kitchen table math, the sequel: Eat, Poop, and Play

Tuesday, April 20, 2010

Eat, Poop, and Play

I caught one of my kids, who hasn't done a lick of work all year, diligently working through a math problem the other day. Astounded, I walked over to compliment him on his efforts. He admonished me not to expect this type of exertion all the time as it "was not his life". I was surprised at his candid offering and inquired as to what exactly he thought his life was. His response?

"Eat, Poop, and Play! That's my life!"

This, even more candid assertion, set me back on my heels a bit and I was curious to know how he had come to this frame of mind so I asked him what he thought his parents life consisted of. His response?

"Parents? My parents work for me!"

At first, this exchange simply tickled me in the same way that any number of open, uninhibited kid conversations do, but this one was different. It's been rattling around in my head for a week now and I can't shake it. It's one of those throw off comments that has more truth than humor in it, I suspect.

It's made me ponder the entire relationship of middle school kids to school and more specifically the role of education from their perspective. In this modern world, how many people really need math anymore? You can get a job at McDonald's and have the register do all the math you'll ever need for work. You can get a phone that has a sophisticated calculator in it. You can get computer programs to perform all manner of things that we used to accomplish only through painstaking pencil and paper scribbles.

So I'll pose a simple thought experiment to you all. How many people actually need a math education in the modern world that goes beyond knowing the number system and the names of simple geometric shapes? What careers would people be limited to if their mathematical knowledge was this limited? What careers would they be truly excluded from pursuing?


Have my kids figured out what they need from their perspective and is our push to have them all learn algebra simply a throwback to a time when it was more important in a world bereft of gadgets? Are my (disadvantaged) kids making a perfectly rational decision when they don't apply themselves to math?

How much education do you really need for a life of: eat, poop, and play?

111 comments:

Anonymous said...

"How many people actually need a math education in the modern world that goes beyond knowing the number system and the names of simple geometric shapes? What careers would people be limited to if their mathematical knowledge was this limited? What careers would they be truly excluded from pursuing?"

Most/all STEM careers are out. No sciencey work, no engineering, no mathematician (big surprise that one).

I *think* that firefighter is out, as is paramedic ... you need to scale things (like drug dosages), if nothing else, and I'm pretty sure that you have to be able to do it in your head (even if you have a calculator).

Doctor and nurse are out ... even if the career doesn't require math, I think the education hurdle does.

I'm hoping that becoming a teacher is out.

A career in the military might be out, too ... certainly officer is out.


Another approach would be to turn this question around. What jobs that pay more than $15/hour can be acquired and held by someone who can count, do simple four-function operations on whole numbers and recognize the common geometric shapes?

Some careers that are in:
*) Waiter/waitress
*) Athlete/Entertainer in general
*) Teller
*) Day laborer
*) Construction worker?
*) Some manufacturing jobs (but not all -- modern SQC often requires more math than specified)

Yet a third approach might be to get a list of the 100 most populated jobs and try to figure out which ones are excluded.


-Mark Roulo

Anonymous said...

A start on what jobs are populated:

http://www.sscnet.ucla.edu/soc/faculty/mcfarland/soc157/femocc.txt

It is probably out of date ... lists 3.9M women employed as secretaries in 1980 ... I don't think the US has that many (by a lot) today.

-Mark Roulo

Anonymous said...

I suspect that many sales jobs can be acquired and held without much math. And talented salescritters can make a lot of money.


-Mark Roulo

MiaZagora said...

I think to be a waitress you would need to know math, as well as a teller, and for sure a construction worker.

Anonymous said...

"Are my (disadvantaged) kids making a perfectly rational decision when they don't apply themselves to math?"

They might be ... but I'll ask another question: what skills do they plan to bring to the party when they go looking for employment?

Unskilled across the board is not good ... it never has been, and those are the jobs that have been going off-shore for several decades now. No math, but great verbal skills is probably viable. No math, but good artisan skills (furniture/cabinet making, for example, or plumbering) probably work quite well ... but I don't know how good you can be at some/all of these without more than 4-function whole-number math. It might be that you are fine, or it might not.

But 18-years old, unskilled and not college bound seems like it might be a bad place to be ...

How difficult is it to get a job doing road construction, for example? Or delivering the mail?

Maybe this is what your kids are planning, and it might work out fine. One thing that they might have going for them is a willingness to actually work with their hands/bodies instead of typing at a keyboard. An 18-year old with poor math skills, but a willingness to work hard and sweat might well be better positioned in the job market than a 22-year old with a degree in worthless studies who can't even imagine breaking a sweat working hard.


-Mark Roulo

Anonymous said...

"I think to be a waitress you would need to know math, as well as a teller..."

I'm not arguing that these people shouldn't know math ... but I think the modern electronic cash registers do all the math for you ... you just enter the price of the meal (or, at places like McDonalds, the button for the meal itself). Then you present the total that the machine gives you.

"...for sure a construction worker."
I'd be willing to be that there are a lot of construction jobs (e.g. hammering 2x4s together, welding, ...) that don't require much/any math.


-Mark Roulo

forty-two said...

Construction worker-wise, my cousin, an intelligent 10th grade dropout, has a good job working construction; I don't know what his math skills are, but class-wise I don't think he got past Alg 1. But while he can do the job fine, he got it because of his dad's connections (my uncle is a bricklayer) - I'm not sure how easy it would be for a dropout with no connections to get their foot in the door.

Jean said...

I should think that if you don't know a reasonable amount of math, people would be able to cheat you all the time. You'd buy the finance plan with the low monthly payments, you wouldn't notice when you were stiffed on your waitressing tips, you'd end up with a ton of credit card debt and an interest-only mortgage. Wait, this is sounding familiar...

Perhaps it will cheer you all up to hear about my husband, who uses linear algebra and calculus in his line of work. A few years ago he was helping my youngest sister and her friend with their calculus homework and commented, "Oh, this problem is just like the one I was working on today!" and he showed them his work on penetration of radioactivity. They were blown away. "You actually use this stuff??"

Allison said...

Come on, in the future, everyone will be a SEIU employee at the Federal Health Care office, be they a non-physician health provider or the bureaucrat who won't approve your treatment. they won't need any math at all.

Paul B said...

For my kids, the future is (maximum) next week. It's pretty well known that kids under stress don't believe they have a future in the same sense that you and I might.


If you think about it, lots and lots of good jobs require only rudimentary math skills like figuring out a tip, commission, bonus, and such. No algebra! Lots of trades require trade specific skills like reading a ruler or knowing the mysteries of a framing square. No algebra! Fire, police, EMT, Nurse, all have specific instrumentation or PDA type devices that do the math.

I think kids are quite aware of this and mine in particular are in an environment where there are no engineers, doctors, scientists, or mathematicians. Do you think this is true of most kids? I do.

If you've never lived in a math nerd's house and everybody around you is all about EPP it would seem natural to me to blow off math instruction.

Allison said...

Or in other words,

if you see that you're going to be getting federal mortgage vouchers with your food stamps debit card, and there's no STEM career path in your future anyway, why bother?

Tell me again who the suckers are here? The one who knows his life is just play? Or the one whose wealth is being spread around?

le radical galoisien said...

abstract thinking. you can't comprehend cause and effect. Mathematics goes beyond the use of numbers. Understanding things like correlation and probability...

I've worked in McDonald's and math is more than about cashiering (despite being a cashier).

Take being a shift manager. You need to calculate off the top of your head -- how much time people under your command will take, how long a task will take, what's the best way to allocate people around the store for maximum profit. And even if you aren't shift manager (a job that pays 10-14 dollars an hour), you have to decide: put fries down in the fryer now, or fries down later?

If you put fries down now, you might waste them later if not enough customers arrive; if you don't put them down now you'll make them wait too long and then they'll be pissed. So then you have to essentially hedge your risk and estimate what's a good amount of fries to put down, if any. And you have to do some mental calculations, given that fries expire after being 7 minutes out of the fryer and that lots of customers now doesn't mean lots of customers later.

There's an inherent sense of a "second derivative" involved. Rates of rates.

Being successful in McDonald's requires a good sense of math, if you ever want to be paid 25 cents beyond minimum wage.

Allison said...

Of course your kids can't see past next week. But even if they did, we've extended adolescence into our 30s. 40 is the new 20, so why should they worry about "the future"?

The serious problem is what careers will remain for those who have no skills past algebra 1, and even those with skills past will be competing globally. The trades aren't as outsourceable as the knowledge jobs (which, incidentally, are what the whole constructivist worldview is supposed to be creating: children who will grow up to have knowledge jobs), but the trades are totally outsourceable to illegal immigrants.

There aren't going to be any private sector service sector jobs in the US when those kids grow up. Those will have vanished into other countries and into automation, based on the very same technology that instrumented away from their needing to do math at all.

Allison said...

--Being successful in McDonald's requires a good sense of math, if you ever want to be paid 25 cents beyond minimum wage.

but it doesn't require an American, so that job won't exist anyway.

Tech and automation will solve all of those scheduling problems--it's just not that hard to implement a scheduler on site in every place. It will be much cheaper than hiring people.

Drive thrus now use Indian order takers--in INDIA! The call is wired to them, they type it in accurately, and it's sent to the body in the mcdonald's here who needs no higher order thinking to process it. The same will happen with how to handle when do cook fries, etc.

Allison said...

Jean,

The people pretending their home was an ATM for buying HDTVs by using IO loans to buy homes they couldnt' afford didnt' get cheated.

Those of us who pay our mortgages got cheated. The suckers were the ones who did the math and said "I can't afford a home 18 times my annual income."

le radical galoisien said...

well the store is essentially a factory.

at least where I worked at, it's good to have a chemistry between your order takers and your food makers.

