Is It 2 More or 2 Less?
Algebra in the Elementary Classroom
February 3, 2006
This article originally appeared in Connect, January-February 2006.
For many adults, algebra means symbolic notation and equations. We may have done well or poorly in high school algebra, but either way, we often have no sense that those equations we manipulated had any meaning beyond what was required to pass the course. Yet, in collaborative work with teachers over the past five years, we have found that students in kindergarten through fifth grade are thinking about ideas that are at the heart of algebra. In the course of working on arithmetic, students notice regularities that might, in later years, be expressed with symbols and equations. These ideas offer opportunities for rich mathematical investigation and discussion.
Examples from the classroom
Grade 4: Is it 2 More or 2 Less?
If you are mentally trying to solve 145 – 98 = ⟡, you might use the easier, related subtraction problem 145 – 100 = ⟡ to help you. You know that 145 – 100 = 45, but is the difference between 145 and 98 two more or two less than 45? In a fourth grade class, students were considering this problem. The context for the problem was related to a science project in which the class weighed apples in grams as they dried out.
Brian drew a closed shape representing an apple divided into two parts to represent 145 – 100 = ⟡: "See, this is the apple at first," he explained. "And you take some away [the part to the right of the dotted line] and have some left [to the left of the dotted line]. Then you take away 98 grams instead, so it’s over here [the part to the right of the solid line is now the part that is subtracted; left of the solid line is what remains]."
With the presence of this picture to focus the discussion, more students joined in, using the representation not only to reason about the particular numbers, but to state and justify a more general claim. Rebecca said, "It’s like you have this big hunk of bread and you can take a tiny bite or a bigger bite. If you take away smaller, you end up with bigger." Then Max stated: "The less you subtract, the more you end up with, AND in fact the thing you end up with is exactly as much larger as the amount less that you subtracted."
Grade 2: Switching Around the Numbers
But what about students in the primary grades? Aren’t they "concrete" thinkers? Young students, too, notice regularities about the work they do as they count, compare quantities, and learn about addition and subtraction. In one second-grade class, the teacher asked students to generate combinations of two addends with a sum of 25. As they listed these, students soon noticed—as they had before in their computation work—that they could "switch around" the numbers and still get the same sum, for example, if 23 + 2 = 25, then 2 + 23 = 25.
The teacher had in mind that this idea would come up and had planned follow-up questions. She asked, "Suppose I asked . . . if you could prove that or explain it better to me . . . if we take the 2 and put it first, do we still get 25?"
Nikki demonstrated with a stack of 23 cubes and a stack of two cubes. She moved the two-cube stack rapidly and repeatedly from the right side to the left side of the 23-cube stack. "It doesn’t matter," she said, "if you keep on just switching it around, it will still make 25 . . . you’re not taking away or adding to it . . . it will still be the same number." Again, in this example, the use of a representation that embodies the operation enables the students to reason about the general claim. Although Nikki is holding particular quantities—23 and 2—her reasoning, that "you’re not taking away or adding to it," applies to any pair of numbers.
Once all the students seemed quite convinced that the order of any pair of numbers in an addition expression could be changed without changing the sum, the teacher asked the students if the same is true for subtraction. From her experience, the teacher knew not to assume that students thought that the "switch around" rule applies only to addition. Several students offered their ideas, using 7 – 3 and 3 – 7 as an example:
Nikki: If you have 3 take away 7, but 3 doesn't have 7. . . . You can only take away 3 to make zero.
Alita: You can't use the 3 because after you use the 3—3, 2, 1, 0, 0, 0. . . the zero’s going to keep on repeating itself.
Edward: It wouldn't be zero. It would be negative 4 . . . That means you're going lower. If you're going lower than zero, that means negative 1, negative 2, negative 3 . . . .
Although these students did not yet have all the number experience necessary to understand this idea, the teacher noticed that they were making important observations about the differences between the properties of addition and the properties of subtraction. She planned to return to this discussion as other opportunities arose.
