[This is an old post but I don't think it's lost any relevance]
Michelle Rhee was on Marketplace yesterday. She's worried about all the coddling and empty praise we've been heaping on our children.
We've lost our competitive spirit. We've become so obsessed with making kids feel good about themselves that we've lost sight of building the skills they need to actually be good at things.
I can see it in my own household. I have two girls, 8 and 12, and they play soccer. And I can tell you that they suck at soccer! They take after their mother in athletic ability. But if you were to see their rooms, they're adorned with ribbons, medals and trophies. You'd think I was raising the next Mia Hamm.
I routinely try to tell my kids that their soccer skills are lacking and that if they want to be better, they have to practice hard. I also communicate to them that all the practice in the world won't guarantee that they'll ever be great at soccer. It's tough to square this though, with the trophies. And that's part of the issue. We've managed to build a sense of complacency with our children.
Take as a counterpoint South Korea, where my family is originally from. In Korea, they have this culture that focuses on always becoming better. Students are ranked one through 40 in their class and everyone knows where they stand. The adults are honest with kids about what they're not good at and how far they have to go until they are number one. Can you imagine if we suggested anything close to that here? There would be anarchy.
This is an old and much loved refrain in the reform movement, but when you look closely and take the time to disaggregate the data the argument completely collapses. The problem is that the culture of esteem-building Rhee describes is essentially a suburban phenomena. In poor neighborhoods, the situation is exactly the opposite:
Hart and Risley also found that, in the first four years after birth, the average child from a professional family receives 560,000 more instances of encouraging feedback than discouraging feedback; a working- class child receives merely 100,000 more encouragements than discouragements; a welfare child receives 125,000 more discouragements than encouragements.
Other words, our best performing schools are filled with kids who, according to Rhee, should be complacent and lacking competitive spirit while most of the kids in our worst performing schools have received little of the empty praise that so concerns her.
Perhaps we should worry less about the exaggerated self opinions of students and more about that of our pundits and commentators.
No, seriously, let's play Risk, or at least talk about playing Risk. In case you're not familiar with the game, here's the Wikipedia summary
Risk is a turn-based game for two to six players. The standard version is played on a board depicting a political map of the Earth, divided into forty-two territories, which are grouped into six continents. The primary object of the game is "world domination," or "to occupy every territory on the board and in so doing, eliminate all other players."[1] Players control armies with which they attempt to capture territories from other players, with results determined by dice rolls.
(invented by the director of the Red Balloon -- who knew?)
Risk can be a good taking off point for a number of lessons and assignments like:
1. Strategic thinking -- break the class into groups, have them write up rules and recommendations for the game then test these ideas in a tournament
2. (not math but why should that stop you?) -- what modern countries and provinces correspond to the regions on the board? What are they like? You might add naming these countries as a condition for conquering a region.
3. Probability -- what are the chances of taking a country with j attackers and k defenders? What's the expected strength of j if there is a conquest, k if the attack is repulsed?
But the main lesson I'd like to suggest is an introduction to graphs. We have other graph based games and puzzles in our tool chest such as doublets and the six degrees game but Risk is probably the most familiar to the kids. More importantly, it requires actually working with graphs as part of a larger problem.
After the students are acquainted with the game, explain the basic terms of graph theory then show them something like this subgraph of North America
Ask the students to do the following (After checking to make sure I got this right) :
1. Fill in the other countries
2. Explain what's special about nodes like Alaska and Greenland
3. Draw subgraphs of each continent
4. Find the shortest path between various pairs of countries
4b Find the shortest path between various pairs of countries when certain territories (particularly Russia) are blocked
4c (advanced) With randomly placed armies, see which path would be easiest to conquer
5. Play a game on a node and edge board
6. Make up new node and edge boards. Give the territories real or mythical names. Try playing a few games on them.
Even in the tightest, most focused class, there will be some slack time. Maybe you don't have a quorum of students, or you're waiting to be called to an assembly, or you finished a lesson early or you just realized that the kids need a break.
