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10 Mar 2006 - 14:04

another reason to teach your children bar models

Ms. K rarely assigns word problems.

What problems she has assigned have been, frequently, far above the kids' skill level. (examples here: scroll down for the entire list)

The parents do these problems at home, then the kids turn them in. This is an open secret. My favorite Extended Response moment happened last year, before Christopher had moved to Phase 4. One weekend parents all over the soccer grounds were grabbing each other & asking whether anyone knew how to do the latest Extended Response. These were all highly educated Westchester parents with important jobs requiring advanced training.

And they're running around the soccer games accosting people about the latest Challenge Problem their kids have to hand in.

This year there's one student in one of the Phase 4 classes who, last semester, was getting 60s & 70s on his tests and had straight '10s' — the highest score possible — on the Extended Response problems.

NOTE to IMS Math Department:

Look for a pattern!

Cs and Ds on tests, A+ on Extended Response problems — what does this suggest?

The Extended Response problems are assigned because, in the beginning, the Phase 4 students were supposed to be mathematically gifted, and Irvington's pedagogical philosophy where the mathematically gifted are concerned can be summed up in two words: Math Olympiads.

The MATH OLYMPIADS approach to educating the gifted and talented, as far as I can determine, is the following:

• The mathematically gifted child needs, above all, to be challenged.

• The best way to challenge the mathematically gifted child is to give him super-hard problems he's never been taught how to do & send him off to grapple with them on his own.

I have the Challenge Philosophy in writing. The Assistant Superintendent for Curriculum said, in a letter to me, that Phase 4 kids 'need to be challenged' — although he agreed that kids shouldn't be given problems so challenging their parents would have to do them.

He may be right about kids who really are GATE in math, although I can't imagine GATE kids don't need instruction.

And my leaning where GATE kids are concerned is towards acceleration over enrichment.

I may be wrong about GATE kids.

But I'm right about the high achievers.

Kids who do well in math because they're high-achieving don't need Math Olympiad problems.

In fact, I'll go for the Strong Form here:

Kids who do well in math because they're high-achieving are harmed when they spend time on Math Olympiad 'challenge problems' instead of word problems pitched to their level and embodying the concepts they are currently trying to master.

Ms. K assigns Challenge problems, not Instructional problems.

As a result, virtually all of the word problems Christopher has done this year fall under the heading of lost instructional time.

What Christopher needs are brilliant instructional word problems of the kind provided by Action Math.

has Math Olympiads become a national curriculum?

I always saw the Extended Response problems in the accelerated class as an 'add-on.' The mathematically talented kids were taught math like everyone else, only they had to do Extended Response problems, too.

Now I'm wondering whether in fact the 'challenge' approach is simply another manifestation of constructivist math.

Instead of being taught how to do word problems, kids are handed a problem and told to figure it out on their own.

Here's what I see in Christopher's class:

number one: The kids have been given virtually no 'normal' word problems — normal meaning do the odd problems for homework-type problems — all year.

number two: They've been given 9 Extended Response problems, only 1 or 2 of which they could plausibly solve on their own.

number three: To my knowledge, they've been given little-or-no direct instruction in the kinds of word problems that will appear on the state test.

number four: This week, when Ms. K. finally did assign a page of word problems for homework, she gave them no instruction whatsoever on how to do them.

number five: Having read all of the sample problems for the state test, I would be stunned if any problems like the ones Ms. K. assigned this week will appear on the state test next week.

number five: My guess is she didn't demonstrate how to do the problems in class the next day, either, unless the students asked her to. (I'll ask Christopher.) I don't see how she could have. There were 5 problems in all, each requiring a different approach the kids have not been taught, and they spent at least 10 minutes writing in their math journals. That doesn't leave a lot of time for demonstrating and explaining five different word problems in one class period.

update: As I suspected, Ms. K went over 'the problems kids had trouble with,' which means that it was up to the kids to a) know they needed help and b) say so in front of a class filled with peers who, at lunchtime, are going to be calling them 'fat,' 'gay,' and/or 'stupid.' The only problem Christopher remembers her going over in class was 'the runner problem.' (This is two days ago, we're talking.) He has no memory, none whatsoever, of what she actually said about how to do the runner problem.

I'm sure the runner problem came up because no one in the class could do it, so there was no shame in admitting defeat.

Almost certainly most of the kids solved problems 8, 11, & 12 through guess-and-check, and that was that. It's unlikely that any of the 11-year olds Christopher knows would say, "I got the right answer, but I'm wondering whether there's a more elegant and efficient way to go approach this problem."

I'm not privy to Ms. K's thinking, but I know exactly what the effect of her approach to word problems has been on Christopher (and I know he's not the only one):

a) properties, rules, and procedures are learned by rote

and

b) all word problems, including simple, beginning problems in algebra, become Challenge Problems

I'm guessing that this approach is the result of the constructivist pedagogy Ms. K, who is very young, would have been taught in ed school — whether she's aware of it or not.

She teaches the 'basics' in class, the kids memorize what she's put on the board, then the kids discover how to apply the basics to word problems on their own.

