## The difference between game and drill

So in my last post I opined that the optimal mathematics game in the Tiny Games spirit should “incidentally have some mathematics in them and are the sorts of games one might play even outside of a mathematics classroom.” That led to some confusion.

Let me try a do-over:

During a game, when the primary action of the players is indistinguishable from doing traditional homework or test problems, it is a gamified drill.

Gamified drills are not always bad. However, they’re not the sort of thing I’d say counters the notion “that students must learn and practice the basic skills of mathematics before they can do anything interesting with them.” They are what Keith Devlin calls a “1st generation educational game”.

There’s lots of gamified drills. It’s easy to do: just take what you normally would do in a math problem review and tack on a game element somewhere (for me it’s usually Math Basketball). To be integrated the primary actions of the players will require using mathematics in a way that is linked with the context of the game. That 1-2 Nim requires understanding multiples of 3 is inextricable from the game itself and not interchangeable the same way Math Basketball can be easily switched to Math Darts.

## Tiny Games, mathematics edition?

So there’s a Kickstarter project closing today which has me wondering about mathematics potential.

Tiny Games: Hundreds of real-world games, inside your phone.

The concept here is to have games suited for different settings that can be described in only a few sentences.

Could one make an all-mathematics variant — mathematical scrimmages, so to speak? The only games I could think of offhand in the same spirit as Tiny Games were some Nim variants and Fizz-buzz.

1-2 Nim (for two players): Start with a row of coins. Alternate turns with your opponent. On your turn you can take either 1 or 2 of the coins. The person who takes the last coin wins.

Fizz-buzz (for a group): Players pick an order. The first player says the number “1″, and then the players count in turn. Numbers divisible by 3 should be replaced by “fizz”. Numbers divisible by 5 should be replaced by “buzz”. Numbers divisible by both should be replaced by “fizz-buzz”. Players who make a mistake are out. Last one in wins the game.

Anyone have some more?

EXTRA NOTE: One condition I’d add is the games need to work as games and not as glorified practice. “Challenge a friend to factor a quadratic you made” meets the “Tiny” but not the “Game” requirement.

EXTRA EXTRA NOTE: Dan Meyer asks “Aside from the counterexample that follows, what are the qualities that make Fizz-Buzz and Nim gamelike and not, say, exerciselike?” In both cases the games incidentally have some mathematics in them and are the sorts of games one might play even outside of a mathematics classroom. Even though Wikipedia claims Fizz-buzz was invented for children to “teach them about division” (?), my first encounter was from The World’s Best Party Games. (This still doesn’t totally answer the question, I know. A related question is: what is the difference between a puzzle and a math problem?)

## Three design puzzles from The Psychology of Arithmetic

Edward L. Thorndike’s book The Psychology of Arithmetic (1922) is the earliest I’ve seen containing criticism of the visuals in textbook design. I wanted to share three of the examples.

Fig. 47.—Would a beginner know that after THIRTEEN he was to switch around and begin at the other end? Could you read the SIX of TWENTY-SIX if you did not already know what it ought to be? What meaning would all the brackets have for a little child in grade 2? Does this picture illustrate or obfuscate?

Fig. 51.—What are these drawings intended to show? Why do they show the facts only obscurely and dubiously?

Fig. 52.—What are these drawings intended to show? What simple change would make them show the facts much more clearly?

## Animated mathematical notation and the genre of the mathematical video

While it is still common (and frankly, necessary) to rail at the limitations of learning mathematics via watching videos, my personal umbrage has more to do with presentation than with educational philosophy.

The mathematical video genre is still in its infancy. I am reminded of early films that were, essentially, canned plays.

(From L’Assassinat du Duc de Guise in 1908.)

Oftentimes in videos teaching mathematics with notation they simply duplicate what could be done on a blackboard, without fully utilizing the medium.

However, there are techniques particular to the video format which can strengthen presentation of even mundane notation. For instance, in my Q*Bert Teaches the Binomial Theorem video I made crude use of a split-screen parallel action to reinforce working an abstract level of mathematics simultaneously with a concrete level.

For now, I want to focus on applying animation to the notation itself for clarity.

The video is chock-full of interesting animated moments, but I want to take apart a small section at 5:43. In particular the video shows some algebra peformed on $\frac{x}{a} = \frac{a}{c}$.