Particularly for inexperienced teachers, slack time can present a real threat to classroom management. When not being challenged with mathematical concepts, those active young brains will quickly turn to the problem of finding clever and effective ways to test authority.
But slack time can also be an opportunity. In a relaxed setting, doing something that doesn't trigger their math-is-hard defenses, you can often sneak in some useful problem solving and pattern recognition practice. Even more importantly, you can reinforce the alertness essential to being good at math.
Which brings us to this family of puzzles.
Go up to the board (preferably without saying anything), draw a horizontal line, then start writing the sequence A, B, C... or 1, 2, 3..., putting some of the characters above the line and some below based on some property of the letters or digits. Somewhere in the middle of this you stop, turn to the class and ask where the next letter goes.
A B D
______________________________________________________________________________
C E F G
If you get the wrong answer, shake your head, write the character in the proper place then ask about the next. You may have to give a couple of hints but usually someone will let out an "Ooooh" of realization and will start eagerly shouting out the answers.
The property here is topological and you can, if you want, use this introduce topology or even to open a lesson on homotopy equivalence.
The important here though isn't introducing the topological terms; it's getting students to think topologically, to add these properties to the things they look for.
The properties in an above/below puzzle can be anything your students could reasonably be expected to spot, from prime vs. composite for numbers to the phonetic spelling of letter (there's a clever New Yorker cartoon that put the alphabet in alphabetical order but I can't find it on line).
A few years ago I did a stint as an instructor at a large state school teaching, among other things, business calculus. The sections for that course tended to be good-sized, usually running from fifty to one fifty. At one time, I probably would have found the experience a bit intimidating but I was just coming off a couple of years as a TA for a professor who routinely taught sections of more than three hundred so I considered myself lucky to be able to make out individual faces.
With few exceptions, experienced teachers are comfortable addressing large groups and with very few exceptions, effective teachers are comfortable demanding the full attention of those group. Along with knowledge of the subject, strong communication skills, and commitment to the students, a "when I talk, you listen" attitude is a defining trait of an effective instructor.
That doesn't automatically translate to a room full of kids sitting quietly while the teacher drones on. Often the result is just the opposite. Teachers are more likely to have looser classes with more student participation if they feel in control. As a rule of thumb, you should never be more than ninety seconds away from having every student seated and reading quietly. For really good teachers, even the most adventurous lesson plans fall into that ninety second radius.
Put another way, it comes down to authority. A teacher's job is to teach, counsel and objectively evaluate his or her students. A sense of authority is an essential trait for all these tasks but it's an incredibly annoying one to find in a direct report.
Good principals (and I've met some excellent ones) are masters at the difficult art of managing managers. They can exercise their authority in a way that actually enhances the authority of those under them.
Even with the best administrators, however, there is always an element of tension and it only gets worse with less competent principals and superintendents. This is something to keep in mind when you hear about plans to improve education by giving principals more authority to get rid of bad teachers. Sometimes bad doesn't mean incompetent; it means inconvenient. (I don't have the book in front of me, but Diane Ravitch' Death and Life has some notable examples.)
Back in my undergrad days, when I was taking a number theory class, I noticed that all squares that weren't multiples of three were always one more than a multiple of three, never one less.
16 = 15 + 1
25 = 24 + 1
100 = 99 + 1
After I thought about it for a while I realized that this had to be the case and there was a simple algebraic proof that showed it (there's a also a simple geometric proof -- think about cutting the square into two smaller squares and two rectangles -- but that can wait for another day).
I'd assign this with the following hint:
think about (x + 1)(x + 1) and (x - 1)(x - 1)
Here's the proof. All natural numbers can be written as:
I. 3k
II. 3k + 1
III. 3k - 1
We're talking about squares that aren't multiples of three so we can skip the first case and look at II and III.
