kitchen table math, the sequel: How much rote memorization do students do?

Tuesday, June 17, 2014

How much rote memorization do students do?

I was chatting-via-email with Allison yesterday re: rote learning .... which I'd been thinking about  again, in the wake of yet another reference to the horrors of brute memorization in the Times:
The Common Core, the most significant change to American public education in a generation, was hailed by the Obama administration as a way of lifting achievement at low-performing schools. After decades of rote learning, children would become nimble thinkers equipped for the modern age, capable of unraveling improper fractions and drawing connections between Lincoln and Pericles.

Common Core, in 9-Year-Old Eyes By JAVIER C. HERNÁNDEZ | JUNE 14, 2014
There it is again: the problem we don't have (decades of rote learning), being solved by the problem we do have (decades of thinking without knowing). Same old, same old, except they've upped the ante. Nimble thinkers, for pete's sake. At age 9.

For a while now, I've been planning to re-read Dan Willingham's "Inflexible Knowledge: The First Step to Expertise."

Haven't done so yet, but I did pull out his definition of rote learning:
In his book Anguished English, Richard Lederer reports that one student provided this definition of "equator": "A managerie lion running around the Earth through Africa." How has the student so grossly misunderstood the definition? And how fragmented and disjointed must the remainder of the student's knowledge of planetary science be if he or she doesn't notice that this "fact" doesn't seem to fit into the other material learned?

All teachers occasionally see this sort of answer, and they are probably fairly confident that they know what has happened. The definition of "equator" has been memorized as rote knowledge. An informal definition of rote knowledge might be "memorizing form in the absence of meaning." This student didn't even memorize words: The student took the memorization down to the level of sounds and so "imaginary line" became "managerie lion."
Re-reading this passage today, I feel less clarity than I did the first time around 10 years ago.

If rote learning is "memorizing form in the absence of meaning," then it's not clear to me that the words "menagerie lion" lack meaning, even as a definition of "equator."

"Menagerie lion" is the wrong meaning, of course, but it's a meaning, and if you didn't understand the words "imaginary line" when you heard your teacher speak them, but you did understand the words "running around the Earth through Africa," then "menagerie lion" is not a bad guess for the sound string ih-maj-in-air-ee-line.

Slightly off-topic, Jimmy (for passers-by, Jimmy is my oldest son & has autism) has always been echolalic. You would think that echolalia would be the hr-example of rote learning, but if you listen to him, you'll hear that the particularly phrases he's echoing are often directly related to what's going on. (Can't think of a good example at the moment - sorry.)

Now I'm wondering about the word "parroting" -- do we know for a fact that parrots have memorized form without meaning? Having once spent a day with a parrot who probably spoken English (including conjugated verbs), I don't think we do.

Memorizing pi

It strikes me that memorizing digits of pi is a good example of rote memorization, although the issue with pi isn't precisely that you're focusing on form in the absence of meaning. You can understand pi, or at least know what pi is, and still have to rote-memorize the digits. (Or do math people see this differently?)

Anyway, the point is: memorizing digits of pi is hard. Not easy. It's much easier to memorize material that has meaning.

Which raises the question: how much rote memorization -- memorization of form in the complete absence of meaning -- do students actually do?

How much rote memorization did students do in the past, when memorization was seen as a good thing (or at least an essential thing)?

And how much do students absolutely have to do?

I don't know how to answer that question. New vocabulary words in every subject have to be learned by rote because the link between form and meaning is arbitrary. Second language vocabulary has to be learned by rote.

Math, it seems to me, may actually require less rote memorization than any other subject. (Or is that wrong once you get past the elementary grades?) much does it all add up to?


Auntie Ann said...

Our kids have done a little. Some in geography: states, their abbreviations and capitols in 5th grade; oceans and continents in earlier grades. Math facts, of course.

Our kid had to write and memorize a poem at the end of 6th grade, but that was about the only time they had to memorize and recite anything.

Other things fall into the category of learning the subject: date range for human ancestors, main systems in the human body (nerve, muscle, adrenal, etc.), Greek/Roman gods, etc.

Anonymous said...

