## Commentary on “Wrong Answer: The Case Against Algebra II”

There’s a new article by the author Nicholson Baker that is not raising as much a fuss as “Is Algebra Necessary?” from The New York Times last year, probably because it’s at Harper’s behind a paywall. Also, as I write this children are fleeing from algebra all over magazine stands:

The title strikes me as odd, given that the general argument is aimed at all algebra, and the text constantly references algebra as a whole rather than the quite US-specific version of Algebra II. (As far as I’m aware no other country has such a thing; most of them integrate all forms of mathematics together.) What is Algebra II anyway, and where is the cutoff point where Nicholson Baker considers algebra to be too hard to handle? He hints at what he’d like as a replacement, which strikes me as a math appreciation analogous to music appreciation courses:

We should, I think, create a new, one-year teaser course for ninth graders, which would briefly cover a few techniques of algebraic manipulation, some mindstretching geometric proofs, some nifty things about parabolas and conic sections, and even perhaps a soft-core hint of the innitesimal, change-explaining powers of calculus. Throw in some scatter plots and data analysis, a touch of mathematical logic, and several representative topics in math history and math appreciation.

This seems like a truly random collection of topics, but it sounds like Mr. Baker is basing his list off the notion of using The Joy of x by Steven Strogatz as a possible text.

I have had some personal experience with using popular texts; when I team-taught statistics at the University of Arizona we used The Drunkard’s Walk, Fooled by Randomness, and Struck by Lightning: The Curious World of Probabilities. While it led to interesting discussions, there just wasn’t enough meat to have students do problem solving based solely on the text. The math from the books was purely a passive experience. (We still did just fine by augmenting with our own material.)

I still hold forward the absurd idea that students still solve math problems in a math class. If you’re designing a freshman mathematics-teaser course, I might humbly suggest Problem Solving Strategies: Crossing the River with Dogs, which has the virtue of steering away from algebra as the sole touchstone for problem solving.

Back to Mr. Baker’s attempt to define Algebra I:

Six weeks of factoring and solving simple equations is enough to give any student a rough idea of what the algebraic ars magna is really like, and whether he or she has any head for it.

Mr. Baker himself seems to have a confused idea of what algebra is like, but he’s not alone. (See: my prior post on what is algebra and why you might have the wrong idea.) Also of note: some countries don’t bother to factor quadratics. (I haven’t made a map comparing countries, but it seems to be continental Europe that ignores it and just says “use the quadratic formula”.)

I could see solving equations managed in six weeks, but in a turgid, just-the-rules style that would be the opposite of what we’d want in this kind of appreciation class in the first place. Just the concept of a variable can take some students a month to wrap their head around, making me disturbed by the notion that six weeks would be enough for a student to find “whether he or she has any head for it”. (This isn’t even touching the issue of just how much is internal to the student. I’ve heard dyscalculia estimates of up to 7% of all students, but excluding that group a great deal of the attitude seems culturally specific. Allegedly in Hungary [I don't have a hard research paper or anything, this is just from personal anecdotes] it is quite common for folks to say mathematics was their favorite subject in school and the level of disdain isn’t remotely comparable to the US.)

They are forced, repeatedly, to stare at hairy, squarerooted, polynomialed horseradish clumps of mute symbology that irritate them

The article’s invective along these lines makes me wonder again how much the visual aspect to mathematics is the source of hatred, as opposed to the mathematics itself.

This picture is from the Adventure Time episode “Slumber Party Panic” and is supposed to represent the ultimate in math difficulty. However, the math symbolism is strewn truly at random and there is no real problem here. I am guessing a good chunk of the audience couldn’t even tell the difference and for them, any difficult math problem looks like random symbolic gibberish.

This is related to another issue, that of bad writing. Here’s Mr. Baker quoting a textbook:

A rational function is a function that you can write in the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomial functions. The domain of f(x) is all real numbers except those for which Q(x) = 0.

I claim the above definition is simply bad writing, and a cursory check of the Internet reveals several definitions I’d peg as clearer in a students-who-don’t-like-math-are-trying-to-read sense:

any function which can be written as the ratio of two polynomial functions.

a function that is the quotient of two polynomials

“Rational function” is the name given to a function which can be represented as the quotient of polynomials, just as a rational number is a number which can be expressed as a quotient of whole numbers.