(chances are they switch positions every shift or so)

Especially when you have special orders -- it's good for the cashier and order taker to know what's going on in the store, especially at night.

Multitasking is huge. The solution being used now is not to have a dedicated order taker but to just equip an existing cashier with a headset. I've seen foodmakers take orders.

Calculating change is trivial. The worker makes production decisions all the time: how much tea to make, how much chicken to put out, given in mind the delayed events of cooking time and customer arrival. Whenever you put out inventory or even buy it, you are deciding how much risk to hedge. Can the order be completed in a reasonable time or should you park the car? The life of a McDonald's worker is one of constant optimisation, maximisation of profit, and minimisation of food waste.

The better your quantitative instinct is, the more efficient of a worker you are ==> more pay, more hours, more promotions.

Anonymous said...

"And you have to do some mental calculations, given that fries expire after being 7 minutes out of the fryer and that lots of customers now doesn't mean lots of customers later.

There's an inherent sense of a 'second derivative' involved. Rates of rates."


It *IS* possible to determine when to put on the next batch of fries by using calculus. Nevertheless, I strongly suspect that most people (even most successful people) working at McDonalds manage this without calculus. Even the managers. This is the sort of skill that can, and is, often picked up on the job without any formal model. This approach works.

Much like your typical athlete doesn't use calculus to hit or catch a baseball.

-Mark Roulo

le radical galoisien said...

You are not doing formal math, but you are using an instinct that formal math uses a lot. To me the connection is so large that while a shift manager I couldn't help think of graphs of time-domain functions anytime I did the most random job-related thing.

Ideally, there are preset times and everything, but that is ideal. Disasters and deviations from the anticipated happen a lot. Quite routinely. When the fryer or counter freezer messes up (or is simply out for say, a 10 minute clean) or the McCafe machine is at a funny water pressure ....

When the fudge runs out of the sundae machine and you have to melt it a little on top of a warmer before putting it in, but it's melting too slowly and the customer is pissed, you consider all the alternatives and their approximate heat flux profiles, bearing in mind you don't want to scorch said fudge, make it so hot that it will melt way too much ice cream, or worse the plastic it's in or blow the bag up.

If you never moved beyond algebra I, you can't even understand the basic science or basic qualitative principles behind some very quantitative things. Like say, the time distribution of labour % (amount of wages paid to workers per $ revenue received); when you open it can be as much as 55% and at peak it drops to 8% but the distribution varies by season and day of the week (or the fact that a football game's going on) and from a statitical compilation of labour % values for the past 20000 hours the store has been operating you actually have statistical programs on the managerial computer suggesting how to allocate workers based on worker output data.

Which is probably why my general manager ended up hiring a statistician and a former programmer to supervise the evening shift.

le radical galoisien said...

btw, managers are pressured to cut excess workers in the middle of the day if labour % goes above 20%. But they must also have the wits to estimate how labour% will change with time, with a certain degree of uncertainty. Managers do a lot of daily hedging.

If I wanted to suggest changes in products or a routine (which we could -- since as a franchise we had greater freedom), we would have to quantitatively back our suggestion up and explain how we thought it would increase profits or productivity and by approximately how much. Because of automation, we had statistics for every product item and even every condiment item given. You could plot distributions of apple pie sales as a function of time (in hours). This is useful because ... people actually have to spend time (and inventory) baking these pies. You can then optimise worker placement positions (whether you kept a worker or not, or whether you wanted to hedge by asking a worker to stay beyond his scheduled time), optimise production rates, inventory purchasing etc.

To earn your 12 dollars an hour you kind of need to have some quantitative skills that go beyond a four function calculator. They don't have to be precise like finding the value of Lagrange multiplier, but they need to be good at optimising profit.

Allison said...

Retek, Demandtec, Peoplesoft, Oracle, SAS, SAP are all in this space. They all do various pieces of the supply chain optimization, and they are moving to doing the store level versions.

These companies already tell their customers how to vary their prices, their promotions, their staff for a variety of outcomes: whether it's to move excess product or increase marginal profit, to reduce labor needs etc.

Yes, it's good for a manager to have these competencies. But it will be easier to have even less of them, even farther away, supported by automation.

le radical galoisien said...

Let's say I hate hate hate Oracle so much. Their customer support is horrible and they run an awful and laggy SIS system at my school.

Whatever "experts" there are, they aren't present on the telephone and they are resistant to student suggestions. Why oh why didn't my school just hire a bunch of e-school students who would actually be available to fix bugs and do a decent job?

Outsourcing is a great idea. But I think outsourcing short-term labour decisions to an external firm would suck horribly. If I get incompetent people on the phone why should I get competent consulting that will decide on-the-moment decisions when I need them?

le radical galoisien said...

people have to outsource the right things.

the examples of outsourcing I have personally encountered have been quite horrible experiences and there is so much lost chemistry.

I'm sorry, but if I ever went back into foodservice I do not think I could develop a good work relationship with callperson half the world away. At least not as an order taker. Things are so different from store to store -- how do I know I'll get someone who really knows what goes on in my store?

le radical galoisien said...
This comment has been removed by the author.
Paul B said...

Middle school kids don't plan. Mine are highly impulsive and reactive. They respond to the things around them and are experts in reading the moment to their advantage.

I guess my reflection is that 90% of my kids have absolutely no way to experience all of the thoughtful things being portrayed here. If they can't experience it, it doesn't exist.

They live a life of immersion in the nanny state that demonstrates in a very real way, 24/7, that education is not required for 'success', that success being remarkably different than what you might think it is. If you live a life of EP&P and your particular needs are being met then our attempts to prepare these kids for a world they don't aspire to are nothing more than tilting at windmills aren't they?

As much as I might detest the attitude, it makes perfect sense, from a kid's perspective, to blow off the hard work of learning. There's no competing perspective in front of them.

Paul B said...

My school has this Dance Dance Revolution type game that is used in PE at times. My EP&P student, who's a bit pudgy and not athletic in any way at recess, blows away the rest of the school at this game. Hmmmmm?

Turns out he spends a few nights a week in the arcade at Foxwoods Casino. Guess what game he's on? Put him in a McDonald's and he'll figure out the fryer routine just fine without ever thinking of an equation or graph. He'll do it by reacting to the rhythms of the place and he'll do it instintively, providing he NEEDS the job. He'll do it even better if you put some flashing lights on the buttons and fridge and turn them on to tunage piped in by Oracle.

lgm said...

You don't need much education for epp when your caregiver is modeling esd as the goal!

The only real problem I see is that they can't think - everything is done by costly trial and error. Heat runs out, cars need tires, money always out before the end of the month.

My favorite was the guy who shared his opinion of engineers with me..all overpaid and know nothing college boys..his unpermitted deck off his second story dining room fell down about a month later from an overload of partiers. He wasn't even disadvantaged, just blue collar and he makes more than a young engineer due to his union's power.

lgm said...

>>What jobs that pay more than $15/hour can be acquired and held by someone who can count, do simple four-function operations on whole numbers and recognize the common geometric shapes?

Downstate, if they wanted to work at that level above the table with little education, they'd be trying to get on with the city police & fire depts or state corrections dept.
http://www.nyc.gov/html/fdny/html/community/ff_salary_benefits_080106.shtml for ex.

farmwifetwo said...

If he doesn't learn math then the banker, the sales clerk, the waiter etc will take his money for him.

He won't be able to budget or balance his cheque book.

He won't know if he got the best rates on his GIC's, Mut Funds or his mortgage.

He won't know how much to tip or if someone gave him the wrong change.

Yes, this doesn't take upper level math to do it. But the upper level's teach you to think and learn. Truth is, once you master the basics.. they aren't hard. But most... and probably including your kid... haven't... so it's very hard to them and they don't see the point in it.

My eldest is bored in Gr 5 math. This is a child with NLD, and poor short term recall skills. BUT, his Mother redid the Gr 3 curriculum using Classical (Saxon) math. We have done little since focusing on English... But he knows the basics... therefore it's easy.

Paul B said...

LGM:

What is ESD?

Paul B said...

My kids don't have mortgages, mutual funds, 401K, checking accounts, or savings accounts. They don't go to restaurants with waiters nor do they tip.

Actually they don't even care about being ripped off by clerks or finding the best deals on their purchases. Many pay with OPM by means of an EBT card and even if mom and dad work hard for their money the kids don't so where the money comes from really doesn't matter to them any more than where it goes as long as they get their stuff.

OPM == other people's money
EBT == not sure what EBT stands for but it's a debit card replenished by nanny

Allison said...

EBT is the modern way food stamps are distributed.

Paul, you are describing a society of nothing but nihilism.

These kids, and many adults, now live in a world of materialism (in the philosophical sense): there's nothing other than what's material. Ideas don't exist, don't count, don't inform. The future can't exist in any real sense then, and certainly no higher purpose or goals can either. Such materialism leads people inexorably to nihilism. There's nothing that matters, so might as well care only about immediate gratification.

It is an empty meaningless existence.

But one teacher isn't going to be able to hold more than a candle to that darkness. still, the candle is a light.

le radical galoisien said...

when my parents divorced at the age of 10 (my father was the sole breadwinner at the time but boy was he abusive), with my mother gaining full custody, we had EBT and all that.

I was suddenly a lot less careless with how I spent my money. No more spending 1 dollar at the school vending machine for that bottle of coke. I hoarded money. A lot.

Nicksmama said...

My kids don't have mortgages, but they do have stocks (thanks to grandparents and uncles), savings accounts, etc.

They know about the stock market and watch their stocks and follow some of their favorites. Their dad tried to explain calls and puts to them, but he can be a bit long-winded and lost them.

They know that we have a mortgage (the bank currently owns 2/3 of my house) and dad brings home a paycheck that varies based on a percentage of net profit.

