The App

Please point here. Do you see the border around this section? Would you please double-click inside it to launch the app? Has a new window appeared? You don't need to close the app unless it asks you do so.

If you don't see the border, you must be looking at a touch device. It's OK, except you may end up wanting two more of them. On touch devices Album for Beginners renders the section headers as web-links, and the app opens in a tab. If you clicked on a link and nothing happened, please open the tab before deriding me. It must have been updated.

The app has two spaces called boards. The smaller one is for me to display the challenges. The bigger one is for you to take them up. On top of each board is its control bar. In the bottom left part of the window are the displays and controls for the whole app.

The small board is playing the record of the pattern, which you will build in the end of this issue of Album for Beginners. To get in sync, count to 8 (tick-tick-tick-tick tick-tick-tick-tick or tick-tick tick-tick tick-tick tick-tick). Click on the small board to stop the player.

Both 8x8 boards have 64 rounded squares called firepegs. Every firepeg on the big board is subdivided into 9 small squares. The core of every firepeg, the small square in the middle, is the signaling organ or part. It keeps FIREPEG's color and tells the skin to display or to hide it.

The other eight small squares are the messaging devices. I call them links. Firepegs use the links to reach out to their neighbors and tell them what to do.

Let's return to the book's control panel for a moment. Please find it at the top-right corner. The CALL control brings up the section of the book, from which the app was launched last time. If the app has not been launched yet, the button does nothing.

If you did launch the app, you can scroll away from from this section and call it back.

PLEASE DON'T CLOSE THE BROWSER AND THE BOOK'S TAB UNTIL YOU FINISH THE NEXT SECTION.

Link, Fire, Tick

Please point here, make sure you see the border around this section, then double-click inside it for the app. If nothing happened, look for F.ABA title in your tabs and open it.

To view the challenge again, click on the small board. Your goal is to link your firepegs to make a "social" network, that will produce the challenge pattern. To do this, assign the numbers to the links.

The picture on the right shows how your network must look. To place four 1s in the central firepeg, simply click on the empty links. The numbers open the links to the fire.

If you clicked on a wrong spot, just click on it again for a quick fix. If you messed up something, double-click on this section to reload the app with the task.

The second control from the left on the big control bar is labeled mode. Click on it to pull down the menu, and select the second option SET. The board and its control bar will change. Hope you remember that controls in the book do not affect the app.

While in the SET mode, click on the central firepeg. It will fire up. You'll see what it means.

In the SET mode, the third control from the left is labeled timer TICK. Please find it and click. The central firepeg will turn its fire off. The four yellow pegs in the directions of the open links will fire up.

Make one more TICK. All the fires will disappear. Click again on the central firepeg, then on TICK. You may do it as many times as you like. The designs in FIREPEGS do not change on their own.

Let's learn one more control from the book. Look again at the top-right corner. The BACK button lets you reload the app with the previous task, complete with you design. If you launched the app and regretted this, just hit BACK. Or you can switch between two challenges and designs, as well as between two designs for the same challenge. You can only do it while you keep the book open.

Diagonal Links

Diagonal thinking is always more difficult, and for good reasons. Fortunately, the corner links in FIREPEGS only look different. They act just like the side links.

Please double-click on this section to launch or update the app. You previous network will be preloaded. The corner firepegs will be colored orange. If you missed the challenge pattern, click on the small board and watch.

The goal is to fire up all the 8 neighbors of the central firepeg at once.

Please open the corner links by clicking on them, like you did to the side links. When you see 1s in all 8 links, switch to the SET mode, fire up the central firepeg and make a TICK or two.

The Wait Time

In the READY mode, look for the fourth control from the left. It's labeled wait.

Change your network to make the side firepegs fire up after the corner firepegs. Click on the challenge board to see what I mean.

Pull down the wait menu and select 2. Click on the side links to change the numbers from 1 to 2. This makes the links slower. To fix an accidental mistake, just click on the same spot again.

Turn to the second control from the left labeled mode. Switch it to SET like you did before and fire up the central firepeg. Return to the mode selector. This time please switch it to GO!. After a brief delay, your pattern will play out on its own.

From 1 to 8

Please double-click here to update the app, then click on the wait selector to pull down the menu. At the bottom of it is ++, which means increment. Increment is the most important and completely overlooked operation in elementary math.

If the ++ option is selected, every click on a link makes the value of its wait time one bigger. Click by click, it reaches 9. Next it becomes empty (which means, the link is closed to fires), then 0, then 1 again. The bigger is the wait time, the slower is the link.

