Calculators Are Not the Enemy

As others have, I often speculate if we assembled a committee for the sole purpose of rendering a subject near useless if it would possible to do a better job than the public school system has done with mathematics. Public school math has been drilled, beaten and stripped of all possible usefulness. All for what? Students hate learning it. Teachers dislike teaching it (or at the least dislike teaching it to students who dislike learning it) and still we press on. Actually it’s worse; we press on without even knowing whom to point the finger. Lacking even the ability to pinpoint the problem, educators, out of desperation, begin turning their hatred toward small little rectangular objects with numbers on them. Why? It is again the purpose of this article to answer this inquiry.

What is it about these small little calculating devices that make them such easy targets for the failure of math literacy? Is it that these machines are doing the thinking for us? Is it that students are simply learning to punch buttons? I assume it's possible, but I recall an article some time back that concluded the helpfulness of most advice is inversely related to the advice-givers age. As age increases, the helpfulness of advice typically decreases. Certainly this hasn't curtailed my desire to tell the world what I think they should do; after all, at our core, we are a bunch of advice-giving machines. Nevertheless, we must accept that we may suffer from generational lag. I come from a generation of the "Resume". I was taught the importance of the resume in job candidacy. I even outsourced my resume making when I entered the labor force because I believed It was so vastly important. Today we have Linked, Facebook and other networking devices that allow many individuals to bypass the importance of the resumes. Resumes are not as important as they were in the past, but how many of us know this? I believe this generational lag is partially responsible for the blame-calculator bias our educators support.

Let us pretent for a second that the importance of mathematics is simply being able to produce textbook answers to standardized textbook questions. If true, then most of todays calculators could, in theory, displace most todays math educators. A four function calculator, at a cost of about .99 cents, could answer 90% off all questions for grades k-4. A scientific calculator, at a cost of about $14.99, could answer 90% of all questions from grades 5-8. A graphing calculator with a variable solver and factoring application could answer the majority of Algebra, Algebra II and Pre Calc questions. Most of theses same graphing calculators have statistical software capable of handling most first year statistics questions. A slight step up to one of the higher end graphing calculators equips all the tools to make first semester calculus a breeze. In all, for about $180.00 ($100 if used) a student could purchase the majority of textbook answers from kindergarten through college freshman.

Now clearly this says nothing about understanding of the material, but, on paper, would an individual truly be able to tell the difference between the traditional paper-pencil student or the calculator student when taking standardized tests? Let me pose a question, Is it possible for a student to have more understanding of the value and methodology of a problem but be more dependent upon technology? Let me approach this problem from a different angle. In academia, I work alongside many engineers who are also adjunct math professors. They tell me they use very little of their mathematical training in their day job, many have forgotten much of their math training. This says nothing about their abilities as engineers, only that they've outsourced much of their work to software applications. The math training they go through is much more a symbol of their ability and reliability than their productivity. Yet, the question still remains, is it possible for a student to have more understanding of the value and methodology of a problem but be more dependent upon technology? To answer this question we need to discuss the parts of a math problem.

Conrad Wolfram points to four parts of teaching math to which I will purposfully change the titles

1) Asking/Understanding the right question...2) Setting up the question....3) The actual computations...4) Interpreting the answer.

Now, according Wolfram, math students today spend about 80% of their math training on part 3, the part that our technology can do exponentially faster than we can. There are only two possible conclusions here. Either part three, the computation part, is the most important portion of the four parts of a math problem (important enough to spend 80% of class) or the other three parts are being neglected.

Let's try a question from a textbook shall we,

1) Find the interest rate r if $2,000 is compounded annually and grows to $2,420 in 2 years.

2) A = P(1+r)^t The formula for compound interest

$2420 = $2000(1+ r)^2

3) The Calculations

2420 = 2000(1+ r)^2

2420/2000 = (1+ r)^2

121/100 = (1+ r)^2

+/- SqRoot (121/100) = 1 + r

+/- 11/10 = 1 + r

-1 +/- 11/10 = r

r = 1/10 or

r = -21/10

Check the solutions by plugging them in for r

$2420 = $2000(1+ 1/10)^2

$2420 = $2000(1.10)^2

$2420 = $2000(1.21)

$2420 = $2420

$2420 = $2000(1 - 21/10)^2

$2420 = $2000(-1.1)^2

$2420 = $2000(1.21)

$2420 = $2420

4) The rate cannot be negative, so we reject r = -21/10

Now, which of the four parts do you believe took up the most time? Yet, I could have plugged part 3 in my calculator or software and churned out the answers in seconds and devoted that time to parts 1, 2 and 4. Instead of calculating we could have been asking,

Why would you want to know the answer to this question?

