The Identity of Science and Mathematics

Science must fall - or not? #sciencemustfallScience and mathematics are objective, genderless, race-blind subjects that can effortlessly bridge language and cultural differences. But is that really true? This week saw online outcry to the #sciencemustfall proposal, an unexpected sidebar to the decolonization and #feesmustfall campaign currently raging on university campuses throughout South Africa. In a seemingly spontaneous statement (you can watch the video here), a “fallist” addresses the science faculty of the University of Cape Town, and calls for “western” science to be banished from African education.

This begs the question: “Why would this student (or anyone else) feel that something as universal as “western” science has no place in the culture they best identify with?

Karin Brodie, professor of Education and Mathematics Education at South Africa’s University of the Witwatersrand, in her article “Yes, mathematics can be decolonized. Here is how to begin“, hints at a possible reason. Mathematics apparently presents itself as a subject that is not, or should not be, accessible to all people (at least at higher education level). The perception is that, even with significant effort, if you lack the right kind or level of intelligence, you are doomed to be a mathematics outsider. Not everyone believes this, but the perception remains, and has done much to scare the less obviously gifted away from the subject. Possibly the sciences suffer from a similar “elitist” reputation. Professor Brodie says that “as teachers, my colleagues and I need to believe – to know – that all students can do mathematics. This knowledge must be transmitted to them. They must be shown that mathematics is a human enterprise: it belongs to all, and it can be taken forward to transform society.”

If students are to come to understand the universal value of subjects such as science and mathematics, they must come to believe they can own these subjects. They must believe they can be part of the process of generating new knowledge to build on what already exists (and has been gathered by people of all cultures and races). If students view these subjects from a distance, they are more likely to associate them with something foreign and threatening. Reduce the distance and the image viewed will not be so distorted.

Should science (and mathematics) “fall” or be removed from a particular cultural group’s education system, because students see these subjects as discriminatory? Surely a better approach is to change the attitude and perception that people have of the subjects instead of the content of the subjects? Why reinvent the wheel when the wheel doesn’t judge and doesn’t mind who owns it?

 

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FRACTONIA at Amazon

Fractonia by P.R. Lewis at Amazon.com“Fractonia” is now available for Kindle e-book readers. Unlike Google Play, Amazon is not offering a special introductory price for the ebook. However, Amazon has enabled the popular ” text-to-speech” option for this book. “Text-to-speech” is available on the following devices: Kindle Fire HDX, Kindle Fire HD, Kindle Touch, Kindle Keyboard, Kindle (2nd generation), and Kindle DX.

Amazon does offer a preview of the book, albeit disappointingly short. If you are interested in seeing a little more of the book before you decide to purchase, I recommend a visit to Lulu. Lulu is offering a more detailed, downloadable preview of the book in ePUB format. You can also purchase the ebook or the print version of “Fractonia” directly from the Lulu bookstore.

Purchase the Kindle-formatted version of “Fractonia” from Amazon.com.

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FRACTONIA in Print

Fractonia by Pearl R. Lewis“Fractonia” has been available for some time from e-book stores around the world. You can read the book on your computer, your tablet, or your phone. But if you are not a fan of virtual books, then this post brings you good news. Paper rules! “Fractonia” is a available as a REAL, tree-based book. You can now purchase your PRINT (paperback) copy of the book, and turn those pages the old fashioned (best) way.

The 116-page illustrated paperback is printed in the easy-to-handle (and slip into your bag) 6″x9″ format.

If you are new to the title and have not been following the development of this project, you can read more about Fractonia in the book section of my website. The book is suitable for middle school (advanced) readers, high school students, and adults. While prior knowledge of very simple fraction algebra is a plus, it is not essential. If you previously avoided mathematics as if it was the enemy, and have little to no recall of algebra, you are the perfect reader for this book. 