A while back I posted a recommendation for a popular pencil-and-paper game:
On the subject of topology, my game of choice is Sprouts, invented by mathematicians John Horton Conway and Michael S. Paterson at Cambridge University in 1967 (as a general rule, you can't go wrong with a game if Conway had anything to do with it).
The rules are simple:
1. Start with some dots on the paper. The more dots you have the longer the game takes so you will probably just want to start with two or three.
2. Players take turns either connecting two of the dots with lines or drawing a line that loops back and connects a dot with itself.
3. The lines can be straight or curved but they can’t cross themselves or any other lines.
4. Each dot can have at most three lines connecting it
5. When you draw a line put a new dot in the middle.
6. The first player who can’t draw a line loses.
I was thinking about sprouts the other day and a few variations occurred to me. I don't know if they're particularly playable or if they add any interesting aspects to the game, but if you can't put a half-baked idea in a blog, what's a blog for?
Variant 1 -- Free sprouts
Played as above but with the following addition: for the first k moves of a game with n dots, the player, after drawing a line, adds a new dot.
Topologically the result is a game with n+2k dots (keep k small) but with the complication that lines are being drawn without knowing exactly how those lines partition the surface. This is still a game of perfect information but the variation should make it more difficult to think a few moves ahead.
Variant 2 and 3 -- Scored sprouts
Each player starts with a separate sheet of paper and proceeds to connect the dot according to the standard rules. After no more lines are possible, the players score their graphs based on the number of dots.
Score = 6
Score = 7
In variant 1 the player with the highest score wins. In variant 2, the win goes to the lowest.
When I was teaching I would always keep some form of this grid around, either printed out on sheets of paper or in a PowerPoint or some other digital form. Whenever I had to keep a room full of students out of trouble I'd put this up on the screen or pass it out and ask them to count the squares, keeping in mind that a square could be one by one, two by two, all the way up to twelve by twelve.
Take a minute and give it a try. I'll wait...
I like this problem for a number of reasons. It's a problem that's extraordinarily difficult unless you hit upon the underlying pattern at which point it can be solved in about a minute. The insight does not require any background other than fifth grade math. The solution is simple and elegant and only relies on sixth grade math but it often eludes people with considerable mathematics background.
Like many problems this is easier if you pick the right side to start from. Most people will start with the one by ones by multiplying 12 by 12, then will move on to the two by twos. You're more likely to succeed if you start with the twelve by twelve then try the eleven by elevens (or better yet, start with the one by ones then jump to the the twelve by twelve then the eleven by elevens) then the ten by tens. Counting the bigger squares is simpler and the pattern will probably jump out quicker this way.
There is, of course, one twelve by twelve square in the picture, and it shouldn't be difficult to see that there are four eleven by elevens and nine ten by tens. So we want to find the sum of a sequence of twelve numbers that starts with one, four, nine and ends with one hundred and forty-four. Once you have the pattern in mind, you can start asking yourself why it holds. With a little thought you can probably get to this insight: for an n by n square, the bottom left corner has to fall in a square with sides twelve minus n plus one.
Like most good math puzzles, this is very Polya -- you use inductive reasoning to formulate a hypothesis then find a proof that shows the hypothesis must be true -- but that's a topic for a lot of future posts
If you follow education at all, you've probably heard about the rise of online courses and their potential for reinventing the way we teach. The idea is that we can make lectures from the best schools in the world available through YouTube or some similar platform. It's not a bad idea, but before we start talking about how much this can change the world, consider the following more-serious-than-it-sounds point.
Let's say, if we're going to do this, we do it right. Find an world renowned historian who's also a skilled and popular lecturer, shoot the series with decent production values (a couple of well-operated cameras, simple but professional pan and zoom), just polished enough not to distract from the content.
And if we're going to talk about democratizing education, let's not spend our time on some tiny niche course like "Building a Search Engine." Instead, let's do a general ed class with the widest possible audience.
