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Get Them Off Their Fingers And Into Math
Moving towards mastery
Mastering the 45 complements is an important step on the way to easy calculus. Addition is simple, if the concepts are understood. 5 + 7 is the same as 7 + 5 and when 7 and 5 are added together it will always end up in 2… so 17 + 5 and 15 + 7 are easy and students can also see that 37 + 5 is basically the same problem like single-digit problems with tens “only during the ride”. You’d be surprised how many students don’t understand that simple concept. They will get 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 easier: 7 takes 3 from 5 making a ten and two, OR 5 takes 5 from 7 making one ten past two Either way, there are 12, and the best way to do it is what the student likes the most.
This method allows the student to get off their fingers by doing “a ten and a bit” when adding two numbers. As it turns out there are only 45 combinations… once students understand this simple addition of “want to be a ten” algorithms it becomes much 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 concepts at the same time to prevent practice from becoming mind-numbing drill work, which will also turn students off to math.
Using fingers is one step on the way 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 overcome this – manipulators are a good way to move them to “make their heads”. For young learners, using their fingers and hands comes naturally…you can also spot kinesthetic learners because they will rely more on their fingers and be slower to progress with them. This does not mean that they are “slow” or less capable than visual or auditory learners, but that 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 tend to excel. Using your fingers is great! And you need to get past that stage if you’re going to be fast at summation and achieve mastery. Being quick at 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 it to start, and then release it for other courses this is the main stay of the course. Adding more sensory input increases learning, and in the case of reading, hand and eye are fully connected. The point is that you want to encourage students to move through this step when it comes to math NOT discourage or skip the step all together. Some students will naturally NOT use their fingers when doing mental math…for those who do use their fingers later, it will become a practical sleeve. Counting quickly makes math easier, because all math is counting; however, do not confuse computing with mathematics. Mathematics is the use of calculation 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 in understanding mathematics and must be mastered to ensure success. Using your fingers can also cause a loss of accuracy, often children (and adults) get one, sometimes even two.
Verbal practice with additions, building walls and towers, games like Under the Cup, simple story problems and picture worksheets give the student the experience they need to make the transition from fingers to symbols to be able to “in their heads.” Drawing rectangles and other mathematical concepts, as well as making drawings of the manipulators 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 in all of math, which is why you often see me using third- and fourth-power algebra to teach addition and multiplication.
In fact, if you take the concept far enough, they can also remove the symbols so to speak and do it ALL in their heads if necessary, no 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 it and tell you the sides. Or if you tell it the sides (x+3)(x+2) it can tell you the whole rectangle not because it’s seeing symbols but because it’s seeing PICTURES. In addition, you are “consolidating” your addition and multiplication facts in your memory. How much easier is it to see 6 taking a 4 from a 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 “wants to be a ten” concepts. He also gets a ton 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 a few basics 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 on images not symbols, so memorizing the 45 additions and times tables is easier because the mind can store much more images. easily than symbols. So when it’s time to be remembered, 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 keypad in their head? They can even point to the numbers and move their index finger on an imaginary keyboard in the air while they are 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 easier is it to add two numbers than to recite seven to ten digits? Especially if you have a method to view them if you somehow forget?
A simple exercise: ask a student to draw a cow. Then ask if they saw the COW or the picture of a cow? Question what color was it? This lets you know they weren’t seeing symbols. The problem is that with math most students have no idea if it’s algebra or simple addition. The “trick” if there is one is to get the information into long term memory so that it is easily recalled and it is pretty well proven that symbols ie. letters and numbersthey are a hard way to get information there.
Manipulators are the perfect bridge to get information there. After all, storage is never the problem that retrieval is.
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