I read a book by Irene Pepperberg about her completely ridiculous and totally uphill battle to convince academics that parrots (hers was an African Grey) really do assimilate words AND meanings and think.
Oy, I wish I could find the blogger who commented so
trenchantly that it brought into sharp focus how crazy-stupid academics can be. For thousands of years people who have actually LIVED with animals in their natural habitat have understood that the animals had brains and could figure things out. It takes a Western scientist to think that fish and birds run only on "instinct", and that humans are the only thinking feeling creatures.
all this to say, yes, some very small group of Westerners do acknowledge that parrots are not just memorizing.
lizabennett aht yahoo

froggiemama said...

Math *should* require less memorization but the reality is that most students *do* memorize their way through it.

I posted a really great blog post today on FB on why teaching programming is very hard. The author notes that one of the central problems is that programming requires very little memorization, and a lot of problem solving. This of course is really hard for students who have largely memorized their way through school. The most telling part of the post is this "I used to be a math teacher and math has somewhat the same thinking requirements and the same issues. The big difference is the kids would have 10 home work problems a night, 8 of which were very easy to do so they would do those and ignore the hard ones. The result would be an 80%." He then goes on to say "A kid can do pretty well in high school just with an average memory. In math memorization of techniques can go a long way. In programming memorization is a trivial part of the skill set needed to succeed. "
And that is the crux of the problem. Lots of kids can memorize their way to the 80% in math, but since there is no way to do that in computer science, they fall apart when they hit CS courses. In fact, they fall apart if they hit advanced college math classes too, but few of them will ever get there.

froggiemama said...

Sorry, forgot to post my source

Kai Musing said...

Most of what kids need to learn is "inflexible" knowledge, as Willingham says.

You could make an argument that systematic synthetic phonics is rote, because students learn to connect letters to the sounds they make (not much "meaning" there), but one they use that knowledge to begin blending words there is meaning.

All the clamoring against "rote" learning are attempts to convince others that all learning can be natural and fun. In the tradition of Dewey and Rousseau, if it isn't natural or enjoyable or something that comes from within the child, it's a bad idea.

You don't know how many times I've heard "math games" as the solution to math fact fluency, rather than repeated drill over time...

froggiemama said...

There is almost no rote learning in computer science, yet I doubt anyone would claim that learning, say, how to build a linked list solution is natural or fun. In fact, good non-rote learning should be a struggle, but a productive struggle.

Note too, I am NOT an advocate of unguided discovery learning. No one can "discover" their way to deep mathematical or algorithmic concepts. But real mathematical understanding, or in the case of my field what we call computational thinking, is not going to come out of memorization.

Auntie Ann said...

It took some of the greatest geniuses of the millennia to work out what we know about mathematics: Fibonacci, Descartes, Newton, Leibnitz, Euler, Pascal, Lagrange, etc. Yet, those who believe learning has to be "authentic" and self-generated expect every single kid to do what they did.

Catherine Johnson said...

Liza - thank you! If you find the link, let me know!

Kai - absolutely: all phonics has to be 'rote' learning by Dan's definition. The link between sound and letter is completely arbitrary.

One thing I need to figure out: what are the parameters of implicit learning (nonconscious learning -- the kind of learning where you 'pick things up' without conscious memorization).

I've begun reading Roediger's new book, which is very useful, BUT he doesn't seem to address the 'cognitive unconscious' at all.

He appears to be saying that **all** declarative knowledge requires conscious practice; one learns nothing from repeated exposure without intentional 'retrieval practice.'

That can't be right -- (and it's something I need to know, both for myself & for my students).

froggiemama - I think of 'computational thinking' (or any kind of thinking) as an "emergent property," based on Dan Willingham's articles.

I don't know whether that's right, though.

I think Dan says that 'understanding' is actually 'remembering,' right?

Will have to check.

gasstationwithoutpumps said...

In response to froggiemama's statement "There is almost no rote learning in computer science, yet I doubt anyone would claim that learning, say, how to build a linked list solution is natural or fun."

I agree with the first (very little rote learning, and that mainly the names of a few commands and operators), but disagree with the second---learning how to build linked lists was fun when I first learned it 40+ years ago.

I definitely down-play memorization in my courses. The applied circuits course I teach has only one page of things to memorize (and students could rederive half of them if they wanted to). The course is all about learning how to use those few concepts to design amplifiers and filters.

froggiemama said...

ha-ha, learning how to build linked lists may have been fun for you (and I), but I guarantee you it is not fun for 95% of students.

SteveH said...

"After decades of rote learning..."

