22 Dec 2005 - 22:14

take the KTM challenge

Here's a sample problem from the Connected Math for parents material that BeckyC found:

Dario made 3 pizzas which he sliced into quarters. After considering how many people he would be sharing with, he thought to himself, 'each person can have a half.'

a. Is it possible that there was only one person to share with? How?
b. Is it possible that there were 5 other people to share with? How?
c. Is it possible that there were 11 other people to share with? How?

I imagine that as a sixth grader myself, I would have stared at this problem for a while, trying to figure out not what the answer to the problem was, but what the question could possibly be getting at with the 3 pizzas and the cutting them into quarters and Dario and his thoughts about a half of something.

Here's the answer (bold-faces are all Connected Math's):

This question illustrates how the actual amount can vary, but still be called a 'half', depending on the size of the 'whole'.

a. If there was only one other person to share with then Dario's comment means that Dario will have half of the total amount of pizza, and so will the other person. (This would mean one and a half pizzas each).
b. If there were 5 other people to share with than Dario's comment would mean that each person could have half of a pizza. (6 people each getting half of a pizza would use a total of 3 pizzas).
c. If there were 11 other people than each person would get a quarter of a pizza. Dario's comment would have to mean that each person gets half of a half-pizza. (12 quarters would be the same as 3 pizzas).

There's just something so awkward and convoluted about this problem, one has the sense that the same point could have been gotten at in a much more effective and straightforward way. Maybe this is because I learned mathematics all the wrong way in the bad old days and need to relearn it, but you know what? I don't think so.

And therefore I offer the following

KTM challenge: construct a problem that achieves the desired end without being misleading or convoluted.

Back to main page.

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d. Is it possible that Dario is Italian, Puerto Rican, Portuguese, or Filipino? Which ethnic group is supposed to identify with Dario??? Sorry, it's late, that's all I can muster -- I just posted a bunch on a previous topic, that might "support the learning" on this topic.

-- BeckyC - 22 Dec 2005

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Maybe we should play Round Food Fraction Bingo?

What you'd need to fix this problem is some sort of divisible object where half of the main object is a meaningful object in its own right.

Like if Igor had three brains and then severed the corpus callosum and then continued to divide the brains until he had 12 (roughly) equal pieces. Then you could talk about half the brain matter, each person having half of a brain, or each person having half of a hemisphere.

Or if we must use food with fractions, we can talk about granola bars that come two to a package. You could have half the food, half a package, or half a bar.

But unless there's a way for half of one of your macro-objects to be easily viewed as a whole smaller-object, this problem is going to continue to be flawed.

-- RudbeckiaHirta - 22 Dec 2005

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This is ludicrous.

I'm in NO MOOD (I've just scanned in a horrific Prentice-Hall lesson to post....)

-- CatherineJohnson - 22 Dec 2005

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Share?

FOOD FIGHT!!!!!

-- SteveH - 22 Dec 2005

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I agree with Rudbeckia, of course.

But I'll also say that the core problem-with-this-problem is that, as usual, it requires the child to teach himself.

Singapore Math would handle this through sequential direct instruction.

First of all, Singapore Math would present the child with a pristine, easily readable, 'abstractified' image of 3 circles divided into 4 parts.

These circles would not be dotted with pepperonis and onions.

They would be abstract representations of pizza.

I should add that Singapore Math wouldn't come up with something this stupid to begin with. WHY would 'Dario'—good point, Becky—go to the trouble of slicing up 3 pizzas into quarters before he has clue one as to how many people he's sharing with?

If you're going to obsess over pizzas, and I for one am finished with pizza fractions, at least pay the elementary school child the respect of saying that the pizzas came out of the box pre-sliced. So we don't have to go into the problem thinking Dario is a moron.

