WillinghamOnLearningModalities < Kitchen

15 Jul 2005 - 04:25

reading Willingham on learning modalities

From Daniel Willingham on learning modality theory, an explanation of why learning modality theory might make sense from a teacher's viewpoint:

There are two ways that a teacher might see what looks like evidence for modality theory in the classroom. First, a teacher who believes the theory may interpret ambiguous situations as support for the theory. For example, a teacher might verbally explain to a student - several times - the idea of borrowing in subtraction without success. Then the teacher draws a diagram that more explicitly represents that the 3 in the tens place really represents 30. Suddenly, the concept clicks for the student. The teacher thinks "Aha. He's a visual learner. Once I drew the diagram, he understood."
But the more likely explanation is that the diagram would have helped any student because it is a good way to represent a difficult concept. The teacher interprets the student's success in terms of modality theory because she has been told the theory is correct and because it seems to explain her experience.

Willingham offers the following suggestion: teach to the best modality for representing the idea, not to the student's best modality.

But what if there are multiple modalities to choose from, for an idea? More generally, what if there are a whole host of different ways to represent an idea, and the kid's not getting any of them?

I ran into that situation recently, when teaching Ben how to do simple problems by adding and subtracting constants on both sides of an equation. Actually, trying to help Ben get the hang of this has taken quite a bit of effort this week, and I don't think it's a hard idea. I've got kinesthetic, visual, and auditory ways of teaching it, too. I could even sing it, though that's getting a bit ridiculous.

For the kinesthetic learner, you could get out a balancing scale or use Bornstein manipulatives. You could draw pictures of pan balances for a visual learner. You can explain verbally, as I did repeatedly, that what you're doing to solve the problem x + 4 = 13 is to 'undo the addition' of the 4 on the left hand side of the equation. If none of this works, what do you do then?

Try each modality over again, I suppose. Round 2: in case he was a kinesthetic learner, I had him copy each step I made in his own handwriting (laugh, if you will, but it works for me when I do it). In case he was visual, I drew pan balances again, next to the equivalent equation: no dice. "Subtracting the 4 is applying the inverse operation to get the x by itself," I said, auditory-like, but that didn't help either.

All this time, of course, he was able to do the problems by repeating the steps I made; he is a fabulous rote learner (is 'rote' a modality? If not, it should be). But I could tell he wasn't really getting the gist of it. Finally, in exasperation, I said, "Look, Ben, what's the opposite of adding 4?

"Subtracting 4."

"Good! And what's the opposite of subtracting 13?"

"Adding 13."

"Good. All you're doing to get the x by itself is doing the opposite of adding or subtracting the number that's with it," I said, but I didn't even get it all out before he said, "OH! I get it!"

And that's the sound I love to hear.

So, knowing Ben's best learning modality didn't help, and wouldn't have helped. I wish teaching, and learning, were so predictable that all you needed to do to teach a whole class reliably was to know what each kid's best learning style was. But I think that learning is inherently unpredictable. The trick is to be able to hit the teaching problem from a bunch of different angles, and you need to know lots of different ways to present the information. The more, the better (by the way, this is a major part of what Liping Ma's Chinese elementary math experts do with their release time; sit around together, thinking up new ways to teach problems to tough cases).

As an aside, I have never been able to figure out Ben's best learning modality (aside from 'rote'. His raw memory is unbelievable). As a person on the autism spectrum, he's supposed to be a visual learner; this is accepted theory to such a degree that teachers will assume he needs to learn visually, but it's not always the right approach.

What Ben really is, is an unpredictable learner. You never know what's going to be easy, where he'll get stuck, and what will unstick him. He's the kind of kid who keeps a teacher on her toes.

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I could even sing it, though that's getting a bit ridiculous.

You SHOULD sing it!

Singing is bizarrely effective for autistic people, and Temple & I both wonder whether this has to do with music being the language of animals (which is Temple's belief).

I read about one mother who sang everything to her autistic daughter--and her daughter understood.

The girl understood nothing if it was spoken.

After reading that I noticed that in autism schools there is a huge amount of informal singing that goes on.

Sometimes the teachers don't even seem to notice that they've adopted a 'policy' of singing......it seems to have come up naturally.

Andrew had one terrific teacher at BOCES--Brooke Schultess (hope I've spelled her name right)--and as I recall they had all kinds of songs.

I remember a going-to-the-bathroom song!

(I could be misremembering, it goes without saying......but I do know I've heard a whole lot of informal singing at schools for autistic children.)

-- CatherineJohnson - 15 Jul 2005

Good. All you're doing to get the x by itself is doing the opposite of adding or subtracting the number that's with it," I said, but I didn't even get it all out before he said, "OH! I get it!"

That's amazing, because this explanation CONFUSES me!!!!

Although I'm not sure it would confuse me in context.

