Dear Lattice Multiplication,
You are by far the coolest thing I've learned of late. This probably makes me sound like a math geek, and I'm not going to defend that, because to me, finding a completely new way to solve a math problem is, in and of itself, awesome. So let's see if I'm the only math geek around here.
Let's multiply two numbers using the lattice method. 528 x 34 = ? (I've color coded these steps so it's easier to follow). Draw a 3 by 2 grid with diagonals in each box, making a lattice grid. Write one factor, 528, across the top of the lattice grid, and the other, 34, to the right side of it. Now multiply the top and side numbers and write the products in their corresponding grids so that the number in the tens place is on top of the diagonal and the number in the ones place is on the bottom of the diagonal. Then starting on the right side, add all the numbers along each diagonal column, carrying over to the next column the when necessary. The answer is the all the numbers you get reading from left to right. Magical? I'd say so.
This all started when Dear Daughter was having trouble with a multiplication method taught in school. Now I've only ever learned one multiplication method -- the traditional method. I came from a country and culture where rote memory was highly valued, and probably why that part of the world scores high on mathematics standardized tests. By second grade, we had already memorized the entire multiplication table, 1 through 10. Notice I said 'memorized', not 'learned'. I don't remember using math manipulatives, counters, numbers sticks, or number beads. We learned multi-digit multiplication straight from the exact method the teacher modeled for us on the blackboard. We used our multiplication number facts (that we memorized), and we added the products to find the answer. I never knew or questioned why we did that. I just knew that when I follow that method, my answers come out correct and I get a good grade on my test.
Fast forward to graduate school for my degree in education here in the US. Fourteen years ago, in my Math Education class, I was introduced to Everyday Mathematics, a math curriculum developed by The University of Chicago. It was a curriculum development that began before 1990, and through years of writing, field-testing, rewriting, and publishing, it became a solid Pre-K through Grade 6 math curriculum. Everyday Mathematics aims to provide children a strong mathematical foundation through a firm grasp on number sense: really understanding what numbers mean, how they relate to each other in operations, and how they are used for problem solving. Moving away from only rote calculations or simple memorization of number facts, EM teaches children to learn number concepts in everyday situations, using hands-on activities and manipulatives, and teaches a variety of strategies to solve one single problem.
Those were the ideas that set apart this curriculum from the way I learned math as a child. I knew I was supposed to carry over the one (as I later learned is called 'regrouping' in English) when a sum is more than 9, but I really saw the reason why when in my grad school Math Ed class, we were given blocks to solve these simple addition problems. While I probably made that connection before using those blocks, I know I didn't understand it when I learned math from the blackboard. Seeing it taught this way was quite an light bulb moment for me. It turned the dimmer light from dim to very bright, just by connecting those blocks to the number operation in front of me.
When Everyday Mathematics first came out, it was controversial in the sense that the grownups at school and at home were not used to teaching and learning math this way. But many years later, I was happy to find out that DD's school uses EM.
So DD brings home math homework that required solving multiplication by the 'partial-product method'. What the heck is that? I only know how to multiply one way, which was the blackboard way. So I had to visit www.everydaymathonline.com to get my own tutorial for the partial-product multiplication method, which turns out to be just breaking the factors down by place values before multiplying, and then adding up all the products to get the final answer.
(Quick tutorial. If I want to multiply 528 by 34, I break it apart like so: 500 x 30, 20 x 30, 8 x 30, 500 x 4, 20 x 4, 8 x 4, and add those answers together).This method helps with understanding place value, and, the numbers' relationship with one another. It doesn't come logically for some of us who learned it the traditional way, because we never saw the value and function of place values in these operations. We just did as we were told to do, and we knew those answers would come out correctly. But to break them down and understand them means true knowledge of how those numbers operate in solving a multiplication problem.
That's how I came across the lattice multiplication method -- from the online tutorial. And that's when I thought, wow, EM is awesome. That is not to say that I will do multiplication this way from now on. I will probably always do it the traditional way (just like I still recite my multiplication table in Chinese, since that's how I memorized it). But it is a different way to tackle the problem, as is the partial-product method. Not only do these methods build up to the traditional method, one of these methods is going to make sense for a child. For some children, the traditional method may not make sense right away. And that is why I believe a variety of strategies is useful for teaching children how math works. And what about making kids really confused with all these methods, you ask? Well, that's why teachers and parents are there to help. After all, if you go from point A to point C skipping point B, how well can you truly understand point C?
If you think about it, the traditional way is really a 'short cut'. If you do it carefully, abiding all the rules, you will get the right answer. But if you understand the partial-product method, you really understand what multiplying those numbers mean. When a child understands that, then using the short cut, or traditional method, is much more meaningful. Moreover, the lattice method, when you dissect it, is essentially the traditional method that, put onto a grid, better allocates the numbers in a spatial way as not to confuse columns when adding. But for the math geek in me, it is absolutely fascinating!
As for myself, I know that there are certain topics in middle and high school math that I am still unsure of (problem solving with ratios and percentages), and that is because I never truly understood the mathematical concepts behind them. And those physics problems about one train traveling at one speed in one direction and another train at another speed in the opposite direction? I still cannot solve those problems because I missed some key concept somewhere in math to understand it in physics.
Some grownups do not like Everyday Mathematics because it teaches different methods and roundabout ways to get to the answer. But I think these grownups have used the traditional way for so long and it's so ingrained in them that the conceptual way doesn't 'ring a bell' because that understanding was never there in the first place. But while I will continue to use the traditional method to multiply, I am happy that DD has learned the partial-product method so that when she eventually exclusively uses the traditional method, she will know where it came from. Faster and easier is just doing the math; understanding why it's faster and easier is building a firm foundation in mathematical number sense.
So this is what I believe mathematics learning should be for children:
- learning a variety of strategies to solve problems makes the connections necessary to fully understand number operations and problem solving.
- applying math in everyday situations connects numbers to real life problems.
- a combination of math facts (from memorization) and math concepts (understanding number sense) is the best way to build a strong mathematical foundation.
Lattice multiplication, I didn't have the privilege to meet you until thirty-eight years of age, but you and your 'posse' have allowed me to see the beauty in teaching and learning mathematics in a different and better way. Even though I learned about Everyday Mathematics way back in grad school, this -- meeting you -- is my 'real life situation' where I truly understood the benefits of this curriculum. And although I won't be using you for solving multiplication problems, you are still magical and fascinating to me. Your presence has enlightened me in ways only a math geek can understand.