kitchen table math, the sequel: worse before it gets better

Saturday, March 19, 2011

worse before it gets better

I went with a friend to the Celebration of Teaching and Learning conference in New York City.

It was depressing.

Vendors everywhere, technology, no books, Smartboards (it's the 20th anniversary of the invention of the Smartboard!), and, during plenary sessions, constant calls for Parent Responsibility, each one met with thunderous applause. Parents were not a popular group amongst the Celebrants.

During the session on bullying, three teachers asked plaintively, "Why is bullying our responsibility?" "Why is everything on us?" They were aggrieved.

The great and the good (Brian Williams, Cory Booker) thought teachers had a lot to be aggrieved about. Democracy is hanging by a thread, they told us: the only reason we have a country at all is teachers. And yet Americans fail to feel "reverence" toward teachers. What is to be done?

Mehmet Oz said pretty much the same thing; then he showed us a graph charting the rise of obesity in America and said rising obesity is the reason "there's no money for education." We need to lose weight! Because we need more money for education!

Also, the NEA wants the government to pay for college and graduate degrees for teachers. We'll need to lose a whole lot of weight for that.

My friend attended a session where there was a group of young administrators seated in the middle of the room. The teachers booed the administrators. Now that's interesting ---- what was going on? I wish I'd been there.

A fellow from the Department of Education told us that DOE is rolling out "an ambitious 5-year initiative": the moon shot of this generation. Which was.....a website. The moon shot of this generation is a Department of Education website.

We watched a lot of student videos, all created with a product called Adobe-something-or-other: raps about Haiti; a geography class in California making soup. In the soup video, a pretty girl who came to America from Nicaragua complained that nobody knows where Nicaragua is or that a person who speaks Spanish and has brown eyes might be from Nicaragua and not Mexico. Another student in the video said somebody thought "Guatemala" was guacamole.

Maybe the reason students don't know where Nicaragua is or that Guatemala is a country not a dip is that they're making soup in geography class.

A high-energy Brit pitched his Teacher Channel, I think it was called: there will be authentic content!! We watched an authentic video of a grade school class in Florida where the kids scotch-taped together little houses and stuck them in a line on a stage. Then the teacher walked along the stage blowing the houses with a leaf-blower to simulate a hurricane. Some of the houses blew apart and some didn't. Shots of fist-pumping little kids; fade-out.

The Brit told us we had just witnessed "learning" and said there would be many thousands such videos available on Teacher Channel, which was being sponsored or hosted or public-private partnered or some such with WNET, the host of Celebration of Teaching and Learning. Applause!

In the session on how to teach counting using a children's book, the Math for America Master Teacher banned the words "permutation" and "order" because "permutation" and "order" are words, not understanding. He told us, repeatedly, that he makes his high school students spend a full test hour drawing the answers to counting problems in order to show them that multiplying 5x4x3 is more efficient than drawing 60 houses with 1 of 3 pigs inside. At the end of the sessions, he advocated the use of children's books for teaching high school counting problems. "How many handshakes amongst the 7 dwarfs?" That was a good counting question we could base on a children's story, he said.

At one point a teacher said she'd made a counting tree, and the Master Teacher said, a look of mock incomprehension on his face, "Tree? What is a tree? Why do you talk about trees?"

Five minutes later he put up a Powerpoint picture of a counting tree -- an actual tree, with a trunk going down to the ground, and branches pointing up to the sky. I don't know why a real tree is good and an abstract tree is bad. He didn't say. The rule seemed to be that everything the teachers said was old-school and wrong, while everything the Master Teacher said was up-to-date and correct. 

The Master Teacher had no blackboard, whiteboard, or Smartboard, so you had to try to remember everything he had just finished saying while trying to follow whatever he was saying now, and his Powerpoint drawings were confusing, at least to me. He spoke too fast. He told us over and over again that we needed to hold with our students the kind of conversation he was holding with us: i.e., a conversation for understanding.

I don't recommend it. The "conversation" consisted mostly of our Master Teacher eliciting wrong answers and forbidden vocabulary from his class. There were probably 5 people of 30 who could work the problems, so he focused on them and didn't bother with the rest of us.

