### Is there a cognitive psychologist in the house?

The comments to my post on math prerequisites that aren’t being met are the reason that, despite all of the comment spam, I leave these pages world-writable. Some excellent stuff there, and I hope that some curriculum developers in my province are reading what my readers have to say. I have more to say in the prerequisites thread, but first I wanted to address some comments from high school math teachers who assure me that *their* students aren’t leaving their classrooms without knowing what an equation was, how to add fractions, and so on.

I believe them. To look at the BC high school curriculum, it’s hard to find anything specific that is explicitly wrong with the course. (Aside, that is, from the “This book is brought to you by the letters T and I” thing, which I suppose is a big one.) By the time British Columbian teenagers leave grade twelve, they’ve been exposed to fractions, exponents, quadratic equations, graphs, and logarithms - the prerequisites for college math. They’ve even been exposed to questions that are a lot more open-ended than I expected - questions that demand a modicum of creativity.

The problem is that many students - perhaps most - never properly learned this stuff. Others knew it to some degree at some point, and promptly forgot it.

I’ve been tutoring a student in that second category - he can’t remember how to solve a linear equation (when do you divide both sides by the same number? When do you add stuff to both sides?), he can’t remember how to evaluate powers, he can’t remember…much of anything he learned. And nothing makes me feel like a failure as a teacher than hearing that the mathematics I’ve been trying to *teach* has come across as something merely to *remember*.

Can anyone recommend any reading material, accessible to a layperson, on how knowledge is stored? Why, and how, some information gets stored in short-term memory, some finds its way into long-term, and other types of data - a first language, for instance - is not something that people think about as *knowing* and not as *remembering*? Because I don’t think it’s something intrinsic about mathematics that it seems to get stored into the most unreliable section of my students’ brains.

As I was thinking about this post, reader Susan made an insightful connection between math (something students memorize) and language (something students learn):

It’s possible for a non-expert to determine whether a child is literate by asking him or her to read something unfamiliar (ideally both outloud and silently) and to explain what they’ve read in their own words.

We need to define and expect something similar as far as being numerate.

To which I add: not expecting anything similar for math gives students no reason to *process* and *learn* math instead of just remember it.

I’m only now remembering that I actually *did* do something similar last term, although I didn’t draw the connection to reading that Susan did. Last term, after one particularly miserably-done test, I gave my students an opportunity to submit (for credit) corrections to the word problems. I gave them a template to follow: they had to describe, in words, what information they were given in the problem, and what they were missing. They had to describe, in words, how those things were related, and then, based on their previous work they had to provide the equations they needed. A few students actually came up to me and said that that exercise really helped them understand what they were doing with word problems. I was shocked - this was the horrible precalculus class - and encouraged. A month later, however, I found that those same students drew a blank when faced with the word problems on the test.

We’ve got a long way to go.

While I don’t have recommendations for cognitive psychology books, you may want to browse Kathy Sierra’s blog . She’s an award winning computer science trainer and author with a deep interest in “brain friendly” education.

Well, since I’m one of the high school math teachers who responded that my kids leave my class knowing at least bits and pieces of those prerequisites well, I should remind you that my school is an exception. We are among the top schools in the U.S. midwest. Two of our students scored perfect scores (2400) on the new SAT. We have a (truly exceptional) 8th grader who will be taking BC calculus next year, even though we discourage students from accelerating, prefering enrichment over speed. Honors in the humanities are also common. We have our merely-above-average students, of course, but somehow we manage to maintain a culture where kids basically like to learn. In schools were it’s necessary to teach to a lowest common denominator they have to hit the “hows” hard because for so many kids the “whys” are a waste of time.

As for the difference between learning math and learning language, it might help not only to have kids use precise language to describe math, but to keep pointing out that math *is* the language that describes patterns: the expression 7w *means* something if w is a number of weeks. Most kids will see the pattern very quickly when you put up on the board:

1/2 - 1/3 = 1/6

1/3 - 1/4 = 1/12

1/4 - 1/5 = 1/20

even so, they have a hard time with 1/n - 1/(n+1) = 1/n(n+1)

They want to believe that the symbols do something more than describe the pattern. They believe that the notation is the math, not that the pattern is the math, and it is merely described by the notation. Notation represents ideas just like words do, and thus algebra is in a very real sense a primitive language.

