Math education

November 15, 2009 7:28 AM

How should math be taught? The Kumon Math curriculum provides a simple and clear description of one possible sequence of skills. Hung-Hsi Wu decries the bogus dichotomy of basic skills versus conceptual understanding (PDF, Google Docs). David Klein provides a detailed history of US K-12 math education in the 20th century. The NYT describes the 2008 report of the National Mathematics Advisory Panel (full text as PDF).

NYC HOLD, a conservative-leaning math education reform group, has a large collection of links to articles on math education. For the progressive side, see Hyman Bass and Deborah Ball.

posted by russilwvong (71 comments total)

There appears to have been a flamewar between Hyman Bass and Sandra Stotsky, centering around a 1999 open letter to then-Secretary of Education Richard Riley. Deleted post on a recent Sandra Stotsky op-ed.
posted by russilwvong at 7:32 AM on November 15, 2009

Good topic, good post. Surely there's no one right answer to "How should math be taught?" But at the same time there are surely wrong answers.

It's worth noting that there's huge dissension on this subject among distinguished research mathematicians (a category that certainly includes Hyman Bass.) Here's something I just read in the notices of the AMS from the late Andrew Gleason, a professor in the Harvard math department:

"Right now there is debate apparently existing as to how mathematics should react to the existence of calculators and computers in the public schools. What should be the effect on the curriculum?...and so on. Now the unfortunate point of that is that there is even a very serious debate as to whether there should be an impact on the curriculum. That is what I regard as absolutely ridiculous. Let me just point out that... in this country there are probably 100,000 fifth grade children right now learning to do long division problems. In that 100,000 you will find very few who are not thoroughly aware that for a very small sum of money (like $10) they can buy a calculator which can do the problems better than they can ever hope to do them. It’s not just a question of doing them just a little better. They do them faster, better, more accurately than any human being can ever expect to do them and this is not lost on those fifth graders. It is an insult to their intelligence to tell them that they should be spending their time doing this. We are demonstrating that we do not respect them when we ask them to do this. We can only expect that they will not respect us when we do that."
posted by escabeche at 8:05 AM on November 15, 2009 [7 favorites]

It's not about the way it is taught, it's just about the quality of the teachers. A repeat, from the deleted thread, earlier today:

The Royal Dutch Academy of Science looked into this, because there is a lot of criticism in the Netherlands on a popular math teaching method as well.

Turns out, it makes no difference in what way children get their math education, but it makes a lot of difference who teaches them. A lot of teachers lack basic qualities:

3. The key to improving mathematical proficiency lies in the teacher’s competences. Teacher training and post-graduate courses have been seriously undermined.
The Ministry of Education, Culture and Science should subject teaching training programmes to a thorough investigation and encourage post-graduate training in mathematics and mathematics teaching.

source: the summary on page 11 in this Dutch pdf.
posted by ijsbrand at 8:12 AM on November 15, 2009 [1 favorite]

I'm about 10% through the fascinating Wu article and I'm on his side completely - I'm actually rather shocked that many of these ideas were ever allowed to be implemented.

I'm all for discovery and creativity, but we're in the twenty-first century - we have literally millennia of discoveries in the past. It is simply impossible for even the most talented student to rediscover all of them; and for underperforming students, it's almost impossible to turn the random creativity of discovery into the discipline of mathematics.

Something like the Kumon system seems much more effective. When it comes down to it, you have to learn things like your times tables by rote. Please do not attempt to tell me that calculators render your times tables obsolete - you have to understand numbers to function in this world as an adult, and the only way to understand them is to watch them operate over and over.

One of the things that these modern educators never seemed to understand is that rote learning can be a lot of fun for kids if you do it right. I learned my times tables from a lovely record with songs and rhyme games - first they'd do the tables with the results, and then repeat it with sound effects in the place of the number patterns (in the shape of the number word).

Kids learn amazingly complicated songs that way without any help from adults. And it's really good for you.

It's exactly like learning scales on a musical instrument. Creativity is all-important here. If you can communicate with your audience, technical skill is far less crucial.

Yet, almost certainly you have to have your scales mastered. To be a fine instrumentalist, you should be able to play all your scales perfectly, even if you don't ever use them - in the same way that a visual artist should be able to draw representationally well, even if they never make representational images.
posted by lupus_yonderboy at 8:20 AM on November 15, 2009 [7 favorites]

If it doesn't include Cuisenaire rods - the handy maths tool/projectile - I'm not interested.
posted by scruss at 8:52 AM on November 15, 2009 [4 favorites]

Very gracious of you to have added a link to that badly-done post in this corrective one. Well done.
posted by jeffen at 8:53 AM on November 15, 2009 [1 favorite]

Please do not attempt to tell me that calculators render your times tables obsolete

By the way, I'm sure Andy Gleason didn't think this, nor do I. What he says is that calculators render long division obsolete, which is a very different matter.
posted by escabeche at 8:58 AM on November 15, 2009

There was a recent discussion on boingboing about his 1st grader's math homework. I thought I'd throw that into the mix.

http://www.boingboing.net/2009/11/12/do-you-understand-my.html
posted by gelos at 9:04 AM on November 15, 2009

http://www.boingboing.net/2009/11/12/do-you-understand-my.html

Wow, see this is a very very very very very good example of integrating algorithms with pictorals, and of breaking down complex algorithms into simpler ones, and yet you still get ignorant people like in the comments at BB who hold problems up like this as something that is wrong with math. For the love of pete, if you have a 6-year-old, this isn't some Kumbayah and crystal-healing bullshit, this is just laying the groundwork for like introducing the idea of "carrying" into the standard addition algorithm that most people in America use.
posted by 23skidoo at 9:15 AM on November 15, 2009 [3 favorites]

That boingboing this took about five seconds to figure out...it's just a visual way to do overflow adding.
posted by notsnot at 9:29 AM on November 15, 2009 [1 favorite]

I say that music education could learn much from math education. Start teaching all the scales in every key in elementary school. By entry highschool every student should know every scale and mode by heart on the basic instrument families. In highschool, they would learn all the rules of counterpoint and harmony, and the advanced students maybe can get an introduction to conducting an orchestra. In college, if they are particularly adept at playing scales and compositional theory, they can all learn to play some actual music. As graduate students in music they may even learn to improvise or write music, once they have all the basics mastered.
posted by idiopath at 10:05 AM on November 15, 2009

I think the focus on better teachers is the single greatest flaw in education. Any normal profession focuses upon getting more with the personnel they have, or even doing more with lesser personnel. You'll never even see medical researchers complain about how their new technology requires better doctors.

