When I was young, I could not get algebra. In particular, I would encounter magical operations like this:
[latex]x + 5 = 8[/latex]
followed by:
[latex] x = 8 – 5 [/latex]
followed by (AHA!):
[latex] x = 3 [/latex]
Obviously, x was three!
Only it wasn’t obvious to me.
Somehow [latex] + 5 [/latex] on the left side of the equation would turn into [latex] – 5 [/latex] on the other side of the equation. In general, pluses turned into minuses, minuses into pluses, multiplies into divides, and divides into multiplies. The algebra textbooks that I had access to never explained how this counter-intuitive miracle worked nor, it seemed, could any teacher.
And what by the way was the mysterious x anyway? Oh, it is a variable! Okay… And what is a variable?
I was pretty lost. Fortunately, I was a fan of the science fiction writer Isaac Asimov (1919/1920 – 1992).
Isaac Asimov as a Young Man
I had read I, Robot (a collection of short stories nothing like the awful Will Smith movie), the Foundation trilogy, and several other of his works. Asimov wrote his most famous and best science fiction works as a young man in the 1930’s and 1940’s. Asimov had also built a career as a popularizer of science and mathematics later in life. By his own account Asimov had a big ego and was something of a know-it-all:
One of the three took it upon himself to explain to the other two how the atomic bomb worked, and needless to say, he got it all wrong. Wearily, I put down my book and began to get to my feet so that I could join them, assume “the smart man’s burden,” and educate them. Halfway to my feet, however, I thought, “Who appointed you to be their educator? Is it going to hurt them to be wrong about the atomic bomb?” And I returned to my book. This is the first occasion I can remember in which I deliberately resisted the impulse to put my remarkability on display. It doesn’t mean that my character changed suddenly and completely, but it was a step, a tiny first step, in the forging of what I can only describe as a new me. I was still obnoxious to many, I still failed to get along with my superiors, but I began to change. I began to be able to “turn it off,” to not be forever putting my cleverness on display. I answer questions if asked, I explain if an explanation is requested, I write educational articles for those who wish to read them, but I have learned not to volunteer my knowledge, unsolicited.
Asimov, Isaac (2009-12-18). I, Asimov: A Memoir (pp. 120-121). Random House Publishing Group. Kindle Edition.
Asimov wrote dozens of books on physics, chemistry, biology, anything and everything. An avowed atheist, he even wrote a guide to the Bible. His books varied in quality and technical level. Like most popularizers to this day, he usually stopped short of Calculus. Most importantly for me, one of his gems was a short book, Realm of Algebra, on the basics of algebra. This was one of his best popular science and math books, especially in actually teaching the subject, algebra in this case, to the reader rather than simply wowing the reader with buzzwords and vague analogies as many popular science books do.
Algebra remains a difficult subject for many students, perhaps because the textbooks and typical in-class teaching have not progressed much since I was a child. The contrarian City University of New York emeritus professor Andrew Hacker penned an op-ed in the New York Times in 2012 “Is algebra necessary?” arguing for the abolition of algebra as a requirement in high schools. He followed up with his 2016 book The Math Myth: And other STEM Delusions elaborating his argument.
Everyone who is struggling with algebra should read and practice the simple examples in Asimov’s Realm of Algebra, which unfortunately is now out of print, before giving up.
What did Asimov do for me that teachers and standard algebra textbooks failed to do? He explained very clearly and in great detail what a variable was. He explained many concepts and steps in algebra that were skipped over or assumed to be obvious in the standard textbook explanations I had read up until then. And he explained those mysterious operations by which a [latex]+5[/latex] turned into a [latex]-5[/latex] — pluses into minuses, minuses into pluses, multiplies into divides, and divides into multiplies.
What is the trick?
[latex]x + 5 = 8[/latex]
followed by subtracting five from both sides of the equation:
[latex] x + 5 – 5 = 8 – 5[/latex]
remembering that:
[latex]5 – 5 = 0[/latex]
followed by:
[latex] x = 8 – 5 [/latex]
followed by (AHA!):
[latex] x = 3 [/latex]
The mysterious operations turning pluses into minuses, minuses into pluses, multiplies into divides, and divides into multiplies are shorthand for adding, subtracting, dividing, or multiplying both sides of the equation by the same value.
The standard algebra textbooks of the day were skipping over critical steps and concepts. If you study the history of algebra and mathematics you will find that these missing steps, as simple as they may seem when explained clearly took years, even generations, for the best mathematicians of their time to develop. It is unlikely that most students, even exceptional students, can reproduce these missing steps in the course of a typical algebra class (nine months — a school year) without a clear explanation from a parent, a teacher, Asimov’s Realm of Algebra or some other comparable source.
Missing Steps
Algebra is hardly unique. Missing steps and missing explanations of key concepts is a common problem in learning and teaching the technical topics that have come to be grouped together under the overused acronym STEM (Science, Technology, Engineering, and Mathematics).
This problem often seems to get worse with more advanced material, notably post first-year Calculus level — past the Advanced Placement BC level calculus taught in top math and science high schools in the United States — topics in math and science. The practitioners and researchers who produce advanced college and post-graduate level texts and teaching materials have frequently internalized various steps, replacing them with shorthand operations, and concepts and fail to explain them adequately or even at all in some cases. There are very few accessible books similar to Realm of Algebra that can explain post-Calculus level topics and make up for the deficiencies in textbooks, lectures, and other available materials.