A couple of months ago I was trying to get some conceptual understanding of negative numbers.

I quickly discovered that--guess what?--as usual, I am clueless about a fundamental concept in mathematics I had always taken for granted as obvious and simple.

Sigh.

BUT--and I think this is important, hence the red ink--the minute negative numbers were defined as OPPOSITES for me, I got it.

It was just like Ben.

CLICK.

-- CatherineJohnson - 15 Jul 2005

I believe that opposites are a core category in the brain--not just a core 'category,' but a core 'processing mode.'

I'm certain there are cognitive scientists out there who can explain this, but for now I'm on my own.

Back when I was studying structuralism & post-structuralism and all that good stuff (REMEMBER! I HAVE A PHD IN FILM STUDIES! WHAT COULD BE MORE RELEVANT!)

OK, I'll try to get a grip.

Back when I was studying structuralism, Claude Levi-Strauss said opposites and oppositions are core, organizing principles.

(I haven't read him in years, so if that's a stupid summary of what he said, I take it back.)

Anyway, I think he's right.

Left-right
Child-adult
Man-woman
Up-down
Happy-sad
Good-evil

(Hmm. Remember when it was Good-bad?????)

We use opposites all the time, to explain everything.

The Swiss linguist deSaussure wrote that meaning comes from difference.

'Cat' has meaning in relation to dogs, not in relation to other cats.

Cat-dog.

We know what cats and dogs are, because they're--somehow--opposites.

(Think about it: cats and dogs are not opposites. They're both mammals, and they can live together as friends & colleagues. We make them into opposites in our thinking.)

Working through the Russian Math book, certain ideas that have confused and eluded me, like factors-and-multiples, suddenly became crystal clear because Russian Math taught them together, as paired opposites.

So.....I'm going to re-read Carolyn's post, because I think it's important.

Look at how many different strategies she tried (strategies!).....and the one that worked was an 'appeal' to opposition.

A basic principle in my own teaching, now, is to emphasize opposites whenever I can.

-- CatherineJohnson - 15 Jul 2005

One other experience I've had re-learning math:

SAXON MATH & SINGAPORE MATH (and possibly many other curricula, I don't know) both teach inverse operations together as opposites.

I've found that to be incredibly powerful.

I did not learn, as a child, that addition and subtraction were inverse operations.

I did learn the concept: inverse operations.

I knew it.

But I didn't get it.

That's because I was taught the four operations of arithemetic separately, as separate things-unto-themselves.

Then I was taught inverse-operations separately, as a thing-unto-itself.

Re-learning addition and subtraction together, as 'opposites,' and multiplication and division together, 'as opposites,' has been incredibly powerful and helpful.

-- CatherineJohnson - 15 Jul 2005

I'm going to re-post here what Christopher said just a couple of weeks into Saxon 6/5:

Multiplication and division are the big brothers, and addition and subtraction are the little brothers. And multiplication and division are cousins.

I'd put money on it that insight came from re-learning these operations as opposites.

-- CatherineJohnson - 15 Jul 2005

Wow, Carolyn!

Great post!

-- CatherineJohnson - 15 Jul 2005

As an aside, I have never been able to figure out Ben's best learning modality (aside from 'rote'. His raw memory is unbelievable

It really is astounding; you have to see it to believe it.

One time we went to visit Ed's parents, and Andrew became obsessed with the idea that there was something under the sofa we had to get for him.

Well, there was nothing under the sofa.

He got more and more agitated, we spent more and more time down on all fours peering under the sofa then trying to reason with him THERE'S NOTHING UNDER THE SOFA.....

It was the whole day.

Finally someone got the bright idea just to go ahead and move the coffee table (first removing everything that was ON the coffee table--it's always a production) and pull the sofa out from the wall and let him just stand there and stare at the nothingness that we knew was behind the sofa.

There was something behind the sofa.

It was the empty case for a videotape we had rented months and months before, on our previous visit.

Ed's dad had returned the video without the case, because he couldn't find it.

Not only did Andrew remember the exact video we'd rented months earlier, he knew exactly where the misplaced video case was. (There was no way he could have seen it under there, unless X-ray vision also comes with autism.)

We all just had to sit there for a little while, taking that in.

It was liking being in the presence of a person with supernatural powers.

-- CatherineJohnson - 15 Jul 2005

I don't know if it's rote learning, exactly, though it could be....

I HAVE to find a cognitive scientist to write for us.

I'm on the case!

-- CatherineJohnson - 15 Jul 2005

"But what if there are multiple modalities to choose from, for an idea? More generally, what if there are a whole host of different ways to represent an idea, and the kid's not getting any of them?"

I agree. I was thinking to myself as I read Willingham's work that there could be an infinite number or a continuum of modalities based on all sorts of genetic combinations. It also reminds me of the following article originally published in The Onion (wink, wink, nudge, nudge).