We have this fantasy that no expense must be spared educating the children, but this is demonstrably false. A talented person usually avoids teaching high school. How many people get PhDs in literature? How many academic jobs are there? Where are the extras going? Quite clearly not high school teaching. It's far far worse in mathematics.

Education research should focus upon doing more with existing teachers, not fantasy.
posted by jeffburdges at 10:05 AM on November 15, 2009 [2 favorites]

I just wanted to say there's a Kumon tutoring center near me, and their logo puzzles the heck out of me. Shouldn't the face in the "O," who I'm assuming symbolizes my prospective child I would send there, be smiling? Or at least not looking as disturbed and freaked out as it's possible for a face with two dots and one line to look?
posted by drjimmy11 at 10:23 AM on November 15, 2009 [4 favorites]

It is an insult to their intelligence to tell them that they should be spending their time doing this

I disagree. Long division of integers may indeed be a useless skill, but once you'd doing long division of polynomials or other things in the context of calculus or algebra the basic mechanics learned doing division with integers still apply. I hate people who think that manipulation of integers is somehow the be-all and end-all of mathematics.
posted by GuyZero at 10:24 AM on November 15, 2009 [5 favorites]

I should have mentioned that Bas Braams appears to be one of the main contributors to NYC HOLD--I found a number of the links here through his web page.

jeffen: Very gracious of you to have added a link to that badly-done post in this corrective one.

Thanks, I thought there were some great comments in that post (e.g. twoleftfeet, billysumday) and didn't want them to get lost.

I liked the Kumon link because it's so clear. (In comparison, here's the math curriculum here in British Columbia, for kindergarten through Grade 7.) In Stevenson and Stigler's The Learning Gap (which I'd definitely recommend to people interested in education), they describe East Asian teachers as being focused on clarity: the master teachers are the ones who have learned the clearest way to convey particular lessons, and they work with their colleagues to pass on that information. (Similar to the way the Hung-Hsi Wu article discusses how to teach particular algorithms.)

On the subject of long division, I'm a bit dubious that we no longer need to teach Grade 5 students long division because they have calculators. Once they get into polynomials (around Grade 11 or so), they wouldn't be able to understand this kind of material. They don't need to be particularly fast at long division, but they should be practiced enough to be fluent, so that they're not trying to understand long division and polynomials at the same time. And is long division really so hard to learn?

Besides, I'd hope that students would have some understanding of what the calculator is doing, that it's not just a black box performing magic.
posted by russilwvong at 10:24 AM on November 15, 2009 [2 favorites]

Sorry, I should say I hate people who propose simplistic theories of math pedagogy based solely on the manipulation of integers. Or I hate their theories at least.
posted by GuyZero at 10:24 AM on November 15, 2009

Education research should focus upon doing more with existing teachers, not fantasy.

I hear you, but I don't think education deserves to get singled out here. Every company I've ever heard of talks about the need to attract the best people. It's definitely not just an education thing.

You'll never even see medical researchers complain about how their new technology requires better doctors.

Being a doctor is already a highly sought after and competitive position. There are problems in medicine, but I don't think finding talent is an issue.

Education focuses on human resources so much because a great teacher can help students reach their potential, but a poor teacher will tend to fail students, no matter how much well-researched teaching theory you throw at the problem. If you've got bad teachers in the classroom, it really doesn't matter how many new textbooks you print or classroom computers you purchase.
posted by anifinder at 10:35 AM on November 15, 2009 [2 favorites]

Holy shit now I know how to divide polynomials.
posted by rlk at 10:58 AM on November 15, 2009 [4 favorites]

When my son started first grade this year, he was already learning multiplication and fractions at home. Now he's coming home with those ridiculous worksheets like the one in the boing boing link, and his school is ACTIVELY discouraging him from doing real math. They're teaching him to count on his fingers. How ridiculous is that?

I asked, at a teach conference, about getting him some material that was more on his level, and they said, and I quote "Well, we don't want to encourage him to get too far ahead of the kids that are still learning to count." What, what, WHAT?

I wish to god there were an affordable private secular school anywhere near me. Because my son is going backwards in math and science knowledge, and it doesn't seem fair to him to make him do "actual" learning after he's been in school all day. Especially since he comes home with "busy work" homework every day like coloring. Coloring which then gets graded. Ridiculous. Because gods forbid they learn something like grammar or subtraction when they can be coloring in corporate mascots instead.

As experiments with the public school system go, so far, I'm not very happy with the system at all. If it weren't for the social aspect, which he really loves, I'd go back to homeschooling in a heartbeat.
posted by dejah420 at 11:44 AM on November 15, 2009

dejah420: "I asked, at a teach conference, about getting him some material that was more on his level, and they said, and I quote "Well, we don't want to encourage him to get too far ahead of the kids that are still learning to count." What, what, WHAT?"

This is a real danger for your son's math education. I was so extremely bored by my math classes for years that I got by acing the tests and doing none of the homework and passing with a C. By the time the rest of my class was at the level that was challenging me I had pretty much forgotten how to learn math because I had been coasting and waiting for the classes to catch up to me for so many years.
posted by idiopath at 11:50 AM on November 15, 2009 [4 favorites]

I am teaching sixth-grade math right now. Even for someone with an undergraduate degree in mathematics, this debate isn't easy to sort out and I have been thinking about it a lot.