They understand the "credit" part of credit cards. They know how interest can work for you and against you.

They go grocery shopping with me, so they know what food costs. They compare prices and know what it costs to feed our family.

They know how much their karate lessons costs. They calculated the cost of gas to and from lessons as well. They comment on the increase in gas prices.

When they want to buy something, they research product reviews and sales on the internet. They have a savings account to buy goodies such as video games. They know that this savings can be used to buy their first car, so they are careful with it.

They are 12 and 13. There is no reason for young adults to go around thinking of their parents as ATMs and being confused about finances.

We deal with money issues everyday - we consider them teaching opportunities.

Anonymous said...

When does he expect his life to change and to what will it change?

He must realize he will be working for his kids someday.

Jeff Boulier said...

lgm, firefighters need math through basic algebra in order to pass the aptitude test and the classwork associated with firefighter I&II.

That having been said, I am unsure how much in the way of formal calculations happen on a fireground scene -- I don't know if e.g. pump operators do the math on their flow rates or judge them instinctively.

Medics do use formal math at the basic algebra level, to calculate flow rates & so on. I've never seen one use a calculator.

le radical galoisien said...

it's not so much as on-site calculation as familiarity. do you expect something to scale linearly, logarithmically, polynomially, etc.?

Pilots on a plane probably don't pull out a calculator, but they probably need to know the mathematical concepts of velocity and acceleration.

Sara R said...

Sounds like the base problem isn't an unwillingness to work at math. It's an unwillingness to work, period.

When my little kids balk at work, I tell them the story of "The Little Boy who Never Learned How to Work": Once upon a time, there was a smart boy. His mom said, "You are so smart, you don't have to work." So he didn't. His big sisters did his chores. When they grew up and left the house, his mom did.

Now he's 45 and still lives with his mom. He never got a job, because he never learned how to work. He never got married and had a family, because taking care of a family takes work. He doesn't even cook his own food, because that would also be work. He has had a sad life, because the best things in life require work.

Crimson Wife said...

My DH worked at McDonald's at minimum wage as a teen. He now does financial analysis for a living at a salary that many politicians would consider to make the earner "rich" (it's not but that's a whole 'nother post). I'm sure he could give Paul's students quite an earful about the importance of math.

Anonymous said...

"I am unsure how much in the way of formal calculations happen on a fireground scene -- I don't know if e.g. pump operators do the math on their flow rates or judge them instinctively.

Medics do use formal math at the basic algebra level, to calculate flow rates & so on."


For many departments, this is a distinction without a difference because the firefighters are the first-responder paramedics.

My son was *VERY* into firefighters and equipment when he was younger ... and we live near our city's training station. We spent a lot of time at that station (and he got to ride in an airport firetruck on several occasions ... this is supercool). Because there are a lot more medical calls than fire calls, our fire department has a paramedic with each pumper engine. This is one grade up from basic firefighter, so we actually have a lot more *trained* to do this.

The medical stuff does require calculations on-site, under pressure, for things like adjusting dosage. Knowing how firefighters work, even if they do have a calculator, they can't get qualified with one (similar example ... they have maps, but are required to be able to find the address of an emergency without one ... we see them driving through our neighborhood a few times a year, just re-familiarizing themselves with the layout).


Two points:
(1) I doubt that people who can't do these calculations will get hired in our city as firefighters,
(2) Because of the overlap, a lot of modern fire fighter duty is medical.

-Mark Roulo

Paul B said...

I think a distinction needs to be made between 'using math' and 'using algebra'.

I can show a student how to adjust a dosage or flow rate or any number of other APPLIED mathematical techniques without providing the algebraic explanation that underlies its usage. There's a big difference.

All of my kids know how to apply math. In fact you might say they're far more eager to apply it than to know why it works. Isn't this a distinction you would make, for instance, between a technician (who is trained to do a procedure) and an engineer who comes up with the procedure? Perhaps this is the dilemma for the kids I teach. They are far more likely to live in homes of 'technicians' than engineers. They see math used, not derived, and know instinctively that you can succeed at the former without the latter.

My kids, for the most part, aren't lazy. There are a few, of course, that wouldn't lift a finger to sharpen a pencil, but most will eagerly do any menial task I give them in class. No, the unwillingness to work is an unwillingness to do academics that are hard.

They aren't living in an environment where that (hard academics) has a value proposition for them and we don't provide one in school either.

I'm thinking if I grew up today in gadget world I could see myself being an EP&P guy myself. It just makes sense.

Anonymous said...

"All of my kids know how to apply math."

If your kids can handle numbers up to and including fractions, decimals and percents, then a lot of jobs stay open.

Your original question, though, was, "How many people actually need a math education in the modern world that goes beyond knowing the number system and the names of simple geometric shapes?"

I interpreted this as the big-4 operations (+, -, ×, ÷) on whole numbers plus knowing some simple shapes.

Fractions, decimals and percents are kinda the "next level up" ... and my opinion is that fractions (including mixed numbers) are the hardest of these three.

Math up to but excluding algebra probably excludes the STEM majors, but not a lot else.

-Mark Roulo

Anonymous said...

No, the unwillingness to work is an unwillingness to do academics that are hard.

If the prerequisite skills are taught to mastery, then the incremental steps are generally not "hard".

Trying to do skilled tasks at full speed without the training, practice, and mastery of prerequisites that competent performers have is hard. This includes almost every task, even ones that aren't generally considered "skilled" - reading, free throws, playing computer games, arithmetic, cooking, you name it.

Paul B said...

I read an article in Discover magazine that discussed the brains of top athletes. It turns out that even when executing very complex interacting and concurrent skills they don't think much. Their brains aren't calculating angles and estimating distances and required forces. They just do it.

Their brains aren't on fire like ours might be in similar situations and it is because of practice. Through practice and repetition, they get wired differently.

The article didn't push this but I would think the same thing would be true of a top 'anything'. I would also suspect that to get the reps you need passion in your 'thing', whatever it is, and the passion comes from your own personal value proposition for it.

You gotta' love it to master it.

Paul B said...

Yes! I was sloppy on my 'fundamentals'. When I wrote it that I was thinking that the only theory that is necessary to support a competent technician is knowledge of the number system.

With that, every other intersection with math can be trained as an algorithm, computer app., or something to drive a protocol.

Imagine a 13 year old. Mom is a long time waitress. Dad mows lawns and plows snow. The teenager sees them do lots of math. They plug in various numbers to a calculator in a pure algorithmic consequence of their respective occupations. They both have to do percentages, tax calcs, tips, scale up, quotes, etc.

They never use algebra because abundant experience has equipped them with every trick in the book to pursue their craft on a four function calculator (with a percent key).

For this kid, that's math.

Anonymous said...

I'm math challenged. My problem started in 3rd grade private school, then I switched to a public school for 4th grade and those kids were just catching up to what we were doing at my private school. Then in 5th grade I couldn't catch on to what we were doing and struggled with math for the rest of my school career.

I only passed one semester of pre-Algebra freshman year of high school. My junior year the counselor insisted I try Algebra 1 and I was failing miserably and ineligible for athletics because of it. So I dropped and got my second semester of pre-Algebra. Senior year I took "Consumer Math" and thankfully I can figure out interest rates for my loans, balance a checkbook, calculate tips, etc.

By today's standards I wouldn't have graduated high school on time.

I work for a municipality in a civilian support job and I make $49k a year. Most of the tasks I'm responsible utilize Excel and Access and we use programs like Peoplesoft and SAP.

I'm thankful I'm able to do OK without a high level of math skills but I don't want the same thing for my child.

Allison said...

--They aren't living in an environment where that (hard academics) has a value proposition for them and we don't provide one in school either.

They don't have a value for hard work in their peer groups, in their pop culture, in their schools, in their political culture. If it's not in their family, it's nowhere.

And why would it be in their family? how many families still have it, and if so, why do they bother? They are holding back the tide with a broom.

Men are increasingly opting out of college. They are also opting out of marriage and fatherhood. why? for the same reason--why work hard? Marriage and fatherhood are hard work, and because it's a bad bet--make money to support the wife and kids, then the wife can leave, take your money and your kids. Why work hard on that? Why not keep playing video games into your 30s and 40s? Your parents or the state will take care of you.

Allison said...

--It turns out that even when executing very complex interacting and concurrent skills they don't think much. ..I would think the same thing would be true of a top 'anything'.

Absolutely right: this is true for everyone who is an expert. Willingham talks about this: our brains are optimized for NOT thinking. By chunking everything we possibly can into memory, we do every task we possibly can by recalling already known procedures. In CompSci land, this is called "reduce to an already solved problem". And it's a lot cheaper than recomputing--even if the new problem is easier than the priorly solved one was.

Putting things in working memory and manipulating them is expensive, and dangerous--if you're too busy with working memory, you can't be polling all of your sensory systems to keep alive. And if you're busy polling your sensory systems to keep alive, you keep having to swap and in and out your working memory state.

So in experts, hard tasks are hard-wired, and you simply have to keep doing them until you get there. The reward mechanism that makes someone WANT to do batting practice 50 minutes straight a day is probably getting a lot more feedback if you've got base talent, and the base talent is getting a LOT better if you're got the passion.

You're just not going to want to do X if you're not good at X. You might say you want to do X, but you're really not going to be out there, practicing it in every free second you've got. You can only force yourself so far.

rocky said...

it's a bad bet--make money to support the wife and kids, then the wife can leave, take your money and your kids

I think everyone should sign prenuptuals so if they ever divorce, no one gets anything; the loggers cut it all in half with chainsaws.

Hey, where did that article go about the boy who had to row a boat to get a doctor? It just disappeared.

Paul B said...