Parents of Nasal Learners Demand Odor-Based Curriculum

http://www.radford.edu/~thompson/obias.html

Can you imagine scratch-and-sniff math workbooks? What does the number 5 smell like? Blueberries. What does the number 2 smell like? Green beans. What does blueberries + Green beans smell like? Cherries, because I think of cherries when I see the number 7.

Actually, I always assign colors to things. I see the number 5 as blue. The number 4 is brown, 3 is yellow, 2 is green, and 1 is white. This is kind of the same thing as my seeing the state of Pennsylvania as green and Kansas as yellow. Perhaps these colors relate to the first toy alphabet and map of states I ever had. I can't say that I ever suffered by not being taught using my color modality.

Speaking of modalities, our schools sent home a questionaire asking parents how their kids learned. One parent wrote back and said: "Fast".

"Willingham offers the following suggestion: teach to the best modality for representing the idea, not to the student's best modality."

He should say that teachers should START with the best known method (like phonics - forget this modality stuff) for each topic and use whatever other techniques work until all students understand. Isn't this what teachers do anyway? Besides, what is a modality of an idea versus a modality of a person? Now where did I leave my hip boots?

I remember specifically (!) thinking in algebra class that the variable 'X' could have multiple values at the same time. I wonder what modality of teaching would fix that? What is the modality of that idea? I think my problem came from equations with both X and Y; you could plug in different values for X and find different values for Y.

It seems to me that many people get way too involved with teaching methodology and forget about content and skills; content follows methodology. I think that methodology should follow content and skills, like form following function. I like the practical nature of KTM where people discuss how or why one explanation worked for a particular problem rather than another.

I also worry about focusing too much on pedagogical methodology. I really don't want to get into an argument over modalities or constructivism when the real problem is extremely slow curricula that leave kids far behind their peers around the world. My feeling is that schools would LOVE to get into discussions over modalities, just as long as you stayed away from content and skills questions.

-- SteveH - 15 Jul 2005

It seems to me it's all about understanding the equals ( = ) sign!

-- ChrisAdams - 15 Jul 2005

"Re-learning addition and subtraction together, as 'opposites,' and multiplication and division together, 'as opposites,' has been incredibly powerful and helpful."

That's very interesting. I have been using the idea of opposites instinctively with the middle graders I tutor. I extend it to exponents and roots. It's been helpful with teaching how to isolate a variable in equations. I hammer in the notion that you must do on the right side what you do on the left side while I extend my arms to visualize a balance. They get it.

-- KtmGuest - 15 Jul 2005

Mr. Fang had a math teacher who said "My job is to give you three different explanations. Your job is to understand one of them," and I think that's the right idea.

I suspect that people who look into what makes a good teacher are going to find that it's not which method the teacher uses to teach something, but whether she keeps trying different methods until all the kids get it. One good teacher might start with phonics, and another good teacher might start with whole words, but both of them are going to end up coming at the problem in a whole lot of ways until the students can read.

-- CardinalFang - 15 Jul 2005

This is a fantastic site. It's becoming encyclopedic. As it gets more and more voluminous, it gets increasingly more difficult to keep up with reader contributions.

You have the very useful feature "Recent changes". This feature may be well-known to oldtimers but it is not easy to locate by newtimers.

My suggestion is to make "Recent changes" very prominent, maybe at the top of the page in bold and bigger letters. Also, describe its function.

Just a thought to make the site easier to navigate.

-- KtmGuest - 15 Jul 2005

I just noticed that "Recent changes" does not capture all reader posts. For example, it does not list SteveH^?'s and CardinalFang^?'s latest posts.

-- KtmGuest - 15 Jul 2005

SteveH^? is correct. Whatever you do, you must bring all students to UNDERSTANDING. And this is what students really do want. "Understanding" or "not understanding" are the reasons students "hate" or "love" math.

A good math teacher learns how to "approach" a student having difficulty. The teacher has all of these ideas (hopefully) stored away back there some place, ready to be pulled out when needed. But a teacher's most important job is determining where the student's understanding fell apart, identifying where there might be gaps in reasoning, and knowing how to bridge those gaps. This is where choosing the right approach comes in. It might involve reteaching, reviewing a step which is being omitted, or helping a student reason through a difficult story problem.

So a hand goes up, and a student says, "I need help."

(Those are my favorite times of the math hour because it means I get to find the puzzle piece that is needed to make this all fit together in his/her head and give understanding to what I've just taught or to what's needed to solve this problem.)

So I have some choices, but I always look to see what the student has already done or tried. That tells me what to do next.

I then start by having the student read the problem to me (if it is a word problem).

Then I make a choice:

I might say,"OK, draw Bill's house. Now write 'B' on it for Bill. Now, draw the school house; now write 'S' on it. Now, draw the road from the house to the school. Now, look at the problem again to see how far it is to the school (and the student answers outloud 4 1/2 miles). OK, write that number on the map you've just drawn."

