Strategies for basic addition and subtraction facts

In this video I show several strategies for learning the basic addition and subtraction facts:

• finding all the possible sums for a certain number; e.g. for 6, these would be 0 + 6, 1 + 5, 2 + 4, and 3 + 3;
• writing out all the fact families where the sum is a certain number;
• the 9 "trick"; 9 + 7 is the same as 10 + 6 (9 wants so badly to be 10 so that it "steals" one from 7);
• doubles and the accompanying addition facts;
• number rainbows and how to use them to practice subtraction;
• a structured drill for addition facts.
  

Enjoy!



Strategies for Addition and Subtraction Facts - Video

Multiplication vs. addition once more

Keith Devlin has published another column along the lines of multiplication not being repeated addition. I feel quite honored that he mentions THIS blog in his column (scroll down to the end), referring to what I wrote about the issue.

This time he expounds on research results. The research clearly shows that thinking of multiplication as repeated addition hinders students' further understanding of mathematics. It can lead to the misconception that multiplication always makes things bigger. Children need to acquire multiplicative reasoning, which is different from additive reasoning. And so on. Go read it yourself.

Multiplication as many groups of the same size

It's been very good and educational for me to refine my thinking on multiplication vs. addition by reading some recent posts around the blogosphere, especially What's wrong with repeated addition by Denise and Devlin's Right Angle Finale at Text Savvy.

I feel that on some blogs people aren't even exactly talking about the same thing. The subjet we're dealing with - is multiplication repeated addition or not? - is subtle. Some people talk about how to define it - it is defined in some systems as repeated addition, and they feel that closes the issue.

BUT, I tend to agree with what Denise wrote: multiplication is a different operation from addition and somehow we need to get students to view it that way. I've always known that; I've never thought anything different. But yet how we present things to children is not always easy; we may understand the idea but not able to convey it right. Talking about multiplication as repeated addition MAY indeed leave the impression in children's minds that "multiplication reduces to addition" or, as Denise put it, it is a "subspecies of addition".

So, to try to summarize what I've been mulling over in my mind:

* Multiplication is simply a DIFFERENT operation from addition. It's not a "subspecies" of addition or some special kind of addition. AND, we need to stress that in our teaching.

Denise offers defining multiplication for kids as "counting by groups" - should be same-size groups, of course. In teaching that, we need to emphasize the meanings attached to the two factors: in M × N, M would be the amount of groups, and N would be the number of elements in one group. M is called the multiplier, and N is the multiplicand.

We need to emphasize the difference between additive and multiplicative situations in word problems:

"Mark has two baskets and each basket has five appples." => this is a multiplication situation
"Mark has five apples in one basket and five in another." => this is an addition situation.

Joshua says, "They are different ideas, fundamentally. The "processes" of finding a product and finding a repeated addition sum are the same for both problems, but the ideas involved--INCLUDING THE MATHEMATICAL IDEAS--are very, very different."

It does sound simple and clear, right? I hope it does. The IDEAS are different even though the way we find the answer may be the same. But the IDEAS match these situations:

"Mark has four baskets and each basket has three appples." => this is a multiplication situation
"Mark has five apples in one basket and seven in another." => this is an addition situation.

Let's keep going.

• In multiplication, 1 is the special number so that if you multiply by it, "nothing" happens . Also called the identity element.
  
  In addition, zero has the similar role.

• Multiplication: each (real) number except zero has its multiplicative inverse so that if you multiply the number and its inverse, you get 1.
  Addition: Each (real) number has its additive inverse so that if you add the number and its inverse, you get 0.

• Multiplication has an opposite operation called division.
  Addition has an opposite operation called subtraction.

Pretty nice, eh? As we keep emphasizing these distinctions, hopefully we can develop in the students' minds the idea that they're different animals, not the same. One is not a special case of another. We can strive to define multiplication (initially) as "so many of the same-size groups" or counting by groups. Of course we have to FIND products by adding repeatedly, but we can treat them as different operations.

