A few years ago I was involved with a math education project which got me looking at popular Apps. The trend is to be very game-y: cartoon backgrounds and animations, but beyond that, coins, collectables, outfits to buy. The thinking is kids hate, hate, hate learning. So the number one goal is to motivate them, using those tricks. Get them to plow through the standard exercises. And these are education Apps made for schools.

Meanwhile, on the project I was on, I was learning you can research types of traditional exercises, think about what having a computer helps you do well, tweak and rewrite, and use game tricks inside an exercise. With the same type and amount of effort you'd use to make anything else, educational exercises can be pretty good.

Nothing here is directly from the project I was involved with, but is inspired by that "make it more fun by making it more educational" idea.

**Menu/sequence:** There are 44 exercises arranged into 9 categories (across 3 big pages.)
You can freely navigate and select any exercise.
When you're inside of an exercise, double-tapping the upper-right circle returns to the menu.

But you can also play exercises straight through. After "winning" an exercise, Again/Next buttons pop up. Next jumps straight to the next exercise, the same as if you returned to the menu and tapped to start the next one.

**Progress:** Most exercises get more difficult as you get correct answers. In general, no mistakes
goes to the next, one mistake repeats the current difficulty, and two+ mistakes goes backwards.
Some exercises run through a specific sequence, then randomly generate "hard" versions if you keep on.
Some give random "version 1" problems for a while then switch to version 2, then 3
(like 3 mini-exercises in one.) Again/Next will pop up when you've answered everything it has to offer. "Again" will continue with more randomly generated problems and can be done as long as you like (it asks you Again/Next at intervals.)

The progress bar (on the main menu, under each exercise) is just for fun. Playing through everything fills it about 1/2. That's just an incentive to play longer (you could think of it as "you've done all the training examples, now show me you understand it all together.") Filling the progress bar does nothing special. The Options/Menu area has a button to zero-out all progress for all exercises (a short-cut for deleting and reinstalling.)

**Cheating:** There's a way to quickly look through all problems in the current exercise, sort of.
The cheat skips to the start of the next problem, and does nothing else. It won't show the solution to
the problem you were on and if you had one part visible and 2 hidden, this trick won't show the hidden
parts (but you can
solve part 1, see part 2, then get bored and skip ahead.) The only use is to get an idea of how it
progresses. Say you want to know how difficult the 1-10 skip counting exercise gets, cheat to skip ahead a dozen
times and glance at each one, until it seems to repeat.

It's meant to be something no one would ever do by mistake. It also "turns off" real progress gain (until the exercise restarts, going back to your real progress.) Put a finger on each corner of the screen and tap with one of them - 4 fingers total, 3 down, 1 tapping. The tap can be in any corner. I use two hands. The area counting as a corner is pretty large. Once you have it, the 4th finger can tap, tap, tap to quickly skip through.

**Number Tiles:** The flat number tiles have a fast mode. You can always hand-drag them
into the answer slots. Tapping one speaks the number. Dragging them is fun for a while, but soon becomes
tedious. To speed up, tap any blank answer slot. A ? should appear inside.
That's "tap-to-select" mode. Tapping a number tile snaps it into the slot with the ?.
If you get it right, the ? will move to the next. You can now solve number tile
problems by only tapping, once per answer.

Tapping the ? toggles back into tap-to-speak mode. The mode persists through exercises (each new tile exercise will start with a ? until you tap to turn it off).

**Therapy cow:** After a mistake, a "therapy cow" may fall from the sky and jump around.
It doesn't mean anything and can be ignored. It goes away on it's own, or by tapping it, or usually when you answer something correctly.

Dragging blocks to make lines length 1 through 10. There's no counting here. This is just a typical exercise to get used to how numbers represent a quantity. It's not always rows 1,2,3,4 ... the next row jumps around to make it interesting. At the end the blocks stack themselves and let you knock them down. That's just for fun.

A version of "Pink Tower" from Montessori. This one has many blocks from 1x1x1 up to 6x6x6. There's no right or wrong. All you do is stack them in various ways and hopefully notice the 6x6x6 is as tall as two 3's or a 5 and a 1, and so on. The blocks will snap into place (if coaxed,) and the smaller ones leave footprints. For fun, a red thing walks around, attempting to climb them.