I'm actually thinking about writing James Simons a letter.

from the Conference Program
Description: If three pigs live in five houses and each pig lives alone, how many living arrangements are possible? Participants will learn how a children’s book illustrates a simple way to solve counting problems like this without listing all possibilities. Teachers at all levels, from elementary to high school, will learn how students can find the answers without using confusing words like “permutation.”
the Celebration for Teaching and Learning on Twitter


Catherine Johnson said...

It was a very depressing day.

The Math for America session was the ultimate Larry Cuban experience: all the fluff and nonsense of reform math combined with the absolute worst of "teacher-centered" teaching.

He was teaching teachers how to teach, and he had no awareness of any kind that teaching involves finding out whether the students are actually learning.

Or, excuse me, understanding.

He simply assumed that we all understood because he had used the techniques (a "conversation") that produce understanding.

Over and over again, he ordered the teachers to have "conversations" just like the "conversation" he was having with us.

He handed out a list of children's story books that have numbers in them at the end.

Catherine Johnson said...

If the secret to teaching combinatrics was to ban the words "permutation" and "order" somebody would have figured that out a long time ago.

Catherine Johnson said...

I wasn't keen on Cory Booker, either, although my friend was.

His only good moments: he refused to bash Chris Christie and he essentially said educational technology is cr**.

Of course, when Brian Williams said, "So you're saying educational technology is overrated," he answered, "No I'm not saying that."

His solution to low student achievement: extended school year.

That, apparently, is his entire plan.

More "time on task."

This after he talked about schools in Newark where student achievement is much higher than schools just a couple of blocks over.

Same time on task; different achievement.

But his plan is more time.

Catherine Johnson said...

His plan is more time because students lose so much learning over the summer.

The fact that students in Direct Instruction don't lose learning over the summer hasn't been brought to his attention, apparently.

Catherine Johnson said...

What makes me so pessimistic after today about the possibility of improvement was the totality of the's close to being a sealed world.

It was clear that no one in the room thought a parent would be present; no one in the room ever stopped to ask himself how a person whose primary identity was "parent" would react to the constant calls for Parent Responsibility and the aggrieved tone of so many of the comments.

And there were SO few books.

They did have a bookstore, but it was smaller and had fewer titles than the bookstores we used to have at autism conferences.

Catherine Johnson said...

I could also see, today, what is going on with Diane Ravitch.

At the mere mention of her name, during the final plenary session with Cory Booker & Brian Williams, there was a sustained and loud cheer.

She is as beloved as Mehmet Oz; she is a celebrity.

Catherine Johnson said...

The technology question to Booker was about the digital divide. Brian Williams painted a picture of "wealthy" suburbs where teachers have to ask students to remove ear buds - and perhaps power down their iPads etc....Booker was appropriately unimpressed. He opened with the image of computers gathering dust in the backs of classrooms and said students need to learn to read, write, and do arithmetic.

Then pointed out that Newark students are all texting each other in bad grammar.

Catherine Johnson said...

Booker probably would have been fine if he'd been talking to a group of parents.

Or to a group of precision teachers (though in that case I hope he'd be doing the listening).

At the "Celebration" every point he made was softened and essentially negated by little sermonettes delivered to straw men of his construction.

And platitudes.

Check out the Twitter feed...

Grace said...

A college professor friend tells me that her students have difficulty with the definitions of words like "increment" or "default" or "manipulate". Now I wonder if high school teachers think those words are too "confusing" for students to learn.

SteveH said...

"What makes me so pessimistic after today about the possibility of improvement was the totality of the's close to being a sealed world."

Was there anything about G/T or AP classes? How about STEM. That's a big code word these days. (I abhor the term because it's used to form generic, high-level arguments about an area that's all about details.)

Was there any talk about absolute or world standards? Was there talk about differentiated instruction? You would think that many teachers would complain about having to get fifth graders, who don't know the times table, ready to take a state test. Perhaps they just blame the test. Were there discussions about testing?

Was there any indication that these people saw parents as customers? Do they acknowledge that parents might have different ideas about education and levels of expectations? Was anyone promoting Core Knowledge there? Was there any talk about charter schools?

"He simply assumed that we all understood because he had used the techniques (a "conversation") that produce understanding."

This is the "Trust the spiral" mentality. It's designed to work by definition. If kids don't learn, then they just are not ready yet for the material. It's the greatest philosophical cop-out. Throw in authentic learning and you can justify anything.