In fact, it's probably worse than that, since Ms. K. told a friend of mine that she teaches the concepts the day after the kids have done homework on those concepts. My friend said Ms. K told her this in a 'DUH!' tone, as if it should just be obvious a teacher wouldn't teach a new concept before assigning homework on the concept.

I'm wondering whether this is an ed school truism at this point.

Do ed schools teach future math teachers to have the students discover everything first, including rules, properties, and procedures, and then "go over it" later after the kids have discovered whatever they're going to discover?

I don't know.

this is where bar models come in

Here is 1 of the 5 problems Christopher's class was assigned for Wednesday night:

None of the kids has been given any instruction whatsoever in how to set up such a problem algebraically.

Nor have they been given any instruction in the Official Prentice-Hall Problem Solving Strategies:

Wednesday afternoon I was working on these problems with Christopher and his friend M.

Needless to say, neither boy Looked for a Pattern, Guessed and Tested, Simplified the Problem, Made an Organized List, Worked Backwards, Accounted for All Possibilities, Made a Table, Wrote an Equation, Solved by Graphing, Drew a Diagram, Made a Model, Solved Another Way, or Simulated the Problem.

No.

Instead, both boys, working independently, subtracted 280 from 2870 and then stopped. They knew they weren't done, but they didn't know what to do next. They didn't know why they'd subtracted 280 from 2870, either.

I pointed out to M. that one runner went 280 m further than the other. Unfortunately I can't remember what he did with this information. I do know that he ended up with answers that were 280 m different, but added up to a whopping big batch of meters, far more than the original 2870. Then he started getting upset, and insisting his answer was right.

I decided it was bar model time.

In hindsight, this was the wrong decision where M. was concerned. Both of the kids do know how to translate English words into an equation, and M. might have been able to think the problem through using x to stand for one runner's distance.

He was flat-out unwilling even to look at a bar model. 'I don't understand anything about this problem,' he said, and that was that. He was done. My mistake.

Christopher was game. He knew he was getting nowhere doing what he was doing, and he'd had enough experience with bar models to take it on faith that a bar model would work.

Which reminds me: I must stress to Christopher that the point of the bar model isn't to solve the problem, but to show you which operations you need to do in what sequence. Bar models are a way of setting up the problem.

He didn't know exactly where to start, though he did know he should draw two bars, one for each runner.

When I started talking him through he caught on quickly and he was able to label everything correctly and quickly on his own, without prompting.

Here's what he drew (this is my version):

Seeing the problem laid out this way, Christopher again subtracted the 280 from the 2870. Then he got stuck again, in the same place he'd been stuck before.

We've got work to do.

However, when I started walking him through it, asking what we had left now that we'd subtracted the 280, he was able to say that we had the two other segments left, and he was able to say, with prompting, that these two segments were equal.

The instant he said they were equal he realized he needed to divide the difference by 2.

2870 - 280 = 2590

2590 ÷ 2 = 1295

At that point I asked him where the 1295 belonged on the bar model, and he knew.

Then he knew that Runner 1's distance was 1295 + 280 meters while Runner 2's distance was 1295 meters.

So I'm thinking....

a) is this happening in other school districts? are math teachers taking a discovery approach to word problems?

and —

b) the best defense is a good offense

If I had little ones I'd be teaching them bar models just so they have a way to tackle all the discovery word problems they're not going to be taught how to do in the years to come.

My mom used to always tell us to Drive Defensively.

Same thing here.

Teach Defensively.

extended response problem from IL state test
extended response problem 1
extended response problem 2
extended response problem 6
extended response problems 7, 8, 9
direct instruction & the rigor conundrum
Dan's daughter reacts to extended response problem
defensive teaching of Singapore bar models
open-ended problems in math ed
problems that teach - "Action Math"
email to the principal

-- CatherineJohnson - 10 Mar 2006

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"Kids who do well in math because they're high-achieving in general are harmed when they spend time on Math Olympiad 'challenge problems' instead of word problems pitched to their level and the concepts they are currently trying to master."

Absolutely. They're not going to be high-acheiving for long if they aren't taught what they need to advance to the next level. Throwing Math Olympiad problems at them without teaching them the concepts used to solve them in a logical sequence is pointless and will only discourage them, especially if they are high-acheiving and not necessarily gifted. A high-acheiving kid can take all the pieces of information he's given and synthesize them more quickly or more accurately than the average student, but he still needs to be given all the data. A gifted kid may be better equipped to tackle an unknown challenge problem because he has a larger, self-taught data base from which to draw or a greater compulsion to seek out the data he needs.

In my experience, Math Olympiad problems are designed for students who are already accelerated in math. When the problems were recently posted, I immediately recognized that I could solve several of them based on things I learned in jr. high/high school math. Thus, it would make sense to me that the kids who are expected to solve them should be working several grades above in math.

My husband is gifted in math (finished college Calculus III at age 15), and he loved Math Olympiad in grade school. However, it was ALWAYS supplementary to his regular math coursework. Solving only challenge problems would never have given him the foundation he needed to succeed in higher math.