“Fractonia” is an adventure story that demonstrates that mathematics can be visualized as something quite different from a stack of numbers and equations. While not all students think “in pictures”, many who are turned off from more traditional ways of approaching mathematics can benefit from exploring topics in an image-centered way. Even though this book is advertised as a children/teen book in many places, this book is a good way for parents and teachers to explore the concept of visualization in mathematics.

Go on – give it a try. If you discover that you cannot connect with the strange characters or that the odd reference to a mathematical term is frying your brain, you can always donate the book to your local school library. Take a break from whatever you have planned this weekend, and go on a mind adventure – you know you want to do it.

 

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What Sane Person Writes a Novel about Fractions?

fractions“Mathematics is boring. Why do we have to study it anyway?”

“Do I REALLY have to finish all my Math homework? It’s just the same stuff over and over, and it makes no sense to me.”

“I hate doing these algebra exercises. They are SOOOOO boring.”

“I don’t understand this. It’s stupid. Why can’t I do something useful with my time?”

If you are a parent or teacher, then you have probably heard it all. The whining. The complaining. The angry outbursts. For a logical and emotionless subject, Mathematics has an uncanny ability to draw passionate responses from young people. It’s seldom a “YAY, I have Math homework” kind of response. No, it’s more like “ARGHHHH, I HATE Math!”

So why would anyone choose Mathematics as a starting point for a youth novel? And note that we are not talking about some mystical and captivating mathematical subject like String Theory or Equations of Relativity. No, sirree! We like a challenge, don’t we? Out with the exciting stuff, so we can sink our teeth into a common, garden-variety subject: fractions. Yes, you read that correctly. FRACTIONS. Not eye-popping fractal mathematics, mind you. Just regular fractions with numerators and denominators: those little number beasts you encountered way back in grade three of four.

Fractions in all their simple glory were the starting point for “Fractonia”. So is this a story about fractions? (Are you yawning and shaking your head in disbelief?) Yes, but probably not in the way you think. When I was at school and fractions were introduced to the class, the teacher talked about picking apples from a tree. (That was in the days when children actually went outside and climbed trees, so students could relate to the image of apples hanging from a tree.) More recently, while doing research for a new project, I took a look at some junior school materials focused on fractions. The apples were gone from the chapters about fractions. In their place, the reference to pizza slices appeared more often than anything else. (It seems the “an apple a day” phrase has been replaced with “a slice of pizza a day“.) Imagine this pizza being cut into pieces. Your friend eats one slice. You eat five slices. What percentage of the pizza is left? Is this stimulating your imagination and encouraging learning, or is it just making you think that you are hungry? Is there a different way to visualize fractions?

I set out to create a story that would give readers an entirely new perspective on fractions. Why? So they could better understand fractions? No – so they could know it is possible for something as “boring” as fractions to become interesting just by changing our perspective. I wanted to paint an imaginative picture over those sad pizza slices with no story to tell – a picture so unexpected that it would encourage readers to create their own imaginative ways to view subjects they found “boring”. In my experience, an interested student is more likely to learn and overcome learning difficulties than a student who is bored with the topic they are studying. In her article entitled, “How the Power of Interest Drives Learning“, Annie Murphy Paul says the following: “When we’re interested in what we’re learning, we pay closer attention; we process the information more efficiently; we employ more effective learning strategies, such as engaging in critical thinking, making connections between old and new knowledge, and attending to deep structure instead of surface features. When we’re interested in a task, we work harder and persist longer, bringing more of our self-regulatory skills into play.” If we learn better when we are interested, why not find a way to make what we have to learn interesting and engaging? It seems like a simple, common-sense way to ensure we learn more and enjoy doing it.

At its core, “Fractonia” is less about Mathematics and more about attitude. Yes, fractions are part of the story, but no, the story is about exploration, discovery, and possibility. It’s about taking responsibility for our own learning. Our learning is not our parents’ or our teachers’ responsibility – it is OURS: yours and mine. We don’t have to wait for our teacher to make the subject exciting or justify why we should study it. We don’t need to be entertained before we can learn something. No, the process of learning is an adventure that can be created and hosted inside your very own imagination. Go on the adventure, or stay home and mope about how boring everything is.

