Alternative Math Methods

The Count — Thu, 17 Sep 2009 04:04:30 +0000

A lot of math is broken down into processes. Start with a problem, do this, do that and, voila, you have the answer. Most people in America learn the same processes for doing most basic arithmetic, and it’s easy to forget there might be other ways to solve these problems. The positional decimal system developed by the Indians (of India) is the standard taught in every school I’ve ever been in, but it’s not the only way to multiply two numbers1. The following is a description of another method I like for its enforced structure and visual guides.

The Lattice method of mulitiplication2 was published in Europe by Fibonacci in the early 13th century. The lattice, though time-consuming to create for variable lengths of multiplicands, brings a helpful bit of structure when teaching multi-digit multiplication to young students. Use the animation below as a guide when reading the instructions, and I think you’ll find this method very intuitive and interesting.

Draw the lattice with the same number of squares per side as the number of digits in each multiplicand. Add a diagonal line from upper-right to lower-left for each square, extending the line out the bottom-left of the left-most and bottom-most squares. For students, I’d recommend pre-creating the lattice for them and photocopying.
Place the digits of the multiplicands across the top and down the right side of the lattice.
In each of the squares, multiply the multiplicand digits at the top and far right of the square, placing the ones-digit of the product in the bottom-right of the divided square, and the tens digit (0 if none) in the upper-left portion of the square. Repeat for all squares in the lattice.
Note how the triangular sections of adjacent squares form diagonal “rows”. Starting with the lower-right triangle, sum these rows, placing the ones digits of the sums in the spaces formed by the extension lines outside the lattice. If a tens digit exists, add it to the next row’s total. Note the color-coding of the rows in the example image, which would help a new user of this method. Perhaps shading every other row would be appropriate when pre-making the lattice for your students?
When completed, the final product can be read down the left side and across the bottom of the lattice.

How does it work? Well, a little study will show that this is really no different than the long multiplication we all know, only some of the operations have been re-ordered; essentially, all of the carrying has been moved to the end of the process. What I find so elegant about this method is the complete lack of “spacing”, padding, or “shifting” that we use in long multiplication which is so difficult to get students to understand and remember. Long multiplication can also be confusing when attempting to place carry values from two different operations in the same space, which can’t happen using the lattice.

Now, I’m not saying that this lattice method should be the new, one-and-only method of multiplication you should use. The lattice is obviously a little cumbersome to draw, especially for larger numbers, though it doesn’t take much more room than long multiplication would. I wouldn’t necessarily expect students to perform multiplication faster using the lattice, but they shouldn’t perform slower, either. As I said, these methods are really very similar when broken down to their elements. While you could certainly use this as a primary teaching method for long multiplication, you could also just present it as an example of the fact that, while math is universal, the methods for performing math are not.

The Count

Making more math geeks

The Count — Sat, 01 Aug 2009 22:41:40 +0000

You may have noticed that I like math. This is not a new thing for me. Math has been a part of my life for as long as I can remember, and not in the organic sense of counting blocks or birthdays. My parents, both with degrees in mathematics, spent a good amount of time making sure I had no troubles in school in any field, but especially math. Now that I have kids, I’ve been doing my own part to make sure they never need to spend valuable school time figuring out the math on the board, they already got it at home. Note, this doesn’t exactly make me a favorite parent with their teachers; making kids “bored” in school doesn’t endear me to them, but teaching them is such fun I can’t really help myself.

Counting from 1 to 10 is something kids get from many sources; television, children’s books, etc. I won’t cover that here, and my kids could count a bit higher than that when I started with these exercises. The following are several good methods to used to give kids the tools to solve most any math problem they can think of.

Number Line: The first basic math technique you should teach a child is the number line1. This gives kids a handy way to compute sums and differences without breaking out the fingers and toes. Try creating several worksheets where each row has its own number line followed by a single-digit addition problem.

Go through several of the problems on the sheet with your child, explaining the process of using the number line to solve the sum:

Find the first number being added on the number line and place your pencil on it, making a mark.
Starting on that first number, move your pencil one number at a time, counting up to the 2nd number being added.
When you finish counting, your pencil will be on the answer to the problem!

This technique will have your kids doing easy sums very quickly. You can start this before your kids can write their number; just have them circle the answer on the first few. Progression from this most basic sheet can be done by making the number lines longer (to 20 or more), as well as using a single line for multiple problems (which helps disconnect the line from the specific problem). Even farther out, this makes a great tool for teaching subtraction, too!

Verbal quizzing: In the car running errands, or at dinner in a restaurant? Here’s a handy technique that will have your kids learning instead of babbling at each other or staring blankly at SpongeBob: Doubles and Halves. Simply put, just ask your children “What’s the double of [insert number here]?”. Start small with single digits (0 to 5). Your kids will be happy you’re involved with them during what could be a boring time, and they’ll be less inclined to act up. Progression for this technique can be the obvious use of larger numbers, though keep in mind that double-10 is probably easier than double-8 for your kid. You can also switch it up with “What’s half of [insert number here]“, starting with small even numbers and moving on to larger or odd numbers.

Number-based Games: Games are an easy way to keep kids interested in math, especially ones where they have to do math to see who won! Dice games like Yahtzee2 and Farkle3 or card games that have scores, like Uno4, keep children amused while they play, and give them great practice adding larger numbers when computing the score. Some of these games also teach basic multiplication (such as Yahtzee’s 3 dice showing 5′s gives a point score of 15).

Mistakes: Mistakes are a part of life, and a big part of learning math. How you deal with these errors can have a strong impact on how long your kids maintain interest and how often they’ll make the same mistakes. Don’t be quick to correct errors. It’s important to let the child know that their answer was incorrect, but do so in a manner that both isn’t triumphant and doesn’t give away the correct answer. Kids take much more pride from an answer they got on their own, even if it takes a few tries. If they get frustrated or start to “guess”, be ready to divert them to another problem saying, “We’ll come back to that one,” or offer a better path to the solution (such as breaking the larger numbers into smaller ones).

So there they are, a good set of tools and techniques for turning your kids into hellions at school giving your kids a head start in math. I hope you use them well. Of course, these aren’t the only math instruction techniques out there; they’re just a few that I’ve found worked with my own kids. Please feel free to comment with ideas and stories of your own, I may just use them!

The Count

Number Line on Wikipedia
Yahtzee on Amazon
Farkle on Amazon
Uno Card Game on Amazon

Discrete Ideas » kids

Alternative Math Methods

Making more math geeks