## The Standard Algorithm of Addition: Part 3

In the blog “Standard Algorithm of Addition: Part 1,” we learned how to set up addition problems in a vertical format so that we can manually calculate their sum. We also learned the mathematical terms for the components of arithmetic expressions of additions. In the blog “Standard Algorithm of Addition: Part 2,” I used a {visual model} to explain the concept behind this fundamental algorithm using green square blocks and blue rods to represent the addends. In this blog, I will show you how the standard algorithm works {number model}.

First we set up the arithmetic expression in a vertical format (A). Now we will first focus on the ones column and work our way from right to left (B). We mentally add the digits in the ones (9+7=16). Note the sum is greater than 9. This sum is equivalent to 1 unit of ten (blue rod) and 6 units of ones (green squares) (C). I write the digit for the 6 units of one in the ones column under the underline (D). I place the 1 digit of ten at the top of the tens column (E).

Working our way from right to left, we move to the tens column (F). We add the digits in this column noting that we are actually adding multiples of ten (G). Since the sum is not greater than 99, we simply write the sum of the digits in the tens column underneath the underline (H). Therefore, the sum of the addends, 29 and 47, is 76 (I).  It is the most common technique of addition taught to math students, but it is not the only one.  We’ll look at some other addition algorithms in future blogs.

## Standard Algorithm of Addition: Part 2

I believe that math students of all ages learn abstract notions by first experiencing them concretely with shapes they can actually see, touch and manipulate {visual model}. we will use of concrete teaching aids such as base-ten blocks help provide insight into the creation of the standard algorithm for addition. In this blog, we will use them to find the sum of two whole numbers, 29 and 47 (A). I will be using two types of blocks (B). A green square block has a value of 1 unit, and a blue rod has a value of 10 units. I represent the values of the two addends using these two blocks. I make sure that they are grouped in the appropriate columns according to the structure of the decimal number system (See “Decimals”). I then add all the blue rods together in the tens column, and add all the green squares together in the ones column (C). Notice that we have more than ten green squares. I group ten of these squares together (D) and replace them with a blue rod (E). This blue rod gets moved from the ones column to the tens column (F). Now we have seven blue rods and six green squares, which represent the sum: 76 (G).

We can use these two blocks to visually add numbers 1-99. Additional blocks can be added that represent larger multiples of 10 so that larger whole numbers can be used.

## Standard Algorithm of Addition: Part I

As part of our primary education in mathematics, we have all been taught a standard method for adding two or more numbers together. Students learn how to add a series of whole numbers by following a step-by-step process until they can apply it without making an error. As part of the math education, it is also important that math students understand why this step-by-step process, the standard algorithm of addition, works.

First we need to set up our arithmetic expression of addition properly {number model}. In the blog, “The Vocabulary of Addition,” we learned the math terms for the different components of arithmetic expressions and equations involving addition. In that blog, the arithmetic expression and equation was presented in a horizontal format. Such expressions and equations can also be presented in a vertical format. This is the format that is used to teach and implement the standard algorithm of addition.

The arithmetic expression, 89 + 47, is expressed in a vertical format.

Note how the digits in the one’s column and ten’s column are aligned vertically in this format.

The numbers listed in this format are the addends of the arithmetic expression of addition.

The plus sign, the symbol representing the operation of addition, is placed to the left of the addend at the bottom of the list.

The underline is used to separate the addends from the sum that is about to be calculated. It serves the same function as an equal sign.

When listing the addends associated with the arithmetic expression, the digits associated with the 1s column must line up vertically. The digits associated with the 10s column must also line up vertically. (See the blog “Decimals” for more details). This will make it easier to execute the algorithm properly. All math students should be encouraged to set up these addition problems properly and write them neatly to avoid confusion.

The most common way to gain insights into the validity of algorithms is to express its abstract concepts to the math student in visual concrete ways {visual model}. We’ll explore this concrete model of addition in Part II.

## Expressing Mathematically – Part 2

In one of my earliest blogs (“Expressing Mathematically”), I introduced the idea that mathematical concepts and skills can be expressed in different ways. Our senses identify natural and artificial objects {world model} that convey math concepts. We use language to identify and communicate our understanding of the sensed math concept written and spoken ways {language model}. We also use geometric curves, shapes and volumes to represent these objects {visual model} in a 2D or 3D mathematical world. We can also plot our visual models on a coördinate plane or volume {graphical model} to generate numerical data about the geometry. Finally, we can express mathematical concepts and skills using symbols, numbers, variables and equations {numerical model}. Keep in mind that all these models are different aspects of what we generally refer to as mathematics. I will use these terms often throughout future blog to explore math concepts and skills through these unique perspectives.

