As good a time as any to go re-read "A mathematician's lament" [0] which begins:
A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory. “We are helping our students become more competitive in an increasingly sound-filled world.” Educators, school systems, and the state are put in charge of this vital project. Studies are commissioned, committees are formed, and decisions are made— all without the advice or participation of a single working musician or composer.
Very interesting read. It's not only a problem with mathematics though, it's nearly every subject in school, whether it's science, languages or math. It seems to me that the problem is that to teach well, you need good, autonomous teachers, and that just doesn't scale. Like the author said, you can't teach teaching.
I'm of the unpopular opinion that digital learning is the only possible solution for this. I'm talking about a digital tutor with infinite patience, with the best qualities of the best teachers. I'm not talking about a robot that passes the Turing test, just one that passes the test for the narrow field of teaching a specific subject.
People say you need real teachers, you need the human touch, but if the average teacher sucks (and they do), then I'd rather be taught by software. And I think it's just one of those things people say because it sounds true: "you need the human touch", just like people said they prefer real books over e-books, face to face conversations over texting, navigating by feel over using the GPS etc. These sentiments almost always turn out to be wrong.
Yes you can. I mean sure there is an explanatory gap with things like empathy and identifying scaffolding opportunities. But that just means you can't teach ALL of teaching with a book, experience and guidance is also required.
> I'm talking about a digital tutor with infinite patience...
One that can empathise with a students particular background/learning style and understand what might work to convey a novel concept better?
> People say you need real teachers, you need the human touch, but if the average teacher sucks (and they do), then I'd rather be taught by software.
It's not about the human touch. It's about empathy, and expert diagnosis of learning conditions, which the worst teacher does a better job of than the best computer/software. Just because most of our teachers suck doesn't mean software is the better solution. Better teachers are the solution, perhaps that is better education for teachers, or perhaps it's more communication from the realm of pedegogy down to the teaching curriculum that teachers are taught from.
Software can assist teachers, and can even replace certain aspects of teaching. But you will leave a lot of students behind if you try to replace teachers completely.
My prediction is that data can replace empathy, just as Watson can beat human competitors most of the time. Students struggle with new concepts, but do they all struggle in uniquely different ways? Aren't there patterns? I think the data from millions of interactions, and a curriculum guided by the best teaching minds the world has to offer will be better than the average teacher, up to a certain point. Note that I mean this in a very narrow sense, when constrained to a specific topic, and especially when explaining something intuitively to a student and answering students' questions. I agree though that for the foreseeable future there still need to be teachers.
I'm all for better teachers, but I think it's just not going to happen. They haven't become better in the past few decades or even century, rather possibly the opposite, with the need for many more teachers than there are competent people to fill those positions. The only solution I can see is to make teaching one of the most highly paid and highly respected professions. There doesn't seem to be any incentive for this, and it would require a huge cultural change. And even if that was the case, there would probably be the a concentration of teaching talent in the big cities at top schools, while rural schools will have to do with the scraps. Software at least is democratizing.
Of course what I'm suggesting would require some very sophisticated software that doesn't exist today, but I would say that for me, if I would have followed the videos on Khan Academy (for science and math) instead of the education I received, I would probably have been better off. I grew up in a rural area, and I know how bad it can be. After getting an engineering degree, I now see how little those "teachers" actually knew or understood. It's shocking really. But I'm not surprised when you look at the kind of people that go into teaching today.
You can't teach teaching in isolation. A math teacher has to know math and teaching, but that's not really how they're trained - an undergraduate Math Education curriculum diverges from the Math curriculum shortly after calculus, which is far too soon.
An undergraduate math education is more than enough to teach the current high school math curriculum.
From my point of view though, if my statement above is true, that is very sad. As I don't think it should be. The math curriculum is horribly deficient. But software doesn't seem to be the answer to fixing the curriculum.
> An undergraduate math education is more than enough to teach the current high school math curriculum.
Is that an undergraduate Math degree, or a Math Education degree? If the former, you don't get training in pedagogy. If the latter, you don't get enough proof-based classes to properly understand what math is or to fully understand the subjects you'll teach.
I just graduated high school, I always thought Math was amazingly interesting and I couldn't understand why I wasn't doing so good in school and why topics that I thought would be very interesting appeared boring after the teacher told I had to learn some rules and apply them over and over again to get good grades (I'm oversimplifiyng but you get the idea). We were thought the 'what' and not the 'why'. I was in a computer science high school so we had subjects that you'd find in a CS curriculum in any university - EE, C, OOP and a bunch of other random stuff. So for example when it was time our math teacher had to tell us about sin and cosine the explanation was merely 'oh here's a bunch of ways to calculate your angles, remember them for the next exam'. It was in some other class (something related to EE) that the teacher went deeper and and explained all the magic behind it. I mean, seriously, the Wikipedia page has animations that make it easier to understand than my math books.
