We hear a lot about the importance that all children master algebra before they graduate from high school. But what exactly ** is** algebra, and is it really as important as everyone claims? And why do so many people find it hard to learn?

Answering these questions turns out to be a lot easier than, well, answering a typical school algebra question, yet surprisingly, few people can give good answers.

First of all, algebra is not “arithmetic with letters.” At the most fundamental level, arithmetic and algebra are two different forms of thinking about numerical issues. (I should stress that in this article I’m focusing on school arithmetic and school algebra. Professional mathematicians use both terms to mean something far more general.)

Let’s start with arithmetic. This is essentially the use of the four numerical operations addition, subtraction, multiplication, and division to calculate numerical values of various things. It is the oldest part of mathematics, having its origins in Sumeria (primarily today’s Iraq) around 10,000 years ago. Sumerian society reached a stage of sophistication that led to the introduction of money as a means to measure an individual’s wealth and mediate the exchange of goods and services. The monetary tokens eventually gave way to abstract markings on clay tablets, which we recognize today as the first numerals (symbols for numbers). Over time, those symbols acquired an abstract meaning of their own: *numbers*. In other words, numbers first arose as money, and arithmetic as a means to use money in trade.

It should be noticed that counting predates numbers and arithmetic by many thousands of years. Humans started to count things (most likely family members, animals, seasons, possessions, etc.) at least 35,000 years ago, as evidenced by the discovery of bones with tally marks on them, which anthropologists conclude were notched to provide what we would today call a numerical record. But those early humans did not have numbers, nor is there any evidence of any kind of arithmetic. The tally markers themselves were the record; the marks referred directly to things in the world, not to abstract numbers.

Something else to note is that arithmetic does not have to be done by the manipulation of symbols, the way we are taught today. The modern approach was developed over many centuries, starting in India in the early half of the First Millennium, adopted by the Arabic speaking traders in the second half of the Millennium, and then transported to Europe in the 13^{th} Century. (Hence its present-day name “Hindu-Arabic arithmetic.”) Prior to the adoption of symbol-based, Hindu-Arabic arithmetic, traders performed their calculations using a sophisticated system of finger counting or a counting board (a board with lines ruled on it on which small pebbles were moved around). Arithmetic instruction books described how to calculate using words, right up to the 15^{th} Century, when symbol manipulation began to take over.

Many people find arithmetic hard to learn, but most of us succeed, or at least pass the tests, provided we put in enough practice. What makes it possible to learn arithmetic is that the basic building blocks of the subject, numbers, arise naturally in the world around us, when we count things, measure things, buy things, make things, use the telephone, go to the bank, check the baseball scores, etc. Numbers may be abstract — you never saw, felt, heard, or smelled the number 3 — but they are tied closely to all the concrete things in the world we live in.

With ** algebra**, however, you are one more step removed from the everyday world. Those

*x*’s and

*y*’s that you have to learn to deal with in algebra denote numbers, but usually numbers

**, not particular numbers. And the human brain is not naturally suited to think at that level of abstraction. Doing so requires quite a lot of effort and training.**

*in general*The important thing to realize is that doing algebra is a **way of thinking** and that it is a way of thinking that is ** different** from arithmetical thinking. Those formulas and equations, involving all those

*x*’s and

*y*’s, are merely a way to represent that thinking on paper. They no more are algebra than a page of musical notation is music. It is possible to do algebra without symbols, just as you can play and instrument without being ably to read music. In fact, traders and other people who needed it used algebra for 3,000 years before the symbolic form was introduced in the 16

^{th}Century. (That earlier way of doing algebra is nowadays referred to as “rhetorical algebra,” to distinguish it from the symbolic approach common today.)

There are several ways to come to an understanding of the difference between arithmetic and (school) algebra.

- First, algebra involves thinking
rather than numerically.*logically* - In arithmetic you reason (calculate)
numbers; in algebra you reason (logically)*with*numbers.*about* - Arithmetic involves
reasoning with numbers; algebra involves*quantitative*reasoning about numbers.*qualitative* - In arithmetic, you
**calculate**a number by working with the numbers you are given; in algebra, you introduce a**term for an unknown number**and reason**logically**to determine its value.

The above distinctions should make it clear that algebra is not doing arithmetic with one or more letters denoting numbers, known or unknown.

For example, putting numerical values for *a, b, c* in the familiar formula

in order to find the numerical solutions to the quadratic equation

is not algebra, it is arithmetic.

In contrast, deriving that formula in the first place is algebra. So too is solving a quadratic equation not by the formula but by the standard method of “completing the square” and factoring.

When students start to learn algebra, they inevitably try to solve problems by arithmetical thinking. That’s a natural thing to do, given all the effort they have put into mastering arithmetic, and at first, when the algebra problems they meet are particularly simple (that’s the teacher’s classification as “simple”), this approach works.

In fact, the stronger a student is at arithmetic, the further they can progress in algebra using arithmetical thinking. For example, many students can solve the quadratic equation *x*^{2} = 2*x* + 15 using basic arithmetic, using no algebra at all.