If you are having difficulty learning mathematics, whether algebra or a more advanced topic, it is important to consider that there may be missing steps that have been telescoped into a single line or operation or even omitted completely. Similarly concepts and definitions may be missing or not really defined. Just saying x is a variable in algebra explains nothing without a clear explanation of what a variable is. If you cannot find a clear explanation of an operation or a skipped step, it can take a surprisingly large amount of trial and error and thought to figure out what was skipped over; it often took years for the original discoverer or discoverers of the missing steps to figure them out and even more years for the shorthand version now in use to be invented.
Conclusion
If you are having trouble learning algebra, definitely read and practice the simple examples and concepts in Asimov’s Realm of Algebra. There is a good chance it will clear up many sources of confusion and problems as it did for me many years ago.
In general, in mathematics and other STEM fields look for missing steps and missing explanations of concepts or definitions of terms if you are having difficulty learning the subject. If a definition is simply in terms of something else that has not been defined or explained adequately, it is really no definition. Even a single unidentified missing step in an explanation of a mathematical topic can prevent any understanding, basic competence, and especially proficiency in the topic.
© 2016 John F. McGowan
About the Author
John F. McGowan, Ph.D. solves problems using mathematics and mathematical software, including developing gesture recognition for touch devices, video compression and speech recognition technologies. He has extensive experience developing software in C, C++, MATLAB, Python, Visual Basic and many other programming languages. He has been a Visiting Scholar at HP Labs developing computer vision algorithms and software for mobile devices. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.
Maybe I just had good math teachers in Junior and Senior High.
We were always taught to balance the equation/s, keeping both sides
equal. Don’t think I heard of a function until college calculus. We
always thought of x etc. as unknowns. to be solved for.
Attended a public high school, in fact the same one as Lee
DuBridge, on the academic track. So maybe it was the teachers
at that time (early 50s), maybe I was just lucky, or maybe things were
(as the Caltech saying goes”intuitively obvious”) so I was never
mystified.
One thing that puzzles me: why aren’t kids trained to read x + 5 = 8 as, “Some number plus 5 is 8. What is that number?”
This natural language statement puts students on more comfortable ground. First, reading the above as “Eks plus five equals 8” is abstract because it puts so much pressure on the mysterious ‘x,’ a seemingly protean being whose value appears to change from problem to problem, to the consternation of many students. How can x be 6 over here and -22 over here? Mysterious indeed.
Whereas reading “x” as “Some number” or “what number,” etcs., puts things into more familiar context. Oh, you want to know what number I’d add to 5 in order to get 8? That’s easy! It’s obviously 3!” Abstraction and intimidation gone or diminished. Problem solved.
Of course, as things get more complicated (e.g., 7x – 17 = 25, which requires more than one “guess” or recall), answering by inspection may become difficult or impossible. But the idea doesn’t change. 7x – 17 = 25 is equivalent to, “What number multiplied by 7 and then reduced by 17 leaves / yields 25? That can be reconstructed as What number times 7 = 42. The answer is clearly 6.
Once we ramp the complexity up even a little, we can get students to realize that it won’t suffice to just solve by inspection; we need a method that is sure to get us to what we need to know. So the notion of creating equivalent expressions/equations can be introduced in a purposeful context. Balancing additions and multiplications (and their inverses) may start to seem obvious.
My son was in fifth grade when he came home asking me if math with letters was what algebra was. On further inquiry, I learned that he saw 5 + x = 8 as the same question as 5 + ? = 8 or 5 + some shape = 8, and wondered how these were different now that there were letters? Isn’t that algebra? When I agreed that it was, he wanted to know why most of the other students seemed to think it was so hard. I said that if he saw it as being the same and hence neither hard nor weird, he was miles ahead of the math game and would likely continue to do well. Which he did.
He’s bright, but he’s not a genius. He simply saw what he saw. Why we don’t help all kids see things so clearly eludes me. Mathematical fear on the part of many of our K-6 teachers is likely a factor. US beliefs and attitudes about math (particularly anything with names like “algebra,” “trigonometry,” “calculus,” etc.) is another culprit. And then there are kids like my son who don’t know to be afraid and so simply see the similarity to what they have been doing successfully for years.
Try Introduction to Mathematics: Alfred North Whitehead, 1911 (Russell and Whitehead)
Google shows an online link to Asimov’s Realm of Algebra at:
https://www.arvindguptatoys.com/arvindgupta/asimov-algebra.pdf
It is illuminating how the first few chapters illustrate a variety of steps in manipulating a single variable algebra equation. And the concluding chapter “Putting Algebra to Work” in Asimov’s Realm of Algebra brings out life to a single variable equation with the story of Cavendish weighing the Earth.
Somewhat related to the topic of identifying “missing steps” in algebra equation manipulation is a puzzle proposed by Terence Tao when he proposed an idea to gamify algebra a few years ago.
What if the player can choose from a finite number of given steps to be applied to the algebra equation?
A mobile game app (Algebra Game) is available at:
https://itunes.apple.com/us/app/algebra-game/id1175728990?mt=8
Another mobile game app (Algebra Maze) that turns this into a maze game is available at:
https://itunes.apple.com/us/app/algebra-maze/id1177827392?mt=8