## You can use pictures to explain column addition!

Intended Audience:  Teachers, prospective teachers, and parents (public, private, homeschool).

In this video, 7-year-old Autumn explains the addition algorithm (column addition) using place value tables. We address the story of a frustrated dad who wrote a “place value table” on a check and how his frustration is not with place value tables.  We discuss good ways to use place value tables to teach the addition algorithm and when place value tables have overstayed their purpose. Enjoy!

Last year (when Autumn and I made the video), a frustrated dad wrote a “place value table” on a check and asked whether or not the elementary school would cash it:

Feel free to read the story here (the picture above is linked from that page).  I’m not going to speak negatively about the father: His frustration was real.  There could have been a number of reasons for the frustration (the particular curriculum that was used, overuse of place value tables, incorrect use of place value tables, wrong timing, etc.), but none of them were likely due to place value tables themselves.

Place value tables are simply a wonderful device for showing visually how column addition works to your children.  Autumn’s presentation in the video shows several things that make teaching and learning column addition enormously easier:

• She shows pictorially how to convert 10 one-valued chips into a single ten-valued chip.  This is exactly the type of exchange one makes when you give a cashier ten pennies (10 one-valued chips) and they give you back a dime (1 ten-valued chip).  This exchange or regrouping is at the heart of much of arithmetic and mathematics in general.
• Autumn relates what is happening with the pictures directly to the marks she makes in addition problem.  For example, after converting 14 ones into 1 ten and 4 ones, she shows the 1 ten as a 1 in the tens column and says “1 ten” (NOT “10 ones” or even “14”).
• By placing the “1” for 1 ten  on the horizontal line, it is much easier for her (and your students) to still see the “14” from the calculation 6+8 than in the old addition algorithm where the 1 is placed above the 5:

• Throughout the discussion, both Autumn and I model and practice precise language.  In the discussion pictured below, we don’t say, “8 plus 7 is 15,” instead we say, “8 tens plus 7 tens is 15 tens.”

Can there be too much of a good thing?  Of course.  Forcing children to continue to draw place value table pictures after they understand the process can get pretty laborious.  Teachers and parents need to assess and reassess each student’s work to determine when the pictures are no longer needed.  The beauty of a well designed curriculum is that students will often announce to teachers and parents when they have had enough, stating, “Can I just add using the numbers only—all that drawing takes too long!” If your students say they want to work with numbers only and you know they understand the process, then let them!

A word of caution.  One way the father could have become frustrated is that it may have seemed to him that the teacher was requiring his son to use pictures of place value tables after his son understood the process, when the teacher was actually using the pictures to explain a new process (or a generalization of the process).  If you are a parent, please be careful about jumping to conclusions about teaching the arithmetic algorithms, i.e., column addition, column subtraction, column multiplication, and long division.  These algorithms–especially long division–are among first nontrivial algorithms the students will learn as they grow up. In fact, part of the reason for learning arithmetic algorithms is to learn what algorithms are in general (computers and software are full of very complicated algorithms, for example).  Even though the arithmetic algorithms are some of the easiest algorithms on Earth to learn, it’s always good to remember that they were not dreamed up by children: the arithmetic algorithms were designed by very smart adult mathematicians in the distant past.

Teaching an arithmetic algorithm to a child requires great care; learning is often broken up into stages and often those stages occur in different grades.  Each time a new stage is reached, place value tables are usually brought out again briefly to show how previous knowledge of the algorithm can be generalized to include a bigger set of numbers, like generalizing the whole number multiplication algorithm (182 x 3) to include decimal numbers (1.82 x 3).  It is during this generalization period that the place value tables become a bridge between the known and the unknown.  As a parent, don’t get upset if you see pictures of place value tables show up in later years–they are likely being used to explain new concepts.  Instead, I encourage you to contact your child’s teacher to find out how place value tables are being used and to set up a plan with him or her to help your child recognize when they no longer need the pictures to do arithmetic.  Always remember that your child’s teacher and you share the same goal: that your child can fluently and competently do numbers-only arithmetic with understanding!

BTW: The curriculum the father was complaining about in the article was the not the  Eureka Math/EngageNY curriculum.  Examples of U.S. curricula that use place value tables poorly go back decades and easily predate the Common Core State Standards.  See this for more information.

CHANNEL: Growing up with Eureka
© 2017 Autumn Baldridge and Scott Baldridge

Distinguished Professor of Mathematics, LSU. Geometric topologist: gauge theory, exotic 4-manifolds, knot theory. Author: Elementary Mathematics for Teachers.
Video | This entry was posted in Engineering School Mathematics, Growing Up With Eureka and tagged , , , , , , , , . Bookmark the permalink.

### One Response to You can use pictures to explain column addition!

1. Teresa says:

Interesting.I would like to comment that this type of pre algorithm work is important.
But I would say that there are many many children who will not learn the algorithm until they actually do it.
The pre work is important, but on its own will leave many children lost and confused and they will experience this as “too much talk” and get restless. Autumn is a bright, verbal young girl, but many many students will struggle with just the pre work.
I would suggest some pre work and then fairly rapid practice with the algorithm as they will re inforce each other.Many children will not understand, unless they combine the understanding with the procedure. And many will get it, but it may come some weeks or months later.
I say this as I have seen this work many times. The last 14 years of my career were spent teaching Grades1/2 and I have tutored for the last 5 years.
I would go as far as to say that the weaker the child’s verbal skills, the more early practice they need with understanding and algorithm combined.But even in the case of verbal bright children, understanding and procedure reinforce each other and I would not wait for full or even nearly full understanding.
Also, use of the standard algorithm combined with the explanation gives students a feeling of competence fairly quickly.
As you know there are many adults who like to get the theory, but also to practise at the same time. Makes ’em feel good and then they ‘get it’ a bit faster.
Cute video and I enjoyed the demonstration.

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