When would this come up in real life?

What would have happened if we incorrectly set the problem up.

Would it be in the interest of the loaner for the borrower to be ignorant of this math?

We could have played around with different scenarios of interest, initial investment and return

Why do we reject negative interest rates?

Could negative interest ever exist?

What would negative interest mean to the lender or the borrower?

Now, one might say that we could still have asked these questions, but could we? Do we? How much could we really discuss within a 60 minute class if we are spending access time doing computations? If an educator wanted to complete, say, 10 of these types of problems after a lecture, they would surely be stressed for time when engaging the other three parts of a math problem. This is normally the case. Educators tend to devote the majority of their time toward praticing computations while the student is left not understanding the value of the question, how or when to set it up or what the solution truly means. Yet, this is just some simple algebra. Imagine if we were using computations to solve something more tedious? Here is the formula of the general form of a 4th degree equation (commonly called a Quartic)

ax4 + bx3 + cx2 + dx + e = 0

Quartics have 4 roots.

The 4 roots can be represented this way:

The four formulas are equal except for a "+ or -" sign at the beginning, and a couple of "+ or -" signs at the far right end of the images. Note that the imaginary unit "i" does not appear in any of these formulas. The formulas on top were obtained with Wolfram's program Mathematica.

How many of these problems do you think we could do within a 60min block of time without the help of technology? One? Are we starting to see what are students might truly be loosing with our unfounded calculator bias?

From here the arguments start become warped. Many old-school math learners were forced to use paper and pencil and thus anything newer and faster must, by definition, be lowering our math intelligence. I think this attitude could be summed up as the "order of invention bias". In other words, the order of the innovation matters to the students development. Since the parents didn't have software and advanced calculators when they were growing up they must not be needed, or worse, they might be detrimental. The fallacy is of course rooted in the technology of previous generations, what did they lack? Look at the history of computing (http://en.wikipedia.org/wiki/History_of_computing_hardware) and how far we have come. One wonders if these same anti-technology arguments were made over the course of time? And before these devices, were they applied to the history of paper? Were the generations who wrote on Papyrus more or less privileged than those who wrote on forms of paper that came after? And what of those who engaged in math before recording was available? They must have performed all computations within their mind but were they better off? The fact of the matter is order of invention is not important to the study of mathematics. We don't need to pretend like calculating devices are not available the same way we don't need to pretend that writing/recording devices are not available. We need to instead invest more time asking the right questions, understanding and setting up the questions and understanding our solutions, not mindless computations.

Yet, why don't we?

Though I don't agree, I can understand why most math educators would be anti-calculator. After all, it's probably within our interestes to limit the public of these devices in order to keep demand for our skills high. Yet, why are other individuals, most of whom grew up prior to the calculator revolution so anti-calculator? Normally their reasoning has to do with some story of the power going out while standing in a checkout line and the clerk who couldn't do a percentage on paper. Or, some myth about how calculator users aren't thinking. It used to be that it wasn't fair to the students who couldn't afford calculators. Of course, its hard to make that argument today with the cheapness of most calculators. I've also heard the argument that standardized tests don't allow calculators and thus more reason why they shouldn't be used. It is my opinion that all of these arguments are weak.

It may be true that todays standardized tests are normally anti-calculator but this alone is not sufficient enough a reason to not explore their use. Instead, educators should be pushing for standardized change (The new GRE will allow basic calculator use) not accepting the status quo.

Where is a happy median?

I once had a math instructor tell me that our generation is the last generation to know arithmetic. My response was "or spell, but does it matter"? Someone once stated that if you want a perfectly written paper eliminate the delete/backspace key. What is meant by this is that people would be so extremely careful about what they type if they didn't have the ability to correct their mistakes (Can you imagine if you had to start a paper all over again because you misspelled a word?). Certainly we would make less mistakes, but would we be more efficient?