-- AndyJoy - 10 Mar 2006

Let's assume that the gifted student is the most brilliant natural mathematician ever born. Is it reasonable to assume that he will, overnight, intuit 200 years of progress by less-brilliant mathematicians just to solve a homework problem?

How much less reasonable is it to assume that for a kid less brilliant -- even if he's really bright?

-- DougSundseth - 10 Mar 2006

The thing about smart kids is that they'll often have to be taught things only once or twice, not that they never have to be taught anything at all.

-- DougSundseth - 10 Mar 2006

Andy

You know, I knew these things were a 'FWOT,' but until I looked at the Action Math website I didn't realize how terrible this course really is.

He's more than halfway through 6th grade and he's learned virtually NOTHING about word problems or applications.

-- CatherineJohnson - 10 Mar 2006

You know, I'm very interested in gifted kids at this point....I don't know what to make of them.

I'm thinking some of them must 'intuit' concepts the rest of us have to be taught...intuit, or 'see'.....or, alternatively, it may just look that way to the rest of us. (When Bill Honig recanted about whole language, I think one of the things he said was that good readers read so fast no one perceived that in fact they were decoding words phonetically.

Anyway, I don't know what makes a mathematically gifted child tick.

I absolutely know that Christopher isn't gifted in math but is bright and reasonably motivated AND HE NEEDS TO BE TAUGHT.

He's not going to discover algebra.

-- CatherineJohnson - 10 Mar 2006

He's certainly not going to discover modular arithmetic.

-- CatherineJohnson - 10 Mar 2006

now that I've taught myself more math, the Math Olympiad problems (grade school & elementary) 'look fun' to me

but they sure as hell don't look like a curriculum

-- CatherineJohnson - 10 Mar 2006

I guess I see them the same way you do; I see them as problems I can solve now given what I know

I don't see them, remotely, as problems I would have been able to solve without a lot of study & instruction beforehand

-- CatherineJohnson - 10 Mar 2006

Doug

None of these people seems to think about cultural evolution.

Why would we want a brilliant child to spend his time reinventing the wheel?

We want our most brilliant kids to absorb everything their forebears have done and use it to move ahead

-- CatherineJohnson - 10 Mar 2006

Of course, if you truly believe, as I think constructivists do, that the only 'real' learning is the concepts you've discovered for yourself, then this route makes sense.

BUT I don't know why you'd make that assumption about a highly gifted child!

A highly gifted child is going to be highly gifted no matter what you do.

You can directly instruction 24 hours a day; your highly gifted child is going to lap it up and zoom on.

-- CatherineJohnson - 10 Mar 2006

You guys have got to take a look at Action Math.

It's unbelievable.

I'm going to start having Christopher watch (and possibly do) the problems.

He'll be happy to hear that.

-- CatherineJohnson - 10 Mar 2006

The other thing that's just making me NUTS is that Ms. K does not seem to have assigned any word problems of the type they will be tested on next week (except for the problems in the Glencoe test prep book. Thank God for Glencoe; that's all I can say.)

The 5 problems she assigned last night are all 'off-topic.' Not one will be on the test.

Meanwhile I discovered last night, while having Christopher go through the sample test again to see if he can do the problems he couldn't a couple of weeks ago that there are going to be two-step equations on the test.

Two step equations meaning equations like this:

2b + 3 = 11

Christopher has never been taught how to solve a two-step equation.

He can solve one-step equations:

b + 3 = 11

2b = 11

He can't do two-step, and it's not obvious to him how one would start.

It's going to be easy for him to learn BUT SHE HASN'T TAUGHT THIS EVER. SHE HASN'T SHOWN TWO-STEP EQUATIONS IN CLASS AND SHE HASN'T ASSIGNED A SINGLE TWO-STEP EQUATION EVER

But they've got time to write in their math journals and work algebra problems before they know any algebra.

At this point I'm so outraged I'm not even able to communicate with her or with the school, because I can't find a calm, respectful, reasonable tone inside myself.

-- CatherineJohnson - 10 Mar 2006

After school today I'll use the Glencoe Parent and Student Study Guide to teach him two-step equations.

-- CatherineJohnson - 10 Mar 2006

"Absolutely. They're not going to be high-acheiving for long if they aren't taught what they need to advance to the next level. Throwing Math Olympiad problems at them without teaching them the concepts used to solve them in a logical sequence is pointless and will only discourage them, especially if they are high-acheiving and not necessarily gifted."

I've noticed in out parts that the TAG and GATE groups have a fuzzy cutoff line and many parents use the gifted program to get something (ANYTHING!) more that what they have. This is a way to get something more by appealing to the "special" needs of the childs, when in reality, the need is just normal or average. This is not necessarily wrong, but it is a political approach to a system that cannot or will not do more for the average or above student. This does not, however, mean that the TAG/GATE curricula are any good!!!

-- SteveH - 10 Mar 2006

You might want to take a closer look at the TAG classes.