On the one hand, there is constructivist pedagogy, which emphasizes inquiry, discovery, and understanding why things work the way they do. As a child I naturally learned this way - even if taught an algorithm by rote I would try to understand its inner workings. I think I turned out to be pretty good at math. More to the point, I want my students to see math not as a random set of rules but as something that makes sense logically.

On the other hand, my students come to me with little in the way of foundational understanding or number sense (this is in a low-income neighborhood with some pretty atrocious schools feeding into mine). Teaching them to construct understanding from scratch would sometimes be too long and arduous for the time we have. Sometimes reasons why something works overload their brains when they are just trying to grasp a basic concept. They will often be more successful (and feel better about what they are doing) if I give them a clear algorithm and time to practice it.

So my teaching is walking a fine line. Sometimes I tell them why something works, sometimes I ask them to figure it out though discovery, sometimes we just do a set of steps and it gets to the answer because I say so. In an ideal world it wouldn't be that way but so it goes.

On the topic of long division - I don't think it's a waste of time to teach children to divide large numbers. I think it's dangerous to let calculators have a monopoly on knowledge. Long division may or may not be the best algorithm for division - on the one hand it is really unintuitive, on the other hand other algorithms (like partial quotient) don't work well with decimals.
posted by mai at 11:57 AM on November 15, 2009 [9 favorites]

What is the purpose of mathematics education? Is it practical skills? Is it how to think logically? Is it what to do if your calculator breaks? Is it aesthetic?

Does everyone need to learn the same exact stuff? Was it a waste that I learned the square root algorithm as a kid? How about if I tell you I no longer remember it? Is it a waste to learn logarithms for calculation? Is it a waste to learn how to use a slide rule?
posted by Obscure Reference at 12:13 PM on November 15, 2009 [1 favorite]

maybe this is broadening out the discussion a bit too much, but i thought this was somehow encouraging: duncan, gingrich and sharpton on meet the press today talking about educational reform and the race to the top fund.

btw, in a clip, the head of sidwell friends mentions finland and singapore.

also fwiw...
Russia's Conquering Zeros: The strength of post-Soviet math stems from decades of lonely productivity

oh and logicomix :P

cheers!
posted by kliuless at 12:24 PM on November 15, 2009

I work part time as a Kumon tutor and I'd be happy to answer any questions people have about the method or my center in general.
posted by kylej at 12:53 PM on November 15, 2009

My kids aren't in school yet and I take heart in the fact that grades don't really matter until high school.
If my kids get sent home with stupid busywork (who grades coloring? My wife (an MD), is still irritated she got a bad grade because she used Magenta instead of Red because she had the larger crayon set),
I'm going to send it back to the teacher with comments.
Kids need to be kids and just because they are playing doesn't mean they aren't learning.

This may be why my wife says I won't be allowed to talk to the teachers. :)
Fortunately the elementary school done the street is supposed to be pretty good.

If you read the comments way down in that boingboing link I posted, the co-author of "the case against homework" weighs in on the discussion.
posted by gelos at 12:58 PM on November 15, 2009

I'm all for completely integrating teaching computing and mathematics. We approach math in such a way that it appears completely useless from the get-go, and once students are faced with trigonometry and algebra they completely hate what they perceive (in many cases, rightly) as a waste of time. Instead we should teach math to the extent that it is relevant.

Start by teaching kids what a computer is and how it works. Integrate programming into lessons, teach kids how to write programs to do their homework for them. All the math will follow naturally.
posted by mek at 1:26 PM on November 15, 2009

I want to share a recent discovery that I, a life-long math-a-phobe, and my child (who I don't want to follow in my footsteps) have made. It is a series called Life Of Fred. It is completely unique in that it starts after multiplication and division skills and uses a story to teach math concepts. Fred has to use math to solve everyday puzzles and live his life as a gifted 5-yr. old, teaching Math at Kittens University. My daughter is enthralled and loves to engage in math now that she has Fred. His series currently goes through College Statistics and there are more on the way. They are wonderful as a full curriculum or to just have a fun way to cement the facts. And he throws in all sorts of neat info and twists that keep the child wanting more. I'm running out the door or I would link some of the wonderful reviews that people have given Fred. A quick google will find them if you are interested. I'm very excited to have found a way to make math more accessible to my very verbal daughter. She loves Fred and doesn't yet realize how much she is learning and how quick her math brain is becoming.
posted by pearlybob at 1:30 PM on November 15, 2009 [7 favorites]

I think the focus on better teachers is the single greatest flaw in education. Any normal profession focuses upon getting more with the personnel they have, or even doing more with lesser personnel. You'll never even see medical researchers complain about how their new technology requires better doctors.

We have this fantasy that no expense must be spared educating the children, but this is demonstrably false. A talented person usually avoids teaching high school. How many people get PhDs in literature? How many academic jobs are there? Where are the extras going? Quite clearly not high school teaching. It's far far worse in mathematics.

Successful businesses recognize a need to get the best possible people they can. For some jobs, the best possible person is pretty easy to find—it's the one who shows up consistently, on time, and hustles until their shift is over. It's relatively easier to find someone like that, and their wage reflects that. I work for a software development company, and software development being a relatively intellectual process, it is VERY difficult to find qualified people. Unfortunately, the company I work for is unable to afford to pay as highly, so highly qualified candidates tend to go elsewhere.

Teacher pay is godawful. If you had a degree in mathematics, and you could either get a job for $50,000 at an engineering or finance company, or get two jobs to scrape together a decent income while teaching, you would only choose the teaching job if you really liked teaching. This means that teachers are effectively self-selected not based upon their qualifications, but upon their love of children and teaching. This is not the way to get the best-qualified applicants to go into teaching. Love of children and teaching is an important asset, but it is not a measure of one's ability to teach mathematics.

What is the purpose of mathematics education? Is it practical skills? Is it how to think logically? Is it what to do if your calculator breaks? Is it aesthetic?