(btw, I think I'm seeing a stupid-child meme in U.S. math textbooks. It's blatant in TRAILBLAZERS; the children are constantly standing around arguing about What's more, 1/2 or 3/6? But I think it runs through a lot of these texts. Just something as subtle as writing a story problem about a boy who slices up his pizzas before considering how many people will be sharing....I don't think this is an accident. I don't think it's done consciously, but I do think that the people writing these texts unconsciously list toward stupid-kid imagery....)

Now that I have that off my chest, Singapore Math would start with the case of 3 pizzas sliced into 4 pieces to be shared between 2 people.

It would illustrate the answer: each person gets 1 1/2 pizza.

Then it would move on to 3 pizzas sliced into 4 pieces to be shared amongst 6 people.

Answer: each person gets 2/4, which is 1/2.

And so on.

At some point, after the concept had been clearly and directly taught, Singapore Math would ask the student to practice finding these answers—and it would begin by offering students already-drawn images of the answer.

Step by step.

Ultimately the student would be asked to move beyond the even division of 3 pieces divided into 4s amongst multiples of 2 or 3 to more complicated problems (although I'm not remembering at the moment how this would occur).

major Singapore Math principle: start easy, practice, practice, practice, then expand to difficult

I've mentioned that when I taught the ratio & proportion material to Christopher, I had trouble doing the exercises by the time we got to the 4th problem. The Singapore Math problems are far more challenging than ours, but the teaching and practice problems are far easier.

-- CatherineJohnson - 22 Dec 2005

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This problem is also hideously confusing (and no doubt embedded in lots of distracting images of pizza delivery people of color).

'What if there were 1 other person; what if there were 5 other people'—I can guarantee you that a 6th grader isn't going to realize: hey. We're talking about 2, 6, and 12.

It took me awhile to see that; I started out thinking I was going to have to divide quarters into 5ths and 11ths.

A problem like this is deliberately confusing—and yet neither challenging nor deep.

-- CatherineJohnson - 22 Dec 2005

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blech

-- CatherineJohnson - 22 Dec 2005

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Actually, that's something I hadn't thought about.

I'm going to have to start another 'Lost in Translation' chart.

What I see in Phase 4 is 'too hard' being equated with 'rigor.'

This problem equates 'confusing' with 'challenging.'

Yes, this problem, as written, is plenty challenging.

But what's challenging is reading the damn thing.

-- CatherineJohnson - 22 Dec 2005

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I'll have to look later on & see whether Primary Mathematics has anything like this....

I know the 3rd & 4th grade fraction lessons pretty well, since I taught those to Christpher and one of his friends last summer.

I don't remember seeing anything like this...

-- CatherineJohnson - 22 Dec 2005

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off-topic: thank God we've done as many bar models as we have.

Christopher was given a batch of do-it-yourselfers last night that he had no clue how to do. Ed says most of the parents in the class won't know how to do them, either. It took him quite a long time to do the first problem, a simple algebra problem, and he got the answer wrong. Funny thing: the Teacher's Edition has the answer wrong, too.

And it's the same wrong answer.

I, too, got the wrong answer, as did Christopher (all of us getting the book's wrong answer—making it 3 for 3).

I only saw that my answer was wrong when I had Christopher draw a bar model. Then it instantly 'popped' that the Teacher's Edition had to be wrong.

-- CatherineJohnson - 22 Dec 2005

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I left out my main point.

Christopher ended up with some idea of how the problem is solved & what it means thanks to bar models.

This was the first time Ed saw how brilliant those bar models are.

It was interesting, because he's quite good at math, though radically out of practice, and he was having the same problem I had for quite a long time as I worked with bar models: he doesn't really see the fact that when working many (most?) problems you're trying to divide quantities into equal parts.

(I realize that's not clear. I'll post the problem.)

-- CatherineJohnson - 22 Dec 2005

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Prentice Hall Pre-Algebra problem:

Carla spent 1/3 of her money at the amusement park. Afterward, she had $15 left. How much money did she have originally?

-- CatherineJohnson - 22 Dec 2005

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x - x/3 = 15

(2/3)x = 15

x = 22.5

-- KDeRosa - 22 Dec 2005

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Hi Ken—glad you signed in—

That's the answer I got.

