## Monday, March 19, 2007

### My Learning Process

If time had permitted I would have liked to dive further into this research to learn more about these methods that have proven to be the most successful. I would also like to research more invented strategies as understanding how children reason will give me great insight for when I begin my teaching career.

Overall, my journey into multiplication and, on a broader level, teaching mathematics in general, is just beginning. There is still much out there to learn and I am certain that with experience, I will become familiar with more strategies and with methods that I will find most useful when teaching others.

### Useful Resources

As a teacher, finding valuable resources to supplement and enhance your learning environment is essential. Several resources I have gathered with respect to multiplication prove to be very useful for teachers and for students. They include:

- http://illuminations.nctm.org/LessonDetail.aspx?id=U109

This website contains a unit of lesson plans on multiplication. These lesson plans are designed to help construct an understanding of the process and properties of multiplication.

- http://www.dositey.com/muldiv/multip.htm

This website is composed of many different resources which can be useful when teaching multiplication. There are games, lessons, problem solving examples, and step-by-step instructions on how to go about multiplying double digits.

- http://www.multiplication.com/classroom_games.htm

This site provides various activities and games which can be used to strengthen one's understanding and abilities relating to multiplication. Many of these games can be played as a whole class or can be played independently by students.

### Invented Strategies

It is important that children are given the opportunity to devise their own strategies to help them gain understanding of the concepts they are learning. Van de Walle and Folk (194-196) discuss many of the invented strategies that are common amongst many students when dealing with larger numbers. They include:

**Useful Representations:**Children will often use a visual model to represent the problem they are presented with. This is often shown by using arrays.

**Complete-Number Strategies:**Students who are not comfortable with breaking a number down into its tens and ones, will resort to other methods when multiplying larger numbers. For example, they may use addition (23 x 6 = 23 + 23 + 23 +23 + 23 + 23 = 138).

**Partitioning Strategies:**When given higher number to multiply, students will sometimes break the numbers down in a variety of different ways. For example, some students may divide the numbers into tens and ones (32 x 3 : 10 x 3 = 30; 10 x 3 = 30; 10 x 3 = 30; 2 x 3 = 6; 30 + 30+ 30 + 6 = 96). Others may decide to partition by decades ( 30 x 3 = 90; 2 x 3 = 6), while others may find even more ways to divide the numbers.

**Compensation Strategies:**Children often find ways to manipulate numbers to allow for easier calculations (48 x 3 : 50 x 3= 150; 2 x 3 = 6; 150 - 6 = 144).

**Using Multiples of 10 and 100:**When presented with multiples of 10 and 100, students will often use the beginning part of the number to find the product. For example, for 300 x 12, students will often first multiply 3 x 12 and then use that to help them figure out 300 x 12. It is important to ensure students are not simply adding zeros to the end but are actually understanding why they are doing that.

Reference

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

### Possible Strategies

Multiplication facts can be mastered by relating new facts to already existing knowledge. While it is generally agreed that children should be given the opportunity to find their own methods of learning the facts, there are several strategies that could be introduced. They include:

**Doubles:**Multiplication by two should not prove to be a great problem to students who know their addition facts.

**Fives Facts:**Students should become familiar with counting by fives; thus, multiplying a number by 5 should be related to this counting by fives.

**Zeros and Ones:**Children should be given the opportunity to find their own reasoning as to why a number multiplied by zero is equal to zero and that a number multiplied by one is equal to that number. They should not be explicitly told these rules but should be able to explain why this is so. Upon gaining this knowledge, they will know 36 facts of single-digit multiplication.

**Nifty Nines:**The nines facts are among the easiest set to remember and they are often fun patterns to find. "Two of these patterns are useful for mastering the nines: (1) The tens digit of the product is always one less than the "other" factor that is not 9, and (2) the sum of the two digits in the product is always 9" (Van de Walle and Folk 151). Upon adding these two facts together, one will get the product of the two numbers. Another strategy which many children find both interesting and very helpful is one in which you use ten fingers to find the product. For example, your fingers are numbered 1-10 starting with your thumb on your left hand. You put down the finger representing the number you are multiplying by 9. Then, the number of fingers standing on the left of that finger represents the number of tens in the product and the number of fingers to the right represent the ones of the product. A diagram of this strategy can be found at http://www.multiplication.com/teachold/teach11o.htm.

These strategies are successful in helping students learn 75 of the possible 100 multiplication facts for single-digit multiplication! (Van de Walle and Folk 149).

To help with the 25 remaining facts, students can use the facts they already know along with mental addition. For example, the 4's can be learned by doing doubles and doubles again and the 3's can be learned by using doubles and adding on one extra set. (Van de Walle and Folk 152).

Reference:

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

### Some Properties to be Familiar With

**The Order Property (Commutative Property)**: Often, it is not evident to children that the order in which the numbers of a multiplication sentence are ordered does not impact the solution. For example, at first children are not aware that 5 groups of 9 and 9 groups of 5 are the same. One method that can be used to help children see this property is the use of arrays. (Van de Walle and Folk 130).

**The Role of Zero and One in Multiplication**: Children often have trouble with the fact that anything multiplied by zero is equal to zero. It is, therefore, much more beneficial to help children realize this in the form of problem solving rather than telling them the rule. By solving a problem, such as asking “how many grams of fat there are in 7 servings of celery with 0 grams of fat in each serving” (Van De Walle and Folk 131), they are more likely to understand the idea of why the answer would be zero. Multiplying something by one also tends to cause some confusion. Children should not be explicitly given the rules for these instances but should, instead, be encouraged to come to this conclusion on their own.

Reference:

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

### The Use of Calculators

One activity presented in by Van de Walle and Folk that involves using the calculator to help enhance the relationship between addition and multiplication is called "The Broken Multiplication Key" (130). This activity requires students to find products on the calculator without using the 'x' key. For example, 4 x 3 can be found by pressing + 3 = = = = (Pressing = will add 3 to the new product each time; you began with zero and added 3 four times.)

Reference:

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

### Including Word Problems in Instruction

It was found to be beneficial to structure a lesson around one to three problems as it develops and increases multiplicative thinking in students because time can be spent on discussing strategies, models and reasoning. (Van de Walle and Folk 128).

Reference:

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

## Monday, March 12, 2007

### Using Arrays to Increase Understanding

**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.

## Tuesday, March 6, 2007

### Opposing Views on Teaching Methods

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.