It is my opinion that math skills are incredibly important to the future of the United States. With manufacturing moving overseas, the major thing of real value we can produce here is science and engineering. How many people do you know who like engineering or science, but who didn't go into it, or switched out of it because they "couldn't handle the math". This starts in grade school. By failing so badly at teaching math, we're not only depriving these students of the ability to pursue their goals, we're hurting America's future by making it less likely that quality engineering and science will happen here.
posted by !Jim at 1:46 PM on November 15, 2009 [1 favorite]

I have to say, I thought that exercise in the BB link was pretty ridiculous. I always aced math all the way through high school, even taking college math classes my senior year, and I couldn't figure out what it wanted. Maybe its because the phrase "Making Ten" didn't mean anything to me, or maybe its because there wasn't an = between 2 sets of problems. I guess it just seems counter-intuitive to me, because I was always comfortable with math as an abstract language and didn't need to turn everything into concretes.
posted by Saxon Kane at 1:50 PM on November 15, 2009

Just read the Wu article, which I thought was very good -- and I expect most people in math education, whether "pro-reform" or "pro-basics," would find little to disagree with there. People who take teachers' unions to be the root of all bad pedagogy might note that Wu's piece was published by the American Federation of Teachers in their journal, American Educator. The current issue is devoted to elementary school math, and features another article by Wu.
posted by escabeche at 2:47 PM on November 15, 2009 [1 favorite]

Something like the Kumon system seems much more effective. When it comes down to it, you have to learn things like your times tables by rote. Please do not attempt to tell me that calculators render your times tables obsolete - you have to understand numbers to function in this world as an adult, and the only way to understand them is to watch them operate over and over.

Can you give me an example of something an "Adult" needs to do that requires the use of a times table? I'm pretty sure I can multiply any pair of 1 digit numbers in my head, and usually I can do most two digit numbers in my head as well, but I don't have a "memorized" copy of the times table in my head. I would argue that knowing a times table isn't really "knowing" numbers any more then memorizing the Latin names of plants means "knowing" plants. For one thing, the times table only applies to the arbitrarily chose base 10 numbering system. There's nothing magic about the number 10 that makes it easy to multiply things by, it's just the fact that 10 is the same number as the base of the number system. If we were using base 16, multiplying things by 16 would seem easy, and multiplying things by 10 would seem hard.

What kids should be doing (I think) memorizing the prime factorization of numbers and memorizing the multiples of some numbers of primes. That would probably be helpful for being able to do math in your head (although I don't know for sure)

The point is, lots of people take the arbitrary set of things taught in elementary school and declare them the "fundamentals" of math when they are not. Obviously there are lots of different sets of fundamentals that people can learn in order to learn what they need to go on to algebra, calculus, combinatorics, classical CS, whatever.

Anyway, the point is while you may rely on your times table a lot, it's perfectly possible to do math without knowing it. It's not very hard to figure out what the answers to questions like 6*3 by adding 6 to 12 rather then having 18 pop into your head right away. In fact, doing it the long way probably requires more understanding of numbers, IMO.

Wow, see this is a very very very very very good example of integrating algorithms with pictorals, and of breaking down complex algorithms into simpler ones, and yet you still get ignorant people like in the comments at BB who hold problems up like this as something that is wrong with math.

I think the problem is that the instructions are not very clear at all. What does the arrow mean? How are you supposed to use the boxes? A paragraph or so would have made it more clear what you are supposed to do, and while a first grader probably can't read that well it would have been helpful for the parents.

I asked, at a teach conference, about getting him some material that was more on his level, and they said, and I quote "Well, we don't want to encourage him to get too far ahead of the kids that are still learning to count." What, what, WHAT?

That's insane. It's like your school is actively making your kid dumber then he would ordinarily be. If I were you I would try to keep teaching him more advanced math at home.
posted by delmoi at 3:37 PM on November 15, 2009

The teacher-pay argument is slightly misleading. I *am* making $50,000 teaching. I have a master's degree and a degree in mathematics from a pretty darn good university, I could be making twice that elsewhere.

It's not that the pay is inadequate such that I would need to take a second job to make ends meet. It's that it's:

a) not equal to what the best candidates could make elsewhere
b) not equal to what people working similar hours (60-65 hours/week) with similar educational levels would make
c) Not enough to raise a family on comfortably, and therefore other jobs start to look more attractive.

In fact, many bright people come to teaching and leave within five years, before they ever become experts. This is partly because of pay but also the working conditions they experience and the fact that for beginning teachers, the level of responsibility they have is not really commensurate with their abilities - it's often a sink-or-swim experience.

Better teachers are not born, they are not hired. They grow from experience and are shaped by the culture of their school.
posted by mai at 3:59 PM on November 15, 2009 [2 favorites]

I should add that in many Asian countries with very high math scores (China, Singapore), the most important quality they look for in potential teachers in enthusiasm, and the curriculum is completely standardized (and, I might add, way better and deeper and more thoughtful and challenging than most of what we use here).
posted by mai at 4:01 PM on November 15, 2009

Start by teaching kids what a computer is and how it works. Integrate programming into lessons, teach kids how to write programs to do their homework for them. All the math will follow naturally.

An interesting idea. Based on my own experience as a math teacher I'd have to say "Ha!!", or "Naturally my ass!!". I'm not saying it's impossible, just impractical.

It's all about numeracy. We need to make our kids comfortable with the language of numbers, symbols and equations. They need to truly understand how numbers relate to each other, and the only way they'll get there is by memorizing their damn times tables and sweating through long division. All this "but they have calculators" is nonsense. It's like saying that we don't need to teach kids how to spell words and construct proper sentences because they'll just do everything on a computer or a cell phone, and can use spellcheck and short forms (i.e. lol, lmao, etc.). We need to take the fear out of math, and we'll never do that until we fully embrace getting down and dirty with numbers and cranking out calculations the old school way. We're trying to teach our children to be logical, rational problem solvers, and mathematical thinking is key to this effort.
posted by Go Banana at 4:13 PM on November 15, 2009 [3 favorites]

The Boing Boing link asks: "Making Ten: Draw counters to solve. Write in the missing numbers."
The trouble is they do not state the problem. What is the question?
They are not "making ten".
There is no visual representation of the question. The shapes mean nothing.