I’m not sure quite how to use that to our advantage other than to repeat it over and over to the students. But teaching math and teaching language ought to have a lot in common.

The problem is the way the material is presented in the book. After the introductory paragraph most students have questions that they would like the book to answer or the teacher to answer, but what happens is that the teacher and the book start expounding definitions upon definitions and lemmas upon lemmas. By the time the book or the teacher gets to the question the student was thinking of she is already to tired to think and is drowning in all the definitions and proofs, that seemed like they had nothing to do with the question the student was asking. And the way the last proof is proved, which was the one the student wanted proved in the first place, goes like this :

Using Lemma 1.1 with Lemma 1.2.3 we can deduce “this” and with this using Lemma 1.9 we can deduce “this” which matches the definition of what we wanted to prove.

It would be hard for anyone to learn anything from such a presentation of any nontrivial material.

Thank you for the compliment.

You have implied something more than I originally wrote and which is likely much more significant and profound as far as an explanation of what is the matter with students. Understanding and using a natural language is not the same as being literate in some particular language. I am not a linguist but I think linguistics teaches that one of the crucial aspects of language use is the ability to produce and understand sentences that one has never heard before. Literacy, by contrast, is the ability to read and write a language and presupposes language knowledge. It is quite possible to understand and use language and be illiterate but not the reverse. (Of course, acquiring literacy generally increases one’s language understanding.)

So if I stick with my previous analogy, then “numeracy” is the ability to read and write mathematics using a combination of natural language (word problems) and special notation. But, drawing a parallel with the relationship between literacy and understanding language suggests that numeracy is not possible unless one already understands mathematics. Other commenters

have made the same point in various ways. Polymath makes the distinction between using notation, which relates to numeracy, and seeing patterns, which relates to understanding.

I think linguists believe that there is some sort of young age (perhaps 8 or 10 years) after which the brain is no longer plastic [?] enough to learn language. (I mean learn some first language; obviously people can learn additional languages past that point.)

So now I’m wondering what are the parallels between understanding and using natural language and understanding and using mathematics? Are there certain items essential to an understanding of mathematics that can only be grasped in early childhood? Is there something roughly like dyslexia where an intelligent person understands mathematics but has great difficulty reading and writing mathematics?

David Karapetyan - I’d agree for upper-level math texts, but for the high school math texts I use, you sure don’t see piles of definitions and lemmas and theorems. There are examples upon examples - so many, that every example that a student will ever see on a test or quiz appears in the text. Rather than being innundated with abstraction, students don’t even get an opportunity to work through abstract ideas and relate them,

on their own, to familiar concepts. Lately I’ve been reading some recreational-ish math texts - the one on my nighttable right now isGeometric Inequalities- and in the introduction to every such book is a memo to readers that they should have a pen and paper with them as they read, so that they can draw their own diagrams and grapple with the equations on their own. This idea is FOREIGN to most high school students; their texts are all about spoon-feeding. (And pretty pictures. And completely contrived “applications” - if I can get my hands on this one particular grade 12 text, I’ll have more to say about that one later.)Is there something roughly like dyslexia where an intelligent person understands mathematics but has great difficulty reading and writing mathematics?My father was an intelligent man who couldn’t do math on paper. As long as the problem was cast in rhetorical terms he had no trouble. A logic problem or mental arithmetic was easy. The simplest equation, or even long division, he wouldn’t attempt.

He said that he had some kind of psychological problem with math. He attributed it to an over-zealous parochial education. That is, that back in the 1930’s the nuns somehow instilled in him a hatred of math. I don’t completely discount his explanation, but the same nuns taught him history, English, etc. with great success.

