Contributed by Monica Neagoy
Teaching Operations in a Technological World—An Enigma?
Do you ever wonder why we spend so much time on teaching computational algorithms, memorizing number facts, and checking for correct answers, in a world full of technological tools that compute faster, more efficiently, and more accurately than we? The bar has been raised: the greater purpose is algebraic thinking in particular and deep mathematical thinking in general. In 2000, the National Council of Teachers of Mathematics [NCTM] described algebra as a way of thinking that cuts across all mathematical content and unifies the curriculum. In 2010, CCSSO made Operations and Algebraic Thinking the first content standard for grades 1-5 (the second for grade K) of the Common Core State Standards for Mathematics (CCSSM).
Algebraic thinking in the early years is a new movement that has received growing attention over the past two decades.
The “early algebra” movement is not about teaching traditional school algebra early.
Rather, it’s about fostering ways of thinking, doing, and communicating about mathematics, of teaching and learning mathematics with understanding, and of cultivating mathematical insight; it’s about making connections, analyzing relationships, and noticing structure; it’s about conjecturing, justifying, and generalizing. These are critical habits of mind for all of mathematics. However, they stand in stark contrast with “manipulating symbols to solve for x (often meaninglessly),” the common definition of algebra in the minds of many adults.
A Daunting Charge for Teachers and Teacher Leaders
Elementary teachers were not educated to teach algebraic thinking and, what’s more, they may barely remember their high school algebra—after all, who does? Some may even harbor unpleasant memories of their algebra experience. So how can elementary teachers learn to cultivate algebraic habits of mind in their students? In particular, what does it really mean to teach operations and algebraic thinking? What does algebraic thinking and doing look like in the early grades? Let’s take subtraction as an example.
An Example: Looking at Subtraction Algebraically
Teachers in different parts of the world often tell me of parents who come in and say, “My child knows her addition (or subtraction) facts; she needs harder problems with bigger numbers.” While bigger numbers might seem challenging to children, simply moving from smaller numbers to larger numbers misses opportunities:
Children can develop sophisticated mathematical thinking working with small numbers too.
1. Subtraction is more than just “take away”
While a program director at the National Science Foundation, I had the privilege of hearing Bob Moses, a civil rights leader and education activist, share his latest algebra research. I still recall his final words: “The take-away model won’t help them later on, in algebra, and algebra is the gateway to higher math and STEM-rich lives!” That day back in 2004, Bob Moses convinced me that we must plant mathematical seeds early. His talk catalyzed my professional move from algebra work at the high school and college levels to algebra in the early grades.
2. We must explore other models for subtraction
The take-away model is not incorrect: every one of us has used it to explain the “–” sign to young children: Eat away, pop away, give away, fly away, and throw away are at the heart of stories we tell. They begin with a start number; then something happens; they end with a smaller number. I call it: picture → video→ picture. In math lingo, we use “change to less” or “change minus.” The point is we must also offer other models to students. The comparison model is more challenging and was scarce for many years in textbooks: “Carla has 6 ribbits and Darnice has 10. How many more ribbits does Darnice have?”
3. From difference as comparison to difference as distance
A 5-year-old can count 6 and 10 cubes in different piles but will be hard pressed to say, “10 is 4 more than 6.” If given connecting cubes, however, he may make two rods and easily see the difference (this context actually gives meaning to the word “difference”). As he gets to Grades 2 and 3, he can visualize this difference as the distance between 6 and 10 on the number line and make a 4-unit rod to symbolize it (Fig. 1). Removing the 10- and 6-rods (blue and red), he can travel up and down the number line with the 4-cube difference rod (grey) and find many subtraction equations sharing the same difference: 12 – 8 = 4, 9 – 5 = 4, 7 – 3 = 4, 4 – 0 = 4, and even 0 – –4 = 4! The algebraic explanation? The infinite equations are transformations of the equation 10 – 6 = 4 by simply adding or subtracting the same number n to both minuend and subtrahend: (10 + n) – (6 + n) = 4 or (10 – n) – (6 – n) = 4.
4. A Child-unfriendly subtraction problem
With this understanding of difference, a 3rd or 4th grader can transform the child-unfriendly problem below (Fig. 2) to a child-friendly one by either mentally shifting the “difference rod” to the right (on the number line) 3 units, or by adding 3 to each number. The first is thinking geometrically; the second is thinking algebraically. This is sometime called the “same change method.”
5. Solving equations by pausing to think … not rushing to compute
At many a Math Night I ask my audience to solve an equation such as, 198 – 55 = 203 – __. Parents share their strategies. The majority performs two operations: first, 198 – 55 to get 143; and second, 203 – 143 to get the answer 60. I then reveal a child’s algebraic thinking and talking: “I just counted up from 55 to 60.” (Fig. 3). Perplexed, parents ask for clarification. So I share fourth grader Aanya’s entire explanation: “203 is 5 more than 198 so I slided 198 – 55 to the right 5 steps and I just counted up from 55 to 60.” Aanya understood subtraction profoundly. Another important aspect of algebraic thinking is exemplified by her understanding of the “=” sign as equivalence between two expressions rather than a mental trigger to compute. Needless to say, parents are amazed: Aanya performed zero operations; she simply counted up. Parents learn a valuable algebraic approach to any problem: pause and observe structure before rushing to compute.
6. The Bridge to High School Algebra
Much of secondary mathematics takes place on the x-axis or better yet, on the xy-plane, meaning on two axes (or number lines): the x-axis and the y-axis. The first notions of a formal algebra course centers on proportional and linear relationships. Their graphs in the xy-plane are straight lines. Proportional reasoning and the concept of slope (of a line) are among the hardest concepts for students to grasp. One explanation for students’ difficulty with slope, said Bob Moses, is because the formula for slope (m below) is made up of two differences, each representing a distance on a number line: y2 – y1 on the y-axis, and x2 – x1 on the x-axis (both in red in Fig. 4). These differences represent none other than distances on number lines between two points! “So take-away just won’t cut it in algebra!” Bob insisted.
This Example embodies many algebraic habits of mind:
- Pausing to think before rushing to compute
- Looking for and making use of Structure (MP7 of the CCSSM Practices)
- Working with and connecting multiple representations
- Thinking about the general when working with the specific
- Distinguishing what varies from what remains the same
“One of Humanity’s Most Ancient and Noble Intellectual Traditions”
Perhaps, if we believe, as research shows, that children are capable of sophisticated algebraic thinking, and if we honor them by tackling challenging problems and discussing powerful ways of thinking and doing, they will grow up truly seeing that “even the most elementary mathematics involves knowledge and reasoning of extraordinary subtlety and beauty.”1.
1. Rand Mathematics Study Panel (2003, p. 3)