Textbook answer: 45.

Can you believe it?

45 was the answer all 3 of us got when we first did it. I'm assuming we must have all skimmed the problem and concluded that 1/3 of her money was $15, so 3/3 would be $45.

This is another useful aspect of the bar models I hadn't thought about: they help you read the problem in the first place.

Because the bar model is visual, it stays put; you aren't trying to hold the numbers in working memory. (This is true of algebraic equations, too, of course, but Christopher doesn't know how to set those up.)

You can really get out of practice with math. It's kind of incredible.

It took Ed quite a long time to figure out how to set up the problem algebraically (and we both set it up in a more complicated fashion than you did).

That's why Ed said most of the parents in the class aren't going to have a clue.

They're just not going to be able to remember, on the spot, how to do these things.

I'm not completely sure of that.....my math education was poor, but what I did learn was overlearned....

I think the problem for Ed and me, last night, was that we had an Authoritative Wrong Answer in the Teacher's edition.

If we hadn't had a Teacher's Edition Wrong Answer we would have plugged the answer back into the original problem and discovered it didn't work. (Which, of course, is basically what happened with the bar model.)

-- CatherineJohnson - 22 Dec 2005

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Prentice Hall Pre-Algebra:

ROTE and WRONG

-- CatherineJohnson - 22 Dec 2005

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rote and rong

-- CatherineJohnson - 22 Dec 2005

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The answer to this question is 45:

Carla spent 1/3 of her money at the amusement park. She spent $15. How much money did she have originally?

(1/3)x = 15

x = (3)(15) = 45

-- KDeRosa - 22 Dec 2005

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"45 was the answer all 3 of us got when we first did it. I'm assuming we must have all skimmed the problem and concluded that 1/3 of her money was $15, so 3/3 would be $45."

I remember in all of my math classes being greatly influenced by the expectation that all answers are going to be "nice". I expect that they wanted it to be nice and nobody carefully checked the problem.

What bothers me is that they probably expect you to solve it by just thinking. Of course, what you want to teach (even for pre-algebra) is to use a variable, define the equation, and solve it, just like what Ken did. Algebra is a whole lot easier if you start with simple equations that you could do in your head.

-- SteveH - 22 Dec 2005

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The answer to this question is 45:

Carla spent 1/3 of her money at the amusement park. She spent $15. How much money did she have originally?

Right.

That's the question all 3 of us answered.

Us and the textbook authors.

-- CatherineJohnson - 22 Dec 2005

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I remember in all of my math classes being greatly influenced by the expectation that all answers are going to be "nice". I expect that they wanted it to be nice and nobody carefully checked the problem.

Yes, that's a big problem.

Let me tell you, Russian Math breaks you of that expectation the hard way.

About 80% of those problems end up in numbers I never even heard of.

-- CatherineJohnson - 22 Dec 2005

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117 23/201

That kind of thing. Which you arrive at via extended pencil and paper calculation.

I've never heard of that number.

I hope I never do again.

-- CatherineJohnson - 22 Dec 2005

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What bothers me is that they probably expect you to solve it by just thinking. Of course, what you want to teach (even for pre-algebra) is to use a variable, define the equation, and solve it, just like what Ken did. Algebra is a whole lot easier if you start with simple equations that you could do in your head.

It is cr**.

-- CatherineJohnson - 22 Dec 2005

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If it's not Scottish it's crap

-- CatherineJohnson - 22 Dec 2005

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-- CatherineJohnson - 22 Dec 2005

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from eduwonk: Does the US face an engineering gap?

-- CatherineJohnson - 22 Dec 2005

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WAY too much stuff to get pulled up front......

-- CatherineJohnson - 22 Dec 2005

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By making more specific comparisons, US competitiveness, as measured by newly minted engineers, is not eroding as fast as many say - if it's eroding at all, according to a Duke University study released last week. "Inconsistent reporting of problematic engineering graduation data has been used to fuel fears that America is losing its technological edge," the study states. "A comparison of like-to-like data suggests that the US produces a highly significant number of engineers, computer scientists, and information technology specialists, and remains competitive in global markets."