I could have drawn that little map for the student, and might do it under certain conditions, but having a student draw the map involves his sight and his movement (and mouth from speaking and ears from hearing his own voice) and it involves more importantly his BRAIN. (I've got to make sure his BRAIN is working and focused on the problem so he can "understand".)

So I would continue, "Start at Bill's house with your pencil; now walk to school. OK how far did Bill just walk? ('4 1/2 miles') OK, write that down. Now, he's at school, but he wants to come home, so have Bill walk back home. How far did Bill walk to get home? OK, write that down under the first number. How would you know how much he walked to school and back on that one day? ('add the two numbers together.') Good, do that. OK, but that is just one day. Now, let's read the problem one more time and let's see what the questiion was. (How far does Bill walk in a week going to school and back?) OK, now how could we figure that out?"

Many times the problem just works itself out in the student's brain as they begin to draw out a picture of the problem.

Or, if I've checked over his work and seen that he's added 4 1/2 miles 5 times for the 5 days of the week, I can see that he's overlooked a part of the question. So I have the student reread the question. Many times, the student will catch his own mistake when he hears his voice repeat the words "to school AND BACK". If not, I have him read just that part again.

Something really important: for some students it's just a matter of not knowing "how or where to get started". There are gaps in processing the information and gaps in understanding.

Not only does he need to know where to start, but he needs to know that where he is starting will get him going in the right direction and will help him get the right answer. This is very important to a student's confidence. If a student doesn't know "where to start" or isn't sure "if he's going about solving it properly", a teacher's trying to find the right modality isn't necessarily the answer.

This is where an "constructivist" approach is so devastating to the student. That student wants to be able to KNOW what to do to get the right answer.

It's terribly upsetting and deflating to a student not to know "where to start" and "if they're taking proper steps to solve the problem correctly." To leave this student to come up with his own idea isn't helping him.

Hopefully, though, a teacher will NOT just repeat the instructions that were given initially (if any were given). If a student didn't get it the first time, at least try a different approach.

One of my former students said of another teacher, "Why should I ask her for help? She always just repeats the same instructions that didn't make sense the first time." This student, smart as a cookie, just wanted to understand the entire process and to know how to work to get the right answer.

-- CarolynMorgan - 15 Jul 2005

Carolyn,

beautiful comment! Great ideas!

-- CarolynJohnston - 15 Jul 2005

It's interesting, I'm having a similar problem with my LD son. It seems so obvious to say you walked this far to point A, how far to walk there and back. I keep doing these big physical moves with my arms, sweeping from an imaginary point A, then back to point B, thinking he can clearly see that my arms are moving the exact equal distance, but it is still not clicking even though it seems like common sense to me. I was going to draw him the picture as you described, but now I think I'll let him do it.

As the Brits say, you never know what makes the penny drop.

-- SusanS - 15 Jul 2005

Yes, try having your son draw the pictures. Be sure and break each little step down into sub steps. If he doesn't get it, get objects to represent point A and then the original point B. Have him label those points (home, school, whatever)using a folded index card. Have him stretch out a long rope to represent the distance and have him label the distance using another card. Then, have your son walk to point A and stop. Have him record the distance he has just walked. Then have him turn around and return to point B and record his distance again.

For some students the movement causes the brain activity to fire up and helps.

The mistake we adults make is that once a student has done it once, we assume he's got it. Not necessarily.

So, you have to repeat this with several other things, but make them interesting. "How far is it from the kitchen to the front door and back (or whatever works at your house)?)

How far would Mom walk from the front door to the mailbox and back?

You'll know when he's got the procedure down.

Then try a story problem without the activity. That may come a day or two or ten down the road. But watch his face when he recognizes that kind of problem and knows he knows it!!!

-- CarolynMorgan - 15 Jul 2005

I always assign colors to things. I see the number 5 as blue. The number 4 is brown, 3 is yellow, 2 is green, and 1 is white. This is kind of the same thing as my seeing the state of Pennsylvania as green and Kansas as yellow. Perhaps these colors relate to the first toy alphabet and map of states I ever had. I can't say that I ever suffered by not being taught using my color modality.

That is fascinating!

Do you think this is common for 'mathies'?

Boy, it's hot today.....what is the word for what Steve does?

Oh--it's 'synesthesia.'

I think Steve is experiencing a variant of synesthesia.

It's not exactly the same thing, I don't think, but I bet it's related.

-- CatherineJohnson - 15 Jul 2005

Well, I think some of this is that teaching is a different form of expertise from mathematics.....and we're all trying to gain (and some of us already have) teaching expertise in mathematics.