Click on the small board to view the challenge. The fire starts in the center, follows to the top, travels clockwise and disappears in the top left corner. To achieve this, you need to program the links of the central firepeg accordingly. The link pointing to the top must have 1. The link pointing top right must have 2, etc. Finally, the one pointing top left must have 8 on it.

Starting with an empty link, mentally count from 0 to your target value. If you missed the target value, just keep clicking. When all 8 links are ready, please switch the mode selector to SET. Then, click on the central firepeg to fire it up but DO NOT TICK. Instead, switch the mode selector to GO! and watch. The timer TICK button will start blinking on its own, and the show will begin.

To restart your pattern, stay in the GO! mode and click on the central firepeg. Remark that the ensuing red blink can be long or short, even too short to see it, depending on the time elapsed between the system tick and your click. This time subtracts from the time of your blink because the latter always ends at the next system tick.

In the GO! mode, if you click on a yellow firepeg, it will blink, but no other peg will follow. The yellow pegs are not causing other pegs to fire up. The central firepeg runs the show.

Link a Track

One can also get this pattern linking the firepegs so that they form a track of fire. It may look slightly confusing on a 3x3 board, so let's trace it. The fire will hop from firepeg to firepeg through the open links. First hop will be from the red central firepeg to the top yellow firepeg, then to the top-right-corner yellow firepeg, and so forth, until there will be no link available.

Double-click to update the app, make the connections, switch to the GO! mode and click to fire up. You may start with the central red firepeg, then try every other one. Except for the top-left-corner peg, they all cause something.

I assume, your board is still in the GO! mode. Please find the show selector on the right, pull down the menu and pick the option STATES. Firepegs can be in one of three states: 0 - sleeping, 1 - waiting and 2 - glowing. You can see them now, but if there is no fire on the board, all firepegs are sleeping.

Remember, you can switch your board to SET mode and make it tick by hand. Either way, when the central firepeg fires up, the other firepeg in the direction of the open link assumes the waiting state, or state 1. At the next tick, the central firepeg will return to the sleeping state 0, but the top firepeg will fire up and send the waking up message to the top-right-corner firepeg. Firing up the red firepeg, you started a wave!

The time diagrams may help to understand how it works. Time on them flows from left to right. Firepegs are listed from top to bottom in the order they fire up.

Track       Central

2000000000  2000000000
1200000000  1200000000
0120000000  1120000000
0012000000  1112000000
0001200000  1111200000
0000120000  1111120000
0000012000  1111112000
0000001200  1111111200
0000000120  1111111120

A waking up message includes the value of the wait time on the link. The second diagram is for the previous task. The number of the 1s in the lines is equal to the number of the time units firepegs were told to wait. If you return to the centralized network and switch the show selector to COUNTS, you will see how firepegs count their wait times.

Change the colors

In FIREPEGS, layout of colors and network of links are two different layers, completely independent from each other. Please double-click to update the app and start playing with colors.

On the challenge board, you must see that 4 spots now blink orange. It will take some attention - not for the last time in FIREPEGS - but I am sure you will be able to identify them. To change the color of those firepegs on your board, use the paint firepegs control, which is only available in READY mode. It's so big, you can't miss it.

Pull down the menu and select ORANGE. With the ORANGE color selected, click on the colored centers of the firepegs you want to change. If you missed, even if you clicked on a link, just click on the same spot again.

Select any other color and apply it. You don't have to match the colors of the challenge in FIREPEGS. Building layouts of colors alone is for toddlers (that's MAKE THEM MAD if you did not see it yet).

Glow Time

Please double-click to update the app. Watch the challenge. The colors are slightly different, the network appears to be the same as before, but firepegs no longer turn off. In the end, the whole population of the board glows together.

The trick is easy. In SET and GO! modes, you have control labeled glow time. You may click and type a number in it. To get it accepted, press Enter or click somewhere else. If your input was found unacceptable, the glow time just wouldn't change.

The < and > buttons allow you to adjust an existing number decrementing or incrementing it, but you can't possibly reach infinity this way. To make firepegs glow forever, double-click on the input. Do it, fire up your pattern and watch the board filling up. To get back to finity, double-click again.

The fired up firepegs remember their glow time allowances. You can't turn them off by changing the glow time for the board, but you have the fire extinguisher control, which is always at hand in the left.

The OFF control brings you to the READY mode, where you can change your design. From READY mode you can always switch to SET or GO! and fire up the pattern with a new glow time.

FIREPEGS with unlimited glow time and couple minor modifications can teach small children build layout of colors spot by spot, like I did in MAKE THEM MAD. That's what I created this toy for. But then, I found that I could...

Link an Oscillator

One step is left to create a self-sustaining fire. Did you update the app? It must display a new color scheme and the same network as before. The fire on the challenge board hops around until you click to stop it. You may click again to restart the pattern.

