The Joy of Mathematics

Prof Arthur Benjamin: The Joy of MathematicsThat the mysteries of mathematics can put a smile on your face may seem far-fetched to some. It is not, however, a strange idea to  Professor Arthur T. Benjamin of Harvey Mudd College. He loves mathematics and is certainly the most enthusiastic mathematics teacher I have ever seen in action. I purchased “The Joy of Mathematics” for a family member, but ended up sitting down to watch the lectures with the gift recipient. I don’t recall ever seeing a maths teacher with so much energy and eagerness to share what he knows. If you are not terrified by numbers (and even if they do scare you a little), you are ready for the magic of mathematics presented in a way only a passionate and down-to-earth mathematician can do it.

The Joy of Mathematics” consists of 24 half-hour lectures that celebrate the sheer joy of mathematics. The lectures are taught by a mathematician who is also a magician, with numbers and the odd rabbit in the hat. Professor Benjamin is renowned for his feats of mental calculation performed before audiences at schools, theaters, museums, and conferences. Yes, you will be in awe of him as you watch him process numbers at speeds that will make you dizzy. But unlike many other magicians, the professor isn’t selfish with his mathematical magic tricks. In this course, Professor Benjamin lets you in on some of the secrets of his wizardry. You may be astounded to learn how easy it is to perform some calculations you once thought would need more than your pocket calculator. By the end of the course, you may decide your new bag of tricks has turned you into something of a minor maths wizard yourself.

Who can benefit from this course? This course is a great refresher course for those who have lost touch with their high school mathematics training and want to feel more comfortable around numbers. It’s also a great warm-up to college entry-level mathematics, and precursor to advanced algebra and basic calculus. For those high school students who feel threatened by mathematics, time spent working through the examples given during the lectures will deepen understanding and boost confidence. Even if you have a degree in mathematics, this course is sure to work out your mental muscles in a way that is both enjoyable and enlightening.

While the complexity of the material presented is not beyond the abilities of average high school students in beginner algebra courses, be warned that at times Professor Benjamin’s enthusiasm causes him to accelerate. If he starts presenting new concepts too quickly, don’t be afraid to reach for the remote and rewind… over and over again, until you feel comfortable enough to move on.  In fact, Professor Benjamin is well aware that his audience is not a university class and will remind you to use your remote from time to time. Take his advice to get the most out of these lectures. Take each lecture at your own pace. Repeat each lecture a few times to let the explanations sink in. I also recommend keeping a pen and the notes that accompany the DVDs handy. Add your own personal notes to those provided with the lectures. You will learn a lot during each short lecture, and personal notes will help you reflect better on what you have learned before you start a new topic. Don’t move on to a new lecture until you feel like you “got it”, or at least got most of it. If you progress too quickly without understanding and internalizing the material, the more advanced lectures may be lost on you.

The Joy of Mathematics” is currently available from The Great Courses, a company that strives to make high quality teaching and the skills of university professors accessible to the general public. (Photo credit: Richard Faverty)


Graphing Can Be Fun

Desmos Free Graphing CalculatorIf mathematical graphs are generally not your cup of tea, perhaps you should try a cup of coffee instead? How about a cup of coffee that is sketched from a collection of equations your students might recognize? That’s not quite your textbook flavor of graphing mathematics, is it?

Desmos offers a remarkable and FREE online graphing calculator that allows you to visualize mathematical equations quickly and easily. Imagine turning equations into images everyone can identify and associate with? According to Desmos, “learning is a process of exploration and discovery, not a series of answers”.  The students who explore tend to be the students who learn, and they have fun doing it. The creators of this extraordinary online calculator believe that “everyone can learn and enjoy math, given the right environment.” The website creates just that environment. Using the available tools. you can upload your own graphs so they are displayed on the screen. The option to change colours aids students in relating the mathematical equations to what they see on the screen.

Why not make your next lesson about graphs an exercise in discovery learning for your homeschooled child or classroom students? If you believe that “math is beautiful and surprisingly fun”, you will enjoy playing with the free online graphing calculator that Desmos has made available. Give it a try and let me know how you experienced the calculator.


Greater Than Less Than Game

Have you played the “Greater Than Less Than Game“? Most people have played some version of the game, but probably don’t recall what the game was called. This game has a myriad of different names, but the one I like best is The Geetee Eltee Game. It is a basic game with huge potential for the classroom and home learning environments. While you can invest some cash and purchase specialist flashcards to play the game, all you really need is a deck of playing cards. While the directions below are written for a classroom environment, they can easily be adapted for homeschooling odd or even-sized groups of students. Yes, this game even works if you are homeschooling a single child. All you need is a student, a willing Dad or Mom, and a deck of cards.

What is the purpose of The Geetee Eltee Game?