I know that I, personally, need to learn from teachers AND from mathematicians (mathematicians in the broad sense, meaning anyone whose expertise & career lie in mathematics-related industries, as well as people like Barry who majored in math & have high levels of math knowledge and comprehension.)

Where I see the danger in focusing too much on constructivism is in something Temple taught me, which is that we must 'audit outputs,' not 'inputs.'

I've been meaning to write about this, but just haven't gotten around to it.

Temple can show--she can prove--that when you are charged with monitoring quality you absolutely should not focus on how people are doing what they're doing.

You must focus on the outcome.

If our goal is to persuade our schools to teach math more effectively, what is important is: evaluating outcomes to our children's learning.

Do our children know the math they need to know when they need to know it?

It is not important how teachers, schools, and administrations achieve that goal.

Even worse, it is destructive to micromanage the people and organizations whose work we are auditing or evaluating.

(Temple can prove this, too.)

So I agree with Steve in one realm, but disagree in another.

In the realm of changing mathematics education in the schools, what we need to focus on is exactly what Steve says.

Outcome.

Are our children learning math, and are they learning it quickly enough to keep up with the rest of the world?:

In the realm of our own attempts to teach math we do need to think about constructivism, about direct instruction, about guided inquiry, about cognitive science, behaviorism and all the rest.

At least, I do!

-- CatherineJohnson - 15 Jul 2005

I'll do a real post about Temple's approach to auditing animal welfare in meatpacking plants SOON.

-- CatherineJohnson - 15 Jul 2005

It seems to me it's all about understanding the equals ( = ) sign!

Hi Chris!

As soon as Christopher gets back from camp, I'm going to test this out on him.

I'm actually not sure whether he's encountered the do-the-same-thing-to-both-sides idea.....and I'm going to be interested to see whether understanding the equals sign is good enough.....

That's assuming he understands the equal sign 'well enough.'

He definitely understands it to the extent that Carpenter talks about in the beginning of his book.....

Let's see.

Here it is:

Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School, page 9 (pdf file).

This is worth looking at.

They asked kids in grades 1 thru 6:

What number would you put in the box to make this a true number sentence?

8 + 4 = __ + 5

Fewer than 10% of the students gave the correct response of 7, and, strikingly, performance did not improve with age.

6th graders did slightly worse than younger kids.

Typical responses were 12 and 17.

Back to Christopher: I gave him this question awhile back, and he was initially confused by it.

He had the standard 'surface feature' confusion I struggle with constantly.

But he got the answer, and fairly quickly, too.

What this tells me is that he does understand the equals sign--but I'm going to be curious whether understanding the equals sign transfers to a do-the-same-thing-to-both-sides problem.

-- CatherineJohnson - 15 Jul 2005

That's very interesting. I have been using the idea of opposites instinctively with the middle graders I tutor.

That's another thing: I TRY to notice what I'm doing instinctively--and I'm VERY interested to hear what other people are doing instinctively.

All of us, I believe, have core knowledge of how to teach--because we are social animals, who learn thru observation.

And because we are a social animal that has experienced extreme cultural evolution, which can only be kept going thru the transmittal of one generation's knowledge to the next.

So I believe we all have 'core,' gut, instinctual knowledge of how to teach.

That does NOT mean WE KNOW HOW TO TEACH!!!

(Don't ask me to explain.)

I think one reason parents in particular get so upset by constructivism is that it violates a gut knowledge we already possess about our children and what they need.

Even if constructivism worked brilliantly, in other words, I think it would still be tough on parents. We'd have to tell ourselves: OK, it looks all wrong, but it works.

So.....paying attention to the things we do instinctively is one useful source of information, because fairly often those insights need to be expanded, refined, built upon, etc.

I'm just beginning to read the NRC's book on learning, and it looks like cognitive science probably does find that 'metacognition' helps.

Metacognition in this case would be making yourself conscious of the things you do that work (in this case using opposites).

-- CatherineJohnson - 15 Jul 2005

Mr. Fang had a math teacher who said "My job is to give you three different explanations. Your job is to understand one of them," and I think that's the right idea.

I LOVE IT!!!

Naturally I have a whole post on this very subject all written-up in my head......

I need a whole new life, with 60 or 70 hours in the day.

-- CatherineJohnson - 15 Jul 2005

This is a fantastic site. It's becoming encyclopedic. As it gets more and more voluminous, it gets increasingly more difficult to keep up with reader contributions.

You have the very useful feature "Recent changes". This feature may be well-known to oldtimers but it is not easy to locate by newtimers.

My suggestion is to make "Recent changes" very prominent, maybe at the top of the page in bold and bigger letters. Also, describe its function.

Just a thought to make the site easier to navigate.

ktm guest: yes yes yes yes yes!!!!

