Author Archives: liadelp

Math Teaching in America: Where is it going?

No one who has gone through the American education system would deny that our math teaching approach is broken. Anyone can look in any newspaper and see that math test scores are below acceptable and not getting any better. The Utah State Board of Education released the latest standardized test scores just a week ago and to no surprise, the overall proficiency rating for all Utah students in mathematics was only 38.7%.1 Jo Boaler, a British education author and Professor of Mathematics Education at the Stanford Graduate School of Education, states that the most surprising and appalling statistic in recent years is that “60 percent of the 13 million two-year college students in the U.S. are currently placed into remedial math courses; 75 percent of them fail or drop the courses and leave college with no degree.”2

In the educational system, we suffer from a wide range of stereotypes that effect our expectations and teaching styles and can alter a child’s life path. That is hefty stuff which, unfortunately, American society at large treats very lightly. The educational system tells students from the time they are in 2nd grade that math is not for everyone but only for “math people.” The profile of a “math person” is a white male, from a middle to upper class family, according to a study done at the University of Washington.3 A study published in the March/April issue of Child Development, suggests that, for girls, lack of interest in mathematics may come from culturally communicated messages about math being more appropriate for boys than for girls.3 Furthermore, Boaler states, This thinking, as well as the teaching practices that go with it, have provided the perfect conditions for the creation of a math underclass. Narrow mathematics teaching combined with low and stereotypical expectations for students are the two main reasons that the U.S. is in dire mathematical straights.”2  

So as you can see, things are not going very well, but what are we doing to change it? A lot of stock has been placed in the hope that the new Common Core Standards will help bring the knowledge and fill in the missing gaps that weren’t taught to our students in their younger years. According to the Common Core website, the Common Core is a set of clear college and career ready standards for kindergarten through 12th grade.4 These standards help guide teachers and districts in their curriculum planning. Jo Boaler’s book, What’s Math Got to Do With it?, is filled with research on how to reach out to students by broadening our perspective of math to include a focus on problem solving, reasoning, asking questions and representing scenarios in multiple forms.2 When we bring these other focuses into the forefront of our classrooms, we help students perform at higher levels which leads to more students taking advanced mathematics. Boaler strongly believes that the “Common Core mathematics is more challenging than the mathematics it will replace. It is also more interesting for students and many times closer to the math that is needed in 21st-century life and work.”2 The increase of all students’ interest in mathematics decreases the achievement gap by engaging students of all races, socio-economic status and gender.

One of the main goals in the Common Core will be for students to spend less time practicing isolated methods and more time solving applied problems that involve connecting different methods, using technology, understanding multiple representations of ideas and justifying their thinking. There are good foundational reasons for this as justification and reasoning lie at the heart of mathematics. “Scientists work to prove or disprove new theories by finding many cases that work or counter-examples that do not. Mathematicians, by contrast prove the validity of their propositions through justification and reasoning,” writes Boaler.2 “Math people” that go on to be mathematicians are not the only people who need to engage in justification and reasoning on a daily basis. These skills allow successful people in today’s workforce to discuss and reason about possible solutions, to make mistakes and have the ability trace their work in order to correct those mistakes. Employers need people who can reason about approaches by estimating and verifying results, and connect with other people’s thought processes.2

While the Common Core still has many flaws, it is a huge and necessary step towards the equality of mathematical education in America. The Common Core promotes creativity and thought rather than focusing on mundane procedure and how fast a student can complete that procedure. The Common Core brings focus to the needs of mathematicians today. The United States needs; people who are confident with numbers; can justify, reason and communicate using mathematical models and predictions; and most importantly approach an unknown problem and take logical steps to solve it. Boaler’s conclusion was that, “We need a broad and diverse range of people who are powerful mathematical thinkers and who have not been held back by stereotypical thinking and teaching. Common Core mathematics, imperfect though it may be, can help us reach those goals.”2

Image: Bill Ferriter, based on Where I Teach by Todd Ehlers.

Image: Bill Ferriter, based on Where I Teach by Todd Ehlers.

Source 1: http://www.sltrib.com/news/1743116-155/percent-students-scores-utah-science-sage

Source 2: http://www.theatlantic.com/education/archive/2013/11/the-stereotypes-that-distort-how-americans-teach-and-learn-math/281303/

Source 3: http://www.sciencedaily.com/releases/2011/03/110314091642.htm

Source 4: http://www.corestandards.org/about-the-standards/frequently-asked-questions/

The Golden Ratio is All Around You

You have been told from the time you started school that math was important because math is everywhere. Did you ever believe that? The point of this post is to prove that statement. Math is everywhere, specifically the golden ratio.