There is a great divide here and I think it's all about academic and philosophical turf, ... and money. The worst part is that kids and parents are held captive. It's one thing for affluent parents to have choice, but not those who might take money away from them. Even if they know that there are big problems, they still don't want to see the money go away. I've had teachers tell me that public education has to teach "all kids", but private school kids are all "pre-selected". If you are affluent, you can get a better education, but if you are poor, they won't let your money go.

SteveH said...

Here are the STEM panels and talks:

First Person Solvers: Mathematics Education in the Video Game Era

From Poop to Plastic

1001 Inventions – Uncovering a 1000 Years of Science and Technology

Common Core State Standards: Math

Materials Science: Maker Faire, the Maker Movement and NOVA’s "Making Stuff"

Get Out! Get Real! Authentic Problem Solving and Adventure for Students and Teachers Outside of Class

Bringing Learning to Life: Engaging and Motivating Students with Service-Learning

PMI's "Smart" Approach to the End of Course Algebra Exam

Utilizing Math Software on a SMART Board

Enhance Your Science Teaching With Video!

Dinosaur Train – Get into Nature with Young Children

Nanotechnology - Tiny Things Bring Big Changes

How Should We Elect Our Favorite Ice Cream Sundae?

A Pedagogical Foundation for Teaching ELLs Mathematics

Geocaching in the K-12 Classroom

Intel Teach - A Pathway to 21st Century Skills and Meeting the Common Core Standards

Assessment and Data Analysis Using Student Response Systems

Sid the Science Kid - Science Resources for Young Children

Cloud Computing with the.News: Interactive & Engaging Current Events

Ping-Pong Balls and Lipstick: Teaching Problem Solving Using Complex Estimation

Progressive Math Initiative's Smart Approach to Elementary Math

Turning the Big Apple Green

Simple Science in the Classroom: An Interdisciplinary Approach Using Objects from Nature

Using Technology in Assessment: Implications and Potential Pitfalls

Challenging Kids to Engage in Science

Engaging Students in Real-World Issues with National Geographic

PSI and PMI: Using SMART Technology to Teach Science and Mathematics

Free Resources to Empower Classroom Transformation and Address the Common Core Standards

Nanotechnology in Your Classroom

Teaching the Curriculum Through the Lens of Sustainability

The Twenty Coolest Things You Can Do from a SMART Board

Enhance Your Mathematics Teaching with Media Resources from WNET!

Extra! Extra! Read All About the Universe!

The Skills and Tools for Virtual Education

Landscapes for Learning Science

Using Children's Literature to Count Complex Tasks Without Counting

Google(ing) in the Classroom

Exploring the Wealth of PBS Education Resources

Enhancing Financial Literacy with Video

Dinosaur Train – Get into Nature with Young Children


They are mostly about engagement (it's the student's fault and/or responsibility) or about selling something.

How about discussions like:

"If you don't get math right in K-6, you can kiss a STEM career goodbye."

"Colleges don't care about Project Lead the Way if your math grades stink"

"The top 25 STEM careers and the level of math and science you need to get in high school"

"What are the SAT scores you need for the top 25 STEM careers"

"Excelling in STEM with paper and pencil"

"Engagement in math is like eating mathematical Twinkies"

Only the Common Core State Standards talk seemed to deal with the real issue, but these standards still allow for curricula like Everyday Math.

Hard work is not sexy and doesn't sell. What sells is engagement without drill and kill. They will keep spending lots of money to find it and many people will be glad to tell them what they want to hear. At the same time, parents will be lectured about critical thinking and understanding.

ChemProf said...

"These are mostly about engagement"

and "docent science," which is what my colleagues and I call it. Science appreciation is another good term. I guess I can see that they figure that most students won't actually go on and do STEM in college, but they are really limiting their students' options. Plus, I can't tell you how many students come to college thinking that there are careers in science appreciation.

Anonymous said...

"The Math for America Master Teacher banned the words 'permutation' and 'order' because 'permutation' and 'order' are words, not understanding;"

This is extra interesting because the Russian math sequence introduces mathematical notation quite early. Words I had never seen before, like "addend" and "subtrahend." The 3rd grade book is very good about the difference between a "line" and a "line segment."

I wonder what the reasoning is behind avoiding precise terminology in the quest for understanding of math?

-Mark Roulo

lgm said...