-- CatherineJohnson - 15 Jul 2005

and furthermore, I, too, have discovered that 'recent changes' and the search don't seem always to capture the comments, which are often the most important part of the thread.....which is why I've been frantically trying to get things logged into the index, too.....

we've got some USABILITY BRAINSTORMING TO DO

-- CatherineJohnson - 15 Jul 2005

Here's the other weird-interesting-Twilight-Zoney thing: last night I wrote down on a 'door-note' that today I had to put up a post reminding people about the 'recent changes' page--

And now here's ktm guest saying the exact same thing in the thread!

-- CatherineJohnson - 15 Jul 2005

encyclopedic!

OK, now you've got me started.

ktm started out as a bliki,

then it became a blooki,

now it's turning into a World Blooki

-- CatherineJohnson - 15 Jul 2005

I was going to draw him the picture as you described, but now I think I'll let him do it.

I think drawing is DEEPLY important, and I have no idea why.

I just believe in it.

This is one of those 'up from the deep' instincts that I'm simply going to carry on listening to no matter what anyone else says, does, publishes, or thinks.

That's why I'm making Christopher do every single bar model story problem in SINGAPORE MATH, start to finish.

-- CatherineJohnson - 15 Jul 2005

get it inside your hand

-- CatherineJohnson - 15 Jul 2005

btw, this is another area where Temple has influenced me profoundly.

She has told me over and over and over again (ok, I'm talking about an autistic person here: she has told me HUNDREDS of times) that her students who have never learned to do architectural drawing by hand cannot make architectural drawings on the computer.

Period.

They can't do it.

-- CatherineJohnson - 15 Jul 2005

The mistake we adults make is that once a student has done it once, he's got it. Not necessarily.

Absolutely not.

People make that mistake, period, not just with teaching.

I was at a big PTSA meeting last fall, and we were discussing paying for kids to take part in activities if their parents couldn't afford to.

The president said the PTSA always pays in such cases, and the teachers know about it.

Then several room parents spoke up and said, 'My child's teacher doesn't know about it.'

The president insisted that they did know about it.

Then more room parents said, no, they don't know about it.

Finally the president said, impatiently, 'We sent them an email!'

I had one of those people-don't-have-a-clue moments.

The whole essence of learning and memory is:

spaced repetitions.

Basically, no one learns anything from a one-time exposure.

So here we are spending 24-hours a day volunteering for our kids' school, and we don't even know that ONE email sent to the teachers ONE time at the beginning of the school year when their brains are overloaded with a zillion things to remember is not gonna do the job.

I figured it was going to be a miracle if any teachers knew the PTSA would pick up fees for kids whose parents couldn't.

-- CatherineJohnson - 15 Jul 2005

Thanks Carolyn,

I will definitely take a shot at some of those suggestions and see what happens. I think because he is older I assume he is going to get it more quickly, but of course, I must never assume.

As far as drawing, it is difficult for me to teach that because it is actually one of the few things I am fairly talented at. It is very organic to me so I can't figure out why others can't do it. I sometimes find teaching the thing that I'm best at more difficult than teaching something that I had to work at.

Of course, being an obsessive doodler might explain some of my ability to draw. Even though that part of my brain has probably shrivelled up quite a bit from lack of use, it's something that feels very organic to me. I wish I had the same attitude towards math that I do towards art. I don't know what that is, a self-posession of sorts, I guess. Even if I drew something that turned out wretched, it wouldn't faze me because I know it's within me to improve it immediately. I do have a harder time teaching that or even understanding why others might have difficulty with it.

Maybe that's what the math people feel towards math. I only have that kind of relaxed confidence if it concerns the arts and things of a visual nature. Otherwise, my processing feels very slow.

My inability to hook quickly into the language of math was one of the most frustrating things for me as a child, and probably for other math-phobics. As an adult, I recognize that certain words used in math are clear as a bell to those using them. Math teachers used to use certain words quickly and then move on leaving me trying to figure out what they meant exactly. I never had the confidence to just come out and ask what certain words really meant. I always keep an eye out for both children being confused by the way math is written about in texts or spoken of in the classroom.

-- SusanS - 15 Jul 2005

Maybe that's what the math people feel towards math. I only have that kind of relaxed confidence if it concerns the arts and things of a visual nature. Otherwise, my processing feels very slow.

Ed and I were talking about this today.

I'm wondering whether 'math-heads' ever experience the same confusions I do.

I know that people who are good at math experience confusion, and must study, practice, work hard, etc.

But I wonder if they experience the particular confusions a 'non-math-head' does?

Ed's crash-and-burn experience with math happened at Princeton, when he took a calculus course for math majors.

He had been good at math in high school, and the Princeton counselor thought he should take calculus for engineers, not 'calculus for poets.'

Ed was utterly lost.

He went to the professor's office all the time, and the professor would explain and explain and explain; this guy was committed to teaching calculus to his students, whatever the level of their understanding.

But he couldn't do it.

