# Doing Math: from Left to Right

Last week we had over 80 parents, along with administrators, thinking, doing and talking about math. The event was a School Planning Council meeting sponsored by our district, facilitated by Richmond educator and consultant Carole Fullerton. She has developed an excellent blog, Mathematical Thinking.

She began by asking parents what they wanted for their children. They responded with similar ideas across the room:

to enjoy and be engaged by math (not to fear math)

to connect with their learning

to understand math (not just memorize procedures)

to be able to use what they know about math to solve real problems

A couple of big ideas that stuck with me:

1. Teach Math from Left to Right
When we do mental math (add, multiply, divide, or estimate in our heads) we tend to perform the calculations using the largest numbers first. For example, if I needed to add 86 books to 73 books, I might think 80+70 = 150 and 6+3 =9, therefore my answer is 159. There are of course, other ways to think of this, but that is one way.

Why then, when we add or subtract numbers on paper do we begin with the ones column and move from right to left? When parents posed this question to  Carole, she cited convention. It is simply the way we’ve always done it, but it is not the best way according to Carole.  She added it is the only thing we do in this order in English . . . Carole modeled how teachers and students can add, subtract and solve various problems by thinking and working from left to right – just as if we were solving the problem in our heads.  The beauty of thinking and teaching in this way is that the mental process mirrors the written process, and facilitates a much stronger number sense.

So back to our example, instead of thinking (and saying to students) first we need to  add 6+3 and then 8+7, we are now thinking about 80+70 and 6+3. This is a huge and important shift. (And if we had to carry, we might say or hear students say in a think-aloud, “and carry the 1”  when the number in question is actually ten….) Yikes! No wonder I struggled when I was first teaching these concepts.

2. The Equal Sign means “is the same as”
This point may seem terribly obvious to you. However, it resonated with me as a parent and a teacher. As Carole noted, the equal symbol = means “is the same as,” not the answer comes next, or put the answer here. This is an important concept, whether in algebra class or pre-algebra class, such as grade one.

Carole emphasized the importance of collaboration, multiple pathways to seek solutions and the importance of making thinking visible – a concept that applies to all learning.

Check out Harvard’s Project Zero Visible Thinking website HERE and excellent article on Making Thinking Visible HERE

Carole shared two key messages for parents helping children at home:

1. Parents are not expected to be fluent with K-12 curriculum. Instead, they should ask good questions:

What does this remind you of?

Can you think of another way to do this?

What connections are you making so far?

What would a similar problem look like?

2. One purpose for homework is to practice skills or concepts learned in class. If your child is struggling with math homework, and you have asked good questions, write a note to the teacher explaining that you tried to work the problems out together and got stuck. The teacher can then follow-up and chances are – you are not alone!

Top photo: Creative Commons from Flickr

### 6 comments to Doing Math: from Left to Right

• Amy,

Thanks for including this post on your blog! I am glad that the ideas addressed in the session resonated with you.

It’s amazing how the brain works – especially when it comes to mathematics. It craves forward motion – addition more than subtraction, multiplication over division – and seeks out the most important elements of a problem. When we add from left to right, we focus on the part of the number that is most important, doing what our brains do best. By starting from the left, our initial step provides us with a partial solution that is close to the final answer!

Let me explain.

456
+879

from left to right, then I’d start with the hundreds and work through the tens and ones, writing each partial sum, like so:

1200
120
15

Right off the bat, we know the sum is more than 1200… and that’s extremely helpful when it comes to assessing reasonableness of our sum. The rest is simple addition.

The brain seeks meaningful chunks of information like this. What better way to add (or subtract, or multiply or divide) with meaning than to establish – and then operate on – the most important part of the number?

Carole

• Amy Newman

Thanks for clarifying with this example Carole!

• Selina

I think that you’ve captured the essence of Carole’s presentation. Here are a couple of other thoughts that I might make:

1) With the addition example, use numbers where the “carrying” has to take place. This will help to make the “argument” stronger. E.g. 86 + 77 = 80 + 70 = 150 and 6 + 7 + 13; 150 + 13 = 163 Otherwise those hard core critics will argue that your sample is fine when you don’t have to regroup but it will confuse the kids when we start regrouping.

2) This type of addition algorithm is not just to according to Carole, N. America is one of the few places in the world that completes addition starting from the right. Carole mentioned this as emphasis for the parents to realize that N. America is the anomaly when teaching and learning addition algorithms. Not to take anything away from Carole, but she wanted parents to realize it isn’t just because she said so.

3) Parents want to help their child(ren), particularly when they are struggling. Sometimes forget that dissonance is an integral part of achieving success … that satisfying sense of accomplishment when one struggles to attain understanding. There is a fine line between the struggling and suffering … but when the adult can show support and scaffold by asking the questions you’ve outlined, our kids can make do the math with confidence.

Glad that you were struck with the math bug!

Selina

• [...] her new blog, Amy Newman also writes about this. As well, she shares Carole’s key messages for parents helping [...]

• Great advice to parents! I would add that parents should start by asking kids to explain what they do understand, rather than ask “Where are you stuck?” or “What part of this do you not know?”.

Right now, my daughters and I are able to spend time together reading to one another, drawing fairies & unicorns, singing & dancing to Selena Gomez, baking cookies & cupcakes, playing road hockey or kicking a soccer ball around, going for sushi & tea or siders & milkshakes, etc. I’m not looking forward to the day when these rich social (and learning) activities are replaced with factoring polynomials, conjugating verbs, & balancing chemical equations at the kitchen table.

I like to think that I carefully chose questions to assign for homework when I was in the classroom. Now that my oldest daughter is in school, I think I will be taking an even closer look at homework (reinvent it?) when I return to the classroom. Amy, your research brief is where I will start.

• For the record, “siders” was a misspelling of “sliders” not “ciders”.