Paradoxically, or so it may seem, however, those better students may find it harder to learn algebra. Because to do algebra, for all but the most basic examples, you have to ** stop** thinking arithmetically and learn to think algebraically.

Is mastery of algebra (i.e., algebraic thinking) worth the effort? You bet — though you’d be hard pressed to reach that conclusion based on what you will find in most school algebra textbooks. In today’s world, most of us really do need to master algebraic thinking. In particular, you need to use algebraic thinking if you want to write a macro to calculate the cells in a spreadsheet like Microsoft *Excel*. This one example alone makes it clear why algebra, ** and not arithmetic**, should now be the main goal of school mathematics instruction. With a spreadsheet, you don’t need to do the arithmetic; the computer does it, generally much faster and with greater accuracy than any human can. What you, the person, have to do is create that spreadsheet in the first place. The computer can’t do that for you.

It doesn’t matter whether the spreadsheet is for calculating scores in a sporting competition, keeping track of your finances, running a business or a club, or figuring out the best way to equip your character in *World of Warcraft*, you need to think ** algebraically** to set it up to do what you want. That means thinking about or across numbers

**, rather than in terms of (specific) numbers.**

*in general*Of course, the need for algebra does not make it any easier to learn — though I think that spreadsheets can provide today’s students with more meaningful and fulfilling applications than problems about trains leaving stations or garden hoses filling swimming pools, that my generation had to endure. But in a world where our very national livelihood depends on staying ahead of the technology curve, it is crucial that we equip our students with the kind of thinking skills today’s world requires. Being able to use computers is one of those skills. And being able to use a computer to do arithmetic requires ** algebraic** thinking.

In future postings I’ll describe the growth of algebra through the ages.

You say that a macro is where algebra happens on a spreadsheet.

But don’t you think that algebra happens just making and analyzing a spreadsheet with “input” cells and “output” cells. The input cells are independent variables and the output cells dependent variables, and the whole thing is a function. And there’s lots more to be said.

In fact, perhaps spreadsheets are a good way to help move a student from arithmetic to algebra, but still being relatively concrete — or at least with things coming alive before your very eyes. Of course, we shouldn’t stop there — we still need to arrive at fully symbolic algebra — but a spreadsheet work might be a valuable part of the transition.

I agree entirely. I mentioned macros purely as a very rich example of algebraic thinking with explicit formulas. Computer spreadsheets are of course all about algebraic thinking.

> Computer spreadsheets are of course all about algebraic thinking.

Well, they are when used dynamically, for “what if” problems. A simple ledger, for example, is not algebra — it’s just static accounting, fancy arithmetic.

But the transition from a static spreadsheet to a dynamic one is very simple in practice, easy for a student to “see”, and entirely analogous to moving from arithmetic to algebra. So doing that with a spreadsheet might be a really effective way of moving a student to algebraic thinking. I have to believe I’m not the first to think of this. Yup, here’s tons — http://bit.ly/rHt0tq

I have taught algebra with spreadsheets before. It can do a lot but it has its limits in terms of what students are asked to reproduce on our standardized tests.

My favorite comment from a student at the time (after realizing how fast he could make a certain calculation with a spreadsheet) was “Algebra is like cheating for math!”

Love that student’s remark. I wonder if he eventually figured out that all of mathematics is “like cheating for understanding and doing useful things in the world.”

Great post. As much as I like the history and the explanation, I think the analogy to Microsoft Excel is great. I use to tell my students that algebra was like running tires in football practice. You will never see tires on a football field, but you do them to train your legs how to run well. Similarly, algebra is learning how to think logically and to problem solve. No boss will ask you to factor a polynomial, but they will ask you to solve a problem logically.

Actually, it seems to me that algebra is more than problem-solving — it is *symbolic* problem-solving, that is, working with abstract, non-linguistic symbols.

I’ve been thinking this morning about why so many students get geometry but struggle with algebra. I think it has to do with this issue of abstract symbols. There are actually not a whole lot of those in the standard geometry course, except where you mix in algebra, of course. The symbols are quite simple and usually relatively concrete, like letters for angles or n for number of sides (labels are a relatively primitive form of symbol, really more like arithmetic or language), things you can put on a diagram in a very solid way. Even geometry proofs are relatively concrete in that you work with a diagram (which can get you into trouble, of course) — the abstraction there is of a different sort than in algebra.

The trick with Excel is making even the more abstract symbols of algebra seem concrete. That happens because you can change the input cells and see the output cells change. So you get to visualize a temporal or a changing meaning of a symbol in ways you otherwise have to visualize mentally in algebra. So maybe that’s one way of looking at the difference — symbols in algebra can, by their very nature, stand for different things on different occasions. They have a change/time dimension.

In fact, that is also true of proper geometric proofs, except that in the interests of student success, we eliminate the issue of change — we try to draw a single, typical diagram (no spurious relationships) and reason with that.

