TABLE OF CONTENTS
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What if, rather than waiting for our toddlers to grow up and learn this…
The FORWARD link is at the bottom right.
we parents will start building with them two-dimensional spatiotemporal patterns of colors, using any suitable toys?
To go forward, you may also click on the header.
Use the ruler on top of it to jump to any page.
The table of contents is on page 00.
Links to learn more are on page 22.
Spatiotemporal patterns are patterns changing and repeating themselves in space and time. The word looks scary, but there is nothing to be afraid of. Think of a simple pattern and unfold it, like I did here…
Building spatiotemporal patterns teaches to understand the rules causing those pattern to happen. Stereo learners expect a certain color to appear at a certain moment in a certain spot. They can fill a void or notice and correct a misplacement. Importantly, the children who build in two dimensions learn to count in two dimensions.
Watch how the pattern develops. The final fit of building four "spinners" next to each other must be familiar to a 4-year-old stereo learner, but placing the first spinner right requires the cool new skill of counting to 10 on the plane.
Without counting, stereolearners are helpless in open two-dimensional space, and they are acutely aware of this shortcoming. Counting to navigate, instantly puts them on par with the big people. The toys become the mathematical tools. The places and the pieces on the board become the tokens to count.
On a 10x10 board, knowing how to count to 10 is enough to build and move small shapes by coordinate numbers. What's next? Fifteen? Twenty? Never mind. To build big, stereo learners need to count at least one hundred items!
Seriously, can 4-year olds learn to count to 100? Of course they can. They can count to much bigger numbers. Just give them what to count, and a reason why.
If counting is not going smoothly,
your computer is busy doing something else.
Dealing with such big numbers, stereo learners grow to appreciate the positional notation, in which one token stands for one hundred of them. When we count one hundred tokens in the positional decimal notation, we leave 99 of them behind. The following pages will show you how.
Meanwhile, don't stop at one hundred. Keep counting! Follow a simple rule: When you get one token too many, pass it to the left and clean up the overflowed position. If the token passed to the left happened to become one too many again, pass it to the left and clean the position again, immediately.
Click to pause the counter at any time. Click to make it slower or faster while it's running.
Stereo learners who understood quantity start solving small problems one by one. This makes them ready to learn big numbers. Positional counting is the gateway to this magnificent world.
The counter you just watched, works in real time because it counts the ticks of the computer clock. It adds one token per tick. If the tick triggers an overflow, a complex event takes place: blinking red, passing a token to the left, cleaning up and repeating this as many times as required. When the work is done, the counter has made only 1 step forward.
Counting with tokens requires numerals. Imagine such a lesson, and you will realize that the parent and the child can't even discuss their progress without naming the quantities. Stereo learners quickly surpass one thousand, and count on.
The biggest number in 10-digit counter is 9,999,999,999. At the rate of one count per second, it will take 317 years to reach.
Years before their schoolteachers will tell them that ten is the smallest two-digit number, stereo learners know that passing to the left and cleaning up does not just have to happen when they step over 9. The forbidden number, at which the building up stops, is called the base, and it can be anything bigger than 1. Ten and 10 are not the same!
The bottom counter goes 0, 1, 2, 3, 4, 5, 10..., which is known as base-6, or heximal. The top counter goes 0, 1, 10..., which is base-2, or binary. Bases smaller than 10 are excellent for learning long operations.
Pause, slower and faster are with you.
Important: This presentation is not for teaching. Passing the 10th token to the left has little to do with the meaning. The procedure only makes sure that the token was counted. Somebody has to explain what's going on.
Cleaning up is the best part. Getting rid of accumulated quantity by exchanging it for more powerful tokens, instead of hauling it on forever—this is what makes a positional notation so helpful. The counters reveals its two-dimensional soul. Every position is a separate number line. A regular number—a string of digits—does not teach this.
Even with tally marks, such counting cannot be played out on paper. Understandably, though, there is no market of toys for teaching preschoolers arithmetic, yet. Some pegboards and mosaic sets are usable, but hardly any of them is good. Devices similar to thumb switches would be very desirable, but industrial designs are not for kids.
Links to the websites are on Page 22. Contact email address is on page 26.
This and the following green pages make a presentation in presentation. The subject is worth digging deeper. The main Table of Contents and the ruler point to this page.
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Playing with counting, stereo learners understood that a positional notation is the same everywhere. They learn to count up and down, starting from any quantity and at any position. For them, the fear of big numbers, carefully cultivated by educators, makes no sense.
Another useful exercise is counting through a given number of ones and stopping when all of them are spent. Adding 3 to 5, we take 1 from the number above the position and put another token on top of the stack until we have no 1s to add.