I am sure you already realized that to get the fire loop forever you only need to link the top-left green firepeg diagonally to the central red firepeg. Such link is called feedback, and feedback enables oscillation.

Make the connection, switch to the GO! mode and start the fire. Switch to the SET mode to pause it. Return to the GO! to continue.

While the pattern is running, try one more control available in SET and GO modes. It is labeled speed. Until now, it was set to 1.

Time in FIREPEGS is measured in ticks. The speed control ties tick to seconds. The bigger is the number of ticks per second, the faster is the action. If you are in the mood to make your student mad, ask what fractional values mean and how do they work.

Remember, you can use the fire extinguisher control to stop the fire and return to READY mode, where you can change your design.

Splits and joins

Modern construction toys became completely wordless. A child may build every model and never learn what part is called a nut. I hope, you will rely on concepts and words to meet FIREPEGS challenges.

FIREPEGS is a parallel system, and our mind is not. Even a simple pattern may defy human imagination. In real world we deal with parallel systems all the time, and we keep them under control using abstractions. Your car's engine oscillates in a convoluted dynamical pattern. We can't think of it in whole, we just say it's running.

Please double click and watch the challenge, but not too hard. It's eyes squinting (or, at least, bewildering) because the fire splits in two fires and joins to become a single one.

Splits and joins in FIREPEGS are nothing special. If a fire has two links to go through, it goes through both, and if two fires hit the same firepeg at the same time, they just fire it up.

Here is the recipe. Starting from the central firepeg in the bottom row, you link straight up until you reach the central firepeg in the top row. Here the track splits. One new track goes to the left, another one to the right. At the top-left and top-right corners respectively, both tracks turn down. At the bottom-left and bottom right corners they turn toward each other and enter the bottom-center firepeg, from which you started.

By the way, if you don't like so much red, change it to something else. I would suggest blue because I call this pattern Fountain. Just pull the paint firepegs menu, pick another color and hit the color ALL button on the right. To transfer the color to the challenge board, find the Copy the Colors control in the CONTROL CORNER and click on the left part of it.

To start the pattern, switch to the GO! mode and click on any linked firepeg. Never mind not linked ones. And you know how to use the speed controls, do you?

Isotropic Thinking: Left to Right

Isotropic roughly means being the same in all directions. Imagine yourself in the middle of nothing.

The space we are in is believed to be isotropic - provided, of course, that there is nothing in it. More often than not, the media behind our abstract models - consider a clean sheet of paper or a plane - are isotropic too.

In FIREPEGS, every angle is a multiple of 45°. The board has two different isotropies. It stays the same if we rotate it multiplying 45° by an even number (90°, 180°, 270°, etc) or by an odd number (45°, 135°, 225°, 315°, etc). A regular grid rotated by 45° changes dramatically: It looks like 2 regular grids. To build the same pattern, we have to disregard one of them.

Some concepts, like feedback, were borrowed from anisotropic settings. In the loop of firepegs, there is no way to tell which link goes back.

Our children suffer through education because their bodily experience is markedly anisotropic. Their left is different from their right and their far is not that their near.

In MAKE THEM MAD I tried to address this issue and develop isotropic thinking as early as possible. For your convenience, here is a set of challenges presenting two similar, but different networks, each rotated to 0, 90, 180 and 270 degrees. On a tablet computer, don't let your student to rotate the device. The goal is to learn to do it mentally.

Watch how your student build the networks, note the difficulties and take measures if necessary. You know what to do, right? You double click, go to the app, watch the pattern, build the network, fire it up and compare.

Isotropic Thinking: Right to Left

I am sure you realized that I am not a native English user. I settled in the US over 15 years ago and made several American children. I had kids before, they went to schools, some teachers were good, some bad, but, by and large, the education was working. I only had to supplement it.

To my big disappointment, I found that US education was not working for my kids. Their English was fine. The schools could not teach my students math. In fact, they were scaring them off it.

This prompted me to take supplemental education very seriously. After several years and several coincidences I came across something I've never heard of before. No matter what school and what king of teaching, the biggest impediment to cognitive development is our naturally anisotropic, self-centric thinking. Addressing this problem, it is possible to enable children to learn very quickly.

And it's not just math. An educated Westerner can turn a clean sheet of paper into a letter, a map, a plot, a time diagram, a chart, or perform a computation on it. Starting from arithmetic and reading, all our wisdoms are spatial.

Isotropic Thinking: Top to Bottom

Go on with the building, please. I am just trying to entertain you while your student is busy. Did you notice that at some moment I started talking about fires as actors? They are not quite alive yet, but what is life, after all? Firepegs belong to immensely interesting class of mathematical models called cellular automata or CA. The most popular variety was named just Life, but it is tempting to think that CA are the ultimate building blocks of physical reality.

