kitchen table math, the sequel

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An analysis of life span memory identifies those variables that affect losses in recall and recognition of the content of high school algebra and geometry courses. Even in the absence of further rehearsal activities, individuals who take college-level mathematics courses at or above the level of calculus have minimal losses of high school algebra for half a century. Individuals who performed equally well in the high school course but took no college mathematics courses reduce performance to near chance levels during the same period. In contrast, the best predictors of test perform ance (e.g., Scholastic A ptitude T est scores and grades) have trivial effects on the rate of performance decline. Pedagogical implications for life span maintenance of knowledge are derived and discussed.

Lifetime M aintenance of High School M athem atics Content
Harry P. Bahrick and Lynda K. Hall
Journal of Experimental Psychology
1991, Vol. 120, No. 1, 20-33

Bahrick, Hall, and Baker have a brand-new book out on long-term retention.

more from Barry's article on the Common Core:

This approach not only complicates the simplest of math problems; it also leads to delays. Under the Common Core Standards, students will not learn traditional methods of adding and subtracting double and triple digit numbers until fourth grade. (Currently, most schools teach these skills two years earlier.) The standard method for two and three digit multiplication is delayed until fifth grade; the standard method for long division until sixth. In the meantime, the students learn alternative strategies that are far less efficient, but that presumably help them "understand" the conceptual underpinnings.

Once again, knowledge stored in memory is entirely different from knowledge stored on Google.

Biological memory is a biological process that requires a period of time during which new memories are consolidated:

Memory consolidation refers to the idea that neural processes transpiring after the initial registration of information contribute to the permanent storage of memory.
Memory consolidation, retrograde amnesia and the hippocampal complex
Lynn Nadel* and Morris Moscovitcht Cognitive Neuroscience

I don't know how much time the brain requires to consolidate memories, but I recall John Medina suggesting that the figure may be as long as 10 years. (That would jibe nicely with the 10-year rule for development of expertise, wouldn't it?)

The "consolidation lag" between first learning a new skill and really knowing that skill explains why "just-in-time" learning is so crazy. There is no such thing as just-in-time learning. The brain doesn't work that way. No matter how smart you are, if you are 17 and you don't know how to do long division, you can't just have your professor show you how and then start doing it. Knowledge has to be consolidated before you can use it well, and consolidation takes time.

Here is James Milgram on his experience teaching Stanford students who had not been taught long division:

What happens when you take long division out of the curriculum? Unfortunately, from personal and recent experience at Stanford, I can tell you exactly what happens. What I'm referring to here is the experience of my students in a differential equations class in the fall of 1998. The students in that course were the last students at Stanford taught using the Harvard calculus. And I had a very difficult time teaching them the usual content of the differential equations course because they could not handle basic polynomial manipulations. Consequently, it was impossible for us to get to the depth needed in both the subjects of Laplace transforms and eigenvalue methods required and expected by the engineering school.

But what made things worse was that the students knew full well what had happened to them and why, and in a sense they were desperate. They were off schedule in 4th and 3rd years, taking differential equations because they were having severe difficulties in their engineering courses. It was a disaster. Moreover, it was very difficult for them to fill in the gaps in their knowledge. It seems to take a considerable amount of time for the requisite skills to develop. [emphasis added]
Transcript of R. James Milgram
1999 Conference on Standards-Based K-12 Education

There is no just-in-time learning, and you can't catch-up.

For the sake of argument, say it takes two years to consolidate the skill of adding and subtracting double-digit numbers. (I'm guessing it takes more than two, but I don't know.) If a child learns to add and subtract double-digit numbers in second grade, he or she will be proficient in fourth grade.

Delay teaching the algorithms until fourth grade and now you have a cohort of students who won't be proficient in addition and subtraction until 6th grade.

That's the way it works. Two years is two years.

and see:
Eide Neurolearning explains elaborative rehearsal


I've just read the transcript of Hawkins' talk, which is posted along with the video.

Maybe this is overstating matters, but on a quick read-through of the transcript my impression is that pretty much everything Hawkins says it as odds with pretty much everything constructivist educators believe:

Intelligence is not behavior.

Intelligence is not computation.

Intelligence is memory.

Memory is memory of sequence.

The point of memory is to predict what comes next.

"[T]he neocortex is just memorizing."
"You cannot learn or recall anything outside of a sequence."
"[I]ntelligence is defined by prediction."
"[P]rediction of future inputs is the desired output."

The education establishment has for many years denigrated both memory and sequence in favor of critical thinking, problem solving, spiraling and history taught as themes instead of narratives.