I always thought I was stupid for not having good math grades, I question the sharpness of my brain to this day.
Essays like this make me want to pick up Math again and this time learning it the proper (whatever that means) way... learn to think analytical and in terms of pattern making/matching etc etc... not even sure where to start though.
If you're motivated / curious enough / have enough time, I recommend Spivak's Calculus as a way of learning calculus the, as you say, 'proper (whatever that means) way.' It's basically a treatment of calculus from the perspective of real analysis, having the 'no ad-hoc rule teaching' policy at its foundation. You prove things, build up a non-fragmented edifice slowly, and end up being introduced to analysis such that you can then pick up other things (it's a rather extensive treatment of calculus indeed.) Or so goes my narrative in the midst of frustrations regarding self-motivating to continue individually progressing through the book. :)
The 3rd and maybe the 4th editions can be found online by doing an internet search for pdf/djvu files.
I majored in mathematics at the University of Chicago and Spivak's Calculus was my first-year calculus textbook. It confirmed what I had long believed: what I was being taught in high school wasn't really mathematics.
Frankly, his parody of a "music education" sounds a lot like my music education.
My orchestra teacher used to drill us on the circle of fifths and how many sharps were in this or that key. But I didn't give a damn about any of it until a friend showed me how to improvise over the 12 Bar Blues.
"See, if you play this note in this key, and if you wiggle your finger like that, it sounds really cool." "Ohhhhh..."
The problem with this metaphor is that math is not only a form of art. Only for a very small fraction of population, math is art. For the rest, some basic maths are necessary skills for surviving in this society, like the skill to following rules/ play under rules. While music is much less on this necessity side.
I did. And part of my comments are even from the later part of his article on the criticizing of high school geometry. he talked from the real subjects of geometry side, and looked down the formal language presentation in the textbooks. But the latter is an important skill training, which is arguably more important than the mere geometry pattern recognizing itself for most of people.
> ... numbers in general, not particular numbers. And the human brain is not naturally suited to think at that level of abstraction.
This is so wrong (at least as a stereotype).
All throughout my early education I HATED arithmetic, and found almost everything about it mind-numbingly boring and repulsively repetitive. At that point in my life, I hated math. The moment I encountered algebra though, it was "love at first sight", and ever since I've absolutely been fascinated and engaged with every type of high-level math I encounter (the more abstract, the better). And not just "fascinated" in the "I like it" sense -- math, CS, etc. is more easy/natural to me than most humanities subjects, by far.
So although I can only speak for myself, I quite disagree with any claim that the brain isn't naturally suited to abstract thinking. While I know not all people think the way I do, certainly quite a few do.
His area of expertise is mathematical teaching/learning. He was undoubtedly talking about the average person, I doubt it was supposed to be an absolute neurological-level statement. And as a general statement/stereotype, there are expected to be exceptions. I doubt he got to be a Stanford math professor without seeing some gifted math students himself. His statement can still stand, despite yourself as a counterexample.
A little more about him: His CourseEra course is "Introduction to Mathematical Thinking". It isn't about math, it's about how to think mathematically. He commonly talks about the pitfalls people make with basic mathematical approaches. He works with helping them understand approaches to math and how to deal with thinking abstractly and purely logically. Some people pick up all that stuff implicitly with little effort, some people never really master it. Given his position, I think he sees a pretty raw view of the average person's approach toward math.
Watching my kids play DragonBox, methinks a major problem with teaching algebra is the insistence on forcing steps from arithmetic to algebra, bogging down in numeric & non-numeric symbols which students have little or no cognitive relationship with at that age. Starting with pure algebraic concepts, devoid of explicit numeric meaning, may be much easier to absorb then transition into meaning-laden symbols.
Is that learning mathematics though or is it just learning symbolic manipulations by rote.
Yes some rote learning is necessary - and I'd warrant very useful in maths. However, generally in order to build on what you're learning you need to understand why you should perform certain actions.
There seems little point in learning to simply mechanically do the actions necessary to solve an equation. It is the meaning that is the reason for doing the learning and I worry that the last transition will be missed and make the entire prologue void of worth.
generally in order to build on what you're learning you need to understand why you should perform certain actions.