If you'll hold that thought for a moment...
A few years ago, while channel surfing in the middle of the night, I came across what looked like Harvey Korman giving a history lesson. It turned out not to be Korman, but it was a history lesson, and an extraordinarily good one by a historian named Eugene Weber, described by the New York Times as "one of the world’s foremost interpreters of modern France." Weber was also a formidable teacher known for popular classes at UCLA.
The program I was watching was “The Western Tradition,” a fifty-two part video course originally produced for public television in 1989. If you wanted to find the ideal lecturer for a Western Civ class, it would probably be Eugen Weber. Like Polya, Weber combined intellectual standing of the first order with an exceptional gift and passion for teaching. On top of that, the Annenberg Foundation put together a full set of course materials to go with it This is about as good as video instruction gets.
All of which raises a troubling question. As far as I know, relatively few schools have set up a Western Civ course around "the Western Tradition." Given the high quality and low cost of such a course, why isn't it a standard option at more schools?
Here are a few possible explanations:
1. Medium is the message
There are certain effects that only work on stage, that fall strangely flat when there's not an audience physically present in the room. Maybe something similar holds with lectures -- something is inevitably lost when moved to another medium.
2. Lecturers already work for kind words and Pez
Why should administrators go to the trouble of developing new approaches when they can get adjuncts to work for virtually nothing?
3. It's that treadmill all over again
You probably know people who have pinned great hopes on home exercise machines, people who showed tremendous excitement about getting fit then lost all interest when they actually brought the Bowflex home and talking about exercise had to be replaced by doing it. Lots of technological solutions are like that. The anticipation is fun; the work required once you get it isn't.
This is not a new story. One of the original missions of educational TV back in the NET days was to provide actual classroom instruction, particularly for rural schools.* The selection was limited and it was undoubtedly a pain for administrators to match class schedules with broadcast schedules but the basic idea (and most of the accompanying rhetoric) was the same as many of the proposals we've been hearing recently.
Of course, educational television was just one of a century of new media and manipulatives that were supposed to revolutionize education. Film, radio, mechanical teaching machines, film strips and other mixed media, visual aides, television, videotape, distance learning, computer aided instruction, DVDs, the internet, tablet computing. All of these efforts had some good ideas behind and many actually did real good in the classroom, but none of them lived up to expectations.
Is this time different? Perhaps. It's possible that greatly expanded quantity and access may push us past some kind of a tipping point, but I'm doubtful. We still haven't thought through the deeper questions about what makes for effective instruction and why certain educational technologies tend to under-perform. Instead we get the standard ddulite boilerplate, made by advocates who are blissfully unaware of how familiar their claims are to anyone reasonably versed in the history of education.
* From Wikipedia
The Arkansas Educational Television Commission was created in 1961, following a two-year legislative study to assess the state’s need for educational television. KETS channel 2 in Little Rock, the flagship station, signed on in 1966 as the nation's 124th educational television station. In the early years, KETS was associated with National Educational Television, the forerunner of the current PBS. The early days saw black-and-white broadcasting only, with color capabilities beginning in 1972. Limited hours of operation in the early years focused primarily on instructional programming for use in Arkansas classrooms
The famous game designer,* Sid Sackson, had over eighteen thousand games in his personal collection so making his short list was quite an accomplishment, particularly for a game that almost nobody has ever heard of.
On this alone, the Blue and the Gray would be worth a look, but the game also has a number of other points to recommend it: it only takes about three minutes to learn (you can find a complete set of rules here); it is, as far as I know, unique; it raises a number of interesting and unusual strategic questions; and for the educators out there, its Turn-of-the-Century** origins provide some opportunities for teaching across the curriculum. My only major complaint is that it requires a dedicated board, but making your own shouldn't take more than a few minutes.
