Year: 2014

Two-Column Proofs that Two-Column Proofs are Terrible

I’m not entirely sure that I completely agree with the author of this post (http://mathwithbaddrawings.com/2013/10/16/two-column-proofs-that-two-column-proofs-are-terrible/), but he certainly provides food for thought and so I’m happy to link to it. Among the most provocative quotes from this post:

In a good proof, each individual step is obvious, but the conclusion is surprising.  In many two-column proofs—especially those taught earliest in a geometry course—each individual step is mystifying, while the conclusion is obvious.

From “Reshaping High Schools”

A colleague pointed out the following article to me: Put Understanding First, by Grant Wiggins and Jay McTighe. A sampling:

Unfortunately, the common methods of teaching and testing in high schools focus on acquisition at the expense of meaning and transfer. As a result, when confronted with unfamiliar questions or problems (even selected-response problems on standardized tests), many students flounder. Consider a high school algebra question that was included on state tests in New York and Massachusetts:

To get from his high school to his home, Jamal travels 5.0 miles east and then 4.0 miles north. When Sheila goes to her home from the same high school, she travels 8.0 miles east and 2.0 miles south. What is the measure of the shortest distance, to the nearest tenth of a mile, between Jamal’s home and Sheila’s home? (Students were provided with a grid they could use to plot the answer.)

Fewer than 40 percent of New York 10th graders correctly answered this item, despite the fact that the requisite knowledge is “covered” in every Algebra I class in North America. Test results such as these reveal not a failure of coverage but a failure of transfer.

Out-of-context learning of skills is arguably one of the greatest weaknesses of the secondary curriculum—the natural outgrowth of marching through the textbook instead of teaching with meaning and transfer in mind. Schools too often teach and test mathematics, writing, and world language skills in isolation rather than in the context of authentic demands requiring thoughtful application. If we don’t give students sufficient ongoing opportunities to puzzle over genuine problems, make meaning of their learning, and apply content in various contexts, then long-term retention and effective performance are unlikely, and high schools will have failed to achieve their purpose.

Engaging students: Multiplying and dividing rational expressions

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Kayla (Koenig) Lambert. Her topic, from Pre-Algebra: solving proportions.

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B) Curriculum: How does this topic extend what your students have learned in previous courses?

Multiplying and dividing rational expressions extends so many topics because the students have to use what they have learned up to multiplying and dividing the rational expressions. For example, this topic extends multiplying and dividing fractions. For multiplying and dividing fractions the students need to multiply across the numerators and multiply across the denominators and then simplify when possible (Multiplying Rational Expressions). Students also use factoring, which they should have learned before getting to this topic. When factoring, the students should remember different ways to factor. Some different ways are finding the greatest common factor, factoring by grouping, and finding the perfect square. They should also remember how to factor polynomials of different degrees.

The students also need to remember how to divide numerical fractions because they use the same method when dividing rational expressions; multiplying by the reciprocal. Another topic students should have previously learned is how to simplify rational expressions and how to multiply polynomials. Lastly, the students should also remember what a term, coefficient, constant, degree of a term, degree of a polynomial and should remember different types of polynomials (monomial, binomial, etc.). I could keep going with what topics are used when multiplying and dividing rational expressions all the way down to counting, addition, and subtraction. There are obviously so many different topics students have learned in the past that are extended when multiplying and dividing rational expressions.

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D) History: What are the contributions of various cultures to this topic?

We can break multiplying and dividing rational expressions into many different mathematical subjects. In order to accomplish multiplying and dividing rational expressions, basic algebra and other basic mathematics had to come first. Methods of multiplication were documented by ancient Egyptian, Greek, and Chinese civilizations (Multiplication-Wikipedia). Around 1800 BC, Egyptians were the first known to use fractions. In 1600 BC, the Babylonians already knew solutions to quadratic equations and also solutions to equations to the third and fourth degree (Mathematics History). Egyptians used papyrus to make papers and used these to “calculate fractions” (Mathematics History).