Before knowing why you should perform certain actions, it helps to know that you can perform certain actions. Methinks getting these basic concepts ("combine something with its inverse and it disappears", "thing over same thing is 1", "1 'times' something is that something", ...) into a kid's head very early is a good thing - may not yet understand why, but it's a mental tool that can be applied. Don't underestimate the value of having tools even if you don't know why/how they work; give a kid a hammer and he'll figure out it's for pounding nails.
>give a kid a hammer and he'll figure out it's for pounding nails. //
How old is the kid? I think you'll just end up with everything broken, unless you present it with the nails then the combination of the two is unlikely to happen naturally I feel until the child is quite old (if then).
I don't think the basics of algebra will have such results, but both are indeed tools that a kid will figure out a good use for if such tools are on hand and have been played with enough.
I could credit a lot of my creative skill to spending inordinate time as a kid just fiddling with a broad range of tools in the basement, regardless of whether I understood them at the time.
What you are saying makes a lot of sense. However, I often used that philosophy in my five years of teaching high school math/physics and found that most students had a very hard time transitioning into the symbolic versions of things. Even with exact parallels it was still almost impossible.
The difference between categorical and analytical modes of thought is not always clear to categorical thinkers, but is usually well-known to the analytical. I'm heavily in the analytical camp, went straight from high-school level algebra to the theory of formal systems and Godel numbering, and didn't 'click' with my scholastic maths education until linear algebra.
Linear algebra was exactly and precisely when I realized that "algebra" was a special case of algebras, which was what I needed to contextualize it. Before that it was a bunch of wasted rote effort.
Yep, I agree. I think that mathematics curriculum in High School should be divergent. Setting up kids, who after being introduced to Mathematical topics still hold disdain for it, to be allowed to check out statistical-based mathematics (with applications to personal finance, etc.)with introductions to basic probability theory and applications as well possibly (I have seen John Baez mentioning this before as well).
Although, in today's age there is no such thing as "practical mathematics" if you are going into a non-mathematical-touching field. Technology basically has you covered.
BUT for those who have an obvious love of Mathematics they really do need to be introduced into abstract thought which basically equates to questioning, generalizing, and enhancing notions/ideas the student has already come across.
Compare the article's premise to the HN-presented app DragonBoxhttps://www.hnsearch.com/search#request/all&q=dragonbox which abstracts the concepts of algebra into a tile-based game which young children (3yo even!) can learn and enjoy.
I will second this. My kid was able to play it at 3, and do decently, despite the concepts involved being (approximately) identical to solving equations.
My favorite way to define "algebra" to my students is to go back to its etymology, from the Persian textbook "Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala", or "The Compendious Book on Calculation by Completion and Balancing".
Specifically (borrowing from Wikipedia):
"The al-ğabr (in Arabic script 'الجبر') ("forcing " or "restoring") operation is moving a deficient quantity from one side of the equation to the other side. In an al-Khwarizmi's example (in modern notation), "x^2 = 40x − 4x^2" is transformed by al-ğabr into "5x^2 = 40x"."[1]
I'll also tell them that Algebra is solving equations through the use of inverses. I admit that both definitions are reductive but mostly to the point.
I am not sure of the pedagogical benefit of what is being described here - a separation of arithmetical and algebraic thinking at the level of school mathematics.
By the distinction that Prof Devlin tries to make, primary school subtraction is typically taught/learned in an 'algebraic' way (logical reasoning to invert addition).
This makes it difficult to understand what he is trying to say.
I most strongly doubt the claim that students who are strong in arithmetic find it harder to learn algebra. It's obvious that students with good arithmetic skills are more quickly able to find value and purpose in algebra.
It may be taught that way, but do kids learn it that way? Kids learn a bunch of "facts", and are tested on their capability to repeat those facts in a timely manner. Then they are taught rote-level procedures for combining those facts in order to do arithmetic on multi-digit numbers. The fact that subtraction is the opposite of addition gets demoted to "interesting bit of trivia".
For most school kids, algebra is the first place where the art of mathematics comes into play: where you are given a problem, and are not told specifically how to get from here to a solution. You have to figure that out yourself with the aid of mathematical tools.
For most people, who aren't used to the art of logical thinking, that is crushing.
Slightly tangential, but every time I work with someone on basic kinematics and they say, "Oh, well, if I just flip the sign from positive to negative, I get the right answer! I'm done!" a little piece of me dies.
It's like... you're not even trying to understand the concepts, are you?
I think this is hard because it's not immediately obvious that direction is such an important concept in kinematics.
It's probably the most important thing to get across when teaching the subject, but you have to battle against prior knowledge:
Math students (that is, everyone) earlier learned that this sign-flipping strategy is ok and perhaps an encouraged shortcut when e.g. subtracting.
