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!
- Download the activity sheet with warm-up and follow-up problems.
Build Mathematical Skills by Delaying Arithmetic, Part 4
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
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
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.
Sample The Moscow Puzzles
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.”
More Free Math from Dover Publications
Olympic Logic
I love logic puzzles! Nrich Maths offers four fun Olympics Logic puzzles. And be sure to check out the rest of their Nrich Olympics Math as well.
Medals Count
Given the following clues, can you work out the number of gold, silver and bronze medals that France, Italy and Japan got in this international sports competition?
- Japan has 1 more gold medal, but 3 fewer silver medals, than Italy.
- France has the most bronze medals (18), but fewest gold medals (7).
- Each country has at least 6 medals of each type.
- Italy has 27 medals in total.
- Italy has 2 more bronze medals than gold medals.
- The three countries have 38 bronze medals in total.
- France has twice as many silver medals as Italy has gold medals.
Princess in the Dungeon Game
Yet more fun from Rosie at Education Unboxed. I found these while looking for videos to use in my PUFM Subtraction post. Rosie says:
This is seriously embarrassing and I debated whether to put this video online or not because this is NOT my normal personality, but my girls made up this game and will play it for over an hour and ask for it repeatedly… so I figured someone out there might be able to use it with their kids, too.
If you know me, please don’t ever ask me to do this in public. I will refuse.
Princess in the Dungeon, Part 1 – Fractions with Cuisenaire Rods
How Long is 10! Seconds?
A fun exploration for upper elementary or middle school students, from Numberphile:
Thinking (and Teaching) like a Mathematician
Most people think that mathematics means working with numbers and that being “good at math” means being able to do (only slower) what any $10 calculator can do. But then, most people think all sorts of silly things, right? That’s what makes “man on the street” interviews so funny.
Numbers are definitely part of math — but only part, and not even the biggest part. And being “good at math” means much more than being able to work with numbers. It means making connections, thinking creatively, seeing familiar things in new ways, asking “Why?” and “What if?” and “Are you sure?”
It means trying something and being willing to fail, then going back and trying something else. Even if your first try succeeded — or maybe, especially if your first try succeeded. Just knowing one way to do something is not, for a mathematician, the same as understanding that something. But the more different ways you know to figure it out, the closer you are to understanding it.
Mathematics is not just memorizing and following rules. If we want to teach real mathematics, we teachers need to learn to think like mathematicians. We need to see math as a mental game, playing with ideas. James Tanton explains:
Continue reading Thinking (and Teaching) like a Mathematician
Why Every Proof that .999… = 1 is Wrong
Vi Hart repents with an update to her last video: “Take that, mathematics!”
Denise Gaskins' Let's Play Math 