If you have never had a teacher show you HOW to create your very own learning adventure, sit down and read. But don’t read to be entertained – read to discover. Read so the book can become your teacher and show you how to create your own learning adventure. “Fractonia” is my adventure with something as simple as fractions. Other authors will take you on different adventures. You may not be ready to write your adventure in a book, but you are ready to have an adventure. There are no age limits on learning – we never outgrow a good adventure. What will your next adventure be?

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The Dot and the Line: A Romance in Lower Mathematics

The Dot and the Line by Norton JusterLove stories abound, even in the world of Mathematics. Mathematics may not be the first thing that springs to mind when you think of romance, but it does boast its fair share of romantic dramas. Norton Juster took the time to document one such love story in “The Dot and the Line: A Romance in Lower Mathematics”. As in all love stories, there are well constructed characters, although these particular characters cannot claim to be multi-facetted. Meet the “him”: a boring, straight line. There is the “her”: a gorgeous dot. And then there is “the competition”: the bad-boy squiggle. Line falls for dot, but squiggle gets in the way. What is line to do when dot gets tangled up with squiggle? Mathematics holds the key, and line is determined to unlock the solution to his problem.

“The Dot and the Line” was published in 1963, and turned into a short film (shown below) in 1965. Apparently, Norton Juster found inspiration in the mathematics fiction classic “Flatland: A Romance of Many Dimensions” (1884).  “Flatland: A Romance of Many Dimensions” was written by an English school teacher named Dr Edwin Abbott. The story plays out in a two-dimensional world where women are simple line segments and men are the more complicated polygons. It may sound like the kind of geometry lesson that will put hairs on the chest of any women’s libber, but you won’t know for sure until you read it. “Flatland” is a lot more than mathematics in an easy-to-read, story format, yet it remains best known for how it opens up the concept of dimensions and challenges us to explore new perspectives. Sadly, this quaint book went largely unnoticed until Albert Einstein’s general theory of relativity (in which the fourth dimension of time plays a significant role) was published in 1915. Thankfully, someone mentioned this extraordinary book in the same Nature article as Einstein was mentioned (1920), and “Flatland” rose to join other mathematical works of note. I discovered “Flatland” many years ago in the basement of a university library, and hope many more students had the curiosity to pull this book from the dusty shelf after me. If you have not read “Flatland: A Romance of Many Dimensions”, do yourself a favor and borrow or buy a copy.

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Questions and answers are simply mathematics

If you find the study of mathematics dreadfully boring, it is time to play a little. In “Amusements in Mathematics” by Henry Ernest Dudeney (published by David Gaddy, Nov 2011),  plenty of mathematical fun is crammed into 640 pages. According to the author, this collection of puzzles and mathematical problems was created so that the user of the book could tap into the pleasure of “doing math”.

Henry Ernest Dudeney (1857-1930), an English mathematician, is best known as a master of logic puzzles. The author views mathematical puzzles as perplexing questions begging our answers. The reader is drawn into the hunt for solutions and answers to these questions. Asking and answering questions is a part of human life, and comes naturally to us all. When mathematics is viewed as the process of asking and answering questions, we allow ourselves to bypass any existing “number” prejudices and start to enjoy what comes naturally.

Amusements in Mathematics” also includes a discussion on the psychology of puzzles and the application of math in our daily lives.  It is an excellent resource for mathematics teachers seeking a readily accessible collection of “questions” that will spice up a lesson. However, this puzzle book is just as useful to anyone seeking a little mental stimulation – after all, we can all answer questions and should not shy away from the challenge of doing so often. This extensive collection of puzzles and problem-solving exercises is now available from Lulu.com.  The book can be purchased at a saving of 20% until the end of February 2012 using the following coupon code: 20% off books – Enter code FEBBOOKS12 – Save up to $25 – Offer ends 2/29/12
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Stop blaming the parents and teachers for students’ choices.

