Multiplication Memory Nightmares? Let's Set the Record Straight
While having lunch with Laura one weekend, we discussed the constant pressure our students with dyslexia face to memorize their multiplication facts. Our kids hear it from teachers, parents, peers, and themselves. They hear that they would be better at math if they only knew their facts (meaning, they were able to recall abstract ideas like 4 x 8 automatically). I'm here today to dispel this notion that quick recall of math facts equates to high grades in math class and long-term successful mathematics learning. Okay, it might for some, but for most of our children, this is not the case. The memorization of mathematics facts usually is slow, painful, and anxiety-ridden. Kids quickly learn to hate math, to hate the pressure, and to hate themself.
Let's be honest, memorization can be quite difficult for people with dyslexia. Quick recall of facts can be quite difficult for people with dyslexia. It turns out that the part of the brain that is most greatly impacted by dyslexia is in fact a major player in where the brain processes multiplication, recalls things, and houses our memory.
I am not saying that our children cannot learn their multiplication facts. Of course they can. Our kids can memorize their facts. They can slog through the process of looking at a card over and over and over and over again until the fact sticks. The questions are: Will they remember the fact the next day? The next week? The next month? For the all-important AZMERIT test? Maybe. Okay then, let me ask you this...do your kids remember their spelling words on Monday after they passed the test on Friday? Most of you probably just shook your head no. Most of our kids memorize their words for the Friday test and then forget the spellings as soon as the test is over (or soon after). So, why in the world do we expect it to be any different when learning math facts?
How about we take a different route? How about we focus on helping our kids learn what multiplication means and how the different facts relate to each other so that they have strategies for figuring our facts when they get stuck? I can guarantee that focusing on strategy building will lead to fluency and automaticity over time. It will also lead to confidence in, excitement for, and a love of mathematics that might not be seen otherwise. We focus on strategy development when teaching reading and spelling to our children with dyslexia and it works wonders! Finger spelling, tile movement, tapping syllables are all strategies for learning to read and spell. These strategies take time to develop, but they guide our kids on their journey to learning the 96 spelling rules of the English language. So why not do the same for mathematics? Why not focus on developing strategies that will guide our kids through the process of learning multiplication facts?
Here are a few strategies that Laura and I strongly believe in:
1) Using a multiplication fact that one already knows to find the answer to an unknown multiplication fact.
2) Using the commutative property to reduce the number of facts that someone has to learn.
3) Using the distributive property to turn unknown facts into known facts.
4) Focusing on knowing the 0s, 1s, 2s (doubles), 5s, 10s, and square facts, so that one can use those to figure out other facts.
These strategies are really helpful, but they work best when we understand multiplication to be about groups. When we "times something by something else", we are making groups. For instance, if given the fact 3 x 4 (often said as, "3 times 4"), we should think in our heads, 3 groups of 4. Using this new verbiage, we can create a physical model of 3 x 4. We can make and see 3 groups of 4 items on the table and, more importantly, in our mind. So often when our kids with dyslexia hear "3 times 4", they think about a clock. When presented with grouping as the definition of multiplication, they have a tangible reference point for making mental models. They have all made groups in their lives or been put into groups. Grouping makes sense. It is also very explicit.
Now that we have a common definition of multiplication, let's explore how the four strategies are quite useful when learning multiplication facts...especially when trying to figure out facts that are troublesome for most.
One of the hardest multiplication facts for kids to remember is 6 x 7, or six groups of seven, so that is the one that we are going to tackle here.
My first thought is that I know 7 groups of 7 (7 x 7) is 49. Square numbers seem to come easy for me. I can picture the square that they make and they seem to have a rhythm when said aloud. Since 6 x 7 is just one less group of 7s, I know that I can subtract 7 from 49 to solve 6 x 7:
6 x 7 = ??
I know --> 7 x 7 = 49
I also know --> 6 groups of 7 is 1 less group of 7 than 7 groups of 7
OR 6 x 7 = (7 x 7) - (1x 7)
Therefore --> 6 x 7 = 49 - 7
So --> 6 x 7 = 42
In this first example, I used my knowledge of Squares (7 x 7) to figure out what 6 x 7 equaled (Strategies 1, 3, and 4).
But, maybe subtraction is difficult for some people to compute mentally, which is a truth for many of our dyslexic students because of how dyslexia impacts the brain. Well, in that case, we can use the commutative property to figure out the unknown fact. The commutative property says that I can multiply numbers in any order. Therefore, I could switch 6 x 7 to 7 x 6. Now, instead of working with groups of 7, I am working with groups of 6! I know that 6 groups of 6 is 36. Since 7 groups of 6 is one more group of 6 than 6 groups of 6, I can add 6 to 6 groups of 6 to solve 7 x 6:
7 x 6 = ??
I know --> 6 x 6 = 36
I also know --> 7 groups of 6 is 1 more group of 6 than 6 groups of 6
OR 7 x 6 = (6 x 6) + (1x 6)
Therefore --> 7 x 6 = 36 + 6
So --> 7 x 6 = 42
In this second example, I used my knowledge of both the commutative property and squared numbers to figure out what 6 x 7 or 7 x 6 equaled (Strategies 1, 2, 3, and 4). The commutative property is also very important to know because it reduces the number of multiplication facts that a person has to learn by half. If one knows that 4 x 2 is 8, then they know that 2 x 4 is also 8, because you can multiply numbers in any order.
The last strategy, using the distributive property to figure out unknown facts, is a little less obvious, but it is one that many people use all of the time when doing mental math. Let's go back to 6 x 7 for our exploration of the distributive property and how it helps people learn their facts.
Let's say that I do not know my Squares. I don't know what 6 x 6 is or what 7 x 7 equals. Instead, I know my 2s (also known as Doubles) and my 5s. Brilliant! I can now figure out what 6 groups of 7 equals! Here's how my 2s, 5s, and knowledge of the distributive property helps me figure out 6 x 7:
6 x 7 = ??
I know --> 7 = 5 + 2
I know --> 6 groups of 5 = 30 (same as 5 groups of 6)
I know --> 6 groups of 2 = 12 (same as 2 groups of 6)
I know --> 6 x 7 is the same as 6 x (5 + 2)**
**I just switched 5 + 2 for 7
Distributive property --> 6 x 7 = (6 x 5) + (6 x 2)**
**I distributed the 6 across the 5 and 2
And now --> 6 x 7 = 30 + 12 = 42
Using the distributive property looks like a lot of steps, but I have a feeling that most of you use this property quite often when multiplying and/or adding numbers.
In the third example, I used my knowledge of 2 and 5 facts and the distributive property to figure out a difficult multiplication fact (Strategies 1, 3, and 4).
Again, I ask you, why do we force our kids to learn multiplication facts with flash cards? We should be capitalizing on what our kids know already. Thankfully, there are so many other methods to use that center on what kid's know than flash cards. Plus, these other methods help our kids build mental models of multiplication, see relationships across the facts they are learning, and gain confidence in their math knowledge. There is no rule stating that flashcards is the best way to learn multiplication facts. In fact, there is lots of evidence showing the complete opposite. So, why not try something new? Why not look at alternative methods for working on facts? Why not make math meaningful and enjoyable?