Click to start counting. You can also do it step by step.
On this step, we are going to count up 8 times starting from 37. Or, if it matters, we will count 8 times by 10 starting from 370.
Click to start counting. You can also do it step by step.
An overflow is indicated with a red light on top of the 9th token. The rule is the same as before: clean the position and pass the token to the left. In case of another overflow, do it again.
The yellow light marks the position, where the counting is taking place. On overflow it blinks red and splits. Another light travels to the left and blinks red if there is no room in this position, too. When the overflow is settled, the counting resumes at the initial light.
We are about to count up 9 times starting from 9994(or 9 times by 100 starting from 999400).
Click to start counting. You can also do it step by step.
On the 6th count we encountered an overflow. The yellow light turned red and spawned another yellow light to the left, where we tried to count up again. No luck, another overflow. Then again and again. Finally, we found the place to rest the overflow token. Our attention returned to the initial position under the yellow light, where we found 3 more counts and performed them.
Congratulations, you just finished counting forward 983 times starting from 999375. You did it in 3 steps, and it was incredibly easy. Care to do it again?
Click to start counting. You can also do it step by step.
Another name for what you have done is addition:
999375 + 983 = 1000358
Let me stress, adding 983 you counted forward 983 times. Please think about counting 983 one by one. Arithmetic does this for us. And it was not even multiplication. Or was it?
Long addition is positional counting broken into short and easy pieces. On this step, you can add any two 9-digit numbers. Here is how.
Click on a token to make it the top token. Click on the top token to zero the position. Click on the red index to copy the addend to the digital register, and enter the other addend.
Start adding or do it step by step.
Learning long operations with tokens is easier in many ways. For example, the adding app, which you just tried, uses only two numbers and builds the result on top of one of them. It doesn't rely on any knowledge of "math facts," and it handles overflows immediately. Without deferred carries, one can instantly see how much is left to add, and what we are going to add it to. Best of all, this algorithm is reversible. It's equally good for subtraction.
Unlike the counter, the adder app does not have to run in real time. It slows down on overflows. Step-by-step mode allows us to see every move.
Like everything else here, the adder app was not made for teaching, and it's not the right place to start. It works with digits and tokens, which is more challenging than using tokens only. Counting down looks better, but it is harder to do than counting up by the number in the digital register.
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Rather than memorizing "facts," "methods," and "strategies," stereo learners can derive and reinvent them. To add big numbers without "math facts," they make and use addition tables. Remember the problem of placing the spinner? Check Page 05 if you don't, and then return to Page 13. An addition table is another instance of two-dimensionally countable space.
After all, what is better for memorizing "math facts": to practice adding 10 ten-digit numbers or 100 one-digit numbers?
A hundred years ago, educators might have thought they had good reasons to eradicate such folklore math. Why do it now? Young kids are no rebels. They naturally follow the big people. Numerals and math facts are junk knowledge nowadays. Common Core is offered at every corner. Nurturing understanding, meanwhile, is nowhere to be seen.
Converting the knowledge of long addition from tokens to digits on paper may happen immediately, or take a few more lessons. The learners are motivated. In writing, long addition is difficult to understand, but much easier to perform.
Stereo knowledge is an educational supplement. The home lessons take about 3 hours a week, they teach the fundamentals, and they are not intended to upset school. Teachers may stuff stereo learners' heads with "math facts to five," "number sense," or "regrouping," get paid for this, and take pride in their achievements.
Please click on a number in the table to see how multiplication has produced it. Every number (or product) is the number of the squares in the rectangle between this number and 1 (which stands for 1x1). Rectangles are made out of the bars. The multiplicand is the number of squares in a bar. The multiplier is the number of bars.
Recognizing this basic pattern in situations like "a car has 4 wheels, a wheel has 5 screws and a screw has 6 sides" may not be easy. Again, stereo learning helps…
For stereo learners, multiplication is a way to find the number of elements in the elements of a pattern. This one has three kinds of shapes, and the total of
3x5 + 2x4 + 2 = 25
squares of every color. We have an option to multiply these numbers by the number of the colors at any level.
To sort out this pattern, select red, blue, yellow or green. Every next color layout is rotated 90°. Otherwise they are the same. You may click REPLAY to see them all.
Another, and more interesting, source of multiplication challenges is combinatorics.
The rules of elementary algebra can be visualized in space. They explain arithmetic, and they are easier to learn.
Here is how distributivity can look:
(R + G)x(Y + B) = RxY + RxB + GxY + GxB
R is red, G is green, Y is yellow and B is blue. That's how stereo learners get accustomed to letters standing for numbers.