It's hard to find good numbers on this but a quick search shows about 11M scientists,engineers, and mathematicians in our 154M person workforce. About 15%, or 1.65M, are foreign born and educated. So we have about 9.35M of these folks educated here. This is about 6% of the workforce or 3% of our population.

Presumably these are the people who have the need to go to the most depth with math. Even if you double or triple this percentage to account for 'other' occupations that might need depth, you still don't get to a very big number. I would submit that the claim that everybody needs math in depth is bogus.

Even assuming that you could get every student an in depth mathematical background, the data says the vast majority of them will never use it. Could it be that kids already know this, just from simple observation of their world.

What irks me is the knowledge that we may be throwing enormous amounts of time and energy into a nonsensical goal with a guaranteed minimal payback. All the while, we are diluting what we deliver to the 10% or so who can truly do the work and will receive a payback for their efforts.

Lose, lose, a perfectly understandable outcome delivered by bureaucrats who make more than you do.

le radical galoisien said...

I'm foreign-born and half-foreign-educated. Where would I fall in?

Anonymous said...

I suspect that the data that say the vast majority of people don't use math is inaccurate. I suspect that every small business owner is constantly using basic algebra. We have a family farm and constantly, daily use algebra. My contractor calcuated angles, areas, amount of paint, flooring, labor and materials necessary to remodel our house. Our banker uses a calculator, but easily and obviously has the mathematical fluency to have a real time conversation about prices, quantities, exchange rates and whether certain data sources are reliable. The sales people we deal with calibrate machinery to determine how much material we need to use on a field. Of course, I check their calibration calculations as well. Yes, we all use tools, but without the understanding of how the math fits together, the tools would be black boxes and we wouldn't know when a number didn't make sense. And we are constantly figuring out, how much seed to put on a field, if a chemical needs to be applied at a certain flow rate and dilution, how fast does the tractor move. If we can x crop and it costs z to grow and we might sell it for a range of y1 - y2, Which crop should we plant.
Jane

Crimson Wife said...

If I use the BLS statistics, there are roughly 19.5 million people employed in occupations classified as "business and financial operations", "computer and mathematical science", "architecture and engineering", or "life, physical, or social science". That's about 14% of the total workforce. Not all of them use math beyond basic arithmetic, but presumably there are people in other occupations that do (most notably in the legal and education fields).

lgm said...

>>firefighters need math through basic algebra

Yes. In NY all unclassified students have to take Integrated Algebra I to get the high school diploma. They can take it over the course of two years, in double period each year. Maybe they pass before they reach the drop out age, maybe they don't.

>>What is ESD?
EPP for grown ups that have no education, little money, no land to farm, and are just passing the time. Eat, Sh*t, and Die. Every day is the same.

The other reason low income people aren't marrying is for the cash and bennies. Single mom and child(ren) get more from the state than a married couple with child(ren). Disability payments are lowered if the spouse has income.

Allison said...

You think people in the legal fields use math? ?!?!?

In what capacity? Do you mean beyond arithmetic?

When I meet lawyers under 50, I'm overwhelmingly aware of how little math they know and how proud they are of this fact. I know several who are proud that they can't balance their checkbook. At least the lawyers over 50 have enough sense not to be proud of it.

The vast majority of business owners use no math beyond arithmetic. Our contractor couldn't have calculated any angles--he did everything by the rules of thumb he'd been taught. A room of X sq ft is 2 gallons of paint; a kitchen Y size will take 2 guys 4 hours to install cabinets. Calculate amount of flooring? Yes, he could multiply and divide. Is that what you call using math?

Business owners use Quicken or Quick Books or a CPA. The barber shop my son goes to orders "the same amount of shampoo" he did last time, which is the same as it was the time before that, which is the same as it was when his father ran the business.

Allison said...

Paul,

I'm totally in agreement with you about educational romanticism.

The question is for those jobs, how many will be here in the US and have rising wages 20 years from now?

· Primary Sector. Produces raw materials and basic goods, such as: agriculture (both subsistence and commercial), mining, forestry.

· Secondary Sector. Manufactures finished goods, such as: automobiles, chemical and industries, energy utilities and construction.

· Tertiary Sector. Provides services, such as: retail sales, transportation and distribution, entertainment, restaurants, insurance, banking, healthcare, and law.

· Quaternary Sector. Provides intellectual services, such as: libraries, education, and information technology.

· Quinary Sector. Provides high-level analytics and decisions, including, science, universities, healthcare. The distribution of US workers using December 2007, by sector is:

•Primary: 2%
•Secondary: 20%
•Tertiary: 38%
•Quaternary: 17%
•Quinary: 23%

That second group is going to shrink even more. The third can shrink too--retail sales through distribution can be outsourced more and more by tech, and outsourced overseas. no need for much of the fourth sector to be in the US, either. The fifth depends entirely on the math skill set of our country, as those jobs can be anywhere in the world.

Wage stagnation is a real problem(and it can't be looked at honestly without confronting the problems of legal and illegal immigration.) Even for those sectors that are still here, how many will see rising salaries? How much demand will there be relative to supply?

Paul B said...

Le Radical G

Since this is about (primarily at least) U.S. education you don't fall in :>}

I was trying to get at what we need to do here in the United States. Foreign born and educated folks are like gifts from heaven. We get to utilize them but we don't have to create them.

Paul B said...

You can always argue minutia about who does and does not use math. The fact is everyone uses it to varying degrees, from the engineer using calculus to find the volume of fuel in an airplane wing to the farmer figuring out how to mix chemicals.

The debate needs to be on; to what degree does this ripple back into the education system as a set of criteria to drive curricula. I maintain it is nonsense (and most kids know it) to attempt to drive every single student towards algebra and calculus. We do so at our peril. It is indeed Utopian as Allison has said.

My son is a contractor who builds high end homes; one of the best in the business as far as I can tell. He sucks at math. But, he knows exactly how to use all the tools at his disposal from calculators to framing squares with little (to me) unfathomable markings on the side. He figures out roof angles, stair rise/run, and compound miters that most mathematicians can't compute. I know this because I also have a son in law who is a physicist and he can't do simple miters, never mind compound ones.

Before my son in law figures out the compound miter with myriad equations in trig and algebra, my son will have cut, installed, and finished the moldings in his house.

One knows the math in depth but lacks application knowledge. The other excels at application knowledge but knows absolutely nothing about the math underneath what he's doing. I love 'em both but only one of them will build my house. The other will design the nuclear reactor for my off the grid home.

Allison said...

Here's another wayto look at it

http://cafehayek.com/2010/04/son-i-want-you-to-grow-up-to-be-a-pipefitter.html

We used to want more for our children than to eat, poop and play. There's the nihilism again.

There needs to be a clear headed view on what education can accomplish, and then move from there to how to create value. It's not all going to be by doing calc.

kids at the top are certainly not getting educated well enough right now by our schools to compete with the top of the international world. Making everyone suffer to fix that is insane.

Anonymous said...

I guess I am older than dirt (I graduated from HS 35 years ago) but when I was in Junior High and High School my teachers did not focus on how we would use math in the "real world." My math courses did include real world word problems (and the teachers presumed that if we know "the math" we would be able to complete the problems) but the teachers focused on the the beauty and logic of math and its role in building logical thinking skills. They seemed to convey their deep love of the system of math as a way of understanding and ordering the world. We knew if we were going to get into the college of our choice we needed to learn the material. End of discussion. Maybe in the quest to make math more relevant we have removed the sense of admiration for the material.

Allison said...

Paul,

Have you read Real Education ? If I say the author's name, multiple heads will explode :)

Bostonian said...

Allison said:

"Have you read Real Education ? If I say the author's name, multiple heads will explode :)"

Hint: he also co-authored the Bell Curve. I've read them both and think most educational policy is based on a fantasy world far removed from the real world described by the author.

Paul B said...

I haven't read Real Education. Went to Amazon and read reviews. Sounds intriguing and familiar to this thread.

Check out my next post. It's a follow up to this one triggered by this discussion.

Catherine Johnson said...

I haven't read the comments, but I think it's time to re-post the link to A Trip to the Number Yard

Catherine Johnson said...

People in the legal fields don't use calculus but they use a boatload of arithmetic.

Also, prosecutors have to unravel financial fraud.

Catherine Johnson said...

I keep reading that all the data mining is going to involve calculus and beyond.

Facebook is hiring statisticians rather than people with Ph.Ds in computer science.

I think that was in....WSJ or the Times.

Paul B said...

The argument isn't that there aren't jobs that require in depth knowledge of math. It's a numbers game. The argument is how many people actually have it and use it. If it's not a significant portion of the population than you can make the argument that it's a ludicrous goal to set that says we have to get every student there.

This thread poses the argument that the kids have already figured this out, long before the adults.

BTW my physicist son in law works in the financial services industry as a software engineer. My chemist daughter works for a medical services software firm that computes insurance payouts. My EE daughter is about to get a degree in PT from Mass General. So when you say career xyz uses math you need to factor in the career changers working in xyz with a degree in abc.

Lawyers, as a discipline, are not trained in mathematics anymore than history majors. That doesn't mean there aren't mathematically astute lawyers (or history majors). It's just that they're the exception not the rule. I've also shared office space with engineers that stink at math.

OT but important context...

I draw a bright line between arithmetic and mathematics. One is a science. The other is memorization.

Catherine Johnson said...

Have you read Liping Ma or Ron Aharoni? (Or Wu?)

I don't think it's right to say that arithmetic is memorization...

Erin Johnson said...

If arithmetic were taught properly in the US, the transition to abstract algebra would be seemless for the vast majority of students. (e.g. Singapore teaches arithmetic in a manner that properly combines foundational concepts with computational fluency.)