(see also that Khan Academy video - "first do the multiplication, then think about the sign")
I think the claims that X is not necessary for everyday living ignores one of the purposes of education, which is to expose you to a diverse array of subjects and teach you ways of thinking about things.
Maybe you took algebra and calculus, but you never actually use them as you did in math class in your everyday life. But you learned problem solving in instances where you don't have all the data, you learned things like limits, infinite sums, and the relationship between lines, equations, areas and sums.
Although you don't use the explicit rule based symbols, perhaps your brain got wired from the overall experience to think somewhat differently when encountering certain situations.
Perhaps you'll understand your mortgage a little bit better because of your exposure to geometric series. Perhaps when looking at your finances, the concepts of slopes and tangents will re-emerge.
The same goes for learning history, or philosophy, or english literature. It's not that you have to "use" your known of the Civil War or of Shakespeare in everyday activities, but as a functional citizen, having been exposed to those things, perhaps when you are asked to evaluate what's happening on the nightly news, you will have a deeper perspective to draw on?
Personally, I use math all the time. I was a math major and I love it, so of course, it's my standard tool. But I don't just use it in coding, I use it thinking about art and lots of other things. I tend to "see math" all throughout the universe and human experience.
I can say that while this covers a few algebraic topics I strongly believe that the class title should be changed to: Practical Mathematics.
If High Schools truly wanted to introduce Algebra to students in a "pure" form then Math teachers would need to know/understand Abstract Algebra. And then be able to teach introductory ideas to Abstract Algebra which, in all honesty, anyone can come to understand.
Btw I encourage everyone to learn at least some basic Abstract Algebra to see the beauty behind what pure algebra is. :)
EDIT: It seems the curriculum is generalized to "set" the students up for future multiple areas of Mathematics at once. Ranging from analysis & differential equations to more advanced algebraic topics. But it certainly introduces students to basic topics (sets, polynomials, functions) albeit in a rather restricted (and not necessarily well taught) sense.
EDIT2: Not only should the title of the class be changed, but possibly the class curriculum should focus even more on "Practical Mathematics" as well??
The best place to look at what is being taught as "Algebra" at the high school level is to go to the (new) source: the Common Core State Standards Initiative[1]. 45 states (and DC) are using the standards. The best places to start are Math Appendix A[2], which explains the Initiative's recommendations for a High School curriculum, and the Introduction to the Algebra Strand[3], which describes (in brief) the goals for actual algebra instruction.
>If High Schools truly wanted to introduce Algebra to students in a "pure" form then Math teachers would need to know/understand Abstract Algebra. And then be able to teach introductory ideas to Abstract Algebra which, in all honesty, anyone can come to understand.
They don't. Not on the general level. You're worrying overmuch about semantics. The name "Algebra" isn't ever going away; no high school guidance department wants to explain to every single college that their program teaches the same thing as a normal "Algebra 1" class but just calling it "Practical Mathematics". (Incidentally, "Practical Mathematics" would imply much more basic mathematics, your traditional "Home Economics" class with taxes, investments, credit cards, balancing checkbooks, etc.)
Beyond that, abstraction is much, much, much, harder for the average student than you realize. Students have trouble seeing the relationship between the Distance Formula and the Pythagorean Theorem. Some have trouble even manipulating basic formulas, such as solving the Ideal Gas Law for a given variable. They will insist on plugging in the numbers into PV=nRT every time, and then solving the equation over and over again. Dividing by 5 is tangible and easily visualized to these students, dividing by R is not.
Math teachers take the job because we love math and want to share that with our students, but practical concerns come first. Symbolic manipulation is much more widely used in the average high school student's future education than pure math. I, and many teachers, include facets of pure mathematics in our courses. When my Honors Geometry students begin working with infinity, I have them read Strogatz's excellent piece on Hilbert[4], which is one of the most popular assignments each year. (And yes, there is next to no Geometry in there, but you have to keep minds sharp somehow.) I also throw in some basic Real Analysis when discussing the concept of rigor in proofs. When my Algebra 2 classes have to trudge through a brief review of Algebra 1, we spend the time talking about why closure matters, and what number systems are and are not closed over what operations. My PreCalc classes do a decent amount of Number Theory.
