by Karl Mattle | Feb 18, 2016 | Math
Factoring Polynomials: Learn It or Die!
Ask any adult if they know how to factor a polynomial. Most will say “What?”. Some will say “I learned that in high school, but I couldn’t tell you how to do it now”. If you happen to be asking an engineer or a scientist, they’ll probably say “Sure! It’s fun, but not really useful”. So why do we, as math educators insist that students must not only know how to factor polynomials but must master the subject as if their very lives depended upon it?
Most math educators would agree with the engineer’s first statement. It is fun. But why is it fun? The answer is that it is a puzzle, a mental exercise that yields satisfaction when solved. But let’s examine its utility. Factoring polynomials does lead to the Fundamental Theorem of Algebra (Wow! That sounds regal). It does lead to completing the square to solve a quadratic polynomial, which, in turn, leads to the quadratic formula. Are some adults required to solve a polynomial equation? The answer is yes. engineers, scientists, actuaries, programmers, and others are sometimes required to solve polynomial equations. But rarely is the solution even rational, much less an integer. Since the solution must be approximated, the clever maneuvers learned in high school are not applicable.
So why bother to teach factoring at all? The answer is that we, as educators, have been charged with teaching our students to think, to enjoy thinking, and to trust the results of their thought. We cannot possibly teach our students everything they will need to know to be productive citizens. But we can train their minds to empower them to figure it out. Factoring polynomials should be taught once, early in the math curriculum, not as a useful life skill, but as mental exercise to hone the mind. The trend in math education is to offer students various gimmicks or procedures (the X method, slide and glide, etc.) to allow students to factor a polynomial without really thinking about it. The mind is not honed. The procedures are robotic and forgotten as soon as they are regurgitated on a test.
That being said, here is a link to a great way to factor a polynomial.
by Karl Mattle | Dec 1, 2015 | Math
Mathematical Proof: What’s the Point?
The sum of two odd numbers is even. Two pictured triangles are similar. There are infinitely many prime numbers. Tell these facts to a student and the overwhelming response is “ok”; ask them to prove it, and the overwhelming response is “What’s the point?” In History class, my teacher doesn’t ask me to prove that Hannibal crossed the Alps. My literature teacher doesn’t ask me to prove that Herman Melville wrote Moby Dick. Even in Science, the subject most closely aligned with, and dependent upon Math, I’m not asked to prove Charles’ and Boyle’s laws. Why do I have to prove things in Math? Tell me what I need to learn, how to use it, and I’ll do it.
In a way, Mathematics has over-succeeded. It has been so successful in explaining, and, horror of horror, quantifying, so many natural, physical, economic, and yes, human relationships, that our modern Math and Science curriculum has become obsessed with conveying the many successes of Mathematics without conveying how those successes were achieved. Most students are well aware of the story of Isaac Newton observing an apple falling and being inspired to quantify gravity. But little or no time is devoted to how such a common observation could lead to the Law of Universal Gravitation, and its proof. Educators will lament, there is simply not enough time. We have to move on to the application. This argument is probably correct, reflecting the fact that we are trying to teach too little about too much. We should not be teaching, or trying to teach, our students every application of Mathematics. Rather, our Mathematics curriculum should emphasize the beauty of Mathematics with the emphasis on thinking, that is, proof.
Where in the busy Secondary Mathematical curriculum should proof be emphasized? The answer is in the first form of Mathematics in which rigorous proof was emphasized, Geometry. But the modern curricular movement is to teach practical application rather than proof. While Heron might be proud to know that modern High School students were being compelled to memorize his formula for the area of a triangle, in reality, few, if any, of the students are ever going to be required to calculate the area of a triangle using only the lengths of the sides. If they ever are confronted with such a problem, they will Google it, and get the procedure and formula from the internet. Heron would be far more proud if students were being instructed on the thought processes he employed to derive his formula and prove that it worked, every time.
Rare is the gainfully employed adult who is compelled to employ any of the Mathematical maneuvers, practical or otherwise, learned in High School. However, all productive citizens will, multiple times in their lives, find themselves having to make a point, or refute a point, using critical thinking and logic. In other words they will have to use proof!
by Karl Mattle | Sep 5, 2015 | Math
Calculators: A gift from the angels or agents of evil?
Here is a true story: A high school student was working an algebra problem that required use of geometry to determine the value of x. He had reached an impasse and so he asked his teacher for help. His teacher assisted him to the point where x could be evaluated. The teacher then prodded the student with the question “Now what is 10 + 6?” The student got a befuddled look on his face and proceeded to punch the arithmetic in to his calculator. After a few seconds the student declared “60” and proceeded to write the erroneous answer on his paper. Besides the disgusted amazement of his teacher, what’s wrong here?
The error analysis is simple, the student accidentally multiplied 6 and 10 rather than adding them. But that is an understandable mistake. Is the problem that the student feels he needs a calculator to do simple arithmetic? In part. But far more alarming is the willingness of the student to accept as fact a woefully incorrect answer. The student had no “number sense”, no concept that the sum of 10 and 6 is nowhere near 60. Many brilliant mathematicians have made, and continue to make similar mistakes. But their sense of in what vicinity of the real number system the correct answer lies, tells them immediately that they have made an error, and they correct it. This may be an extreme example. But it should be an alert to the fact that we as a society, cannot let this tech-savvy generation pass in to adulthood without a sense of what’s greater than what.
Still, it’s not time for a calculator burning party. If the problem our geometry/algebra student was working required him to determine a distance that was miles, he would have to use a calculator to approximate the square root of 10. Telling a police dispatcher that you had witnessed a crime miles back, is likely to get you arrested yourself. But what our student and our crime witness, and indeed all citizens need is a sense that the square root of 10 is somewhere between the square root of 9, or 3, and the square root of 16, or 4. Further, they need to be able to instantly reason that is closer to than it is to because 10 is closer to 9 than it is to 16. Then, if our student, or witness, or good citizen accidentally punches in to a calculator and gets 10, they would immediately realize they had made a mistake and correct it.
So what’s to be done? Calculators do, in fact, make life more productive by freeing our minds for higher order thinking. But basic arithmetic is not higher order thinking. While memorization is anathema to modern educational philosophers, it does breed confidence. Our student should have been certain that 10 + 6 is 16, not because he reasoned it out, or because he counted on his fingers and toes, but because the fact had been burned in to his brain while still in Elementary School. This “number sense” is not acquired by doing Differential Calculus, or Linear Algebra, or Euclidean Geometry. Number sense should be established before tackling these subjects. Memorization is tedious. But its very tedium makes students appreciate the power of calculators once they are permitted use them. Until that number sense is established, calculator use should not be permitted.
We, as tutors, should be very reluctant to permit students to use calculators. However, when it comes to checking an answer, we should be enthusiastic about using calculators. If, for instance, a student from Booth Middle School, uses linear interpolation to approximate and gets 3.1, He or she should be shown that a better approximation is 3.1623, found by using a calculator. Now the student can see that they have done well (after all, they are just getting an approximation), they get a confirmation of their number sense, and they are compelled to think on a higher level in reasoning why their method works but is less precise than the calculator.
