The Math You Don't Learn is Harder Still

[Moonbear]

We also have to remember that the vast majority of high school students are not going to go on to advanced math courses. They don't need to learn calculus or how to do proofs; they need to know how to balance their checkbook and figure out how many cans of paint they need to paint the house.

 

[physics girl phd]

If it were your slide rule...

Well, I inherited it. Had to fight Mom for it.

But everyone in the sisterhood rules. Smart women are ALL sexy.

 

[Bystander]

The "Dummies" and "Idiots" series don't assume a "complete" background of the reader --- they review/summarize the necessary background (when properly done --- the half dozen or so I've skimmed), and tie a subject into an integrated picture for the reader rather than treating it in isolation.

 

[Evo]

In my job, if I didn't have a calculator, I'd have to do long division several times a day. Luckily I do have a calculator, but knowing that if I didn't I could actually do the long division is a nice thing. Sometimes if it isn't that long, I'll do it on paper. The calculator saves me time. How can anyone think that knowing how to do long division can be a bad thing?

I have to create legal documents that have to be 100% accurate. I cannot guess at the numbers I quote. A wrong answer can result in a lawsuit. No one checks my numbers for accuracy when it goes to legal. Now there's a scary thought.

 

[whitay]

We also have to remember that the vast majority of high school students are not going to go on to advanced math courses. They don't need to learn calculus or how to do proofs; they need to know how to balance their checkbook and figure out how many cans of paint they need to paint the house.

School has taught me a minimum amount of calculus. I wouldn't mind learning proofs. I can balance check books, I can do quite a variety of accountancy work as I've done a fair bit of voluntary work experience at 2 accounting firms. However I don't know how many cans of paint i'd need to paint a house, mainly because I am not into manual work.

 

[Moonbear]

However I don't know how many cans of paint i'd need to paint a house, mainly because I am not into manual work.

But if you can read a label (each paint can tells you the approximate square footage of coverage) and work a tape measure, you can be sure the painter you hire isn't ripping you off by telling you you need to pay for twice as many cans as are really needed for the job.

I got a good deal on the car I recently bought because I could do math in my head faster than the salesman...he was trying the old negotiate the trade-in and new car price simultaneously trick, but it backfired when I gave him the figure I'd pay for the car with the trade-in value he was asking, and he did the counter-measure of trying to meet me in the middle, so I countered back with "if you increase the trade-in value to X, I'll go with that amount on the car." I increased the trade-in value by the amount he tried increasing the price of the new car (minus about $6 so the number didn't easily round off in his mind), and he didn't realize it until he agreed to it and then started punching the numbers into his calculator.

 

[0rthodontist]

One exercise that I think is good for jr high/early high school: give a student a ruler, a calculator, a random size photograph and a piece of paper, and tell them to enlarge or shrink the photograph as needed so that they can draw it as large as possible to fill the piece of paper (and no marking up the photograph, and it's best if the photograph is moderately complex so they can't copy it just by eye). I did this in art a few times (8th or 9th grade I think) and that's how I got the hang of ratios. The bonus is that you immediately get to see the result of your work. I tutored a couple of kids in high school and tried to get them to try it, but I don't think they ever did.

 

[arildno]

I have a notion that all learning is rote at first.

I disagree somewhat.
A learning process is also to coax out in the open (i.e, as writing down) ideas and ways of thinking the kid already is using.

For example, most of the axioms of addition are very "trivial" in the sense that just about any kid would agree that the properties hold for, say, for sticks of differing lengths. Similarly for axioms of multiplication.

 

[Gokul43201]

We also have to remember that the vast majority of high school students are not going to go on to advanced math courses. They don't need to learn calculus or how to do proofs;

I disagree. If nothing else, these things are valuable towards developing any semblance of logical and critical thinking.

Honestly, I find it quite mind-numbing to hear that you can "learn" math without learning "to do proofs"! What the hell kind of math is this anyway?

 

[Astronuc]

I disagree. If nothing else, these things are valuable towards developing any semblance of logical and critical thinking...something I find grossly lacking even among the educated folk here.