It's not just math, it's the rest of education also. 'Syllable' is gone from the elementary LA material for ex. The reason seems to be that the words are a barrier for some students in the fully included classroom....more general words are put in their place.

Addend and subtrahend were in the New Math I had in the 70s in elementary school. Permutation is still here in NY, but the associative, commutative, and distributive properties have other names in elementary school.

Lisa said...

Dinosaur Train?! Does that mean my three year old is on track for a STEM career?

Anonymous said...


These are great titles. The content fits a MSMI talk I gave and another I'm setting up. Can I steal your titles for my slides for the future ones?

Anonymous said...

"I wonder what the reasoning is behind avoiding precise terminology in the quest for understanding of math?"

It seems obvious that it is useful to have words to describe precisely what you are doing when you are learning something new, right? If you are learning about permutations you want a word for it.

But this reminds me of my days as a TA teaching calculus to unwilling freshmen. I noticed a stiff resistance to using the terminology -- especially from the weaker students. Everything was "the answer". They would not "calculate a derivative". The would "find the answer". The would not solve for a parameter, work out an integral, or any such thing: "find the answer". I think that they did not want to use the language as psychic rebellion against the possibility of actually learning the material. Each "find the answer" was a supplication to their deity along the lines of "please get me through this ordeal and I will pour many libations of mocha javas on your altar. Or whatever. And on to the next semester!"

Taking appropriate terminology out of the curriculum is a sop to this attitude.

SteveH said...

"These are great titles. The content fits a MSMI talk I gave and another I'm setting up. Can I steal your titles for my slides for the future ones?"

Yes, please! I'm always amazed how many educators don't see the critical importance of K-6 math.

Catherine Johnson said...

Was there any indication that these people saw parents as customers?

That was one of the depressing things. Parents were mentioned **only** in the context of calls for greater "Parent Responsibility." Huge applause for every Parent Responsibility line.

Powerful feeling in every audience that the problem is parents.

If parents took responsibility, the kids would be learning.

Along with that, there was lots of pandering to feelings of aggrievement. The famous people who spoke all said the public doesn't appreciate teachers - and that policy makers and pundits think they know something about education when only teachers know anything about education & policy makers & pundits should just butt out.

Interestingly, there **wasn't** a lot of talk about education not having enough money, apart from Mehmet Oz, who told teachers the reason "there isn't any money for education" was that rising obesity was sucking up all the dollars.

As I recall, at least two of the famous speakers - Jon Meacham and, I think, Brian Williams - pointed out that we spend a lot more on education than other countries.

What seems to be happening is that instead of famous speakers saying teachers are underpaid, they say that teachers are under-loved -- which may be worse.

Catherine Johnson said...

Do they acknowledge that parents might have different ideas about education and levels of expectations? Was anyone promoting Core Knowledge there? Was there any talk about charter schools?

No charters; no Core Knowledge.

Kumon and CollegeBoard had booths, though.

The CollegeBoard gals gave us copies of Springboard, the new pre-AP curriculum --- but they don't look great so far.

Catherine Johnson said...

It seems obvious that it is useful to have words to describe precisely what you are doing when you are learning something new, right? If you are learning about permutations you want a word for it.

No kidding.

What am I supposed to say --- "the 3 little pigs problem"?

Of course, as a writer I was appalled.

Catherine Johnson said...

I came to the session late, so maybe I missed something ... but for a time after I got there Master Teacher kept asking whether anyone got an answer of 243.

I have no idea where he came up with that or why he thought we would have come up with it.

The correct answer is 60.

Which I would have gotten if I hadn't overcounted. (I figured 60 x 3)

I got the next 1 or 2 questions right, then fell out of the "conversation."

Michael Weiss said...

I wonder if you have had a chance to take a look at the list of workshops at next month's NCTM Annual Meeting and Research Presession. In my experience, the Annual Meeting presentations are often very thin on substance -- lots of "How to use a Smartboard" and "Use children's literature to generate story problems" stuff -- but the Research Presession has considerably more substance to them. Both programs are online and searchable.

Anonymous said...

243 = 3 × 3 × 3 × 3 × 3

I still don't know how to get this as an (wrong) answer for the problem in question.

-Mark Roulo

allison said...

Catherine, is that really how the problem is phrased exactly?

Not "the three pigs live in *any of* 5 houses; each pig lives alone and in exactly one house."