The object of the game is to be the first to get your general to the center by moving along the designated path while using your soldiers to block your opponent's progress. Since soldiers can capture each other, the game has two offensive options (capturing and advancing) compared to one defensive option (blocking). (Something I learned from developing my own game was the more the designer can shift the focus to offense, the better and faster the game will play.)
I don't know of any attempt to do a serious analysis of the Blue and the Gray. Might be fun to look into. If someone out there finds anything interesting, make sure to let us know.
* Yes, I did just use the phrase, 'famous game designer.'
** I'm going off memory here about the age of the game. You should probably double check before building a lesson plan around this. (see update)
The first is a Gamut of Games by the great game historian, designer and collector, Sid Sackson. As Wikipedia puts it:
Many of the games in the book had never before been published. It is considered by many to be an essential text for anyone interested in abstract strategy games, and a number of the rules were later expanded into full-fledged published board games.
I once saw an alphametic in an SAT question -- simpler than this one but with the same basic principle. My first thought (after, "Was that an alphametic?") was what a great question.
Of course, solving alphmetics is a completely useless skill. No one has ever or will ever actually needed to do one of these. It is that very frivolousness that makes it such a good question for a college entrance exam. It requires sophisticated mathematical reasoning but it comes in a form almost none of the students will have seen before.
For comparison, consider a problem you would not see on the SAT*, factoring a trinomial that wasn't the square of a binomial (this is another skill you'll never actually need but it's not a bad way for students to get a feel for working with polynomials). Let's look a two students who got the problem right:
Student one hasn't taken algebra since junior high but understands the fundamental relationships, finds the correct answer by multiplying out the possibilities;
Student two was recently taught an algorithm for factoring, doesn't really understand the foundation but is able to grind out the right answer.
Obviously, we have a confounding problem here, and a fairly common one at that. We would like to identify understanding and long term retention but these can easily be confused with familiarity with recently presented information (particularly when certain teachers bend their schedules and curricula out of shape to teach to the test). The people behind SAT have partly addressed this confounding by including puzzle-type questions that most students would be unfamiliar with.**
All too often, the people behind other standardized tests deal with the issue by pretending it doesn't exist.
* Not to be confused with the SAT II, which is a different and less interesting test.
** The type of kid who reads Martin Gardner books for recreation would generally do fine on the SAT even without the familiarity factor (though the prom may not go as well).
It's one of the fundamental questions of teaching, especially teaching mathematics: how much do you explain and how much do you leave for the students to figure out on their own. There are compelling arguments on either side. The end goal is to produce students who are independent and resourceful, who can work independently, thus making the case against spelling things out. On the other hand, discovery is a slow and often frustrating process. Given the material to cover and the danger of students just giving up, simply leaving them to their own devices is not an option.
This leads to obvious questions (if not obvious answers) about when we should and shouldn't teach algorithms, step-by-step procedures for solving specific classes of problems.
It takes a great deal of thought to come up with an algorithm and to understand why it works, but actually performing one should be an almost entirely mechanical process. The whole point is to get the answer reliably and quickly with an absolute minimum of thinking.
This isn't a bug; it's a feature. There are situations where you want people operating on autopilot. Thought is slow, unpredictable and distracting. You probably don't want your tax preparer stopping to reflect on the subtleties of economic distortions while filling out your 1040 and if you're an administrator, you certainly don't want students thinking about the nature of numbers while doing long division on a standardized test that determines your next bonus. You could even argue that most of the progress of mathematics over the past three centuries is due to notation that makes much of the work thought-free thus allowing mathematicians and scientists to focus on more important matters.
So, to be clear, I'm not opposed to sometimes just coming out and telling students "these are the steps for solving this kind of problem" rather than always having the students think through everything on their own. What I am opposed to is teaching without thinking things through.
Only a small portion of what we learn in math classes (or most classes, for that matter) will show up in any sort of significant way in life after graduation. Keep in mind, my day job is writing programs, building models and mining data. It's math-centric work but I use almost none of what I learned in junior high algebra. How often do you actually need to factor a polynomial?