That it is typically taught using memorization contributes greatly to our students lack of understanding of the foundational concepts present in all mathematics.

Allison said...

That WSJ article was a great example of The Murray Gell Mann effect: even when they got the facts right, they didn't have any idea what they were talking about. There's no big difference between statisticians and lots of cs phds.

The field of Computer Science is large. There's an entire swath of it called " cs theory" which is all math--the math of probability, combinatorics, statistics, game theory, graph theory, discrete math, markov chain monte carlo methods, etc. Other CS folks that cross the line are numerical methods folks, computer algebra folks, and those who worry about how to compute matrix operations quickly.

Such phds are in high demand anywhere and everywhere and they can do data mining. Other phds, the kind that are used for building faster servers or who know how to handle race conditions--ie software or hardware engineers-- are sometimes farther away from cs theory people than math or stat phds are. Statisticians know R and SAS, and are in demand, but any CS person can learn R in a few days. The idea that these two fields are far apart in skill set is silly.

The point, poorly made in the article, is that Facebook is moving away from infrastructure to worrying about finding value for advertisers.

Such people are in all sorts of fields, in CS, math, stat, econ, physics, etc.

And such people will be WRITING the data mining algorithms and the software.

But they give the dials to the rest of the folks--again, this is what SAS and SAP and whoever Oracle bought and Retek and Demandtec and the rest do: the give sw to suppliers and producers and retailers to help them data mine. Their endusers won't be doing phd level algorithms. And neither they nor their enduser need be in the United States.

Paul B said...

Arithmetic is computation with numbers. You can learn it by memorizing a handful of algorithms and two ten by ten tables.

I'm not saying this is the right way to learn it but reduced to absurdity that's what it is. It is largely what is embedded in a four function calculator. Most people don't need more than that to function in good careers provided they are also provided with various trade specific gadgets to handle higher order things.

Again, I'm not advocating this at all. I'm trying to put forth the idea that this is a typical middle school perception of the world. Absent anyone presenting another reality, this is why EP&P is their dominant attitude.

Remember I'm talking about kids with one or more family members absent or in jail. I'm talking about kids (>80% by our stats) who have experienced some form of trauma. I'm talking about kids who have few if any role models that work every day.

One thing these kids are very good at is self preservation and that means knowing when they are being had. Bottom line, the math programs we are putting in front of them don't make any sense (to them).

Erin Johnson said...

Paul B,

Middle schoolers, particularly those with a heavy dose of realism, are great at cutting to the chase.

And frankly, with arithmetic, the battle for building the foundational concepts essential to arithmetic are already lost by 6th or 7th grade if arthmetic has been taught as just computation in the earlier grades.

Arithmetic is not just computation with numbers or else a calculator would suffice and there would be no need to learn to compute by hand. It embodies all the foundational concepts of mathematics that are essential for understanding algebra, geometry, trig, calculus, etc... The concepts of equal units; place value, distributive property (not the formal definition but the practical application), etc. that are essential for "doing" all abstract mathematics.

That most American math programs suck (both conventional and reform), I would agree. It does not have to be so. If you are so inclined, please check out the Singapore math books. From a mathematical standpoint, they are a work of beauty.

J. D. Salinger said...

That it is typically taught using memorization contributes greatly to our students lack of understanding of the foundational concepts present in all mathematics.

Math requires memorization; sorry. There's no getting around that. Even Singapore relies on that. There's nothing wrong with mastering procedure and reaching fluency with it. While some books in the conventional arena may present computation problems in isolation from word problems, they are most likely books written in the last two decades. A look at books written in the 40's, 50's and 60's shows explanation of what's going on with the math behind the computation procedure and even uses some of the same techniques that Singapore Math does. While there could be improvement, it is a gross mischaracterization to say that math was taught using "rote" methods.

See Hung-Hsi Wu's article BASIC SKILLS VERSUS CONCEPTUAL UNDERSTANDING: A Bogus Dichotomy in Mathematics Education.

Catherine Johnson said...

Bottom line, the math programs we are putting in front of them don't make any sense (to them).

Paul - have you looked at Lemov's book yet?

He talks about how teachers get kids - the population you're talking about - to value what they're learning.

CassyT said...

This week I've been attending the NCSM & NCTM conferences in Sandiego. That's <a href="http://grou.ps/mathedleaders/files </a> and the Nat'l. Council of Teachers of Mathematics and I have seen some interesting speakers.

Yesterday was Professor Wu, who spoke on why we shouldn't teach fractions like poetry and he spoke explicitly about memorizing information.

Wednesday was Dr Fong & Dr. Yeap of Singapore, on teacher preparation in their country. Lots more to come today & tomorrow, but I will post about those two sessions and others when I've had time to synthesize (and find all my notes!) over the weekend.

Erin Johnson said...

Memorization is a technique for learning. It is not the end unto itself.

Is it necessary to be fast and efficient at calculations: absolutely! But is it necessary to learn through memorization: the answer is no.

There is an opportunity cost associated with requiring speed before understanding that may not be beneficial in preparing students to understand abstract mathematics.

This element of learning (developing understanding before requiring speed) is probably the only part that the Fuzzy Math guys got right. (They got everything else wrong, including what fundamental math concepts need to be taught, how to teach mathematics and the pace of instruction.)

J.D. Salinger said...

This element of learning (developing understanding before requiring speed) is probably the only part that the Fuzzy Math guys got right. (They got everything else wrong, including what fundamental math concepts need to be taught, how to teach mathematics and the pace of instruction.)

I believe it was Paul B who said he didn't fully understand arithmetic until he got to algebra. Procedural fluency leads to understanding. There obviously needs to be some explanation of what it is one is doing. But even Singapore waits til 7th grade to explain why invert and multiply works (using the algebraic symbology that 6th grade students lack).

Erin Johnson said...

Mastering the foundational concepts, procedures and structures of arithmetic leads to understanding. Simply memorizing procedures does not by itself always result in the comprehensive understanding of arithmetic needed to easily transition to abstract mathematics.

Procedural fluency is one great indication of understanding but it is not always a sufficient indicator. Using the concepts in an unfamiliar procedure or problem is another indicator of understanding.

The underlying concepts of arithmetic are non-trivial and non-obvious and certainly we do not have a strong tradition of teaching arithmetic is a comprehensive manner. That the foundational arithmetic concepts are missed by a large fraction of kids is completely understandable. For those fortunate few who are able to intuit the underlying conceptual structure by practicing procedures, then great. But a whole lot of kids do not intuit those foundational concepts that are essential for understanding and becoming proficient with abstract math.

Wouldn't it be better if elementary schools were able to develop the foundational arithmetic concepts necessary for transitioning to Algebra using procedures as they should be used: practical techniques useful by themselves but even more valuable in providing the extensive practice needed to understand and apply the foundational concepts of arithmetic.

J.D. Salinger said...

Using the concepts in an unfamiliar procedure or problem is another indicator of understanding.

Which is the purpose of many word problems. This idea that students must be able to apply prior knowledge to new or unfamiliar procedures is an oft-cited goal, but which practically speaking is pretty damned hard to accomplish, even for adults. Willingham has written about this extensively.

Well developed multi-step word problems can accomplish this in part.

I agree that simply memorizing procedures does not always result in comprehensive understanding. Knowing how to invert and multiply in order to divide fractions will not help a student know how to apply it. But as I've said, I have not seen many programs that emphasize procedures devoid of word problems or what the process does.

On the other hand, the student who knows how to set up and solve the problem of finding the number of 1/16 inch segments in a 7/8 strip of metal, is in a good position to later learn why invert and multiply works. The student who is forced to understand and explain or justify why one inverts and multiplies may end up memorizing the reason and not understanding a word of it.

Erin Johnson said...

"The student who is forced to understand and explain or justify why one inverts and multiplies may end up memorizing the reason and not understanding a word of it."

Thus my reluctance to use memorization as a tool to develop mathematical abilities.

In every problem and in every procedure there is a great opportunity to reinforce those foundational concepts that are helpful in understanding the current issue but will also be essential later on in abstract math. Why waste that opportunity?

If you tell students to derive any procedure from foundational principals each and every time they need to solve a complex problem, it doesn't take long at doing that laborious work until they are starting to look for a more efficient method for doing those calculations. And lo and behold those efficient methods are the standard algorithms.

The only program (in English) that is worth using is Singapore math. The foundational arthimetic concepts are developed quite well along with procedural fluency and problem solving abilities. And Singapore math, especially in the Primary grades, does spend a considerable amount of time developing the foundational concepts of composition and decomposition of a number (number bonds), place value, equal groups, distributive property, etc.. and then uses those concepts to understand and become proficient at procedures and complex problem solving.

Barry Garelick said...

If you tell students to derive any procedure from foundational principals each and every time they need to solve a complex problem, it doesn't take long at doing that laborious work until they are starting to look for a more efficient method for doing those calculations. And lo and behold those efficient methods are the standard algorithms.

The have to go back to first principles every time? That sounds like Investigations. Many parents have commented here at the lunacy of asking students to find three ways to add two three-digit numbers. It adds nothing to understanding and results in a lot of confusion and frustration. Certainly Singapore doesn't do that. They provide the foundations, true, but once the procedure is presented as an outcome of the foundation, the focus is on mastery of the procedure, as it should be. The focus of Singapore is on procedural fluency, but there is a foundational context. The student is not expected to fully understand the reasoning behind the procedure. Some get it; some don't. But mostly all have mastery of procedure and problem solving. Singapore doesn't explain the invert and multiply rule for division of fractions. It is presented in 6th grade. The explanation appears in 7th grade.

SteveH said...

"It's a numbers game. The argument is how many people actually have it and use it. If it's not a significant portion of the population than you can make the argument that it's a ludicrous goal to set that says we have to get every student there."


