We hear a lot about algebra teaching and the importance that all children master high school algebra. 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 algebra question.
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. Algebraic thinking is different from arithmetical thinking. Those x’s and y’s that you have to learn to deal with in algebra denote numbers, but usually numbers in general, 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.
Is it worth the effort? You bet. In today’s world, most of us really do need to acquire that skill. 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 high school math. 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 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 in general, rather than in terms of (specific) numbers.
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. (Many students can solve the quadratic equation x2 = 2x + 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.
What that amounts to, is thinking logically rather than numerically. Ultimately, it’s still about numbers, but in algebra you use analytic, qualitative reasoning about numbers, whereas in arithmetic you use numerical, quantitative reasoning with numbers.
It is precisely because qualitative, analytic reasoning plays such a major role in today’s society that former President Bush’s National Mathematics Advisory Panel recommended in 2007 that all US high school students should become proficient in algebra.
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. QED.
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While I agree that algebra requires analytical reasoning, the distinction between arithmetic and algebraic reasoning is drawn too starkly in my opinion. Students with good quantitative reasoning are able to transfer that to the relational reasoning that algebra requires. That is why we spend so much time in the early grades teaching students to understand and use number relationships as well as to develop fluency. A student who knew that 5 squared was 15 more than 2 times 5 can much more easily be shown how to factor, graph, or make a table to find the solution, as well as the additional answer of -3.
If you have actually taught mathematics to a wide range of people, including young people, you will notice many who are really good at arithmetic and then hit a wall with algebra. There are fewer people who are sort of marginal with arithmetic, primarily because of sloppiness, but who are able to do the algebraic logic. Arithmetic insights are not the same as algebra insights even though they are closely related. The closest explanation for this is a phrase that Carl Menninger used in his very fine book, “The Calculator’s Cunning,” Meninger used the term “The Number Sense” as a goal for youngsters studying arithmetic. He begins each problem with an estimate of the order of magnitude, and ends the problem with a “check” such as “casting out” nines or elevens to verify the result. A good elementary algebra student would likewise end a problem by “checking” the result by substitution, by reviewing the order of operations and maybe checking against division by zero.
The students who are apparently using arithmetic to solve their algebra problems might actually be reasoning in some kind of individual “semantic algebra.” A long time ago, algebra problems and solutions were written out in words, but it is easy to identify it as real algebra. One possible general definition of basic algebra is “a logical director for deciding what arithmetic calculation to do” in order to solve a certain kind of problem. In history, such director algebras can be implicit or tacit as they were in the Egyptian and Babylonian mathematics.
Let me close by adding the observation that a similar “wall” exists between algebra, geometry and trigonometry and the Calculus and Analytic Geometry courses.
One of the great unsolved problems in mathematics is to create a continuity of narrative from beginning to end. It’s like breaking up a long novel into ten unrelated short stories.
Yours sincerely
Norman Woldow
I am not a teacher of math but was certainly a student. When comparing my math education with today’s I find a that there is a huge gap in the learning. Students today are taught not to memorize math “facts” addition, subtraction, multiplication. Instead they “estimate” answers.
I attended HS 40 yrs ago and I still can do algebra problems and remember some of my geometry theroms. My college students have difficulty multiplying in their head.
I understand the need for us to understand math concepts but starting out with the “old fashion” reading, writing, arthmetic I feel was much more successful. Teachers today have to spend the first half of the year reviewing the previous years’ material because so much is crammed into a years’ curriculum. The students do not fully understand or have the ability to complete a concept before they are off on the next.
Logic and analysis can be integrated into the basic math skills but it is my opinion that the basic skills should come first.
“Arithmetic insights are not the same as algebra insights even though they are closely related.”
I wholeheartedly agree with Norman’s observation. Ditto for Keith Devlin who quipped on NPR that arithmetic works with number computations, but algebra is reasoning “about” those numbers. I teach accounting in college, and minimize bookkeeping aspects. It’s the “relationships” between organization assets and claims on them (creditors versus owners) that causes beginning students to stumble. There is a stark contrast between students who “get” this relationship early on versus those who never quite see it. It’s a matter of confusing the things that are the wealth of an organization versus to whom that wealth belongs.
Luca Pacioli, a truly Renaissance man (tutor to Da Vinci), published a seminal text on algebra. He illustrated it with one of the more practical applications of mathematical theory–the Venetian accounting equation! This simple algebraic relationship earned Luca the contemporary title “Father of Accounting”. He didn’t create this knowledge but simply illustrated the relationships algebraically. I’ve started my own booklet aimed at jumpstarting this understanding among beginning students to get them on the other side of the wall ASAP.
The narratives I’ve observed in many principle textbooks fall a tad short for many students. Getting the “logic” in place must precede getting the numbers in place. If not, the numbers just remain numbers.
I teach algebra to advanced students in grades 6, 7 and 8. It will be interesting to read these articles with them in class.
Bravo! Professor Devlin!
Brilliant post. I disagree with Mr. Woldow above. As a math teacher for the past twenty years, I have seen algebra open the door to mathematics for some students. The variable behaves like a number so the rules and principles which govern that behavior become more important than any specific value. Until sixth grade or so, students often mistakenly believe that immediate recall of facts and rapid calculations are what make someone “good at math.” Arithmetic facts are to spelling as math is to writing. Beginning algebra paves the way to great literature. Spelling errors stemming from a poor visual memory may thwart an author’s early attempts, but that will hardly be an obstacle in the end.
Algebra is way to too complicated. If I spent hours on end learning this stuff, I wouldn’t have time to do my other homework. I think that has to be a major problem with students that don’t pick up on it right away.
Algebra has been my nemisis. I didn’t understand it in highschool and by the time I was taking calculus my senior year the teacher told me my calculus was fine but my algebra was appaling and he didn’t know how that was even possible. I understand a lot more, now, about what I wasn’t understanding then trying to teach my own children and seeing them struggle with similar issues. To make a sweeping statement, algebra teaching lacks “the big picture”. Learning arithmetic does not elude to “why” you would want to look at a problem algebraically. It’s not merely a matter of giving examples of where and how algebra can be used it has to relate one-to-one. The child needs to “see” it for him/herself. If that connection can not be made the math education is no more than a rote memorization of patterns and systems. The memory of which will dissolve over time. As homeschool educator, who has used, numerous curriculum they tend to jump right into having students get patterns up and running minus giving them a reason why. And even if why is indicated it tend not to be personalized leaving the student, still, only able to deal remotely with the material-not realy “own” the concepts. This vacuum needs to be filled and fast. There are many right-brained students who struggle with ordering things with words and so have that avenue of understanding closed to them or very inefficient as an entry point for information. These kids desperately need a way to “picture” concepts/bits. And in order to picture bits they have to have an overall “why” or framework to hang the “bits” on.
This comment is totally on point about the education of young children.
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