What is algebra?
profkeithdevlin.org
profkeithdevlin.org
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
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.
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.
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'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.
The author was only targeting at talent people.
http://www.maa.org/external_archive/devlin/devlin_03_08.html
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..."
See: http://www.amazon.com/Paint-Number-How-To-Craze-Nation/dp/15...
Sorry for my English.
The 3rd and maybe the 4th editions can be found online by doing an internet search for pdf/djvu files.
The book might seem very alien at first, though.
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.
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.
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.
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.
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.
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.
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.
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.
[1]http://en.wikipedia.org/wiki/Compendious_Book_on_Calculation...
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.
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.
It's like... you're not even trying to understand the concepts, are you?
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")
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??
>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
[1]http://www.corestandards.org
[2]http://www.corestandards.org/assets/CCSSI_Mathematics_Append...
[3]http://www.corestandards.org/Math/Content/HSA/introduction
[4]http://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert-...
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).
1=2 | false
0 * 1=0 * 2 |Multiply both sides by 0
0=0 | true
Again, this specific example is contrived, but this does come up annoyingly often.
http://www.jimloy.com/algebra/two.htm
So multiplying by zero isn't always a legitimate operation. That's why the multiplicative identity is one, not zero.
If only pascal had won that particular syntax battle.
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.
Edit: Of course one has to take a different approach for pedagogical purposes.
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
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
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.
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.
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."
> 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.
--Designer guy who is not super great at math
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?
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?
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!
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.
I'm really not sure where you're going with this, or what it is that you see as "run[ning] off the rails".
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."
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.
[0]https://en.wikipedia.org/wiki/Igor_Shafarevich [1] https://encrypted.google.com/books/about/Basic_Notions_of_Al...