Finally, most schools offer Discrete Mathematics (i.e., an introduction to Pure Math) as a senior-level math elective. However, it's competing for the brightest minds with AP Calculus and AP Statistics. If you'd like to see more students study Pure Mathematics, the best way to do that would probably be to gather a group of like-minded educators and petition the College Board to create an AP course covering said material
This touches on a semantic point which confuses many new programming students: in algebra, the "=" is a specification of truth; in programming, the "=" is an operation of assignment. There is a difference, as in "2 * x = x + x", which is perfectly sensible in the former but pointless/absurd in the latter context. Also, "x = 2" in algebra is the notion of "when it is true that x is the same as 2, other truths occur", a different notion than "store value 2 in variable x, then see what the consequences of other operations are." Alas, the nuance is hard to grok until much later in learning both.
But algebra does not preserve truth values. It guarantees that if you start with a true statement you will end with a true statement. If you begin with a false statement, it is still valid to end with a true statement. Simple examble:
If you replace your numerical values with algebraic symbols, you may see the problem with your analysis -- its use of a zero allows 1 to equal 2, as shown in this classic example:
Multiplying both sides by zero is always legitimate (if rarely useful). In the example linked, the error is in dividing by zero. More specifically, in dividing by (a-x). Formally speaking, this operation is only defined when (a-x)!=0. Mutipling by zero is defined for all real numbers.
I think you're having trouble with the definition of "preserve". If you begin with a false statement, there isn't actually a truth value available to preserve.
True, but it is still valid algebra. There are other examples where one might turn a false statement into a true statement without doing something as pointless as multipling by zero. A common example is squaring an equation. Consider a system of equation in which it can be shown that x=3. Within this system, the statement x=-3 is false, however the statement x^2=9 is true.
Again, this specific example is contrived, but this does come up annoyingly often.
That's probably not far from the best definition, and also not far from the translation of `algebra` from the original writings, "breaking and combining".
The words al-Khwarizmi used -- al-jabr and al-muqabala -- actually meant "restoring" and "balancing", and referred to adding or subtracting some quantity from both sides of an equation. Muslim mathematicians of the day did not deal with negatives, so today we would consider al-jabr and al-muqabala to be variants of the same operation.
Nevertheless, the original writings were what suggested to me that the important bit of algebra was doing things to equations to change them into some easily solvable canonical form (al-Khwarizmi's work dealt primarily with quadratics).
Thanks for this, I blended al-khwarizmi's book title with "Kitāb al-Jamʿ wa-l-tafrīq bi-ḥisāb al-Hind (The Book of Bringing_together and Separating According to the Hindu Calculation)" mentioned in its wikipedia page.
Well, he only talks about school algebra. I'm not familiar with the US high school curriculum but I doubt it goes much beyond polynomials…
FTA (3rd paragraph):
> (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.)
That's almost clear from the question. If this was about higher math, the question would likely have been "What is an algebra?", as "What is algebra?" has the simple but almost content-free answer "The study of algebras." (http://mathworld.wolfram.com/Algebra.html)
To that point, does anyone know if there is a formal definition of what it means to be a number? I know we have number systems in which the Integer-like numbers do not behave the same as the Integers, where the numbers of that system are defined completely independently of the Reals.
Oh yes, some outstanding thinkers have gone into mind boggling depth on the meanings of fundamental maths concepts. This started getting a proper treatment in the early 20c. (see https://en.wikipedia.org/wiki/Principia_Mathematica ) and presumably has come a long way since then.
Edit: Of course one has to take a different approach for pedagogical purposes.
It should be said that way, as a choice and not an arbitrary concept tied to numbers. For pupils so many why's can rise from coupling the properties of what we call algebra to a projection on numbers.
Hmm, I think this is a fine essay for people who have already internalized the notion of algebra per se, but it's not great for a student who has to yet to do so and is asking the titular question "What is algebra?"
Here's my attempt for, say, a relatively smart high school student. Consider this a rough draft; I'm writing it from start to finish without editing. I'd love, love, love feedback, though.
Today even schoolchildren understand arithmetic. We have these things called "numbers." There are different types of numbers like natural numbers, integers, rational numbers, and irrational numbers. We have rules for manipulating these numbers like addition, subtraction, multiplication, and division.
This wasn't always the case, however. I don't mean that humans couldn't always add, but I do mean that it took humans thousands of years and multiple false starts to come up with a sensible way to represent numbers and these operations. Think about trying to do division with Roman numerals, for example. It'd be a nightmare!
There was a time when numbers like "4/5", "-2", "0", and "√5" made people freak out because we didn't have the symbols to represent them and it wasn't obvious what they corresponded to "in real life," if anything. Imagine yourself living in a world like the ancient Greeks, for example, where you represented numbers by talking about lines of a given length. It was very hard for you to talk about numbers per se without drawing a shape that somehow encoded that number. Now, pop quiz: how do you represent something like "0" or "-2" in this world? If you were a Roman using Roman numerals, how would you represent "4/5"?