Algebra for all students
The work of generalizing and justifying in the elementary classroom has the potential of enhancing the learning of all students. The teachers with whom we have collaborated for several years have realized this potential in their classrooms. Teacher collaborators report to us that students who tend to have difficulty in mathematics become stronger mathematical thinkers through this work. As one teacher wrote, "When I began to work on generalizations with my students, I noticed a shift in my less capable learners. Things seemed more accessible to them."
When the generalizations are made explicit—through language and through visual representations—they become accessible to more students and can become the foundation for greater computational fluency. Furthermore, the disposition to create a representation when a mathematical question arises supports students in reasoning through their confusions. Brian (in the grade 4 example above), a tentative learner in mathematics, created a representation that illuminated an important idea. In the second grade classroom, in an urban center with a historically large proportion of underachieving students, a range of students offered important ideas about how addition is and subtraction is not commutative.
At the same time, students who generally outperform their peers in mathematics find this content challenging and stimulating. The study of number and operations extends beyond efficient computation to the excitement of making and proving conjectures about mathematical relationships that apply to an infinite class of numbers. A teacher explained, "Students develop a habit of mind of looking beyond the activity to search for something more, some broader mathematical context to fit the experience into." In the fourth-grade example above, Max, one of the most mathematically successful students in the class, listened carefully to his classmates’ explanations and then enjoyed the challenge of formulating a precise statement of the generalization. And Edward (in the grade two example), who knew more about numbers than his peers, was able to seed the conversation with a new idea about numbers less than zero.
Early algebra is fundamental
Underlying these kinds of discussion are what one of our mathematician advisors calls "foundational principles"—principles that connect elementary students’ work in arithmetic to later work in algebra. For example, the idea explored by the fourth graders (the less you subtract, the more you have left) can be represented as, "If a – b = c, then a – (b – x) = c + x," or, more concisely, "a – (b – x) = (a – b) + x." A discussion among middle schoolers similar to that in the fourth-grade example could provide an opportunity to consider why the associative property does not apply to subtraction, and to articulate a rule that does. The second graders do not yet have the experience with negative numbers to allow them to completely make sense of 3 – 7, but they are nevertheless engaged in reasoning about foundational ideas, in this case, that addition is commutative, but subtraction is not: a + b = b + a, but c – d ≠ d – c. In later years, they will come to see that there is a regularity here, that if c – d = a, then d – c = -a, or c – d = - (d – c).
Underlying the notation are ways of reasoning about how the operations work. This reasoning—about how numbers can be put together and taken apart under different operations—not the notation, is the central work of elementary students in algebra. In the course of our work and through the insights of teachers and the thinking of their students, we have concluded that work in early algebra is fundamental to the experience of young students. Early algebra is not an add-on. The foundations of algebra arise naturally throughout students' work on number and operations. Considering general claims provides the opportunity for students to learn about the power of representation as a basis for mathematical reasoning. This work anchors students' concepts of the operations and underlies greater computational flexibility.
Susan Jo Russell is a principal scientist at the Educational Research Collaborative, TERC. She co-directed the development of the K–5 curriculum, Investigations in Number, Data, and Space.
Deborah Schifter is a senior scientist at the Education Development Center, Newton, Massachusetts, where she has directed major teacher development projects, including the project that created five modules in the professional development series, Developing Mathematical Ideas.
Virginia Bastable is the Director of the SummerMath for Teachers program at Mount Holyoke College in Holyoke, Massachusetts.
Drs. Russell, Schifter, and Bastable are currently working together to develop two new modules focused on early algebra in the Developing Mathematical Ideas series and to develop an algebra strand for
Investigations in Number, Data, and Space.The work described in this article was supported in part by the National Science Foundation through Grant No. ESI-0095450 to TERC and Grant No. ESI- 0242609 to the Education Development Center. Any opinions, findings, conclusions, or recommendations expressed here are those of the author and do not necessarily reflect the views of the National Science Foundation.
Further reading
Schifter, D., Monk, S., Russell, S. J., Bastable, V., and Earnest, D. Early Algebra: What Does Understanding the Laws of Arithmetic Mean in the Elementary Grades? (Draft of a chapter being prepared for a volume edited by James Kaput and David Carraher.) This draft can be found at: http://investigations.terc.edu/resources/papersPubs.cfm#Schifter.