missed out on higher education because I was under-achieving in Math(s). When it came to Math(s), I “wouldn’t get it“. My grades suggested it. My teachers projected it. And society punishes me for it by denying me opportunities for self advancement. By the end I am hopelessly reconciled to an embarrassing truth: “I’m not smart enough. That’s all there’s to it. For Math(s) you need to be smart. ….and I’m not. ”

This is not an autobiographical “I”. It is a positional “I”, an abstract middle seat that keeps this blog pinned down to a center.You can sit on this seat and get a better view of my protestations as a Math teacher.

So, to continue.

I could also be among those who perform brilliantly in any of the non-Math(s) areas I am lured into, like: business, marketing, sports, the performing arts, journalism, creative writing, the behavioural sciences, education, and complex historical studies. Despite my poor Math scores I am able to express, articulate, analyse and synthesise facts, information and knowledge with varying degrees of insight, knowledge and precision. I can display a perceptive understanding of forces sweeping across societies and affecting lives in strange, often predictable, ways.

One would conclude that it takes some thinking to do that. Or brains, whatever. So, even though I cannot analyse complex graphical data or solve quadratic equations I can still be among the many who possess the brains to be successful and content. But I grew up ostracized as “not smart enough to do Math”. It’s a stagnant truth that curdles into a sour abbreviation: “Not smart enough”. Period. A toxic recipe for adolescents caught in the throes of identifying themselves via feelings of self-esteem.

Assuming that you are at least a few centuries old (mentally, as a historical abbreviation of humankind personified and all that sort of thing), you might recall that centuries ago (before the advent of Arabic numerals (0,1,2,3,……), multiplication and division was an extremely convoluted process. Do you remember multiplying and dividing using Roman numerals LXXXVCX and CXVII. Or Greek numerals. Which was superior, I would wonder, sometimes, while wrestling with a post-division headache.

Tobias Dantzig, in his book “Number: The language of Science” replies as follows:

“A far more important question is: how well is the system adapted to arithmetical operations, and what ease does it lend to calculations? In this respect there is hardly any choice between the two methods.: neither was capable of creating an arithmetic which could be used by a man of average intelligence...

…from the beginning of history, until the advent of our modern positional numeration (or the place-value system of writing numerals), so little progress was made in the art of reckoning. Not that there were no attempts to devise rules for operating on these numerals. How difficult these rules were, can be gleaned from the great awe in which all reckoning was held in these days. A man skilled in the art was regarded as endowed with almost supernatural powers… And to a certain extent this awe persists to this day. The average man identifies mathematical ability with quickness in figures.”

So going back to those times, you and I, and folks like us, possessing average intelligence, would be turfed out as people lacking the mental competence to multiply and divide (Add and subtract? Perhaps, yeah, okay, we could do that, maybe).

Would I protest? Would you? Would we?

Well, take the ancient English Tally-Stick bearing notches cut into it of various sizes: the small ones representing Pounds Sterling, the larger ones 10 pounds, the largest 100 pounds etc. Tobias reports that “the English tally persisted for many centuries after the introduction of modern numeration made its use ridiculously obsolete”. A very sarcastic Charles Dickens , in an address on Administrative Reform in the English Parliament, complained about the continuing “savage mode of keeping accounts on notched sticks…. much as Robinson Crusoe kept his calendar on the desert island”. He reminded his audience that: “in the reign of George III an enquiry was made by some revolutionary spirit whether pens, ink and paper, slates and pencils being in existence, this obsolete adherence to an obsolete custom ought to be continued, and whether a change ought not to be effected. All the red tape in the country grew redder at the bare mention of this bold and original conception and it took until 1826 to get these sticks abolished”. (Tobias Dantzig)

I did protest. I know I did because my brain is programmed that way. Yours too, perhaps. You may have been the “revolutionary spirit” that Dickens was referring to. But despite our passionately strident protests, the reforms came long after you and I were dead. Which is too bad, because when we protest and all we see is the red tape “getting redder”, many of us give up and loose the courage to make defiant statements like: “You need exceptionally gifted minds to multiply and divide numerals??!! If that’s so, then we need to find better ways to multiply and divide so that ordinary folks like us can help ourselves to the benefits of knowing how to do it simply, accurately and swiftly“.

And if you were a true visionary, you would gaze into the crystal ball and tell them what you see: “There will come a day when 7-8 year olds possessing average intelligence, even less, will be able to multiply and divide with greater speed, accuracy and confidence than you old farts!”

The entire history of Mathematics education proves over and over again that we reformist spirits were always right. There is always a better way. Today, yes, right now as we speak, Mathematics is being taught in schools throughout the world, using approaches and strategies, methods and materials that have been made very complex and convoluted by well-meaning people who simply don’t know any better. The resulting system victimises one set of people with average intelligence while glorifying another. Victims from the first lot have their normal intelligence denigrated. It is made to appear low and unworthy of learning Mathematics.

Why?

Because throughout the ages, inquiring minds “fail” to blindly accept the manipulation of inconsistent rules as the foundation for learning Mathematics. So the sooner we dispense with the popular myth about Math and its dubious association with the “brainy” for success, the sooner we can cross over to the other side.

“The greatly increased facility with which the average man today manipulates numbers”, writes Tobias Dantzig, “has been often taken as proof of the growth of the human intellect. The truth of the matter is that the difficulties then experienced were inherent in the numeration in use, a numeration not susceptible to simple clear-cut rules. The discovery of the modern positional numeration did away with these obstacles and made arithmetic accessible even to the dullest mind”.

Likewise, the difficulties being experienced by today’s under-achieving Math learners has much to do with difficulties….(”limitations” might be a better word)……. inherent in the obsoleteness of the curriculum design, the redundancies in the teaching materials and the complete absence of intelligent and coherent educational design. Today’s failures in Math(s) learning should no more be allowed to invite the traditional knee-jerk response: “Insufficiency of the Average Intelligence of the Masses”.

It’s why we came up with CLSO-MATH. It’s a programmed response that goes beyond protest …even though my protesting will never cease.

More next week!

Shad