This is just a story to highlight that although we take our numbering system and arithmetic for granted, this was not the case for most of human history, even most of recorded human history. The way we do arithmetic was an invention that both helped us do arithmetic and helped us understand WTF a number even is.
By the way, if you freak out at the idea of imaginary numbers or the idea of a number i which satisfies the property i^2 = -1, this is no different than the kind of freaking out the ancient Greeks did when they first encountered √2 or other cultures tried to make sense of negative numbers.
Now, let's think about what we really did by inventing arithmetic as we understand it today. You can't point to the number "5" anywhere in the world, right? Even the symbol 5 isn't five per se, any more than "five", "fünf", "|||||", "V", or "五" are five per se. But with one symbol "5" we can represent this abstract thing.
Then something like "5 + 4" might represent the length of a line segment made from concatenating a line segment of length 5 and a line segment of length 4, the age of a 5-year-old in 4 years, the volume of water made from pouring five buckets of water into a pool and then four buckets of water, the number of apples shared between a 5-apple basket and a 4-apple basket, and so on. So, these abstract things we call "numbers" and "arithmetical operations" can represent many more concrete things.
Let's call this process "abstracting." Algebra is what you get when you treat numbers as the concrete thing and apply this same process. With arithmetic, we want to talk about numbers divorced from a particular concrete realization. That is, we want to talk about the number 5 without having to talk about a basket of five apples. With algebra, we want to talk about numbers per se divorced from a particular concrete number.
Remember, when we invented arithmetic we had to invent a bunch of symbols to represent the abstract thing. We do that in algebra, too. Often we use single-letter symbols like x and y, but we could use anything like ☃, ☂, or zorpzop. These symbols "stand in" some number in the same way that the symbol 5 "stands in" for all the things 5 could possibly represent in the world. We can talk about 5 without talking about the things it might represent.
So, we say things like "let x be a number" or "let x be a positive number" or "let x be a rational number." What can we say about x in each of these situations?
For example, if x, y, and z are all standing in for some number, we can say the following:
x + 0 = 0 + x = x regardless of what number x is
x + y = y + x regardless of what numbers x and y are
x + (y + z) = (x + y) + z regardless of what numbers x, y, and z are
x*1 = 1*x = x regardless of what number x is
x*(y + x) = x*y + x*z regardless of what numbers x, y, and z are
These are true because of what we mean when we say "number" and what we mean when we say "addition." It's not as if these are true for some numbers and not all, nor is it as if we know these are true because we've "checked all the numbers." That's impossible because there are an infinitude of numbers.
So, now we might ask things like, "Are there any numbers x such that x^2 + 1 = 0? How about x^2 + x - 1 = 0?" These questions might be hard to answer, but we have now at least invented a language where we can ask them, whereas before "abstracting" arithmetic into algebra we had no easy and succinct way of asking them.
This is no different than not being able to easily ask, "Can we construct an equilateral triangle with side lengths of π?" before abstracting from more concrete things into numbers. Without a symbol for π we have to say things like "the constant that is the ratio formed between the circumference and diameter of a circle." This is how mathematics was done for thousands of years. It was tough going, as you can imagine, and we missed many things that would seem "obvious" to people using our notation.
You can continue this process further, by the way, and abstract further from algebra. This is what mathematicians call abstract algebra (http://en.wikipedia.org/wiki/Abstract_algebra). In this context there are multiple algebras and the "algebra of numbers" becomes the concrete thing in this new system. Linear algebra is a different algebra, for example, with a different sets of "numbers" and a different set of "arithmetical operations" that don't always correspond 1-to-1 with the numbers and operations we find in arithmetic.
Often, when presented with a new physical system of objects that interact in a certain way, we can try to abstract these objects and operations into symbols and derive rules about these abstract symbols and operations that correspond to the workings of the physical system.
We might call this symbolic system "an algebra." For example, Claude Shannon invented an algebra for relay and switching circuits that allows us to understand how they operated and how to combine them without actually building physical circuits. See http://www.cs.virginia.edu/~evans/greatworks/shannon38.pdf
I like this explanation. High school students are surprisingly smart and a little perspective -- that "algebra" is just manipulations in a symbolic system (and not even a special one, just very widely used) -- is sometimes all it takes to help them "get it" a little more.
It's kind of funny, because students are taught how to use these things like numbers and operations and variables, but it's never explained in an abstract way what "algebra" is or why it's even called that. Hell, I didn't know until I took linear algebra in college.
