From Memorization to Understanding
It is imperative to teach children the meaning of what they are doing when they manipulate numbers during arithmetic computations. Meaning not only increases the chances that information will be stored in long-term memory, but also gives the learner the opportunity to change procedures as the nature of the problem changes. Without meaning, students memorize procedures without understanding how and why they work. As a result, they end up confused about when to use which procedure. Teachers who use primarily a declarative approach emphasize not only arithmetic facts but how they are related to each other and connected to other concepts the students have already learned. They use elaborative rehearsal and provide for cognitive closure.
Are We Teaching Elementary-Grade Arithmetic for Understanding?
In some schools, we teach too much arithmetic through procedural approaches and very little with declarative methods. Could it be because that is how most teachers learned arithmetic themselves? Could this explain why arithmetic instruction in the primary grades has not changed very much over the years? We teach students a procedure for solving computation problems, which they then repeatedly practice (procedural memory). But the practice does not result in computational fluency because we rarely talk about how and why the procedure works. Consequently, when we give the students a problem to solve, they reflexively draw on their knowledge of the practiced procedure and apply that procedure quickly and efficiently, but with little understanding of the mathematical concepts involved.
Of course, students need to learn some basic procedural activities, such as memorizing a short version of the multiplication tables along with a few number facts. But the emphasis should be on showing students (at the earliest possible age) why they are performing certain arithmetic operations. The more arithmetic we can teach through declarative processes involving understanding and meaning, the more likely students will succeed and actually enjoy mathematics.
Example of a Declarative-Based Approach
A declarative-based approach focuses on capitalizing on the students’ innate number sense, intuitive notions of counting by finger manipulation, and an understanding of a base-10 model for expressing quantities. It includes allowing students to create their own procedures for arithmetic computations so that they truly understand the algorithms involved. Researchers have long recognized that students in the primary grades are capable of constructing their own methods of computation (Carpenter et al., 1998; Fuson et al., 1997). In doing so, the primary-grades students pass through three predictable developmental levels.
- At the first level, students deal with all the quantities in a problem. To add a group of objects, they count out separate groups of objects, combine the groups, and then recount the total. To subtract, students count out and separate a group, and then recount what is left.
- At the next level, students consider all parts of the problem before solving it. They demonstrate this ability by counting on from, or back to, a quantity to determine an answer.
- At the most advanced level, students use abstract knowledge and consider quantities in flexible ways. They make use of what they already know to solve new problems. For example, students might use their prior knowledge to realize that 6 + 7 is equal to 6 + 6 + 1, or that 7 + 9 is equal to 6 + 10, by decomposing and recombining tens and ones.
Understanding the development of mathematical thinking in young students allows teachers to anticipate procedures that students are apt to invent and find ways to support students as they progress through the different levels. When teachers encourage students to invent alternative problem-solving strategies, the learning objectives are different from those that result from instruction using standard memorization procedures. The emphasis is on making sense and finding meaning in the methods that students create and successfully use (Scharton, 2004).
Mathematics educator, Susan Scharton, has been a strong advocate for giving primary-grade students opportunities to solve computational problems, to create their own procedures for solving them, and to explain their methods to others. She found that this approach improved the students’ accuracy as well as their understanding of the methods they had created. When she asked students
to explain their methods, their understanding of their own procedures deepened as a result of this elaborative rehearsal. Listening to the methods that others had used prompted some students to experiment with other students’ methods of computing.
Carpenter, T. P., Franke, M. L., Jacobs, V. R., Fennema, E., & Empson, S. B. (1998, January). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29, 3–20.
Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I., Carpenter, T. P., & Fennema, E. (1997, March). Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education 29, 130–162.
Scharton, S. (2004, January). I did it my way: Providing opportunities for students to create, explain, and analyze computation procedures. Teaching Children Mathematics, 10, 278–283.