FIREPEGS, at least in this prototype, has never been meant to imitate the world around us. The idea was derived from my observations of a child's attention.

Firepegs are more complex than the classical CA. This makes them more predictable, understandable, manageable and, probably, not interesting for mathematicians. Well, they are for kids.

Isotropic Thinking: Bottom-Left to Top-Right

Are you still building the fountains with your student? I hope you do, and you have noticed that you are facing a new kind of them. The fountains with the diagonal stream can be harder to link. Good thing, the builder knows where to start. The straight fountains required to find the center.

For only five firepegs this could be intuitive. Not so for 7, 9 or 11. Or how to find the middle of the board?

Building two-dimensional patterns makes children spatially intelligent, but they remain crippled until they learn to navigate the boards by counting. Never mind they call it Cartesian plane. My last student learned this before she turned four.

Not sure if you saw STEREO LEARNING yet. This is an online presentation which I made recently. Working on SIX PROJECTS for several years, I was thinking that they were about using conceptually the same toy from early development to arithmetic and through FIREPEGS to computer programming. There were toys, indeed. You are playing with one of them. The most important part, however, was the knowledge, what the toys could deliver.

Isotropic Thinking: Bottom-Right to Top-Left

Remember I made the first fountain red? It was because it looked like a blood circulation diagram with two bilateral (like in fish) or functionally different (like in amphibians) circles. Yes, there is no heart, but it does not really matter. The fires can be moved with peristaltic.

The discovery of the blood circulation was about the heart, but most of all it was about the dynamical pattern in general. I began to think whether there might not be a Motion, As It Were, In A Circle wrote William Harvey (in Latin, of course). He did not know how the blood returns to the heart. He did not have a microscope to find out.

Another famous example is the great debate of heliocentrism and geocentrism. Neither Copernicus, nor Kepler knew what was preventing the celestial bodies from falling and making them move. The whole issue was about dynamical patterns. FIREPEGS, by the way, could help to understand it too.

Isotropic Thinking: Top Right to Bottom-Left

In this part of the world, dynamical patterns are everywhere and they are everything. We people are aggregates of dynamical patterns, yet I have never heard of them being studied or taught per se. Every trade struggles with dynamical patterns on it's own.

Virtually the same device migrated from gunnery to electrical engineering, to physics, to electronics, to medicine, every time emerging under different names. A fountain pattern can be found in an electrical circuit, a dishwasher, a lubrication system or a hurricane. Again, you don't need batteries, motors, pumps and warm water because every firepeg has it's own rechargeable power source.

Being omnipresent, dynamical patterns are notoriously hard to investigate, Few people have a chance to play with them. I hope, FIREPEGS will make dynamical patterns available to every learner.

Isotropic Thinking: Top-Left to Bottom-Right

Meanwhile, you have arrived at the last challenge in this infinite series. Please do it. Next you are going to build a square of fountains. I just wanted you to make sure that your student is prepared for this task.

Spray Around

Please double-click to update the app. Your 6x6 board is divided into four 3x3 squares colored blue, yellow, blue, yellow. I linked the blue quarters to create two little diagonal fountains (the ones you built before were 5x5 big). The main streams are directed from the inner to the outer corners.

You may switch to the SET mode and fire up two outer blue corners. Switch to the GO! next and watch the fountains at work. Your goal is to link two yellow fountains spraying outward in the same manner.

When you are done linking, select the SET mode and fire up all 4 corners of the board. Then find the sync START control on the top of the challenge board and click on it. You speed and glow time will be automatically adjusted to match the record.

If the running patterns are not the same, try to find out what's wrong. Click on the OFF control to extinguish the fires, then fix your network.

To create a new pattern, try to change the glow time. Then, try to fire up the network differently - for example, the blue quarters in the inner corners and the yellow quarters in the outer ones. You may use different colors and change the network itself too. There are many networks similar to this, and usually they are veritable kaleidoscopic patterns generators.

Term Project

The final pattern in this issue of Album for Beginners looks like a corolla. It comprises four identical elements. If you double-clicked to update the app, you must see one of them linked. It's just a loop in a 4x4 square.

The goal is to link the other 3 petals. They all must connect an inner corner with the outer corner (and back), and the direction of the rotation of the fires must be the same (clockwise). The "spinners" in the center are just an apparition. They only seem important. To match the challenge pattern, you'll need to set the glow time to 2.

I hope Album for Beginners has demonstrated few basic concepts and skills of FIREPEGS, and delivered the most important message: Like in real life, in FIREPEGS things are not what they seem. Thank you for your attention.