And what have they been doing for the last two decades? At some point they have to start accepting their share of history. Apparently, the Common Core allows them to reset the clock on that nugget.

"If rote learning is "memorizing form in the absence of meaning," ..."

But of course, mastering math skills does require learning of meaning. One cannot pass math with just rote knowledge, unless the teachers are completely pathetic test writers. Traditional math education is not defined by rote learning. Rote learning just defines bad teaching.

In seventh grade, my son (for fun) memorized pi to 120 digits and he memorized the periodic table. Memorizing pi had few to no benefits, but memorizing the periodic table had very many. Other knowledge attached itself along the way. He learned about the organization of the table and valence electrons - and a whole lot more.

Building a framework for understanding using lists or facts is not for the weak of pedagogy, but it is a very powerful and efficient tool. It is not something you attempt half-heartedly. Memorizing the list of presidents can have a great payoff, but not if you do it poorly. One cannot learn scales half-heartedly. There is no halfway or self-discovery "Royal Road" to Carnegie Hall. Avoiding memorization is an educational admission of defeat.

And then there is memorization that's good for the soul, like memorizing poetry or the Firesign Theatre.

"Yes, and at the last possible moment, he stopped on a dime!" ... "Unfortunately, the dime was in Mr. Rococo's pocket."

Allison said...

I know the numbers from 1 to 10 in Korean. I was taught them rote by my father when I was 6, and I still know them.
Il, i, sam, sa, oh, yuk, chil, pal, coo, ship.

I learned to say the sounds.

Do I know what 7 is? No. I have to start at 1, il, again, and count to 7.

That is what most people refer to when they mean "rote learning" : they mean literally having learned a verbal stream of utterances without really understanding the meaning behind them.

I have met 8th grade students who knew their times tables rotely, and who could not compute in their heads 13x11 even though they could recite 12x11. I have heard many stories of children rotely learning numbers for 10, 0 and 10, 1 and 9, 2 and 8, etc. but who are stumped when asked "if your current number bond for 10 is 2 and 8, and I add one more to the 8, what do I have now?"

This kind of rote learning IS NOT effective in mathematics. It isn't what is necessary for success.

Now, the word "rote" has too many negative connotations to be useful here.

What IS necessary is learning to automaticity. Automaticity means knowing something accurately and immediately. My knowing Korean numbers from 1-10 rote never meant I learned the numbers themselves to automaticity. More, automaticity comes in layers, building upon itself.
More on this in a bit.

Re: the issue of comparing just arithmetic to other fields, it is still not like other subjects, because after a handful of assumptions and conventions, everything else children are taught MUST BE TRUE.

This is utterly different than, say, history. In history, the fact that Wash DC is the capital, not Philadelphia, or the fact that Lincoln, not Douglas, won, may have reasons, but those outcomes were not *logically required*. Mathematics is true because it is derived from other truths that MUST be true. So the facts learned must relate to each other, and none can be changed without collapsing the rest.

Real understanding in math is not remembering. Real understsnding is a kind of "forgetting", where it is so true to you that you no longer need to think about it at all.

More on automaticity's layers tomorrow.

et's just consider arithmetic for now. To be successful at learningtudents must have mastered, to the point of being automatic, their sums and differences to 1

Anonymous said...

Of course we who actually paid attention know that rote learning went out much longer ago than last year. We've had two or three generations already with little or no rote learning. It was already unheard of when I was a tyke in the seventies, at least in the better part of the States. In some schools and in some other countries, memorizing for declamation continued, leaving certain elders with vast recall of classics. That is the baby we threw out with the bathwater.

That said, there's a problem with the definition of "rote" as "memorization without meaning," as if anything that has meaning is thereby saved from roteness. Anything you memorized that proved useful can be shown not to be rote, making the term "rote" meaningless in itself. "Rote" becomes only a word you use to make fun of other people or other cultures; "rote" becomes a marker of ignorance.

Anybody who persists with their education, no matter how good their natural memory is, will reach the point where memorization for memorizations' sake must be confronted. The lists and flash cards come out sooner or later. But when they do it, it's rote, and when we do it, it's not?

As Catherine indicates, it's a silly, mistaken, historically inaccurate argument.

I feel for people who have to engage with schools and educational experts, because such battles seem the most dispiriting, quaggy, and pointless.

SteveH said...