Who makes this decision and when do they make it?


"This thread poses the argument that the kids have already figured this out, long before the adults."

What, exactly, do they figure out? Do they come to the conclusion that they are just not good in math; that they don't like math? Is that really the case or is that because of bad teaching and curricula? KTM is all about fixing the problem of K-8 math and making sure that kids don't come to the wrong conclusion. It's about keeping educational doors open. Besides, up to a point, I don't care what my son likes or doesn't like. He has to do the work. I will decide which doors to close, and I will thank the school and other adults for not making that decision for me. I would also hope that they would not make that decision for kids who do not have parents who protect them from do-gooders and societal bean counters.

Katharine Beals said...

"If you tell students to derive any procedure from foundational principals each and every time they need to solve a complex problem, it doesn't take long at doing that laborious work until they are starting to look for a more efficient method for doing those calculations. And lo and behold those efficient methods are the standard algorithms. "

Two questions:

1. What are the foundational principles you are talking about?

2. How often have you seen students develop the standard algorithm for long division after repeatedly applying those principles?

(Beginning with higher-level college math, the foundational principles are those of set theory, and they are used to prove, e.g., that 0 * 0= 0--a proof that is generally not left up to students to discover it on their own).

Katharine Beals said...

"Simply memorizing procedures does not by itself always result in the comprehensive understanding of arithmetic needed to easily transition to abstract mathematics."

"does not by itself always" shouldn't rule out using this pedagogical tool.

I'm a programmer, and often find, when learning a new language, that repeated use of chunks of computer code I've mindlessly cut and pasted from elsewhere eventually leads me to conceptual understanding of that code.

Same is true in learning another language in an immersion context: you find yourself memorizing chunks of unanalyzed speech, and then, through repeated trial and error, you eventually figure out the underlying linguistic concepts and how they fit together. Yes, you could also go the route of first understanding the underlying syntax and semantics of the language in question, but this route doesn't work for everyone, and many people like to start out with something more concrete.

Catherine Johnson said...

I mentioned the other day that I'm trying to figure out the relationship of 'basal ganglia' learning to giftedness (and to everything else).

Basal ganglia learning is:

* associative & probabilistic - you learn the associations in your environment, which means you learn how to predict your environment. "Where there's smoke, there's fire" is associative learning.

* skill learning - which includes cognitive skills - AND there is a huge amount of skill learning in any field or discipline. Thus, learning to ride a bike is skill learning AND learning to play chess or "do math" is skill learning.

You can see from this (probably) why 'meaning' and 'skill learning' are connected, or likely to be connected. The same part of the brain figures out which things go with which AND handles procedural learning.

Another part of the brain handles the learning of facts and meanings. In other words, remembering the date of your birthday isn't the basal ganglia's job. I think the hippocampus handles facts and other non-skill content.

As far as I can tell at the moment (this is what I surmise based in what I've read thus far), 'basal ganglia' learning does not involve conscious memorization. Basal ganglia learning seems to happen after repeated exposures to content or after repeated practice.

Where am I going with this?

Good question.

I guess I'm saying (surmising) that memorization and "cognitive skill learning" are two different things inside the brain. Both appear to involve "meaning" and "understanding."

Also, both forms of learning depend upon spaced repetition (or distributed practice). Generally speaking, you don't memorize a fact after one exposure (except under certain conditions...). Nor do you acquire a cognitive or motor skill after one exposure.

I realize this isn't especially clear ... as I get a better grasp of the field I'll post more.

Catherine Johnson said...

One more thing, just in terms of memorization, procedural proficiency, etc.: an awful lot of expertise is unconscious.

When it comes to cognitive skills, most experts can't tell you how they do what they do. That's why the field of psychology that studies the "cognitive unconscious" came into being. iirc, psychologists were concerned about doctors transferring their knowledge to medical students. They wanted to make sure knowledge was passed on. Their first thought was simply to ask doctors how they did what they did -- and soon found that the doctors didn't know and couldn't say. (Or, alternatively, experts can believe they are doing something they are not.)

Also, creative experts - this includes mathematicians - often come up with their insights and hypotheses (and elements of proofs, presumably) via unconscious processes, such as sleep. (There is research showing that people solve problems much more creatively after they 'sleep on it.')

The fact that so much of our intellectual life is unconscious implies to me that both the benefits of having kids consciously explain what they are doing AND the negatives of having kids memorize are probably overstated.

One way or another, you need to get 'content' and 'skills' into long-term memory, where the cognitive unconscious can work on it.

Erin Johnson said...

The fact that pretend-math programs, such as Investigations, have co-opted the term “conceptual understanding” does not negate the fact that a thorough understanding of arithmetic requires conceptual understanding, procedural fluency and the ability to problem solve by applying foundational principals to unfamiliar problems. Liping Ma’s book clearly demonstrated why both are needed. Singapore math does develop these concepts quite well without the formalism used in more technical mathematics.

While Investigations has obvious defects in the lack of procedural development, what is less apparent and more pervasive is the promotion of “concepts” that are misleading or wrong (e.g. “number decomposition can replace place value in addition and subtraction” or “the fraction ½ is a concept” or “division is repeated subtraction”). Please do not credit Investigations (or any other Fuzzy pretend-math program) with developing any of the foundational concepts needed to master arithmetic.

“1. What are the foundational principles you are talking about?”

Some foundational arithmetic concepts include: number decomposition and composition, place value, regrouping, equal grouping, equivalency, transformability of expressions, distributive property, commutative property, fraction as a number, fraction as division, etc.

“2. How often have you seen students develop the standard algorithm for long division after repeatedly applying those principles?”

All the time.

Unlike the claims by fuzzy proponents students are unlikely to “discover” efficient algorithms by themselves. So the question then is: what are the foundational concepts needed to both understand and be proficient with long division? Foremost concepts are place value, regrouping and equal grouping. Proficiency proceeds the quickest with conceptual understanding, doing long division the laborious way, understanding how long division is a short-cut notation for multiple conceptually dense steps, and tremendous amount of varied practice.

“Long Division” the laborious way: Decompose the dividend into place value components (Visually done by using place value coins and a place value table). Identify the number of equal groups that the divisor represents and draw out boxes of equal size that represents the number of the divisor. If of sufficient quantity, distribute the highest value place value components of the dividend into equal groups. If there are extra place value components regroup into the lower value component. Repeat until all of the dividend is equally distributed into the divisor boxes. (Obviously, using this terminology with elementary school children would be confusing and counter-productive.) Lots of practice. Followed by demonstrating why the long division algorithm is a short-cut compared to laborious method. Lots more practice. Concepts reinforced: place value, regrouping, division as distributing among equal groups. All concepts that are essential for abstract math as well.

Certainly, memorization is a very viable technique for learning. And certainly having memorized scripts in one’s head can be quite useful at some later point in time when trying to refine one’s own understanding.

But unlike language, arithmetic is well characterized and it is not necessary to delay that leap to understanding while developing proficient procedures. We know the concepts and how those concepts are embedded in the procedures. A well designed math program can build arithmetic understanding by properly developing the concepts that are intimately intertwined with procedural fluency. It is easier to learn and more satisfying for the student to really understand what they are doing while they are doing it.

Katharine Beals said...

"But unlike language, arithmetic is well characterized" I'm a linguist (in addition to being a programmer), and can say with confidence that no linguist would agree with that statement.

For many people, including linguists, learning a language's superficial features can precede learning its underlying structure--without any pedagogical cost. In fact, this happens all the time with our native languages. The superficial features then become the data on which the rules about underlying structure are based.

le radical galoisien said...

language is very mathematic-like actually...from sound processing to syntax

Erin Johnson said...

"For many people, including linguists, learning a language's superficial features can precede learning its underlying structure--without any pedagogical cost. In fact, this happens all the time with our native languages."

Several authors have suggested that academic language is quite unlike the conversational language learning that is aquired by young children. And given that academic language is rarely spoken but largely encoutered in written form, an explict teaching of that specific language is necessary. Could you comment on the quote from Schleppegrell in "The Language of Schooling"?

"Students who encounter academic registers in contexts outside of school may be able to draw on this exposure and the implicit knowledge about language it engenders for success in school-based tasks. Students from other background, however, may need an explicit focus on the form language takes to raise their awareness of how different grammatical choices are functional for achieving particular goals in their writing and for analyzing the points of view that are naturalized in the texts they read. ... an explicit focus on the linguistic features of the language of schooling can raise students' awareness of the choices available to them for writing the texts of schooling and can help reveal the layers of meaning in the texts they read."

This would seem to imply that for academic language acquisition, explicitly teaching the structure of academic language (as opposed to just the superficial elements) is necessary for students to fully master the content domain language .

Niels Henrik Abel said...

Catherine: Also, creative experts - this includes mathematicians - often come up with their insights and hypotheses (and elements of proofs, presumably) via unconscious processes, such as sleep. (There is research showing that people solve problems much more creatively after they 'sleep on it.')

My personal anecdote confirming this observation:

I remember being stumped on a topology problem in grad school - I could get the rest of the problem set, but that particular one eluded me. I finally gave up and handed the homework set in. After returning home and taking a nap (much needed due to having stayed up into the wee hours of the morning), the solution occurred to me while I was in that not-fully-asleep-yet-not-fully-awake state. I woke up immediately and phoned the professor, explaining that I had just handed my homework in but now I knew the solution to the unfinished problem. I outlined the solution over the phone, and he let me write it up for full credit :)

Katharine Beals said...

"This would seem to imply that for academic language acquisition, explicitly teaching the structure of academic language (as opposed to just the superficial elements) is necessary for students to fully master the content domain language ."