Part of the problem with smart people who are bad at math might be that math as presented through high school is a very narrow business. Arithmetic, Algebra, “Geometry,” Algebra, Calc. All very heavy on symbolic manipulation and very light on logical reasoning. I think unless the student is precociously good at that narrow (clerical?) subset, he never gets to the point where he can take a real math course.

Sanctimonious Hypocrite -

Also light on

motivation. I just started tutoring a business student who “was never any good at math”; she wanted me to work with her before she began the service course she’ll be taking in the fall. I spent our first session giving an overview of what she could expect in the class, and WHY that course was part of the program. It was clear that she’d never thought about math that way - she thought that I’d be spending our sessions going over, say, how to solve equations, and that would be it. She also interrupted me frequently with questions that anticipated where I (and the course) would be taking the material - obviously, she’s no dummy. But then she would APOLOGIZE for interrupting, and tell me to just go on, continue with what I was saying. I tryed to excise that mindset - trying to see where the material was heading, I told her, meant she was thinking about the material, and that was what active learning was all about. I encouraged her to continue.Susan -

I don’t know (but I’m curious); however, there are a lot of building blocks of mathematical reasoning that little kids pick up on naturally, and for some reason math teachers and parents seldom try to build on those, choosing instead to start math, as it were, from scratch. (See the umbrella example (#6) from my last post; also, and I know I’ve mentioned this before, but there’s NO EXCUSE for five-year-olds who know that two dogs plus four dogs equals six dogs, growing up into twelve-year-olds who can’t simplify the expression 2d+4d.)

I’m a cognitive psych graduate student, so I may be able to shed minimal light on some of the issues brought up. Concerning a basic exposition of memory processes: any undergrad-level cognitive psych textbook will have at least one chapter on memory, and often two or three. There are any number of good books out there; the best thing to do is head to a university bookstore and leaf through a few to find one you like. I’d strongly recommend getting a neurobiologically-informed book, since most contemporary activation-based models of how memory works are heavily influenced by our understanding of the brain. Short-term and long-term memory have proven to be rather fuzzy concepts; they’re nothing like the digital computer-based models that were common 50 years ago that posited physically distinct stores in which you literally ‘put’ memories.

As far as practical advice goes, while it’s probably doing many of the people I work with a disservice to say this, I feel much of memory research can be summed up as follows: people tend to remember things that they (a) rehearse a lot, (b) think about deeply, and (b) are interested in. Given a choice, (b) is preferable to (a); i.e., people remember things better when they use what’s called ‘elaborative’ or ‘deep’ encoding (thinking about how a fact relates to other knowledge in their nomological network) than when they use ’superficial’ encoding (i.e., sheer memorization, or attending only to surface characteristics). Hands up if you find that surprising…

It goes without saying that if you can get your students genuinely interested in the subject matter, they’re going to do better. But don’t feel bad about yourself if you can’t get everyone interested, or if some people just can’t do math! Mathematical ability is, unfortunately, rarer than most other intellectual abilities–particularly the more spatial branches such as calculus, which begin to draw heavily on both quantitative and visuospatial abilities (in addition to many abilities that, as you surmise, appear to share much in common with the linguistic faculty).

As far as ‘mathematical dyslexia’ goes, sure. People have argued that particular brain regions (mostly in the parietal lobe, towards the back and top of the brain) are specialized for transformation between linguistic and symbolic information (or more generally, between different representation systems). It’s quite possible for someone to be relatively high on what’s called general or fluid intelligence (i.e., to have sophisticated abstract reasoning and problem-solving ability) but be relatively deficient on some other lower-order ability that’s important for math. A really nifty extreme example of this sort of thing is a condition called acalculia, which typically occurs after stroke-induced damage to a region called the angular gyrus. Acalculic individuals can retain virtually all of their cognitive faculties, including complex abastract reasoning, with one striking exception: they can no longer manipulation quantities in any way. Even single-digit addition is beyond the capacity of some patients. Put simply, cognitive ability is enormously diverse, and different individuals can vary along any number of dimensions. As someone who presumably has a natural faculty for math, it may seem to you introspectively like the rate-limiting factor is some monolithic form of intelligence, but rest assured that there are any number of other potential bottlenecks an individual may suffer from first. As someone who works in close proximity with psychologists and neuroscientists, I think the old myth that many people go into the social and life sciences because they can’t do math is truer than most care to admit….