I thought exactly what you thought, going in — and I was told, explicitly, by the Math Chair, that this class was not constructivist.

She said she couldn't use a 'project approach' with the phase 4 kids because 'they have to learn too much.' That is a DIRECT quote.

I have no idea whether Ms. K. thinks she's teaching a fuzzy course.

But in reality, when you experience it, there's very little effective direct instruction and a HUGE amount of discovery.

I'm beginning to think that 'challenge' is code (possibly an unconscious code) for 'discovery.'

-- CatherineJohnson - 10 Mar 2006

Students learn less when they are challenged with problems above their instructional level, This is simply not an efficient way to transmit knowledge.

Challenging brightstudents with difficult problems once they know the material may be ok, but before that stage it's just silly.

The concept of building up knowledge through incrementally more difficult steps is lost on modern educators.

In fact, it's probably worse than that, since Ms. K. told a friend of mine that she teaches the concepts the day after the kids have done homework on those concepts. My friend said she told her this in a 'DUH!' tone, as if it should just be obvious a teacher wouldn't teach a new concept before assigning homework on the concept.

This is so wrong, it's not even wrong. It'd be one thing if the students were required to teach themselves the material first using a competent textbook so that the students bring something to the table and the teacher can expand on what has been learned. We call this college. Or at least we used to. It is doubtful that middle schoolers are ready to be self-learners yet and current math texts in use are far from competent teaching tools. To put it more simply, this is just poor teaching, the magnitude of which can only be achieved when the teacher has been taught absolutely nothing in ed school on how to effectively teach math. It takes a special effort to get it this wrong. Such poor teaching does not occur by happenstance.

-- KDeRosa - 10 Mar 2006

One of my 11th grader's good friends is gifted--not just in math, but every subject. Plus, she's an incredibly hard worker. I thought it might be helpful to paraphrase some of her thoughts about learning.

1. Regarding tests--If the goal is that you want students to be as prepared as possible, then why put surprises on the test? There's a difference between being able to analyze and anticipate harder questions and being surprised by concepts which you haven't yet been taught.

2. You can't assume that students intuitively get difficult subjects. While some students do get information intuitively, others can develop it. It's not a random skill, there is a process by which intuition can be learned. Effective teachers connect the dots for students by teaching them both the content and the process.

3. There is a natural frustration element that is going to exist when you're learning content that's new and difficult. There is a difference between putting students in a situation in which they are taught a skill versus forcing the students to reach the outcome of that skill by requiring them to "sink or swim."

4. When she doesn't get information intuitively (when she has to struggle), she uses reference points from past knowledge, or seeks out experiences from her own life to help her make the connections. Putting it into a context helps make the principles sink in for her. She also thinks that she probably learns the material more deeply and more thoroughly than if she had processed it intuitively. She thinks that when students process information intuitively they sometimes incorrectly mistake that for mastery and move on before they have deeply learned the material.

-- KarenA - 10 Mar 2006

The concept of building up knowledge through incrementally more difficult steps is lost on modern educators.

It's worse than lost.

It's almost as if they're actively against it.

And of course many are.

-- CatherineJohnson - 10 Mar 2006

It'd be one thing if the students were required to teach themselves the material first using a competent textbook so that the students bring something to the table and the teacher can expand on what has been learned. We call this college.

Right. Exactly.

And that's completely different.

That's not remotely the case here.

The kids are just handed Stuff To Do. Then they go do it, or don't do it, or their parents do it, or God knows what.

Now that I've dealt with this age for awhile, I would never arrange a middle school course so that the kids FIRST have the responsibility of reading a text.

Number one, they're not going to do it.

Number two, they don't have the critical reading skills TO do it, even if they did.

Number three, HORMONES.

Christopher is such a nut these days, I hardly recognize him. He is OBSESSED. His brain is churning, and not with thoughts about math.

I totally see where the whole 'brain periodization' idea came from!

-- CatherineJohnson - 10 Mar 2006

It'd be one thing if the students were required to teach themselves the material first using a competent textbook so that the students bring something to the table and the teacher can expand on what has been learned. We call this college.

Right. Exactly.

And that's completely different.

That's not remotely the case here.

The kids are just handed Stuff To Do. Then they go do it, or don't do it, or their parents do it, or God knows what.

Now that I've dealt with this age for awhile, I would never arrange a middle school course so that the kids FIRST have the responsibility of reading a text.

Number one, they're not going to do it.

Number two, they don't have the critical reading skills TO do it, even if they wanted to.

Number three, HORMONES.

Christopher is such a nut these days, I hardly recognize him. He is OBSESSED. His brain is churning, and not with thoughts about math.

I totally see where the whole 'brain periodization' idea came from!

-- CatherineJohnson - 10 Mar 2006

It's almost as if they're actively against it.

Teaching basic skills in a stepwise fashion is the antithesis of constructivism.

-- KDeRosa - 10 Mar 2006

Karen That's very interesting.

When she talks about 'intuitive' processing.....what kind of thing is she talking about?

