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11 Oct 2005 - 13:27

## Wayne Wickelgren on why math is confusing

Carolyn's post on teaching Ben unit conversions reminded me of a passage in Math Coach:

One of the reasons math is so difficult for children is not the number of facts to be learned, but that many of those facts are very similar to one another. To see this, let's first look at what your child has to learn. It's all in a table with the numbers 0 to 10, the numbers to be added or multiplied, on each side....In such a table, there are just 11 x 11 = 121 facts. This isn't much. Everyday, a child learns far more than 121 facts of this size about other life events well enough to correctly answer a question about them the next day. To remember most facts longer than a day requires just repeating them maybe a day, a week, a month, and a year later. It doesn't require the ten to thirty minutes of daily studying over many weeks that children need to learn 121 addition or multiplication facts. What's the difficulty here?

### spaced repetition

I think this is so important, I'm going to pause & repeat the basic principles:

• Everyday, a child learns far more than 121 facts...about...life events well enough to correctly answer a question about them the next day.

• To remember most facts longer than a day requires just repeating them maybe a day, a week, a month, and a year later.

• [Remembering ordinary, non-math facts] doesn't require the ten to thirty minutes of daily studying over many weeks that children need to learn 121 addition or multiplication facts.

### why is remembering what you've learned about math hard?

Wickelgren:

It's the similarity between the facts. That is, the fact 3 + 5 = 8 is not so different from 3 + 6 = 9. They both contain 3's; they both contain +'s, and they both contain single-digit numbers....

Thus, to a child beginning to learn such facts, the facts overlap in the brain, creating a blur that makes it easy to confuse them and difficult to remember any single answer. In cognitive psychology, this "blur" is called associative interference, which occurs when one idea, A, is linked in the mind to two or more other ideas. It's like static on the radio, which often occurs when other stations or electrical impulses interfere with a radio station's music or speech. When the child sees 3 + 5 = ___, all the facts involving 3 and 5 get activated in the mind, and the wrong answers create interference for the right answer.

Indeed, such interference is probably the main reason why all mathematics is harder to learn than other subjects. In every area of math, unlike nonmathematical subjects, a relatively small number of basic concepts are used to express a large number of facts or more advanced concepts. This situation creates interference because each basic concept activates many other facts or concepts, which in turn interfere with one another.

### another pause

• [Math] facts overlap in the brain, creating a blur....called associative interference.

• When the child sees 3 + 5 = ___, all the facts involving 3 and 5 get activated in the mind, and the wrong answers create interference for the right answer.

• [Such] interference is probably the main reason why all mathematics is harder to learn than other subjects. In every area of math, unlike nonmathematical subjects, a relatively small number of basic concepts are used to express a large number of facts or more advanced concepts.

• [Each] basic concept activates many other facts or concepts, which in turn interfere with one another.

### Ben's associative interference

Back to Carolyn:

Well, Ben has been consistently getting his unit conversions backward. He'll convert, say, 13 meters to .013 millimeters, or 12 meters to 12000 kilometers.

Classic.

First of all, the numbers 13 and .013 are extremely similar. Think how much gunk is getting activated in your brain when you contemplates the numbers 13 & .013. Not just math facts, but Friday the 13th, horror films--I'd hate to see the CT scan of a kid staring at 13 and .013.

Second, Ben's right: in the past small numbers like .013 (as opposed to 13) have been associated with small measures like milliimeters (as opposed to meters). And now, all of a sudden, .013 meters is supposed to convert to 13,000 millimeters (is that even right? I'm getting confused myself), yet another number with a big fat 13 in it......who came up with this system, anyway?

Math stinks.

OK, I didn't say that.

Math doesn't stink.

But math is hard, and that, I think, is a core truth about math & about teaching math, which tells me we should constantly be looking for ways to do exactly what Carolyn did:

[Unit conversion] is a reliable procedure that will get him through these problems, and hopefully work around that rut that was forming. We're going to practice it to automaticity.

And as he goes through school, tricks like this -- using dimensional analysis -- will get him through a lot more than unit conversions, too.

### back to the magical number 7plus or minus 2

Needless to say, Wickelgren is a huge fan of practicing to automaticity.