Pardon for being off the subject of how knowledge is actually stored — I have no idea. But I will speak as a person now attempting to make up for past failures to come to grips with this stuff.

I think there is at least one distinct difference between math and language learning, insomuch as the structures of math are far more explicit when attempting to do the work. As most people to identify the gerund in a sentence, and you’ll get either giggles or blank stares. Similar to actually defining split infinitives, dangling participles, and other more complicated constructions of language…yet people use it all the time. Aside from basic araithmatic, the math doesn’t get used this way. Each problem requires explicit acceptance of components before moving on to a solution. Of course, that’s just my uneducated thought on it.

But it would lead me to this, possibly useless, thought: make the kids teach each other. I don’t know how this might get played out in an actual class, but it comes from my own experience. In trying to learn statistics, some game theory, microeconomic theory, etc., I found a study group that would break up the work, then gather to “teach” the other people their portion of a subject. This meant having to try answering other people’s questions, or following up with explanations to the group if the “teacher” didn’t know the answer at that point. We’d even swap topics so that we all heard the same subject from different viewpoints. It made a HUGE difference in getting a subject cemented in my head if I knew I was going to have to explain it to someone. Even know, sitting in math classes I should have taken YEARS ago, I think to myself how I would tell someone else about what I was learning. If I can’t, I realize I don’t know it well enough myself.

Not to be cliched, but I think there is some R. Feynman quote about it. Something about how, if he couldn’t explain some effect or phenomenon of physics in a way that freshman physics students could grasp the idea, then they didn’t understand the issue well enough themselves.

From a cognitive psychology standpoint the most likely problem is that the material has not been overlearned sufficiently to facilitate mastery of the material and automatic recall. Students learning is most likely still in either the inflexible knowledge stage or in the merely learned stage (whereupon it is quickly forgotten). The student then bumps into the 7 plus/minus 2 limitation (i.e., the number of things that the human brain can keep in active memory) because he is expending so much effort remembering the old material and is unable to learn new material. Basically, because math is cumulative, once the student loses the thread of understanding, from that point forward their math knowledge becomes increasingly inflexible.

Rehearsal to mastery is the key to success

Holy crap, there’s actually a cognitive psychologist in the house! Two, even! Damn, this internet thing is cool. Next up:

Is there a sugar daddy in the house?In-house cognitive scientist -

Heh, I’ve found that people are actually seldom shy when it comes to divulging their inability to do math, and will readily admit [above]. But, it doesn’t seem to me that the rate-limiting factor responsible for my students’ (and others’) failings is, as you say, some monolithic form of intelligence; I’ve had many many students who couldn’t do math, and considerably fewer who I would consider dumb. If I had to ascribe students’ mathematical inability to one cognitive function (bearing in mind that I am very much not an expert on this, and that I know that there’s more than one cause), it wouldn’t be lack of intelligence; it would be something more akin to an inability to focus and think deeply about a certain subject for a sustained period of time. The decline in attention spans in this hemisphere is well-documented; and that’s important when it comes to doing more complex mathematics. Insofar as external causes for mathematical inability, the list is huge: the omnipresence of calculators in the classroom, the inability of teachers and texts to get students interested in the subject (I’m guilty of this one; in my defense, I love math and can’t imagine a single less interesting or less useful collection of high-school-level mathematics than the mathematics that is actually taught in high school), the philosophy (on the parts of educators) that it’s better to expose students to lots of material rather than expect proficiency in a subset of it…

kderosa - it’s possible that we mean different things by “recall”, but what you say here was part of my point:

But what frustrates and upsets me is exactly that - that for my students, it’s a matter of

recalling, rather than a matter ofknowing. (I generally teach with the aim of showing a method or a topic, so that the students can use that to derive, say, the formulas that I then show on their own. This is a very, very unrealistic expectation, though - as soon as I slap a formula on the board, students seem to go into “must memorize this!” mode.) By “overlearning”, do you mean the process that puts this new knowledge into a less fickle part of the brain than short-term memory?Next up: Is there a sugar daddy in the house?I’ll have to pass on that one… graduate student stipends aren’t what they used to be *cough*.

