PUFM 1.3 Addition

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

Study Teaching Materials

We introduce addition in preschool or kindergarten, and our children will practice working with numbers and developing a gradually deeper understanding of the operation, until the addition algorithm is completely mastered in 3rd-4th grade, depending on your curriculum.

In this section, our textbook only refers to Singapore Primary Math 3A p. 22. It seems to me that the following would also be helpful, if you have these books: 1A pp. 16-37 and 62-65; 1B pp. 32-41 and 77-81; 2A pp. 22-42; 2B pp. 6-14; 3A pp. 18-38; 3B pp. 6-7. Page numbers are from my 3rd edition books. If you have a different edition or another textbook series, look for headings like these:

• Number Bonds
• Addition
• Addition and Subtraction
• Mental Addition, or Mental Calculation

Stage 1: Introduction

At this stage it is the variety of models and contexts, not the difficulty of the problems, that builds understanding.

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

The important thing at this stage is to make sure that our children work lots and lots of word problems. For young children, word problems are much easier than abstract, numbers-only worksheet calculations, because word problems provide a mental image — ducks or flowers or candies or whatever — which children can manipulate in their imagination to find the answer.

If you haven’t read my blog post about oral story problems, please click over and read it now. This is the absolute best way I know to make sure your children never suffer from math anxiety:

• Tell Me a (Math) Story

A wonderful tool/toy for this stage is a counting rope. It’s flexible, portable, and fun to use, and it reduces a child’s reliance on finger-counting. The pattern of alternating colors in groups of five beads makes visible the fingers-and-hands pattern of our number system. Check it out:

• Easy-to-Make Counting Rope

Stage 2: Development and Practice

This is where students begin to deepen their understanding of numbers by mastering the properties of addition. Young children work their way through several stages:

• Push all the items together and then count them all, starting at 1.
• Count on from the first number.
• Count on from the bigger number.
• Use logic: number bonds and mental math strategies.

Important point: Our goal at this level is NOT for our children to memorize a series of math facts, but to develop confidence in working with numbers. In fact, if we stress fact memorization too early, we will short-circuit the child’s learning process. Once children “know” an answer, they don’t bother to think about it — but it is in the “thinking about it” stage that they build a logical foundation for understanding all numbers.

The approach of focusing on strategies rather than ‘the basic facts’ … serves to minimize memory load, allows children to proceed quickly to larger numbers and more interesting problems, and simultaneously teaches skills that will be useful later.

All children will begin addition by counting-on, but beyond that each will have favorite strategies and will be ahead on some methods and behind on others … With practice, they rely more on memory for 1-digit additions and reserve the strategies for use with larger numbers.

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

Pay close attention to the thinking strategies listed on pp.15-18 of the textbook. These are important. Shelley Gray has written a good blog post offering tips on teaching several Mental Math Addition Strategies. You can see several of these thinking strategies in action in my post Mental Math: Addition.

Which thinking strategies are your favorites?

• Counting on.
• +0 doesn’t change anything.
• +10 is easy with place value.
• Use the facts you know: doubles, and number bonds for 10.
• Start on the left (highest place values first).
• Add in chunks, bit by bit, until you get the answer.
• “Make It Easier.”

Our textbook authors call that last strategy “compensation”, by which they mean that you can adjust one number up and the other number down (as if the numbers were piles of blocks, and you were moving pieces from one pile to the other) until you get an easier sum. Usually this involves creating a multiple of 10:
17 + 28 = 16 + 29 = 15 + 30 = 45

But the principal applies more widely. Whenever you have difficult numbers to add, you may be able to adjust them by thinking of an easier option. For instance, many students find the doubles relatively easy, and they can use those facts to figure out the “doubles plus one” or “doubles minus one”:
7 + 8 = double 7 + 1 = double 8 – 1 = 15

Or to add 97 to a number, a student can just think: “Add 100, that’s easy, and then take back 3 of them.”

Exercise 3.3 on page 18 looks like a good way for our children to practice thinking strategies: Have them take any workbook page and “make it easier” by marking the appropriate mental math strategies above each problem.

Compensation:

Add in Chunks:

Make It Easier:

Stage 3: Algorithm

This lesson didn’t talk about developing the algorithm. We’ll get into that in chapter 3. I found that my students needed more practice on the algorithm than the Primary Math books provided. My son, especially, got to be so good at mental math that he rarely needed the pencil-and-paper algorithm to solve anything in the workbook. I printed out worksheets with big numbers from The Math Worksheet Site and had him work one of each type of calculation every day until they came naturally.

Homework Set 3

Homework problem 5b was fun! I assumed that the counting order (+1 facts) is already well known and that the +10 facts (except for 10 + 10, which is a double) are just a place-value issue that doesn’t need to be memorized. That gave me just the doubles and tens combinations to learn. I counted 13 facts.

Then I printed out a (0 + 0) to (10 + 10) addition chart and tried to find the absolute minimum number of facts to learn. I figured out a way to find all 121* facts using the thinking strategies mentioned in our chapter and only seven memorized facts. Wow!

How many addition facts did you need?

*[Not 100. The paragraph at the bottom of p. 16 in our textbook is a typo.]

What’s the Problem with the Equal Sign?

Several people wondered about the common student error mentioned on page 18 and homework problem 7 — that is, students writing something like:
3 + 5 = 8 + 9 = 17 + 2 = 19
Clearly, this would be nonsense in a formal paper, and certainly a teacher should not write such “stream of consciousness” math. But I can understand how students would see this as a common-sense way to write down what goes on in their heads as they work through a problem.

What do you think: Is this really a bad habit that we should try to correct? Or is it okay on scratch paper but not for public consumption?

Points to consider:

• The equal sign problem
• The Meaning of the Equals Sign
• Student (mis)understanding of the equals sign