Then later, students will encounter multiplication of fractions and of decimals, and leave behind the idea that they solve multiplication by repeated addition. Yet, the basid properties of multiplication hold true:

• Its identity element is 1.


• Every number except zero has a multiplicative inverse (a.k.a. a reciprocal number)

I also found an interesting study quoted at Text Savvy. I quote:

Two alternative hypotheses have been offered to explain the origin of the concept of multiplication in children's reasoning. The first suggests that the concept of multiplication is grounded on the understanding of repeated addition, and the second proposes that repeated addition is only a calculation procedure and that the understanding of multiplication has its roots in the schema of correspondence. . . .

Pupils (mean age 6 years 7 months) from two primary schools in England, who had not been taught about multiplication in school, were pretested in additive and multiplicative reasoning problems. They were then randomly assigned to one of two treatment conditions: teaching of multiplication through repeated addition or teaching through correspondence. . . . At posttest, the correspondence group performed significantly better than the repeated addition group in multiplicative reasoning problems even after controlling for level of performance at pretest.

I should note here that, although it may read that way above, the ultimate aim of this study was not to compare the effectiveness of the correspondence and repeated addition treatments; it was to test two hypotheses about the "origin of the concept of multiplication in children's reasoning." Obviously, one of the hypotheses says that the origin is in repeated addition, and another says that it is in correspondence.

In other words, the "significantly better" performance of the correspondence group over the repeated addition group was taken by the researchers not as evidence of the superiority of the correspondence treatment, but as evidence of the fact that children begin to think about multiplication NOT as repeated addition but as a "one-to-many correspondence."

Well, I'm trying. Here's the way I changed one page in my Multiplication book to read for now. What do you think?

Click on the link to download the one page (PDF).

A new book for the Blue Series: Math Mammoth Add & Subtract 4

I have a new book to add to the Blue Series collection of worktexts:

Math Mammoth Add & Subtract 4

Math Mammoth Add & Subtract 4 covers the following addition and subtraction related topics which are approximately for the third and fourth grade level:

* mental addition and subtraction strategies;
* adding and subtracting in columns;
* Roman numerals;
* the connection between addition and subtraction, missing addend problems, and solving word problems with bar diagrams;
* order of operations;
* bar graphs, line graphs, and money problems.

Price: $5.00 Buy now

Samples pages include:

Contents and Introduction
Addition Review
Adding Terminology and Practice
More Subtraction Review
Subtract in Columns
Word Problems and Bar Diagrams
Rounding
Line Graphs

Math card game for all four operations (elementary)

I'm sure youngsters will like this card "game"! I'm going to try it with mine.

Actually it's not really a game, but a fun way for kids to write some of their own math problems using a deck of cards. She even provides a free printout:

Little Blue School: Math Card Game for Addition, Subtraction, Multiplication, Division

HT: This week's Carnival of Homeschooling

Number rainbows to learn subtraction facts

I thought some of you (those who teach second grade) might enjoy my NUMBER RAINBOWS. The idea is that you connect two numbers with an arc if they add up to the particular number, such as 13.



Then, the child can use it as a reference when subtracting from 13 or when doing subtraction drill. You could first drill subtraction facts WITH the rainbow (such as 13 − 4, 13 − 7 etc.) and then without.



You would also ask the child to reproduce the rainbow - and color it, of course! These make for quite pretty math facts practice, don't you think!

I'm going to add these to my Add & Subtract 2-A book.

Does the child need to add completing the ten?

Question:

...have an other question about using your worksheets for my daughter. She is six years old - home schooled.
Currently, we are using Addition & Subtraction 2

She is able to add and subtract any numbers from 1 to 100 but when I try to explain "complete the ten" concept, she doesn't like thinking about it. She would rather solve 9+6 by counting on the finger 6 after 9 ... . She complains that I have to first subtract then add to make 10.

My question is: Should I let her complete the exercises without making her think in this manner?