Step 1: Multiply the left side by $a$. The variable “falls from the sky” and is enlonged to convey the gravity of motion.

Step 2: Once the variable $a$ has fallen, the equation “tilts” to show how it is imbalanced. A second $a$ falls onto the right side of the equation.

Step 3: The equation comes back into balance, and the two $a$ variables on the left side of the equal side divide.

Step 4: The $a$ variables on the right hand side start to multiply, conveyed by a “merge” effect …

Step 5: … forming $a^2$.

Here’s a much more recent example from TED-Ed:

When adding matrices, the positions are not only emphasized by color but by bouncing balls.

When mentioning the term “2×2 matrix” meaning “2 rows by 2 columns” the vocabulary use is emphasized by motion across the rows and columns.

The second matrix is “translated up a bit” by doing a full animation of the matrix sliding to the position.

When the video discusses “the first row” and the “the first column” not only are the relevant numbers highlighted, but they shrink and enlarge as a strong visual signal.

When discussing the problem of why matrix multiplication sometimes doesn’t work, the “shrink-and-enlarge” signal moves along the row-matched-with-column progression in such a way it becomes visually clear why the narrator becomes stuck at “3 x ….”

These are work-heavy to make, yes, but what if there was some application customized to create animation with mathematics notation? At the very least, there’s a whole vocabulary of cinematic technique that has gone unexplored in the presentation of mathematics.

## What is Algebra? (and why you might have the wrong idea and why it is important)

I’ve been frustrated lately reading definitions of algebra along the lines of this:

Look: the mere usage of variables or symbols does not immediately indicate algebra. Compare two ways of writing the Celsius to Farenheit formula: $C \cdot \frac{9}{5} + 32 = F$ vs. “Multiply by 9, then divide by 5, then add 32.” Mere calculation is going on. This is arithmetic.

Keith Devlin gets the essence of the problem right, succinctly, with:

In arithmetic you reason (calculate) with numbers; in algebra you reason (logically) about numbers.

Taking the Celsius to Farenheit formula, and using reasoning to transform it into a Farenheit to Celsius formula –

\begin{aligned} C \cdot \frac{9}{5} + 32 &= F \\ C \cdot \frac{9}{5} &= F - 32 \\ C \cdot 9 &= 5 \cdot (F - 32) \\ C &= \frac{5}{9} \cdot (F - 32) \end{aligned}

– now that is algebra. However, the symbols are not required.

To get from Celsius to Farenheit, you multiply by 9, then divide by 5, then add 32. To get from Celsius to Farenheit, you need to do the inverse operations in reverse order. Hence, you subtract 32, multiply by 5, then divide 9.

As Keith Devlin points out, people were using algebra for 3,000 years before symbolic notation.* The two are not equivalent.

Symbolic notation is a massive convenience and once learned it should be used. However, there are good reasons that students in the process of learning should use the real definition of algebra, not the artificial one defined by symbols.

1. You can reason using algebra with words.

The Celsius / Farenheit conversion already given is an example. Most students naturally understand the logic where reversing “add 5″ requires “subtract 5″ and reversing “add 5 then multiply 6″ requires “divide 6 then subtract 5″. Moreover, in this fashion students tend to understand the logic of inverses, not just the mechanics behind a raw procedure.

The students afraid of mathematics tend to like words. It is a comfortable segue for them.

2. You can do algebra without variables.

As practice, it is extremely helpful to perform algebra — that is, reason about arithmetic, not just do arithmetic — with no variables at all. I see many textbooks that introduce the distributive property like this:

Here are two ways to find the value of 6(29 + 24).
Method 1: 6(29 + 24) = 6(53) = 318
Method 2: 6(29) + 6(24) = 174 + 144 = 318
Thus, 6(29+24) = 6(29) + 6(24). This illustrates the Distributive Property of Multiplication over Addition.

The exercises that immediately follow, however, dive straight into variables:

Write each product as a sum or difference.
22. 4(7-4r)
23. (3c + 9)15
24. 3y(7y – 8)

Students can linger on pure numbers for a while, thinking intuitively and using geometric models. The rush to variables seems to occur because of the feeling that without variables it isn’t algebra yet. Google is wrong. Variables are not algebra.