She's saying that she has consciously looked for 'hooks' to other material she's learned, things from her life.....while other kids are just kind of 'getting it'.....and she thinks kids are 'fooled' by 'intuitive getting it' into believing they have more mastery than they actually do?

-- CatherineJohnson - 10 Mar 2006

and she thinks kids are 'fooled' by 'intuitive getting it' into believing they have more mastery than they actually do?

A budding Willingham.

-- KDeRosa - 10 Mar 2006

"Challenge" can be good in the right circumstances. Sometimes a kid should face a problem hard enough that the answer doesn't instantly pop fully formed into his head, where some work and thought is necessary to find the answer. Difficult enough to be engaging, but not so difficult to be discouraging.

And for some topics, it might be worth it to have a kid struggle with a problem to the point where he thinks something along the lines of, "The methods I know now are pretty lousy for solving this problem." But then use this as a motivation for learning some spectacular new technique. (Followed by practicing the technique and applying it to such problems.)

My diagnosis (from afar): there is too much stuff being covered in one year. She is teaching much like how I teach when I'm given too much stuff to cover too fast.

In a few years she may be able to figure out how best to rob Peter to pay Paul.

-- RudbeckiaHirta - 10 Mar 2006

"Challenge" can be good in the right circumstances. Sometimes a kid should face a problem hard enough that the answer doesn't instantly pop fully formed into his head, where some work and thought is necessary to find the answer. Difficult enough to be engaging, but not so difficult to be discouraging.

Unfortunately, I just don't know enough (yet) about either pre-algebra OR math ed to have a coherent view or even perception of this.

Clearly - it's just common sense - 'challenge' is a good thing.

Anyone who's good at what they do likes a challenge, and can manage a challenge.

But there's something very wrong here - and if 'challenge' is being used to mean 'discovery' that's bunk.

-- CatherineJohnson - 10 Mar 2006

My diagnosis (from afar): there is too much stuff being covered in one year. She is teaching much like how I teach when I'm given too much stuff to cover too fast.

ALTHOUGH......from what I've read on your blog (which I don't get to often enough) I'm not so sure that's the case.

Yes, definitely, she's been assigned a Death March through Prentice Hall.

I can see where I myself would have to fall back on teaching one procedure a day and asking the kids to remember it until the test (although I suspect there are ways to Rob Peter to pay Paul, as you say....and this is one of the things I want to know & learn).

I remember your description of your students not reading the book before class, and of what you were doing with the book in class (I wish I remembered where that was)—it was amazing.

You were making connections all over the place; basically, from where I sit, you were doing a brilliant edit job on a published textbook IN A CLASSROOM LECTURE.

I don't think that's the case here, ALTHOUGH it's true that Christopher never complains that he doesn't understand his teacher (and I have reasonable trust in his perception & reports on this. If he abjectly didn't understand a word coming out of her mouth I think he'd know it and say so....)

I'm eager to learn ways teachers at any level can compensate for the situation they and their students are in.

I do think that Ms. K. could make her life a lot easier using & assigning mini-problems of the kind Kathy talked about, and there's no reason for her not to be doing distributed practice throughout the year.

-- CatherineJohnson - 10 Mar 2006

One thing I speculate about is that the kids may actually have received enough distributed practice at this point just by the nature of mathematics (i.e. you keep using negative numbers in every chapter, not just the chapter on negative numbers) so that they're starting to have an easier time....

-- CatherineJohnson - 10 Mar 2006

short and sweet

Definitely, there's too much material.

No question.

It's ludicrous.

I'd cut this course in half. (At a minimum.)

-- CatherineJohnson - 10 Mar 2006

More spiralling nonsense: my neighbor, who SUFFERED ACUTELY all through this class last year, tells me that the Phase 4 math class in 7th grade is almost all review.

They learn the same stuff again, with some statistics as well.

-- CatherineJohnson - 10 Mar 2006

Modulating "challenge" is on of those advanced teaching techiques that most teachers aren't any good at. Most teachers don't know what effective teaching even is let alone how to incorporate these advanced techniques into lessons. It's not like these teachers are a bunch of Engelmanns. Engelmann knows all about "challenge":

Rule 3: Always place students appropriately for more rapid mastery progress. This fact contradicts the belief that students are placed appropriately in a sequence if they have to struggle— scratch their head, make false starts, sigh, frown, gut it out. According to one version of this belief, if there are no signs of hard work there is no evidence of learning. This belief does not place emphasis on the program and the teacher to make learning manageable but on the grit of the student to meet the “challenge.” In the traditional interpretation, much of the “homework” assigned to students (and their families) is motivated by this belief. The assumption seems to be that students will be strengthened if they are “challenged.”

This belief is flatly wrong. If students are placed appropriately, the work is relatively easy. Students tend to learn it without as much “struggle.” They tend to retain it better and they tend to apply it better, if they learn it with fewer mistakes.

...

Placing students at the edge of their ability to perform, however, means placing them where the students are “working very hard” and where they will make a high percentage of mistakes. This placement effectively negates good teaching.

...