On the other hand, I am concerned that a vast majority of students do not develop a sufficient level of logical and critical thinking.

Honestly, I find it quite mind-numbing to hear that you can "learn" math without learning "to do proofs"! What the hell kind of math is this anyway?

Math for dummies?

 

[arildno]

I disagree. If nothing else, these things are valuable towards developing any semblance of logical and critical thinking.

Honestly, I find it quite mind-numbing to hear that you can "learn" math without learning "to do proofs"! What the hell kind of math is this anyway?

In general I'd agree with you.

However, I might accept to dispense with rigorous proofs insofar as the properties of the "intuitive" model does not conflict with the properties gained from a more rigorous construction.

For example, to think of the real numbers lying on a line, and "deducing" properties about numbers by explicit reference to this model is probably a better way to learn about (and understand!) our numbers than a careful construction of equivalence classes of sequences of rational numbers.

The line model captures the properties of the real numbers very well..

Another example would be to use "geometric definitions" of the trig. functions, even though for rigour, we should regard sine and cosine as solutions of a particular boundary value problem.

 

[ChemGuy]

This problem is not new. It has been a problem for years. As a student in a "good" high school (in the 80's) I specifically took classes that had been outlined as "Pre-med". I supposed it should have been pre-pre-med! It included chemistry and physics. When I graduated and went off to college I had to take algebra on up to get up to speed. It was very depressing. I finally got a BS in Chemistry and Physics but it took me an extra year to do it manly due to the remedial math I needed. I thought I was getting prepared for college in high school. I guess not.

In grad school I saw many many students who were not prepared for college level math. I am not talking about calculas I am talking about algebra.

However now that I have sixth graders I can say that their math assignments are what I think is on track. They are doing multiplication and long division of decimals and are doing it the old fashion way...by hand. They have calculators which are a real benefit to them. They do a problem and show the work. They can then check the answer on the calc and if they disagree they can go back and figure out how to correct it. They learn a ton this way. In addition they are doing a ton of word problems. This eally helps them understand the concepts and application.

 

[0rthodontist]

I disagree somewhat.
A learning process is also to coax out in the open (i.e, as writing down) ideas and ways of thinking the kid already is using.

For example, most of the axioms of addition are very "trivial" in the sense that just about any kid would agree that the properties hold for, say, for sticks of differing lengths. Similarly for axioms of multiplication.

But they can't prove it. That just means that they "memorized" it already from real-world examples. I agree that it's good to draw on prior knowledge, but if you spend all your time hoping your students will have a flash of insight you'll never teach anything. Give them the raw materials first, which means teaching them memory. Any student should have a good enough memory for symbols to be able to do everything mentally, if given enough time. Only once they are familiar with the operations can you make them understand it. A lot of students feel alienated from math because it seems to them like a whole bunch of unfamiliar symbols. If you make the symbols more familiar by having them develop a memory for them, they won't have that problem.

Also, I think (tentatively) that a good part of the curriculum to introduce a student to proofs is to have them memorize good proofs so that they can recite them. Give homework assignments including parts like "handwrite each of these proofs 5 times." Then get them to present memorized proofs in their own words. This would be in addition to having them work problems.

 

[russ_watters]

Once you've figured it's about 52, and want more detail, is it really that hard to multiply 27 by 50 (1350) and subtract from 1479 (129), at which point you know it's more like 54.5?

Well now we're up to what - 6 steps? I don't know, maybe I'm not that adept at it because I never try it, but to me it seems like a lot more trouble than its worth. Yeah, maybe we're talking 12 seconds for that vs 2 for adding two numbers, but since I'm almost always in front of a computer (or have a cell phone on me), it is faster (more importantly, more accurate) to do it electronically.

 

[physics girl phd]

I have a notion that all learning is rote at first.

I disagree. I'm mostly thinking about early cognitive development in children -- where learning is very much about experimentation -- testing different situations and analyzing responses. This includes language development and spatial conceptualization (which I'd count as early math). This includes testing related situations... if the stick goes into the jug and I can peek into see it inside, will the rock do the same? I don't think that is rote.