Because the question as written makes no sense.

If three pigs live in five houses and each pig lives alone, how many living arrangements are possible?

How does a pig live in five houses? All of them live in five? Oh, but they live alone. Does each pig live in a house, but some live in more than one? How does a person live in more than one house at a time? What?

So, if you couldn't read or make sense of this, you could decide the first house could have pig A B or C, the second house could have pig A B or C, the third...the fifth could have A B or C. 3 choices 5 times.

but this question isn't even properly phrased to get the answer of 60.

Anonymous said...

Ah-ha! What Allison says!

243 is what you get when you do permutations with replacement.

Any one of three pigs can live in the first house. Then any one of three pigs can live in the second house. Then ...

= 243 permutations.

If we replace pigs with three differently color jellybeans and draw from a bin, replacing the jellybean chosen each time ...

-Mark Roulo

Ethan said...

In case you didn't know, Jim Simons is one of the richest people in America -- he made billions running a hedge fund in which he used high-level math (machine learning, I'm guessing) to exploit patterns that no one else saw. Get this: I saw a video of him lecturing at MIT, and the MIT math prof who introduced him said that Simons' hedge fund was "the best math/physics department in the world."

Michael Weiss said...

As it happens, I am currently tutoring an 11th grader in Algebra 2, and they are taking their test on the combinatorics chapter this week. So I gave him the pigs / houses question. His first words: "Would this be a permutation or a combination?"

My opinion? That's not the right question he should be asking. He should be asking, "How many choices does the first pig have? Then, how many choices does the second pig have? Then, how many choices does the third pig have? Okay, so the answer is 5*4*3 = 60." Whether the right word for this is "permutation" or "combination" is completely irrelevant, and may even be a distraction.

Maybe that's why the "Master Teacher" thinks terminology is a bad thing? Because some students get distracted by the words, to the detriment of the basic counting principles?

None of which, of course, supports the use of badly-worded, ambiguous problems, or of silly "draw the houses" approaches.

allison said...

Simons didn't do machine learning. He did math. He was a differential geometer and his particular stuff was basically related to string theory. His people do all sorts of things, like statistical mechanics/MCMC, represenation theory, fluid dynamics, string theory.

Simons is brilliant, and he's an expert in his field. Experts tend to think that since they are so smart at one thing, they are so smart at other things. And they often fail to remember what it's like to not understand. Being brilliant and rich doesn't mean he's not wrong about what is actually happening in math ed.

allison said...

This is another expert-novice problem, I suspect.

I didn't even LEARN the word "combination" in my undergrad program, which had 1 discrete math course, 1 probability course by a combinatorist, 1 combinatorics course, and several stat mech courses where we did counting.

I didn't see that word in my discrete-math-for-CS course, where I was a TA in any of the several terms I TA'd it.

But how can you not use the word order? or permutation?

We had defined "n choose k" which meant "permute n things, and then divide out the permutation of k things and n-k things." We had defined things like "boxes and bins problem", where boxes were labelled and order in bins was unspecified, "flagpole problems" where poles(boxes) were labelled and order on them did matter, etc. Permutation was a primitive that had to be mastered, period. Recognizing when order matters and when it doesn't is another skill that must be developed.

So I tentatively "agree" with the teacher that the words can lead back over and over to a non-grokking student desperately trying to find the right formula instead of thinking about the problem. But you aren't going to get them to think about the problem if you can't even phrase it well.

My guess is these teachers listening weren't expert enough to understand how to define the problem properly let alone answer it without some word clarity leading them to know which model to use either.

allison said...

In a nutshell, an expert recognizes that a novice "bitterly clings" so to speak to the terminology in order to desperately grab at whatever formula they were taught. No actual thinking, just hear a word, turn off brain, and grab the formula. Expert then says "See, reliance on terminology bad."

But an expert then thinks that drawing houses will lead to mastery because when they draw the picture, the aha happens for them.

But drawing houses doesn't lead to mastery or that aha unless you ALREADY KNOW what you're trying know. You've got to start with better models, better thinking. Yes, drawing some pictures will help. But you need some primitives.

ChemProf said...

"No actual thinking, just hear a word, turn off brain, and grab the formula."

Sure, and that's a common novice strategy. But Allison is right -- teaching everyone to draw a picture first doesn't solve the problem, it just teaches them a new novice strategy that they can use whether it helps or not.