Meanwhile actual experts persist in knowing stuff and in organizing the stuff they know in coherent sequences.

On the other hand, constructivists have picked up on the idea of 'pattern' and 'prediction' in a way instructivists arguably have not....but the injunction that students must 'look for a pattern' (math) and 'make predictions' (reading) seems often to be a means of avoiding sequence (math) and the kind of ordinary nouns-come-after-prepositions-type prediction Hawkins is talking about.

I don't know whether Hawkins is right, of course. Reading the transcript, I wanted to hear him talk about cognitive illusions and the invisible gorilla. (I don't remember whether he discusses illusions in his book with Sandra Blakeslee.)

From the transcript :

So what is the intuitive, but incorrect assumption, that's kept us from understanding brains? Now I'm going to tell it to you, and it's going to seem obvious that that is correct, and that's the point, right? Then I'm going to have to make an argument why you're incorrect about the other assumption. The intuitive but obvious thing is that somehow intelligence is defined by behavior, that we are intelligent because of the way that we do things and the way we behave intelligently, and I'm going to tell you that's wrong. What it is is intelligence is defined by prediction.

[snip]

The AI people said, well, the thing in the box is a programmable computer because that's equivalent to a brain, and we'll feed it some inputs and we'll get it to do something, have some behavior. And Alan Turing defined the Turing test, which is essentially saying, we'll know if something's intelligent if it behaves identical to a human. A behavioral metric of what intelligence is, and this has stuck in our minds for a long period of time.

Reality though, I call it real intelligence. Real intelligence is built on something else. We experience the world through a sequence of patterns, and we store them, and we recall them. And when we recall them, we match them up against reality, and we're making predictions all the time. It's an eternal metric.

[snip]

You're all being intelligent right now, but you're not doing anything. Maybe you're scratching yourself, or picking your nose, I don't know, but you're not doing anything right now, but you're being intelligent; you're understanding what I'm saying. Because you're intelligent and you speak English, you know what word is at the end of this -- (Silence) sentence.

[snip]

You still have that alligator brain. You do. It's your emotional brain. It's all those things, and all those gut reactions you have. And on top of it, we have this memory system called the neocortex. And the memory system is sitting over the sensory part of the brain. And so as the sensory input comes in and feeds from the old brain, it also goes up into the neocortex. And the neocortex is just memorizing. It's sitting there saying, ah, I'm going to memorize all the things that are going on: where I've been, people I've seen, things I've heard, and so on. And in the future, when it sees something similar to that again, so in a similar environment, or the exact same environment, it'll play it back. It'll start playing it back. Oh, I've been here before. And when you've been here before, this happened next. It allows you to predict the future. It allows you to, literally it feeds back the signals into your brain; they'll let you see what's going to happen next, will let you hear the word "sentence" before I said it. And it's this feeding back into the old brain that'll allow you to make very more intelligent decisions.

[snip]

So what is the recipe for brain theory? First of all, we have to have the right framework. And the framework is a memory framework, not a computation or behavior framework. It's a memory framework. How do you store and recall these sequences or patterns?

Well, thanks to Texas Republicans including the words "critical thinking" in a statement no one outside a tiny group of public school obsessives actually understands (or cares to understand, apparently), we now have precious NYTimes real estate going to the celebration of non-memorization in schools.

Here, courtesy of the Times, we have the thoughts of a recipient of the Presidential Award for Excellence in Math and Science Teaching and the Distinguished Fulbright Award in Teaching:

I’ve worked for many years with students of varying demographics and learning abilities and what I’ve learned over and over is that nearly all kids love to learn – even those who would like us to believe they hate school. But what they need from their education is more than the memorization of facts – they need great teaching, foundational knowledge, problem solving skills, and the understanding of current issues.

What is a Good Teacher Worth?
By ANDREW C. REVKIN
July 6, 2012, 10:03 AM

So they're going to acquire "foundational knowledge" but they're not going to memorize any facts. Or not many.

How exactly do you pull that off?

And please don't tell me 'they construct their own knowledge.'

Speaking as a writer, I have constructed knowledge any number of times -- and then promptly forgotten what it was I constructed. For nonfiction writers, forgetting your own ideas is a common occurrence and an occupational hazard. That's why writers keep notebooks.

I do recall, I think, Willingham once saying that we remember knowledge we've figured out for ourselves somewhat better than we do knowledge we've been told by someone else. Assuming that's the case, I surmise that the mechanism is the amount of time you spend trying to figure something out, which amounts to a form of practice or rehearsal as well.