I agree with anon in that these arguments are pointless. Educators use the term in a pejorative fashion and are not interested in anything other than supporting their position. They are not about to read this thread and think "Gee, I never really thought about that."

They don't care about anything outside of their box. They don't care what parents think. In the end, it doesn't matter unless parents really become stakeholders in the process. That will never happen. We've won the blog battle, but so what?

The real battle is about school choice. Some choices may be bad, but I will place my long-term bets on parents, not educational pedagogues, knowing what's best for their individual kids.

Catherine Johnson said...

Anon wrote: << That said, there's a problem with the definition of "rote" as "memorization without meaning," as if anything that has meaning is thereby saved from rottenness. >>


I REALLY need to finally master the literature on memory.... (there are good sources available for people like us).

I don't think 'rote learning' is much of a concept in cognitive psychology EXCEPT that, and this is interesting, most of cognitive psychology's research on memory has involved rote learning (of letter sequences, etc).

One part of me questions whether rote learning **ever** happens in schools, including 19th century schools; the other part thinks everything is rote learning BECAUSE all knowledge has to be expressed in a form, and the learning of form is 'rote' almost by definition.

That said, I think it's possible there's an exception for math (that's intuition speaking, nothing more).

Here's an example of what I mean.

How does one 'know' a poem?

I think there are only two ways:

* read it enough that you instantly recognize the INDIVIDUAL words and lines when you re-read (i.e. you have 'recognition memory')
* memorize it (recall memory)

A poem is nothing without its form; a poem, in a real sense, is form.

I'm coming to think that is true to a somewhat shocking degree with verbal disciplines, but I haven't worked it out in my own mind yet.

I've been thinking about this a lot because Katie Beals and I are writing a 'writing supplement' to accompany Ed's textbook on European history...and I more or less do, now, see how and why "writing is thinking."

Writing isn't just thinking, though; writing is knowing.

That's what ktm has done for me over the past 10 years: it's allowed me to think (the social aspect of the comments thread was critical there, obviously) and it allowed me to **know** what I'm seeing.

Catherine Johnson said...

recognitiion versus recall

It just occurred to me that when you translate educators' antipathy to memorization, what they are really saying is that recognition memory is sufficient.

Psychologists have made a distinction between recognizing something when you see it again & being able to retrieve it from long-term memory.

Retrieval is hard, obviously.

When I put it that way, I'm not as scandalized by constructivism. (I have Steve H to thank for the oft-repeated insight that constructivists aren't actually **against** kids storing knowing in long-term memory. They just don't want any formal, conscious memorization taking place in the school.)

If I have time today, I'm going to take a look at what my books have to say about retrieval versus recognition.

btw, 'anonymous' commented that Core Knowledge is all about throwing **tons** of knowledge at kids and hoping they remember 10% of it after the school year.

I've heard that complaint before, and I don't gainsay it, as attached as I am to Hirsch and his ideas.

However, I'm thinking that it's actually possible to gain a huge amount of recognition memory.....and that for a lot of what you learn in school, recognition memory is probably perfectly acceptable.

But I have no idea whether this is true!

Hainish said...

Catherine, I've noticed that teachers believe that students will forget almost everything they learn.

Cognitive psychologists believe that, once learned, nothing is ever forgotten. You may not be able to recall something, but you can recognize it. (Even if you can't recognize it, you can re-learn it much, much more easily on the next pass.)

I'm sure that when teachers talk about remembering, they are talking about recall memory.

Anonymous said...

"I'm sure that when teachers talk about remembering, they are talking about recall memory."

Probably. This is what *I* (as a non-teacher) mean.

And for a lot of stuff, this is what I want out of K-12 (of K-16) education. I don't want my kid to recognize Ukraine on a map. I want him to recall where it is (on a map).

Ditto for a lot of history.

And math.

And science.

The long standing "fire and forget" approach to schooling pretty much ensures that very little learning is recallable. A bit more spaced review would go a long way to correcting this(*). Because we know this and don't correct it, I assume that kids graduating while being able to recall very little is okay with the folks in charge.

-Mark Roulo
(*) Example: In 5th grade I "learned" the US states on a map. And after the big test never did anything with this (not even review). Years later I had forgotten many of the state locations. A bit of review in 6th, 7th, 8th, etc. would have (I think) cemented that knowledge. It wouldn't take much time in each 6-12 grade. But we didn't do it. My conclusion is that I was "taught" something that the schools don't consider worth learning. I don't know why they do this.

