By 
Moving towards mastery
Mastering the 45 addends is an important step on the way to making the calculation easier. The sum is simple, if you understand the concepts. 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 problem as single-digit problems with tens “just for the ride.” You will be surprised how many students do not understand that simple concept. By adding 15 + 7, they will get 21 or 23 instead of 22. You can also use the simple algorithm “I want to be a ten” to make it easier: 7 takes 3 of 5 to make one ten and two, OR 5 takes 5 of 7 doing one ten and two. Either way there are 12, and the best way to do it is the one that the student likes the most.
This method allows the student to relax by doing “a ten and a little more” when adding two numbers. Turns out there are only 45 combinations … once students understand this simple “I want to be a ten” algorithm, the addition becomes much easier and they can tackle larger problems on their own. So 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 the practice from becoming a mental numbing exercise that will also divert students from math.
Using your fingers is a step on the way to mastering addition facts, unfortunately many students remain stuck in this step until adulthood. For kinesthetic learners who use their fingers and hands IT IS IMPORTANT: this is how they learn, And you should help them get through this – Manipulators are a great way to get them to “do it with their heads.” For young students, using fingers and hands comes naturally … you can also detect kinesthetic students because they will rely more on their fingers 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 fast or faster than those with other learning styles. We also found that when it comes to sports and other activities that require hand-eye coordination (such as arts and crafts), they often excel. Using your fingers is great! And you need to get through that stage if you are going to be quick on the sum and achieve mastery. Being quick in addition leads to easy mastery of multiplication as an added bonus. They might 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 get started and then leave it for other courses. This is the main permanence of the course. Adding more sensory information increases learning, and in the case of reading, the hand and eye are integrally connected. The point is that you want to encourage students to move forward in this step when it comes to math, DO NOT get discouraged or skip the step all together. Some students will naturally NOT use their fingers when doing mental math … for those who use their fingers later on, it will become a handy hat. Counting quickly makes math easier, because all math is counting; however, do not confuse computing with mathematics. Mathematics is the use of computer skills and critical thinking to solve problems and express reality numerically.
Addition and subtraction, as well as multiplication, are counting quickly. They are among the first steps to understanding mathematics and must be mastered to ensure success. Using your fingers can also lead to a loss of precision, often children (and adults) lose one, sometimes even two.
Verbal practice with addends, building walls and towers, playing 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. and being able to do it “in their heads.” Drawing rectangles and other math concepts, as well as drawing pictures of the manipulators they use, help the student understand 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 throughout 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 too can step out of the symbols, so to speak, and do it ALL in their heads if necessary, without paper or pencil. This was perfectly illustrated by a five year old boy 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. Also, you are “cementing” your addends and multiplication tables in your 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 is also quite easy to see 6 + x = 13 or x + 7 = 13, especially if you give them a simple algorithm to solve this concept based on “I want to be a ten”. 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 simple praise.
Once they learn a few basics and understand the meaning of symbols, math becomes easy and even fun. Being able to visualize what you are doing makes a difference, it also makes it MUCH easier to memorize because the mind works on images, not symbols, so memorizing the 45 addends and multiplication tables is easier because the mind can store much more images. easily than symbols. Then when the time comes to be remembered, an image or symbols or just words can easily be 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 obtain information. How much easier is it to add two numbers than to recite from 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 COW or a picture of a cow. Ask what color it was. This lets you know that they weren’t seeing symbols. The problem with math is that most students have nothing to imagine, be it algebra or simple addition. The “trick”, if there is one, is to bring the information into long-term memory so that it is easily remembered and is fairly well proven that symbols, ie letters and numbers, they are a difficult way to get information there.
Manipulators are the perfect bridge to get information there. After all, the problem is never storage, but recovery.