the results of calculating with large numbers. Both of these skills are important even when children use calculators.

At present, many students have not achieved procedural fluency with single-digit multiplication when they begin work on multidigit multiplication. A proper balance in instruction among the strands of mathematical proficiency would serve to diminish the number of such students. Until that balance is achieved, however, such students need help in working simultaneously on a multiplication algorithm and obtaining fluency with single-digit multiplication. Using a table to look up some single-digit products can permit students to participate in classwork on algorithms while perhaps motivating as well as supporting their continued learning of single-digit arithmetic.

As we indicated earlier, relatively little research is available to shed light on how students think about multidigit division or what learning activities might be of most help to them. Sample teaching lessons have been proposed, and preliminary results suggest that students can construct their own procedures that, over time, approximate standard algorithms.^{77} As with multiplication, however, the best that educators can do at this point is to examine some alternative algorithms that are likely to support students’ efforts to develop proficiency with multidigit division.

Common U.S. division algorithms have two aspects that can create difficulties for students. First, the algorithms require students to determine *exactly* the maximum copies of the divisor that can be taken from successive parts of the dividend. For example, in the problem 3129÷46=?, one must first determine exactly how many 46s can be subtracted from 312. That determination is not always easy. Second, the algorithms creates no sense of the size of the answers one is writing, in part because one is always multiplying by what looks like a single-digit number written above the dividend. In the example in Box 6–14, to begin the division process, the student just writes a 6 above the line as the first digit in the quotient. There is no sense of 60, because the student will be multiplying 46 by 6.

The accessible division method shown in Box 6–15 facilitates safe underestimating. Rather than trying to determine the largest number of 46s that can be subtracted from 312, the student can just keep subtracting multiples of 46s until the remainder is less than 46. This method builds experience with estimating (as well as accurate assessment of calculator answers) because students multiply by the correct number (e.g., 50, not 5). It is procedurally easy for those students still mastering multiplication combinations because it