Digit recognition. You get a small, user-movable window to see part of a large number. Slide it around to examine more of the digit, and guess it. Two fonts (Tahoma and Arial.) This feels like a tracing exercise to me. You notice how the curves in 2's, 3's, 5's, 6's, 8's and 9's are all the same. I wonder how well children learn from tracing letters on a screen? I feel like using a pencil is different. A screen is simulating finger-painting. Do children finger-paint words? Is it helpful for later pencil use?

Colors are random. Progress has the window show less of the number. Eventually numbers acquire a small tilt (I thought about spinning them more, but at 90-degree tilt 6's and 9's are identical.) The range starts at 1-5, then snaps to 1-9, then finally 0-9. I feel like children learn 1-9 all at once, but it gives a sense of progress. Zero works here since you don't need to know what it stands for. Maybe you get that's it's also a number, but the exercise works if you think it's a tall O.

Left alone, the window slides around slowly. That doesn't mean anything - just a hint of the draggability. Or you can make your own fun by watching only the auto-slide to discover the number. It's also just for fun how the number grows when you guess wrong.

Matching numbers words to digits ("two" with 2.) This starts showing random 1-10 digits and you drag the words over. Then it flips to showing words and you drag the digits. Then it flips to using incorrect spellings (one, won, wun.) For fun, numbers are in batches of 3-5. Also for fun it alternates between one-at-a-time and all at once.

Simple counting on a number grid, up to 20. Sequences are things like 1 to 10, 10 to 1, evens, odds, 1 to 20, 20 to 1 ... .

Each new number gets a random-ish selection of possible tiles, attempting to guess common mistakes.

Matching digits 11 to 20 with the words. This is the same as the 1-10 version, except no alternate spellings

These are about matching a group of 6 things with the number 6, and vice-versa. The idea is that at first children merely memorize the number sequence. So far 1, 2, 3 is the same as a, b, c. Learning how 3 actually means OOO and when you see OOO you can say "three" is the next step.

Counting items. These are always in a line. At first pushed together. Then with spaces between, then changed shape and color. They come in groups of three to hopefully trick you into thinking "this is 2 more than the last one, which was 6, so it must be... ." They alternate between showing all 3 at once, or one-at-a-time, just for fun. To answer faster, can tap to get the ?.

Counting jumbled different color and/or size items. They can be hand-dragged to help count them. For fun they come in 2 at a time and the areas change between various-sized rings and rectangles.

The theory is to work towards how three is just three. So put identical objects in a line, then change colors ... then finally jumble them up, which this does. Kids learn many things can have three-ness. I also think having to move things around as you count them is a skill.

This one shows you a number and you have to drag cubes into the lines to make it. In edu-speak, the previous were number-to-representation and this is representation-to-number. Tapping the "crystal" pops out extra blocks. Tap each number when you think the blocks are correct. For fun, and to reduce dragging, each line starts with some blocks. Also in groups of 3 (it's surprisingly fun to try to use the extras for one number to finish another one.)

Matching numbers to dot-patterns, from 1 to 10. This works the same as the "match digits to words" from before except it's now counting. The dots are arranged a 5x2 pattern or 4x3 or domino-style. As before, it switches around starting with the digits, or the dots, then later it has you match only dots to other dots in different patterns, then finally random combinations.

Counting items arranged randomly in a grid, up to 20. Tapping them toggles a little dot, just as an aide. In theory this encourages you to think of them as a group of three, a cluster of 6... and add them.

The color patterns, cube sizes and patterns of squares, are random. Sometimes you get some pretty ones.

The same as the pervious "dots 1-10" exercise, except going up to 20. There are no domino patterns for numbers past 12, but this version adds words (you may see "fifteen" and need to find the tile with 15 dots.)

These are repeats of previous exercises, but using zero.

Matching word "zero" with number 0. Like the 1-10 version, except using 0-4 and always a 0. This also eventually uses tiles with dots, where the 0-dot tile has nothing on it.

Counting cubes in lines, using 0-6. The same as the 1-10 version, except one line is always 0, which is just an empty line.

The reverse version number-to-representation, using 0-6 with a guaranteed 0. Drag blocks to make each number; tap the number when you think it's correct. It's a little fun that 0 means "drag them all out."