The word polynomial comes from the Greek work “poly” meaning “many” and from the Latin word “binomium” meaning “binomial” and was introduced in Latin by a French mathematician, Franciscus Vieta (Polynomial-Wikipedia). The history of algebra goes back to ancient Egypt and Babylon where people learned to solve linear and quadratic equations. Also, Islamic mathematicians were able to multiply, divide and find the square root of polynomials.  The Hindu-Arabic numerical system was first described by Brahmagupta who gave rules for addition, subtraction, multiplication and division. In orient mathematics, algebra “ultimately evolved from arithmetic” (Mathematics History). Nicole Oresme, from Normandy, was the first person to use fraction and exponents. Many cultures have contributed to multiplying and dividing rational expressions, but I would have to say that the Egyptians, Babylonians, Chinese, and Arabic have contributed the most.

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E) Technology: How can technology (YouTube, Geometers Sketchpad, graphing calculator, etc.) be used to efficiently engage students with this topic?

Rational functions are used for many things including:

  • Fields and forces in physics
  • Spectroscopy in chemistry
  • Enzyme kinetics in biochemistry
  • Electronic circuitry
  • Aerodynamics
  • Medicine concentration
  • Wave functions for atoms and molecules
  • Optics and photography to improve image resolution
  • Acoustics and sound

Since the above topics are a little too advanced, I could show the student a video on YouTube to introduce the topic and to show them what multiplying and dividing rational functions are used for in the real world. After this, I would explain to the students that many other careers use rational functions like architects, foresters, and chemists. After talking about the topic, I could them give them a problem like the one below and ask them to graph the rational function with their calculator and can use their calculator to set up tables of values for their rational function. This will make it easy for them to see the maximum and minimum of the function and to see how the function behaves.

Example 9 from PreCalculus:

A rectangular page is designed to contain 48 square inches of print. The margins at the top and bottom of the page are 1 inch deep. The margins on each side are 1 ½ inches wide. What should the dimensions of the page be so the least amount of paper is used?

Works Cited

Larson, Ron, and David C. Falvo. “Precalculus – Ron Larson, David C. Falvo – Google Books.” 7 Feb. 2012. http://books.google.com/books?id=JRzhE6yqeFcC&pg=PA125&dq=what+are+rational+functions+used+for&hl=en&sa=X&ei=1lo1T9zDN-GusQLcrpyuAg&ved=0CFwQ6AEwBQ#v=onepage&q=what%20are%20rational%20functions%20used%20for&f=false.

“Mathematics History.” ThinkQuest : Library. 7 Feb. 2012. http://library.thinkquest.org/22584.

“Multiplication – Wikipedia, the free encyclopedia.” Wikipedia, the free encyclopedia. 7 Feb. 2012. <http://en.wikipedia.org/wiki/Multiplication&gt;.

“Multiplying Rational Expressions.” Purplemath. 7 Feb. 2012. http://purplemath.com/modules/rtnlmult.htm.

“Polynomial – Wikipedia, the free encyclopedia.” Wikipedia, the free encyclopedia. 7 Feb. 2012. http://en.wikipedia.org/wiki/Polynomial_Functions#Polynomial_functions.

“Who Created Fractions | Ask Kids Answers.” AskKids Answers | AskKids.com. 7 Feb. 2012. http://answers.askkids.com/Math/who_created_fractions.

Volume of solid of revolution

In Calculus I, we teach two different techniques for finding the volume of a solid of revolution:

  • Disks (or washers), in which the cross-section is perpendicular to the axis of revolution, and
  • Cylindrical shells, in which the cross-section is parallel to the axis of revolution.

Both of these could be expressed as either an integral with respect to x or as an integral with respect to y, depending on the axis of revolution. I won’t go into a full treatment of the procedure here; this can be found in places like http://www.cliffsnotes.com/math/calculus/calculus/applications-of-the-definite-integral/volumes-of-solids-of-revolution or http://mathworld.wolfram.com/SolidofRevolution.html or http://en.wikipedia.org/wiki/Disk_integration or http://en.wikipedia.org/wiki/Shell_integration.

A natural question asked by students is, “If I have the choice, should I use disks or shells?” The correct answer, of course, is “Pick the method that gives you the easier integral to compute.” But that’s not a very satisfying answer for novice students who’ve just been exposed to integral calculus. So, over the years, I developed a standard reply to this query:

That’s an excellent question, and it’s one of the classic conundrums faced by mankind over the years.