I sometimes try to get a student to “see” a theorem by imagining the pieces of the diagram moving, given the constraints, and observe what ahs to stay the same. For example, increase one angle of a triangle and see that the other angles (or maybe just one) have to decrease by the same amount. That establishes intuitively that the sum of the angles is a constant. But what constant? Well, increase one angle until it is almost 180, and the other two angles are now almost zero — voila, it’s 180 degrees. This also works with polygons in general, and even the general case — increase n-2 of the angles to near 180 and the final two have to be zero!

But I must say that this idea usually does not appeal to students (as much as it appeals to me. If I could do it in a geometric “spreadsheet” like geometer’s sketchpad it might go over better). So that’s more evidence that the time/change issue is the new element that makes things harder for students to grasp. And change is what algebra embodies in a big way — just look at the word — variable! So it’s more than “thinking logically”.

Sorry — lots of random musings today.

–David

I agree that higher algebra (sometimes called modern algebra) is intrinsically abstract-symbolic, but school algebra, the focus of my original post, does not have to be. Indeed, it was done in a rhetorical fashion for thousands of years until the symbolic approach was introduced (by Viete) in the 16th century.

I examined the cognitive problems the human brain encounters in dealing with abstraction in my 2000 book “The Math Gene”. If the goal is to teach classical algebra (call it algebraic thinking if you like), the abstract symbols can be downplayed, and I think spreadsheets provide a good way to set about doing that.

Of course, there is value too to being able to handle abstract symbolic structures. But whereas spreadsheet use is now surely a useful skill for everyone to have, mastery of abstract-symbolic systems is I think less critical — which is just as well since many people find it difficult or impossible to acquire. So maybe we need to delay heavy symbol use until after students have mastered algebraic thinking.

I’ll talk more about these issues in my promised future blogs on the topic.

Thanks. A copy of The Math Gene is on my shelf, in my reading queue.

There is a fascinating discussion about algebra in high school at http://blog.mrmeyer.com/?p=12148

I loved this article, I wish I found it several years ago!

For several years have I been using arithmetical thinking, trying to solve algebra, linear algebra, discrete mathematics and so on – noone ever told me that algebra, or the other paradigms, worked with numbers and symbols in general. I had to discover it on my own, after constantly having tried to solve every problem with pure arithmetic. I feel ashamed over that fact – I didn’t understand math, but I understood counting. And I was really good at it. Now I understand math – and it has opened up a world to me that I had always felt but never been able to visit. Now I bathe in its presence every single day, enjoying the adventures of algebra!

Zolomon, if there are any feelings of shame it should be on the part of the education system that has turned a beautiful and powerful way of thinking into a dull collection of symbolic rules, and never bothers to tell — and show — students what algebra really is. Glad you were able to make the breakthrough.

I like that example on substituting values on the quadratic formula as an arithmetic exercise. The following link shows example of an algebraic solution to an arithmetic task. It shows that algebra (algebraic thinking) can be taught using arithmetic tasks. http://math4teaching.com/2009/10/13/arithmetic-vs-algebra/

Erlina, Thanks for writing. I like the brief discussion in that article you link to. Nice way to approach the distinction between algebraic and arithmetic thinking.

Microsoft Excel is one thing that should be removed from standardised teaching (in the UK at least). Inadvertent advertising/approval of it is best avoided too, for anyone who recognises Microsoft’s horrible track record of anti-competitive and downright aggressive business practices.

I agree Microsoft products have their faults, as does the company, though that can be said about any product or organization. Since those products are ubiquitous in the world students live in and do or will work in, I’d suggest it is a disservice to them not to have them available. I don’t think we should try to provide education in a sterile bubble. And it serves them well to be aware of problems and faults. Having said all that, I *do* appreciate your sentiment.🙂

“numbers first arose as money, and arithmetic as a means to use money in trade.”

do you really believe that? what about unary numbers, presented as drawn mamoths on cave walls or what have you?

other than that, nice article, i supose.

You raise an interesting and subtle point. The kind of thing you describe did not involve numbers. It was an early form of tally system. Tally systems were common in our early history, but they just require one-one correspondences. “I have this many of those.” Numbers arise when you abstract them as intermediaries between collections of objects and their associated tallies. We see the same stage of abstraction when young children learn. At first they get the idea of one-one correspondence, and only later to they acquire a concept of number.

Very well written article! Nice job highlighting the differences between arithmetic and algebra. Now do that with calculus.

“As you wish.” See http://www.youtube.com/watch?v=8ZLC0egL6pc

wondering if someone could be really good in algebra but suck at arithmetic?

Many professional mathematicians (including me) show some tendencies that way. In practice, it’s probably due to (i) lack of interest in arithmetic and (ii) mathematicians almost never use arithmetic, so any prowess they developed at high school tends to die away through lack of use. On the other hand, arithmetical fluency requires a fair amount of memorization, but in algebra you can figure everything out using logic, so there may be more going on with mathematicians.

algebraic thinking is not taught well in schools is because the teachers do not really have them. In my school, many kids working hard just on manipulating symbols, but without understanding what they are doing, they often have big trouble to come up with an equation for a problem.

however, I thinking strong arithmetic skills paves a way to mastering algebra.