But the argument of didactic versus constructivist education aside -- there is the question of teaching a formalism. Thinking back to the analogy of language -- How often do I diagram a sentence? (Never) Does this mean the skill of diagramming sentences should not be taught? I use this analogy because maybe because I view long division as a deconstruction of the division process in the decimal system... and diagramming as the deconstruction the sentence structure in our language. I distinctly remember that my understanding of language/grammar increased when I was taught the technique of diagramming. Now maybe some would argue that grammar is not requisite to communication. But I think we need formalism (math itself is a formalism, right?)... and I feel skills that increase the understanding of the formalism should probably be taught to students, even if those skills will not be daily skills in the students' future lives.

Now, I'm not sure -- is sentence diagramming taught in other educational systems than that which I experienced (US)? I put that up as an inquiry... I'm personally a big fan.

 

[Bystander]

If the EEs have done their jobs correctly --- original 486, couple quirky HPs, don't know 'bout TIs, Commodores, or others.

 

[russ_watters]

I got a good deal on the car I recently bought because I could do math in my head faster than the salesman...he was trying the old negotiate the trade-in and new car price simultaneously trick, but it backfired when I gave him the figure I'd pay for the car with the trade-in value he was asking, and he did the counter-measure of trying to meet me in the middle, so I countered back with "if you increase the trade-in value to X, I'll go with that amount on the car." I increased the trade-in value by the amount he tried increasing the price of the new car (minus about $6 so the number didn't easily round off in his mind), and he didn't realize it until he agreed to it and then started punching the numbers into his calculator.

Nice.

Along a similar veign, though, I'm a little bit anal about loose change (pennies, especially) - I like to get rid of it. I'd say that unless a cashier can enter the number into his register, there is no hope of getting proper change by giving them $5.07 for a $4.82 purchase. So don't get me wrong - a certain amount of doing-it-in-your-head is useful.

 

[0rthodontist]

I disagree. I'm mostly thinking about early cognitive development in children -- where learning is very much about experimentation -- testing different situations and analyzing responses. This includes language development and spatial conceptualization (which I'd count as early math). This includes testing related situations... if the stick goes into the jug and I can peek into see it inside, will the rock do the same? I don't think that is rote.

You lost me with the stick/jug/rock thing.

Experimentation is good, but it's not the first step. Students should be given tough problems that they will have to experiment to solve. But before you can experiment, you need to know the basics. If you had no idea what a stick, a jug, or a rock is, you'd have trouble doing whatever it is you're talking about. How can a student with a shaky grasp of the manipulations of basic algebra do any experimentation at all with it?

But the argument of didactic versus constructivist education aside -- there is the question of teaching a formalism. Thinking back to the analogy of language -- How often do I diagram a sentence? (Never) Does this mean the skill of diagramming sentences should not be taught? I use this analogy because maybe because I view long division as a deconstruction of the division process in the decimal system... and diagramming as the deconstruction the sentence structure in our language. I distinctly remember that my understanding of language/grammar increased when I was taught the technique of diagramming. Now maybe some would argue that grammar is not requisite to communication. But I think we need formalism (math itself is a formalism, right?)... and I feel skills that increase the understanding of the formalism should probably be taught to students, even if those skills will not be daily skills in the students' future lives.

I agree that long division should be taught, but I don't see how it increases understanding of division. It increases manipulative skill and number memory, which are important things.

 

[turbo]

My wife does all the shopping now, but back when I was doing some of it, I was amazed how many cashiers would just "blank out" when I did an on-the-fly calculation to rid myself of pennies while getting a more useful coin (dime, quarter). Some of them would catch on to the dime, but overpaying to get a quarter seemed to throw them. That is a really low-functioning person (mathematically) to trust running a cash register.

 

[physics girl phd]

I don't think that ALL learning is experimentation. I just disagree with you that "it all starts with rote" and builds up from rote skills.