I believe you're confusing underlying linguistic structure with rules of style. Linguistic structure is highly abstract, and most non-linguists don't learn it explicitly. You don't need comprehensive knowledge of a language's underlying linguistic structure in order to become fluent in that language--even in the case of a second language--or in order to master the formal/written register of the language.

Erin Johnson said...

"Linguistic structure is highly abstract, and most non-linguists don't learn it explicitly."

Schleppegrell asserts that these academic linguistic elements are not elements of style but are essential components of content discourse and thus are needed to become proficient within the subject matter domains. And additionally, because these academic language structures are not typically enountered in everyday conversation, they need to be explicitly taught. Is she mistaken?

And back to arithmetic: Liping Ma asserts that procedural experience is insufficient to provide the experience necessary for students to intuit the underlying conceptual structure of arithmetic. And in fact she postulates that teachers are critical in making the conceptual connections to the underlying arithmetic structures so that students can truly master and become proficient at arithmetic. Is she mistaken?

Barry Garelick said...

Conceptual understanding and procedural fluency are not mutually exclusive. Wu in his paper about this ("A Bogus Dichotomy") emphasizes that they work in tandem. Many people mischaracterize traditional math as ignoring the conceptual underpinnings. The conceptual context is there, and procedure is rarely presented in isolation of its application in word problems.

Many times, mastery of the procedure allows students to understand the underlying concept. I recall in some math courses, like differential equations, reading the theoretical explanation for a certain type of equation, but not following very well. I then started working the problems, imitating the procedures outlined in some of the worked examples. After getting to a certain level of proficiency, I was then able to re-read the explanation and understand what was going on. If this is what you are talking about then I agree. If, however, you believe students must experience the laborious version in order to understand the "shortcut", I disagree.

Bostonian said...

Part of the reason algebra is taught is that it is useful. However, another reason that algebra and higher math such as calculus is required for many professions is that they function as an IQ test filtering out people who are not smart enough to grasp those subjects (and also as tests to see who will put forth the effort to learn those subjects). Many medical schools expect students to have taken a year of calculus, but I'd be very surprised if the courses in medical school use calculus. Certainly doctors do not in their daily work. Medical schools are implicitly using calculus requirements as tests of IQ and effort.

le radical galoisien said...

academic writing is simply, another register.

though whenever I try to pronounce academic words (especially those I learnt early in life) sometimes they come out funny, based on how I thought it should be read as a 10-year-old.

Katharine Beals said...

'Schleppegrell asserts that these academic linguistic elements are not elements of style but are essential components of content discourse and thus are needed to become proficient within the subject matter domains. And additionally, because these academic language structures are not typically enountered in everyday conversation, they need to be explicitly taught. Is she mistaken?"

If she thinks that "academic linguistic elements" are the same as "abstract linguistic structure," then she is mistaken.

I don't know what she/you mean by "academic linguistic elements". Do you know what I mean by "abstract linguistic structure"? Unless we know we're talking about the same thing, it's silly to keep talking about it.

Erin Johnson said...

"Many times, mastery of the procedure allows students to understand the underlying concept."

Of course, some students can intuit the concepts from learning procedures. But is this the most efficient manner in which to reach mastery of arithmetic for a large cohort of children?

Certainly, the Singapore math program, Liping Ma, etc. would argue against that postiion.

The entire point about the bar graphs in Singapore math is to make visual the conceptual elements of the problem. Drawing out bar graphs takes time, it is slow and laborious at first, but it allows the students to conceptualize the problem before solving. And yet that visualization/conceptualization allows a greater efficiency, understanding and transferability in solving complex problems.

"I recall in some math courses, like differential equations, reading the theoretical explanation for a certain type of equation, but not following very well. I then started working the problems, imitating the procedures outlined in some of the worked examples. After getting to a certain level of proficiency, I was then able to re-read the explanation and understand what was going on."

You are recalling what a fairly proficient student can do after developing a substantial base of knowledge. Arithmetic is taught to fairly young children who are not highly self-aware (nor should they be asked to be) and so the structure of the program needs to be quite different.

"If, however, you believe students must experience the laborious version in order to understand the "shortcut", I disagree."

Laborious does not mean slow. Even learning to memorize the procedures take time. My experience is that the laborious, conceptual route takes significantly *less* time to master the algorithms than just introducing the algorithms by themselves, memorizing the steps and practicing enough for fluency. I don't think that students should be wasting a lot of time with algorithms and finding the most efficient route to mastery should be the goal of any arithmetic program. Learn why algorithms they work, how they work, become proficient with their use and move on to the more important elements of arithmetic: solving complex multi-step problems.

"I don't know what she/you mean by "academic linguistic elements". Do you know what I mean by "abstract linguistic structure"? Unless we know we're talking about the same thing, it's silly to keep talking about it."

Agreed and given that I did not write that monograph, perhaps I should defer that more explicit definition to you.

I think on the larger scale, what we are discussing falls under the the debate of the whole-to-part vs part-to-whole approach to learning. Certainly, my experience is that, if possible, part-to-whole works more efficiently for the largest number of children. But to teach that way all the parts need to be well defined (and in arithmetic they have been extensively if not completely characterized). For many elements of learning, particularly in language, that "part" structure is poorly defined or non-existant. (Vocabulary learning is largely best learned by whole-to-part while decoding is best learned part-to-whole.)

le radical galoisien said...

I would turn off from a "how to write/read academically" seminar, so much. I expect most of my peers would too.

I think it's a register that can be acquired naturally, i.e. by writing correspondence.

Erin Johnson said...

Schleppegrell's assertion is not that there is "one academic register" but multiple subject specific registers with subtle but important language differences necessary for effective communication in the content areas. I would imagine that this would be of interest to people who are teaching children to communicate in the content areas and particularly those teachers who are working with children who have not experienced those academic language registers outside the school setting. As she states: "A more nuanced understanding of the role of language in schooling recognizes that students' difficulties may be related to inexpereience with the linguistic demands of the tasks of schooling and unfamiliarity with ways of structuring discourse that are expected in school."

"I think it's a register that can be acquired naturally, i.e. by writing correspondence."

If this were true for all students then why is it that so many students fail to become proficient at academic reading and writing?

le radical galoisien said...

they don't have enough out-of-classroom practice.

really it's the same with French or another foreign language

investing a little effort into writing a poem or short story of your own accord out of your own interest makes you discover idioms, discuss usage with people, etc. that you wouldn't know otherwise

it helps to know vocabulary and to read Wikipedia, Google Scholar etc. on your own time

Katharine Beals said...

Who is Schleppegrell and why should we care what she "asserts"? All sorts of people (with and without PhDs) have asserted (and even published) all sorts of strange ideas about language and "texts" and "contexts"; unless it's more than armchair theorizing (i.e., in this case, backed up by empirical studies of how children acquire different "subject specific registers", complete with a definition of key terms, including this last one) I see absolutely no reason to pay attention to her.

Erin Johnson said...

She has a PhD in linguistics and is currently a professor at University of Michigan specializing in academic language development.

Her work:
"The Language of Schooling: A Functional Linguistics Perspective"
"Reading in Secondary Content Areas"

You certainly don't have to pay attention to her work if you choose not to. But as a linguist that is also interested in education, you might be interested in the application of linguistic theory towards learning in the content areas.

Katharine Beals said...

As a linguist who is also interested in education, I am indeed always thinking about connections between learning and language. Here, the most interesting and evidence-backed theories are in psycholinguistics. Once people start making proposals about how children learn, and recommendations about how teachers should teach, empirical research is key. I've taken a "look inside" the first of the Scheppegrell books you list via its Amazon page (the second book doesn't allow me to look inside), and, on the topic of "academic registers," it doesn't look promising on the empirical front. Instead of references to experiments, we have references to what other people "suggest" and "argue":

"Veel (1997) suggests that academic registers construe "distinctive and favored ways of viewing the world; ways which we recognize as 'scientific,' logical,' and 'rational'" (p. 161) Similarly Lemke (1987) argues that what we call "thinking logically" is for the most part simply using language... according to genre patterns ... to teach"

The empirical research she cites seems to pertain more to well-known facts about language development in general (e.g. increasing sentence complexity); once she moves into "genre patterns," we seem to have entered a house build on sand.

le radical galoisien said...

I think the whole issue of too much theorising and not enough experiment may apply to Ed School "paradigms" as well

besides I don't know how you'd implement a program to teach people how to read/write academically. I think it'd be rather awful. Neither people who are already good nor people who are "bad" would tolerate it very well. I really think it's a vibe you pick up ... you try to sound precise, neutral, etc. with language reflecting as little personal judgment, prejudice, etc. as possible and it just flows out that way.

And half of the other issue is just jargon specific to the field being specialised in.

I think a 12-year-old could write academically. Now instead of say, physical chemistry or chromosome rearrangements, get her to try neutrally analyse the rhetorical techniques of Miley Cyrus and her appeal, and we lose most of the jargon issue.

Erin Johnson said...

"...once she moves into "genre patterns," we seem to have entered a house build on sand."

So would you say, then, that there are no differences in language (vocabulary, syntax, grammar, textual structures, ellipsis, background knowledge, etc....) between genres/subject matter domains or that those differences are inconsequential in student learning? Her point is, of course, that those subtle differences between academic discourse can be quite confusing to students who do not have a home life that is filled with academic language. Hardly a radical position, inconsequential position.

What I find bizarre in your comments of whole vs part learning is that your GrammarTrainer is so well developed in capturing the essential parts that are needed to understand language. Not only does it partition out the needed subskills in learning so that kids can access the substructure of language instead of just the superfical aspects, it also sequences the learning to carefully build the whole (language). So why are you such a strong advocate of "whole language" when your program necessitates delineating the parts while building the whole?