It reminds me of being in Grade 3 and them trying to teach me how to count. I kept looking at what we were doing, assuming we were going to doing something about summing of series. I remember the humiliating memory of realizing that I had been put in a special group because they thought I couldn't count TO TEN.

Most of my elementary education was filling out sheets of paper titled with the word "solve". Usually they involved bizarre assignments like this. It went on for years.
posted by niccolo at 4:19 PM on November 15, 2009

I'm a big fan of experiential learning and my kid goes to a "free school" that we love where there is little in the way of formal instruction and almost no rote memorization. Still, I wish I'd had a better math foundation and because of that I'm augmenting my kid's math education through rote memorization at home. I try to make that memorization process fun, because the reason I never learned mine in the first place was that I had a mean spirited teacher who made the memorization process miserable and humiliating.

Periodically I've run into situations where things would have been easier for me if I'd learned a little more basic math early on: figuring out what percentage of a whole something is, calculating tips, etc. When I decided to go to nursing school a few years ago, I had to take the algebra class that as a high school dropout I'd managed to avoid the first time around. I sure as hell wished I'd known my times tables by heart for that class. Once I got into a clinical setting, having better numeracy would have really helped me in calculating drug dosages - a real and important task.

In general, when you look at elementary education, I think you can find many, many examples of lessons that will never be revisited and have questionable if any long-term value. A lot of people think there's a value to just learning how to sit still and do what you're told: I don't. But in the case of math, I think a solid foundation is pretty helpful long-term, if taught in an encouraging and fun way. And there's no reason you can't have a balance of rote memorization and more conceptual learning.
posted by serazin at 4:40 PM on November 15, 2009

One thing that made a huge difference for my siblings and I, that we were very lucky to have, were video games by The Learning Company. We were completely addicted to the stuff. I wonder who is making those sorts of interactive education games now.
posted by mek at 4:56 PM on November 15, 2009

I think the problem is that the instructions are not very clear at all. What does the arrow mean? How are you supposed to use the boxes? A paragraph or so would have made it more clear what you are supposed to do, and while a first grader probably can't read that well it would have been helpful for the parents.

A paragraph for the parents would have totally eaten up like half the page. I think that people are so afraid of math (and they're so afraid of math because they're afraid of being wrong) that they won't even take the time to look at something and figure it out. These types of problems are totally figureoutable for adults, even with NO instructions, if they look at the pictures and see how they relate to the numbers that are on the page.
posted by 23skidoo at 5:20 PM on November 15, 2009

The reason children learn multiplication tables and long division isn't so they can replicate the behavior of a cheap calculator, it's to train the mind to focus on a sequence of symbolic manipulations, a skill that can be applied to many other contexts. This kind of mental training might be possible in other ways, but arithmetic has the advantage of allowing feedback around the correct answer.
posted by twoleftfeet at 5:34 PM on November 15, 2009

original texts, seminar style. That's how.
posted by MNDZ at 5:52 PM on November 15, 2009

The reason children learn multiplication tables and long division isn't so they can replicate the behavior of a cheap calculator, it's to train the mind to focus on a sequence of symbolic manipulations, a skill that can be applied to many other contexts. This kind of mental training might be possible in other ways, but arithmetic has the advantage of allowing feedback around the correct answer.

And in the process, make them "hate math". Sorry, that's pure conjecture. First of all you're making the mistake of assuming that people are doing what they do for a good reason, and not simply following tradition, traditions that were started before computers were widely available and people really did need to do lots and lots of math by hand.
posted by delmoi at 8:55 PM on November 15, 2009

The answer to (really not very good, generic question) "how should math be taught" is "in whatever way that is most interesting and satisfying for the most students."

Once you create the impression that math is hard instead of fun, a chore instead of an exploration, there is *no* way to teach math.

That's the important part of the answer. The rest is up to the creative teacher.
posted by Twang at 10:41 PM on November 15, 2009

Can you give me an example of something an "Adult" needs to do that requires the use of a times table?

No. In fact, during the brief time I was teaching Math to high school students, I realized that the most honest answer to the ubiquitous "when are we going to use this" question was "never." I've come to believe that most adults can live out their entire lives comfortably and effectively without more than a mastery of mathematics that a good 2nd/3rd grade student achieves. If you know what the basic operations of arithmetic are and have a vague idea corresponding to the concept of a variable (simple blanks or boxes will do), you can work around the rest of the gaps in your knowledge with modern tools, some social skills, and maybe a few simple heuristics. Unless you choose a technical field, the chances that you will need any given bit of math knowledge are pretty slim. And the ROI of many technical fields is a bit iffy, really.

So, I'll see you your skepticism about the utility of multiplication and raise you a general skepticism about the individual value of math education in general. However, this may not be the best metric for the value of a given concept or practice.

I'm pretty sure I can multiply any pair of 1 digit numbers in my head, and usually I can do most two digit numbers in my head as well, but I don't have a "memorized" copy of the times table in my head

Can you explain the process you believe you go through? It seems really unlikely to me that you don't do any lookups, and the fewer you use, the more nuanced understanding of arithmetic and its associated algebra you need.

you're making the mistake of assuming that people are doing what they do for a good reason, and not simply following tradition, traditions that were started before computers were widely available and people really did need to do lots and lots of math by hand.

I'm not sure it matters whether a teacher's reasons are tradition, or a belief in focusing on symbolic manipulations, or because aliens made them with their mind control rays. It either helps students develop certain skills or it doesn't.