3. You can do algebra with alternate representations.

Elementary teachers are familiar with the question mark substitution

5 + ? = 8

which gives the start of sensing (as John Derbyshire puts it) “a simple turn of thought from the declarative to the interrogative”. However, the question mark is still a symbolic representation.

Rather bolder steps can be made with algebra-as-geometry (for example, tape diagrams, which are now fairly standard in elementary school but usually forgotten by the time high school algebra rolls around):

or even algebra-as-graph-theory-puzzle (solid lines mean multiply, dotted lines mean add):

It is bizarre that something as simple as a definition can restrict thinking, but after reading many textbooks I’m starting to be convinced it is the main obstacle to opening new frontiers in the explanation of algebra.

## Current Kickstarter mathematics projects

Recently a commenter plugged their Kickstarter project, which made me curious how many mathematics-related projects were going on these days. I found 8 currently in progress; as of this writing (2013-3-6) all these projects are still looking for money:

The Adventures of Zelza Zero and Friends

88 Backers, $10,272 pledged of$400,000 goal, 3 days to go

Zelza Zero also teaches kids a fun way to add and subtract on a basic level through her interactions with her friends. Every episode will have what we call a “Math Moment”, which is when one of the characters performs a mathematical function, by adding or subtracting little friends to find answers.

The Number Hunter

27 Backers, $871 pledged of$2,500 goal, 10 days to go

We’ve got ideas for a season of 12 episodes lined up. Each episode introduces one topic in mathematics and explores it in an original, adventurous way – covering every corner of the planet. What Bill Nye did for Science, we’re going to do for math. Throw in a little “Crocodile Hunter” (we’re going to be exploring in hunting gear and speaking with Aussie-esque accents) and you’ve got The Number Hunter.

Children’s Wallet Cards: Color & Shape, Numbers, GO, Wallet

80 Backers, $4,277 pledged of$24,950 goal, 23 days to go

Counting is a skill that is mastered through practice and exposure. As we speak, my boys are learning to count. We count everyday, but grasping the concept is still confusing to them. It has been difficult to find affordable educational materials that are durable and simple for them to reference. I really look forward to working with them with these new cards and finding familiar objects in our own home to count and match with the artwork on the cards. I hope you and your children find these cards as exciting and useful as ours will.

The Monster Numbers Book

119 Backers, $3,954 pledged of$1,000 goal, 28 days to go

The Monster Numbers is a 10-page board book that combines legendary monsters and the numbers 1-10.

Robotic painting for complex geometries

3 Backers, $540 pledged of$4,500 goal, 8 days to go

I am creating this Kickstarter project to help me develop a mechanical means for transferring computer renderings to paper, wall, or other media. The machine as I am calling it, will be a three axis cnc with work-space size of roughly 5.5′ x 10′. I will be constructing several attachments to work with different mediums, namely spray paint and pen.

One Million Monsters Childrens Mathematics Book

1 Backer, £10.00 pledged of £5,000 goal, 25 days to go

We will incorporate the concept of fun monsters into each question. For example if one monsters egg weighs 500 grams and another monster egg weighs 400 grams. How much do the two eggs weigh together?

The Math Board Game – Aligned to the Common Core Standards

16 Backers, $1,013 pledged of$17,500 goal, 9 days to go

- Game is designed to be played by 2-4 players per game board.
- Flexible starting and ending points, so the game can be played in about 15-minute, 30-minute, or 45-minute timeframes.
- Game is so unique that students will enjoy learning math just by playing.

Chump Genius Card Game App

5 Backers, $216 pledged of$50,000 goal, 29 days to go

The best part? Players learn along the way—without feeling like it’s school. Game play is rich with teaching interactions like quizzes, puzzles and storyline choices. Players get plenty of reading, science, math, and history all served up in small bites that leave them wanting to learn more.

## Inverse problems in education

Forward problems are problems with a well-defined answer: throwing a fair die, what’s the probability of getting a 4? Inverse problems look at data and create what necessarily has uncertainty: looking at this data, was it generated with a fair die?

Most problems given in mathematics classes (outside of statistics) are forward problems, with well-defined answers. Yet, most real-life problems are inverse problems. We don’t know the actual equations of the world, and even if we did, our measurement of reality would have uncertainty.

