Problem-Solving Poll: What’s Your Answer?

Posted on by Denise Gaskins

[Photo by Alex E. Proimos via flickr.]

Patrick Vennebush, author of Math Jokes 4 Mathy Folks (the book and the blog) wants to know how you and your children would answer a tricky math problem.

He’s taking a poll:

I have often heard that, “Good teachers borrow, great teachers steal.” So today, I am stealing one of Marilyn Burns’s most famous problems. She takes this problem to the streets, and various adults give lots of different answers. When I’ve used it in workshops, even among a mathy crowd, I get lots of different answers, too.

What’s your answer?

“A man buys a truck for $600, then sells it for $700. Later, he decides to buy it back again and pays $800 dollars. However…”

Go to Patrick’s blog to read the whole problem and submit your answer. Let everybody in the family try it!

Update: Patrick posted the solution and percentages correct for students of various ages.

Lockhart’s Measurement

Posted on by Denise Gaskins

After watching the video on the Amazon.com page, this book has jumped to the top of my wish list.

You may have read Paul Lockhart’s earlier piece, A Mathematician’s Lament, which explored the ways that traditional schooling distorts mathematics. In this book, he attempts to share the wonder and beauty of math in a way that anyone can understand.

According to the publisher: “Measurement offers a permanent solution to math phobia by introducing us to mathematics as an artful way of thinking and living. Favoring plain English and pictures over jargon and formulas, Lockhart succeeds in making complex ideas about the mathematics of shape and motion intuitive and graspable.”

If you take any 4-sided shape at all — make it as awkward and as ridiculous as you want — if you take the middles of the sides and connect them, it always makes a parallelogram. Always! No matter what crazy, kooky thing you started with.

That’s scary to me. That’s a conspiracy.

That’s amazing!

That’s completely unexpected. I would have expected: You make some crazy blob and connect the middles, it’s gonna be another crazy blob. But it isn’t — it’s always a slanted box, beautifully parallel.

WHY is it that?!

The mathematical question is “Why?” It’s always why. And the only way we know how to answer such questions is to come up, from scratch, with these narrative arguments that explain it.

So what I want to do with this book is open up this world of mathematical reality, the creatures that we build there, the questions that we ask there, the ways in which we poke and prod (known as problems), and how we can possibly craft these elegant reason-poems.

— Paul Lockhart
author of Measurement

Rate × Time = Distance Problems

Posted on by Denise Gaskins

I love how Richard Rusczyk explains math problems. It’s a new school year, and that means it’s time for new MathCounts Mini videos. Woohoo!

Continue reading Rate × Time = Distance Problems

Sample from the Introduction to Mathematical Thinking Class

Posted on by Denise Gaskins

I’m really looking forward to Keith Devlin’s free Introduction to Mathematical Thinking class, which starts in mid-September. There are more than 30,000 nearly 40,000 students signed up already. Will you join us?

These days, mathematics books tend to be awash with symbols, but mathematical notation no more is mathematics than musical notation is music.

A page of sheet music represents a piece of music: the music itself is what you get when the notes on the page are sung or performed on a musical instrument. It is in its performance that the music comes alive and becomes part of our experience. The music exists not on the printed page but in our minds.

The same is true for mathematics. The symbols on a page are just a representation of the mathematics. When read by a competent performer (in this case, someone trained in mathematics), the symbols on the printed page come alive — the mathematics lives and breathes in the mind of the reader like some abstract symphony.

— Keith Devlin
Introduction to Mathematical Thinking

Build Mathematical Skills by Delaying Arithmetic, Part 4

Posted on by Denise Gaskins

To my fellow homeschoolers,

While Benezet originally sought to build his students’ reasoning powers by delaying formal arithmetic until seventh grade, pressure from “the deeply rooted prejudices of the educated portion of our citizens” forced a compromise. Students began to learn the traditional methods of arithmetic in sixth grade, but still the teachers focused as much as possible on mental math and the development of thinking strategies.

Notice how waiting until the children were developmentally ready made the work more efficient. Benezet’s students studied arithmetic for only 20-30 minutes per day. In a similar modern-day experiment, Daniel Greenberg of Sudbury School discovered the same thing: Students who are ready to learn can master arithmetic quickly!

Grade VI

[20 to 25 minutes a day]

At this grade formal work in arithmetic begins. Strayer-Upton Arithmetic, book III, is used as a basis.

[Note: Essentials of Arithmetic by George Wentworth and David Eugene Smith is available free and would probably work as a substitute.]

The processes of addition, subtraction, multiplication, and division are taught.

Care is taken to avoid purely mechanical drill. Children are made to understand the reason for the processes which they use. This is especially true in the case of subtraction.

Problems involving long numbers which would confuse them are avoided. Accuracy is insisted upon from the outset at the expense of speed or the covering of ground, and where possible the processes are mental rather than written.

Before starting on a problem in any one of these four fundamental processes, the children are asked to estimate or guess about what the answer will be and they check their final result by this preliminary figure. The teacher is careful not to let the teaching of arithmetic degenerate into mechanical manipulation without thought.

Fractions and mixed numbers are taught in this grade. Again care is taken not to confuse the thought of the children by giving them problems which are too involved and complicated.

Multiplication tables and tables of denominate numbers, hitherto learned, are reviewed.