He is also a fan of chunking:

The human mind possesses a way around this problem, namely to create a new idea that binds together a set of constituent ideas or facts into "chunks"--making them hang together in logical ways, like notes in a song.

I love that!

notes in a song: lovely

That's going to be my conscious goal from now on. I want Christopher's math facts, & my own, to hold together the way the notes in a song hold together.

back to Wickelgren:

Students who are very skilled at math may often do this without explicit instruction, but all students can benefit from instruction that helps them do this.

Using my training in learning and memory, I decided to find a way to help children create chunks in math--to glue mathematical facts together in a manner that creates a kind of mathematical melody that is much easier to remember than a sequence of disconnected notes. The key is forming as many connections between the ideas and facts as possible.

source:
Math Coach by Wayne Wickelgren, page 84

### procedural memory: another kind of glue?

Early on, Carolyn wrote a couple of posts that have stayed with me: Swoop and Swoop and The Craft of Math.

Here is Carolyn on teaching Ben the cross-multiplication algorithm:

And that was it: he got it: those swooping moves with the pencil and the crossing numbers. That's what the standard algorithms are: they are moves that you learn how to make. Those moves get into your fingers, just like learning the piano or the violin or typing, and eventually you can do them completely mindlessly.

What Carolyn is talking about here, I think, is procedural memory, which one usually sees defined as motor skills memory. (I'm not sure that's all it involves, but that's what you see. I'll check Willingham's text.)

Cognitive scientists and their fans always invoke the concepts of overlearning and automaticity when discussing math ed.

But Carolyn is the first person I've seen talk about the importance of procedural memory.

I think Carolyn may have finally solved a problem I've been puzzling over, which has to do with something Temple has told me many, many times. Temple says that students in her architectural drawing class who've never learned to make scale drawings by hand, but only on the computer, can't do it.

I've spent quite a bit of time musing over this. It made instant sense to me, but why?

Why should a person who never learned to draw by hand not be able to draw on CAD?

I've come up with different answers at different times. Mainly, these fall into two categories.

I've thought, Montessori. Multiple learning modes, multisensory this-and-that.....

I've also thought, lack of conceptual knowledge, which is undoubtedly correct; Temple's students don't seem to understand what a scale drawing is. They'll click on a 'door' icon on CAD, and not realize that the door is opening into the chute the cattle have to go through, and that, furthermore, the door is wider than the chute itself, and if a chute is so narrow a normal-sized door can't open up all the way, then a 2000-pound cow isn't going to get through it, either. The drawing looks fine on the computer screen, but would be ludicrous in real life.

Now, reading Carolyn's posts again back-to-back with Wickelgren, I'm thinking: procedural memory. Carolyn is enlisting Ben's procedural memory to teach him math, and, perhaps even more importantly, to remember math. Remembering math is hard.

Procedural memory is incredibly sturdy. No one forgets how to ride a bicycle. Which, when you stop to think about it, is pretty astounding. I bought a bike a couple of years ago. I hadn't ridden one in.....20 years?

I hadn't forgotten how. (Though I was rusty, that's for sure, and I felt unstable & even scared when I first got on the thing. That was strange, like being transported back to childhood.)

So....can you get math into motor memory?

I don't know.

But I'm going to assume you can.

One more reason not to rely on computer programs for math facts practice.

### update

I've checked Cognition, The Thinking Animal, & it contains only one brief reference to procedural memory. Along with gazillions of references to concepts that sound exactly like procedural memory.

This is the horror of cognitive science.

There are a zillion different terms for the same thing. It's as if every cognitive psychologist who ever lived re-discovered the same concept everyone else discovered, and named it something new.

Cognitive science is the exact opposite of math. In this respect.

Temple and I used to have nervous breakdowns over this back when we were writing Animals in Translation: Using the Mysteries of Autism to Decode Animal Behavior. Temple would actually comb through books & articles and draw up Equivalence Charts. Implicit memory is the same thing as cognitive unconscious is the same thing as incidental learning etc. (I no longer remember whether that particular equivalence is correct; I'd have to do endless checking to see. Which I'm not going to do.)