Should I choose Coke… or Pepsi?

McDonald’s… or Burger King?

Ginger… or Mary Ann?

Disks… or shells?

The answer is, it just takes a little practice and experience to determine which technique gives you the easier integral.

If you don’t get the cultural reference, here’s a reminder. As of 10 years ago, I could still tell this joke to college students and still get smiles of acknowledgement. But, given the passage of time, I’m not sure if this same joke would fly college students now.

The Law of Averages

Colloquially, the Law of Averages dictates that what ought to happen does happen if it happens long enough. If a gambler plays a casino for a very long time, he is almost certainly guaranteed to lose. If a weak sports team plays a stronger team in a multiple-game series, then it is almost certain to lose the series.

However, if the gambler plays in the casino for only a little while, then there is a realistic (though less than 50%) chance of coming out ahead. And a weak sports team may defeat a stronger one if only one game is played… hence the appeal of the NCAA basketball tournament and, on a larger scale, the knockout stages of the World Cup.

In my statistics class, I use a simple simple spreadsheet to illustrate that \hbox{SD}(K) = \sqrt{n p(1-p}) for a sample count, but \hbox{SD}(\hat{p}) = \displaystyle \sqrt{ \frac{p(1-p)}{n} } for a sample proportion.  Here is one image from the spreadsheet:

coinflip1

The user can change the bright green cell to be any positive integer up to 5000. This number represents the number of simulated coins that are flipped. In the above example, ten coins are flipped. Column B shows the results of the simulated coins, while column C shows a running count of the number of heads that have appeared. In the above example, 7 of the 10 flips are heads, for an observed error of +2 (two more heads than the expected number of 5) and a percentage error of 20%.

In class, I would run the spreadsheet several times, and students will see that the observed error usually is in the range of -2 to +2, and the percentage error is usually in the range of -20% to +20%.

By contrast, look what happens when the number of flips increases to a large number, like 5000.

coinflip2

There is now a larger absolute error — in this case, -28. Of course, an absolute error of that size is impossible with 10 coin flips or even 50 coin flips. However, at the same time, the percentage error is now significantly smaller (only -0.56%).

This example gives evidence for the counter-intuitive result that the absolute error grows like \sqrt{n} while the relative error decreases like \sqrt{n}.

Day One of my Calculus I class: Part 6

In this series of posts, I’d like to describe what I tell my students on the very first day of Calculus I. On this first day, I try to set the table for the topics that will be discussed throughout the semester. I should emphasize that I don’t hold students immediately responsible for the content of this lecture. Instead, this introduction, which usually takes 30-45 minutes, depending on the questions I get, is meant to help my students see the forest for all of the trees. For example, when we start discussing somewhat dry topics like the definition of a continuous function and the Mean Value Theorem, I can always refer back to this initial lecture for why these concepts are ultimately important.

I’ve told students that the topics in Calculus I build upon each other (unlike the topics of Precalculus), but that there are going to be two themes that run throughout the course:

  1. Approximating curved things by straight things, and
  2. Passing to limits

I’ve then quickly used these themes to solve two completely different problems: (1) finding the speed of a falling object at impact and (2) finding the area under a parabola. I can usually cover these topics in less than 50 minutes, sometimes in 35 minutes. Again, because I’m not immediately holding my students responsible for the contents of this introduction, I feel freer to move a little quicker than I would otherwise in the hopes of showing the forest for all of the trees.

I then ask the obvious question: what do these two questions have to do with each other. One involves the distance-rate-time formula. The other involves the areas of rectangles. At first blush, these two questions seem completely unrelated. And at second blush. And at third blush.

I tell my class that these two apparently unrelated questions are indeed related by something called the Fundamental Theorem of Calculus. Somehow, the process of finding the area under a curve is intimately related to finding an instantaneous rate of change. I then make a bold, eye-catching statement: The Fundamental Theorem of Calculus is one of the greatest discoveries in the history of mankind, period. And, at the ripe old age of 17, 18, or 19 years old, my students are now privileged to understand this great accomplishment.

This ends my introduction to Calculus I. I’ll then begin the more mundane development of limits on the way to formally defining a derivative.