The Greater Than Less Than Game familiarizes students (K-1) with number ranking and the concepts of “greater than” and “less than“. The non-threatening environment of the game encourages children to rehearse the skill of number evaluation and comparison for extended periods of time, often far in excess of what they would commit to when completing worksheets or similar activities. Often avoidance of the extended practice of basic mathematical skills is what keeps students from developing competence in those skills. By cloaking the skill development in a game, most students have fun and are totally oblivious to the fact that they are doing basic mathematics.

What will I need to play the game?

One deck of playing cards per two students in the class.

How do we play the game?

1.  Remove the jokers, aces, jacks, queens, and kings from the pack. Number values can be allocated to these cards and they can be reintroduced at a later stage, but it is best that the students become comfortable with the game before adding this extra level of complexity.

2.  Have everyone in the class pair up. (Where necessary, use a group of three so everyone participates.)

3.  Allocate a pack of cards to each pair of students, and divide the pack equally between the two students. The task of shuffling and dividing the pack in half can be given to each group, if time allows. Students must keep their half-pack of cards face downwards. No peeping at the cards before or during the game.

4.  The students take turns to count to three. At the count of 3, each student flips their top card to reveal its number. The student with the highest value takes both cards and puts them at the bottom of their pack. What happens if both students reveal cards with the same number? In the case of a tie, each student returns their card to the bottom of their pack and the game continues.

 5.  The game ends when one of the two students has taken all the cards. If you see that you will run out of class time before the majority of students have finished the game, stop everyone 5 minutes before the end of the class. Have each student count their cards. The student with the most cards in each group is the winner.

Warning: Be prepared for a little noise. Students may forget that they are “learning” and become quite excited while playing.



Learning Math the Natural Way

"Memory Tips for Math" by D. YatesLet’s be honest. Not every child is a natural mathematician. Encouragement and praise is great, but the help it provides can never substitute for real, hands-on, practical help that makes the mathematics learning experience effective. Many children try their best, yet their efforts leave them caught in a constant struggle to grasp the mathematical concepts. Their frustration is evident as they become tongue-tied with the heavy math jargon. How can a parent or teacher reach those children who are not natural logical or mathematical learners? How does one bring the best out of the kids who don’t naturally thrive on numbers and logic problems? In her book, “Memory Tips for Math, Memorization and Learning Styles: The Successful Way to Teach K-5 Math” Donnalyn Yates proposes a practical and creative solution that will take a lot of the “ouch” out of math class.

Memory Tips for Math, Memorization and Learning Styles” recognizes that the three most common perceptual learning styles are visual, auditory and tactile/kinesthetic. Learning activities in the book focus on providing these categories of learners with stimulation that leads to effective learning. Acronyms, pictures, rhymes, and stories help students to develop vocabulary and retain mathematical procedures. For example, think about how you learned the relationship between the gallon, quart, pint and cup. Now imagine if you had discovered this relationship through a story of fantasy. Imagine the Kingdom of Gallon in which lived three queens of the family of Quart. Each Queen Quart lived in a castle with a young prince and princess – they’re the Pints. Prince and Princess Pint don’t have children but each of the Pints has 2 cats – the cats are the Cups. Imagine how much fun you might have had in Math class if you learned using the tools provided in “Memory Tips for Math, Memorization and Learning Styles: The Successful Way to Teach K-5 Math”. As you read the creative examples, don’t be surprised to find yourself conjuring up a few inspired examples of your own to help your child or student learn more effectively.

This book can be purchased at a discount of 30% for a limited period. Use coupon code FEBRUARYCART305USD at check out. The coupon expires on 19 February 2012.



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  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


Are Puzzles Too Old Fashioned for Modern Kids?

Puzzle building is a lost art, pushed aside by electronic gaming and dvd watching. Should you as a parent or teacher make any attempt to resurrect this lost art? Experts in the field of early learning tell us that young learners will benefit significantly from opening that puzzle box and putting the pieces together.

According to the article “Puzzles and Games for Preschoolers” by Alvin Poussaint, M.D. and Susan Linn, Ed.D., puzzles serve various educational functions in the development of young learners. Standing head and shoulders above the other advantages of playing with puzzles, is the fact that puzzle-building helps kids develop problem solving skills. And who doesn’t want a child who can think for herself and figure out solutions to every day problems?

Problem solving is a skill that goes well beyond the realms of mathematics and science. Without the ability to problem solve, relationships become dysfunctional and workplaces become a source of deadly stress. According to psychologist Dr. Jeffrey Bernstein in his article “Two Essential Skills for an Emotionally Healthy Life“, the ability to problem solve is critical for effective management our lives. Why then would anyone withhold opportunities for their children or students to develop this vital skill?

Poussaint and Linn suggest that trying to fit the puzzle pieces together helps children “learn the value of flexible thinking and persistence”. Moving and placing the pieces develops fine motor skills. Puzzle building also stimulates deductive reasoning and inference. The process of assembling a puzzle helps children understand that big things can usually be broken down into smaller parts. This realization is a critical element in successful problem solving.