I know for a fact that 'discovering' and 'constructing' your own knowledge is absolutely no guarantee that you will recall your own knowledge later on.

Not even close.

There's only one route to Carnegie Hall.

instructivist weighs in
think

Larry Summers, economist:

Suppose the educational system is drastically altered to reflect the structure of society and what we now understand about how people learn.

Richard Clark, Paul Kirschner, & John Swellers, psychologists and education researchers:

Our understanding of the role of long-term memory in human cognition cognition has altered dramatically over the last few decades. It is no longer seen as a passive repository of discrete, isolated fragments of information that permit us to repeat what we have learned. Nor is it seen as having only peripheral influence on complex cognitive processes such as critical thinking and problem solving. Rather, long-term memory is now viewed as the central, dominant structure of human cognition. Everything we see, hear, and think about is dependent on and influenced by our long-termmemory.
Putting Students on the Path to Learning
Richard E. Clark, Paul A. Kirschner, and John Sweller
American Educator | Spring 2012

Here's a thought.

Before Larry Summers writes a NY Times op ed invoking "what we now understand about how people learn," he should do a little nosing around and find out what we now understand about how people learn.

Hint: what we now understand about how people learn turns out NOT to be that in a world where the entire Library of Congress will soon be accessible on a mobile device with search procedures that are vastly better than any card catalog, factual mastery will become less and less important.

Long-term memory is a biological entity with cognitive functions.

Internet archives are storage facilities.

Those two things are not the same.

P.S.: A Commenter reminded me of the Clark/Kirschner/Sweller article the other day, when I was trying to recall where I'd read the passage about long-term memory having a cognitive and biological function that Google does not. Unfortunately, I can't remember who it was, but thank you!

AND SEE:
Larry Summers has a really bad idea
Look it up

Barry's new article. In my comment, I left a passage on memory consolidation from John Medina's Brain Rules:

It takes years to consolidate a memory. Not minutes, hours, or days but years. What you learn in first grade is not completely formed until your sophomore year in high school.

In today's Times, Larry Summers weighs in on the question of what college students ought to learn in college.

Larry's answer: not too much, because the entire Library of Congress will soon be accessible on a mobile device with search procedures that are vastly better than any card catalog!

Larry bases his novel and highly original thesis (to wit: "factual mastery will become less and less important") on "what we now understand about how people learn."

(Does Harvard have node chairs, I wonder? Sounds like no.)

OK, I'm going to go look up calculus on the internet. I've always been interested in calculus, so now that I've received a mobile device for Christmas, I'm going to look it up. Then I'm going to collaborate with some friends who also looked up calculus on the internet to figure out what to do about the 21st century global world meltdown.

I'm going to do this because I've noticed that economists use calculus in their collaborative group papers.

[pause]

There is a reason why students must commit content to memory as opposed to looking it up on a mobile device with search procedures that are vastly better than any card catalog.

That reason has to do with working memory.

More anon.

What You (Really) Need to Know by Lawrence A. Summers

update: Why students have to memorize things
and see: Extremely fast learning & extended working memory

AND SEE:
The founder, chair, and CEO of Netflix has a really bad idea
Larry Summers has a really bad idea
Wash U professor on Reed Hastings' really bad idea
Barry Eichengreen has a really bad idea
President Obama has a really bad idea

David Brooks has a really bad idea
David Brooks has a really bad idea, part 2
David Brooks has a really good idea

The Daily has a really bad idea

This looks interesting:

Implicit learning as an ability.
Kaufman SB, Deyoung CG, Gray JR, Jiménez L, Brown J, Mackintosh N.
Cognition. 2010 Sep;116(3):321-40. Epub 2010 Jun 22.

Looks like math is more heavily dependent upon explicit learning than language --- which makes sense, given that language is innate and math isn't (or not so much, at any rate).

Yet more evidence that the K-12, constructivist preference for tacit or implicit learning as opposed to direct, focused, conscious learning is a very bad idea when it comes to math.

I re-took another Blue Book math test I'd taken some time ago. I again had no conscious memory of any of the problems, but did them all (save one) fast, finished early, and got everything right.

The problem I couldn't do was the last one in the set and thus the most difficult. My conscious thought was that I had no idea how to do it, a conclusion I arrived at after having in fact done the problem and finding that my answer wasn't amongst the choices. I left it blank.

After the timer rang and I had checked my answers, I went back to the last question.

Turned out my solution was right. It was my arithmetic that was wrong.

Not only do I not recognize the problems, it appears that I don't recognize the solutions, either, even a solution I have just written myself.

I'm going to re-read Arthur Reber, I think. One of my favorite books.

