Multiplication Woes

Multiplication and multiplicative reasoning can make a world of difference in a student's mathematical development, however, for many it becomes a stumbling block that slows them down, sometimes to the point of hindering their studies in high school and limiting their post-secondary options.

There are many different ways to understand multiplicative expansion (the word "growth" can become confusing when students begin to multiply fractions and integers). One way is to see it as a series of addition problems, such as:

2
2+2
2+2+2
2+2+2+2
2+2+2+2+2

and so on. As we move down each row, the multiplier increases by 1 so that:

1x2 is 2
2x2 is 2+2
3x2 is 2+2+2
4x2 is 2+2+2+2
5x2 is 2+2+2+2+2

 You can also say it as "3 two's make 6" or, "3 groups of 2 equals 6".

Another way to represent this is to use grouping. Students can make piles of manipulatives such that each pile has the same number of items in it. This time, let's use multiples of 7. In this case, each pile would have 7 pieces. Let's say they wanted to know how many 4 groups of 7 are, or, 4 x 7. Students can use various strategies to determine the total.
They can:

• count up the total number of pieces by either counting each individual item in all of the piles
• start counting-on using their understanding that the first pile has 7, then continuing counting the remaining piles from 8 onward
• count each group "by 7's", also known in some circles as "skip counting"
• count the number of groups and use their remembered answer for 4x7

Each of these stages show a different level of mastery of the concept.

However, piles of manipulatives, or circled pictures of groups of objects on a worksheet have a limited usefulness when it comes to visualizing the patterns that are common to multiplication.

For this reason, we can try and move to an area model as shown below. It is called the area model, because the solution to the multiplication problem also represents the value of the area of the rectangle. Area is another way in which multiplication can be visualized, and it also shows a practical application of the concept.

In our example the number of units in each row is 7, while the number in each column is 6. We have 6 rows of 7, or 6 x 7 units in the rectangle.

We can look at this model in two ways. We can look at the columns (7 columns of 6 units each, as shown on the left), or we can look at rows (6 rows of 7 units each, as shown on the right). We have simply lined up the groups into columns or rows to make counting, as well as visual representation, easier.

The grid lines in these pictures don't have to be there for the model to work. Simply knowing the base and the height of the rectangle gives enough information so that the multiplication problem can be solved, and the area found.

Adding the grid lines helps us see the groupings involved, and makes the visual representation of the problem clearer, particularly for students who have not yet reached mastery. Approaching the same problem using groups of rows and repeating it using groups of columns helps reinforce the key principle of commutativity, in which the order of the numbers multiplied does not change the final product.

The grid model can be used with manipulatives that allow for columns of units to be connected, such as unifix cubes, multi links, Lego, or square tiles. These columns can be put together to form the rectangle that represents the problem.

Once the multiplication concept has been explored, students will eventually need to learn to access those facts quickly. There are a number of options that can help with this including:

• learning to skip-count (counting by a number, such as 3-6-9-12-15-18-21-24-27-30-33-36-39-42-45 etc.
• classroom games
• finding number patterns to follow (even numbers for 2's, ends in 0 for 10's, digits add to 9 for 9's etc.)
• recognizing patterns in daily life (eggs come in 2x6=12; a case of canned vegetables has 4x5=20 cans, etc.)
• intensive answering, such as with regular timed tests, Mad Minutes, etc.
• rote repetition
• written tables such as the one to the right, which are given blank for students to complete and mark for patterns
• musical chants /songs
• classroom charts
• calculators 

Each of these has its place at various times, however, students who can spend less effort to retrieve these facts do better as more concepts are introduced in higher grades.

The over-use of rote methods, and the dawn of Bloom's Taxonomy, have made the task of having students memorize their times-tables unpopular in the classroom. This is slowly changing.

The problem is not so much that students spend time memorizing these facts, which is admittedly a lower-level task, but that if they do so without an understanding of how multiplication actually works, the knowledge of "facts" will have limited value as more complex mathematics are introduced.  Students who understand the language, the commutative property, the groupings, and that multiplication is an advanced form of addition, can show how multiplication patterns continue, and likewise, how using the inverse operation of division causes the pattern to reverse, will be well equipped to apply it to fractions, decimals, integers, algebra, etc. They will also be better able to handle related concepts such as area and volume.

What is 4?

I'm currently reading The Glass Wall: Why Mathematics Can Seem Difficult by Frank Smith.

Early on, he describes how number is not the same as quantity. He uses 4 as an initial example, but then moves on to use a large number, somewhere over 7 hundred million, to demonstrate that the number is valid whether you have an associated quantity of something that it represents or not. Number is number. However we develop number sense, he argues, it occurs separate from natural language which tends to be ambiguous where math, buy its nature is not ambiguous (at least not to those who understand it!).

So, what in fact is 4? How do we truly understand the concept of 4 (or any other number)? Smith argues that a number can only be put into context when it is compared with other numbers. He sees mathematics as a separate existence than the rest of the world.

I'm not quite sure if this rings true for me or not, but it is an interesting thought to explore.

This also made me wonder what certain young mathematically inclined students might think about it, which led to the following idea for math enrichment.

Lesson Plan Idea: What is 4?

Students are challenged to brainstorm how they would explain the concept of 4 to people who had no numeracy (aliens, young children, etc.). They then move together into groups and share their ideas. The group chooses several to share with the class. One person (either the teacher or another student) plays the role of the learner while the students attempt to explain the concept. The learner should do their best to avoid using any previous mathematical knowledge and base their "understanding" purely on the information given by the students.

Class discussion should include:

•  counting--number/object connection; meaning of each number name
•  quantity--did they use concrete items (manipulatives) to demonstrate their ideas, and how successful might this be in getting the idea of number across without ambiguity (were the shape of items, colour, function or other characteristics confused with sense of number)
•  use of geometry and/or other drawings or models
•  other ways of relating the concept
•  would their system of explanation work for very large numbers, fractions, decimals, negative integers, zero, etc., and if not, how could they adapt or change it so that it will
•  was this difficult, and if so, why do the students think it was

The activity is open-ended and intended to provide deeper insights into the complexity of seemingly simple mathematical concepts.

Other related lessons might include:

• working with different number systems
•  working with different bases 
• writing computer programs to solve very basic mathematical problems (using a machine-based language or something without built-in mathematical algorithms that the students can access)
• a study of the historical use of "zero" and what it really means in mathematics (hint: it has a more complex meaning than simply "nothing")

For more math activities, see the Lemonade Math Page