Previously by Jeremiah Dyke: A Word-Problem World: A World of Mathematical Meaning
As others have, I often speculate if we assembled a committee for the sole purpose of rendering a subject near useless, if that committee could possibly 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 that calculate numbers. 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, however, I recall an article some time ago where the author concluded that the helpfulness of most advice is inversely related to the advice-givers age; as age increased, the helpfulness of advice typically decreased. Certainly this hasn’t curtailed my desire to tell the world what I think they should do; after all, at our core, we are simply a bunch of advice-giving machines. Nevertheless, we must accept that we may suffer from this generational lag and our distaste for calculators may simply be a function of this. For example, 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, studies show that networking devices such as Linked and Facebook allow many individuals to bypass the importance of building a resume by networking. It's been shown that resumes are not as important today as they were in the past, but how many of us know this? How many of you who read that previous sentence cringed and shook your head in disbelief? I did, when I first read it. This inability to place our beliefs under scrutiny creates vast problems in our advice giving. I believe this generational lag is partially responsible for the u2018blame-calculator' bias our educators support.
Let us pretend 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 today's calculators could, in theory, displace most of today's math educators. A four function calculator, at a cost of about $0.99 cents, could answer 90% off all math questions for grades K-4. A scientific calculator, at a cost of about $14.99, could answer 90% of all math questions from grades K-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 these same graphing calculators have statistical software capable of handling most first year statistics courses. A slight step up to one of the higher end graphing calculators equips all the tools to make a first semester of calculus a breeze. In all, for about $180.00 ($100 if used), a student could purchase the majority of textbook answers from kindergarten through their freshman year of college.
Now, clearly this says nothing about understanding the material, but, on paper, would an individual truly be able to tell the difference between the traditional paper-pencil student and the calculator student when taking standardized tests? Let me pose a question, is it possible for a student to have more understanding of the significance and methodology of a problem but be more dependent upon technology to solve it? Let me approach this problem from a different angle. For instance, 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 and many have mentioned that they have nearly forgotten much of their math training. This says nothing about their abilities as engineers; they’ve simply outsourced much of their work to software applications. The math training they went 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 significance and methodology of a problem but be more dependent upon technology to solve it? To answer this question we need to discuss the parts of a math problem.
Conrad Wolfram, in a wonderful talk about using computers to teach mathematics, points to four parts of teaching a math problem to which I will purposefully change the titles:
1) Asking/Understanding the right question
2) Setting up the question
3) The actual computations
4) Interpreting the answer.
Now, according to 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. In fact, Wolfram has calculated that "on an average school day in the world, we use up about 106 average lifetimes learning hand calculating." Now, 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 classroom time, or 106 average lifetimes per day) or the other three parts are being neglected.
An example may serve,
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. r = 1/10
Now, which of the four parts do you believe took up the most time? Yet, I could have plugged part 3 into my calculator or software and churned out the answers in a matter of seconds. We could have been devoting 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 you encounter this question in real life?
- What would have happened if we incorrectly set up the problem?
- How would being ignorant of this math be in the interest of a lender?
- 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 80% of our time doing computations? If an educator wanted to complete, say, 5 of these types of problems after a lecture, he or she 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 practicing 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. We can see this whenever we slightly change a problem. When a problem, or wording of a problem, is slightly changed, students instantly become confused. This is partially due to the fact that they never understood the question in the first place; they were simply parroting their teachers' computations.
Yet, the problem above is just simple algebra. Imagine if we were using computations to solve something more tedious? Let's call upon the same problem Wolfram used in his talk. 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 which can be represented in the following equations – we will produce one of the four below (click here for the other three):
(Equation 1: First Root (of four))
How many of these problems do you think we could compute within a 60 minute block of time without the help of technology? One? Are we starting to see what our students might truly be loosing with our unfounded calculator bias?
From here the arguments start becoming 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 invention matters to the students' development. Since the parents didn’t have software and advanced calculators when they were growing up, these technologies must not be needed, or worse, they might be detrimental. The fallacy is, of course, rooted in the limited technology of even previous generations. What did the parents of our parent's lack? Look at the history of computing, and how far we have come. One wonders if these same anti-technology arguments were made over the course of time. For example, 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 do not need to pretend that calculating devices are not available the same way we don’t need to pretend that writing/recording devices are not available. Instead of mindless computation we need to invest more time in asking the right questions, understanding and setting up the right questions, and understanding our solutions.
Yet, why don’t we?
Though I disagree, the skeptic in me understands why most math educators would be anti-calculator. After all, it’s probably within our interests to limit the public from 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 add in their head. Or, some myth about how calculator users aren’t thinking. Before this, it used to be that it wasn’t fair to the students who couldn’t afford calculators (of course, it's becoming harder to make these types of arguments today). 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 and lacking. They are reaching at best.
Before offering a potential median for the public school math classroom, let me put forward my thoughts as to how I will personally teach mathematics to my two sons. Foremost, I will never drill multiplication/division tables (or anything for that matter) into their minds. I personally don't care if they ever remember them. At the earliest age possible I will place a calculator in their hands and free them from all this barbaric drill. I will do this so that we can jump into the more advanced topics sooner (see my blog for ways to teach math). Moreover, we will only, and I mean only, engage in real life problems where the solutions are meaningful.
- When discussing functions we'll build small bridges in my shop
- When discussing probability we'll play card games and risk.
- When discussing who between my sons is truly better at their first-person shooter video games we will gather data and analyze it.
- How angles are used in a cut to fall a tree.
- How building expressions can help you bid on landscape projects
- The blast radius of a firecrackers
- And so on…
With all this in mind, what are math educators to do? It is here that my advice is limited. That is, to simply do your best, under the circumstances, with the time you have. There are things that I would like to do in my college math classes that I haven't the time to do. There are methods of teaching that I know are inferior to others, but I haven't the time to change them. But, given the few hours I have with my students per semester, I do my best to engage as many application problems as I can. I do my best to word my test questions so that students must demonstrate mastery. I do my best to outsource the computation process to technology so that I can concentrate on the more important parts of the problems. We can only do our best…