Allison said...

Teachers talk a lot about what is lost over the summer, but since they don't see the same kids the following September, it makes less of an impression than it should--they know there is loss, but have had little way to quantify it, let alone do something about it.

MAP testing is the one product where schools take data on what kids know in spring and fall and spring again. (but the anti-testing movement is gonna fix that!)

Mark, I think it isn't that the school didn't value it, as much as the entity called the school didn't have any goals per se; teachers did. So the 5th grade teacher taught it; the 6th grade teacher assumed you knew it, and went on accordingly, without any thought to the need to continue to cement it.

If I had to guess, I'd even say the 6th grade teacher had probably cemented it, and didn't even notice what wasn't known.

This is what we see in high school math all the time: an assumption by the teacher that the students really know the arithmetic and fractions they should know, and they proceed to teach accordingly. Except the kids don't know it to mastery, even if they can recall it, so this teaching methods is wrong for the student. At best, the teacher says "my kids don't know this", but they still don't really understand what it means they know and don't know. Teachers and students both are often tricked by students' familiarity with a topic into thinking they have mastery.

MommyPenguin said...

froggiemama made an interesting point about memorization in CS classes. I did very well in school, largely because I have a very good memory. I did okay in CS classes through about the 300 level, which was where I faltered, and in math through about linear algebra (same issue). I'm thinking that, while I never realized it before, I'd reached a point where memorization was less helpful than logical thinking and problem solving and I just wasn't as good at those.

I'm reading an interesting book about the teaching of elementary mathematics, and one of the things they talk about is multiplying two numbers with several digits. A number of the teachers understood the process, but when asked how they would help a child who had a particular misunderstanding of the problem, their resulting explanations showed that none of them actually understood how the multiplication process was working! They'd all memorized the process without understanding the math behind it. And these were math teachers!

I think there's benefit in rote memorization, though. I memorized the presidents in order when I was a teenager, and it might sound crazy, but I can't tell you the number of times I've found that useful! I memorized where all the states were on a map, and that's been useful, too (not making mistakes like that congresswoman who said that Arizona didn't have problems with illegal immigration because they didn't share a border with Mexico).

I think another benefit of memorization is that it strengthens your memory and your ability to memorize. Most jobs can be made easier by some kind of memorizing. Maybe it wasn't useful to me as a librarian to have memorized various phyla names as a teenager, but strengthening my memory with that made it easier for me to memorize where books are located in the Dewey Decimal system, which made my job as a librarian easier.

I think also that being introduced to lots of things when you are young, even if you don't remember all of the details, is useful. It's sort of like setting up coat racks for future coats. When you learn about Egypt as a 1st grader, you probably won't remember the specific pharaohs that you read about. But when you read about Egypt again when you're older, you'll remember that Egyptian kings were called pharaoh's, that Egyptians worshiped many gods, that they built pyramids, and made mummies. Not having to learn that information again means that you can this time focus on who some of the major pharaohs were and what they did, and you'll have a sense of the progression of Egyptian history.

Anonymous said...

"I'm reading an interesting book about the teaching of elementary mathematics, and one of the things they talk about is multiplying two numbers with several digits. A number of the teachers understood the process, but when asked how they would help a child who had a particular misunderstanding of the problem, their resulting explanations showed that none of them actually understood how the multiplication process was working! They'd all memorized the process without understanding the math behind it. And these were math teachers!"

The book was probably 'Knowing and Teaching Elementary Mathematics' by Liping Ma.

-Mark Roulo

MommyPenguin said...

You're absolutely right, Mark! I didn't have the book handy and didn't want to make a mistake on the title.

SteveH said...

"And these were math teachers!"

Elementary teachers of math are generally not math teachers. That is one of the problems. Our state only requires subject certification starting in 7th grade. This problem is compounded by what is called seniority bumping. When a class section is no longer needed (in K-6), more senior teachers can bump lower teachers out of any grade/class. I remember one year when this caused a chain reaction to four classes where teachers were forced into teaching another grade level. Only in 7th grade did I start to see any level of proper mathematical understanding from teachers.

Add to this all of their fuzzy, hands-on, group learning, anti-skill ideas, and only those kids with basic skill help at home or with tutors will survive.

