Category Archives: Resources and methods

Mathematics: A Human Endeavor (Textbook review)

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This book is amazing.  Written by Harold R. Jacobs almost 50 years ago, it is a blessing that the content will not likely become out-dated.  It presumes nothing more than fluency with arithmetic from the learner, but covers a range of topics that span pre-algebra to advanced algebra.  The subtitle is “A Textbook For Those Who Think They Don’t Like The Subject”, but in my opinion it is simply a textbook for everyone (including those who think they don’t like the subject!)

First and foremost, this book goes beyond interesting into downright fascinating.  It takes the learner on a safari of mysterious patterns, both in abstract and in nature.  It unveils the beauty of geometric solids and mathematical curves.  It inspires awe in large numbers, and delight in mathematical tricks.

Despite all this depth, it is accessible.  The chapters are short and the prose is comfortable and inviting to read.  Much of the instruction happens in the well-crafted exercises.  The learner is allowed to discover many concepts for themselves, so that the learning feels more like inquiry than instruction.

The treatment of logarithms in the fourth chapter is particularly impressive.  In my years as a tutor I have often needed to provide mathematical triage to learners who become hopelessly confused by this or the other explanation of logarithms.  This book puts to rest any question of whether logs can be explained in a clear and intuitive manner.  The author draws on the learners comprehension of arithmetic and geometric series and gradually leads the learner through the use of logarithmic patterns and functions without introducing the obscuring notation until the concept is solidly established.

The book is also filled with “experiments” which range from abstract inquiries to mathematical arts and crafts.  I opened the book expecting to build a class around it, only to find that the class is already built.  All we need is some graph paper, compasses, rulers and a room full of minds ready to learn!

 

Base 10 Toothpicks

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Practicing our number system with a tangible model is an invaluable way for learners to internalize base 10.  I remember how valuable base 10 blocks were for me in second and third grade.  A few years ago someone pointed out that toothpicks work great as a base ten manipulative and I’ve been using them regularly in my math tutoring.  Searching around just now to see if anyone else had blogged about this idea, I found this great blog post about why pop blocks that can be broken apart work better than base 10 blocks as a manipulative for learners who are still mastering base 10.

Here’s how you can do it with toothpicks.

Base Ten Toothpicks Supplies

 

Get about 500 toothpicks for each student who will be using them at one time.  Get some of the tiny hair rubber bands and some larger rubber bands. 

 

 

Ask learners to help you construct the base 10 toothpick set by making some bundles of 10.  Even if you already have bundles made by previous students, it’s great to start learners off with these by letting them construct at least some bundles of their own.

When hundreds are needed, they can be created by bundling 10 of the 10-bundles together with a bigger rubber band.

As with the pop blocks, a great thing about these is that they can be physically taken apart to aid in modeling subtraction and division, or to easily create numbers close to 10 or close to 100.

Plus, they fit in my cute manipulatives bag!             

A Review of “Calculus By and For Young People”

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Calculus By and For Young People by Don Cohen is one of my most frequently used resources these days.

I use this book mainly for two types of learners.  The first group are those interested in math for it’s own sake and who are working at a fairly advanced level for their age.  The second type is any math learner who could use a playful and fascinating way to master fractions.

I love so many things about this book.  And it’s not just me.  Much of what I find delightful in the book matches what kids love and are drawn to.

Kids love the idea of the infinite.  I mean…. who doesn’t?… but for those young minds the fascination is fresh and all the more captivating.  It is tragic to make these fascinated and inquisitive learners wait all the way until college, by which time many of them have been tortured by far too much algebra drudgery, to discover the magic of the infinite mingling with the finite.  Through working one-on-one with kids himself, Don created a series of explorations that are accessible to anyone who can do basic arithmetic.  They guide learners into discoveries of infinite series and limits.

Kids enjoy finding patterns and often find them more easily than adults.  This book takes great advantage of that, leading the learner to discover patterns in numbers and in the world.

Kids love solving puzzles.  A main section of the chapter on functions relies on two fascinating puzzles – the peg puzzle, and the Towers of Hanoi – for exploring different kinds of functions.  It’s great because kids get to solve the puzzles first and then discover that there is a mathematical pattern underlying the moves they have been making.

Kids love being silly.  Scattered throughout the book are playful tidbits.  1/2 is sometimes called “one-twoth” and the chapter on functions is entitled “On Thin Spaghetti and Nocturnal Animals”.

Kids like to be challenged.  The depth of this book is incredible.  You will be hard pressed to find anyone who will have trouble finding a difficult task.  At the same time, no one who can do a little arithmetic will find the easiest challenges to be inaccessible.

There’s a lot more I could say, but go look at the book, it will speak for itself.  Feel free to ask me questions about my experience teaching with it!