OK, it also does matter if it makes them hate the subject, too. I think the psychology of the educational experience is worth balancing with the benefits of a method of practice or discovery, and that enthusiasm for a topic really does matter. But a certain amount of by-hand work, even when it can be tedious, even when it's not the way most practitioners would do things all the time, has a lot of value in bringing you to grips with certain concepts, and someone who can't accept some degree of it is going to be handicapped as a student.
posted by weston at 2:00 AM on November 16, 2009

many bright people come to teaching and leave within five years, before they ever become experts. This is partly because of pay but also the working conditions they experience and the fact that for beginning teachers, the level of responsibility they have is not really commensurate with their abilities - it's often a sink-or-swim experience.

A-men.

I think most people consistently underestimate the problem of the school as a workplace for teachers when talking about the problems of attracting and keeping good teachers. People know when they choose teaching as a profession they're likely choosing a middle class existence at best, they know there are more economically rewarding options out there, and a lot of them pick it anyway as a calling. But there is remarkably little discussion of what it is that often drives people so motivated to abandon ship.
posted by weston at 2:22 AM on November 16, 2009

Can you explain the process you believe you go through? It seems really unlikely to me that you don't do any lookups, and the fewer you use, the more nuanced understanding of arithmetic and its associated algebra you need.

Well, I didn't say I didn't do any lookups. For example, someone mentioned 7*4 in this thread or another one. I actually did happen to know the answer to that, but I could have just doubled 14, for example. If I don't know the exact answer right away I can multiply some other pair of numbers and then add or subtract. So for example if I had 7*8 I could do 8*8-7 = 64 - 7 = 60-3 = 57 and so on. Now I suppose if I had to multiply two digit numbers knowing the single-digit multiplication table would be helpful, but honestly while I can multiply most two digit numbers without a calculator (or paper) if I'm ever in a situation where I can't figure it out, well, I'll just live without knowing. It's usually not important. If I have a calculator (either hardware or software) and I'm working on some specific problem, I probably would have used it before trying.
posted by delmoi at 3:11 AM on November 16, 2009

Also I didn't say I had a more nuanced understanding of 'algebra', I said I had a deeper understanding of 'the numbers', specifically how they are composed from other numbers by addition and multiplication.
posted by delmoi at 3:39 AM on November 16, 2009

...if I had 7*8 I could do 8*8-7 = 64 - 7 = 60-3 = 57 and so on...


so close.
posted by logicpunk at 4:58 AM on November 16, 2009 [5 favorites]

delmoi, I recently read a book about math education, which I can't give the title of this minute because my booklist is on another computer, but it suggested that students who can do the kind of manipulations you do--rather than having every multiplication fact memorized--actually have a better skillset for dealing with more complex math.

I know that I only stopped feeling hampered by my failure to have the 7s and 8s times tables memorized when I started to realize things like 8*7=8*5+8*2, or to do the kind of chunking into tens that the kid's homework referenced up-thread was practicing.

I'll drop the title of the book into the thread later, if I remember. I found it very interesting.
posted by not that girl at 8:11 AM on November 16, 2009

Just because a kid has memorized all the answers to single digit multiplication problems doesn't mean that she can't also understand (and excel at) using various properties of mathematics for working with more complex problems.
posted by 23skidoo at 8:26 AM on November 16, 2009

The book I am thinking of is What's Math Got to Do With It? Helping Children Learn to Love Their Most Hated Subject--and Why It's Important for America, by Jo Boaler (Amazon Link). As a homeschooling mom, I am interested in questions of what to teach and how, and found this book interesting from a theoretical perspective, less useful from a practical one.
posted by not that girl at 8:47 AM on November 16, 2009

Just because a kid has memorized all the answers to single digit multiplication problems doesn't mean that she can't also understand (and excel at) using various properties of mathematics for working with more complex problems.

I don't think anyone said it did. The book I mentioned just talked about the kind of thing delmoi is doing as a useful strategy, and suggested that kids who do not have access to those kinds of strategies have more trouble approaching higher level math; that quick summary doesn't do justice the argument and isn't meant to; I just thought delmoi might find the notion interesting.
posted by not that girl at 8:49 AM on November 16, 2009

not that girl, that's exactly what a lot of "new math" curricula are trying to impart -- but it's so foreign to the parents, who cannot make the leaps from what they know in order to help their kids, that it's foundering.

(And for the record, yes you do need to know how to do long division, people: how else can you compare unit costs in the grocery aisle, or figure out simple banking math?!)
posted by wenestvedt at 10:10 AM on November 16, 2009

Another user of the "delmoi method" here. For example, I never multiply anything by 5, I just multiply by 10 then divide by 2.

And this is exactly the kind of heuristic that pays off if you are trying to solve a novel problem in programming - finding the most efficient composition of the tools available to match the task at hand.

wenestvedt: if it needs to be exact you are better off with a calculator, and if an estimate is OK a newtonian method of estimating, checking the estimate by multiplying and then trying a revised estimate is much faster than long division is, and much easier to do in your head.
posted by idiopath at 10:31 AM on November 16, 2009

And for the record, yes you do need to know how to do long division, people: how else can you compare unit costs in the grocery aisle

Ummm, I mostly use multiplication and addition of rounded numbers. Long division is a pain in the ass to do in your head, but multiplication is a lot easier. Like, 750 grams of crackers cost $1.39. 1000 grams of crackers cost $1.89. Which is has the lower unit cost?

750 x 2 = 1500
1.39 is about 1.40
1.40 x 2 = 2.80
For the smaller box, 1500 grams cost about $2.80.

1000 x 1.5 = 1500
1.89 is about 1.90
1.90 x .5 = 100 x .5 + 90 x .5 = .50 + .45 = .95
1.90 x 1.5 = 1.90 x 1 + 1.90 x .5 = 1.90 + .95 = 2.00 - .10 + 1.00 - .5 = 3.00 - .15 = 2.85
For the bigger box, 1500 grams cost about $2.85.

The smaller box has the lower unit cost.
posted by 23skidoo at 10:43 AM on November 16, 2009

Well, I didn't say I didn't do any lookups.

