Mental computation of students in a reform-based mathematics curriculum

School Science and Mathematics, Oct 1996 by Carroll, William M

Traditional school instruction in mathematics has generally produced students who are poor at mental computation and exhibit a weak sense of number and mathematical operations. In this study,fifth graders who had been in a reform-based mathematics curriculum since kindergarten were given a whole-class test on mental computation problems. Baseline data with students in traditional mathematics curricula were used as a comparison. The students in this reform-based mathematics curriculum performed much higher than the comparison group on all but one problem, and on most problems, this difference was substantial. Additionally, a student preference survey indicated that students in the reform curriculum were more likely to consider the calculator as an option than were the baseline group. They were also more able to recognize problems that did not lend themselves to mental computation. Individual interviews indicated that experiences in the primary grades with "invented" algorithms and discussing alternative solutions led to a better ability to compute mentally and a stronger number sense.

When faced with a problem involving mathematical calculations, various solution methods are possible, including the use of paper-and-pencil algorithms, a calculator, or a mental solution. Although schools have generally emphasized standard written algorithms in both instruction and testing, the calculator has received an increasing acceptance as a regular tool from the primary grades onward (e.g., Wheatley & Shumway, 1992). In contrast, mental arithmetic is underrepresented in instruction and is rarely included on high-stakes tests. This is surprising, given that mental calculations and estimations are more common in everyday life than are the paper-and-pencil algorithms practiced in school (Hope, 1986). Current mathematics reform initiatives have recognized the need to align school and everyday uses of mathematics by including mental arithmetic and estimation as regular components in the curriculum (NCTM, 1989).

Besides better preparing students for the mathematics encountered outside of school, experiences with mental computation should help to develop flexible mathematical reasoning, estimation skills, alternative ways of expressing numbers, and the relationship between operations. Rather than focusing simply on the product, (i.e., the correct answer), mental computation allows greater opportunity to view mathematical problem solving as a process in a domain of interconnected knowledge sometimes called number sense (Sowder, 1992). For example, when adding 98 17, people with a good number sense may simplify the problem to (98 2) IS or 100 20 - 5. They may also note that the sum will be about, but not quite, 120. Experiences with mental arithmetic also help students to recognize when a problem lends itself to a mental calculation and when it is better to use a calculator or pencil, metacognitive skills such as these are important in mathematical problem solving (Schoenfeld, 1992).

Perhaps most importantly, use of mental computation is motivating for many students. For example, in a recent survey (Carroll & Porter, 1994), fourth graders were asked to compare the mathematical knowledge of those who used a mental, paper-pencil, or calculator solution. Nearly all fourth-grade students surveyed felt that the student who used mental computation were better at mathematics.

Good mental calculators use procedures not necessarily learned in school (Dowker, 1992; Hope & Sherrill, 1987; Reys, Bestgen, Rybolt, & Wyatt, 1982) but which build upon number and operation relationships in an efficient manner. For example, good mental calculators often add from left-to-right, recombining numbers as they go along, thus reducing the load on memory. However, without frequent opportunities to develop mental solutions or modeling by the teacher or other students, many students do not develop these efficient techniques. Instead, they attempt to apply standard pencil-and-paper algorithms mentally (Hope & Sherrill, 1987). Due to the strain on short-term memory, these right-to-left calculations with "carrying" and "borrowing" are error-prone when used mentally.

A recent study of by Reys, Reys, and Hope (1993) attempted to set benchmarks in mental computation for comparative purposes. Results on this whole-class test, which tested classes using traditional mathematics curricula, highlighted weaknesses of students in mental computation. For example, only about one-third of fifth graders could mentally add 47 and 29 or could solve a story problem which required subtracting 65 from 100. Few problems were solved by more than half of the students. A preference survey indicated that students chose pencil-and-paper as their preferred method on most problems, even those that lent themselves to mental calculation. The calculator was the least preferred method on all items.

While traditional mathematics curricula emphasize practice of paper-and-pencil algorithms, a number of new mathematics curricula have attempted to incorporate ideas recommended by the current reform movement (NCTM, 1989, 1991; NRC, 1989). In these curricula, the development of estimation skills and number sense of answers is given more emphasis than paper-and-pencil computation. Primary students are often not taught specific algorithms but instead are encouraged to develop and share their own methods. Often these "invented" algorithms are mental and reflect methods used by expert mental calculators, e.g., adding left to right or decomposing numbers (Carroll & Balfanz, 1995; Porter & Carroll, 1995). One such curriculum in wide use is the University of Chicago School Mathematics Project elementary program, Everyday Mathematics. In addition to inventing their own algorithms, students in this curriculum are encouraged to share multiple solutions during group work and whole-class discussion. Manipulatives and calculators are used regularly during problem solving, and computation problems are nearly always presented in an applied situation or in a game. Mental arithmetic is emphasized in the curriculum, both implicitly by encouraging invented algorithms, and explicitly by encouraging the sharing of alternative strategies and short mental arithmetic exercises.