Moving Towards Mastery
Mastering the 45 addends is an important step on the way to making computation easy. Addition is simple, if the concepts are understood. 5 + 7 is the same as 7 + 5 and when 7 and 5 get together it’s always going to end in 2…so 17 + 5 and 15 + 7 are easy and students can also see that 37 + 5 is basically the same problem as the single digit problems with tens “just along for the ride.” You would be amazed at the number of students who don’t get that simple concept. They’ll come up with 21 or 23 instead of 22 when adding 15 + 7. They can also use the simple “want to be a ten” algorithm to make it easy: 7 takes 3 from 5 making one ten and two, OR 5 takes 5 from 7 making one ten and two. Either way it’s 12, and the best way to do it is the way the student likes best.
This method allows the student to get off their fingers by making “a ten and some more” when adding two numbers. As it turns out there are only 45 combinations…once students understand this simple “want to be a ten” algorithm addition becomes a lot easier and they can tackle bigger problems on their own. Then it just comes down to practice and repetition. Use a wide variety of problems to practice this skill and teach other concept the same time in order to keep the practice from becoming mind numbing drill work which will also turn students off to math.
Using their fingers is a step on the way to mastery of addition facts, unfortunately many students remain stuck at this step all the way into adulthood. For kinesthetic learners using fingers and hands IS IMPORTANT: that’s HOW they learn, and you need to help them move past this: manipulatives are a great way to move them into “doing it their heads.” For young students using fingers and hands is just natural…you can also spot the kinesthetic learners because they will rely more on their fingers and be slower to move on from them. This does not mean they are “slow” or any less able than visual or auditory learners, they grasp concepts just as fast or faster than those with other learning styles. We also find when it comes to sports and other activities requiring hand eye coordination (like arts and crafts) they often excel. Using your fingers is great! AND you need to get past that stage if you are going to be fast at addition and attain mastery. Being fast at addition leads to easy mastery of multiplication as an added bonus. They may even like math, why wouldn’t they if it’s fun and easy?
Many speed reading courses incorporate the use of the finger to guide the eye along the page, some use this to start, and then drop it for other courses this is the main stay of the course. Adding more sensory input increases learning, and in the case of reading the hand and the eye are integrally connected. The point is you want to encourage students to move through this step when it comes to the mathematics NOT discourage or skip the step all together. Some students will naturally NOT use their fingers when doing mental calculations…for those that do use their fingers later it will become a handy-cap. Counting quickly makes math easier, because all math is is counting; however, don’t confuse computation with the mathematics. The mathematics is the use of computation and critical thinking skills to solve problems and express reality numerically.
Addition and subtraction as well as multiplication are just counting quickly. They are among the first steps to understanding math, and they should be mastered to ensure success. Using fingers can lead to a loss of accuracy too, often children (and adults) are off by one sometimes even two.
Practice with the addends verbally, build walls and towers, play games like what’s under the cup, simple story problems and work sheets with pictures give the student the experience they need to make the transition from fingers to symbols to being able to do it “in their heads.” Drawing rectangles and other math concepts as well as making drawings of the manipulatives they use, help the student make sense of the symbols and see what they are doing. It also adds variety, and helps students (and teachers) see that you use the same skill sets all through the mathematics, which is why you often see me use third and fourth power algebra to teach addition and multiplication facts.
Indeed if you carry the concept far enough they can also get off the symbols as it were and do it ALL in their heads if need be, no paper or pencil. This was illustrated perfectly by a five year old who is able to factor trinomials in his head because he can see the pictures when he hears expressions like x^2 + 3x +2, he can see it and tell you the sides. Or if you tell him the sides (x+3)(x+2) he can tell you the whole rectangle not because he is seeing symbols but because he is seeing PICTURES. Further he is “cementing” his addends and multiplication facts into his memory. How much easier is it to see 6 taking a 4 out of a 7 to make 13 when presented with a problem like x = 6 + 7 than to do algebra? It’s also quite easy to see 6 + x = 13 or x + 7 = 13, especially if you give them a simple algorithm to solve these based concept of “want to be a ten.” He also gets a ton of positive reinforcement because people think he is a little genius which motivates children to do more. Never underestimate the power of simple praise.
Once they learn some basic concepts and understand what the symbols mean math becomes easy and even fun. Being able to visualize what you are doing makes all the difference, it also makes it MUCH easier to commit to memory because the mind works in pictures not symbols, so memorizing the 45 addends and multiplication tables is easier because the mind can store pictures much more readily than symbols. Then when it is time to be recalled, a picture or the symbols or just words can easily be retrieved from that place we call the long term memory.
Have you ever known anybody that remembers phone numbers by picturing the keypad in their head? They may even point to the numbers and move their pointer finger on an imaginary keypad in the air as they are recalling the number. This is a visual kinesthetic way of storing long numbers. The brain works with pictures and this makes it easier to get the information out. How much simpler is it to add two numbers together than recite seven to ten digits? Especially if you have a method for visualizing them if you somehow forget?
A simple exercise: ask a student to picture a cow. Then ask if they saw C O W or a picture of a cow? Ask what color was it? This lets you know they weren’t seeing symbols. The problem is with math most students have nothing to picture whether it’s algebra or simple addition. The “trick” if there is one is to get the information into the long term memory so it easily recalled and it’s pretty well proven that symbols, that is letters and numbers, are a difficult way to get information there.
Manipulatives are the perfect bridge to get information there. After all, it’s never storage that’s the problem it’s retrieval.