Implicit Learning and Tacit Knowledge: An Essay on the Cognitive Unconscious (Oxford Psychology Series)

The Official SAT Study Guide, 2nd edition

FedUpMom on remembering area vs circumference:

I had the following brainstorm at an embarrassingly advanced age:

For a long time, I knew there were two formulas that were somehow relevant to circles, namely 2πr and πr², but I could never remember which one was area and which one was circumference.

I finally realized that πr² must be the formula for area, because area is described in square units.

Tell your kids.

re: memorization is a dirty word, Magister Green offers this advice:

I have found that using "automaticity" in place of "memorization" lets me get around the stigma of the latter word while retaining its basic meaning. And "automaticity" sounds cool, so people don't press.

Guerrilla teaching forever!

Guerrilla teaching or marketing --- !

So now we have our public schools setting no-recall standards.

Meanwhile, back on planet Earth, the rest of us are trying to remember where we put our car keys, one of the few remaining factoids you can't find on Google.^*

Which probably accounts for the appearance in the Sunday Magazine of a 6000-word book excerpt on the World Memory Championships.

[pause]

Oh, look!

The world memory champion is from China!



^* Speaking of where I put my car keys, the iPad has a finder! I need a finder-for-everything, and I'm often amazed that a finder-for-everything doesn't exist and can't be purchased on Amazon. The Sharper Image had a finder-for-everything gadget out a few years back (actually a finder-for-four-things-of-your-choosing), but the one I ordered didn't work.


I remember nothing

Educators rely heavily on learning activities that encourage elaborative studying, while activities that require students to practice retrieving and reconstructing knowledge are used less frequently. Here, we show that practicing retrieval produces greater gains in meaningful learning than elaborative studying with concept mapping. The advantage of retrieval practice generalized across texts identical to those commonly found in science education. The advantage of retrieval practice was observed with test questions that assessed comprehension and required students to make inferences. The advantage of retrieval practice occurred even when the criterial test involved creating concept maps. Our findings support the theory that retrieval practice enhances learning by retrieval-specific mechanisms rather than by elaborative study processes. Retrieval practice is an effective tool to promote conceptual learning about science.

Most thought on human learning is guided by a few tacit assumptions. One assumption is that learning happens primarily when people encode knowledge and experiences. A
related assumption is that retrieval—the active, cue-driven process of reconstructing knowledge—only measures the products of a prior learning experience but does not itself produce learning. Just as we assume that the act of measuring a physical object would not change the size, shape, or weight of the object, so too people often assume that the act of measuring memory does not change memory (1, 2). Thus most educational research and practice has focused on enhancing the processing that occurs when students encode knowledge – that is, getting knowledge "in memory". Far less attention has been paid to the potential importance of retrieval to the process of learning. Indeed, recent National Research Council books about how students learn in educational settings (3–5) contain no mention of retrieval processes.

It is beyond question that activities that promote effective encoding, known as elaborative study tasks, are important for learning (6). However, research in cognitive science has challenged the assumption that retrieval is neutral and
uninfluential in the learning process (7–11). Not only does retrieval produce learning, but a retrieval event may actually represent a more powerful learning activity than an encoding event. This research suggests a conceptualization of mind and learning that is different from one in which encoding places knowledge in memory and retrieval simply accesses that stored knowledge. Because each act of retrieval changes memory, the act of reconstructing knowledge must be considered essential to the process of learning. Most prior research on retrieval practice has been conducted in the verbal learning tradition of memory research (12). The materials used have often not reflected the complex information students learn in actual educational settings (13).

Most prior research has not used assessments thought to measure meaningful learning, which refers to students' abilities to make inferences and exhibit deep understanding of concepts (14, 15).

[snip]

[B]oth elaborative concept mapping and retrieval practice are active learning tasks, and our results make it clear that whether a task is considered "active" is not diagnostic of how much learning the task will produce.

The Critical Importance of Retrieval for Learning
Jeffrey D. Karpicke* and Janell R. Blunt
Sciencexpress Report
www.sciencexpress.org / 20 January 2011 / Page 1 / 10.1126/science.1199327

Although I read the research that came out a couple of years ago on learning-via-testing, I hadn't absorbed the idea that testing is a form of studying.

Years ago, I learned the distinction between recall (remembering an actor's name, say) and recognition (recognizing an actor's name when you hear it). Recall is harder.

I thought the learning-testing effect was a simple matter of working on recall as opposed to recognition.

Looks like it's not. Remembering always means reconstructing.


news flash: There's no such thing as regurgitating facts.