That does not mean that these topics weren't useful to me. They taught me how to approach problems. They helped me develop learning strategies and study skills. They provided an intellectual framework for other ideas. They enhanced mathematical intuition. You could even make the case that some of those problems I would never see again were the most important and useful ones I studied.
This raises some big questions about when to teach detailed, non-intuitive algorithms. I put in the non-intuitive condition because some algorithms do illustrate broader points. For example, you can lay out a simple and intuitive algorithm for approximating square roots through a split-the-difference iterative process which ties in nicely to a number of important ideas and builds intuition at the same time.
But let's take something like synthetic division, an alternate algorithm for performing polynomial long division. Keep in mind the following:
1. It is very likely that you will never have the occasion to perform polynomial long division;
2. The traditional method (modeled after regular long division) is easy to remember, intuitive, and it helps illustrate the relationship between algebra and the more familiar arithmetic that preceded it.
3. The time savings from synthetic isn't that great.
That's an extreme case but it's not all different from more familiar examples like factoring trinomials, things you will probably never use outside of a high school math class. Most of these techniques do have value as part of a larger framework of ideas, but that value depends on having the students actually think about what they're doing. That's not what algorithms are good for.
This is one of the fundamental tensions of teaching. We can't wait for students to rediscover hundreds of years of mathematics (even if we hold their hands). Sometimes you just have to tell them "this is how you do it." The trouble is, when you tell someone an answer, you take away the chance for them to figure it out on their own.
It's been a few years since I read this but it made a good impression on me at the time and a few of the sections (on voting paradoxes, cryptography and topology, particularly the role it played in the side effects of thalidomide). I'm not sure how well the computer science section holds up, but it's definitely worth a look.
I don't know exactly what happened to Agon, but I'm pretty sure those damned orthogonalists had something to do with it. The game was all the rage in Victorian England where it was appreciated for its simple rules but surprisingly complex strategies. (the Victorians were also big fans of Lewis Carroll's Doublets, thus showing remarkably good taste in diversions.) The game was and is remarkably challenging and enjoyable but early in the Twentieth Century it faded away, perhaps due to the hegemony of orthogonal game boards.
You can find a complete set of rules on my game site. You can also buy boards there but obviously waiting for delivery would undercut the whole 'here's a game for this weekend' concept so I've included two JPEGs that you can print off if you can't find a suitable substitute (lots of games use a 6x6x6 hexboard so locating one shouldn't be that difficult).
The game is extraordinarily easy to learn. Each player starts out with six pawns and a queen spread out around the edge out the board.
A piece can either move around a concentric hexagon or go toward the center. The object is to get your pieces arranged like this (black wins):
'Capturing' is done in the style of many older games by placing two of your pieces on either side of the opponent's piece. I put the word in quotes because a captured piece is not removed from the board. Instead, it is moved back to the outer ring. Agon is therefore entirely a game of position. Novice chess players have a tendency to play for points and measure how well they're doing by how many of their opponent's pieces are lined up by the side of the board. Learning Agon can help break them of some bad habits.
I first came across Agon in David Parlett's Oxford History of Board Games -- an excellent resource if you're thinking about teaching a math class and not a bad read if you just enjoy games. Parlett is also a game designer of some note so he brings a lot of insight to the discussion.
When I was teaching high school math, I would often take my classes to the computer lab so the students could use Excel (or an open source alternative) to apply some of the concepts they were covering. Here's an example that would be appropriate for 8th grade* and up. It employs the following skills and concepts:
Area of a circle
Pi
Area of a square
Inscribed figures
The Pythagorean Theorem
Proportion
Symmetry
Basic algebra
Basic computing
Monte Carlo techniques
Start with the following
The area of the circle is pi-r-squared.
The area of the square is 4pi-squared.