Pure mathematics is important, but I maintain complete allergy to error is unhealthy and gives a distorted view of mathematics. Consider, for instance –

This is my favorite of Dan Meyer’s videos.

If you go through the calculations correctly for working out how long it takes Dan to get up the stairs, the answer comes short by about a second and a half.

Yet, this is still a perfectly valid problem. Where did the extra time come from? This is a useful discussion and matches the sorts of discussions scientists and engineers have often.

(Note the long step, which would naturally extend the time slightly.)

There’s also inverse problems where nobody could truly know the answer (but we can get a pretty good idea anyway with mathematics). I’ve mentioned previously my favorite problem from teaching statistics:
 

Based solely on the number of wrecks, is there anything mystical going on in the Bermuda Triangle?

By its very nature, “is anything mystical going on?” is a unfalsifiable claim, hence the problem is necessarily an inverse one. The students used a shipwreck database to decide if the number of wrecks in the area is abnormally high. (They found it was safer inside the Bermuda Triangle than outside it.)

The teacher can also manufacture an inverse problem where the teacher knows the answer but the students are not given enough to make a truly definitive answer.

For example, here are two excerpts from 19th century American humorists:

EXCERPT A
Under favorable circumstances the Roller-Towel House would no doubt be thoroughly refitted and refurnished throughout. The little writing-table in each room would have its legs reglued, new wicks would be inserted in the kerosene lamps, the stairs would be dazzled over with soft soap, and the teeth in the comb down in the wash-room would be reset and filled. Numerous changes would be made in the corps de ballet also. The large-handed chambermaid, with the cow-catcher teeth and the red Brazil-nut of hair on the back of her head, would be sent down in the dining-room to recite that little rhetorical burst so often rendered by the elocutionist of the dining-room—the smart Aleckutionist, in the language of the poet, beginning: “Bfsteakprkstk’ncoldts,” with a falling inflection that sticks its head into the bosom of the earth and gives its tail a tremolo movement in the air.
On receipt of \$5 from each one of the traveling men of the union new hinges would be put into the slippery-elm towels; the pink soap would be revarnished; the different kinds of meat on the table will have tags on them, stating in plain words what kinds of meat they are so that guests will not be forced to take the word of servant or to rely on their own judgement; fresh vinegar with a sour taste to it, and without microbes, will be put in the cruets; the old and useless cockroaches will be discharged; and the latest and most approved adjuncts of hotel life will be adopted.

EXCERPT B
On the fourth night temptation came, and I was not strong enough to resist. When I had gazed at the disk awhile I pretended to be sleepy, and began to nod. Straightway came the professor and made passes over my head and down my body and legs and arms, finishing each pass with a snap of his fingers in the air, to discharge the surplus electricity; then he began to “draw” me with the disk, holding it in his fingers and telling me I could not take my eyes off it, try as I might; so I rose slowly, bent and gazing, and followed that disk all over the place, just as I had seen the others do. Then I was put through the other paces. Upon suggestion I fled from snakes; passed buckets at a fire; became excited over hot steamboat-races; made love to imaginary girls and kissed them; fished from the platform and landed mud-cats that outweighed me—and so on, all the customary marvels. But not in the customary way. I was cautious at first, and watchful, being afraid the professor would discover that I was an impostor and drive me from the platform in disgrace; but as soon as I realized that I was not in danger, I set myself the task of terminating Hicks’s usefulness as a subject, and of usurping his place.
It was a sufficiently easy task. Hicks was born honest; I, without that incumbrance—so some people said. Hicks saw what he saw, and reported accordingly; I saw more than was visible, and added to it such details as could help. Hicks had no imagination, I had a double supply. He was born calm, I was born excited. No vision could start a rapture in him, and he was constipated as to language, anyway; but if I saw a vision I emptied the dictionary onto it and lost the remnant of my mind into the bargain.

Which one is Mark Twain? I gave another known Mark Twain excerpt to the students and had them do statistical analysis to justify their answer as A or B.

It’s a messy and “impure” problem and even can be partly reckoned with via English class skills. Statistics deals with such worries all the time, yet many American students never see such a problem until possibly their senior year and often not until college.

Even ignoring statistics and just considering modeling problems like the first one, mathematics teachers seem deeply uncomfortable with the possibility of error. Mathematics is only infallible when contained within its own world.