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.

Why I Hate Flashcards

Way back in the 1970's, I was a student in a grade 1-2-3 split class. It was not a small school, so I would guess that it was one of the many educational experiments going on during that time.

There are two things I remember well about my grade 1 teacher: how hard she spanked and her incessant use of flashcards. The words, "if you're so smart..." still set my teeth on edge. I didn't think I was smart at all. Before we could read actual stories, we each needed to recite the current list of flash-words correctly, even though they were completely out of context. Only then, when we were tired and bored out of our minds, would we have the "privilege" of reading aloud in front of the class a page from the classroom reader. That classroom reader, "Just For Fun", was decidedly NOT fun for me!

It may be that I have an unfounded bias against flashcards based on my experiences with that teacher.

But aside from those experiences, there is a lot to be said against "flashcard" style instruction. Rote and memorization techniques do not involve any higher cognitive functioning; there is no deep thought involved in memorizing a set of facts. There is no "grey area" to be explored or probed; there is no room for reflection or making connections with other areas. It is a sort of "desert learning" where only that particular piece of information is deemed relevant. For those familiar with Bloom's Taxonomy, you will recognize that flash cards represent remembering, which sits at the bottom of the list as the lowest of the lower-order skills.

Yes, there are some facts that simply beg for memorization, such as the alphabet, musical note names, capital cities, etc. For these purposes, flashcards may not be a bad choice. Likewise, some older students find writing facts out on index cards and reviewing them to be a useful study tool that combines kinesthetic (writing) and visual cues.

For math facts though, it is crucial that students work through the actual quantities to understand the patterns involved. These do not show up on flashcards; nor do "real life" applications of the concept. 9 x 9 can be solved in many different ways, and for students to gain numeracy they must be encouraged to explore those ways for themselves. Some ways they might do this can be found here. Only once they have gained the concept, the "understanding" and "applying" on the next levels of Bloom's Taxonomy, is memorization relevant. This doesn't mean that a student who learns the concept quickly should need to continue working with manipulatives ad nauseum, but that they should be able to demonstrate the concept behind the fact before moving on.

The same goes for reading. Using flashcards to memorize words deprives students of many of the other different ways to decode and comprehend words and phrases. There are many aspects used in reading including contextual clues, phonics, word shape, structural patterns, linguistic patterns, and many others as well. Not many of us have the mental capacity to memorize every word we will ever read; we clearly need to  use a variety of strategies in order to become fluent readers and gain higher level literacy skills. Isolating one skill and focusing on that alone (usually either flashcards or phonics) does not reflect the reality of how readers process print. It is true that it might be useful for students with specific needs to reduce the visual or auditory component for a very short time in order to allow them to focus on specifics, but the students who require this sort of breaking down tend to have special needs related to sensory issues; and their needs are quite different from the needs of the general student population.

Going back again to the musical note example, could we not improve the experience by making it richer with a wider variety of sensory input? Imagine a simple program that not only shows the note written on the staff, but also plays the tone. Now the student has an opportunity not only to learn the note visually, but also to gain an auditory sense of it at the same time. This is not a complicated thing to program, in fact, there are many websites, programs and apps that do this that have been around for years. If you want to avoid electronic technology, you can always do this with a piano or other instrument as well. Of course, you can also take this much further by reversing the process, and add composition, duets, etc. to the mix. None of this is new.

This is only one example. I challenge those of you who use flashcards to find a replacement or means of enriching the experience.