As for memorization leaving one high and dry in math at some (higher/college) level, I don't believe it. Something else is going on. One cannot do well in high school math with just memorization unless the testing is incompetent. What is more likely for many is that the gaps and shallow understandings catch up with them.

Understanding is never black and white. I remember struggling in Algebra II before I found a way to get the correct answer any which way I approached a problem. Why was I determined to do that? There are classic examples of teachers showing how 1 equals 2 and asking students where the mistake is. You can understand a lot and still not find the error. What causes students to persist in this process? It's not logic or memorization, and it's not engagement.

Computer science is a matter of individual persistence. It's not so much a matter of lack of logic, but the inability (or desire) to deal with an incredible amount of detail. Some want to balance a checkbook to the penny, while others have no such (psychotic?) desire to do so.

My old CS students were bright and logical enough, but some surely lacked persistence. Handling details is very hard work. When I was in school, I knew that computer science classes were even more difficult than math classes, where you could slip by with a few minor misunderstandings. However, programs had to be perfectly correct. You can't just fix most of the errors and expect to get a 90 in the course. I knew that all of the programs I wrote had to be 100% correct. For math and CS, once you start to slip down from the ideal peak, it's very difficult or impossible to recover. Unfortunately, this happens to many by the time they hit 7th grade.

Auntie Ann said...

Math requires doing lots and lots of problems, but most math teachers and curricula do not stress quantity. I went through all the "Math Journals" for 5th grade Everyday Math and found only 60 fraction problems which required finding a common denominator--and most of those were simple like 4 & 8 or 2 & 3. There wasn't a hard denominator to be found.

Our 8th grade girl's algebra teacher assigns tons of problems. At the start of the year he told the class that a couple years ago he tried assigning fewer, but the students then couldn't pass his tests. Each problem introduces slight variations and each problem is an opportunity to focus on keeping your minus signs straight! I've told our 7th grader that the problems I'm giving him now (pre and early algebra) are all about tracking the minus signs.


Off topic, re: Egypt. As for knowing the pharaohs, which our now 7th grader did last year, I kept being dumbfounded by the pronunciation of pharaoh's names that his teacher used. Somehow Hatshepsut was coming out as: hat'-che-put. The teacher had obviously never watched Discovery Channel, or bothered to look up the pronunciation.

momof4 said...

IIRC, one of the first things that Wise and Bauer's Classical Curriculum has kids do in history is to start a timeline, which will be expanded and deepened over the successive years. The curriculum structure also is built to create continuity; in history, the 4-year sequence of ancient, medieval, early modern and late modern is repeated three times - in the grammar stage (1-4), the logic stage (5-8) and the rhetoric stage (HS). The literature, art/architecture/music, econ etc. follow the same sequence. Kids learn Aesop's Fables, mythology etc. and ancient history (and geography) in first grade, but they get more and deeper material in 5th and 9th. It's always seemed to me to be a very good approach

Froggiemama said...

You absolutely can do well through high school math on just memorization. Not the kind of rote memorization that you would use when memorizing president's names, but memorization of process. It is weird, but when you memorize an algorithm or process, you even feel like you understand it. It isn't until you have to do something related but not exactly the same, or as in the example with the math teachers, explain it to someone who doesn't get it, that you realize you didn't *really* understand. The way that high school math is taught, including AP calculus, various processes are demonstrated, and then students work a bunch of problems that are exactly the same. When the teacher makes it harder, it is usually by mixing up types of problems, or maybe giving problems where a couple of the memorized algorithms must be used. But that just requires the student to recognize that this is a problem that requires algorithm A rather than B. When my husband was teaching calculus at an elite SLAC, he said that students came in routinely with good grades in AP calculus, and would immediately bomb out in Calc III, where they were asking kids to apply algorithms to larger, messier problems.

I also totally disagree that persistence is the only thing in CS. I have taught CS at every level for about 20 years now. I have seen many many students persist and persist, but still fail because they can't come up with the algorithm to do something. And I see other students who get it with little effort. You need persistence in the debugging stages, but persistence doesn't help you when designing the problem solving strategy. My favorite example is the classic problem in which students are asked to write a program that sums the individual digits. It is really easy, unless you never understood place value - and most of my students do not.

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SteveH said...

"Not the kind of rote memorization that you would use when memorizing president's names, but memorization of process."

I disagree. There are too many variations in problems where process will leave you nowhere.

