“The current changes in mathematics education focus more on questions of what students get out of studying mathematics, (mathematical reasoning) than on specific new content” (American Mathematical Society).

Single Digit Multiplication and Division

Facts and Fact Families:

Many elementary school classrooms focus on memorization of facts and fact families when it comes to multiplication and division. Single digit facts are often taught through flashcards, songs, and timed tests. Many teachers and students get frustrated when facts are not easily memorized. Although memorization is not the only key to learning multiplication and division, recall becomes critical as multi-digit multiplication and division make math more difficult.

Memorizing single digit facts is important. Research shows that they way to do this is through repeated practice. Increasing speed of performance occurs through practice of an individual item (Cohen, Servan-Schreiber, & McCelland, 1992) and increasing the amount of drill and practice is often considered to be the most effective approach to improve learning {Chase & Symonds, 1992). There are many resources available to help students with practice of multiplication and division facts.

Fact families help students understand the relationship between multiplication and division. Division is the inverse of multiplication like subtraction is the inverse of addition. This realization does not come naturally to students. Although fact families are good resources for students, the meaning of the inverse needs to first be taught concretely and then they can more fully understand this abstract idea. An example of a fact family can be seen below.

6×3=18 3×6=18 18/3=6 18/6=3

The following websites can be helpful in teaching and learning multiplication and division facts.

http://themathworksheetsite.com/mult_single_vert.html

http://www.geocities.com/smilecdg/asmultipli.html

http://www.songsforteaching.com/math/multiplicationsongs.htm

Grouping:

Although it is important for students to memorize their facts it is much more important that they understand what multiplication and division are. Multiplication is different than addition and subtraction in that the numbers mean two different things. One number is the amount of groups while the other is the amount in each group. Manipulatives and pictures should be used to introduce these concepts. This will foster the students understanding of division and multiplication and help them to grasp the concepts.

There is only one type of multiplication. This is where the number of groups are multiplied by the number of items in each group. There are two types of division, measurement and partitive. Measurement is when the number of groups is unknown and partitive is when the number in each group is unknown. (See examples below).

Grouping teaches students how to solve multiplication and division problems when they cannot recall the math fact. Students should first be taught using manipulatives to allow concrete knowledge to grow. They can then be moved into drawing pictures. Introduce multiplication using a launch that allows students to figure out the problem themselves. Students will automatically make groups or do repeated addition. This allows the concept of grouping to come from them instead of from direct instruction.

Multiplication: Amanda had 3 boxes of crayons. Each box had 8 crayons in it. How many crayons did Amanda have?

Measurement Division: There are 8 crayons that come in a box. How many boxes of crayons does Amanda have if she has 24 crayons?

Partitive Division: Amanda has 3 boxes of crayons. If she has 24 crayons, how many crayons are in each box?

The following website provides practice in multiplication and division:

Resources

American Mathematical Society. (2008). *Association resource guide: Standards 2000. * Retrieved December 16, 2008 from http://www.ams.org/sitesearch/?cx=006070101833274801941%3Acyhu7o79tee&cof=FORID%3A11&q=nctm+multiplication+#991

Chase, D. H,, & Symonds, P. M. (1992). Practice vs, motivation,* journal of Educational Psychology, 84.* 282-289.

Cohen, J- D., Servan-Schreiber, D,, & McCelland, J. C, (1992), A parallel distributed processing approach to automaticity,* American journal of Psychology, 2,* 239-269.

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