On three occasions, we had the opportunity to split groups that were fairly homogeneous in performance and to place half the group at the beginning of D and the other half at the beginning of C, where they would learn the facts and operations that are assumed by Level D. The strategy for these students was to make sure they performed according to the ideal percentages of first-time performance and to move as quickly as possible. If students were clearly firm on something, we would either direct the teacher to skip it in half the lessons or present the problems as independent work. As soon as the percentages started to drop, we would return to presenting full lessons and continue at that pace until it was clear that the students could be safely accelerated. (Note: We tend not to skip material when we accelerate students. We simply go through the material faster. We’ve discovered that when teachers start skipping material, they often skip too much or skip material that should not be skipped even if students perform at acceptable percentages.)

In all cases, groups that started in C performed much better and actually passed up groups that started in D. In two cases, this occurred before the end of the first year. For the last case, it occurred in the middle of the second year. The students who started in D tended not to perform near the ideal first-time percentages. They often failed the ten-lesson tests, and teachers had to spend a great deal of time reviewing and reteaching things the students were expected to have learned. In contrast, the students who had been placed in C were able to do more than one lesson a day (until they reached about lesson 30 in D) and had a very high rate of passing the ten-lesson tests. For these students, the sequence of the program was congruous with their skill level, and the steps in the program were small; for the students who started in D, the program steps were too large and the climb too steep. The overall effect was that the D-starting students didn’t like math as much as the other students did and had far less confidence about their ability to learn math. We later adopted the practice of starting all students with marginal understanding in Level C, not D.

-- KDeRosa - 10 Mar 2006

Catherine: I'm not teaching like that now because I am refusing to teach the bat-out-of-hell course until it's fixed. (Of course I'm enough of a nobody that I don't really have the option to "refuse" things at work. I can, however, suggest that more people have experience teaching it, and I can find gullible other people to volunteer for it.)

-- RudbeckiaHirta - 10 Mar 2006

It may be that Christopher is understanding the teacher's lecture (such as it is in constructivist teaching), and things look easy when they do a couple of examples in class. But by the time he gets home, he takes a look at the homework and says, "Uh-oh. I don't remember how to do this. How did she say we were supposed to do this?"

I had my fair share of that in high school -- mostly in chemistry, although I had a much keener sense that I didn't quite grasp the concepts. That class was one of the only C's I ever got in school.

-- BrendaM - 11 Mar 2006

I have never used them. Shame on me. Sadly, until I came to the Table, I have never seen one. Anyway, my question is should the question mark above Runner 1 extend over the entire bar or just to the end of where Runner 2's bar?

When I look at it, I can see how you can replace the ? with an x, but if the bracket with the question mark includes the +280, then it may result in some confusion.

-- SmartestTractor - 11 Mar 2006

My daughter (fourth grade) is very good at math. I don't know if she should be considered "gifted" or not, and I don't worry about that.

The fact that she can quickly--perhaps intuitively--solve many problems, including some challenge type problems, sometimes hinders the effort to teach her new stuff. Her math teacher and I have had to press her a little to get her to write things down when doing problems. For story problems in particular, I want her to write down what she is solving for in an intermediate step so she knows that she hasn't yet gotten to the final answer.

Also, when I want to explain some subtlety that could lead to a shortcut or something halfway through the problem, I have to lead up to it by showing her how I set up the problem. This often doesn't resonate with her because she didn't really go through a setup step in her mind. She just kind of leapt to the answer--even if it meant using some brute force along the way.

So counting on and continuing to feed this apparent intuitiveness might actually lead to a dead end in that the student will have to learn to back away from some of that intuitiveness before learning systematic ways to solve problems.

-- DanK - 11 Mar 2006

Hey Dan,

My son is a year ahead of your daughter, but he is very similar. We are constantly trying to balance his joy of using his natural intuitiveness with "the way things are done." One of the reasons I hang out here is to get a clearer picture of why things are done a certain way so I can pass it on to him.

Pre-Algebra was what finally made him realize that he was going to have to write everything out, but it was very tough going. Algebra has been easier.

Another thing that helped this year was his tutoring other children. Realizing that others don't think like him has been an eye opener. He's been more accepting of learning what he sees as the "long way" because he realizes that other kids might need it. I don't get the fight as much.

-- SusanS - 11 Mar 2006

I'm not teaching like that now because I am refusing to teach the bat-out-of-hell course until it's fixed. (Of course I'm enough of a nobody that I don't really have the option to "refuse" things at work. I can, however, suggest that more people have experience teaching it, and I can find gullible other people to volunteer for it.)

LOL!

You have great political sense.

I remember your advice on Mrs. R: make it so she wants Christopher out of her class.

heh

-- CatherineJohnson - 11 Mar 2006

RH

Which course is the bat-out-of-hell course?

This was an introductory calculus course, right?

Was it aimed at some specific group?

-- CatherineJohnson - 11 Mar 2006

Ken

I LOVE IT!!

I was remembering some of those quotes....but wasn't sure which one of Engelmann's writings had them.....definitely getting those pulled up front.