The proportion of the square that's shaded is (area of the circle)/(area of the square)
Do a little algebra and you get p = pi/4 or pi = 4p
If you picked points in the square at random, the number in the circle divided by total number would converge on p
Since the figure is symmetric along the vertical and horizontal axes, the shaded part of a quadrant should also be p.
Now pick a radius. I used r = 3 here but make sure to mix it when assigning this project and use different radii (but not one -- you don't want a radius that equals its own square when presenting examples).
Have the students create x and y coordinates using
=rnd()*3
Then use a conditional based on the Pythagorean Theorem that takes the value 1 if the point is in the circle.
Your estimate of pi is four times the average of that field.
As with all Monte Carlo based lessons, have the students start with a small sample and move up until they start getting reasonable answwers.
I realize this may seem like a bit much but remember:
1. These spreadsheet skills (functions, conditions, random number generators) should already be familiar to the students.
2. Kids have a way of surprising you (and sometimes in the good sense)
* Some people out there are probably saying this is too advanced for 8th graders. You know your kids best but I would encourage you to give it a try. They might surprise you.
I'm pretty sure I'm going to be making this claim repeatedly so I might as well take a few minutes to put it down in a linkable form for future use.
Of all the subjects a student is likely to encounter after elementary school, mathematics is by far the easiest to teach yourself. With the appropriate attitude and assumptions, adequate motivation and a simple and easily mastered set of skills the majority of students can take themselves from pre-algebra through calculus.
What is it that makes math teachers so expendable? Part of the answer lies in mathematics position on the fact/process spectrum. Viewed in sufficiently general terms, all subjects start with giving the student a set of facts and ideally end with the student performing some process using those facts. In subjects like history and to a slightly lesser extent, science, most of the early stages of mastering the subject center around absorbing the facts. On the other end of the spectrum, subjects like music, writing and mathematics involve a relatively small set of facts*. Students studying these subjects tend to focus primarily on process almost from the beginning.
Put another way, at some point all disciplines require the transition from passive to active and that transition can be challenging. In courses like high school history and science, the emphasis on passively acquiring knowledge (yes, I realize that students write essays in history classes and apply formulas in science classes but that represents a relatively small portion of their time and, more importantly, the work those students do is fundamentally different from the day-to-day work done by historians and scientists). By comparison, junior high students playing in an orchestra, writing short stories or solving math problems are almost entirely focused on processes and those processes are essentially the same as those engaged in by professional musicians, writers and mathematicians.
Unlike music and writing, however, mathematics starts out as a convergent process. It doesn't stay that way. By the time a student gets to upper level math courses like real analysis or applied subjects like statistics ** there are any number of valid proofs for theorems and approaches to problems, but for anything before, say, differential equations, most math problems have only one solution and students are able to quickly and accurately check their work. Compare this to writing. There is no quick or accurate way to gauge the quality of a piece of prose or, worse yet, verse. Writers spend years refining their editing skills and even then they still generally seek out other critics to help them assess their own work.
This unique position of mathematics allows for any number of easy and effective self-study techniques. One of the simplest is the approach that got me through a linear algebra section taught by the worst college level instructor I have ever encountered (and that, my covers some territory).
All you need is a textbook and a few sheets of scratch paper. You cover everything below the paragraph you're reading with the sheet of paper. When you get to an example, leave the solution covered and try the problem. After you've finished check your work. If you got it right you continue working your way through the section. If you got it wrong, you have a few choices. If you feel you basically understood the solution and see where you made your mistake, you might simply want to go on; if you're not quite sure about some of the steps in the solution, you should probably go back to the beginning of the section; if you're really lost, you should go back to the preceding section and/or the previous sections that introduced the concepts you're having trouble with.
Once you've worked through all the examples, start on the odd numbered problems and check your answers as you go. If you're feeling confident, you can skip to the difficult problems but if you make a mistake or get stuck you should probably go back to number 1.