You lost me with the stick/jug/rock thing.

I think "learners" learn what a stick/jug/rock is by experimentation... I think they first learn that they can grab and "feel" sticks and rocks, but that these things are not good for eating... but that they are tangible and solid, and stay there as an "object" (although they don't have the vocab of the object -- just the concept of it). Repeated experimention seals this as a "rote" concept. Then they experiment and discover it's interesting to make objects "disappear" in a "jug," etc.). 1-2 year-olds love the rock/stick/jug thing... or the sock/box thing...

Now -- on to the "algebra" thing. I'll start at something earlier than algebra -- subtraction. I think you learn about subtraction before you know how it's defined (try to take some candy -- "objects that are good for eating", from a 3 year-old -- he/she will know some of it has been subtracted). After it starts... I think learners do need rote learning to build up skills that will be useful for future engagement with the learning environment (they need to know numbers to count to quantify how many pieces of candy were taken). So with a more developed skillset you get more developed learning experiences... so learners are going to need a balance of rote types of learning and experimentation.

So to algebra. I think students are "experimenting" with early algebra by solving word problems before they know what algebra is -- there's unknowns, relationships, and ways to move things around and set up the problems. -- I noticed this before I knew the algebra formalism. But because I never took "pre-algebra" and am basing this on what I remember from MANY years back... I'll have to look at a pre-algebra curriculum sometime and how that engages the learner with concepts.

I agree that argument is that learning is a building process... but I happen to think if you move down the skill set on how the skills and formalism all build up... it starts with experimentation... and maybe instinct... engagment of the learner with the environment... not rote.

Long division should be taught, but I don't see how it increases understanding of division.

I think long division increases understanding of "division in the decimal system" -- It's easy to understand the division of 6 by 3, but what about 3 by 6? Well.. carrying down the 3 in the long division process, we have 30 "tenths". Divided by 6, we have 5 "tenths." So we learn the logic of getting decimals from division, and then if we notice and 5 is 1/2 of 10... we have a "half." we learn about what five tenths is and about what 1/2 -- all kinds of mutual feedback connecting understanding of decimals, fractions and division.

What if we were using some other system... like hexidecimal? Long division in hex would be fun to think about. I want to see someone do that. Maybe I did in a computer science class way back... way back.

 

[0rthodontist]

Long division doesn't show you much about what division actually is. It just says to follow a process, which is no more insightful about the value of 3/6 than than reading the result off of a calculator. It's good because it teaches number manipulation skills, especially if it's done mentally rather than on paper. But it doesn't show you much about the number. Long division is useful for teaching polynomial division but I don't know of any other good application.

Also, I don't like the word "rote." Memorization need not be rote. If you learn to do basic algebra mentally, it drives you to develop a good memory for certain symbols and numbers. But it's hardly rote. Not that rote is bad, when appropriate--it's just that rote is the very first step in learning to remember things. You don't learn your multiplication tables by experimenting with rocks. But memorization through practice is the next step. Not all memorization is rote.

According to a "Scientific American" article a few months ago, a good memory for your subject is an essential part of being skilled about your subject. Chessmasters can remember board positions far better than lesser players. Other things I have read and experienced support this. Furthermore, memory is not a terribly difficult skill to teach. It's just a matter of the student practicing a lot; almost anyone can develop a phenomenal memory for anything, with enough practice. And in certain subjects, math being one of them, a phenomenal memory equals much better skills.

I completely disagree that most students are doing any kind of algebra before they learn the symbols. If you ask a child who has not taken any basic algebra to find 2 numbers a and b such that a + 2b = 30 and a + b = 12, they probably wouldn't be able to solve it. I can remember being presented with these kinds of things in 4th grade or so--guess and check was all I could come up with. I doubt that I was an unusual case. This only becomes more true as the math becomes better, because it really has no analogue to everyday experience.

