Have you seen Dragon Box? Once you do, you will be a believer in the power of technology for learning. I wasn’t before, I am now. My son is 6 and after about 4 hours of fun he can solve simple one-variable equations. Here’s how it works.

In the first level of Dragon Box you see a screen with two halves, “This side” and “That side.” There is a box on one side and some cards with random pictures on them. Your job is to isolate the box on one side, i.e. remove all the cards from the same side of the box.

This is very simple at the beginning because the only cards on that side are these funky vortex cards and all you have to do is touch them and they disappear. Vortex cards represent zero, but only you know that.

Later, other cards start appearing on the box’s side but then you learn something new: every card has a “night card” which graphically is represented by a card with the same picture but in negative exposure. Negative. If you slide a card onto its night card (or vice versa) the card turns into a vortex which you then dispatch with a subsequent tap.

Later again it happens that cards appear on the same side of the box but with no night card. But then you learn something new. You have cards in your deck and you can drag them onto either side of the screen. A card in your deck can be turned into its “night” version by tapping. Thus, you can eliminate a card on the box’s side by taking the same card from the deck, “nighting” it and then using it to vortex the offending card.

*But any card you drag from the deck to one side of the screen you must drag to the other side also.* This represents adding or subtracting a constant from both sides of an equation. After you have isolated the box on one side you have shown that the box *equals *the sum of all the cards remaining on the other side. But only you know these things.

Later still, cards appear with “partners,” i.e. another card right up next to it with an inexplicable dot connecting them. If the box has a partner you can eliminate the partner by dragging the corresponding card from your deck below a line which magically appears below the partners as you drag.

Dragon Box requires that whenever you drag a card from the deck below the line of any card, you must drag the same card below the lines of all card-groups on both sides of the screen. Once you have done that you can drag the card that is below the line onto its duplicate above the line and they together turn into a card with looks like a die with one pip showing. Such a card can then be dragged onto the box leaving only the box.

Here’s a demonstration (by me of an early level.)

The partners represent multiplication, the line represents division, the die with one pip represents the number 1 (i.e. the identity) and 1 times the box is just the box. After you have isolated the box you have shown that the box equals the sum and/or products of cards that appear on the other side. But only you know this.

Finally, the box mysteriously becomes the letter x. The cards lose their pretty pictures and become numbers and other constants. Night cards are now negative numbers. The vortex becomes zero and the die becomes the number 1. In the dividing zone between the two sides of the screen eventually appears an equals sign, and all the operations the child has learned now take their more familiar form and by pure sleight of hand he has been tricked into porting the very very simple logic of combining symbolic operations into the otherwise tedious world of “solve for x.”

I personally am astounded.

A few final thoughts.

- The reason a six-year-old can learn algebra with Dragon Box but could not before is that Dragon Box unbundles algebra from arithmetic. You don’t have to know what crazy-frog times lizard-fish equals to know that Box = CF times LF. Simplifying the right-hand-side is beside the point. Conventionally algebra comes after arithmetic because you need arithmetic to simplify the right-hand side.
- Actually what you learn from this is that algebra is far more elementary than arithmetic. My son can add numbers (up to one digit plus two digit) but just barely grasps the concept of multiplication. He has no idea what division is.
- Someone who already knows arithmetic can still learn algebra faster (and have more fun in the process) because Dragon Box shows how all the arithmetic can essentially be saved for the very end, modularizing the learning.
- Dragon box also rewards you if you solve the equation with the precise number of operations recommended. (This is usually the minimum number but not always.) This is a clever addition to the game because all of my kids refused out of pure pride to move on until they had solved each one in the right number of moves. Imagine asking a kid learning algebra to do that.

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January 21, 2013 at 6:00 am

Dr. Henry BorensonYour description of the app is an accurate one and you explain how the app “unbundles” algebra from arithmetic. I would like to question, however, your statement that “a six-year old can learn algebra with Dragonbox.”

A first point: you say that “Dragon Box requires that whenever you drag a card from the deck below the line of any card, you must drag the same card below the lines of all card-groups on both sides of the screen.” This statement may not exactly convey the meaning that once the child has placed a card from below the deck unto one of the two sides, the app freezes until the child places the same card in the required locations as determined by the app. An indentation appears to inform the child where else the card must be placed. Hence, the child can simply follow the app prompts without distinguishing when a card needs to be placed once on the other side vs. when it needs to be placed next to or under each term on both sides.

A second point: even when the pictures change to letters toward the end of the app, the card outlines are still visible. Hence, the child can solve these puzzles just as he did previously without being cognizant of the operations that are displayed. He still may not realize that it is only under the operation of addition (and not that of multiplication or division) that you place the card only once on the other side (even if there are several terms on that side).

A third point: In a number of respects the hidden algebra behind the puzzles is a deficient algebra. For example whereas a day card “divided” by day card will give you the one-cube, the app does not enable you to simplify a night card “divided” by a night-card to also give you a 1-cube. In other words, the app will enable a/a to simplified to 1, but not (-a)/(-a) to be simplified to 1. Likewise, although you can “multiply” a card by 1, nothing happens when you attempt to do the reverse. In other words, it recognizes 1(a) =a but is not “aware” that (a) (1) is also a.

