## Build Mathematical Skills by Delaying Arithmetic, Part 2

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:

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

## How Crazy Can You Make It?

And here is yet more fun from Education Unboxed. This type of page was always one of my my favorites in Miquon Math.

### Update:

Handmade “How Crazy…?” worksheets are wonderful, but if you want something a tad more polished, I created a printable. The first page has a sample number, and the second is blank so that you can fill in any target:

Add an extra degree of freedom: students can fill in the blanks with equivalent and non-equivalent expressions. Draw lines anchoring the ones that are equivalent to the target number, but leave the non-answers floating in space.

Or don’t draw lines. Let the kids create a worksheet for you to solve. After they finish their expressions, can you figure out which ones connect to the target number?

## PUFM 1.4 Subtraction

Photo by Martin Thomas via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education.

When adding, we combine two addends to get a sum. For subtraction we are given the sum and one addend and must find the “missing addend”.

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

Notice that subtraction is not defined independently of addition. It must be taught along with addition, as an inverse (or mirror-image) operation. The basic question of subtraction is, “What would I have to add to this number, to get that number?”

Inverse operations are a very fundamental idea in mathematics. The inverse of any math operation is whatever will get you back to where you started. In order to fully understand a math operation, you must understand its inverse.

Photo by Luis Argerich via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education.

The basic idea of addition is that we are combining similar things. Once again, we meet the counting models from lesson 1.1: sets, measurement, and the numberline. As homeschooling parents, we need to keep our eyes open for a chance to use all of these models — to point them out in the “real world” or to weave them into oral story problems — so our children gain a well-rounded understanding of math.

Addition arises in the set model when we combine two sets, and in the measurement model when we combine objects and measure their total length, weight, etc.

One can also model addition as “steps on the number line”. In this number line model the two summands play different roles: the first specifies our starting point and the second specifies how many steps to take.

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

## What to Do with a Hundred Chart #27

[Photo by geishaboy500.]

It began with a humble list of 7 things to do with a hundred chart in one of my out-of-print books about teaching home school math. Over the years I added a few new ideas, and online friends contributed still more, so the list grew to its current length of 26. Recently, thanks to several fans at pinterest, it has become the most popular post on my blog:

Now I am working several hours a day revising my old math books, in preparation for publishing new, much-expanded editions. And as I typed in all the new things to do with a hundred chart, I thought of one more to add to the list:

(27) How many numbers are there from 11 to 25? Are you sure? What does it mean to count from one number to another? When you count, do you include the first number, or the last one, or both, or neither? Talk about inclusive and exclusive counting, and then make up counting puzzles for each other.

Can you think of anything else we might do with a hundred chart? Add your ideas in the Comments section below, and I’ll add the best ones to our master list.

## How to Conquer the Times Table, Part 5

Photo of Lex times 11, by Dan DeChiaro, via flickr.

We are finishing up an experiment in mental math, using the world’s oldest interactive game — conversation — to explore multiplication patterns while memorizing as little as possible.

Take your time to fix each of these patterns in mind. Ask questions of your student, and let her quiz you, too. Discuss a variety of ways to find each answer. Use the card game Once Through the Deck (explained in part 3)as a quick method to test your memory. When you feel comfortable with each number pattern, when you are able to apply it to most of the numbers you and your child can think of, then mark off that row and column on your times table chart.

So far, we have studied the times-1 and times-10 families and the Commutative Property (that you can multiply numbers in any order). Then we memorized the doubles and mastered the facts built on them. And then last time we worked on the square numbers and their next-door neighbors.

## How to Conquer the Times Table, Part 4

Photo of Miss Karen (and computer) times 3, by Karen, via flickr.

If you remember, we are in the middle of an experiment in mental math. We are using the world’s oldest interactive game — conversation — to explore multiplication patterns while memorizing as little as possible. So far, we have studied the times-1 and times-10 families and the Commutative Property (that you can multiply numbers in any order). Then we memorized the doubles and mastered the facts built on them.

## How to Conquer the Times Table, Part 3

Photo of Javier times 4, by Javier Ignacio Acuña Ditzel, via flickr.

If you remember, we are in the middle of an experiment in mental math. We are using the world’s oldest interactive game — conversation — to explore multiplication patterns while memorizing as little as possible. Talk through these patterns with your student. Work many, many, many oral math problems together. Discuss the different ways you can find each answer, and notice how the number patterns connect to each other.

So far, we have mastered the times-1 and times-10 families and the Commutative Property (that you can multiply numbers in any order).

## How to Conquer the Times Table, Part 2

Photo of Eeva times 6, by Eric Horst, via flickr.

The question is common on parenting forums:

My daughter is in 4th grade. She has been studying multiplication in school for nearly a year, but she still stumbles over the facts and counts on her fingers. How can I help her?

Many people resort to flashcards and worksheets in such situations, and computer games that flash the math facts are quite popular with parents. I recommend a different approach: Challenge your student to a joint experiment in mental math. Over the next two months, without flashcards or memory drill, how many math facts can the two of you learn together?

We will use the world’s oldest interactive game — conversation — to explore multiplication patterns while memorizing as little as possible.

## Game: Target Number (or 24)

[Photo by stevendepolo.]

Math concepts: addition, subtraction, multiplication, division, powers and roots, factorial, mental math, multi-step thinking
Number of players: any number
Equipment: deck of math cards, pencils and scratch paper, timer (optional)

### Set Up

All players must agree on a Target Number for the game. Try to choose a number that has several factors, which means there will be a variety of ways to make it. Traditionally, I start my math club students with a target of 24.

Shuffle the deck, and deal four cards face down to each player. (For larger target numbers, such as 48 or 100, deal five or six cards to each player.) The players must leave the cards face down until everyone is ready. Set the remainder of the deck to one side.