The golden ratio is Φ  = (1 + √5) /2 = 1.61803398874989484820. “This “golden” number, 1.61803399, represented by the Greek letter Phi, is known as the Golden Ratio, Golden Number, Golden Proportion, Golden Mean, Golden Section, Divine Proportion and Divine Section.”1 This number was written about by Euclid in “Elements” around 300 B.C., by Luca Pacioli, a contemporary of Leonardo Da Vinci, in “De Divina Proportione” in 1509, by Johannes Kepler around 1600, and by Dan Brown in 2003 in his best selling novel, “The Da Vinci Code.”1

The golden ratio is obviously found in the world of mathematics. The golden ratio is created when one can divide a line in a unique way. Imagine being presented with a wooden plank to cut. Where should I make the cut? There is one unique place you could cut that would give you the golden ratio. This mean that the ratio of the larger piece to the smaller piece is the same ratio as the larger piece to the entire plank prior to being cut. We could then cut that smaller piece at a certain point and get a piece one and piece two such that the ratio of piece one to piece two is the same ratio as piece two to the whole smaller piece. And the process could continue. So what makes that so special? It is special because this proportion doesn’t just appear in mathematics; it appears in your body, nature, architecture, and the solar system.

Nature and Life

Think of an ant or search of an image for an ant. Their bodies seem a bit odd at first, but look closer. An ant’s body has been distributed by the golden ratio. Think of a moth or a butterfly. In order for their wings to do what they do, they have been distributed in the golden ratio. Think of a snail’s shell or the classic spiral seashell. How is it that the shell can look like a never-ending spiral? It is because that shell uses the golden ratio. Look at your neighbor. The torso to leg, the head to the torso, the sections of your fingers; all of these are examples of the golden ratios in your own body.

Image courtesy of http://www.goldennumber.net, Gary B. Meisner, Copyright 2001-2013.

Image courtesy of http://www.goldennumber.net, Gary B. Meisner, Copyright 2001-2013.

It was always said that beauty is in the eye of the beholder, but is that really true? What if I told you that beauty was based on the golden ratio? Would you believe me? There is sound basis in scientific study and evidence to support that what we perceive as beauty in women and men is based on how closely the proportions of facial and body dimensions come to Phi.2 “For this reason, Phi is applied in both facial plastic surgery and cosmetic dentistry as a guide to achieving the most natural and beautiful results in facial features and appearance.”2

Image courtesy of http://www.goldennumber.net, Gary B. Meisner, Copyright 2001-2013.

Image courtesy of http://www.goldennumber.net, Gary B. Meisner, Copyright 2001-2013.

Art and Design

Because of the constant presence of phi in nature, we, as a civilization, have brought the golden ratio into many of our masterpieces in art. Consider the Egyptian Pyramids. It is said that the ratio between the height, base, and hypotenuse is the golden ratio. Greeks were aware of the golden ratio when they built the Parthenon.1 It is quite obvious if you have seen the movie, The Da Vinci Code, that Leonardo Da Vinci used phi in his classic drawings. The painting of the Last Supper used the golden ratio to determine the placement of Christ and the disciples to the table, walls, and windows around them.1

Image courtesy of http://www.goldennumber.net, Gary B. Meisner, Copyright 2001-2013.

Image courtesy of http://www.goldennumber.net, Gary B. Meisner, Copyright 2001-2013.

The golden proportion is in places where we would never imagine. Next time you buy a bottle of Pepsi, look very critically. How did they decide where to but the circular logo or how big to make the logo or to what proportion the writing should be to the logo? These questions are answered by using the golden ratio. “It’s even being used in high fashion clothing design, such as in the “Phi Collection” announced in 2004 and covered by Vogue, Elle and Vanity Fair.”1

Art and Design 2nd paragraph

Image courtesy of http://www.goldennumber.net, Gary B. Meisner, Copyright 2001-2013.

Do You Believe Me Now?

“In matters of reason, seeing is believing but in matters of faith, it is believing that first opens the door to seeing. The best way to know for yourself where Phi is present and where it is imagined is to explore with an open mind, learn and reach your own conclusions.”1This post was created to open your eyes to being critical of how things came to be around you. One can always ask the questions “Why?”. I challenge you to ask that question and play with the idea that the golden ratio may be the answer. Your opinion is your own. I can only present you with articles that either support or deny the importance of the golden ratio. This particular post was in support of its importance, but please feel free to read the following article that has a difference perspective. http://www.umcs.maine.edu/~markov/GoldenRatio.pdf

Sources:

Meisner, Gary B. (2014, May). Article title. Phi: The Golden Number, September 2014, from     http://www.goldennumber.net

The Divine Proportion : A Study in Mathematical Beauty by H. E. Huntley

Why base 60?

800px-Plimpton_322

Plimpton 322, a text that uses the Babylonian base 60 number system. Image: Public domain, via Wikimedia Commons.

There always seems to be so many “why” questions in mathematics. One of the largest mistakes made by educators today is brushing over those “why” questions. As a future educator I wanted to dissect the reason for base 60, so that I can explain to my future students exactly why or at least give them a better answer than “because I said so”. This is why I have chosen to research the theories behind why the Babylonians chose to work in base 60.

What were they thinking when they chose that base? We are not the first ones to question the Babylonians’ use of base 60. Theon of Alexandria, fourth century AD, and Otto Neugebauer of the 1900s also tried to answer this question.4 The struggle comes with the uncertainty of the past. No historian has been able to present such a convincing theory that it dismisses all other theories. With that being said, read the theories and pick the one that makes most sense to you and just go with it!