Is it time for our children to put aside the gaming console for a little while and pick up an old fashioned box of puzzle pieces? To help young learners face life with well developed life skills, you cannot afford to procrastinate. Unpack that old jigsaw puzzle today.


Can I make my own linear equation jigsaw puzzle?

Converting your “less than exciting” linear equation worksheet activities to puzzles is possible with a small investment in some supplies and a slightly larger investment in time. To get started, find some sturdy, colorful cardboard, a ruler, black marker, a pair of scissors, self-adhesive lamination plastic, an elastic band, and a set of solved linear equations. Dark colored card tends to be difficult to read equations from, so avoid dark colors unless you are working with white or silver metallic markers. While I prefer working with black markers, any colored pens that create contrast with the background card color will work.

Your puzzle can take any form, but simple shapes are the easiest to work with. For beginners, I recommend a square puzzle with no more than 9 pieces. Use the ruler to mark out a square on the card, and divide the square into a table with 3 equally spaced columns and 3 equally spaced rows. If you are feeling less ambitious, start with a 2×2 table. Make sure the lines and boundaries of the puzzle are drawn in bold ink. Neatly insert an equation along a side of a cell that has an adjacent cell. The solution to this equation is filled in across the boundary line in the neighboring cell. Try not to choose equations that will generate the same solution as this may cause confusion for learners who are new to this type of puzzle building. Duplicate solutions can be intentionally incorporated into more challenging puzzles, but should be avoided when first introducing these puzzles to a class or student. Complete the puzzle with equations and solutions, laminate the card for durability, and cut the puzzle into its individual pieces. Use the elastic band to keep the puzzle pieces together.

If you are new to putting the pieces back together, or want to know how to explain the process to your students, read “How to Solve a Linear Equation Jigsaw Puzzle“.


How do I build a linear equation jigsaw puzzle?

Unlike traditional “picture” jigsaw puzzles, linear equation jigsaw puzzles are largely blank. There are seldom patterns or background images to guide you as you put the pieces together. Although traditionally rectangular, some puzzles are designed to take on unexpected shapes when completed. However, these “shaped” puzzles are not usually sold with obvious clues that will allow the puzzle builder to construct the puzzle using only the goal of a particular shape. There are no short-cuts, cheat-sheets, or ways to avoid solving the equations. If you want to build the puzzle, you must first solve the equations printed on the puzzle pieces – they alone hold the keys to putting the puzzle together. If you are new to linear equation jigsaw puzzles, and need some help getting starting, read “How to Solve a Linear Equation Jigsaw Puzzle“.


Are linear equation jigsaw puzzles only for classroom use?

Puzzles are enjoyed by everyone from grandma to your toddler. Classrooms certainly don’t have exclusive rights over them. While perfect for building class spirit and developing team work skills within the classroom environment, linear equation jigsaw puzzles are even more useful at home.

If you home school your children, integrate the puzzles in your home school lesson plan to spice up traditional algebra lessons. Parents with children who show reluctance to do their algebra homework can encourage an interest in the subject by introducing these puzzles as part of a reward system. For example, for every 3 traditional equation worksheets completed, the child could earn the opportunity to complete a linear equation jigsaw puzzle instead of a worksheet.

Does your family enjoy building puzzles together? Take it to the next level by completing a linear equation jigsaw puzzle as a family. This type of family activity helps encourage an appreciation for mathematics, and teaches children that the topics they deal with “in school” are not for exclusive use in school.


Solving Linear Equations is the Game to Play

Linear Equation Jigsaw PuzzlesBad experiences with linear equations can brand algebra as the subject to hate. Many a middle-schooler has adopted this negative attitude during the early stages of exposure to algebra. As countless high school mathematics teachers will attest, changing this perception of algebra as being a “boring and difficult” subject is not easy. But solving linear equations need not be the doorway to mathematical doom and darkness. Repetition is certainly necessary to develop problem-solving skill, but the unexciting repetition that usually kills any interest in algebra can be presented as something fun and challenging. How this goal is achieved is limited only by the imagination of the teacher. A favorite for me is to present the activity of solving linear equations as a jigsaw puzzle. The puzzle may be offered as an individual or a team challenge, depending on the objectives of the teacher.

Linear equation jigsaw puzzles are the game-players’ alternative to solving pages of equations. These puzzles take advantage of all the skill-developing attributes of puzzle building, but do this on top of developing algebraic problem-solving skills. Before the second piece of the puzzle can be laid, a linear equation must be solved. While some students may be hesitant to embrace the challenge of a page of linear equations begging solutions, few will back down from the chance to build a puzzle.

The rate at which the puzzle can be built is primarily determined by the speed at which the student can solve the linear equations. The linear equation jigsaw puzzle therefore has the potential to serve as an informal speed test, but teachers should be cautioned against using these puzzles for formal tests. Since jigsaw puzzle building depends on the hand-eye co-ordination and the spatial perception of the students, timed test results may be indicative of more than just the student’s ability to solve linear equations.