This also adds something new -- sometimes you're given the numbers spelled out, with no digits. That's mostly an excuse to have you see "zero" a little more often.

Counting *some* cubes according to a rule. The rules for
what to count vary: in/out of the square, only cubes or only chips, only small or large items,
only stretched-out ones, or a certain color. Sometimes the big square is there just to confuse you.
Tapping the rule reads it out loud.

This is experimental. It's faster than sorting (you don't have to move them anywhere, or even tap them.) But it's discrimination, which is the point of sorting.

Measuring. You get a rod and 2 blocks to help measure it. Sometimes both blocks are free, other times one is anchored midway (the hope is this will encourage thinking such as "there are 2 to the left of the one I can't move. and I can measure out 4 to the right, giving 7.") Eventually it starts varying the size of cubes - a short rod might be 7 small cubes long, while longer one might only 3 long.

This is from the kindergarten Core "understand 11-19 are ten plus a single digit." As usual you have to fill each area with how many it says. But the areas are now 10x3 and length 10 blocks are added. The sequences tend to things like 2, 12, 22 or 9, 10, 11 (nine is 9 one's, while 10 is only a single ten.) Other than that it tries to avoid too many 1's. There's no way to dispose of cubes - if you pop out too many 10's it might be some work shoving them out of the way.

Sorting, based on a variety of rules. They're the same as before: color, size, chip vs. cube, regular vs. stretched. Some have 3 categories (color, size.) Objects pop in and out of areas as you pull them into the side. You're not allowed to drag into the wrong area, and it counts as a mistake to try (for the progress bar on the main menu.)

Three mini-games involving equal/not-equal, and less/equal/more. It runs through them as you get the old ones right. First decide if two areas are the same or different. For fun it gives a random mix of "same," "different," "equal" and "not equal." After a while it switches from a line to loose objects. Next it switches: the items are always unequal and you have to label them both with greater/fewer (or less/more.) After a while it has you compare dots on tiles.

The last mini-game combines them. First you choose same/different. Then you have to choose from between <, = and >. In theory, having placed more/less it's obvious why a single </> symbol is an improvement. In theory.

.Counting up to 40. By 1's, then skip-counting, but not too hard. Everything is in proper multiples - 5, 10, 15 but never 12, 17, 22.

Skip counting anywhere between 1 and 100, including odd multiples (the 12, 17, 22 example.) The middle numbers are sometimes filled in (like the picture) and sometimes not.

This is the more normal skip-counting. Nothing to drag - you just tap the correct spot. This is faster but it's easier to just tap the pattern without thinking about the numbers. Bunnies also hop across. They have no educational value (if you have disabled the wrong-answer-cow, the bunnies will also stay home.).

It also occasionally has rows like 10-19 instead of 11-20.

Try to make a number by adding 2's, 3's ... . You get srandom of 1 to 5 length cube-strips to try to make a target number. The size of the area to fill and the number below are the same. The number doesn't do anything. You "win" as soon as you fill the area. The equation (on top) is just a decoraction. It's displayed as you add strips.

At first you get random length strips. Sometimes a few singles, but never enough to make the number. Later on you get "puzzles" with only one solution, like making six out of 5,4,3,3 or 5,4,3,2.

Four mini games about making an equation with strips. Part one you're given a sum like 2+4 and have to make it by dragging strips into line. You're only allowed to place the correct strips, in the correct order. Part 2 is the reverse. You're shown a row line a strip of 2 and a strip of 4, and have to drag in the "2+4" equation tile.

The next two parts are the same, but with three numbers. You might have to make 2+3+2; or be shown ABBBBCCC and have to select the "1+4+3" tile.

Introduction to number lines. At first you're shown scrambled 1-12 and need to pick out each one (drag the number line that amount and tap the tile.)

The rest are finding how far apart 2 numbers are. Section one of the numberline is locked to one number. You don't have to, but you can adjust the second section to help figure the distance. It's also used to show wrong answers. Later versions have you you backwards ("how far from 6 to 8" becomes "how far from 8 to 6").

This is the same as "block strips" except with a numberline. At first you're show a locked numberline with two sections. You have to drag the equation it makes. Part two you're given an equation to make, like "4+2," and need to make it using the 2-section numberline.