When my sister (who is currently taking algebra) asked me what "algebra" meant, I told her:
An algebra is a combination of a set of objects and a set of operations. The algebra you're learning has real numbers as the set of objects. You know what the operations are: they're things like multiplication, addition, sqrt, whatever. There are other algebras too that use things other than the reals.
As an aside, this is a great way to teach fractions as well (or at least, I like it a lot). People try to give intuitive explanations with pieces of pie or whatever but students tend to find them very confusing. You would be surprised how fucking confused students can get about fractions, I mean some just can't wrap their heads around it ever. Even as adults. They're not dumb, they try to learn fractions by "intuition" and, frankly, fractions aren't intuitive. The simplest route is to just lay it out: these are objects with one expression and another expression and a line between them, and you can multiply them like this, and add them like this, and "move" expressions from one side of the line to the other by taking the reciprocal like this, and that's it.
Division with Roman numerals being a nightmare was not obvious to me. That was my first friction point here :-)
> There was a time when numbers like "4/5", "-2", "0", and "√5" made people freak out because we didn't have the symbols to represent them
Who freaked out, and how could they freak out if they could not behold these numbers in the first place (i.e. no symbolic representations)?
> without drawing a shape that somehow encoded that number
Recommend you use "represented that number" because encoding has a strong, separate type of meaning to me.
> Now, pop quiz: how do you represent something like "0" or "-2" in this world? If you were a Roman using Roman numerals, how would you represent "4/5"?
For "0" I would have made an empty box, out of strings if necessary. For the -2 I would have placed two of the strings in a different location. For the Roman numeral conundrum I would have placed IV and then a line and then a V below that. Now, I just solved your conundrums. Or didn't I? To really speak to beginners it's important to delve into this stuff. So far I have not discovered why our number system is so great, and I don't feel like I'm freaking out about anything in particular. :-)
Thanks for your writeup, though. I hope you can turn it into a book for people like me.
> Who freaked out, and how could they freak out if they could not behold these numbers in the first place (i.e. no symbolic representations)?
The story goes that Hippasos of Metapontum discovered the existence of irrational numbers like √2 while at sea, and his fellow Pythagoreans threw him overboard, because it proved that there were aspects of the world that could not be represented with rational numbers. That said, the notion of a square root had been established for at least a thousand years by that point (by the Egyptians and Sumerians), so the freaking out was less about their representation and more about their properties.
> For "0" I would have made an empty box...
Your representations are parasitic on the fact that you already have a deeply ingrained representation for those concepts and are comfortable manipulating those concepts. It's like suggesting that you'd reinvent the wheel if you lived in a civilization with no wheels merely because it seems obvious to you looking at it today. Really it's not at all obvious, as no New World civilization ever developed the wheel, and most Old World civilizations borrowed it rather than inventing it independently.
Think of it this way: if you understood fractions, but nobody else did, how would you represent them symbolically such that everyone else would? I'm of the opinion that our typical notation for calculus is spectacular (especially compared to Newton's original notation—the d/dx notation used today comes via Leibniz) but if you don't understand calculus, how does that help you? You understand negative numbers, but to a person with no concept of debt, a person for whom numbers represent 'how many sheep you have', what does that mean, and how do you symbolize it so that they do understand?
> For "0" I would have made an empty box, out of strings if necessary.
That's 4. Because you used four strings. Oh, you do it with one string? It looks like 1. You do it with N strings? That's N. Oh, you're trying to talk about what's inside? Nothing's inside. That's silly. We both know there's nothing in there. Why are you trying to show me nothing?
Are you trying to multiply by the inside? That doesn't make any sense. Why would you multiply by nothing, or add by nothing? What's the point? No, I don't know what it does. Why should I care about your nonsense?
> For the -2 I would have placed two of the strings in a different location.
Now do "square root of negative nine".
> For the Roman numeral conundrum I would have placed IV and then a line and then a V below that.
Oh, so there's 4 on the top and 5 on the bottom? So, 9 total, right? IX?
According to story, Pythagoras freaked out about it. You don't need a symbolic representation of √2 to ask the question "what is the distance between opposite corners of a square with side length 1"?
Great explanation, worth an article! My thinking was more of a struggle a student has to take to correlate with a abstract concepts like algebra while in school. It is almost impossible for a kid to really look meaningful context out her daily maths class. However, all this concepts are born out of "need". If that "need" magically does not gets shown to kids "in their own language" there is going to be lot of trouble in bringing up right math power.
