Advancing towards mastery

Mastering all 45 addends is an important step on the path to easy calculation. The addition is simple, if the concepts are understood. 5 + 7 is the same as 7 + 5 and when 7 and 5 come together it will always end in 2… so 17 + 5 and 15 + 7 are easy and students can also see that 37 + 5 is basically the same thing. same problem as single digit problems with tens “just for the ride”. You’d be surprised how many students don’t understand that simple concept. They will get 21 or 23 instead of 22 by adding 15 + 7. They can also use the simple “I want to be a ten” algorithm to make it easier: 7 takes 3 of 5 making one ten and two, OR 5 takes 5 of 7 making one Ten and two. In any case, there are 12, and the best way to do it is how the student likes it best.

This method allows the student to not worry about doing “a ten and something more” when adding two numbers. Turns out there are only 45 combinations… once students understand this simple “I want to be a ten” algorithm, addition becomes much easier and they can tackle bigger problems on their own. Then it all comes down to practice and repetition. Use a wide variety of problems to practice this skill and teach other concepts at the same time to prevent practice from becoming mind-numbing mock work that also leads students astray toward math.

Using the fingers is one step on the path to mastering addition facts, unfortunately many students remain stuck at this step well into adulthood. For kinesthetic learners who use fingers and hands IT IS IMPORTANT: that’s how they learn and you need to help them through this – manipulatives are a great way to move them to “do it with their head”. For young learners, using fingers and hands is just natural… you can also spot kinesthetic learners because they will trust their fingers more and be slower to move away from them. This does not mean that they are “slow” or less capable than visual or auditory learners, they grasp concepts as quickly or faster than those with other learning styles. We also find that when it comes to sports and other activities that require 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’re going to be fast at addition and achieve mastery. Being fast at addition leads to an easy mastery of multiplication as an added bonus. They may even like math, why wouldn’t they when it’s fun and easy?

Many speed reading courses incorporate the use of the finger to guide the eye along the page, some use it to start and then leave it for other courses, this is the main part of the course. Adding more sensory information increases learning, and in the case of reading, the hand and the eye are integrally connected. The point is that you want to encourage students to move through this step when it comes to math NOT to get discouraged or skip the step entirely. Some students will naturally NOT use their fingers when doing mental calculations…for those who do use their fingers later, it will become a handy cap. Counting quickly makes math easier, because all math is counting; however, do not confuse computing with mathematics. Mathematics is the use of computational and critical thinking skills to solve problems and express reality numerically.

Addition and subtraction, as well as multiplication, only count quickly. They are among the first steps in understanding mathematics and must be mastered to ensure success. Using your fingers can also lead to a loss of accuracy, often children (and adults) get it wrong by one, sometimes even by two.

Practice with addends verbally, build walls and towers, play games like what’s under the cup, simple story problems and picture worksheets give the student the experience they need to transition from fingers to symbols to be able to do it “in their heads.” Drawing rectangles and other math concepts, as well as drawing pictures of the manipulatives they use, helps students understand symbols and see what they are doing. It also adds variety and helps students (and teachers) see that they use the same skill sets across all of math, which is why you often see me use third and fourth power algebra to teach addition and multiplication facts.

In fact, if you take the concept far enough, they can also pull out the symbols so to speak and do it ALL in their heads if need be, no paper or pencil. This was perfectly illustrated 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 them 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 IMAGES. In addition, he is “cementing” his addends and multiplication tables in his memory. How much easier is it to watch 6 get a 4 out of 7 to make 13 when presented with a problem like x = 6 + 7 than to do algebra? It’s also pretty easy to see 6 + x = 13 or x + 7 = 13, especially if you give them a simple algorithm to solve these “I want to be a ten” based concepts. He also gets a lot of positive reinforcement because people think he’s a little genius who motivates kids to do more. Never underestimate the power of a simple compliment.

Once they learn some basic concepts and understand what the symbols mean, math becomes easy and even fun. Being able to visualize what you’re doing makes all the difference, it also makes it MUCH easier to memorize because the mind works on pictures, not symbols, so memorizing the 45 addends and multiplication tables is easier because the mind you can store images much more easily than symbols. Then, when it’s time to remember, an image or symbols or just words can be easily retrieved from that place we call long-term memory.

Have you ever met someone who remembers phone numbers by imagining the keyboard in their head? They can even point to numbers and move their index finger on an imaginary keyboard in the air while remembering the number. This is a visual kinesthetic way of storing long numbers. The brain works with images and this makes it easier to get the information out. How much simpler is it to add two numbers than to recite seven to ten digits? Especially if you have a method to visualize them if you somehow forget?

A simple exercise: ask a student to imagine a cow. Then ask if they saw a COW or a picture of a cow. Ask what color was it? This lets you know that they were not seeing symbols. The problem with math is that most students don’t have anything to figure out, be it algebra or simple addition. The “trick”, if there is one, is to get the information into long term memory so that it is easily remembered and it is pretty well proven that symbols i.e. letters and numbersthey are a hard way to get information there.

The manipulators are the perfect bridge to get information there. After all, it’s never storage that’s the problem, it’s recovery.