NOTHING he said made sense to Ed--nothing.

Ed said the professor simply could not see where Ed was, or why Ed couldn't understand what he was saying.

He wasn't just repeating himself, either.

He was following the 'try something else' rule, and he STILL couldn't explain anything to Ed.

-- CatherineJohnson - 15 Jul 2005

Ed was the only one in the class having that problem (that he knew of).

Everyone else was a math major, and they were learning just fine from the guy.

In other words, this just doesn't seem to have been a 'bad teacher' problem.

It seemed like a DEEP DARK BRAIN DIFFERENCE problem.

-- CatherineJohnson - 15 Jul 2005

I have to say that I immediately fall in love with anyone who has even a passing familiarity with Ferdinand de Saussure (duh so SYOOR--for anyone like me who skips over things he can't pronounce).

Ferdi's--can I call you Ferdi?--writings provide many of the ideological bases behind MathandText^?, specifically the discussions regarding paradigmatic and syntagmatic constructions.

Anyhoo, I wanted to say that the best tactic that worked for me as a tutor was to introduce the ridiculous equation 9 = 9.

Most of my tutees would proclaim immediately that that was not an equation, so the conversation that ensued was productive in and of itself.

But for those who didn't, I would just ask if that was true. I would get a chuckle first most times; then I would say something about the reflexive property, and they would stop smiling.

"Yes," they would say.

So I subtracted 5 from the right side of the equation (the left side was too obvious). "Is that true?"

"No."

"How do we make it true?"

"Add 5 back."

Ha ha ha! Delightful. I've never in all my time heard that ol' gag before!

"What must we do to the other side to make it true?"

"Subtract 5."

"Good."

Then basically I started turning 9 into 4 + 5, then 2 + 2 + 5, until he got it.

Not only did he learn about equations, but he also picked up some learning about expressions--specifically, naming quantities in a variety of different ways (construction and deconstruction).

-- JdFisher - 16 Jul 2005

"I think Steve is experiencing a variant of synesthesia."

Gee, this sounds too much like something to do with senescence.

"In the realm of our own attempts to teach math we do need to think about constructivism, about direct instruction, about guided inquiry, about cognitive science, behaviorism and all the rest."

Don't get me wrong about this. I think it is important too, but I like to take a more pragmatic approach. I like to look at the problems kids are having and come up with ways to solve them, perhaps using ideas from cognitive science or behaviorism (or just plain common sense and knowledge of the material). Content and real kids dictate methods. Form follows function.

However, discussions like Willingham's on modalities strike me as an attempt to define or discuss a theory that covers everything. I don't like the idea of using the latest and greatest cognitive or behavioral theory to define how everything is taught (like the Harkness Table or even LapTop Learning). This forces the content, skills, and kids to fit the theory, not the other way around.

-- SteveH - 16 Jul 2005

Susan (and all), Let me clear up one thing. When I ask a student to draw a house and label it, the house is only a box with a roof on it. Nothing fancy or intimidating.

I don't want Math class to become Art class, or journal writing, or Science class. Math time is too precious and there are huge amounts of material we've got to cover and seemingly never enough time to cover it adequately.

Keep it quick and simple. :)

-- CarolynMorgan - 16 Jul 2005

Right. That would be all I could get from either kid. But simple is usually clearer.

"Math time is too precious and there are huge amounts of material we've got to cover and seemingly never enough time to cover it adequately."

Especially with ADHD. Time is critical. Fatigue sets in so much quicker than with other kids. We rarely get past a third to a half of Saxon 6/5 (He will be an 8th grader) at one sitting. Still, Saxon is set up easier for that kind of attention span over other texts I've tried working with.

-- SusanS - 16 Jul 2005

When I am teaching kids about =, I often say "is the same as. For example, 2+3 is the same as 5. 24 is the same as 12+12 is the same as 3X8. This often seems to solidify the concept.

-- LoneRanger - 16 Jul 2005

Susan: good point & very important

I had no idea that there were different levels of mental fatigue until I read Cathy Crimmins' book about her husband's taumatic brain injury (Where Is The Mango Princess? is the title, IIRC).

I was shocked to learn that people with TBI fatigue mentally in about 5 seconds....and this has huge ramifications for their functioning and rehabilitation.

Until I read your comment this moment, it never crossed my mind that of course kids with any kind of brain issues at all are going to fatigue more quickly--they're burning more energy doing the same thing. (WIth mentally retarded children we know this definitely, from brain scans. But my guess is that ADHD kids would probably be burning a lot of energy forcing themselves to focus & control impulses, though I don't know this for sure, and researchers do speak of 'hypofrontality,' which I believe means that we see less energy being consumed on scans....)

Still, this is a critically important point.

A child with a brain 'issue' (Carolyn J's favorite word!) is in all likelihood going to tire more quickly, either because he's burning up more energy trying to focus, or becauase he's got to use huge amounts of energy to filter extraneous things out, or because of whatever it is that we don't understand yet.