Peter Hilts, in his article  “Is it time to blame the students?”  addresses an issue that many educators feel uncomfortable expressing opinions on.  In this “politically correct” time that we live in, everyone is afraid to step on toes.  Responsibility is commonly shifted from one person to the next, so no-one has to feel too bad for too long.  Hilts boldly looks at that the statement “all students are all good all the time” and critically evaluates it from the perspective of the teacher.  If students are not just to acquire knowledge as they grow in years, but are also expected to “grow up“, shouldn’t they be taught to shoulder some responsibility for their learning?

If attendance, effort, and integrity are part of the problem in education, it isn’t fair to hold teachers, parents, reformers, unions, politicians, or the tooth fairy responsible,” says Hilts.  He is to be commended for making such a bold stand on a sensitive educational issue.   “Students who give partial or no effort to classwork, exams and standardized tests are mostly or exclusively responsible for their behavior.  When a student who can attend skips instead, that student is responsible.”  Certainly any education system has some disinterested teachers, or teachers who simply hate the work they do but refuse to leave it.  Every society has some parents who actively discourage the educational growth of their children, or who simply don’t care enough to encourage it.  But is it always the teachers and the parents fault when children don’t succeed at school?

As Hilts so insightfully points out, “responsibility has two faces”, and this is as true in the classroom as it is anywhere else.  When a teenage student is offered the opportunity to learn and CHOOSES not to, shouldn’t they be the ones to accept responsibility for that choice?

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Parents are the reason that students cannot hack the Math in Physics?

High school kids cannot use fractions.Another physics teacher told me that students cannot hack the math in physics,” says Stewart Brekke in his article entitled, “Urgent Math Crisis in our Nation: Basic Math Deficits Affect Student Performance in High School Physics and Chemistry“.  Is this an unusual observation for a Physics teacher?  Brekke estimates that the USA “may now have over 100,000 high-school students who do not know fractions and decimals well enough to do high-school physics and chemistry successfully, let alone go on to college and pass a physics or chemistry course.”

There is clearly a problem on our hands – many teens cannot do basic mathematics.  Where do we find the source of this problem?

Stewart Brekke speculates that part of the problem may be attributed to the elementary schools placing too much emphasis on reading skills and not nearly enough on basic arithmetic skills. Japanese elementary school students typically spend two to three times as much time on developing mathematical skills as their American counterparts. The result of this shift in priorities is evident.  Stewart also believes that the “lack of a proper foundation at home” is also a significant contributor to the poor arithmetic skills observed in high school students. Sadly, many children enter first grade without being able to count to ten, and their progress in arithmetic skill development is severely hampered.

It is not that parents do not care, for, on the whole, I have seen them show deep concern about their children’s education, but that many of these parents do not take the time to teach their children number facts nor reading skills. These parents must be informed early that their child’s success in school means that they must start educating their children before they enter kindergarten,” says Brekke.

Education systems all over the world invest vast sums of money into remediation of high school students struggling with poor basic skills. Yet high school Physics and Chemistry classes continue to shrink in size as teenagers avoid confronting the issues that stand in their way of understanding these subjects.  Are we trying to solve a problem instead of preventing it?  What would happen if more of the national or state education investment was used for programs aimed at educating the PARENTS of pre-school children, thus effectively equipping them to help their children develop the basic skills needed for future success at school? 

 Can parents make a difference at home?  All indications are that if parents do not participate in the education process BEFORE their child enters the school system, they may in fact be contributing to their child’s future scholastic failure.

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Is kindergarten too young to study Physics?

Studying Physics in a kindergarten classMany parents of young children have vague (and sometimes not so pleasant) memories of studying Physics during their high school years.  These same parents with their somewhat patchy memories of what matter and energy are, and how these “Physics things” interact, would be astounded to learn that their kindergarten-age children are in fact ready to study Physics.  But isn’t Physics terribly complex with lots of formulae, obscure calculations, and plenty of abstract concepts to glue it all together?  How can a kindergarten-age child possibly study Physics?