Anyway, it looks like the answer is yes, procedural memory does cover skills like knowing how to use an algorithm in math.

What I can't tell is whether the 'kind' of procedural memory you use in learning how to use an algorithm has the same sturdiness as the 'kind' of procedural memory you use in learning to ride a bike.

### an illustration of my suffering

Here's a great passage from Willingham's textbook that perfectly captures what Temple and I were dealing with:

An early and important distinction was between procedural and declarative memory (Cohen & Squire, 1980). Procedural memory is memory for skills and is often called "knowing how" memory. For example, if you know how to ride a bicycle, that ability is supported by procedural memory. Declarative memory supports memory for facts and events and is often called "knowing that" memory, such as knowing that George Washington was the first U.S. President.

These hypothetical memory systems are closely identified with the implicit versus explicit distinction, [ed.: "closely"? how close? are they the same thing? not the same thing? or what?] which is a distinction of tasks (Graf & Schacter, 1985). [ed: oh, I get it! closely associated with tasks! not exactly memory per se, but memory for tasks, which isn't exactly the same as knowing-how, but is close...] Explicit tasks are those that directly query memory ("Who was the first President?"), and usually they are supported by the declarative memory system. [ed: usually? but not always?] Implicit tasks do not directly query memory ("Ride this bicycle"); rather, memory is inferred from the participant's performance. [ed: memory is inferred?] Implicit tasks usually are supported by procedural memory. [ed: again with the 'usually'...]

This kind of thing was basically MY LIFE for about two years there.

Don't get me wrong.

I love cognitive science; I majored in cognitive psych as un undergrad. I probably think about Daniel Willingham's American Educator articles once a day at a minimum.

But I don't share E.D. Hirsch's conviction that cognitive science is going to save us any time soon.

### procedural versus declarative, and a web site about math & memory

These two pages are short, to the point, and useful:

Course syllabus & textbook here:

And here's a working document on "Mathematical Memory" by some professors trying to figure out the connection between procedural & declarative memory in math ed.

### oh, swell

The Death of Implicit Memory by Daniel B. Willingham & Laura Preuss (question: is Daniel B. Willingham the same person as Daniel T. Willingham?)

Abstract

It's going to be many years before cognitive science knows enough about implicit memory to help with math ed.

keywords: Wayne Wickelgren associative interference math facts why math is confusing

Wickelgren on introducing algebra
Wayne Wickelgren on algebra in 7th & 8th grade
Wickelgren on math talent & when to supplement
late bloomers in math & Wickelgren on children's desire to learn math
Wayne Wickelgren on mastery of math & on creativity & domain knowledge
Wickelgren on why math is confusing

Confessions of an engineering school wash-out
more confessions of an engineering school washout
the Terminator, or 'the magical number 7, plus or minus 2'
On Having a Math Brain (by Carolyn)
grandmasters and the magical number 7
math brain debunked (by Carolyn)
math professors versus computer science professors

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Catherine,

this is a great post... I'm going to have to think this stuff over for a while.

I was reminded yesterday of an experience I had in high school drama, of all things. I'd been given a part in a short play called ... I can't remember what it was called, but maybe someone else will. Anyway, it was about a bedridden rich woman who overhears, on her phone, a man plotting a murder. She tries calling the police, they blow her off, she tries calling her husband, he blows her off, everyone blows her off, and at the end of the play it turns out to have been a hit ordered on her, by her husband.

So I played the bedridden woman, the center of the action. Other people at the other end of the phone appear in spotlights on corners of the stage.

I was terrified because I could not remember all the darned meaningless exchanges and phone numbers. I was constantly saying things like "operator, please give me Murray Hill 3, oh 6 oh 9", and I couldn't just make them up at random, because I had to say the same ones several times.

In retrospect, I think that's how my anxiety disorder began. Although it worked very well for that part.

If I were an adult now, I'd know it was a huge challenge and I'd have some strategies to apply. At the time, I didn't know what to do, and I had no help, either.

I expect this is how a kid who hasn't memorized the multiplication tables feels, the entire way through school. No wonder they avoid math as much as possible.

