Teaching Computation in the Twenty-First Century, by Linda Levi
When our parents (and some of us) were in school, standard algorithms were taught so that people were able to add, subtract, multiply and divide fairly quickly and accurately. Before the 1970’s, if you didn’t know an efficient algorithm for computation with large numbers, there were many jobs you couldn’t hold and you would have had difficulty managing your personal finances. When calculators become easier to obtain in the mid 1970’s, our need for efficient algorithms drastically changed. For example, teachers no longer needed to add all of a student’s grades for a semester and divide by the number of total points to get a final grade; nurses no longer needed to figure out medicine dosages by multiplying a patient’s weight by the amount of medicine needed per pound (which was usually a number in the hundredths or thousandths); people no longer needed to subtract a check amount from their balance at the grocery store so that they knew their balance before buying something else, and so on.
Some people might conclude that teaching computation is less important now that calculators are available – I strongly disagree with this conclusion. The basic operations of addition, subtraction, multiplication and division are essential to understanding almost everything about our world. Understanding computation is necessary to understanding global, national and local economic issues, environmental issues, social issues and much more. Understanding number and operations is also vital to understanding everyday issues such as how to plan your day to arrive at school or work on time, how to make decisions about spending money, and how to fairly share things with other people.
Unfortunately, in many classrooms, computation is taught in the same manner as it was taught to our parents – students are taught computation as if calculators aren’t readily available. They spend hours practicing standard algorithms. If our goal in teaching computation is to prepare children to make sense of the world, the way we teach computation should reflect this goal. Teaching computation as a preparation for supporting children to understand the world around them is very different than teaching computation for speed and accuracy.
As children move from Direct Modeling to Counting to Relational Thinking, their understanding of how operations work develops. For example, kindergartners Directly Modeling a Join (Result Unknown) problem may have no conception that what they are doing relates to anything more than the problem at hand. After solving many Join (Result Unknown) problems, these children will grow in their understanding of how addition works. They may start to notice that when they join two groups together, it doesn’t matter which group they count first; or they may start to realize that when they are joining objects, grouping objects in different ways doesn’t change the total number of objects. Concepts such as these are crucial in understanding how addition works. These concepts apply as much to Direct Modeling simple story problems as they do to using formal algebra as a tool when solving Calculus problems.
The concepts that dictate how addition, subtraction, multiplication and division work are all derived from the Properties of Operations. Although the Properties of Operations were first published as a subset of the Field Properties in the 1850’s, the Properties of Operations didn’t become commonly discussed in elementary school until they were included in many State Standards for Mathematics documents. (See the table below for an example).
Amazingly, everything we do when we add, subtract, multiply or divide can be justified in terms of these nine properties. These properties explain why it works to count the tens and then the ones when Direct Modeling with Tens; they explain why Invented Algorithms work; and, they explain how the Standard Algorithms work and much, much more.
Students who understand how operations work are empowered to make sense of the world around them. They may pull out their calculators to perform a complex computation, but they will use their calculators differently than students whose instruction focuses on learning procedures. I was waiting in a cashier’s line and overheard the following conversation between a man and his teenage daughter. The dad told his daughter that the sweater she had chosen, which cost $49.49, was too expensive. The daughter said that everything in the store was 40% off. The dad pulled out his phone and hit the screen several times while the daughter waited patiently. The dad reported that even with the sale, the sweater would cost $47.51, which was still too expensive. The daughter replied, “I think you made a mistake. If the sweater cost $50, 40 percent off would make the sweater $30, because 40 percent off $100 would be $60. 50 is half of 100 so this sweater has to be less than $30.” Much to the man’s credit, he smiled and bought his daughter the sweater. This girl had an understanding of how multiplication with decimals works. Although employing the standard algorithm to figure .40 times 49.49 is too complex for most of us to do in our heads, she had a good strategy for figuring out an approximate answer. Her dad, unfortunately, did not seem to have a general understanding of how multiplying with a decimal works or at least didn’t use his understanding at that moment. (Any of you who have shopped with a teenage girl understand why he might understand this concept but not have been able to use it at that particular moment.)
Developing an understanding of how operations work is one of the most important things you can do to support your students to meet your grade level standards for Operations and Algebraic Thinking and Number and Number and Operations in Base Ten. Developing their understanding of Base Ten and Fraction Concepts is also one of the most important things you can do to help your students meet your grade level standards. Developing this understanding not only enhances your students’ learning of the current grade level standards, it also provides a foundation for success with middle, high school and college mathematics.
In a study of all of the sixth-grade students in one small school district, 50% of them used the following strategy to solve 300 – 299 = n.
The standard algorithm is clearly not a good strategy for this problem. Students who understand how subtraction works, will not use a standard algorithm to solve this problem. One way to assess your students’ understanding of how addition and subtraction works is to pose a series of problems and see if they use strategies that are suited to the problem. Some problems you might want to include on your list include the following:
As Cognitively Guided Instruction teachers, you have a rich foundation to base your instructional decisions. You understand both how children’s thinking typically progresses and how the children in your class understand mathematics. When you make decisions about how to teach computation, remember to consider why you are teaching computation. If, like me, your goal in teaching computation is to prepare your students to better understand the world around them, your instruction should support children in developing an understanding of the properties of operations. Your goals in teaching computation should be considered in deciding issues such as:
Will I have my students practice solving many problems, or will I have my students solve fewer problems and spend more time sharing their ideas with each other?
What tools will I provide to my students when they are solving computation problems?
How will I interact with students who make errors when they compute? What errors will I focus on?
How will I know when to accept a students’ strategy and when to support the student to use a different strategy?
How will I choose problems for my students?
Will I insist that students understand each other’s strategies, or will it be enough for students to understand their own strategy?
And more….
This blog post was supported in part by the U.S. Department of Education, through grant award number U423A180115 to Florida State University. The opinions expressed are those of the authors and do not represent views of the U.S. Department of Education.