"It isn't until you have to do something related but not exactly the same"

If you don't get this on tests, then there is something wrong with the testing. This is what homework sets are all about - seeing and working on these variations. However, it's common for fuzzy educational pedagogues to grab onto anything that slightly seems like memorization as an excuse to flip the process around to one of discovery and some sort of "deep" understanding. They get this completely wrong.

"..including AP calculus, various processes are demonstrated, and then students work a bunch of problems that are exactly the same."

They are not exactly the same.

I agree with Auntie Ann that: "Math requires doing lots and lots of problems". My point is that, especially in K-6, kids never get this practice. They never see all of the variations of minus signs and fractions. I saw students who cross-multiplied, but failed when the equation changed slightly. Proper testing would catch that and that usually happens by high school, when it's too late.

The real problem is not memorization, but limited understanding due to not getting enough practice and seeing enough variations. Gaps in understanding will eventually catch up with everyone. If you do not have the persistence (or push) to minimize these gaps as you go along, then you will reach your peak in math. At some point, remediation and tutoring become much less effective. They take the form of putting out fires, not providing a proper foundation for future math.

SteveH said...

"My favorite example is the classic problem in which students are asked to write a program that sums the individual digits. It is really easy, unless you never understood place value - and most of my students do not."

This is a gap in knowledge, not logic, but I've seen many smart, logical kids miss this algorithm. Success on this problem is not a litmus test of logic.

"but persistence doesn't help you when designing the problem solving strategy."

Yes, it does, and experience and practice. So many things in CS are not clear at first, but once you've see a solution and work hard at mastering it, it becomes part of your "toolkit" for solving future problems. While some people seem to naturally handle details better than others, CS is dominated by the need for persistence. Too many of my old students struggled not so much because of lack of logic, but lack of understanding of all of the hours and hard work required. Some might have an easier time than others, but lack of logic was never a reason for failing.

Barry Garelick said...

The same edu-experts who decry memorization, seem to have no problem with the ridiculous "Pi Day" celebrations at schools in which there are contests to see who has memorized the most digits of pi.

Not to mention pie-eating contests.

R. Craigen said...

Allison said, " I have heard many stories of children rotely learning numbers for 10, 0 and 10, 1 and 9, 2 and 8, etc. but who are stumped when asked "if your current number bond for 10 is 2 and 8, and I add one more to the 8, what do I have now?""

I have a PhD in math. If you asked me, "if your current number bond for 10 is 2 and 8, and I add one more to the 8, what do I have now?" I would be utterly confused and it would take some time just to sort out what you're asking for.

First I'd have to work out whether it "your current" and "I add" and "I have now" all attach to the same numbers/number bonds which are "had" by one or the other person under discussion.

Second, I'd have to ponder what is meant by "number bond" because this is a nonstandard and relatively new piece of terminology introduced by some well-intentioned educator ages ago -- a perfect example of the many non-mathematical pieces of jargon introduced into elementary school math education, which will be of no use to children outside of math classes up to perhaps grade 5.

Finally, once it is understood that the questioner is talking about the sum 2+8=10 and "adding 1 more to the 8" then I must decide whether it is the 2 or the 10 that must be adjusted to form equality, and whether another "number bond" is the desired answer, a sum of some sort, or a conceptual statement about the resulting situation.

Among the possible answers that come to mind immediately are

1. The number bond 1 and 9 (i.e. the equation 1+9=10)

2. The equation 2+9 =11

3. A non-number bond "2 and 9" (assuming that one is restricting "number bond" to sums making 10 -- a tenuous assumption considering that "number bond" is poorly defined and each text appears to have it's own conventions in mind. Otherwise this is the number bond 2 and 9)

4. The non-equation 2+9=10

5. I don't know what you have, but you don't have the 1 because you added it to my 8 so now I have it. Unless, of course, you had a spare all along.

There are others, a bit more snide.

But I'll add that this poorly-worded question is, unfortunately, typical of what is found in the fuzzy texts of today. Some call them "rich problems" because they leave a frustrating interpretation problem to the student on top of the math. Apparently confusion is a variety of complexity and complexity, is of course always good.

In a personal discussion with Prominent Nelson Publishing textbook writer Marion Small not long ago she declared to me that "I think ambiguity is a great way to teach mathematics!" This, apparently is what she meant -- those aren't badly worded questions; they're value-added.