I call the "grab the formula" approach treasure hunting, because they'll hunt through their notes for anything with the right variables. But in thermodynamics, which is on my mind because I'm teaching it now, I can give you half a dozen formulas that involve the same variables. THIS one applies to a gas, THAT one applies to any phase change, and this OTHER one applies to a phase change if one of the phases is a gas, but not otherwise. If I want students to figure it out, I need to explain when the equations apply, give them practice in choosing the equation with support, and then remove the support until they can really choose the right solution on their own.

I had a college physics teacher who insisted we needed to draw the picture to figure out the problem, but I found it useless. I eventually figured out that if I solved the problem, I could use my solution to draw the picture, so I'd leave a space and do just that!

Michael Weiss said...

Maybe some additional detail on my current tutee's strategy is in order:

When faced with any combinatorics problem, the first question he asks himself is "Is this a permutation problem or a combinations problem?" He looks for various cues (keywords suggesting order) to identify which it is. Then, having decided, he pulls out his TI-83, navigates to the Math > Prob menu, and selects either "n P k" or "n C k".

In other words, he is (or was, before I started working with him) both completely reliant on the calculator, and completely lacking in any conceptual understanding. And his "find the right word" strategy just reinforces that dependence.

So the first thing I told him? Stop worrying about words, and think about the situation. Imagine making a series of choices to build one possible configuration. How many choices do you have? Now how many choices do you have? When you have finished one configuration, ask yourself: If you scrambled these up, would that count as the same configuration, or a different one? Think about the situation.

I think drawing pictures of little houses with little piggies in them is silly, and completely unwieldy in most problems. But I also think getting hung up on words can be a huge distraction, and banning those words temporarily can help focus students on what actually matters.

I would argue that the expert/novice dichotomy actually runs a different way than allison suggested above. For the expert, a word like "permutation" contains within it a highly compiled set of ideas. It's useful for an expert to use words like that because they know what they mean and when they are relevant. For a novice, the word "permutation" just means "cue up the appropriate command on the TI-83". In my opinion that's a solid reason not to use it.

Once again, I want to be clear think the "Master Teacher" in this episode sounds like a twit. The idea that children's books are somehow more authentic or engaging than... well, than anything is just silly, and the idea that a pretty picture of a real tree is somehow more useful than an abstracted counting tree is, I don't know, crazy? But even a broken clock is right twice a day, and I think there is some truth to the idea that avoiding technical vocabulary temporarily can help students focus on what they are doing, rather than what it is called.

Eventually, though, you need to step back, and say: Look at these six problems that we just did. See what they all have in common? Let's give that a name.

The $64 question is: When do you do that? Many teachers introduce terminology before students know what the words mean: "Today we're going to learn about permutations and combinations. The difference between them is that in permutations, the order matters..." An introduction like that makes almost no sense because the student has no ideas on which to hang the words.

Build the ideas first with a few examples, then generalize to an abstraction, then name the abstraction. What's wrong with that sequence?

allison said...

Michael and Chemprof, I think we're in almost complete agreement.

First, Michael said:
Build the ideas first with a few examples, then generalize to an abstraction, then name the abstraction. What's wrong with that sequence?

This is the only sequence that works. Period. It's the nugget that works in a loop that you have to keep repeating over and over and over. A novice has to be taught something to hang the concept on, HAS to. But all but the most brilliant won't get it after one time or three times. So you keep having to do it, and then keep having to slightly modify the examples to cover more and more cases as you flesh out the abstractions, etc.

I didn't mean to imply that you need to "define" the "word" permutation first. You need to build up to the concept.

But to steal the word away from the concept is to undermine teaching the precision of mathematics. The student must be allowed to reach the point where they have a Precise Definition that they can refer back to, and learn to use deductively. Otherwise it's hogwash, not math.

A student MUST learn that "permutation" means not just some way of putting some things in order, but a specific ordering of n labelled items without duplication. At some point, they must learn that the identity permutation is valid. They must learn something's not a permutation if it only included a subset of the items. And then must understand that it didn't matter whether they started with the first item or the last item, whether they did it recursively or sequentially, that the operation still works.