It is important that she understands the IDEA in completing the ten. I gather that she does indeed understand the idea, but doesn't want to do it, since you'd have to both subtract and add, right? She just likes to do one operation, not two?

The adding though, is really easy, because you add 10 + 5 or 10 + 7 or something to ten. If she's counting with fingers, she's not yet seeing the easiness of this adding. It is also faster to do 9 + 6 by first subtracting one from 6 (getting 5), and then adding 10 + 5, because you can do 10 + 5 really easy: it is one ten and five, or we call that fifteen.

Of course she must understand place value to realize how easy that addition is.

Now, the goal is ALSO to eventually memorize the answers. You might let her know that she can't always count with fingers to find 9 + 6 but a quicker method is needed.

You can let her do some exercises "her way" for now as long as she also is able to understand and do the "complete the ten".

But come back to this idea how completing the ten makes the whole calculation easier and quicker. She WILL need to understand the idea when studying CARRYING in addition, (I mean to totally understand how it works. I realize kids can learn carrying mechanically as well.)

So you can say 1-3 months later reprint some pages and try again.

How NOT to drill addition facts

Some people think "drill is kill", and many people think it's necessary.

And of those that use it, not everyone knows HOW to actually drill math facts effectively.

You know, this is NOT the most effective way:

Shuffle the flash cards and start asking randomly.

Why? Because you are not utilizing techniques that help our brain remember quicker.

For example, it is easier to remember when the mind can tie the fact into something already known.

This is the idea behind silly rhymes such as "five, six, seven, eight - fifty-six is seven times eight."

Besides those, we want to show our children the PATTERNS in math.

So this is how I start drilling math facts (whether addition or multiplication):

I make a list on paper, IN ORDER. For example, lately we've been doing this with my daughter:

8 + 2
8 + 3
8 + 4
8 + 5
8 + 6
8 + 7
8 + 8
8 + 9

We went through the answers and notice how each one is ONE MORE than the next! That's a pattern!

Then I would point to a fact and say the problem so she'd both see and hear it (using two senses). You can additionally MOVE her finger on the chart with yours - so she's using three senses. This should help the auditory, visual, and kinesthetic learners all.

When I'd point to a fact further down the list, automatically she'd know it's more than a fact that is up on the list. It's a visual pattern.

First, I drilled just a few of them, namely 8 + 3, 8 + 5, and 8 + 8 until she remembered those.

After that, I would go first to 8 + 8 which she knew, and immediately after that to 8 + 9, and she was able to deduce it from knowing 8 + 8.

I would gradually add new facts in a similar manner - using the known facts as "stepping stones" so that the new fact was one more or less than a well-known fact.

And so we go "round and round" on this chart.

NOTICE THIS:
= > The chart creates an organized context for the addition facts.

Obviously, the child is also associating the position of the fact on the chart with the answer, and so after this is well remembered, it will still take another effort to remember the facts when they're in isolated context, such as in a game, or in a math book, or on flash cards.

But at least it is a very good start, I feel!

What are your thoughts?

Worksheet generators

Someone pointed out to me how my addition worksheet generator had a bug: they couldn't make missing addend worksheets where the student is completing the next whole hundred.

Well I was able to fix it somewhat so now it works for that situation, as long as you don't choose step 2 for the value 2.

These generators... many people have mentioned how they like them. When I first made them, I didn't realize that putting so many options on them would make them not always work... not for every possible combination of options.

I figure that's why on most websites, math worksheet generators don't have many options!

But I made mine originally with as many options as I could possibly think of, because I figured teachers would like them that way! You know, you can choose the number of problems, how many rows, columns, font size, range of numbers including a 'step', missing addend or factor, switching value 1 and 2, etc.

So then it follows, that if you choose some step for values 1 and 2, AND for the answer, that quite likely the script won't find suitable problems. And similarly for other complicated combinations of various options.

Here's a direct link for generating a random worksheet like the lady wanted:

Missing addend worksheet - complete the next whole hundred.

Refresh the page to get a new one and new one and so on.

Here's the link to the addition worksheet generator itself.