Tapping the ball switches between sections (drag the blue bar, tap the ball, drag the yellow bar.)

This is the same as the previous, except using subtraction (the second yellow bar runs backwards.) After getting enough subtractions correct, it mixes subtraction and addition (and still either "make this" or "decide what this is.")

Given an equation like 3+??=7, use the numberline to find the missing number. As you slide your numberline segment the ?? on the tile changes to your guess. To keep it from being too easy, You're no longer shown the total. When you have it adjusted, tap the tile.

There are four versions: find second addend: "3+""=5", find second subtractand, "5-?? is 3," find first addend, "??+2=5" and find first subtractand, "??-2 is 3." The problem is always just to sliding one segment to the correct length.

As equations, 6+??=8 might be a little advanced, but this is more like "make the picture that represents to equation". I feel like that's one of the things numberlines are for.

Giving regular N-gons with 3-10 sides and sort them. At first they differ by several sides, then later you might get 7,8 and 9 sides to sort. Eventually they're stretched, for fun. Like the other sorting, dragging them against an edge pops in/out, and you can't pull something into the wrong area.

Latin-style number prefixes (the kinds used in n-gons.) Another one matching words to numbers, then numbers to words, then misspellings.

Matching words for regular n-gons from 3 to 10 sides with the shape.

Matching the types of triangles, then later quadrilaterals. Each has two sample shapes (for isoscoles triangles it randomly shows an acute and obtuse version.)

Matching with roman numerals, up to 20. Eventually runs every version o matching: roman numerals dragged to the correct digit, vice-versa, 1-at-at-time with wrong spellings (XV, VX, VVV, XY,) then random missing items.

A clock with roman numerals. The only action is placing the correct number in the slot. The clock moves ahead a few hours each time, doing it all again once the clock is filled.

A sun (with glaring sunlight,) and moon (which brings darkness) spin with the clock. Placing the roman numerals isn't terribly exciting but watching it spin through a few days is kind of fun (and possibly educational "intro to the funny way clocks work").

Groups of 2-addend addition-only flash cards, in set patterns (each set of 4-6 cards is one particular pattern. Pattens are: 10+(random 1-9) and the reverse; all same sum (3+8, 6+5, 10+1); constant plus interval (7+2, 7+4, 7+6, 7+8, 7+10,) and the reverse; both +1 (3+6, 4+7, 5+8, 6+9); and commuting (random pairs: 3+6, 6+3, 1+14, 14+1, 12+6, 6+12.)

I'm not sure how well the "Addition" exercises lead up to straight math like this. I just liked these patterns and put it at the end in Misc, since why not. There are also bunnies that hop around for no reason. I figure everyone has a snooze-reflex to flash cards, so many seeing a bunny would help.

This, and the next exercise, is based on the idea of adding a sequence by rearranging them to get easy sums. In the picture, we're trying to make 10's: combine 1 with 9, and 4 with 6 (dragging a tile over of another attempts to add them.) Once you make the required sums, you finish up by combining them into a single sum, in any order. I thought that made the point that the overall goal is adding these numbers (in the picture we'd create 10, 10 and 8, then combine them to make 28.)

The plus-signs are only a hint. They vanish as soon as you touch anything. The tiles can also be freely placed anywhere - this allows thing like sorting into low/high.

For progress, it starts with ten-pairs, maybe 7,3 and 6 (6 is a dummy.) It goes up to three pairs plus a fake. Ex: 7 1 4 3 6 5 5. Then it moves into creating any multiple of 10. For example: 17, 13, 8, 12 and 19 (the dummy). I feel like this is the same idea "we can find some pairs to combine that make this sum simpler."

The last is making 5's and 15's. For example: 3 2 7 8 9 6. Maybe it's from playing cribbage too much, but spotting pairs that end in a 5 always seems like a good way to add.

As usual, when that's done it randomly picks from any of the previous exercises.

This introduces the idea of splitting a number, and the tool. If you have 8+6 you might regroup it as 8+(2+4), then 10+4. But it also has some variations that didn't fit in the previous one.