 

[Moonbear]

Along a similar veign, though, I'm a little bit anal about loose change (pennies, especially) - I like to get rid of it. I'd say that unless a cashier can enter the number into his register, there is no hope of getting proper change by giving them $5.07 for a $4.82 purchase. So don't get me wrong - a certain amount of doing-it-in-your-head is useful.

I'm not exactly anal about change, but I'll accumulate it for a while, and then when my purse starts weighing too much, I'll remember it's there, and start trying to use change every chance I get, and to really try to minimize how much I'm going to get back of it (though, when it gets that bad, I can usually scrounge up the full 82 cents). But, yeah, I've done that, and they always stare at me like I misheard them or something when I do something like that...they'll often try handing back the 7 cents. It's not until they punch in the numbers that it dawns on them that there was a reason for it.

 

[selfAdjoint]

I often play grub-for-exact-pennies with the cashiers at Cub Foods where I mostly shop, and have never had any static or memorable gaffes either from those long suffering folks.

 

[physics girl phd]

I have a notion that all learning is rote at first.

Also, I don't like the word "rote."...it's just that rote is the very first step in learning to remember things.

Did you then mean to say that "memorization" is the first step to ALL learning? I still disagree. I will agree that a good memory of things improves your skills/capability... therefore memorizing things (both facts and procedures) is important in the learning process. Never disagreed with you there. As an aside, I am familiar with the SA article you refer to (my mother has me on a never-ending subscription from the cash-in of my deceased father's frequent-flier miles), and I have recently been playing chess at the brewpub several times a week with some folks that are high-ranked in state competitions -- and yeah -- they are impressive. I just get them tipsy so I have an occasional win.

My argument has been that "learning" is a building process that begins when you are very young, and that during that time, learning is "first" about experimentation.

It's been a fun debate... but within the next day I'm likely off on a 1400 mile road trip. I'll be sure confuse the undereducated personnel at all nasty fast-food joints by making change in a manner to confuse...

 

[Gza]

I had no idea things had gotten nearly this bad. Does anyone here have any experience of this? It is evidently only recently the increase in this "teaching" philosophy has ceased expanding.

From NYT:http://www.nytimes.com/2006/11/14/education/14math.html"

Note that the NYT prevents you from accessing articles more than 14 days old. I've saved a copy in case the debate continues past that point.

I don't know if this point is even relevant, but I'm a second year physics graduate student working on a project in quantum computing, and I seriously don't remember how to do long division. Why long division constitutes the end-all in mathematical prowess according to the quote from the original poster evades me, I can assure you, there is far more to math than some dumb, memorizable math algorithms such as long division.

 

[0rthodontist]

Did you then mean to say that "memorization" is the first step to ALL learning?

Yes, that's also a good representation of what I want to say. Rote comes first, then more sophisticated memory, then understanding. I just object to the comparison of rote learning to experimental learning--somewhat who is at the stage of being able to perform experiments and judge their results is so far above "rote" that it's not a fair comparison. The negative connotations of "rote" just seemed to be too prominent.

I still disagree. I will agree that a good memory of things improves your skills/capability... therefore memorizing things (both facts and procedures) is important in the learning process.

Do you believe that understanding should come before memory? How can you understand some concept whole if you can't remember its components?

Also, it's not just memorization. A chess master's memory for board positions is not the result of "memorization" but of mental practice.

Never disagreed with you there. As an aside, I am familiar with the SA article you refer to (my mother has me on a never-ending subscription from the cash-in of my deceased father's frequent-flier miles), and I have recently been playing chess at the brewpub several times a week with some folks that are high-ranked in state competitions -- and yeah -- they are impressive. I just get them tipsy so I have an occasional win.

My argument has been that "learning" is a building process that begins when you are very young, and that during that time, learning is "first" about experimentation.

Well, from a physical standpoint when you are very young you are building on instincts that you already have. It's quite a different process from learning something academic. Learning addition is not about experimentation. It's not by experimentation that I know 5 + 7 = 12. 12 is a number that is too large for a human brain to grasp without symbolic help--even if I had learned it through experimentation with pebbles, I would therefore not have been able to remember what I actually saw; I would only have been able to remember the symbols. I only know 5 + 7 = 12 because I memorized it once and then made it familiar through mental practice.

