Shad’s Blog for Week # 5
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