As others have, I often speculate if we assembled a committee for the sole purpose of rendering a subject near useless if it would possible to do a better job than the public school system has done with mathematics. Public school math has been drilled, beaten and stripped of all possible usefulness. All for what? Students hate learning it. Teachers dislike teaching it (or at the least dislike teaching it to students who dislike learning it) and still we press on. Actually it’s worse; we press on without even knowing whom to point the finger. Lacking even the ability to pinpoint the problem, educators, out of desperation, begin turning their hatred toward small little rectangular objects with numbers on them. Why? It is again the purpose of this article to answer this inquiry.

What is it about these small little calculating devices that make them such easy targets for the failure of math literacy? Is it that these machines are doing the thinking for us? Is it that students are simply learning to punch buttons? I assume it's possible, but I recall an article some time back that concluded the helpfulness of most advice is inversely related to the advice-givers age. As age increases, the helpfulness of advice typically decreases. Certainly this hasn't curtailed my desire to tell the world what I think they should do; after all, at our core, we are a bunch of advice-giving machines. Nevertheless, we must accept that we may suffer from generational lag. I come from a generation of the "Resume". I was taught the importance of the resume in job candidacy. I even outsourced my resume making when I entered the labor force because I believed It was so vastly important. Today we have Linked, Facebook and other networking devices that allow many individuals to bypass the importance of the resumes. Resumes are not as important as they were in the past, but how many of us know this? I believe this generational lag is partially responsible for the blame-calculator bias our educators support.

Let us pretent for a second that the importance of mathematics is simply being able to produce textbook answers to standardized textbook questions. If true, then most of todays calculators could, in theory, displace most todays math educators. A four function calculator, at a cost of about .99 cents, could answer 90% off all questions for grades k-4. A scientific calculator, at a cost of about $14.99, could answer 90% of all questions from grades 5-8. A graphing calculator with a variable solver and factoring application could answer the majority of Algebra, Algebra II and Pre Calc questions. Most of theses same graphing calculators have statistical software capable of handling most first year statistics questions. A slight step up to one of the higher end graphing calculators equips all the tools to make first semester calculus a breeze. In all, for about $180.00 ($100 if used) a student could purchase the majority of textbook answers from kindergarten through college freshman.

Now clearly this says nothing about understanding of the material, but, on paper, would an individual truly be able to tell the difference between the traditional paper-pencil student or the calculator student when taking standardized tests? Let me pose a question, Is it possible for a student to have more understanding of the value and methodology of a problem but be more dependent upon technology? Let me approach this problem from a different angle. In academia, I work alongside many engineers who are also adjunct math professors. They tell me they use very little of their mathematical training in their day job, many have forgotten much of their math training. This says nothing about their abilities as engineers, only that they've outsourced much of their work to software applications. The math training they go through is much more a symbol of their ability and reliability than their productivity. Yet, the question still remains, is it possible for a student to have more understanding of the value and methodology of a problem but be more dependent upon technology? To answer this question we need to discuss the parts of a math problem.

Conrad Wolfram points to four parts of teaching math to which I will purposfully change the titles

1) Asking/Understanding the right question...2) Setting up the question....3) The actual computations...4) Interpreting the answer.

Now, according Wolfram, math students today spend about 80% of their math training on part 3, the part that our technology can do exponentially faster than we can. There are only two possible conclusions here. Either part three, the computation part, is the most important portion of the four parts of a math problem (important enough to spend 80% of class) or the other three parts are being neglected.

Let's try a question from a textbook shall we,

1) Find the interest rate r if $2,000 is compounded annually and grows to $2,420 in 2 years.

2) A = P(1+r)^t The formula for compound interest

$2420 = $2000(1+ r)^2

3) The Calculations

2420 = 2000(1+ r)^2

2420/2000 = (1+ r)^2

121/100 = (1+ r)^2

+/- SqRoot (121/100) = 1 + r

+/- 11/10 = 1 + r

-1 +/- 11/10 = r

r = 1/10 or

r = -21/10

Check the solutions by plugging them in for r

$2420 = $2000(1+ 1/10)^2

$2420 = $2000(1.10)^2

$2420 = $2000(1.21)

$2420 = $2420

$2420 = $2000(1 - 21/10)^2

$2420 = $2000(-1.1)^2

$2420 = $2000(1.21)

$2420 = $2420

4) The rate cannot be negative, so we reject r = -21/10

Now, which of the four parts do you believe took up the most time? Yet, I could have plugged part 3 in my calculator or software and churned out the answers in seconds and devoted that time to parts 1, 2 and 4. Instead of calculating we could have been asking,

Why would you want to know the answer to this question?