A fourth point: As you point out, the app does not require or use arithmetic. You present the advantages of this feature. On the other hand, this feature prohibits the presentation of simple linear equations which require combining like terms. Hence examples such as 3x + x = 2x + 6, or even 3x + x = 8 are not presented and cannot be solved on this app.

This is not to discourage anyone from purchasing the app. The app is a fun app and its developers showed creativity in its development. However, it is very questionable to suggest that a child is actually learning algebra in using the app, especially since the child does not once solve an equation on his own, without the prompting provided by the app…

January 21, 2013 at 2:17 pm

jeffThanks for your comments. In general you are right that the app alone will not teach a 6 year old an *understanding* of what he is doing. But a parent popping in from time to time to explain things is enough. I would talk to him about the operations by saying things like “divide by,” “cancel the” “subtract from both sides.” And finally when he solved it I would say out loud “x equals…(whatever is on RHS)”

In response to your specific points.

1. They pretty quickly learn to multiply all terms and add to both sides as a matter of habit. I don’t see this as a particular weakness of an app. A teacher would also remind the student. What is true is that when a child graduates to paper and pencil they are cut loose from this safety belt and they have a new way to make a mistake. But they learn not to forget.

2. This is where a parent voice over helps a lot. “Oh look its 5 plus y.”

3. You can cancel -5/-5 and you get 1. But you are right that multiplication by identity is implemented only by dragging g the one onto the other card.

4. Not true. In later stages you can add x to x and the app processes it and yields 2x.

To summarize, after a few hours of fun you can say “let’s play Dragon Box with pencil and paper” and the child can replicate all of the operations by rewriting equations line by line and a parent can explain to the child what he’s doing. I said things like “This equation says that x is a number which when multiplied by 5 equals 7. We are going to use the Dragon Box tricks to figure out what that number is. By the way we call those tricks algebra.”

January 21, 2013 at 5:32 pm

AnonymousThanks, Jeff. Any other similar suggestions?

January 22, 2013 at 8:42 am

Kfir EliazI think it’s a great app despite the weaknesses. I can’t figure out how to add users? any help will be appreciated!

Kfir

January 22, 2013 at 1:03 pm

jeffkfir you mean more than 4 users? i didnt see any way.

January 24, 2013 at 6:28 am

twickerSecond method for teaching a six-year-old algebra:

Have one or more much-older siblings (9+ years older) who are learning algebra and have a lot of patience for their younger sibling. Add a curious six-year-old (or 5-year-old, or whatever) who is happy to bother said older sibling with questions about what s/he is doing. Mix well.

That said, Dragon Box sounds like great fun. :)

January 31, 2013 at 1:45 am

Dr. Henry BorensonThank you for your responses. It is true that a parent knowledgeable of algebra or a math teacher can help the child to make algebraic sense of the Dragonbox puzzles and their solutions.

You used the expression “subtract from both sides.” Can you provide one example where the child uses this principle to solve a Dragonbox problem? Is the child able to swipe a card off either side of the display? He can certainly drag a card to each side, but can he do the reverse?

You mention that the app recognizes that (-5)/(-5) is 1. I stand corrected, as I notice that in 4/20 it recognizes (-b)/(-b) as 1. However, in that same problem, the app will not recognize (-b)/b to be -1. Likewise, it will not recognize (-e)/(-f) to be e/f (see 2/20).

Not only is multiplication by 1 not commutative, neither is addition of zero.The app does not recognize that a + 0 = a. If you try to drag the “a” unto the zero, it will be pushed off. (See 2/11). The app is also faulty in not allowing the user to drag the zero (swirl) unto a card before it makes the swirl disappear.

You mention that in later lessons, the child learns that x + x = 2x. Would you please provide the lesson and example number where you observed this? In going through the full app I did not see this once nor do I see how this is possible using the moves presented by the app.

If your 6-year old can make sense of the statement, “A number when multiplied by 5 equals 7″ than he certainly is not a typical child!

You may try giving your child this example: 4x+3=3x+9. Can he solve it? If not, my free app The Fun Way to Learn Algebra: Hands-On Equations 1 Lite will enable him to do so in three short lessons.

February 11, 2013 at 7:51 am

Mõlgutusi » Blog Archive » Kuidas õpetada kuueaastast lineaarvõrrandeid lahendama[...] Ely ajaveeb Cheap Talk kirjeldab, kuidas arvutimäng Dragon Box õpetab lapsi, kes veel aritmeetikat ei tunne, [...]

February 22, 2013 at 4:18 am

wewanttoo (@wewanttoo)Jeff, thanks for the blog post. You will be happy to know that we are working on improving our application with much more maths and transfer. This will also please Henry I guess.

Kfir, the app store version allows 4 profiles maximum. Can you tell us why you would need more ? We are working on a school offer that will allow online profiles e.g. to switch users.

Finally, Henry, thanks for your meaningful feedback. As always, it is taken with great consideration in order to improve our products. We wish you the best with your app! Finding ways to improve the way people learn and perceive Maths is our shared goal!

February 22, 2013 at 4:28 am

Dr. Henry BorensonThanks for your gracious comment.

One other suggestion, since the swirly is supposed to indicate zero, it should not disappear when touched. You want to illustrate the addition property of zero. Hence, it should disappear only when one brings it over to another card, or when you bring another card to it.