-- CarolynJohnston - 11 Oct 2005

oh my gosh!

what an incredible story!

-- CatherineJohnson - 11 Oct 2005

I have some books on memory strategies that are supposed to be fantastic. They're tough to read, because, for me, reading about memory just isn't gripping the way reading about a lot of other things is.

But I've started to use the strategies I have managed to read about.

A few years ago I went to a friend's 40th birthday party, where the only person I knew was her.

I used a visualization device to remember every name of every person there, and pretty soon I was the Party Entertainment; people were standing around watching me Remember Names.

It was hilarious.

-- CatherineJohnson - 11 Oct 2005

One of the issues that constantly gets jumbled up in talking about math ed is the "simple" issue of REMEMBERING math.

You can understand math perfectly, and still not remember it.

They're not separate issues either biologically (I don't think) or 'professionally' (professional meaning teachers figuring out how best to teach). (For instance, I've seen research, which I believe, showing that conceptual understanding helps memory; when you understand a concept you remember it better, too.)

But functionally speaking, in everyday life, I think it pays to see them as separate and distinct.

HAS CHRISTOPHER PRACTICED THIS CONCEPT/ALGORITHM/DEFINITION ENOUGH THAT HE CAN REMEMBER IT?

AND: HAS HE PRACTICED IT ENOUGH THAT HE CAN REMEMBER IT ON DEMAND.

-- CatherineJohnson - 11 Oct 2005

ON DEMAND AND FAST

-- CatherineJohnson - 11 Oct 2005

I like Zig's observation about Hirsh:

Hirsch's case is very weak. He has an underlying message that makes sense, which is that students must learn a base of knowledge if they are to perform on a range of unanticipated tasks. The reason I have taken issue with the details of his orientation, however, is that they are not scientific. They lack respect for the importance of bottom-line classroom data, which show what works and what doesn't. (emphasis mine)

By basing all our rules and decisions on what actually goes on the classroom, we avoid tarnishing our understanding and relying on questionable theories. And that's the whole point that Feynman tried to make in his chapter: The goal of scientific investigation is to be honest and not to be influenced by what we would like to believe, what people have said we should believe, what is politically correct to believe, and above all what seems believable. To yield to these influences is to buy a ticket to the cargo cults.

Zig is of course the grandfather of direct instruction and it has taken him many years to hone his curriculum by constantly testing lesons in the classroom, observing the results, and erfining the lessons until the kids are learning what they are supposed to.

-- KDeRosa - 11 Oct 2005

KDeRosa

Fabulous!

That's exactly the way I feel about it.

I gobble up every cog-sci text I can get to, but when push comes to shove you have to look clearly at what exactly is happening when you're trying to teach or learn.

-- CatherineJohnson - 11 Oct 2005

This is why I'm not a diehard advocate of 'science' & 'research' as the answer to our prayers....

Somebody out there (and it could have been Hirsch himself) drew a terrific distinction between craft & science, or maybe it was between craft & profession.

He pointed out that long before architecture was formalized into a professional discipline, people knew how to build houses. Building houses was a craft.

He wasn't saying that it's a bad thing to formalize a craft into a profession.

He was saying, instead, that when a craft is formalized into a discipline in fact you don't throw out the knowledge craftsmen possess.

Their knowledge is real.

You build on it, and extend it.

Good teachers today know how to teach. It's a craft.

Good research will, in the years to come I hope, extend & formalize the knowledge good teachers possess.

What that tells me is that I need to look at what good teachers are doing.

I need to find those teachers, and pay attention.

And when someone like Carolyn, who did a very good job learning math herself, and who is now doing a brilliant job teaching math to a child with a significant disability, tells me that math has to 'get into a child's fingers,' I pay attention.

-- CatherineJohnson - 11 Oct 2005

Conceptual understanding only gets you so far. Even a great deal of conceptual understanding is no substitute for knowing something on demand and fast. Unless your conceptual understanding is so high that you can in fact recall things on demand and quickly.

I have taken many exams where I have undertood the material conceptually and could even derive material I had not memorized, but it takes a tremendous amount of mental energy to do this and then determine the correct procedure for solving the problems in a timed exam. Ultimately, you are forced to rush, leading to silly mistakes, and increasing the stress level. Better to have mastered the facts and procedure to automaticity. Less mistakes and far less stress. There just is no substitute for repeated practice beforehand.

