Using Arrays to Increase Understanding

When introducing students to multiplication, many teachers often use the array to allow children to visualize what they are learning. An array can be defined as a rectangular set of objects placed into rows and columns.

Using arrays is a good method to use when you want students to realize that the order in which numbers are placed in a multiplication sentence does not impact the answer. For example, students can use the arrays to understand that 2 x 4 is the same as 4 x 2.

We, as teachers, must provide students with different representations of problems to foster flexibility and problem solving abilities. An array is a very useful representation to present to students to help them gain a stronger understanding of what is involved with multiplication; it allows students to create a visual depiction of what the multiplicative sentence represents. (Young-Loveridge 38,39).

Another strategy which may prove to be useful when children are beginning to learn multiplication skills is to have them write addition sentences for the multiplication equation. For example, 4 x 6 = 6 + 6 + 6 + 6. Writing addition sentences will aid in making a connection between multiplication and addition. (Van de Walle and Folk 128).

References:

Van de Walle, John, and Folk, Sandra. Elementary and Middle School Mathematics - Teaching Developmentally. Canadian ed. Pearson Education Canada, 2005.

Young-Loveridge, Jenny. "Fostering Multiplicative Thinking Using Array-Based Materials." Australian Mathematics Teacher 61(2005): 34-40.

Opposing Views on Teaching Methods

When discussing multiplication and how to teach it in the classroom, one can expect to hear conflicting thoughts and views. Some individuals believe the most successful method to teach multiplication is by rote memorization while others feel as though students should take a more constructive lead of their own learning through methods such as problem solving and invented strategies.

Recent research has proven that the most successful methods in fostering an understanding of multiplication in children is not by rote memorization but is by allowing children time to construct their own understandings.

There are many arguments that can be presented to argue for or against rote learning. Rote Learning is a technique that emphasizes memorizing information so that it can be recalled quickly and efficiently. Although it is often the method chosen by teachers when teaching multiplication, it does not encourage taking the time to truly understand what is being learned.
Rote memorization does indeed help students to recall their times tables quickly but are they truly understanding what multiplication is? Are they merely recalling information they have memorized or do they have an idea of how to use different strategies to find an answer? Those who support this type of learning often argue that thinking skills alone will not be sufficient in learning multiplication unless there is already a knowledge-base of memorized skills to work with.

If you decide that rote memorization is what you are aiming for when teaching multiplication, there are different strategies you can use to help your students remember their times tables. Some activities can be found on the website entitled http://www.multiplication.com/. The site proposes using strategies such as:

• Excessive Practice - Students should be given the opportunity to practice their times tables often. Only a few facts should be concentrated on each day and you should always start with some of the hardest to remember (such as "7 x 8, 8 x 8, 6 x 8").


• Flash Cards - Two sets of flash cards should be made for each student so that they may be used when the student has spare time that can be used to practice. It is important to encourage students to utilize these flash cards to ensure they remember their multiplication facts. It can be encouraged that students use these with other peers and with family members who could help them.


• Frequent Tests and Quizzes - It is recommended that when using a rote memorization style of teaching, that students be quizzed on their multiplication skills daily.

Despite the belief that some teachers hold in that rote memorization is beneficial, much of todays research proves otherwise. As is made evident by Levenson, Tirosh and Tsamir in their article entitled Elementary School Students' Use of Mathematically-Based and Practically-Based Explanations: The Case of Multiplication, children should be given the opportunity to relate their learning to their own lives and make new tasks meaningful rather than rely solely on memorization. A study was conducted to determine what type of learning children used when doing multiplication problems. The results proved that before mathematical instruction, students tried to understand multiplication by relating it to their everyday lives. They often drew diagrams to represent the problems they were faced with. However, after learning the times tables and different rules, they began to focus more on this and less on understanding through their own thoughts and experiences. They began to focus on these 'rules' that they had learned and no longer focused on their own understandings to explain their answers.

There are many important issues raised in this study that a teacher may want to consider when deciding which method to use when teaching students multiplication: Is this how you want your students to learn? Do you want your students to do something because they are told to do it in a certain way? Or do you want to give them an opportunity to create their own knowledge?

Reference:

Levenson, Esther, Tirosh, Dina, and Tsamir, Pessia. "Elementary School Students' Use of Mathematically-Based and Practically-Based Explanations: The Case of Multiplication." Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 3(2004): 241-248.

Introduction

As a part of one of my Education courses, I have decided to pursue the issue of understanding multiplication. I felt that this was a very important issue to investigate because there are different beliefs as to how multiplication should be taught in the classroom. Some individuals believe that students should memorize their multiplication tables while others believe that children should learn to understand why and how you multiply. My aim is to provide further clarification on how to help students understand and learn multiplication.