Fabulous article in the WSJ today - !

A lot of people wonder how Chinese parents raise such stereotypically successful kids. They wonder what these parents do to produce so many math whizzes and music prodigies, what it's like inside the family, and whether they could do it too. Well, I can tell them, because I've done it. Here are some things my daughters, Sophia and Louisa, were never allowed to do:

• attend a sleepover

• have a playdate

• be in a school play

• complain about not being in a school play

• watch TV or play computer games

• choose their own extracurricular activities

• get any grade less than an A

• not be the No. 1 student in every subject except gym and drama

• play any instrument other than the piano or violin

• not play the piano or violin.

Why Chinese Mothers Are Superior
JANUARY 8, 2011
By AMY CHUA

Reading through the whole-brain thread I've just come across this observation by Tracy W:

Understanding is good, but it's only momentary. In my work I often spend hours trying to understand a problem, and then finally getting the flash of insight. And I've learnt that when the flash of insight comes I need to write it down and store it somewhere I can find it again, because otherwise I forget.

There's a terrific scene in Mad Men built on this premise. One of the copy writers stays up all night working on a brilliant idea for an ad campaign, and then, the next morning, he can't remember what his brilliant idea was.

The punchline: with Don Draper glowering and Peggy Olsen nervously trying to cover for her colleague, the guy finally cracks and says, "I had a brilliant idea, but I don't remember it."

There's a beat while the audience waits for Don to blow up.

Then Don says, "I hate when that happens."

I've just written Amazon.com review of Dan Willingham's book "Why Children Don't Like School." Here it is:

The Jesuit teachers also knew that fixing the impression was extremely important. This fits in with their stress on memory-work—and it was not automatic memorizing, it had to be thorough understanding. Their manuals of teaching never tire of saying: Repeat, repeat and repeat. They nearly always add that the master must watch carefully and vary his questions to ensure that there is nothing mechanical about this repetition, but then they urge once more Repeat and once more Repeat.

The Art of Teaching by Gilbert Highet
p. 147-148

I checked out Spanish by Association and I can see why you're liking it.

I was raised in a bi-lingual home and so, never experienced learning another language from scratch until I studied German and Italian (but mostly German because it's not Latin-based). I'm not sure if I ever took the time to consider how I organize language until more recently.

Anyway, I had a funny experience where I consciously realized that when I hear a word, particularly Eng/Span, I have an image, a sound (music, voice), or a feeling, etc. running through my mind. These references are tagged with words in the languages I speak and in the case of Spanish I can recall them just as easily as I can in English. These associations are what allow me to be fluent which I am less so in Italian, and even less than that in German. In Spanish the associations are on par with the English while the other are weaker. It only makes sense that finding an efficient way to tag or associate concepts would improve your ability to speak and understand another language, or anything for that matter.

Strangely enough, I recently picked up a couple of mnemonic books (Thirty Days Hath September, Every Good Boy Deserves Fudge) as a result of this feeling that I should be helping my children associate many of the concepts they've been learning so they can more easily retrieve the ideas when they need them.

and see: It's on the Tip of the Tongue by Charles Zanor WAPO March 11, 2008

Anonymous left a link to this new study by Susan Goldin-Meadow and her colleagues:

ABSTRACT—How does gesturing help children learn? Gesturing might encourage children to extract meaning implicit in their hand movements. If so, children should be sensitive to the particular movements they produce and learn accordingly. Alternatively, all that may matter is that children move their hands. If so, they should learn regardless of which movements they produce. To investigate these alternatives,we manipulated gesturing during a math lesson. We found that children required to produce correct gestures learned more than children required to produce partially correct gestures, who learned more than children required to produce no gestures. This effect was mediated by whether children took information conveyed solely in their gestures and added it to their speech. The findings suggest that body movements are involved not only in processing old ideas, but also in creating new ones. We may be able to lay foundations for new knowledge simply by telling learners how to move their hands.

Psychological Science - March 2009
Volume 20—Number 3 p. 267 - 272

I'm keenly interested in this work, and in fact taught one set of gestures to the kids in my afterschool Singapore Math class.

I'm wondering whether this phenomenon is related to mnemonic devices. I've been thinking about this subject lately because I've been making my way through an amazing book: Spanish by Association, which deserves every one of the 5 stars Amazon reviewers have bestowed upon it.

I need to learn more about mnemonics. From what I gather thus far, mnemonic devices work for a different reason than I had thought. More anon.

I'll try to steal a moment to read the Goldin-Meadow study, too. If you'd like a copy, send me an email. cijohn @ verizon.net