Teachers report that certain errors in math appear every year, in almost every class, in spite of repeated explanations.
Although 46 − 18 = 28, we see students write:
46
− 18
32
And although 16 × 18 = 288, we see students write,
16 × 18 = (10 × 10) + (6 × 8) = 148
These same kinds of mistakes persist in algebra, where, although 2(3 × 5) ≠ (2 × 3) × (2 × 5), we see students write 2(ab) = (2a)(2b).
Students who make these errors are not thinking about the behavior of the operations or relationships among quantities. Students who decide that 46 − 18 = 32 are reversing the order of the digits in the ones place, which is acceptable for addition but not subtraction.
Students who claim that 16 × 18 = 148 are also employing symbol patterns that work for addition: To find 16 + 18, you can add the tens, add the ones, and then add the results: 20 + 14 = 34. Why not do the same for multiplication?
Similarly, the algebra error stems from confusing patterns that apply to addition with those that apply to multiplication.
A New Approach
The core of the elementary mathematics curriculum has been and continues to be calculation procedures. At each grade, instruction emphasizes computation with numbers on the edge of students' cognitive abilities. Students reach for larger numbers, then fractions and decimals, then integers.
But the exclusive goal of performing computations on ever more challenging numbers may obscure students' understanding of the operations. This can be true even when instruction encourages students to learn a variety of procedures. On the other hand, investigating the behavior of the operations supports students' computational fluency and understanding.
Since 1993, we have collaborated with Virginia Bastable of Mount Holyoke College to develop the following approach to elementary math:
- Students investigate, describe, and justify general claims about how an operation behaves.
- Students shift from solving individual problems to looking for regularities across problems.
- Students create representations of the operations as the basis for explanation and argument.
- The operations become objects of study rather than instructions for how to calculate.
This approach targets issues that cause many persistent errors in arithmetic and algebra and gets students thinking about what the operations mean rather than the procedures they need to follow to arrive at an answer. The following examples show this process in action.
Example 1. Contrasting Addition and Subtraction
Lysette Peron works with students in grades 3 and 4 who are behind their peers in mathematics. When many of her 3rd graders were making the subtraction error illustrated above (46 − 18 = 32), she took a step back.
First, Ms. Peron asked her students to think about the effect of changing the order of numbers in an addition expression. After some investigation, students were sure that changing the order of addends does not affect the sum. They showed two stacks of cubes and then explained that when you switch the order, you're not putting in or removing any cubes, so the total stays the same.
Figure
Then Ms. Peron asked what happens if you reverse the numbers in the expression 17 − 9. Using what they understood about subtraction, some students thought 9 − 17 must equal zero. After all, if you have 9 things and you try to remove 17, you remove all you have. Some students had a bit of knowledge of negative numbers and used a number line to show numbers below 0. But some students who had not previously encountered negative numbers developed their own ways to think about 9 − 17. For example, Maritza drew this:
Figure
The ovals represent 9; the Xs represent 17. The Xs on top of the ovals show how the 9 ovals are subtracted; the additional Xs show the 8 that could not be subtracted. Maritza concluded from her representation that 9 − 17 = 0. However, some of her classmates began referring to Maritza's extra Xs as "invisible numbers." One student wrote:
60 − 50 = 1050 − 60 = invisible 10.
The point of the discussion was not to introduce the concept of integers. Rather, Ms. Peron wanted her students to recognize addition and subtraction as distinct operations that behave differently. Her intent was to have students reason through why it works to switch the order of the numbers in addition but not in subtraction.
After two class sessions spent on the order of the numbers in subtraction, Ms. Peron again posed the problem, 46 − 18, and immediately one of the students began to subtract in his old, incorrect way: "40 minus 10 is 30, 6 minus 8 is …" Then he paused, and said, "No, that won't work." He thought for a minute, and then, instead of reversing the digits, as he might he in the past, he changed his strategy: "46 − 10 is 36, then subtract 8, that's 28."
The students' difficulties with subtraction were not necessarily fully and finally resolved, but exploring what happens when reversing the terms in addition and subtraction enabled students to not only recognize their error and correct it, but also gain foundational knowledge that will serve them when they work with integers and algebraic notation in later grades.
Example 2. Contrasting Addition and Multiplication
Alice Kaye presented the following poster to her 3rd grade class.
Beyond Computation-table
"7 + 5 = 12 7 + 6 = ___" | "7 + 5 = 12 8 + 5 = ___" |
|---|---|
| "9 + 4 = 13 9 + 5 = ___" | "9 + 4 = 13 10 + 4 = __" |
What do you notice?
What's happening here?
The numbers were not meant to be challenging. The purpose of the discussion was for students to consider what was going on in the pairs of problems.
The students filled in the blanks and talked about what they noticed. Students mentioned that one number changed, one stayed the same, and the last number in the equation changed, too. The discussion went on for a few minutes, until Evan said, "Because 9 + 4 is 13, 9 + 5 has to be 1 more than 13."
After further discussion, a student named Pamela said, "How did Evan come up with that idea?"
Evan responded, "I'm not really sure. I just know it. It seems obvious to me, so I didn't think to think about it before."
At the end of that discussion, Ms. Kaye asked the class to do one of two things as homework: (1) to write down a statement that puts what they saw in these problems into words or (2) to come up with other pairs of equations that work in the same way. On the second day, the class built on their homework to write the following conjecture together:
In addition, if you increase one of the addends by 1 and keep the other addends the same, the sum will also increase by 1.