The spitter tool turns on by tapping a tile. Blocks pop up below and you can slide/tap to select the split, then tap the original tile (now grey) to split. In the picture we've tapped the 5 and selected 2+3 as the split. Tapping the 5 again replaces it with a 2 tile and a 3 tile. The system won't allow you to make non-useful splits (in this case, you have to make a 2, to go with an 8. Or you could split an 8 into 5+3.)

For progress, at first there's no splitting. The first is combining *three* digits into a 10,
for example: 4 5 1 and an extra 2. Then it moves onto using the splitter with two numbers.
That ramps up to two sets and a fake, like 6, 4, 7, 5, 1. In this case 6+4 gives an easy ten,
but we need to split to 7 or 5 to get the other ten. And the 1 is a dummy. After
combining, you'd have 10, 10, 2 and 1.

Finally, for fun, it gives pairs to combine into a "mystery number". If the secret number is 26, you might get 12, 14, 9, 17 and an extra 15. It seems to fit in with the theme of finding good pairs to combine. A trick to make it easy is to start with the highest and lowest number (except one of them may be a dummy. You should drag them near each other and then check the next highest/lowest.) I tried this one after I forgot the rules and it was more fun than I thought.

These are about the very basics in base-10: 1) 942 is like three single digits counting three categories, 2) the ones on the left count for more, 3) the zero's are important as place-holders, and 4) the places are called ones, tens, hundred and thousands.

Sure, we'd like to show the idea how the places are a unique way to form any value, and ten of a places rolls over into the next. But we need addition and regrouping to really show that.

A fairly straight-forward block exercise. Tapping gives you 1x1x1 cubes, and the long 10x1 ten-rods which we've used before (in the 11-29 base-10 exercise), and new hundreds cubes (flat 10x10) and big thousands cubes (sized 10x10x10). Drag them into the area to create the number. It shows the total so far, with each digit turning green as you get it (for 521, the tens place turns green when you drag in two 10-rods.)

You can think of it as simply getting the correct quantity of each item. With the size of the blocks suggesting the power-of-ten relationship. The blocks snap in place. At first the "hundreds" style stacks them -- a user can easily see how several 10's and 1's cover part of the area a single 100 would. Later on, the thousands version goes from left to right (the hundreds on their sides, like folders. Again, showing how several hundreds take less space than a single thousand.

To make them fit and avoid too much busy-work, the numbers are limited (at most 3 thousands, and so on.)

The reverse of the previous -- you see a stack of 1, 10, 100, 1000's cubes and have to choose the number it makes, dragging one digit at a time. The numbers purposely favor 0's (302, 320 or 300). I feel like those help emphasize using 0's as place-holders.

Tapping also shows and says "ones", "tens", "hundreds" and "thousands".

The next three are about sorting. The idea being to show each place is exponentially larger than the previous: 300 < 299, but 389 < 612. This is probably too much sorting, but all three ways seemed interesting, and this is the first time we've sorted lots of numbers, and two of them also review < and >.

Sort by swapping adjacent items. items. Either tap the < between tiles, or attempt to drag a tile. Tap the ?? to check.

This seems simpler since you focus on just two items, and whether they're out of order. The error feedback complains about the first wrong pair. It starts with 2 items, goes up to 5, and sometimes flips to >, meaning they should run from high to low.

It's a little fun since you can make a swap and realize you've moved the "other" number further away from where it should be. Using the smallest number of swaps is like a little puzzle (but not required). The starting order is created based on "swaps from correct order", which tends to get larger.

You're shown one 2 or 3-digit number and have to sort a handful of other numbers into more or less. It tries to fool you with the 1's place -- if the middle number is 35 it might give 19 and 27, and 71 and 82. When it starts using hundreds, numbers are made by changing one or two digits: 345, 145, 385, 341.

It also features the mini-100 grid. Tapping a number animates a small 10x10 grid showing that many tiny cubes.

One the one hand, it's difficult since you can pick any of those numbers. But it's easy since the order doesn't matter and each is a simple comparison between two numbers.

Sorting by picking the numbers in order. This seems harder, since you have to scan all of the tiles. It also flips between < and > and left-to-right and right-to-left. It's a little fun realizing that right-to-left with > is merely low to high.

But it's the same skill as the others: compare digits from left to right.