— L. P. Benezet
The Teaching of Arithmetic II: The Story of an experiment

Continue reading Build Mathematical Skills by Delaying Arithmetic, Part 4

Build Mathematical Skills by Delaying Arithmetic, Part 3

Posted on by Denise Gaskins

To my fellow homeschoolers,

How can our children learn mathematics if we delay teaching formal arithmetic rules? Ask your librarian to help you find some of the wonderful living books about math. Math picture books are great for elementary students. Check your library for the Time-Life “I Love Math” books or the “Young Math Book” series. You’ll be amazed at the advanced topics your children can understand!

Benezet’s students explored their world through measurement, estimation, and mental math. Check out my PUFM Series for mental math thinking strategies that build your child’s understanding of number patterns and relationships.

Grade IV

Still there is no formal instruction in arithmetic.

By means of foot rules and yard sticks, the children are taught the meaning of inch, foot, and yard. They are given much practise in estimating the lengths of various objects in inches, feet, or yards. Each member of the class, for example, is asked to set down on paper his estimate of the height of a certain child, or the width of a window, or the length of the room, and then these estimates are checked by actual measurement.

The children are taught to read the thermometer and are given the significance of 32 degrees, 98.6 degrees, and 212 degrees.

They are introduced to the terms “square inch,” “square foot,” and “square yard” as units of surface measure.

With toy money [or real coins, if available] they are given some practise in making change, in denominations of 5’s only.

All of this work is done mentally. Any problem in making change which cannot be solved without putting figures on paper or on the blackboard is too difficult and is deferred until the children are older.

Toward the end of the year the children will have done a great deal of work in estimating areas, distances, etc., and in checking their estimates by subsequent measuring. The terms “half mile,” “quarter mile,” and “mile” are taught and the children are given an idea of how far these different distances are by actual comparisons or distances measured by automobile speedometer.

The table of time, involving seconds, minutes, and days, is taught before the end of the year. Relation of pounds and ounces is also taught.

— L. P. Benezet
The Teaching of Arithmetic II: The Story of an experiment

Continue reading Build Mathematical Skills by Delaying Arithmetic, Part 3

Cool Fibonacci Conversion Trick

Posted on by Denise Gaskins
photo by Muffet via flickr

Maria explains how to use the Fibonacci Numbers to convert distance measurements between miles and kilometers:

P.S.: Congratulations to Maria for her Math Mammoth program being featured in the latest edition of Cathy Duffy’s 100 Top Picks for Homeschool Curriculum! And Home School Buyer’s Co-op has a sale on Cathy Duffy’s book through the end of July.

Build Mathematical Skills by Delaying Arithmetic, Part 2

Posted on by Denise Gaskins

To my fellow homeschoolers,

Most young children are not developmentally ready to master abstract, pencil-and-paper rules for manipulating numbers. But they are eager to learn about and explore the world of ideas. Numbers, patterns, and shapes are part of life all around us. As parent-teachers, we have many ways to feed our children’s voracious mental appetites without resorting to workbooks.

To delay formal arithmetic does not mean that we avoid mathematical topics — only that we delay math fact drill and the memorization of procedures. Notice the wide variety of mathematics Benezet’s children explored through books and through their own life experiences:

Grade I

There is no formal instruction in arithmetic. In connection with the use of readers, and as the need for it arises, the children are taught to recognize and read numbers up to 100. This instruction is not concentrated into any particular period or time but comes in incidentally in connection with assignments of the reading lesson or with reference to certain pages of the text.

Meanwhile, the children are given a basic idea of comparison and estimate thru [sic] the understanding of such contrasting words as: more, less; many. few; higher, lower; taller, shorter; earlier, later; narrower, wider; smaller, larger; etc.

As soon as it is practicable the children are taught to keep count of the date upon the calendar. Holidays and birthdays, both of members of the class and their friends and relatives, are noted.

— L. P. Benezet
The Teaching of Arithmetic II: The Story of an experiment

Continue reading Build Mathematical Skills by Delaying Arithmetic, Part 2

Build Mathematical Skills by Delaying Arithmetic, Part 1

Posted on by Denise Gaskins

To my fellow homeschoolers,

It’s counter-intuitive, but true: Our children will do better in math if we delay teaching them formal arithmetic skills. In the early years, we need to focus on conversation and reasoning — talking to them about numbers, bugs, patterns, cooking, shapes, dinosaurs, logic, science, gardening, knights, princesses, and whatever else they are interested in.

In the fall of 1929 I made up my mind to try the experiment of abandoning all formal instruction in arithmetic below the seventh grade and concentrating on teaching the children to read, to reason, and to recite — my new Three R’s. And by reciting I did not mean giving back, verbatim, the words of the teacher or of the textbook. I meant speaking the English language.

— L. P. Benezet
The Teaching of Arithmetic I: The Story of an experiment

Continue reading Build Mathematical Skills by Delaying Arithmetic, Part 1

Sample The Moscow Puzzles

Posted on by Denise Gaskins

Dover Publications is offering a free sample chapter from The Moscow Puzzles.

Cat and Mice
Purrer has decided to take a nap. He dreams he is encircle by 13 mice: 12 gray and 1 white. He hears his owner saying: “Purrer, you are to eat each thirteenth mouse, keeping the same direction. The last mouse you eat must be the white one.”

Download the sample chapter from The Moscow Puzzles.

More Free Math from Dover Publications