Multiplication with Understanding
An elementary school principal recently told me of conversations she had with parents about the third-grade mathematics curriculum. The parents felt there should be heavy emphasis on memorizing multiplication facts. To them, third-grade mathematics should include memorizing facts through drill and practice, worksheets, flash cards, and other memorization aids. But this school principal was promoting an approach that encouraged problem solving and understanding. She explained to the parents that this approach would help children remember the processes of multiplication for a much longer time. She recounted from her own experiences that students who had mastered their multiplication tables during third grade were barely able to remember them the following year. Apparently, memorizing multiplication facts during third grade had accomplished little because it did not build understanding of multiplication concepts. Despite having experienced a “back to basics” curriculum, they still did not know what multiplication is.
The Principles and Standards for School Mathematics (NCTM, 2000) states that “learning mathematics with understanding is essential” and that research shows “the alliance of factual knowledge, procedural proficiency, and conceptual understanding makes all three components usable in powerful ways.” NCTM’s Curriculum Focal Points (NCTM, 2006) emphasizes that “Students understand the meanings of multiplication and division of whole numbers through the use of representations.”
Students typically develop the ability to add quite naturally, but multiplication is much more complex than addition and requires guidance to understand the actions that are important elements of the process. By memorizing facts before developing an understanding of multiplication, students get the mistaken impression about the need to understand what it means to multiply and the
situations in which multiplying is the appropriate thing to do.
So what does it mean to understand multiplication? The mathematics education literature suggests that a basic understanding of multiplication requires four interconnected concepts:
(a) quantity, (b) problem situations requiring multiplication, (c) equal groups, and (d) units relevant to multiplication. Most of these understandings can develop from experiences using counting and grouping strategies to solve meaningful problems in the early grades (Smith & Smith, 2006).
- Understanding quantity. The meaning of quantity often gets overlooked in addition, but it provides an important foundation for understanding multiplication. A quantity is a characteristic of objects that can be counted or measured, and it consists of a number and a unit. Seven dollars is an example of a quantity because it includes both the number 7 and the unit, dollars. Number words (e.g., seven) are often used to describe the number portion of a quantity, but other representations, such as pictures (e.g., 7 bills representing 7 dollars), can be used. In addition to the number, a unit must be specified to know the complete quantity. A count is a particular type of number that is part of the quantity characteristic of collections of objects. It answers the question, “How many.” Counting begins with counting by ones and progresses to skip counting using larger, equal-sized units. Students need sufficient experience in counting collections of objects to clearly understand these two aspects of quantities and the various ways of representing them. A measure (e.g., length) is a particular type of quantity that is a continuous characteristic of individual objects. Measuring includes selecting an appropriate unit of measure (e.g., an inch) and determining the number of these units in the continuous characteristic of the object. Thus, to fully understand quantity, students need to understand the differences between discrete and continuous quantities, recognizing they use both different units and different processes (counting versus measuring) to determine the number portion of the quantity.
- Understanding problem situations requiring multiplication. Students need experience interpreting word problems that require multiplication and distinguishing them from other situations requiring addition, subtraction, or division. Students also need to understand the relationships between multiplication and division and be able to find each of the three possible unknown quantities, providing any two of these three pieces of information are given (e.g., 3 × 7 = ? or 3 × ? = 21).
- Understanding equal groups.Students need experience arranging objects into groups to understand the role of equal groups in multiplication and to understand the efficiency of multiplying equal groups instead of counting all of the objects in the problem. Number sense includes the ability to compose and decompose numbers. Reasoning in multiplication includes using factors and multiples as equal groups when composing and decomposing numbers instead of using adding. For example, eight objects can be arranged into groups representing multiplication (one group of eight, two groups of four, four groups of two, or eight groups of one) rather than groups representing addition (one and seven, two and six, four and four, and eight and zero). Visual images are particularly helpful in understanding grouping (e.g., the difference between a disorganized collection of 60 items and the same 60 items organized into 5 groups of 12 items or an array of 6 rows and 10 columns).
- Understanding units relevant to multiplication. Students need experience with counting and arranging objects into groups to understand the differences between various kinds of units that are relevant to multiplication. Addition most often involves the joining of unequal quantities of the same unit (e.g., adding 35 cents and 24 cents). However, the two factors in multiplication most often refer to different units (e.g., multiplying 12 dogs by four legs for each dog). Students also need to understand how units are sometimes transformed in multiplication. For example, adding 7 oranges to 7 oranges makes 14 oranges, but multiplying the same units, such a 7 inches times 3 inches equals 21 square inches.
One way to increase the students’ deeper understanding of the process of multiplication is to show different ways that multiplication can be carried out by hand. Figure 5.8 shows how to multiply a three-digit number by a two-digit number using the traditional method (a) as well as another way known as lattice multiplication (b). Multiplication requires three steps: multiply, carry, and add. In the traditional method, the multiplying and carrying steps are done together, so it is easy to get confused. In lattice multiplication, each step is clear. It was introduced to Europe by the famous mathematician, Fibonacci, in 1202 in his treatise, Liber Abacii (Book of the Abacus).
The process is simple. If we wish to multiply 427 by 36, we write 427 across the top of the lattice and 36 down the right-hand side of a 3 × 2 rectangle (because we have three- and two-digit numbers). We fill in the lattice by multiplying the digits at the head of the columns by the digits to the right of the row. If the partial product is two digits, the tens digit goes above the diagonal and the units digit is written below the diagonal. If the partial product is only one digit, a zero is placed above the diagonal and the unit digit below.
When all the combinations have been multiplied, we add the numbers along the diagonal, beginning in the upper right and placing the sum on the diagonal to the left outside the grid. If the sum is two digits, the tens digit is placed in the top row of the diagonal to the left and added to that diagonal’s sum. Reading the digits outside the grid from upper left down across the bottom gives the final product of 15,372. This approach is not a cure-all, but it does provide novelty, and it may be just what some students need to better understand the process of multiplication.
National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics (NCTM). (2006). Curriculum focal points for mathematics in prekindergarten through grade 8. Reston, VA: Author.
Smith, S. Z., & Smith, M. E. (2006, March). Assessing elementary understanding of multiplication concepts. School Science and mathematics, 106, 140–149.