Theory 1: Maximization of Factors

Theon of Alexandria originally presented this theory for the reason of base 60.4 This theory states that 60 was chosen because it was divisible by 1, 2, 3, 4, 5, and 6. Therefore, 60 is the smallest number that maximized divisors. Because of the vast number of factors 60 is “easy” to work with.1 I put easy in quotation marks because it is only easy if that’s what you were taught from the beginning! Imagine being taught to work in base 10 all your life and all the sudden you switched to working in base 60 when you got to college. I’m sure you would have some other choice words to describe base 60 that did not include the word easy. The theory of maximization of factors is the most popular of the theories. I am assuming this is the case because it is fairly straight forward in its explanation.

Theory 2: Weights and Measures

This theory was presented by Neugebauer.4 He proposed that the Babylonians chose base 60 based on the weights and measures adopted from the Sumerians. The overall idea behind his theory was that a decimal counting system was modified to base 60 to allow for dividing weights and measures into thirds. After research we can find that the system of weights and measures of the Sumerians did use 1/3 and 2/3 as basic fractions. My complaint with this theory is that the systems of weights and measures would have come prior to determining the base. I feel as if the determination of a base would have come prior to the system of weights and measures. But that’s just my humble opinion.

Theory 3/4 : Geometric shapes with 60°

These theories have to do with basic geometry. The Babylonians knew that there were 365 days in a year. However, when creating a circle, they chose to have the degree of a circle equal 360°. This was a choice made out of convenience because 360° was simpler to work with than 365°. The Babylonians then used a standard ruler and compass to construct a hexagon inscribed in a circle. This hexagon allowed for 6 partitions measuring 60° each.3 This theory was presented by the historian of mathematics Moritz Cantor.3 The reasoning behind choosing a hexagon is still fuzzy to me. The other coordinating theory was based on the importance of the equilateral triangle to the Sumerians. Sumerians considered the equilateral triangle the “fundamental geometric building block”.4 Perhaps the reason the Sumerians thought the equilateral triangle was so important was because of the connection between it and the hexagon inscribed in a circle discussed earlier. An equilateral triangle has angle measures of 60°. Was the choice of base 60 as easy as that?

Theory 5: Astronomy

The astronomical theory was quite simple. The number 60 is the product of 5 and 12. Babylonians believed there were five planets at the time; Mercury, Venus, Mars, Jupiter, and Saturn.4 They also believed there were 12 months in a year. Could base 60 have been the obvious choice because of those two important numbers?

Theory 6: Fingers and Part of Fingers

Could it be that they counted the segments of their hands? We are all familiar with the idea of counting fingers and even toes, but counting pieces of fingers? That seams odd. This theory was very confusing for me. When you count the segments you get 12 on each hand. That gives you 24 pieces in total. That doesn’t tell me anything about 60. But apparently there is some further explanation. “One can count up to 60 using your two hands. On your left hand there are three parts on each of four fingers (excluding the thumb). The parts are divided from each other by the joints in the fingers. Now one can count up to 60 by pointing at one of the twelve parts of the fingers of the left hand with one of the five fingers of the right hand. This gives a way of finger counting up to 60 rather than to 10. Anyone convinced?”4 I am most certainly not convinced. Unfortunately, I was unable to identify who originated this method.

Theory 7: Joined Forces

Joined forces is the name I gave to a theory proposed by several historians. This theory states that there were two civilizations prior to the Babylonians. One civilization worked in bases 5 and the other civilization worked in base 12. Another possibility is that one civilization worked in base 6 and the other in base 10. Either way, when the Babylonians joined with these other civilizations, they decided to compromise with the previous civilizations. The compromise was decided by multiply to two previously used bases together to get base 60.3 “One of the nicest things about these theories is that it may be possible to find written evidence of the two mixing systems and thereby give what would essentially amount to a proof of the conjecture.”3 I will not dive into how they chose base 5 or base 6 or base 10 or base 12 in this post.

Theory 8: Committee

The last intriguing possibility is that either a ruler or a committee made the decision to use base 60. I think this theory along with a combination of another theory is very plausible. I believe that there was a committee of scientists and mathematicians that researched base 60. After the research of base 60 and other basses, the committee met with the hierarchy. The hierarchy could have been a political leader or the leader of the educational system at the time. After comparing the pros and cons of base 60 along with other bases, the hierarchy and committee chose the base that would be used in mathematics from then on. There is the argument that changing a civilizations number structure by committee creates a mess. Remember, America trying to switch to the metric system? It didn’t end well.

Nevertheless, math didn’t come from this magical land. It has origins and theories discovered by real people. I believe that if we discuss these origins and the thought process behind the theories more students will have an interest in mathematics.

*Superscripts within the text refer to the corresponding links and resources

Resources:

http://mathforum.org/library/drmath/view/57550.html 1

http://en.wikipedia.org/wiki/Babylonian_mathematics 2

http://www.math.tamu.edu/~dallen/masters/egypt_babylon/babylon.pdf 3

http://www-history.mcs.st-and.ac.uk/HistTopics/Babylonian_numerals.html 4