On the third day, Ms. Kaye challenged the students to use a story context, picture, diagram, or manipulatives to convince somebody else that this conjecture is true for all whole numbers. Many of the students' solutions included a graphic representation like Melody's.
Figure
Some students used specific numbers to describe these representations: "I have 5 yellow cubes and 7 green cubes. 5 + 7 = 12. If I add one cube to the green tower, I get 5 + 8 = 13."
However, Melody described her representation in general terms. "When you add the yellow and green, that's some amount. When you add the blue cube to the green stack, the total goes up by one."
Ms. Kaye asked, "Does it matter how many are in the stacks?"
Melody said, "No. It can be any number of yellow cubes and any number of green cubes. When you add one cube to either stack, the total goes up by one."
Melody had created a representation of addition and used that representation to make a claim that would apply to any numbers. No matter what quantities she used in the representation, the result of adding 1 would always be the same.
The following day, Ms. Kaye shifted the focus to multiplication. She presented this poster and writing prompt:
Beyond Computation-table2
"7 × 5 = 35 7 × 6 = 42" | "7 × 5 = 35 8 × 5 = 40" |
|---|---|
| "9 × 4 = 36 9 × 5 = 45" | "9 × 4 = 36 10 × 4 = 40" |
In a multiplication problem, if you add 1 to a factor, I think this will happen to the product: ________________________________________
Ms. Kaye asked her students to create a story context and representation for the first equation in the top row (7 × 5 = 35); then change the context and representation just enough to match the second equation in each box (7 × 6 = 42 and 8 × 5 = 40). After having time to work in pairs, students came together to share their work. Jacob and Billy presented this story:
There are 7 groups of 5 fish living in the store: 7 × 5 = 35. There are 35 fish in the store.
Figure
The representation on the left shows the 7 groups of 5 fish. The representation on the right shows one more fish added to each group, 7 more fish all together: 7 × 6 = 42.
To represent the second box in the top row, Jacob and Billy again started with 7 groups of 5 fish, but this time they added one more group of 5 fish, as shown on the right: 8 × 5 = 40.
Figure
As different groups presented their stories and representations for various equations, the class worked to understand classmates' thinking and formulate a generalization. They came up with this claim:
If you increase the size of each group by 1, the product increases by the number of groups. If you increase the number of groups by 1, the product increases by the size of each group.
Some students joined these two claims into one:
If you increase a factor by 1, you increase the product by the other factor.
At the end of the lesson, Ms. Kaye said, "A few days ago, we were talking about addends changing by 1 and what happens to the sum. Now we're talking about the factors changing by 1 and what happens to the product. How is adding 1 to a factor different from adding 1 to an addend?"
After a pause, May explained, "When you work with multiplication, you have to think in terms of groups, and that's different from when we were just doing sums."
Understanding the Operations
In the first example, although Ms. Person did not use the term, the students in her classroom were examining the commutative property of addition, in algebraic form: a + b = b + a. They also concluded that subtraction is not commutative: a − b ≠ b − a. Some students in Ms. Peron's class further specified the result of changing the order of terms in a subtraction expression: a − b = − (b − a).
In the second example, the students in Ms. Kaye's class investigated special cases of two fundamental properties:
The associative property of addition: a + (b + 1) = (a + b) + 1The distributive property of multiplication over addition: a(b + 1) = ab + b.
These students are developing a strong foundation in understanding the properties and behaviors of the operations. When they encounter such properties in later years, they will be able to read the algebraic notation with meaning, as expressing ideas they already recognize.
This sequence of activities engages a range of learners, from students who tend to struggle with computation to those who excel. Ms. Peron's students generally have difficulty with mathematics and are frequently unengaged. However, all of her students were extremely involved in the discussions about addition and subtraction.
The students in Ms. Kaye's class are more representative of a typical classroom. Students like Evan, who said the idea about adding 1 to an addend was obvious, may already be in the habit of looking for regularity in the behavior of operations. For them, the challenge is to communicate conjectures with precision and to prove them. Students like Pamela, who have not developed the habit of noticing regularities, learn to think explicitly about the properties on which calculation procedures are based.
All students benefit from representing the operations in ways that strengthen their grasp of what the operations do and how they behave. These activities deepen their understanding of how each operation is different from the others.
Three Pillars
Deep understanding of arithmetic, central to the K–5 curriculum, rests on three pillars:
- Understanding numbers includes understanding written and oral counting; the structure of the base ten system with whole numbers and decimals; and the meaning of fractions, zero, and quantities less than zero.
- Developing computational fluency includes building a repertoire of accurate, efficient, and flexible strategies for each operation and knowing how and when to apply them.
- Examining the behavior of the operations includes modeling these operations, recognizing appropriate contexts for each, learning about the properties of each operation, describing and justifying behaviors that are consequences of those properties, and comparing and contrasting the behaviors of different operations.
This third pillar, which is the focus of this article, is as crucial as the others. Investigating the behavior of the operations in the elementary grades means spending time with familiar numbers—numbers small enough that students can pay attention to patterns across problems. Once students have articulated, represented, and justified these generalizations, they are positioned to apply those ideas to more challenging numbers and, eventually, use the same principles to interpret relationships expressed in algebraic symbols. Examining the behavior of the operations supports all students in developing computational fluency and provides a crucial link to their future study of algebra.
Authors' note: This work was supported by the National Science Foundation under Grant No. ESI-0550176. Any opinions, findings, conclusions, or recommendations expressed in this article are those of the authors and do not necessarily reflect the views of the National Science Foundation. Pseudonyms are used for teachers and their students.