When would this come up in real life?

What would have happened if we incorrectly set the problem up.

Would it be in the interest of the loaner for the borrower to be ignorant of this math?

We could have played around with different scenarios of interest, initial investment and return

Why do we reject negative interest rates?

Could negative interest ever exist?

What would negative interest mean to the lender or the borrower?

Now, one might say that we could still have asked these questions, but could we? Do we? How much could we really discuss within a 60 minute class if we are spending access time doing computations? If an educator wanted to complete, say, 10 of these types of problems after a lecture, they would surely be stressed for time when engaging the other three parts of a math problem. This is normally the case. Educators tend to devote the majority of their time toward praticing computations while the student is left not understanding the value of the question, how or when to set it up or what the solution truly means. Yet, this is just some simple algebra. Imagine if we were using computations to solve something more tedious? Here is the formula of the general form of a

**4th degree**equation (commonly called a**Quartic**)

**ax**

^{4}

**+ bx**

^{3}

**+ cx**

^{2}

**+ dx**

**+ e = 0**

**Quartics**have

**4**roots.

The

**4 roots**can be represented this way:

**First root (of four):**

**Second root (of four):**

**Third root (of four):**

**Fourth root (of four):**

The four formulas are equal except for a "+ or -" sign at the beginning, and a couple of "+ or -" signs at the far right end of the images. Note that the imaginary unit "

**i**" does not appear in any of these formulas. The formulas on top were obtained with Wolfram's program

**Mathematica**.

How many of these problems do you think we could do within a 60min block of time without the help of technology? One? Are we starting to see what are students might truly be loosing with our unfounded calculator bias?

From here the arguments start become warped. Many old-school math learners were forced to use paper and pencil and thus anything newer and faster must, by definition, be lowering our math intelligence. I think this attitude could be summed up as the "order of invention bias". In other words, the order of the innovation matters to the students development. Since the parents didn't have software and advanced calculators when they were growing up they must not be needed, or worse, they might be detrimental. The fallacy is of course rooted in the technology of previous generations, what did they lack? Look at the history of computing (http://en.wikipedia.org/wiki/History_of_computing_hardware) and how far we have come. One wonders if these same anti-technology arguments were made over the course of time? And before these devices, were they applied to the history of paper? Were the generations who wrote on Papyrus more or less privileged than those who wrote on forms of paper that came after? And what of those who engaged in math before recording was available? They must have performed all computations within their mind but were they better off? The fact of the matter is order of invention is not important to the study of mathematics. We don't need to pretend like calculating devices are not available the same way we don't need to pretend that writing/recording devices are not available. We need to instead invest more time asking the right questions, understanding and setting up the questions and understanding our solutions, not mindless computations.

Yet, why don't we?

Though I don't agree, I can understand why most math educators would be anti-calculator. After all, it's probably within our interestes to limit the public of these devices in order to keep demand for our skills high. Yet, why are other individuals, most of whom grew up prior to the calculator revolution so anti-calculator? Normally their reasoning has to do with some story of the power going out while standing in a checkout line and the clerk who couldn't do a percentage on paper. Or, some myth about how calculator users aren't thinking. It used to be that it wasn't fair to the students who couldn't afford calculators. Of course, its hard to make that argument today with the cheapness of most calculators. I've also heard the argument that standardized tests don't allow calculators and thus more reason why they shouldn't be used. It is my opinion that all of these arguments are weak.

It may be true that todays standardized tests are normally anti-calculator but this alone is not sufficient enough a reason to not explore their use. Instead, educators should be pushing for standardized change (The new GRE will allow basic calculator use) not accepting the status quo.

Where is a happy median?

I once had a math instructor tell me that our generation is the last generation to know arithmetic. My response was "or spell, but does it matter"? Someone once stated that if you want a perfectly written paper eliminate the delete/backspace key. What is meant by this is that people would be so extremely careful about what they type if they didn't have the ability to correct their mistakes (Can you imagine if you had to start a paper all over again because you misspelled a word?). Certainly we would make less mistakes, but would we be more efficient?