## 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.

## 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?

## Two teaching paradoxes

Is it possible to explain something too well? That is, something appears very clear to students after it is explained, so they don’t practice (or at least pay attention to their practice because they assume they already understand the topic), and then the lack of practice means they forget what was explained? I’m not meaning “they never learned it in the first place” but rather “they learned it so well that they forgot it because they assumed the memory was permanent”. (This is a slightly different issue than students who assume they learn something but really just keep their misconceptions.)

Are there circumstances where practice can actually lessen understanding; for example, when a student who learns a “trick” that works for an entire worksheet may attempt the same trick in circumstances where it doesn’t work? Thus it may be a bad idea at times to have a student practice a topic without all the special cases? (Specific example: suppose a student practices integer addition using only a positive with a negative number, but doesn’t attempting adding negative numbers with negative numbers until later. Will their earlier practice hinder their learning in the new situation?)

## TED-Ed attempts logarithms

Each of the TED-Ed videos is meticulously animated and represents, I am sure, many many (many) hours of effort. Knowing this made the TED-Ed take on logarithms rather painful to watch:

Oof. Let me attempt to sort my thoughts:

1. The hook baffled me.

A hook should, optimally, be incorporated into the topic being learned. This hook was simply a preview of a future part of the video, and didn’t carry much interest on its own. The “red eyes” made me think it was referring to the eye-bleedingly long numbers being presented.

While my own logarithm video isn’t perfect (also not entirely comparable since it’s about the addition property in particular) I do at least manage a hook that’s useful in the explanation of the topic.

2. “…small numbers and in some cases extremely large numbers leading us to the concept of logarithms.”

Logarithms come out of the inverse concept of an exponential. The numbers don’t have to be large or small. (If you want to get historical, they were often used as a method to multiply quickly by turning the operation into addition.) While a logarithmic scale can be used to handle large or small numbers, I don’t see how that leads to the statement in the video.

3. “the exponent p is said to be the logarithm of the number n”

Math videos often are on the glacially slow side, but this part was presented so fast parts of my brain melted.

Look: Logarithms represent, in essence, the first new mathematical operation students have had to reckon with since grade school. They cause intuitions to fail. I have seen students who have never had problems with mathematics before have them for the first time with logarithms.

It’s worthwhile, then, to spend a little more than five seconds on your definition.

The definition is confusing, anyway; a logarithm is a function. It applies from one number to another number in a specific way. It is not simply an extract from an exponential equation. While the video mentions that (sort of) it waffles on the implications of introducing a new mathematical operation.

4. “…log base 10 is used so frequently in the sciences that it has the honor of having its own button.”

First off, no: the sciences often use base e (given how much continuous growth and decay happens in real life). Base 10 logarithms do still get used for logarithmic scales, but the statement as given in the video is just confusing.

Also, that’s a TI graphing calculator? Which one of has a logarithm button but not a natural logarithm button? Even the TI-81 has one.

5. “If the calculator will figure out logarithms for you, why study them?”

The answer the video gives … is so you can figure out a logarithm base 2.

That’s a terribly weak answer, given a.) yes there are many applications of logarithms where understanding the mathematics is both good and necessary, and the video even goes into one application immediately after making this statement b.) the answer doesn’t really answer the question (since it doesn’t explain where the computer science-related equation came from) c.) with the current operating system, Texas Instruments calculators are perfectly capable of putting in alternate bases without a change of base formula (The video incidentally doesn’t mention the change of base formula even though one of the questions in the post assessment asks what it is.) and d.) The statement presumes the use of a calculator in the first place (computer-based systems are also perfectly capable of doing logarithms with alternate bases).

6. The video then wants to show how useful logarithms are by giving a formula from science.

Based on the post-test, I’m guessing this part is here merely to show how logarithms are used in “real life”.

In the master catalog of Ways to Convince Students Why Something is Useful, “look, a formula that shows up in science!” ranks somewhere between “because math is good for you” and “so you can get into a good college”.

Is it really that bad? Am I just being grouchy here?

## Cooperative learning tasks in mathematics

By cooperative learning tasks I mean giving particular “jobs” to students during group work; here’s a sampling from this website:

Checker: Checks team members for understanding and agreement
Datakeeper: Keeps track of information generated by group
Helper: Gives help in reading, spelling, problem solving, or using materials
Questioner: Asks questions of instructor or other groups
Reporter: Gives oral reports to the total group
Summarizer: Sums up what the group did or the conclusions the group came to
Validator: Paraphrases what is said for clarity
Writer/Recorder: Writes down ideas and records the task

I tried experimenting with them last year (based on the urging of several people) but I’ve been distinctly unhappy. It feels like the jobs segment up the work in a rote sort of way which gives a student permission to “shut down” when they aren’t needed for something in particular. I’ve still had some luck with engineering-like projects which involve building, but this sort of thing fails for me in general. For example, today I’m having my Algebra I students work on these questions in groups:

You have an
a row by b column matrix
and want to multiply by a
x row by y column matrix.
1. When is this multiplication impossible?
2. If the multiplication is possible, what is the size of the new matrix?

3. When multiplying 2×2 matrices, there is an identity operation (just like multiplying by 1 is an identity operation in arithmetic). What is it?
4. What about for nxn matrices?

5. Give an example (with all work) that shows that multiplying 2×2 matrices is in general not a commutative operation.
6. Even though the commutative law doesn’t apply in general there are specific cases where it works. Give an example of a matrix A and B such that AB = BA.

How would working out answers here divide neatly into jobs?

[EXTRA NOTE: I still think matrices shouldn't be taught before vectors. I'm making my best go at the curriculum I need to do, though. Bert Speelpenning's series on matrices (especially this post) is helping me get some motivation out there.]