Heh, I’ve found that people are actually seldom shy when it comes to divulging their inability to do math, and will readily admit [above].Oh, I agree… I just think that people are much less forthright in admitting how much their inability to do math

bugs them!If I had to ascribe students’ mathematical inability to one cognitive function (bearing in mind that I am very much not an expert on this, and that I know that there’s more than one cause), it wouldn’t be lack of intelligence; it would be something more akin to an inability to focus and think deeply about a certain subject for a sustained period of time. The decline in attention spans in this hemisphere is well-documented; and that’s important when it comes to doing more complex mathematics.What you describe actually sounds more like most researchers’ description of fluid or general intelligence. There are various ways to cash out the notion of sustained attention, but one fairly popular and well-supported idea is that fluid intelligence is at bottom the capacity to inhibit interference; i.e., being able to pay attention to something in the face of distractions is at the core of what it means to say someone’s intelligent. There’s no doubt this ability is a prerequisite for math, but it’s also a prerequisite for pretty much any sort of abstract reasoning. There are any number of very smart academics who have no trouble thinking deeply about abstract concepts–just not mathematical ones. So I suspect the bottleneck on math for many people is probably something much lower in the processing stream, e.g., an inability to apply formal rules to arbitrary symbols (as opposed to meaningful words). Mathematical cognition isn’t my area of expertise though, so that’s just (somewhat informed) speculation.

As far as the decline in Western attention spans goes, that’s really more media fear-mongering than anything else. There’s little if any empirical evidence that ability to sustain attention is decreasing; in fact, it’s well-established that average IQ is steadily increasing (and has been for several decades). People often point to increasing diagnosis of ADHD in kids as evidence, but that’s almost certainly an artifact of culture (much as the alleged rise in incidence of autism is largely due to increased awareness of the disorder and broadening of diagnostic criteria).

But what frustrates and upsets me is exactly that - that for my students, it’s a matter of recalling, rather than a matter of knowing.This much is anecdotal, but in my own experience, I’ve found that I end up doing this myself with math. Lacking any aptitude or basic intuition for what the next step is to solve a given equation, I’m forced to algorithmically recall all of the rules I’ve been taught and try to figure out very painstakingly how to apply them. It’s not a lack of interest (I’ve always held math to be a supremely valuable asset for any scientist) or practice; it’s just that, no matter how much I work at it, it doesn’t become easy, and I don’t build intuitions the way I do in non-mathematical areas involving abstract reasoning. To the extent that some of your students find themselves in a similar situation, there’s no reason for you to beat yourself up over their lack of progress…

I’m not a cogntitive Psychologist, but Daniel Willingham is. And, Ed Hirsh has certainly read the literature. But also see Zig Engelmann’s critique.

It sounds like your students’ knowledge is still at the inflexible stage, i.e., their knowledge is still superficial and not fully understood. This is why they have trouble taking what you’ve taught them and applying it to new types of problems.

With respect to your students’ inability to recall previously learned material, this is a result of merely learning to mastery, at best, instead of learning past mastery (i.e., overlearning. Willingham puts it this way:

Due to the brutally cumulative nature of math, overlearning is critical. One system of math instruction that I know of that forces the students to overlearn the material is Kumon with their timed problem sheets. Get too many problems wrong in the allotted time period means your recall of the material isn’t sufficient to progress to the next level. Only when the student has mastered the material is he allowed to advance.

Looks like I mangled my links. Here’s the first two paragraphs again with proper links (hopefully).

I’m not a cogntitive Psychologist, but Daniel Willingham is. And, Ed Hirsh has certainly read the literature. But also see Zig Engelmann’s critique.

