Crewton's Math Manipulative Manifesto

Math Manipulatives

Here is an article I wrote about math manipulatives for my website directing people to this blog. Some of the points here will be covered in more detail in the next parent teacher training.


Math Manipulative Manifesto

"Learning is not the product of teaching. Learning is the product of the activity of learners." ~John Holt

This blog is dedicated almost exclusively to showing you how to use manipulativess with your children at your house whether you home-school or not, and also to get you to use them in your classroom if you are a teacher. Note the lack of advertising, adsense, annoying pop up vids etc. The only things I advertise here are blocks, books, and passwords to vids and info.

While meditating it hit me why there is such a disparity of viewpoint between me and most of my "fans" regarding curriculum and organization. Whereas I see it all as just counting, others think of it in subjects, grades or levels or what have you, because the way you were taught was very compartmentalized.

It's been ingrained in you that FIRST you have to learn how to count, next you add next you subtract then you multiply then you divide next you do fractions etc.

What I'm saying and what this method teaches using math manipulatives, is that ALL that is, is counting. All of it. You're just counting. Since we are just counting, we can use division to teach it or we can use fractions to teach counting like addition and multiplication or we can use algebra to teach counting...most people think that in order to do division the student has to have learned how to multiply and subtract...NO. (Look at me! Breaking the rules: you are supposed to remove the no from the lesson.)

Students don't need to have gone through basic operations (addition, subtraction, multiplication, division) before they can do algebra, in fact algebra will teach basic operations. Division is also just counting and we can use division to teach a lot of math concepts as well as how to add, subtract, and multiply and they learn to multiply and subtract while they are learning to count. It's the same with fractions...or algebra.

Ask what is it we are counting?
"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 mathematical questions boil down to "what are we counting?" When teaching math you can use division to teach counting or addition to teach counting or algebra to teach counting...you can focus on whatever you like but it doesn't need to be structured such that addition comes first and then subtraction, in fact you will quickly see it's very hard to separate the two and trying to do so may actually be harmful to their understanding.

I have sat in awe in grade schools while lessons on multiplication were taught without any reference to area, or division, or in some cases any reference to repetitive addition...just learn 7 x 8 = 56...and to this day it is amusing that 99.99999999999% of people I meet are wholly unaware that that division "thing" we use is just shorthand for a rectangle. Or what the equals sign means. Or where that percent sign comes from. Or that square numbers are square...but I digress.

The Mathematics is a language and it all goes together and shouldn't be segmented or compartmentalized to the point that the student is unaware that multiplication and division are inverse functions, manipulatives make that visually obvious. They will be able to see and understand this naturally...and then it's easy to point out to them, this is part of "directed discovery" you know what you want them to learn but you let them discover it. You facilitate this by putting them in a math rich environment.

The point is that while we are teaching basic operations we are just counting but don't get confused: basic operations is computation and computation is how we do math but math is much more than just computation. (The fact that I have to write that statement is a clue to the complete failure that is the current state of math education.)

There isn't anything more fun than seeing your own students or children learn something and have fun at the same time, and in knowing you are the one responsible for it. This is why some teachers are addicted to teaching no matter how poor the pay is and no matter how lousy the conditions are.

Math is the language of logic and reasoning combined with critical thinking skills. I have met many students who are great at computation but very,very poor at math. (Some of them go on to get degrees and then, ironically, attempt to teach others the math they don't really understand.

If you put the child in a situation where they CAN NOT FAIL, expect and allow for error, allow for self correction with direction...direct their discovery in a math rich environment: they learn math. They can't help it. AS an added bonus they have fun learning with math manipulatives and/or base ten blocks.

Give the child an algorithm for addition (wanna be a ten) that can be applied to multiplication and suddenly almost effortlessly (much to the amazement of parents and even veteran teachers) the students are adding and multiplying...multiplication is the first milestone because it allows the child to count quickly, once computation is easy the math becomes easy too because computation is how we DO math but again, it's not the math itself.

With computation mastery, math concepts that are understood can be applied to problem solving and answers are fun and easy to see. The combination of computational mastery and conceptual mastery combined with visualization is very powerful. The playful, curious attitude that problems are just games or puzzles removes fear and tears, and learning is seemingly effortless because they are having fun doing it.


Avoid the problems in the first place.

"Intellectuals solve problems, geniuses prevent them." - Albert Einstein

 
There is no greater joy for a parent (or any teacher) than seeing the ah-ha moments and light bulbs go off...it is especially poignant if the child had formally been struggling with math, but I prefer to avoid the problems in the first place.

The question should be "what kind of counting do you want to do today?" Or what kind of counting do you want to teach today? If you are a teacher. 1st graders as an example can learn division and thereby learn a lot about counting, and multiplication and subtraction...as well as addition...they can learn about fractions and thereby learn about addition, multiplication and division as well as concepts like SAME...the way we currently teach it children somehow come to the conclusion that rules have changed when we add fractions. They haven't. We can teach them problem solving, which at first is just recognizing same and "hero zero" and then "no fun get back to one"...all of which is just simple COUNTING even if they are little.

Please click on this link for more on "hero zero" and "no fun get back to one" and removing no from the lesson the other teaching concepts referenced in this manifesto. (Rant. Diatribe).

I've done lessons for 2nd graders on square roots...in order to do square roots we had to count the squares and the sides...so we learned skip counting, used addition skills to count 7 sevens which was more work than counting 4 fours and figured out what the symbol means--just count one side. We also learned about economy of symbol instead of 4 + 4 + 4 + 4 = 16, or even 4 x 4 =16 we could just use 4² =16. I wasn't just teaching square numbers I was teaching lots of counting. NONE of it would be possible without the use of base ten blocks. People are amazed that I go on to do lessons on Pythagorean theorem but they don't understand the part about it's easy when they can see it. I wouldn't even attempt to teach them any of this without math manipulatives.

Further students often never understand the relationship between Pythagorean theorem, distance formula and the first basic trig identity sin² +cos² = 1, it's just a bunch of stuff they memorize long enough to be tested on and then forget--never having made any of the  leaps of understanding or connections. My students avoid this because they've been playing with these concepts for years and baby step their way there instead of it being all new and confusing in all at once in high school.

Shift your paradigm.

What I have to do is get YOU to change YOUR paradigm and understand the way you were taught via compartmentalized, separate subjects, grades and levels might (just maybe, quite possibly) NOT be the best way to teach math, and is part of the reason the teaching of mathematics in general in the USA is an utterly abysmal failure. Most of you recognize that it is an utterly abysmal failure which is why you are here in the first place then the first thing you ask me is, "couldn't you teach and organize your blog and website more like the way we know doesn't work and is an utterly abysmal failure?"