I guess the teaching style has to change. Was also wondering, folks in this thread, talking great things about math are teachers? Should be less, I guess!
I'm confused about what algebra actually means. Wikipedia states "It follows that algebra, instead of being a true branch of mathematics, appears nowadays, to be a collection of branches sharing common methods."
Like any word, it has evolved into meaning a bunch of different things and it's not easy to define what it actually means. The original article "What is algebra?" seems to talk about elementary algebra.
Forget about the Wikipedia article, which is trying to define algebra "in general." I know of no definition that encompasses all uses, both technical and colloquial and I'm not sure why we'd want one. A better question than "What is algebra?" might be "What does it mean when Person X says algebra?"
For most people it means high school algebra, which is what I was trying to explain until the very end. (High school) algebra is to arithmetic as arithmetic is to the more concrete and often physical things I mentioned. Put another way, algebra is what we get when we go through the same process we applied to invent arithmetic as we understand it today, but treat particular numbers as the more concrete thing.
Speaking precisely high school algebra means "the study of the real numbers under the operations of addition and multiplication as we typically understand them." Linear algebra and the algebra of the complex numbers sometimes make an appearance, too, which often confuse students because we're now equivocating and calling all these things just "algebra."
From the article: “(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.)”
From the article, which runs off the rails after this "In today’s world, most of us really do need to master algebraic thinking."
In practice its a class based (several meanings of class...) sorting/filtering path for students... this group will successfully take orders from above to handle unusual logical problems with some stability, discipline, and a self directed very short term plan (aka future supervisor material) and this group of kids can't/won't and will end up working underneath the first group as proles.
Unless you think most people will become supervisory workers or above, its just not necessary.
The explicit example in the article was "Creating formulas in a spreadsheet" (paraphrasing a bit). Is that really a task that will never be required of non-supervisory workers?
I'm really not sure where you're going with this, or what it is that you see as "run[ning] off the rails".
Alpha children wear grey. They work much harder than we do, because they're so frightfully clever. I'm really awfully glad I'm a Beta, because I don't work so hard.
Algebra is an abstraction over arithmetic, which is an abstraction layer over counting. Or at least that is the abstraction I am using today after reading the article.
As a nitpick, I am not convinced that arithmetic is prior to geometry - e.g. Stonehenge and celestial navigation require geometric abstractions but not arithmetic. "Fabricate bricks until I tell you I have enough" is a more reliable logistical model for the construction of Rome's aqueducts than "provide x bricks for each of the y miles."
This is funny, because the text you quoted wasn't referring to the construction of the aqueducts, only the fabrication of the bricks used in the construction.
Tangentially related to the article, but for many years I struggled with understanding what algebra was an more importantly why it was of interest. On one hand you had groups and rings that sort of seemed related, but they weren't algebra's. Then there were special techniques like linear algebra and I could see that they were all sort of similar but why they were of interest and all considered to be part of the same subject escaped me. Sure they were interesting in a way and group theory was sort of cool but what was the point? Analysis on the other hand made sense, it was the study of how continuous things changed. As a result I though of algebra as a sort of collection of things that you got when you discretized continuous objects. Lie algebra's and groups came as a bit of a shock, but they were basically still thought of algebra as discrete.
Many of my friends raved about its beauty, but I still had no intuition as to why it was interesting. I went on a bit of a reading binge, and eventually ended up bumping into some papers of Shafarevich[0] which in turn lead me to his delightful book on algebra [1]. In it he defines algebra as the construction and study of systems of measurement. For example counting is the simplest such system and gives the natural numbers. Attempts to describe and measure the diagonal of the unit square gave irrational numbers. As we explored the world we needed to construct new systems. A more recent example is provided by quantum mechanics, where numbers are insufficient to describe our observations, however Hilbert space provides a natural setting. Attempts to describe what a simultaneous measurement of two quantities is find a natural description as commuting operators. He provides many other examples in his very readable book. For whatever reason this definition of algebra resonated with me and if this was a Zen koan I would say I was enlightened.
This is really just a very long winded way of saying that I think Shafarevich's book in which he defines algebra as the study of measurement is lovely. You should read it if such things interest you.
I really like the "algebra is like cheating for math" epiphany. It gets to the core of both higher mathematics and the problems in our educational approach.
A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory. “We are helping our students become more competitive in an increasingly sound-filled world.” Educators, school systems, and the state are put in charge of this vital project. Studies are commissioned, committees are formed, and decisions are made— all without the advice or participation of a single working musician or composer.
[0]: http://worrydream.com/refs/Lockhart-MathematiciansLament.pdf