Hardy’s A Course of Pure Mathematics, page 38:

the quotient of one polynomial by another

Godfrey Harold Hardy admittedly goes on from there, but his text is written for mathematicians. High school texts have no such excuse.

In a similar vein, the article later quotes a 7th grade Common Core standard:

solve word problems leading to equations of the form px + q = r and p(x+q)=r, where p, q, and r are specific rational numbers

which I suppose is meant to be horrifying, but in this case the standards are written for the teachers and aim to remove any ambiguity. Here’s problems that matches the standard:

1. You bought 3 sodas for 99 cents each and paid 10 cents in tax. How much money did you pay?
2. You bought 4 candies for 1.50 each and paid 6.20. How much was tax? The standards can’t simply produce many examples and gesture vaguely. They must be exact. Just because 4th grade specifies that students “Explain why a fraction a/b is equivalent to a fraction (n×a)/(n×b)” does not mean students are using variables to do so. (In case you’re curious what it does look like to the 4th graders, Illustrative Mathematics has tasks here and here matching the standard.) There’s lots more to comment on, but let me leave off for the moment on this quote, because I’m curious… Math-intensive education hasn’t done much for Russia, as it turns out. …is this statement (in the last paragraph of the article) accurate? ## Guess the year of the quote Let’s say plus or minus 5 years: Teaching is more than telling and explaining, and learning is more than imitating and memorizing. During the last 60 years teachers of mathematics have gradually sensed that, above all else, their pupils should learn the meaning of mathematical terms, principles, operations, and patterns of thought. EDIT: Answer in the comments. ## 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. ###### * He seems to be excluding Diophantus’ Arithmetica from the 3rd century. However, the symbolic notation therein wasn’t really picked up until the 16th century, so his claim still holds. ## 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.

## Computer Based Math Redux

I’ve posted before about Conrad Wolfram’s efforts to remove calculation from the curriculum and make everything computer based. There is now a website devoted to the initiative (http://computerbasedmath.org) and Conrad Wolfram’s blog recently announced their first country interested in taking up the curriculum: Estonia.

Estonia isn’t too surprising a choice; they recently put programming in the standard curriculum starting at first grade.

However, they’re not diving into axing algebraic manipulation from the curriculum yet; rather Computer Based Math (abbreviated CBM) is planning to “rewrite key years of school probability and statistics from scratch”. This is a reasonable first step given statistics is often taught computer based or at least calculator based these days (my colleague who teaches AP Statistics next door does so) and it does feel very silly to work through a passel of “figure out the standard deviation” problems by hand.

However, I’m going to play devil’s advocate again with a thought experiment. Since algebraic manipulation is not being removed at this time, these questions aren’t going to be applicable to Estonia yet, but presuming Computer Based Math continues working with them it should come up soon.

Suppose you are in a curriculum where you are used to algebraic manipulations being done by a CAS system. You are learning about statistics and come across these formulas:

Mean for a probability distribution
$\mu = \Sigma[x \cdot P(x)]$

Variance for a probability distribution (easier to understand)
$\sigma^2=\Sigma[(x-\mu)^2 \cdot P(x)]$

Variance for a probability distribution (easier computations)
$\sigma^2=\Sigma[x^2 \cdot P(x)] - \mu^2$

Standard deviation for a probability distribution
$\sigma=\sqrt{\Sigma[x^2 \cdot P(x)] - \mu^2}$

[These are incidentally off page 208 of Triola's Elementary Statistics, 11th Edition.]

What is necessary to use the formulas conceptually? What understandings might someone lack by not having experienced the algebra directly? Is it possible to understand the progressive nature and relations with these formulas just by looking at them? Is it necessary (to be well-educated in statistics) to do so? If it is necessary, what specific errors could somebody potentially make in a statistics calculation? Could this be mitigated by the text? Could this be mitigated by steps taking during the CAS portion of the education that while not leading to lengthy practice in “manipulate the algebra” problems will still allow understanding of the text above?