-- KDeRosa - 11 Oct 2005

"And when someone like Carolyn, who did a very good job learning math herself, and who is now doing a brilliant job teaching math to a child with a significant disability, tells me that math has to 'get into a child's fingers,' I pay attention."

Golly. Like Gomer said. Thank you!

I really do think that the standard algorithms and algebraic manipulations need to become part of motor memory, like riding a bicycle, in order to become automatic. Doing them by hand is an aid to this process; that's how I expect the processes that pull something into motor memory are triggered.

I have no hard evidence for that, however.

Here's an interesting idea for an experiment. Take someone with demonstrable expertise in math, with selective brain damage to parts of the brain other than motor memory, where procedural knowledge resides (frontal lobes? Catherine?). How much of the ability to do those grade-school algorithms do they retain?

-- CarolynJohnston - 11 Oct 2005

Although the experiment would be all messed up if they'd lost their domain knowledge of numbers. You need both.

-- CarolynJohnston - 11 Oct 2005

The great book on Carolyn's experiment (in the realm, at least) is:

I still haven't read it, probably because I find it too upsetting.

We loved the movie about the guy with no short-term memory, though.

What the he** was its name???

I'll think of it.

-- CatherineJohnson - 11 Oct 2005

Here's a short write-up on Patient H.M.

-- CatherineJohnson - 11 Oct 2005

Brilliant, brilliant film.

-- CatherineJohnson - 11 Oct 2005

Here's the Memento flash site

-- CatherineJohnson - 11 Oct 2005

AND O MY GOD--HERE'S ANDY KLEIN, A FRIEND FROM GRAD SCHOOL WHO I HAVEN'T TALKED TO SINCE, WRITING ABOUT MEMENTO AT SALON

oh boy

that was weird

I'm going to have to go lie down

-- CatherineJohnson - 11 Oct 2005

Not everyone may wish to go quite as far as I have -- four theatrical viewings, three of them with copious note taking; a fifth viewing on videotape, with lots of whipping back and forth to check for differences in "repeated" shots, and slo-mo attention to quick-cut subliminal moments; reading the published script and comparing it to the film; reading the short story, "Memento Mori," written by Nolan's brother Jonathan and credited as the film's source; and a few trips through www.otnemem.com, the film's official Web site, also by Jonathan Nolan. More than anything, I'm grateful to everyone who posted ideas about "Memento" in the movie conference of the Well -- you know, "America's pioneering online community, see www.well.com" -- a whole gang of enthusiastic, contentious, brilliant, pigheaded and articulate fans, who have more than once opened up for me some movie that I simply did not get.

wow

that sounds like Andy

-- CatherineJohnson - 11 Oct 2005

can't wait to show this to Ed

-- CatherineJohnson - 11 Oct 2005

ok, i don't need to lie down

i need to go for a run

-- CatherineJohnson - 11 Oct 2005

"CAD students can't draw" "I think Carolyn may have finally solved a problem I've been puzzling over, which has to do with something Temple has told me many, many times. Temple says that students in her architectural drawing class who've never learned to make scale drawings by hand, but only on the computer, can't do it."

What is "it"?

If you are talking about drafting, then these students haven't been taught. They have been taught how to use a program, like AutoCAD?. Old-time designers spent a long time learning how to draft and some of their drawings are like works of art. Not only that, they are drawn so that others can understand them; different line thicknesses and styles, proper dimensioning, and many detail views. This can be done in CAD, but it takes time and that is not what is taught in school. I also don't think the solution is to first learn drafting by hand. CAD doesn't preclude good drafting, but it is pretty good at distracting people from the real subject. Think about people learning to write who spend more time on learning how to use a word processor (pictures and fonts), than learning how to communicate. I don't think there is a need to write with paper and pencil first. I just think that the tools become an end in themselves and reduce the amount of time applied to what you really need to learn. Some students might become good CAD jockeys, but not very good architects. (They could have gone to a technical school.) Colleges have to be very careful about getting distracted by the tools.