Distributivity ushers in long multiplication, if we teach it right. Stereo learners can get long multiplication before school, but their understanding may be rather shallow. The illustration could mean 53x44.
And now, look at this: 6, 12, 18, 24 and 30 squares of each color are being divided between 6 elements of the pattern.
Division has its basic pattern too. The app on the right progressively divides by 6 every number from 0 to 48. Think about dividing 48 coins among 6 pirates, one coin at a time.
The orange bar is the modulo (or remainder). You've seen this rising and falling pattern in counting. Division is the mother of all arithmetic. Positional counting is division.
Stereo learners can get exponentiation, but this would be too much. Instead, let me spend a page on modulo, as an example of what we could teach, if only we cared.
Modulo is the easiest periodic function. It is available to preschoolers who've learned division. If plotted, modulo produces the pattern of a sawtooth wave. Remember, stereo learners discover Cartesian coordinates as soon as they learn to count.
You can pause the pattern at any time.
Combining modulos, one can build other integer periodic functions, much as real periodic functions can be made out of sines. Modulo is quintessential for computer programming. It covers the space with periodic patterns. We could even use it to teach discrete calculus.
Stereo knowledge is like bicycling or swimming: Once picked up, the skill lives on. Arithmetic is the only difficult part of the elementary school curriculum, so stereo learners have more time to learn.
Arithmetic creates a different kind of learner. Early literacy leads to easy reading. Literacy plus numeracy enables preschoolers to acquire deep and rich knowledge, and an early start is an exponential advantage.
In the world, where "high achievers" are considered to be the worst kind of deviants, every child, without the distinction of being a natural-born weirdo, gravitates to the center of mass. Few stereo learners in a class will be able to change this.
Life after school is what matters, though. More than ever before, we have no idea what usable knowledge to teach today, but we can challenge developing minds, and mathematics is the best source of intellectual challenges.
This presentation was made in FIREPEGS: the computer toy for building "social" networks. Recurring patterns appear as networks oscillate. Designing and reverse engineering them can be made easy for preschoolers or challenging for their parents.
The fire in FIREPEGS passes between the bordering pieces. The idea was derived from observations of the first stereo learner. FIREPEGS grew from a toy pegboard to become a source of challenges like none education has ever seen before.
FIREPEGS was augmented with CODEPEGS: the system for learning mainstream computer programming. FIREPEGS was made to require scripting. Stereo learners see, touch and use real code before they know. They solve puzzles at first, then create their own patterns, and the more they learn to code, the cooler are the things they can do.
FIREPEGS networks are made of colors and numbers. Firepegs fire up and kick their neighbors, telling them when it is their turn to follow. A preschooler can get it and start learning the core of science and engineering: space, time and causality. By the way, you can pause the pattern at any time.
FIREPEGS has a presentation of its own. The example here was borrowed from it, to show the system's internals. The pattern—named Inchworm—runs on a centralized network, where the red peg kicks every other peg. Click to load another network, in which 4 orange pegs are kicking each other, and each of them kicks a yellow peg to produce the same Inchworm pattern.
CODEPEGS is about learning real computer programming, by creating and changing the colorful layouts of symmetries and patterns. It's truly elementary—the angles are right and the numbers are whole—and truly fundamental, because the elements are oversized pixels. A learner is not expected to be an aspiring animator or video games designer. No talent in arts, rarely seen together with a propensity for STEM, is required.
Arithmetic, algebra, tessellations, polyominoes, moires, affine transformations, cellular automata, methods of computer graphics and animation, together with all the realm of STEM, provide tons of challenges, hard and easy. They teach STEM subjects and programming at the same time.
As a scripting tool, CODEPEGS can switch the states of firepegs, make their behavior smarter or randomized, modify and reload the networks on the fly and do this in response to events, like timer ticks, mouse clicks or firepegs turning on.
CODEPEGS on FIREPEGS is a process control software because FIREPEGS creations are more mechanisms than computer programs. Even the simplest of them demonstrate such interdisciplinary concepts as signal, reaction, delay, hysteresis, feedback, oscillation, wave, memory, network, emergence, etc.
In Greek, στερεός means "solid" or "firm". In modern usage, "stereo" means enhanced by one dimension. Stereo sound is two-dimensional, stereo image—three.
This presentation is not for learning or teaching. It only explains SIX PROJECTS.
SIX PROJECTS were created to make STEM subjects learnable and their students teachable. They were published as demos because the sole author might not live long enough to turn so many intentions into products. Currently, every piece of software is a prototype. All e-books are demo excerpts, some of them not even proofread yet.
The purpose of this publication is to solicit feedback. If you have something to ask or tell, please direct it to:
Get the text.
© 2016 Georgiy Kuznetsov