It sounds like your students’ knowledge is still at the inflexible stage, i.e., their knowledge is still superficial and not fully understood. This is why they have trouble taking what you’ve taught them and applying it to new types of problems.

Well, it’s certainly reported more; I’ve had conversations (anecdotal, again) with teachers who’ve been in the classroom a long time (>25 years), and they all report it. Though whether we can identify an exact cause and effect, I’m not sure: I know that high school students in Ontario, for instance, no longer have to write research essays; they’re never given much chance to hone their concentration skills.

Thanks for the info on overlearning; I’ll definitely read over it closely when I have my own computer back and am not restricted to an hour of computer time a day in an overheated public library.

“Rather than being innundated with abstraction, students don’t even get an opportunity to work through abstract ideas and relate them, on their own, to familiar concepts.”

Right on! This summer I am teaching a Transition to Advanced Math course. One of my students today complained that the textbook examples bear no relation to the exercises. Of course the exercises are mostly proofs and appications of concepts (like functions) to unusual situations. One example I can think of is a simple exercise requiring the student to test whether a function meets the criteria for a metric, which are given in the introduction to the exercise. This student has problems applying the definition of a function to relations on small finite sets. She wants to be a math teacher!

Many of my students will skim the text but seem to have no ability to wrestle with it.

I think attention spans in the classroom are lower but not because of increased ADHD. Students are capable of watching a movie intently for two hours that makes me fidget. We no longer favor instructional methods that build the student’s attention span - we accomodate their inattention. Inattentiveness in class is a matter of habit, learned skill and most importantly will.

The single best short article on remembering what you’ve learned is Daniel Willingham’s, “Practice Makes Perfect–

But Only If You Practice Beyond the Point of Perfection” at http://www.aft.org/pubs-reports/american_educator/spring2004/cogsci.html

What your students don’t have, which they need to have, is ‘overlearning.’

here’s an excerpt:

Although practice takes on a different character for the longer-term, it is no less important. Studies show that if material is studied for one semester or one year, it will be retained adequately for perhaps a year after the last practice (Semb, Ellis, & Araujo, 1993), but most of it will be forgotten by the end of three or four years in the absence of further practice. If material is studied for three or four years, however, the learning may be retained for as long as 50 years after the last practice (Bahrick, 1984; Bahrick & Hall, 1991). There is some forgetting over the first five years, but after that, forgetting stops and the remainder will not be forgotten even if it is not practiced again. Researchers have examined a large number of variables that potentially could account for why research subjects forgot or failed to forget material, and they concluded that the key variable in very long-term memory was practice.*(see below *) Exactly what knowledge will be retained over the long-term has not been examined in detail, but it is reasonable to suppose that it is the material that overlaps multiple courses of study: Students who study American history for four years will retain the facts and themes that came up again and again in their history courses. [end of quote]

There’s quite a lot of interesting material on learning, understanding & memory in cognitive science research.

For instance, conceptual understanding does help us remember material, as seems intuitively correct, so some (many?) of your students probably would be remembering the math they learned better if their conceptual understanding had been more advanced in the first place.

Basically, though, forgetting stuff is what we do. Everyone forgets everything. Period. The miracle is probably how much your students remember, not how much they’ve forgotten.

Since your students have forgotten things they really must remember in order to take any college-level math at all, I think the solution for you would be to find a book they can work through that give them vast amounts of structured practice.

For elementary math, I think Mathematics 6 by Enn Nurk and Aksel Telgmaa (http://www.perpendicularpress.com/math6.html) is probably the single most powerful elementary mathematics textbook around. There are probably 10,000 individual problems in the 300-page book, and they’re structured to teach the concepts as well as produce overlearning. It’s brilliant.

I don’t know (yet) what might work for algebra, geometry, trig & high school calculus….

You can try asking Chris of Mixing Memory - an excellent cognitive science blog.

Excellent, excellent - thank you, cogsci readers. And this -

- just supports my suspicion that the TWELVE or so years that students spend in math classes before they graduate high school, is almost certainly spent doing something other than learning. Sigh…