If you are talking about design (engineering, form, fit, and function), then many students are not taught that either. The holy grail of computer-aided design is 3D modeling first and then derive the construction information from that automatically. Solid modelers are now set up to automatically generate dimensioned, exploded views and to automatically generate dimensioned traditional 2D scale drawings. Unfortunately, many of these programs leave a lot to be desired and the students aren't taught to focus on the end product and not the process. If the program can't do it, then they won't learn it. Once again, students are distracted from the real topic of architecture. They may be able to produce a 3D model of a building in record time and generate nicely rendered views set against a digital background of the building site, but the building might be a failure.

The problem is not learning how to draft by hand first. The problem is that they are not taught what they need to know. There are too many distractions. CAD (the process) becomes the end and not the means to an end. CAD distracts; it doesn't impair.

One has to be careful about thinking that everything has to be done by hand first before using the calculator or computer. However, let me state categorically what I have said before. I see no need to use a calculator for math before 6th or even 8th grade. I could probably come up with some good examples, but it sure isn't necessary. I have also said that using computers or calculators at any level of schooling should raise expectations, not lower them. Many modern math curricula see calculators as avoidance tools, not as a means to tackle more complicated real-world problems that couldn't be done by hand.

I have been writing CAD programs for 30 years and I have dealt with customers at both ends; the older designer who learned everything by hand, and the younger person who only knows the computer ways. Many of the younger people are beginning to understand that there is a big difference between being a CAD jockey in an architectural office and being the lead architect. And, many of the older designers are realizing that their lack of CAD knowledge limits their ability to be competitive and to try new things.

As for the process of learning, hand drafting (even CAD drafting) isn't the goal. I have met some amazing designers who couldn't draw a straight line with a ruler.

-- SteveH - 11 Oct 2005

"But math is hard, and that, I think, is a core truth about math & about teaching math, which tells me we should constantly be looking for ways to do exactly what Carolyn did: ..."

-- SteveH - 11 Oct 2005

Teaching is not yet a mature profession. They talk a good talk when it comes to the science of teachiing and cognitive issues, but they stubornly refuse to abandon their fads and cognitively unsound practices. No amount of empirical data or scientific evidence will dissuade them. In fact, I don't even think they can distinguish between real science and junk science. Accountability, fixed goals, and government sinecures are part of the problem, but the real problem lies much deeper. In mature professions, practioneers cannot afford to become enamored of failed theories because products need to be developed, built, and shipped in working condition by fixed deadlines.

-- KDeRosa - 11 Oct 2005

Steve -- From what Temple tells me, her students don't know how to do proportional scale drawings, if that's the correct term.

They'll click on icons, put them on the screen, and then assume from looking at the image that their design works.

They'll make mistakes like creating a chute that's only 1 foot across.

I imagine they also don't know how to design (assuming I know what 'design' means)....but from what Temple says, these students are making elementary errors in math, measurement, and proportion.

-- CatherineJohnson - 11 Oct 2005

I have also said that using computers or calculators at any level of schooling should raise expectations, not lower them.

I think I should put that in 'Wit and Wisdom.

Also, I'm looking for something you said one day (I wish to heck I'd bookmarked it that minute): engineers aren't looking for the right answer, but for the best answer.

Do you have any idea where you left that Comment?

(I haven't done a Comments search yet--if I find it, I'll tell you.)

-- CatherineJohnson - 11 Oct 2005

I found it:

-- CatherineJohnson - 11 Oct 2005

Catherine,

You're way too fast for me. You've come and gone before I know what is going on.

-- SteveH - 12 Oct 2005

carolyn's play sounds like sorry, wrong number.
movie-ized with barbara stanwyck, iirc. memento rocks.

-- VlorbikDotCom - 12 Oct 2005

I think "Sorry Wrong Number" is correct.

It's a sad commentary on the state of my memory that even when prompted -- I can't be certain.

-- CarolynJohnston - 12 Oct 2005

WebLogForm
Title: Wayne Wickelgren on why math is confusing
TopicType: WebLog
SubjectArea: CognitiveScience
LogDate: 200510110926