## Addition and the Ishango Bone

In today’s blog, we are going to start a steady and cumulative process of exploring the concepts and techniques of mathematics in an evolutionary way. You will soon realize that even the most basic concepts can be seen in different and interesting ways. As always, and whenever possible, I’ll point you to some interesting places, events and people in the history of mathematics.  Lets start with the concept of addition. Addition evolved out of a very fundamental desire to count and know the quantity of similar objects we have.

Suppose we begin with two sets of objects (1). Set A consists of six elements (blue discs) and Set B has three elements (green discs). These two sets have no elements in common. The intersection of sets A and B is the empty set. Thus, they are referred to as disjoint sets. The union of sets A and B consists of both the six blue discs and the three green discs. In the Hindu/Arabic number system, there are numbers that are used to represent the quantities of discs in each set (2). In this case, it is the numbers “6” and “3.” The union of two disjointed sets is a visual way of representing the sum of two numbers. The addition symbol (+) is used to represent the operation of addition in a mathematical expression or equation. The sum of these two numbers is equivalent to counting the number of blue and green discs in the union of two sets (3).  In this case, six plus three is equal to nine.  One should note that counting is nothing more than the repeated addition of 1 to generate a set of natural numbers.

In 1960, a Belgian geologist, Jean de Heinzelin de Braucourt (1920-1998) was exploring an area of Africa near the headwaters of the Nile River at Lake Edwards, called Ishango. At the time, the area was part of the Belgian Congo. He discovered a large number of tools, artifacts, and human remains.  In the world of archeology, there have been numerous discoveries of prehistoric animal bones that have notches carved into them. These artifacts are generally known as tallies or tally sticks. Some of these notched bones have proven to be more interesting to the mathematical community than others. One of these interesting bones was discovered at Ishango by Jean de Heinzelin de Braucourt . Today, this artifact is referred to as the Ishango bone.  It is considered the second oldest mathematical object. However, its exact purpose has yet to be determined.

The Ishango bone is currently on display at the Royal Institute of Natural Sciences in Brussels, Belgium.  There, you will find a Flash-based website on the Ishango archeological site, which will give you a detailed explanation of the bone’s markings and its possible uses. Explore this site and you will find out how far back the history of mathematics stretches.

## Word Problems

One of the most common forms of problems in mathematics are word problems.  They are essentially mathematical exercises presented in the form of a hypothetical situation that requires an equation to solve.  They are usually expressed in a narrative form.   I consider this a verbal (literary) form of expressing math (See the blog “Expressing Mathematically”).  There are two insights I want to share with you about word problems:

Hypothetical Situations
First, when I solve word problems, I am actually executing at most five steps, not necessarily in the order given: (1) I translate the hypothetical situation into a visual expression, like a diagram using geometric concepts, or straight into an equation or expression. (2) I find the values that we are given in the  word problem, and (3) I find the variable(s) we need to solve for (the “unknown”) to find the solution to the hypothetical situation. (4) I then decide which concepts and/or equations we need to apply to the hypothetical situation. (5) Finally, I apply the proper skills from our math toolbox to find the solution to the word problem.

Actual Situations
Second, a traditional word problem explicitly supplies the students all the necessary information to do steps (1),(2),and (3).  When I am presented with  a problem in a real-world situation, I usually have to define the situation in a way that I can solve it mathematically and express it as a word problem, or a labeled diagram.  Many times, I have to conduct measurements in order to establish the “given” values.  Usually, the “unknown(s)” end up being those values I need to know that I cannot measure in a practical or direct way.  It usually takes some work to properly define the situation that is usually given in a classroom-type word problem.

Word problems are one of many ways we can test the competency of a math student.  They are also effectively used to help the math student achieve mastery in applying certain skills.  I understand that it is impractical to consistently present students with real-world situations to solve.  However, it is just as important for students to know how to properly generate a problem as it is to solve it.

## Expressing Mathematically

In many ways, mathematics is a language with its own alphabet, words, grammar and syntax.  Mathematics expresses itself in multiple ways. Lets see how.In my home office, I have a coaster illustrating a Siamese cat (A).  I can specify the shape of this coaster verbally, using the proper word (B).  A different word would come to your mind if your native language is not English.  I can also specify the shape of the coaster visually, by drawing a circle of the given radius (C).  By envisioning such a circle on a 2-dimensional coördinate system (D), I can identify specific points on the edge of the coaster numerically (via ordered pairs of numbers).  Finally, I can specify the shape of the coaster in the form of an equation (E).

In math, we must translate a given scenario into one of these translation so that we can successfully generate the requested solution.  We will explore how we specifically do this in future blogs.  This is a source of frustration for math students who have not yet mastered these fundamental, and often overlooked, skills.  It can impede their progress to understanding more complex mathematical concepts.

Do you believe that there are forms of expressing math that does not fall under one of the five categories shown above?