It comes in several patterns: rearranging 2 digits (704, 407, 740 ... ), rearranging one digit (404, 440, 44, 400 ... ), three scrambled (456, 645, 564 ... ), and the old "fool with tens and one place" pattern (555, 399, 721).

Flash cards with base-10 exercises. None of these have a regroup -- they only involve adding one's to one's and ten's to ten's. As before, they aren't random, and each set has a pattern. They are:

A) Only the tens place: 40+30, 20+20. B) Merely combining tens and ones: 30+6, 50+2. C) Adding the ones place: 21+6, 94+2, D) Adding the tens place: 34+20, 70+19, E) pairs adding same to tens and ones: 42+3 / 42+30, and F) pairs with a digit flip: 23+16 / 23 + 61.

This is the first use of the big sliding numberline (the area just below the tiles, beside being a "you are here" inset, also allows fast slides). The tens (and fives to a lesser amount) look different. I don't love it, but having them all the same was worse.

If you're wrong a few times in a row, it auto-sides to put the correct answer near the center.

Spelling for 20 to 99 (45 is "forty five").

Fix slight hiss/hollow-ringing on audio.

Showing adding with regroup using the N-cube rods (anything combined longer than 10 is converted into a 10-rod, with the extra lopped off).

No information is collected by this App. No names, ages or locations; there's nothing to buy so there's no financial data.

A better App would at least collect scrubbed usage statistics (for example, which exercises no one ever spends more than 15 seconds on.) This doesn't even do that.

Every exercise has the rule that it won't just say "wrong" and move on. You have to keep trying until you get it right. I think this works out better than the alternatives. It lets you give a subtle not-so-scary error message (you know your answer wasn't right, since you didn't go to the next problem.) And every problem ends with you definitely knowing the correct answer, and with a happy sound.

One problem is avoiding "try stuff until you get it." The trick is to give a longer delay after a mistake, so it's clearly faster to try to get the correct one. Another is scoring, but that's not a problem. After someone gets it wrong (or wrong twice or whatever you like,) you can count it as a mistake while still having them need to get it right.

**Q:** Why the cutesy name in an otherwise non-cutesy App?

**A:** It took me 2 weeks to name my cat. Finally settled on "Abby." For this I was thinking "Early Numeracy,"
or "Math Game_056." It's hard thinking of a name you don't hate.

**Q:** Clearly the first two "correct" sounds are a small bell being dropped on a hammer. But what is the
last one?

**A:** The third "correct" sound is conceptual: a glass of apple juice being poured and drunk. It seemed more fun than the usual cheering or trumpets.

**Q:** Why do the number sounds sometimes mess up?

**A:** There are two sets of 1-9 sounds: normal, for taps, and fast for when the computer does a
count-off. To mix it up, there's a third set of slow numbers They rarely, randomly play instead of the
normal ones.

**Q:** Where can I find a real rug with that pattern?

**A:** If you're serious, I'm sorry - it's not based on any specific rug. But if you're mocking it, let me tell you that
is a great rug. It goes in the dining room and is a beautiful non-ironic red plush. The gold is patterned like trim or curtain
rods with knobs, or filigree. It classes up your meals. Then it can be streets, or the fat parts can be cities for when you
play under the table with your hot-wheel cars. It's a great rug and you know it.

**Q:** Are there details on how the cow works?

**A:** Yes. The cow randomly appears after any mistake (the chance increases slightly after each mistake and is lower if you've recently been cow'ed.) Where it falls is random. The little jumps it makes are also random (both the timing and the force, which is why it sometimes seems to stampede.) When you move a block or number tile, the cow will face it, enabling you to lead it around.

It will leave after a minute or so, or randomly after being tapped. Tapping the cow also shoos it away from your finger.
There can only be one cow at a time. There are 4 moo sounds (none of which are from a real cow,) played semi-randomly.

The cow is made with simple game-physics joints. The head-bob, tail wag and leg flex are the result of tweaking settings, and involve no coding. Even the way it turns itself feet-down is code-free - it just naturally does that when air-borne since the feet are heavier than the head. If you've used Unity3D, it's all just character joints and settings, especially the Spring numbers. They are tricky, and in my first try it was un-cow-like junk, but it can be done.

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