OK, I'm a lot more sanguine about that. It isn't impossible to get along without any of them, but I think it's hard to be efficient, and I suspect without at least a partially memorized times-table, a lot of arithmetic becomes more tedious, not less.

...if I had 7*8 I could do 8*8-7 = 64 - 7 = 60-3 = 57 and so on...

Maybe lookups have advantages. :)

(8 - 1) * 8 = 64 - 8 = 56

or

7 * (7 + 1) = 49 + 7 = 56

I'm being a bit pedantic -- in general I agree it's probably more useful to know your squares plus a good number composition/algebra sense.

I didn't say I had a more nuanced understanding of 'algebra', I said I had a deeper understanding of 'the numbers', specifically how they are composed from other numbers by addition and multiplication.

I think we're talking about the same thing. Maybe your terminology is better.

I recently read a book about math education, which I can't give the title of this minute because my booklist is on another computer, but it suggested that students who can do the kind of manipulations you do--rather than having every multiplication fact memorized--actually have a better skillset for dealing with more complex math.

I buy this, and it's in line with some of what I was taught in Math Education courses. But I think the way that you get there is by doing some significant practice without a calculator on hand.

I'm pretty sure this is how I got there. I had a linear algebra teacher who told us not to use calculators on homework and wouldn't let us use them on tests. This sometimes seemed cruel (particularly, say, on determinants of 4x4 matrices without numbers contrived to be nice), but partway through I noticed that my general number sense was getting much, much better.

Of course, by the time someone is taking a linear algebra course, it's possible they're a bit more motivated about the subject than your typical fifth grader who might be assigned problem sets containing 100 long division problems. If we're talking about drilling children for reflexive accuracy, I can agree with Gleason and others as critics, and maybe we could do better things with that time. But if we're talking about eliminating the teaching of long division or generally cutting out anything a calculator can do from the curriculum, I'm a lot more skeptical. Practice doing calculation isn't just about learning do to calculation.
posted by weston at 10:54 AM on November 16, 2009

Can you give me an example of something an "Adult" needs to do that requires the use of a times table?

No. In fact, during the brief time I was teaching Math to high school students, I realized that the most honest answer to the ubiquitous "when are we going to use this" question was "never."

What. Are you kidding me? I was at a bar this weekend. I enjoy beer, yet I am also poor, consequently, I set a budget to limit my bar spending. I used my multiplication tables to calculate the cost of buying beers. A beer was $4. I wanted to spend no more than $15. Since I know my multiplication tables, I knew that 3 beers would cost $12, leaving me 15 - 12 = 3 bucks left over to tip the bartender.

Budgeting is a thing that "Adults" do that uses the multiplication table.
posted by taliaferro at 11:24 AM on November 16, 2009

(Yes, I realize I made a mistake up there. I subtracted 7 when I should have subtracted an 8. But the point I was trying to make was that for most day to day activities it works well and usually gets the right answer, or an answer that's close enough. For anything important, it will probably be done in a spreadsheet or some custom software. Of course people need to understand the concepts of multiplication and division, but we're teaching kids the math they would need to be an office worker in the 1960s. It's a huge waste of time, and we'd be better off teaching algebra much earlier, IMO.)
posted by delmoi at 11:45 AM on November 16, 2009

What. Are you kidding me? I was at a bar this weekend. I enjoy beer, yet I am also poor, consequently, I set a budget to limit my bar spending. I used my multiplication tables to calculate the cost of buying beers. A beer was $4. I wanted to spend no more than $15. Since I know my multiplication tables, I knew that 3 beers would cost $12, leaving me 15 - 12 = 3 bucks left over to tip the bartender.

Uh, Except I can do that without memorizing the whole multiplication table? What would have done if you wanted to buy a few rounds of beer for multiple people and needed to use numbers greater then 10? Just given up?

Anyway, if you really want to do mental math well, it's probably a good idea to learn the whole table. But I've learned the multiplication for a lot of small numbers just by experience. that 4*3 = 12 is really obvious and something that comes up all the time. 7*8 is less common, and multiplying by 9 involves a 'trick' where you just multiply by 10 and subtract the number again, so you don't need to keep each multiple memorized. Multiplying by 5 is easy too, you just divide by two and multiply by 10. Maybe learning these tricks counts as memorizing the table.

And while I don't advocate anyone not be able to figure out 4*3, I guess you probably have a cellphone with a calculator.

I'm not saying people shouldn't learn their multiplication tables, I'm just saying drilling kids on it won't help them too much in algebra, calculus, etc.
posted by delmoi at 11:56 AM on November 16, 2009

It's a huge waste of time, and we'd be better off teaching algebra much earlier, IMO.

For about 10 years or so, algebra has been integrated into math curricula earlier and earlier. Second graders typically start to do baby-bullshit-algebra, and they keep introducing more and more each year.
posted by 23skidoo at 1:10 PM on November 16, 2009

Yeah, my daughter is basically being asked algebra questions in grade 4. Unfortunately they're not teaching it very rigorously.
posted by GuyZero at 1:36 PM on November 16, 2009

Are you kidding me?

No.

A beer was $4. I wanted to spend no more than $15. Since I know my multiplication tables

Basic addition and subtraction would have gotten you by just fine, and you probably have a calculator on your cell phone, or in a pinch, you could have asked someone else.

and we'd be better off teaching algebra much earlier

If calculation is useless for most people, algebra is doubly so, despite the fact it's probably ten times as useful for the small segment of the population that actually needs it to do their job.

I'm not saying people shouldn't learn their multiplication tables, I'm just saying drilling kids on it won't help them too much in algebra, calculus, etc.

And I'm not also not really saying math is useless or that students shouldn't understand how to arrive at answers other than via rote memorization. :)

I'm mostly saying I find it completely credible based on my experience as a student and limited experience as a teacher that practice doing computation by hand develops both a number sense and symbolic manipulation skills that will likely help anyone with algebra and calculus.
posted by weston at 3:10 PM on November 16, 2009

escabeche, thanks for linking the second article by Wu--I read it today, and it's pretty awesome.