In some ways, experts say, today's debate over engineers reflects the cold-war controversy over the so-called missile gap in which the Soviets' advantage in missile numbers was counterbalanced to some extent by the quality and accuracy of America's nuclear arsenal.

"During the 'missile gap' and post-missile gap until the fall of the Berlin Wall the same sorts of issues were being raised about Russia as are being raised now about China and India," says Frank Huband, of the American Society for Engineering Education in Washington.

-- CatherineJohnson - 22 Dec 2005

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I wouldn't be at all surprised if they're right.

The 'engineering gap' meme has been seeming nutty to me for awhile (which doesn't mean it's wrong, necessarily).

Any time you've got a huge bestseller about the 'flat world,' you have to look askance.

-- CatherineJohnson - 22 Dec 2005

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AND let me add that this is meaningless to me as far as the issue of having decent math teaching in our schools.

ON PRINCIPLE, we need good curricula and good teaching.

-- CatherineJohnson - 22 Dec 2005

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regardless of what China is or is not doing

-- CatherineJohnson - 22 Dec 2005

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I love it!

"Business groups have been very smart about trying to change the subject from outsourcing and offshoring to the supposed shortfall in US engineers," says Ron Hira, an outsourcing expert at Rochester Institute of Technology. "There's really no serious shortage of engineers in this country."

-- CatherineJohnson - 22 Dec 2005

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Last year, the US awarded bachelor's degrees to 72,893 engineering students, according to the American Society for Engineering Education. But using India's more inclusive definition, the Duke study finds the US handed out 137,437 bachelor's degrees last year, more than India's 112,000. The US number is far more impressive in rela-tive terms, since India has more than three times as many people.

-- CatherineJohnson - 22 Dec 2005

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China's numbers are more problematic because its government does not break them down. In its revised figures, the National Academies reduced the Chinese total from 600,000 to 500,000. The Duke study pegs the total at 644,106, as reported by the Chinese Ministry of Education. But the study also points out that, as with India, the Chinese total includes engineering graduates with so-called "short cycle degrees" that represent three years or less of college training.

Of course, I think we've all seen that YEARS OF EDUCATION does not equate to QUALITY OF KNOWLEDGE AND PERFORMANCE

This is more INPUTS analysis.....

-- CatherineJohnson - 22 Dec 2005

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-- CatherineJohnson - 22 Dec 2005

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I think Rudbeckia won the KTM challenge: with 2 granola bars per package, six packages in a box.

I agree with Catherine that Singapore builds from the simple to the complex -- my kids and I did 3B equivalent fractions yesterday. It's a beautiful thing.

-- BeckyC - 22 Dec 2005

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it will be a shame if a renewed push on math and science grads is the next big thing in education at the expense of addressing the equity problem which threatens to make America look a lot like a Third World country over time. The latter seems a real risk since it's a lot easier for business leaders to run around saying we need more math and science grads than go to state capitals and force legislators to deal with the more thorny and hot-button equity-related questions about, for instance, licensing for educators, school finance, collective bargaining agreements, or pluralism in service provision and educational choice that need fixing but piss a lot of people off.

This is Eduwonk, and this is something I object to.

He's right, I believe, that it's easier for business & community leaders to focus on math and science than on teachers' unions, etc.

He's wrong, though, to continually imply that the 'real' problem in U.S. education is equity.

The real problem is extremely low quality education period. As Mickey Kaus says:

Unionized teachers stand in the way of the educational changes that might ameliorate our twin education crises (inner city disaster and suburban mediocrity)

-- CatherineJohnson - 22 Dec 2005

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Additionally, I now believe we have a very serious problem with boys and school.

I don't generally like being apocalyptic or declinist, which is why I haven't been on board for anxiety about the engineering gap for awhile now.

