“There is a great danger in the present day lest science- teaching should degenerate into the accumulation of disconnected facts and unexplained formulae, which burden the memory without cultivating the understanding.” ~J. D. Everett [In the preface to his 1873 English translation of Elementary Treatise on Natural Philosophy by A Privat Deschanel. (D. Appleton and Co.)]
Unfortunately this is EXACTLY what has happened with mathematics education. The whole of mathematics has been broken down into tiny parts and because of this the whole picture is little understood. The more I am forced to teach out of text books whose well meaning authors have fragmented math beyond the basic operations into such disjointed parts that students just memorize enough to past the test and then forget it, the more I see this is true. This is compounded by teachers who are pressured into “teaching for the standardized tests” not teaching for deeper understanding. The students understand it well enough to pass the test…often with a low “C”.
Fundamental concepts like, math is a language, numbers are made up of other numbers, and all we can do with math is count, are lost. Concept based teaching using manipulatives makes math fun and easy and highly understandable, to the point that even very young children can do so called “higher math” with out problems. Find Crewton Ramone’s Blog Of Math and there you will see six and seven year old students doing algebra, making change for $100.00, videos and more. Best of all they don’t know it’s hard.
The idea is to teach math like a language, using basic concepts that build a solid foundation for the next set of concepts that build a solid foundation for the next set of concepts and so on. Working your way through math using concepts that increase in degree of difficulty slowly but surely while the student is in a situation where they cannot fail is a sure fire method for success. Because most concept based teaching uses manipulatives and because text does not lend itself to mathematical notation trying to describe it in detail in a short article like this is like trying to describe blue to a blind man. You have to see it to really get a grasp of what I’m talking about. There are several systems that you might look into to assist in he math education of your child. Any Montessori based system like Mortensen More than Math or Math-U-See will generally be concept based, but the primary complaint I hear from parents and teachers especially with Mortensen Math is that they don’t know how to use these systems. This is often the case with many base ten manipulative sets: they don’t come with enough instruction so that they can be used to their fullest extent.
At Crewton Ramone’s House of Math you can SEE how to use base ten blocks to do a whole lot more than just teach counting. You will find many concept based lessons for teaching basic operations, algebra and more that can be used with any set of base ten blocks.
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The mathematics begins with counting. How do we teach counting? Start with patterning. Just say the numbers from one to five, then one to ten then ten to twenty. Repeatedly. Start with your fingers, at bed time is a good time but any time is fine.
With most beginning students I make them write out the five basic concepts. With older students counting is redundant, they can all count, and you can make a joke of it. After they write down “Math is the study of numbers and all we do with numbers is count”, or some variation thereof, ask them to count to twenty. Then ask them to count backwards from twenty.
“See, I can teach you math, we are one fifth of the way there…you can even count backwards.” Then ask them to recite the abc’s backwards.
“Hmm, looks like you’re better at math.”
Although I am almost illiterate mathematically, I grasped very early in life that any one who can count to ten can count upward indefinitely if he is fool enough to do so. ~Robertson Davies, “Of the Conservation of Youth,” The Table Talk of Samuel Marchbanks
Two tens and three units has a name, twenty three…what’s important is understanding the concepts.
We count the big ones first. One hundred, one ten and one unit are one hundred eleven. 111. With manipulatives this is visually obvious but when writing 111, little kids often think THREE…can you see how much easier it is to teach place value?
You will at Crewton Ramone’s House of Math!
Teaching a child to count properly takes MONTHS, you can teach them to pattern and memorize by counting from one to 20 over and over again, which is fine; however this can lead to some confusion. Ever hear, “my kid can count to 10 but gets a little confused in the teens”? I have. A lot. Make sure they see all the teens are are just ten and some more, or specifically ten and one through nine units more, and that the pattern repeats with two tens and one through nine units, and three tens and so on. This way they get the concept and the pattern and link the vocabulary to the number.
I can’t tell you how many times I got a student that was failing algebra that had to use their fingers to add numbers. Simple numbers like nine and three. Asked to multiply they didn’t have their tables memorized either without think hard about it. OF COURSE THEY WERE FAILING ALGEBRA: they had no foundation upon which to stand. Back to basics. No magic. Then again, when the basics are mastered correctly the magic begins. First counting, then addition, then multiplication seems obvious.
“Obvious” is the most dangerous word in mathematics.” ~Anon
So first thing: get them “off their fingers.”
Addition and multiplication are just way of counting very quickly. In order to get there you start off crawling then you walk then you can run, then you can hit the hyperspace button…or at leaste turn on the after burners. Teach counting the easy way, visit Crewton Ramone’s House of Math for FREE SOFTWARE that helps teach counting and much more.
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The Five Basic Concepts:
1. Mathematics is the study of numbers and all we can do with numbers is COUNT
2. The concept of SAME. The highest number we can count to (in base ten) is 9, the numbers tell you how many the places tell you what kind. Before we can count they must be SAME.
3. We form rectangles to facilitate counting.
4. O
Hero ZERO.
5. 1
No Fun Get Back To One. NFGBTO.
Math text books are always just variations on these five themes depending on subject matter. The trick (if there is one) is knowing how and when to apply them.
My students hear “hero zero, no fun get back to one” during problem solving and solving equations algebra constantly…and they realize very quickly just how easy it is when you apply these concepts. The reactions range from relief to joy to rage. Relief for some when they realize they aren’t stupid after all, joy for others who go from “F” to “A” so instantly that the teachers accuse them of cheating, rage by others who get mad when they realize all that fear, frustration and anxiety was for nothing.
Algebra as it turns out isn’t really that hard if you understand some basic concepts and can see what it is you are doing with the help of manipulatives. Even Algebra becomes child’s play: Once you understand the concepts doing the math is easy…and fun. At my website, Crewton Ramone’s House of Math students can actually see what they are doing because we do use base ten manipulatives to teach math concepts and along with detailed instructions and even some video parents, students and teachers alike can see what they are doing and it makes sense.
“Mathematics may be defined as the economy of counting. There is no problem in the whole of mathematics which cannot be solved by direct counting.” ~Ernst Mach
All math is is counting. Every math problem counts something. Believe me I’ve had this argument in seminars with professors and teachers across the USA. If you aren’t counting something you aren’t doing math. Period. So one of the most basic concepts I teach ANY student is to ask “what are we counting?” when confronted with a math problem.
“Although I am almost illiterate mathematically, I grasped very early in life that any one who can count to ten can count upward indefinitely if he is fool enough to do so.” ~Robertson Davies, “Of the Conservation of Youth,” The Table Talk of Samuel Marchbanks.
In the mathematics there is a concept known as economy of symbol. That is, we use the fewest symbols possible to express an idea making the mathematics a beautiful concise language.
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