Use Algebra To Teach Basic Operations With Base Ten Manipulaives.
There are many kinds of base ten blocks that you can purchase, but base ten blocks are not all the same and some are much more powerful than others. First off, make sure they ARE base ten blocks, I have seen block sets where they are base seven and even base three…where seven of the x are same as an x^2 or just three y are the same as the y^2, while still quite workable doing the larger problems becomes somewhat difficult because it takes too long to count out the units. This is very much akin to getting lost in computation…many students lose track of the problem they are doing because even if they understand the concepts they can’t do the math or it takes them a long time to do the math and they forget what they were doing or trying to solve in the first place.
The mind wants to work very quickly counting out individual units from the various manipulative sets takes too long. Sometimes even counting out tens or x’s takes too long. So while sets that have base ten blocks with tens and individual units are quite useful they become tedious if you want to use them to demonstrate many problems in a row or even just a few larger problems. By larger, I simply mean problems that will have a lot of pieces to get out. Showing 60% of 80 for example or even a simple problem like (x+6)(x+7) will require getting out 42 units…you already have to get out 13x, counting out 42 units takes a lot of time, and the student can often get lost (and bored) because of it; however, if all you have to do is get out an x-square, 6 sevens (or 7 sixes) and you can grab 7x and 6x, the problem can be built very rapidly. At Crewton Ramone’s House of Math you can actually SEE what it is I’m talking about. It also becomes easy to teach counting, addition and multiplication using algebra because the child has to count out and add up the blocks in order to build the rectangles needed to solve the problems. At this point you begin to see why a picture is worth a thousand words, and if a picture is worth a thousand words having your hands on the blocks and then being able to draw them is worth a thousand explanations. Further, this is the way the subconscious mind works, using pictures NOT symbols and we have come to realize over the last 100 years that most learning takes place in the subconscious mind not in the conscious mind.
Students who benefit from base ten blocks most are the kinesthetic learners, because they literally can get their hands on algebra and work with the problems. They can see and touch the problems in a very real sense. They can see that x times x is an x square, they can touch it…they see and touch the distributive theory of multiplication because they can see an x+3 is just a number with two parts (a binomial) and if I have 3(x+3) spoken three times x plus three I get 3x+9 and they can see it. Getting this problem out is still easy even if the blocks you have are all individual units, but it’s still easier to get out 3 threes…and by the way we can note than 9 is a square number when we build it this way. As soon as you do a problem like 5(x+8) counting out the 40 units is going to slow things down and actually take away from the lesson and the concept that the 5 multiplies the x and the 8. Manipulative sets that use rods to represent numbers are again quite useful but will have serious drawbacks when attempting to demonstrate algebra or square numbers and square roots because most often 3, 3-rods do not make a square. Simple concepts like x + x = 2x are easy for any student who is just learning to count. Base ten manipulatives or base ten blocks can be used for so much more than just teach counting or to teach addition. When used properly very young students can lean to count and add WHILE they learn algebra and vis-versa, and counting, adding and subtracting lead to multiplication and division.
“We used to think that if we knew one, we knew two, because one and one are two. We are finding that we must learn a great deal more about ‘and’.” ~Arthur Stanley Eddington, as quoted in A Dictionary of Scientific Quotations” (1991) by Alan L. Mackay, p. 79