 [1]Marxen in her article “Push, Pull, Toss, Tilt, Swing: Physics for Young Children”, explores the role of Physics in the learning process and problem-solving skill development of young children.  Marxen comments that there are “similarities between how children think and learn and how scientists work. Children, like scientists, are theory builders. When children are allowed to construct knowledge by acting on their environment, they expand their understanding, which in turn contributes to their intellectual development.”  So your children are little rocket scientists in disguise, how exactly are they learning and building these theories?

Marxen explains that young children’s Physics experiences usually involve the movement of objects.  For most parents and teachers, “movement of objects” is synonymous with play.  The action is primary and the observation is secondary. Children typically make discoveries about matter and energy through creative play and simple discovery activities in the classroom and at home. For example, something as simple and inexpensive as some small balls and a few sheets of cardboard (that can be folded into ramp-like structures of varying steepness) can invite children to explore concepts that will only be translated into detailed formulae and complex concepts many years down the road for them.  Playing and learning to ask the question “why does that happen” gives these children the opportunity to acquire valuable learning experience.  This experience can be built upon to create a practical knowledge base which will later provide a sturdy foundation to which more complex, abstract Physics knowledge can easily be added.

Are kindergarten children too young to study Physics?  Absolutely not!  Teachers and parents alike can introduce young children to Physics discovery and learning with play-based activities without fear that the children may be overwhelmed or turned off Physics.  Plan playtime or classroom activities that focus on getting the children to experiment and make observations about the world they live in, and you will be well on your way to stimulating a life-long interest in, and appreciation for Physics.

[1]        Carol E. Marxen; Childhood Education, Vol. 71, 1995.

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Engaging the reflective mind

The image of a good problem solver is one of an intelligent person. But years of teaching Physics to intelligent young people has convinced me that intelligence isn’t the only criterion for a successful problem solver. Common sense is perhaps more critical than many have recognized. Without it, an intelligent individual may have a lot of knowledge  and the capacity to make complex connections, but may simply lack the practical wisdom to apply it appropriately. Unfortunately, recognition of the fact that common sense is critical to effective problem solving is where most people stop. Possibly this resistance to dig deeper is due to an underlying belief that if you don’t have a whole lot of common sense to begin with, you are never going to get more.

Daniel Willingham, in a recent article, raises the question of whether common sense can indeed be taught. Willingham debates this question from a psychological perspective and eloquently references psychologist Keith Stanovich who, in his new book What Intelligence Tests Miss, offers a way to understand the difference between intelligence and common sense.   Stanovich sticks to a more traditional definition of intelligence that focuses on the ability to solve problems and make effective decisions. Stanovich suggests that there are three components to the cognitive system that handles these functions: the autonomous mind (which engages in quick thinking based on simple associations and past experiences), the algorithmic mind (which processes information by making comparisons and combining concepts), and the reflective mind (which interprets goals and beliefs and determines appropriate actions to achieve those goals). What most people don’t realize is that typical intelligence tests measure the efficiency of the algorithmic mind, but fail to consider the moderating effect of the reflective mind.

To problem solve effectively, you don’t only need to decide which facts should be combined to generate a solution. You have to test and adapt that selection (made by the algorithmic mind) to the situation at hand. In other words, the solution needs to fit into the environment of the problem, or the solution will never be practical. And this is the job of the reflective mind. According to Willingham, “You need to see your environment for what it is, you need to set realistic goals, and you need to select actions that move you towards those goals.” Intelligent people (categorized this way by typical intelligence tests) don’t always successful engage their reflective minds (the source of common sense) to determine the appropriateness of their solution. The result? Intelligent people are not always naturally good problem solvers. But could they become good problem solvers? To the critical question, “can common sense be taught?” Willingham’s response is “To some extent, yes. With sufficient practice, people can come to recognize the types of errors the reflective mind makes, and learn to avoid them.”

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