Regarding the times tables, he has this to say:

The essence of the addition algorithm, like all standard algorithms, lies in the abstract understanding that the arithmetic computations with whole numbers, no matter how large, can all be reduced to computations with single-digit numbers.

... it leads seamlessly to the explanation of why students must memorize the multiplication table (of single-digit numbers) to automaticity before they do multidigit multiplication: in the same way that knowing how to add single-digit numbers enables them to add any two numbers, no matter how large, knowing how to multiply single-digit numbers enables them to multiply any two numbers, no matter how large.

It's not that hard to learn the times tables--see 23skidoo's answer on AskMe.
posted by russilwvong at 8:20 PM on November 16, 2009

Basic addition and subtraction would have gotten you by just fine, and you probably have a calculator on your cell phone, or in a pinch, you could have asked someone else.

Really? You're going to ask someone how many beers you could buy because you can't multiply 2x4 in your head? That's ridiculous.

Multiplication facts are just necessary things that need to be learned in order to do upper level math and function in everyday life.
posted by kylej at 9:11 PM on November 16, 2009

Really? You're going to ask someone how many beers you could buy because you can't multiply 2x4 in your head?

Why not? People have been asked stranger questions in bars. Yeah, it might be a little embarrassing, but I'd bet someone with good social skills could even pull it off smoothly, particularly if it's addressed to the bartender. If that stings too much and you can't do multiplication, use iterative addition/subtraction/counting to fake it, or pull out the calculator. You still don't need to do any multiplication in this kind of situation.

Multiplication facts are just necessary things that need to be learned in order to do upper level math and function in everyday life.

That's the claim anyway, but the longer I look at how most people live their lives, the less it seems true.

Studying mathematics might confer advantages, and it might be enriching intellectually, but necessity? There are other ways to manage.
posted by weston at 10:51 PM on November 16, 2009

What a tragedy, the unnecessary advantage. Surely an efficient educational system would confer only the necessary, leaving additional advantages up to the student!

I bought a sandwich a few years ago from a guy who, I decided later, must have been functionally illterate. He asked me what sort of "sauce" I wanted on my sandwich, and I asked what the choices were. "Well, there's ... um ... ranch!, and ... hmmm ... no idea here." He showed me the label on the other bottle: mayonnaise. He had excellent social skills; he was very friendly and cheerful and personable; he was never working that sandwich counter again when I went back to look for him.

A century and a half ago, an illiterate person could more or less fully function in society. Today that's not true. It's still possible for an innumerate person to fully function in society. That makes any specific answer to "when will I need to know this math tool?" essentially refutable. There's the old joke where you get to the gates of heaven and St. Peter asks when the train heading east from Denver meets the train heading west from Chicago. There's how many beers I can buy with the cash in my pocket. There's how much tip to leave the waiter, how much money is left in my checking account, whether the grocer gave me the right change, whether the accountant did my taxes right, whether my gas mileage has started going down, how many bags of sand can I haul in my one-ton truck, do I want an exploding-arm mortgage. An innumerate person with a friend or a calculator or good social skills or ambivalence about losing money could answer any of these questions; a person with adequate math education will, on average, answer more of them.

If I'd known the sandwich guy as a teenager, and I'd told him that one day he'd lose a job because he couldn't identify the mayo, he would have laughed at me. Better go to the dictionary! Better study my Ms! How little can someone know about mathematics before their ignorance is a handicap, rather than just an inconvenience? That's a backwards and terrible way to think about education.
posted by fantabulous timewaster at 5:54 AM on November 17, 2009 [1 favorite]

How little can someone know about mathematics before their ignorance is a handicap, rather than just an inconvenience? That's a backwards and terrible way to think about education.

And you've arrived at my ultimate point: hard absolute necessity is a terrible metric for the value of any concept.

It's not hard to get by without knowing your multiplication tables at all, just like it's not hard to get by without most math you learn after 2nd grade. Does that mean memorizing them is a waste of time?

(And literacy is a terrible parallel to numeracy. It's so much more useful that I'm pretty sure if most people who already have both had to give one up, they wouldn't even have to think about which.)
posted by weston at 12:41 PM on November 17, 2009

After learning about Vedic math, I am surprised it is not taught in the US.
posted by idiopath at 4:19 AM on November 18, 2009

I thought I was disagreeing with you, so one of us is misreading. I'm with you that good number sense is more useful to an adult than a corpus of memorized facts. You are wrong that good social skills compensate for not being able to multiply small numbers. Even in bars.

I think the parallel between literacy and numeracy is obscured because, nowadays, most people are literate but not numerate. I think this will change over time.
posted by fantabulous timewaster at 5:08 AM on November 18, 2009

I mostly use multiplication and addition of rounded numbers. Long division is a pain in the ass to do in your head, but multiplication is a lot easier... Like, 750 grams of crackers cost $1.39. 1000 grams of crackers cost $1.89. Which is has the lower unit cost?

But long division is so much faster, if you can learn to do it in your head!

750 is 1/4 less than 1000, so if 1/4 less than $1.89 is also less than $1.39, then we'll know the lower unit cost.

1/4 of $1.89 = (about) $0.42

$1.89 - $0.42 = $1.47

$1.47 > $1.39

On the other hand you could add 1/3 of $1.39 to $1.39 and come to the same conclusion ($1.85), but my point is, don't knock long division.
posted by jabberjaw at 5:17 PM on November 18, 2009

But long division is so much faster, if you can learn to do it in your head!

1) I can do simple long division like this in my head.
2) I don't actually think through all the steps that I showed in my solution, I was just explaining a way to do that problem to someone who said it had to be done using division. Division is not the only way to solve the problem, and not even the first way I'd want to use.
3) Ummm, you made a mistake when you divided.
posted by 23skidoo at 5:18 AM on November 19, 2009

See also: Lockhart's Lament (direct link to the PDF).
posted by sillygwailo at 2:09 PM on November 29, 2009 [1 favorite]