Block Math Conclusion

Last week I left the definition of 'midway' hanging.  To define midway, I'm going to use my old friends 'imaginary blocks'.  Let's say you have a chipped-stack, that is all rows have the same amount of blocks except the top row.  We can fill in the top row with imaginary blocks as so:

XXYYY
XXXXX
XXXXX

Here the 'Y' blocks are imaginary, we've used our imagination to fill in the top row.  Now we compare the amount of X blocks in the top row with the amount of Y blocks by stacking them:

YYY
XX

It would appear that the X block amount is less than the Y block amount.  My definition of 'midway', is when the X block amount is the same size as the Y block amount.  If the X block amount is greater than the Y block amount, we can say the top row is greater than midway.  Similarly if the X block amount is less than the Y block amount, we can say that the top row is less than midway.

So why is midway useful?  In the real world, when we need a clean stack, one isn't always available.  In fact, for stacks with a large amount of blocks in each row, it is easy to imagine that one is rarely available given an arbitrarily sized block-bag we are trying to arrange.  By comparing the real blocks with the imaginary blocks on the top row, we can say that the top row is more real than imaginary and consider it whole, and we can say if the top row is more imaginary than real, we can disregard it.  Additionally, assuming that the top row will have one amount with equal probability to any other amount, on average the number of times that we consider the top row by filling in the blanks will be equal to the number of times we disregard it, which allows things to even out in the end.

Bear with me for one more scenario, and then I will summarize and get off my soap box.  Imagine that Costco has a sale on garden boxes.  The garden boxes come in many sizes, but all of them are square in shape, that is each side is of equal amount.  You got a bag full of corn seeds from your Aunt for Christmas.  You need to figure out what size garden box to buy so that you can plant all your corn.  Since each seed needs to be planted some distance from the other, you devise a scheme where you can use some of your blocks to represent where you are planting your corn seeds in an imaginary arrangement, whereby each block represents a size equal to the distance needed between seeds.  So you find a block bag with an equal amount of blocks as the amount of corn seeds in your bag.  Now you need an arrangement strategy for finding the smallest 'square-stack' needed for all the blocks in your bag.  One strategy might be to arrange in a 'spiral' pattern, always trying to maintain the 'squarest' shape.  As you pull each block out of the bag, here's the arrangement:

X

XX

   X
XX

XX
XX

XXX
   XX

XXX
XXX

XXX
XXX
X

XXX
XXX
XX

XXX
XXX
XXX

You continue this process until you run out of blocks.  Now you can remove the blocks in the longest side to represent the amount of 'corn-seed-distance' units needed to allow you to plant all your corn.  You then make a stick that represents the 'corn-seed-distance' unit, take it down to Costco, and find the garden box that will allow that stick to be placed end-to-end the correct amount of times, and purchase that size of garden box (hopefully it will fit in your car).

So what have I done here?  Not much, really, most of the problems I've solved with these operations seem trivial.  Some of the scenarios are rather arbitrary (how often does Costco have an assortment of square garden boxes of different sizes on sale?) What I've done is to think about the processes involved and come up with a new simple vocabulary to describe them.  That allows me to think more clearly about the processes, and not be saddled with my prior knowledge sneaking in and providing short-cuts that may not be intuitive to other people.  It allows me to communicate better with other people that don't share my knowledge and, I hope, makes it easier for other people to understand.  In short it makes me a better teacher.

Much of the vocabulary used in today's math is foreign to the rest of our language, and so words like 'roots' and 'fractions' become scary unknowns to most individuals.  The arcane symbology used to describe all but the simplest formula is enough to send most people running. Much of what we are taught of math is from one perspective: sterile abstract operations such as addition, subtraction, multiplication, and division.  Memorizing multiplication tables, and performing long division.  Which makes us into mistake-prone calculator, and a little less valuable then the cheap electronic calculators you can find at the dollar store.  We should be teaching by saying first, heres's a problem, (pause to let the problem sink in) and second, here's a solution and how it works.  Instead we are teaching by saying: here's a solution, and oh-by-the-way, there are problems out there that might be solved with it, trust me.

Playing with blocks, making arrangements, talking about types of arrangements, and how they can be used to solve problems seems to me to be a simple and intuitive way of teaching the fundamentals of mathematics.  In the past three posts I have described addition, subtraction, multiplication, division, ratios, roots, negative amounts, and rounding, without ever using a single number and with a limited vocabulary of common words.  The blocks allow us to think abstractly with representations, and to visualize these process as they are occurring.  If, when teaching math, we lay that as a foundation, we can then teach the tradition constructs and formulas of math and have something concrete to relate it to.

For those of you with only the tradition constructs and formulas of math in your knowledge set, and without the benefit of higher math which forces you to think about the processes involved, it may be hard to reconcile block math with real math.  So here's a cheat sheet.  The scenerio of the block bank robbers teaches about division.  You have some amount of blocks, and you need to divide them among your gang.  There's even a remainder (remember those?).  The scenerio of the siblings buying a new game and pooling their money teaches about addition, subtraction (or negative numbers) and comparison.  You need to add the amount of blocks, compare their amount with the imaginary amount (a negative number), and subtract the imaginary amount.  The scenario of paying the waiter a tip teaches multiplication, ratios (fractions), and rounding. Building the clean-stacks is dividing and multiplying: the first stack is the bill divided by some number, and the second stack is the height of the first stack multiplied by some other number.  Building a pair of clean stacks with equal amount of stacks, but unequal amount of blocks in each row is a ratio, and evening out a chipped stack to a clean stack is rounding.  And finally the square garden box scenario teaches about square roots (appropriate to talk about gardening when discussing roots).  The size of the box you need is the square root of the amount of corn seeds that you have, rounded up to a whole number.

And lastly for you mathematics fans out there, I have nothing against your specialized vocabulary and arcane symbology.  Unambiguous definitions and rigorous proofs for the most part require them.  But when trying to communicate with someone that doesn't understand the vocabulary, it might be easier to explain things using a common vocabulary even if you're not being as precise as you might be if you were communicating with one of your own kind.

Block math can be used to describe succinctly operations that involve integral things.  A thing is integral if it can't be cut in half and still be useful.  You can't cut a computer in half and have two half computers.  You can't cut a penny in half and have a monetary amount of half a cent.  You will have two pieces of junk.  It can't be used to fully describe things that can be cut in half.  A yard of cloth can be cut in half to make two half-yards of cloth.  These are very distinct ideas, and yet our temptation is to use one concept 'numbers' to represent both ideas.  Block math avoids some of the pitfalls by making clear the distinction.  Perhaps in another series of posts I can talk about the other half of the number problem.  Until then, have fun playing with your blocks!