You need a framework to experiment on. In math this framework comes largely from developing a good symbolic memory. You need to learn all of the rules--addition, multiplication table, etc.--before you can do much with them.

It's been a fun debate... but within the next day I'm likely off on a 1400 mile road trip. I'll be sure confuse the undereducated personnel at all nasty fast-food joints by making change in a manner to confuse...

 

[Gza]

Yes, that's also a good representation of what I want to say. Rote comes first, then more sophisticated memory, then understanding. I just object to the comparison of rote learning to experimental learning--somewhat who is at the stage of being able to perform experiments and judge their results is so far above "rote" that it's not a fair comparison. The negative connotations of "rote" just seemed to be too prominent.

Wow orthodontist, from talking to you today, I've found the first person I've met (relative to me) at the polar opposite of the cognitive spectrum. I can't believe you'd put rote memorization as coming first before any kind of actual, creative thought. You must be a chemist, or some other similar discipline (perhaps dentist ) where simply knowing a fact is more important than knowing where that fact came from, and knowing how that fact can build many more facts that lead to more complex ideas. Rote memorization does not provide that sort of understanding; the understanding that leads to new ideas.

 

[0rthodontist]

I'm not a dentist. I'm a computer science and math major. I certainly believe in the value of creativity; I believe that nothing is more important than creativity. But I also believe that creativity is impossible without a firm foundation, which comes chiefly through developing a memory for the subject. In addition, you can't really teach creativity. But you certainly can teach memory, with easily measurable results.

I've made the distinction before, but I'd like to make it again, that developing a memory for a subject is completely different from, but builds on, memorizing the basics of the subject.


Part of this stems from my memories of middle school and high school math texts before calculus being extremely lightweight. One chapter might contain three facts, as compared to a chapter in a standard history text for the same age level which might contain a hundred facts and be covered in about as much time. The only possible reason for such a huge difference in content density is that most kids were unable to remember as many math facts as they could history facts, presumably because history is written in English and is a collection of stories, both of which they have a lot of practice with. I remember seeing laughable "formula sheets" that contained maybe twenty or thirty formulas at the outside. Why does a student need a formula sheet for twenty math formulas that were accumulated over a year or more when he or she can pick up fifty facts about history in an evening just by reading a chapter? It's a question of specialized memory--for math, they don't have it, for history, they do. Why not close that memory gap by training the students specifically to remember and mentally work with mathematical symbols, and then make the math texts as dense in information as the history texts?

 

[arildno]

I don't know if this point is even relevant, but I'm a second year physics graduate student working on a project in quantum computing, and I seriously don't remember how to do long division. Why long division constitutes the end-all in mathematical prowess according to the quote from the original poster evades me, I can assure you, there is far more to math than some dumb, memorizable math algorithms such as long division.

What's your problem?
First of all, long division isn't "dumb", secondly, if you actually claim to understand the concept of division, you'd be able to develop that algorithm in about two minute's time.

 

[Gza]

"Originally Posted by Gza
I don't know if this point is even relevant, but I'm a second year physics graduate student working on a project in quantum computing, and I seriously don't remember how to do long division. Why long division constitutes the end-all in mathematical prowess according to the quote from the original poster evades me, I can assure you, there is far more to math than some dumb, memorizable math algorithms such as long division."

What's your problem?
First of all, long division isn't "dumb", secondly, if you actually claim to understand the concept of division, you'd be able to develop that algorithm in about two minute's time.

I don't have any sort of disdain for long division, and meant it was "dumb" in the way its taught as some sort of memorizeable math trick, and yes, i'll go out on a limb here and "claim to understand the concept of division." And as far as me developing the algorithm for long division, why would I need to do that when there's a nifty little invention called a calculator/computer/cellphone/watch/etc, that is faster and more accurate than i am at such simple computations. I suppose you still handwash all of your clothes, use a slide rule, and have to build a fire every time you cook something.