REMIND ME TO GET THIS COMMENT PULLED UP FRONT!

(I have GOT to get poor Andrew to the video store....)

Steve H

I hear you.

I'm reading the National Research Council book on the brain and learning, and all through the book you see Really Bad useages of solid empirical findings in the classroom.

I haven't written about this yet, but the motto, in my mind, for my own perception of constructivism is lost in translation.

Over and over and over again you'll see constructivists take a perfectly valid and reasonable idea, or even a very well founded research concept, and completely sc*** it up in the translation.

I'll be posting some of these things.

But do you remember that hideous final page of the 5th grade Trailblazers text? (You can find it easily in the CompareAndContrast topic thread.)

That page, which sounded insane to all of us, is clearly based in serious research into metacognition.

But their application of research on metacognition is awful.

So what I should say is that, for me personally, reading the research (and theories) is incredibly helpful and always has been.

It's helpful not just in terms of figuring out what to do, it also helps me drop prejudices that aren't sound.

Like probably most of us here I was assuming group work was bs, but when Carolyn posted the study showing it's good I went back and read some of the literature on observational learning and completely changed my mind. J D

Now that is interesting.

Unfortunately, I don't know much about de Saussure, except that the meaning-comes-from-difference basically changed my life!

What do you think is happening with 9 = 9?

Hmm.

That seems like a cool way to go about it.

Is 9 = 9 an example of using 'difference'?

It may be, because if a child thinks 9 = 9 isn't really an equation, then you are definitely using opposing or sharply contrasting categories to jolt him into new 'seeing.'

I'm definitely going to use this with Christopher.

By the way, I've spent a lot of time bugging Christopher to do mental subtraction, thanks to Singapore Math.

I had no idea, before reading Parker & Baldridge, that 'compensation' in addition and subtraction are opposites.

I didn't know that you make a mental subtraction problem easier by doing the same thing to both sides, while you make a mental addition problem easier by doing the opposite!

I still don't really get this!

But I love it, and I use it myself, mentally, all the time.

I'm thinking mental subtraction could possibly be a bridge....

-- CatherineJohnson - 16 Jul 2005

Another display of fragmented knowledge:

I didn't really know that '5 + 3' is another expression for '6 + 2' or '8' etc.

All the various curricula these days seem to talk about 'naming' & renaming numbers: '5 + 3' is another name for '8' and so on.

Speaking only for myself, I found that incredibly helpful, liberating, and....'deepening'???

In other words, I felt like I knew A LOT more about maths after encountering the 'renaming' idea.

-- CatherineJohnson - 16 Jul 2005

What a great idea, JD. I love that 9 = 9 idea. I too plan to use it in my teaching. Thanks for sharing it.

Wow, what a great thread!

-- CarolynMorgan - 17 Jul 2005

Fatigue has definitely impacted my son's learning on every level and I imagine will seriously impact his abilities in the world as an adult.

When he was young it just seemed like immaturity. His inability to focus past 5 minutes just seemed to me to be connected to that and nothing more. As he has aged he doesn't seem to have much more focus then he did then. There is something to that, but there is clearly a physical component.

From the standpoint of just practice and repetition, unless someone is hovering, he simply doesn't get in nearly the practice of a regular kid. If the teacher walks away, he usually shuts down by fiddling with something else or just freezing and looking into space. If the teacher can get something going that day he has at max 15-20 minutes.

Medication has certainly increased his time hanging in there, but it is no panacea, just a tool that helps some days. I imagine some sort of brain scan would enlighten his father and me, but we haven't gotten around to that. I had a bad experience with one pediatric neurologist in the area and never followed through with some testing that he ordered up. From what I've learned since then, they all seem to have a lousy bed side manner compared to others in pediatrics.

I caught part of a Dr. Phil the other day (doing laundry. I do lead an exciting life.) and it was on ADHD. It was interesting that the expert was pointing to hyperactivity as coming from the child trying to wake the brain up to keep it from shutting down. I had never heard that. So is the brain constantly wanting to shut down and the child has to wake it up, or is it racing around until it has to shut down from sheer exhaustion? Or is it both?

Physically, develomental issues such as hypertonic joints can cause also cause fatigue and help lead to a shutdown even quicker. It is expected that by 6 most kids can sit at a desk for a few hours. When you have hypertonic joints you're just not able to do that. It can look like ADHD because the child has the need to fidget and move since his body can't hold up like the others.

-- SusanS - 22 Jul 2005

WebLogForm
Title:	reading Willingham on learning modalities
TopicType:	WebLog
SubjectArea:	CognitiveScience, ParentsTeachingKids, SaxonMath, SingaporeMath, TipsAndTricks
LogDate:	200507150024

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