That's DIFFICULT work no doubt. It's not going to come in one lesson, nor should it be attempted in one lesson. But the goal must be precision.

wrt permutation, the concept is INCREDIBLY sophisticated, and it rests on mostly unstated concepts of "labelling" and "ordering". Doing away with the words "labelling" and "ordering" doesn't help teach the concept. Building up as you said, starting from examples, working up to the definition, is how you get there.

I believe Master Teacher was making an important point: too often, children do what ChemProf calls the treasure hunt, hearing the word and using that to reach for the calculator or formula rather than thinking. Far too many students are in "gimme the answer" mode rather than "help me learn to think" mode, whether that's due to fear, uncertainty, laziness, despair or confusion.

And yes, moving them away from the word and toward thinking about a problem is right.

But math doesn't lend itself to the conversational. And here is where I think experts go so wrong: they have so much background knowledge as you point out that they've forgotten how DIFFICULT it is to build up intuition. They think they can skip the formula and build conceptual understanding by talking or drawing pictures.

ChemProf said...

Oh Lord, the overly powerful calculators. I have banned programmable calculators in my class (as a cheating device) and it is appalling how many students can't cope with a normal scientific calculator. Why? Because you can't just type in the whole thing and then hit "evaluate". In other words, they can do calculus, but they don't really know their algebra, and can fall apart very quickly.

As you say, Michael, he feels like he can "do" the problem, but he doesn't really understand what he's doing, and if you throw a tiny twist at him, he can't do anything with the problem at all.

allison said...

But as ChemProf pointed out, a novice won't know how to draw the right picture or have a relevant conversation. What's the right picture here? 3 pigs with 5 slots next to them? 5 houses with 3 slots next to them? 3 pigs, 5 houses, and bunch of lines? How do you indicate occupancy? How do you enumerate your choices--are you choosing houses or pigs? A novice won't know.

And this is where Master Teacher went so wrong: he managed to fail to convey to Catherine WHAT leads you to draw the right picture, to use the right model.

Experts often underestimate how hard it is for a novice to do this part. And jumping into word problems isn't the way to do it, having a conversation about a carefully controlled artificial problem might be a good start.

The way you do that is how ChemProf said, in the order you said: you start with examples that are inauthentic, you generalize up to an abstraction, you practice applying the abstraction with scaffolding from the teacher at first, and then slowly remove that scaffolding, repeat with more sophisticated inauthentic examples, etc.

Diving into a word problem breaks that habit and leaves them drowning. For one, it requires far too much working memory. There is NO WAY that anyone learns anything by drawing the 60 houses, because by the time they've done that, they've forgotten all connection to what problem was the issue, what it's got to do with choosing ordering, what things were being rearranged, what things were labelled, etc.

So you've got to start with something concrete-here are three labelled items. how can you rearrange them in a line? here are 4, here are n. Here, these items ARE NOT labelled. How does that change your answer? Now, put 3 of those items into 3 bins. Bins are labelled, balls are labelled, but no order in the bins. how does that work? what if you have 5 bins? what if you have 2 bins? What if the bins are not labelled--how does that change things

You can't do this in one class period. You can't do it in 3. But you won't be able to do it at all if you don't try to arrive at precise definitions that the students can use over and over again to build up mastery.

I wonder, as an aside, if the whole idea of telling the teachers to use ideas from literature was just to try and get them away from relying on the textbook. Maybe it was designed to get THEM to think about the problem. Because I bet most of them can't do these problems without relying on the formula either. Given the horrendous phrasing the Master Teacher appears to have used, however, it's dangerous to have teacheeyrs make up combinatorics problems. They simply don't have the precision themselves to do it well--and it is difficult.

allison said...

As a tutor, the question remains whether you can even get your student to try and think. Without the calculator, most student simply shut down. They have no clue what they are trying to do, and have so many years of not comprehending that it's nearly impossible to back them up to where they want to think about what doesn't make sense to them.

Michael Weiss said...

As a tutor, the question remains whether you can even get your student to try and think.

Oh, as a tutor I'm successful at this all the time. The one-on-one interaction is extremely powerful. I think the real question is how to scale this up to the size of a classroom.

(Actually, for me, the "real" real question is how to teach preservice teachers to get their students to think. Because that's what I try to do for a living.)

allison said...

You do? What content do you try to get them to teach? Send 'em to MSMI!

You can't learn to think in a vacuum. You have to think about something. And your teachers need to know the stuff they want the kids to think about.