I 'discovered' this principal about two seconds after my Singapore Math books came in the mail a couple of years ago.

Those books look 'too easy.'

They're not.

They start super-easy and work up fast.

I opened my first Singapore Math class telling this to the kids, and at least one of them never forgot it. I ran into his mom at the beginning of this year, and she said he often repeats this to her — and he'd moved up from Phase 3 to Phase 4.

She attributed the move to the Singapore Math class.

I think he learned some useful things in that class, but I think the most useful thing he learned was the idea of starting small and building up.

He's taken it to heart.

-- CatherineJohnson - 11 Mar 2006

Hi Smartest Tractor — thanks SO MUCH for the l-shaped perimeter problems!

I'm no expert on bar models, and I'm not quite sure there's as logical a rule about the question mark as there is for variables.

(I'll take a look at my books and see what they say —)

In this case I used the question marks to indicate the final answers the child is supposed to come up with (I hope — let me go check).

[pause]

OK, yes. That's the way I used them.

The first question mark spans the entire bar, because the question asks for the entire distance each runner covered.

I think that's how the PRIMARY MATHEMATICS series would use them....in other words, I don't think the question marks are 'variables,' exactly.

Generally speaking, I've found that the bar models I've drawn exactly duplicate the way I would solve a simple algebra problem using two variables — so you can show kids, each step of the way, 'Look, we just subtracted out the extra distance the one runner ran.'

But I'm just learning there are a few ways to solve two-variable equations.....I was only taught the 'substitution' method, if that's the right term.

So I don't know whether bar models model every approach to solving two-variable equations....

-- CatherineJohnson - 11 Mar 2006

Remember, too, that these bar models are taught to quite young children. They do problems like this one in 3rd grade.

So the goal at that point, I assume, is just to get them setting up the problem in a way that helps them solve it, which means in a way that 'shows' them what the question(s) are.

If you used bar models with older kids might want to start making the question marks into true variables.....or, if you wanted to use them transitionally, you'd get rid of the question mark and label the drawing with a variable......or perhaps use both.

-- CatherineJohnson - 11 Mar 2006

Placing students at the edge of their ability to perform, however, means placing them where the students are “working very hard” and where they will make a high percentage of mistakes. This placement effectively negates good teaching.

This is what we have on a near-daily basis.

As I keep mentioning, last year kids were getting 20s and 30s on the tests in this class.

-- CatherineJohnson - 11 Mar 2006

What I want to know is why Christopher is frequently 'getting it' now.....

I talked to another mom whose two kids have been doing well for quite awhile, and she said that at the beginning of the year they were struggling so much that she was going to hire a math tutor. She was doing lots of reteaching.

I was thinking these kids were brainiacs while Christopher is a Math Peasant who's going to have to do the whole thing by brute force....but in fact their trajectory sounds like his. They're both girls, they're older (Christopher's one of the youngest kids in the class), they're WAY more focused, and they didn't have a Mrs. R. trauma (though they do now....)

So it sounds like Christopher has followed their track only a few months behind them.

I don't get the sense that Ms. K's teaching has changed appreciably, though I do know the principal has pressed her about performing formative assessment.....

I'm thinking there's enough 'logical' repetition in any math course that if your parents get you through the first months you start to have the foundational skills you need.

Here's an example.

Christopher had a horrific time learning how to add & subtract integers. It was a nightmare. I have all kinds of 'out loud' worksheets I made up myself (I think I loaded them to the 'Math Lessons' page; I had him do Out Loud problems from Russian Math, etc.

He learned them well enough to do well on that test; but then in the very next chapter he had to be taught all over again; then in the next; it just went on and on.

I remember at one point, when Ed had gotten involved, so now we had TWO full-time parent teachers in the house, hearing from Ed that CHRISTOPHER DOESN'T KNOW ANYTHING ABOUT SUBTRACTING NEGATIVE NUMBERS.

Of course I insisted he DID know SOMETHING......

Finally, during the lesson on negative exponents, which Ed handled and put A LOT of time into, Christopher suddenly started getting some automaticity — and now he acts like it's an insult if I suggest maybe he ought to practice subtracting negative numbers. (He does still make mistakes.)

So that's my new theory.

If you can get your child 1/4 to 1/3 of the way through an accelerated pre-algebra course (or any of the courses we've had up to now), they start to be able to handle the material on their own if only because all math courses create opportunities for distributed practice by their very nature.

-- CatherineJohnson - 11 Mar 2006

The bat-out-of-hell course is "Math for Elementary Ed."

I am very lucky that Calc 1 is paced reasonably slow (for now! change may be afoot!), so I don't mind teaching it. Unfortunately, this is at the expense of Calc 2 being faster and Calc 3 having to insanely cram in everything else.

Still, when I saw "slow", our Calc 1 is paced slowly for a college course. It goes about twice as fast as the comparable high school course (and covers the material in more depth).

-- RudbeckiaHirta - 11 Mar 2006

It may be that Christopher is understanding the teacher's lecture (such as it is in constructivist teaching), and things look easy when they do a couple of examples in class. But by the time he gets home, he takes a look at the homework and says, "Uh-oh. I don't remember how to do this. How did she say we were supposed to do this?"