In a previous comment you stated that "For many people, including linguists, learning a language's superficial features can precede learning its underlying structure--without any pedagogical cost. In fact, this happens all the time with our native languages."

My original referencing of Schleppegrell was in response to your (above) comment that all linguists would disagree with the assertion that "unlike language, arithmetic is well characterized: I'm a linguist (in addition to being a programmer), and can say with confidence that no linguist would agree with that statement."

I would assume that linguistic discourse is similar to the hard sciences and that real consensus is often difficult to find, thus making statements of "all linguists" or "all scientists" rarely justified.

"I don't know how you'd implement a program to teach people how to read/write academically. I think it'd be rather awful."

Based upon what evidence?

"I think a 12-year-old could write academically"

Based upon what evidence?

"...rhetorical techniques of Miley Cyrus and her appeal"

And this is now considered "academic"? No wonder our schools are in such dire need of reform.

Katharine Beals said...

'"...once she moves into "genre patterns," we seem to have entered a house build on sand."
So would you say, then, that there are no differences in language (vocabulary, syntax, grammar, textual structures, ellipsis, background knowledge, etc....) between genres/subject matter domains or that those differences are inconsequential in student learning? '

First, your conclusion about what I would say is a non-sequitor. I was writing about how recommendations for teaching need to be based on empirical results, not armchair theory.

Second, there are obviously differences in vocabulary and background knowledge from field to field. This is hardly a revolutionary point. But background knowledge isn't language, and, linguistically speaking, vocabulary is a superficial aspect of language. Furthermore, I don't now anyone who would disagree that people need to learn the assumed background knowledge and vocabulary of a given field before becoming an effective reader of its texts.

Third, while academic writing involves different statistical *rates* of different grammatical structures, it doesn't involve grammatical structures that aren't found in non-academic speech.

Fourth, "textual structures" isn't a well-defined linguistic term.

Fifth, different academic fields don't involve different grammatical structures. E.g., you don't find different sorts of elipses in history texts than you find in economics text.

Sixth, the only case where grammar rules need to be taught explcitly are with nonnative Standard English speakers, and children with language delays (that's what GrammarTrainer does).

"What I find bizarre in your comments of whole vs part learning is that your GrammarTrainer is so well developed in capturing the essential parts that are needed to understand language."

First, GrammarTrainer is informed by psycholinguistic findings about language delayed children, and about language acquisition.

Second, I haven't made any comments "of whole vs. part learning."

'"My original referencing of Schleppegrell was in response to your (above) comment that all linguists would disagree with the assertion that "unlike language, arithmetic is well characterized: I'm a linguist (in addition to being a programmer), and can say with confidence that no linguist would agree with that statement."'

None of what you've quoted from Schleppengrell shows a linguist disagreeing with the assertion that language is well-characterized.

"I would assume that linguistic discourse is similar to the hard sciences and that real consensus is often difficult to find, thus making statements of "all linguists" or "all scientists" rarely justified."

There is consensus on basic, well-established notions. There is also consensus among (true) biologists about evolution, and among (true) astronomers about the (round) shape of the earth.

Erin Johnson said...

Perhaps you have a technical definition to "well-characterized" that I am not aware of.

My use of the term was strictly in terms of educational practice, in that academic language acquisition is not well-characterized (unlike arithmetic which is well-characterized), as much about what and how students acquire proficiency in the content areas is not well known.

Your comments regarding memorization first (for both programming and language) and understanding second are commonly categorized as a "whole-to-part" strategy of learning.

le radical galoisien said...

sorry. I must frankly be pained at the sheer. lack. of. science. Linguistics isn't into wishy-washy English lit textual criticism. Linguistics is the science of language: language processing, acquisition, from the canonical forms of syllable structure you can write on paper to FOXP2 genes you can sequence. Thank goodness.

"So would you say, then, that there are no differences in language (vocabulary, syntax, grammar, textual structures, ellipsis, background knowledge, etc....) between genres/subject matter domains"

A few self-evident statements.

1. Spoken language is THE language i.e. written language is dependent on spoken language.

2. Academic language uses grammatical structures already present in a language. It just uses them with a lot more frequency. That is to say, the variance of grammatical forms, etc. is actually reduced.

3. It's hard to get a divergence in actual grammar unless there is a divergence in spoken languages. This can happen among people specialising in different trades that over some period of time become isolated from each other, e.g. "argots". Kind of hard to do it with written literature.

4. For this matter, literature tends to be unity-promoting, not divergence-promoting. Classical Chinese separated from the vernacular dialects because people actually spoke it in daily life, i.e. used it to recite poems and communicate across disparate regions lacking a common language, i.e. it was the equivalent of Latin.

I really can't see that happening with academic English.

le radical galoisien said...

"My use of the term was strictly in terms of educational practice, in that academic language acquisition is not well-characterized (unlike arithmetic which is well-characterized), as much about what and how students acquire proficiency in the content areas is not well known."

I really don't comprehend this sentence.

Now if you're talking about how much we understand about language acquisition versus arithmetic acquisition, the truth is, we don't actually know a lot about either. In fact our theories of learning anything are embarrassingly vague.

Nevertheless, there are interesting phenomena to observe...

Ever heard of the Piraha? Well some anthropologists studied a group sort of like them. There was an interesting article about it. The key finding of the researchers is that the "natural state" of people is to have a logarithmic sense of the number line, e.g. the *absolute* distance between 10 and 11 is smaller than the distance between 2 and 3. (There was a rather clever experimental design to sound out these intuitions.) That's because in "the state of nature" it was much more useful to compare relative sizes rather than to do sort of any absolute counting (is that bear bigger than me? is this fruit tree better to pluck from than this one? do I have enough? etc.) and arithmetic developed only because it was a wa of making sure one wasn't getting ripped off in trade.

e.g. formal math skills developed out of the need for commerce.

kindergarteners start out with a very logarithmic sense of the number line (yes, despite their not knowing what logarithms are), but it steadily "flattens" to a straight line as they proceed to second and third grade.

this is not observed with tribes in Brazil that don't have math classes. They don't have distinct words for anything bigger than five.

still despite this our knowledge of how we learn arithmetic is still very ill-characterised. Certainly we haven't characterised any sort of neurobiological circuit that would act as an adder or a subtractor, as we have done with computers. Heck, we have characterised the neural circuits tht give rise to central pattern generators that create sinusoidal waveforms for constant swimmers in dogsharks because of mutual inhibition.

but we haven't characterised arithmetic acquisition that well -- neurobiologically.

linguistically I think we're actually a little better off because we have historical linguistics data and much more case studies and interesting (very telling) phenomena like creolisation.

Barry Garelick said...

The bar models that Erin talks about provide a mode to solve problems. It does shed some light on the conceptual underpinning, but by and large functions to let the student solve the problem fairly efficiently. I would not deem it as laborious, in the same manner as requiring students to list three ways to add two three-digit numbers, or draw a rectangle model to illustrate what 4/5 X 2/3 is, which some of the reform texts do.

The problem solving method does allow students to see more of what is going on in part/whole relationships. It is not a be-all end-all in and of itself. Singapore works well because of a number of things. Problem solving via bar modeling is one of them, but it is also the sequencing of topics and the building upon mastered material. Students still are required to solve many numerical problems as well as multi-step word problems.

The idea of procedural fluency leading to understanding is done in tandem with concepts, such as Singapore does. But a more complete understanding will probably come later--particularly when they enter algebra when they have more symbolic tools with which to analyze and set up problems.

The example I gave of differential equations was meant to illustrate how you get to the understanding later. In college, there is a shorter time period that allows one to go back and understand the theory because there are more schema than what kids in grade school have. Thus, the conceptual understanding that kids have of why algorithms work, even those who come from the Singapore program, is going to be somewhat limited. But the foundation is there upon which to build, which I think is what you are saying and I agree with that.

Keith Devlin has an interesting article about procedural fluency and understanding here.

Erin Johnson said...

I fully agree with you regarding "reform math".

To clarify our discussion, conceptual understanding can come in two broad strokes: a working knowledge of the concepts and a formal description of those concepts. What students should be developing in arithmetic is a solid working knowledge of the concepts. This does not mean that they have to make explicit definitions of place value or the distributive property, but they should be able to use both in calculations and problem solving. A solid conceptual understanding in arithmetic can be characterized by the ability to use those fundamental arithmetic concepts and skills in solving novel, unfamiliar problems.

Regarding Keith Devlin's article: he makes an interesting point "one of the things that high school mathematics education should definitely produce is the ability to learn and be able to apply rule-based symbolic processes without understanding them." He is, of course, referring to the mathematics learned in high school which by definition is abstract and not concrete.

But arithmetic is by definition concrete. It deals with real numbers and how to solve problems with those numbers. It is not necessary to deal with arithmetic in the abstract as is the case with more advanced mathematics. But is is necessary in arithmetic to lay the conceptual foundation for being able to think abstractly. That is done by developing a working knowledge of the foundational concepts in arithmetic (place value, equivalency, etc...)

I am, if nothing else, extremely practical. If memorization of procedures was the quickest road to mastery, I would have whole-heartedly embrace that approach. But after reading Liping Ma and using Singapore to teach students math, it became exceptionally clear that coupling conceptual development with procedural fluency was *substantially* faster than memorization and led to an almost seemless transition to abstract math (Algebra). Singapore is not procedural fluency coupled with a few concepts. At its core is the idea that both concepts and the procedures are necessary for developing arithmetic mastery.

I realize that the press by the fuzzy math guys has soured many on the idea of conceptual development. And if your only exposure to teaching "conceptual math" came from TERC, I could see why you would be horrified. But we shouldn't let the Fuzzys take away one of the more insightful ideas that has come out of international comparisions:
concepts + procedures + problem solving => fast and efficient arithmetic mastery