why there should be computer based math

7/28/2019 Why There Should Be Computer Based Math

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Teacher: Mrs.Calhoun

Hari Rau-Murthy

Computer Based Mathematics - Preparing the Next Generation for a Global Economy

Traditionally, mathematics is taught through memorization of addition, subtraction,

multiplication and division. This rote process stretches through six years of early childhood

education. Elementary school mathematics is focused on brute force; the ability to recite

numbers is paramount.(Wolfram 2012).

Recently, a new school of thought has emerged: computer based math (CBM). With it,

the focus shifts from memorization and computation to logic. Children are exposed to advanced

concepts at an earlier age, giving them time to learn more math in the classroom setting. CBMs

standard bearer Conrad Wolfram argues, I believe that correctly using computers is the silver

bullet for making math education work. Removing the barriers of memorization allows students

to expand their minds. Calculus, for example, was previously reserved for individuals able to

handle the complex calculations. With the computational barrier removed by computers, young

minds can focus on the theories. One does not need six years of rote memorization to understand

that if you continue to add sides to a polygon, it eventually becomes a circle. This rudimentary

understanding of differential equations can fuel analytical exploration at an earlier age.

(Wolfram 2012)

Schools across the nation should adopt computer based math to allow their students to

have a competitive chance in todays programming based world. Moving away from rote

memorization to CBM in elementary school will give students classroom time with

programming, rather than having programming be the focus only for college students.


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computations(Cox 1992). Todays world is not the world in which rote learning was so crucial.

CBM is necessary to keep the US competitive in research and in the job market.

While CBM has advantages, there are concerns as well. Without the initial memorization

of basic arithmetic functions, there is a concern that children will never learn the fundamentals of

mathematics. CBM may also close the door on discoveries that can be made only without a

computer. Additionally, such a program may not be feasible given the need for nationwide

implementation.

A possible side effect of CBM is missing out on the fundamentals of mathematics -

surface learning. Without the fundamentals of mathematics, theories cannot be challenged. If

Einstein and Ernst Mach had not had a strong grasp of fundamentals, they would not have been

able to challenge such subtle aspects as the notion of length and bounded variation. This lead to

the theory of relativity. This demonstrates the crucial importance of fundamentals in

mathematics to the growth of the field. Major discoveries are not made if breakthroughs are not

made in basics.

Fundamentals are also crucial to practical tasks such as approximation(Wolfram-Closing

Remarks 2012). A computer is not always the fastest route to an answer, and may not be easily

accessible. Simple everyday activities like calculating the tip on a bill, setting aside the correct

amount of time for a task, or estimating the hours remaining until a phone dies are all done faster

mentally than by a computer(Wolfram-TED 2012). Until CBM is tested in the classroom setting

and a clear curriculum is set forth, one cannot be certain that students taught with CBM will be

adept in performing mental math.

Certain fundamentals cannot be replaced through working in a computer based

environment(Wolfram 2012- Closing Remarks). Computer based/assisted proofs are based on


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reductio ad absurdum by exhaustion: that is through taking a tree of possibilities resulting from a

purposely made incorrect assumption, rejecting each of the possibilities and coming out with an

alternative that has to be the truth by definition. In order for the proof to be rigorous, the

boundary value conditions of the trees must be known. These are questions that can only be

answered through the mathematics of fundamentals - Analysis. Thus computer assisted proof

does not take away the need for mathematical analysis and fundamentals.

In fact, in a technology abetted era where numerical integration and computer algebra

systems(mathematica) take away the virtue in the actual analytical solutions to such problems as

partial differential equations, the much greater issue is on finding the boundary value conditions

under which solutions hold, when solutions make sense, and when stronger claims can be made

on solutions(Shatah 2012)- all very important analysis problems. In many senses, the need for

analysis is made even greater in relation to its role in previous years and to other subjects.

There are many barriers that prevent CBM from being implemented in todays education

system. Radical ideas cannot be tested out based on old ideology. CBM necessitates the

retraining of teachers to ensure mastery in programming and understanding of the paradigm shift.

Test questions will need to be remodeled so that the difficulty does not lie in computation but

rather in analysis. This difficulty extends beyond the classroom to standardized tests. If CBM is

to be truly implemented in todays classrooms, a complete overhaul of the educational system is

required. This is unlikely with todays American bureaucracy. However, one could envision

implementation in countries without a current mathematics curriculum (Wolfram 2012- Closing

Remarks).

Another concern is that mathematical elegance is lost in computer assisted proof, a

necessary element of CBM. Solving a problem for an infinite number of cases with a finite


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number of iterations of an algorithm is one of the biggest accomplishments of human society

over the past 30 years leaving barely any room for mathematical elegance. Because of the

infinite number of cases, computer assisted/CBM proofs(Four Color Theorem 2012) are

exclusively reductio ad absurdum. The problem with reductio ad absurdum is that the insight

that one would get in other methods that would then spur on more discoveries is not

obtained(Rudin 1976). All that reductio ad absurdum requires is knowledge of all the

possibilities and whether there is enough information given in order to reach a conclusion. The

property of each specific possibility that made it possible to prove the conclusion is not learned.

Conversely, a successful completion of proof through reductio ad absurdum, which includes

CBM, reveals connection among seemingly unrelated concepts, thereby creating new fields of

study(Rudin 1976). Computer assisted proofs and CBM stand to both detract from and add to

mathematics.

There are problems of elegance and future applicability with proofs that can be executed

via CBM. The bigger pressing problem is that even when theorems are true, the existence of a

reductio ad absurdum proof is not guaranteed. In fact, Godels incompleteness theorem(Godel

1992) guarantees that there are infinitely many proofs that can not be completed reductio ad

absurdum, let alone finding elegant solutions to the ones that exist. Due to limitations of

transistor size originating from the size of the atom that are starting to be realized, one can not

count on Moores law stating that the transistors on integrated circuits double every 18 months.

This suggests that there are many proofs that cannot be carried out via brute force. One such

example is the Poincare recurrence theorem(Marwan 2007) stating that the measure of the set of

all x in a set A that is a subset of X, such that a measure preserving transformation T applied

n>N times is a subset of the set X-A, is zero. The proof of this theorem then in turn guarantees a


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trajectory of a dynamical system coming within a certain neighborhood(i.e. a trajectory repeating

itself) happens almost surely(that is, with probability of one.) However, since many systems

such as the state of atmosphere of the earth would take 20 orders of magnitude longer than the

age of the universe to recur, a brute force algorithm, or even an algorithm with its base in

contradiction, would not be able to prove the theorem. Therefore much of the same artifice

employed in a traditional proof is still required in order for a rigorous proof to be complete.

CBM is important step to implement in bringing technology to the classroom in a manner

not only preserving the logical thought processes but even enhancing and restructure the

curriculum. The wide applicability of learning material earlier through programming based

education rather than education focused on the arcane methods of arithmetic will allow a larger

number of interested students to learn more about the axioms at the heart of mathematics.

Despite the many pitfalls of CBM, it is crucial that it is implemented in elementary classrooms

across the nation. Computer based mathematics is the only answer to a computer based society.

why there should be computer based math

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