Don't get me wrong. I'm not saying this is the only technique, let alone the best, for teaching yourself mathematics. Nor am I suggesting that we make a practice of dumping student in sink-or-swim situations. I think we should provide students with the best teachers and support system possible, but even under those conditions, the internal resources needed to teach yourself mathematics are tremendously valuable to all students and are absolutely essential to anyone who has to use sophisticated analytical reasoning.
Tragic postscript: In what I can only assume is an idiotic attempt raise standards, most books have stopped giving answers to odd-numbered problems. Under the old system you would assign odd problems when you wanted the students to be able to check their work and even problems when you wanted to make sure they weren't just looking answers up in the back of the book. It was a simple, flexible, and effective system that encouraged students to be independent and resourceful. No wonder it was such a prime target for reform.
* I heard a story (possibly apocryphal) of a professor who walked into an upper level math class, wrote properties of real numbers on the board, told the class that was all they needed to know to prove all the theorems in the book, then walked out.
** The relationship between mathematics and statistics is particularly complex, far too complex to discuss in a blog post.
This one grew out of some time I spent recently helping a family first-grader with her subtraction homework. I made up some sheets with numbers and letters. My original thought was just to make the practice sheets more interesting by having the answer spell out a word, but I noticed my tutoree (who naturally has very good math genes) had, without prompting, started using the code key as a number line to figure out the answers.
The more I thought about it, the more I liked the idea of using this approach for a wide range of problems. For the very early grades, it teaches numbers and letters. It has great appeal for kids (particularly if presented with the right air of mystery -- I'd suggest a pirate motif). It reinforces the number line concept and the essential idea of looking things up in tables.
Here are some sample problems
For those learning to read, present it as a straightforward code:
PIRATES LOVE 7-15-12-4
For slightly more advanced students, replace the numbers with problems
PIRATES LOVE
10 - 3
10 + 5
6 + 6
6 - 2
For even more advanced students, make it a more explicit number line exercise
PIRATES LOVE
7
+8
-3
-8
You can even play with functions by using the old SAT trick of using a circle to represent x+1 and a triangle to represent x-1 (or any other functions you can think of).
triangle 8
circle 14
circle 11
triangle 5
I'd suggest using this technique frequently enough to keep the familiarity high. It also offers extensive opportunities for teaching across the curriculum.
It's important to have students think deeply about math in both structured and unstructured ways (I have a guilty feeling that I ought to say more about this, but that will have to wait for a future post). It's the unstructured part that tends to cause problems. That's one of the reasons I liked to make games part of my lessons when I was a teacher.
Games (at least the kind I recommend) require a great deal of focus -- you have to think about what you're doing or you won't do well -- and they encourage exploration and a playful attitude to the material. All of these things help build mathematical intuition.
1. Start with some dots on the paper. The more dots you have the longer the game takes so you will probably just want to start with two or three.
2. Players take turns either connecting two of the dots with lines or drawing a line that loops back and connects a dot with itself.
3. The lines can be straight or curved but they can’t cross themselves or any other lines.
4. Each dot can have at most three lines connecting it.
5. When you draw a line put a new dot in the middle.
6. The first player who can’t draw a line loses.
Particularly from someone who already has a backlog of unfinished posts at another blog. There is a method to the madness here. I've been doing most of my blogging for a site called West Coast Stat Views (formerly Observational Epidemiology) which features analytic takes on various topics including education.
Back in my salad days, I taught high school math and English and college math and statistics and I accumulated a stockpile math games, puzzles, tips, exercises and the like. Some of these fit in at the stat blog but many don't. Besides I wanted to get this material to teachers, parents and math buffs, some of whom wouldn't have much interest in most of the topics covered on the other blog.
This site also gives me a chance to try a different approach to blogging. Since few of these posts will be topical, I'll be making aggressive use of the scheduling option to make sure that there's a steady stream of posts. Some of them will be recycled, others may be a bit on the short side, but every week you'll see at least one or two new posts.