But I think when you're looking at the numbers we're looking at on boys and school, you've got trouble.

Off the top of my head (so I could be wildly wrong) those numbers look like the numbers we used to see for blacks: women far exceeding men in education and, thus, in opportunities.

We can see just how terrific a huge disparity in education favoring women has been for black families and children.

-- CatherineJohnson - 22 Dec 2005

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I agree with Catherine that Singapore builds from the simple to the complex -- my kids and I did 3B equivalent fractions yesterday. It's a beautiful thing.

Beautiful. Yes.

-- CatherineJohnson - 22 Dec 2005

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FOOD FIGHT!!!!!

YEAHHHHHH

-- CatherineJohnson - 22 Dec 2005

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Back to the pizza. The example doesn't even make the supposed point. Each person can have a half would be understood by most people to mean a half of one pizza. Typically, you would say, "you can have half" (no a) when you mean half of all there is.

Plus who would think "each person" when there are only two people?

There's an age when kids like to trick each other with word games based on two different meanings (usually more plausible than this) but it doesn't have any place in math.

This would be funny if it weren't sad.

-- SusanJ - 22 Dec 2005

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I think the educators should have to figure out this problem without having the answer key in front of them. My guess is they wouldn't do very well, either.

-- KtmGuest - 22 Dec 2005

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Re original problem: Each person can have a half? A half of what?! my little brain is saying.

Re RH's problem: You can only sever the corpus callosum once; after that, the left hand doesn't know what the right hand is doing.

-- KtmGuest - 22 Dec 2005

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(hmph. I registered, and tossed my cookies, but I'm still showing up as KTMG. :( )

-- KtmGuest - 22 Dec 2005

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Susan J

Back to the pizza. The example doesn't even make the supposed point. Each person can have a half would be understood by most people to mean a half of one pizza.

Thank you.

-- CatherineJohnson - 22 Dec 2005

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yes

I kept thinking.....well, why do you have to give people 1/4s??????

in real life wouldn't you just start divvying up the pizzas by halves????

-- CatherineJohnson - 22 Dec 2005

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until you started needing 4ths?

-- CatherineJohnson - 22 Dec 2005

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just makin' those real world connections

-- CatherineJohnson - 22 Dec 2005

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"Carla spent 1/3 of her money at the amusement park. She spent $15. How much money did she have originally?"

This style of problem appears regularly in Singapore 4A.

-- LoneRanger - 22 Dec 2005

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oh right, there are TONS of them

AND you can do them!

We're going straight back to the bar models, let me tell you

-- CatherineJohnson - 22 Dec 2005

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"each person can have a half."

"a. If there was only one other person to share with then Dario's comment means that Dario will have half of the total amount of pizza, and so will the other person. (This would mean one and a half pizzas each)."

That's a technically correct but unidiomatic usage of "a half". For that meaning, you'd expect the speaker to use "half", not "a half".

The problem with this is that idiom lies below the consciousness horizon for nearly everybody. To expect a school child to figure this out is unrealistic. The other side of that argument, though, is that kids love this sort of word-play.

Or, what Susan said, since I wrote this before reading all the comments.

8-/

-- DougSundseth - 22 Dec 2005

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WELCOME GoogleMaster!

Glad you registered (and figured out the logoff trick)!

-- CarolynJohnston - 22 Dec 2005

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Dario made 3 pizzas which he sliced into quarters. After considering how many people he would be sharing with, he thought to himself, 'each person can have a half.'

Given "a half" without defining of what, I prefer an answer of 23.

3pizzas x 4qtrs = 12 pizza qtrs.

If each person can only have "a half" that means there must be 23 people + Dario:

12 pizza qtrs x "a half" (.5) = 24 pizza qtrs halfs

What, that's not the right answer?

-- BenCalvin - 22 Dec 2005

WebLogForm
Title:	take the KTM challenge
TopicType:	WebLog
SubjectArea:	ConnectedMath
LogDate:	200512220103