I'm sure that's it.

That happens CONSTANTLY in learning math (or in learning anything).

That's what makes me think she's probably decent at explaining things — what's missing is sufficient guided practice in class and problem sets for homework.

I think he'd be able to tell me if he understood NOTHING in class. (I think he has told me, once or twice, that he didn't understand what she explained to them...)

-- CatherineJohnson - 11 Mar 2006

Dan K

So counting on and continuing to feed this apparent intuitiveness might actually lead to a dead end in that the student will have to learn to back away from some of that intuitiveness before learning systematic ways to solve problems.

That's very interesting.

This is true of ALL CHILDREN UNIVERSALLY, as far as I can tell.

If they've 'got' the answer, THEY DON'T WANT TO HEAR FROM YOU ABOUT SETTING IT UP, WRITING OUT EACH STEP, ETC.

This is interesting, because it hadn't occurred to me that this would be even a bigger problem for a kid who's really good at math.

But of course it would be!

-- CatherineJohnson - 11 Mar 2006

Christopher started to get the message about writing out steps a little bit this year......and I'm forgetting why.

First of all, Ed and I both hammered him — and I think Ms. K. pushed him on it (not sure).

But I remember a couple of times when he missed an answer because he'd forgotten a step.....I wish I'd written it down.

Anyway, it was an object lesson in why writing out steps is a good idea.

He became a little less obnoxious about the whole thing.

A little.

-- CatherineJohnson - 11 Mar 2006

Catherine--

I should mention that the young lady made her comments in the context of discussing upper level math and science courses. I'm not entirely sure what she meant by "intuition." Maybe a better description would have been the "light bulb" or "aha" effect.

I will say that she has very solid foundational skills. Her parents are traditionalists in the sense that they believe that primary education should focus on the teaching the 3Rs to mastery.

I think a lot of what she calls intuition is a result of having a very solid knowledge base combined with the ability to process information quickly.

What I find fascinating is that she is the most academically talented child I know, and her views on education seem to closely align with those of the ktm contributors:

1. Teach me the skill you want me to learn. 2. Let me practice it to mastery. 3. When I've mastered it, let me move on. 4. Don't simply teach me the result. Teach me the process as well so that I have the tools to make the connections myself. That is, to paraphrase Dan, "teach me systematic ways to solve problems."

-- KarenA - 11 Mar 2006

I think a lot of what she calls intuition is a result of having a very solid knowledge base combined with the ability to process information quickly.

That makes sense.

-- CatherineJohnson - 11 Mar 2006

Karen A

she is the most academically talented child I know, and her views on education seem to closely align with those of the ktm contributors:

Teach me the skill you want me to learn. 2. Let me practice it to mastery. 3. When I've mastered it, let me move on. 4. Don't simply teach me the result. Teach me the process as well so that I have the tools to make the connections myself. That is, to paraphrase Dan, "teach me systematic ways to solve problems."

That's why the constructivist - content split doesn't align with left-right politics.

The number of left-wing professors outside of ed schools who have any affinity to constructivism is tiny.

It's probably the case that most people who've worked hard & long to master an intellectual discipline just aren't remotely interested in children doing discovery.

Once you've mastered a discipline you aren't interested in experiencing Radical Equality with a 10 year old.

You ARE the sage on the stage!

-- CatherineJohnson - 11 Mar 2006

The bat-out-of-hell course is "Math for Elementary Ed."

oh

my

god

that scares even me

-- CatherineJohnson - 11 Mar 2006

I am very lucky that Calc 1 is paced reasonably slow (for now! change may be afoot!), so I don't mind teaching it. Unfortunately, this is at the expense of Calc 2 being faster and Calc 3 having to insanely cram in everything else.

Still, when I saw "slow", our Calc 1 is paced slowly for a college course. It goes about twice as fast as the comparable high school course (and covers the material in more depth).

uh-oh.....

-- CatherineJohnson - 11 Mar 2006

The comparable high school course is BC calculus?

-- CatherineJohnson - 11 Mar 2006

BC Calc is calculus 1 + calculus 2.

AB Calc is most of calculus 1 + almost half of calculus 2. AB leaves out all the hard topics.

Where the speed differential comes in is when you count the number of class meetings.

One year of high school calculus (AB or BC -- take your pick) would be 180 class periods.

One year of college calculus (calc 1 + calc 2) (here -- credit varies by institution) would be 112 class periods.

This means that there is an expectation that college students will be responsible for learning a decent fraction of the material on their own outside of class.

-- RudbeckiaHirta - 12 Mar 2006

"Maybe a better description would have been the 'light bulb' or 'aha' effect.
I will say that she has very solid foundational skills."

Chance favors the prepared mind. --Louis Pasteur

-- OldGrouch - 12 Mar 2006

I love that saying!

-- CatherineJohnson - 13 Mar 2006

Rudbeckia oh, I see

thanks

-- CatherineJohnson - 13 Mar 2006