HelpDeskCombination < Kitchen

19 Nov 2006 - 19:03

help desk: combinations

What is the answer to this question?

Phil has a summer reading list of 12 books.
If he has to read three books, how many different sets of books can he choose to read?

ummmm....

Is this a combination or a permutation?

I think it's a combination.

I mention combinations, because combinations aren't in Christopher's textbook.

Ms. K covered combinations in class, but Christopher forgot to tell me, and of course there was no review sheet, because why would there be a review sheet when you've covered 18 or 19 or 20 brand-new topics in probability, some of which aren't in the book, in 2 weeks' time and now you're going to give a test?

Maybe there was something on edline.

If so, it's not there now.

So I had no idea combinations were going to be on the test, so I didn't teach myself combinations and thus couldn't reteach combinations to Christopher & assign practice problems, etc., etc.

So he missed the combination problems.....

....also, there seems to be a new mystery concerning POINTS OFF for crossing out incorrect work on your test paper.

Christopher is under the impression that Ms. K. has told them she has to take POINTS OFF for any crossed-out incorrect work, because the state tests don't allow you to cross out incorrect work.

So: POINTS OFF.

In the middle of chewing that one over (POINTS OFF! FOR FAILING TO OBSERVE MANDATORY STATE TEST STYLE AND USAGE REQUIREMENTS! I'M PRETTY SURE I'M AGAINST IT!) I was seized by an impulse to check.

What are our mandatory state test syle and usage requirements, anyway?

[pause]

As far as I can tell, there aren't any:

Mathematics Scoring Policies, Grades 3–8
Listed below are the policies to be followed while scoring the Mathematics Tests for all grades.
1. If the question does not specifically direct students to show their work, teachers may not score any work that the student shows.
2. If a student does the work in other than a designated “Show your work” area, that work may still be scored. (Additional paper is an allowable accommodation for a student with disabilities if indicated on the student’s IEP or 504 Plan.)
3. If the question requires students to show their work, and a student shows appropriate work and clearly identifies a correct answer but fails to write that answer in the answer blank, the student should still receive full credit.
4. If the question requires students to show their work, and a student shows appropriate work and arrives at the correct answer but writes an incorrect answer in the answer blank, the student may not receive full credit.
5. If the student provides one legible response (and one response only), teachers should score the response, even if it has been crossed out.
6. If the student has written more than one response but has crossed some out, teachers should score only the response that has not been crossed out.
7. For questions in which students use a trial-and-error (guess-and-check) process, evidence of three rounds of trial-and-error must be present for the student to receive credit for the process. Trial-and-error items are not subject to Scoring Policy #6, since crossing out is part of the trial-and-error process.
8. If a response shows repeated occurrences of the same conceptual error within a question, the student should not be penalized more than once.
9. In questions that provide ruled lines for the students to write an explanation of their work, mathematical work shown elsewhere on the page may be considered and scored if, and only if, the student explicitly points to the work as part of the answer.
10. Responses containing a conceptual error may not receive more than fifty percent of the maximum score.
11. In all questions that provide a response space for one numerical answer and require work to be shown, if the correct numerical answer is provided but no work is shown, the score is 1.
12. In all questions that provide response spaces for two numerical answers and require work to be shown for both parts, if one correct numerical answer is provided but no work is shown in either part, the score is 0. If two correct numerical answers are provided but no work is shown in either part, the score is 1.
13. In all 3-point questions that provide response spaces for two numerical answers and require work to be shown in one part, if two correct numerical answers are provided but no work is shown, the score is 2.
Introduction to the Grades 3 –8 Testing Program in English Language Arts and Mathematics (pdf file)

So it's a mystery.

Did Ms. K tell the kids she has to take points off for crossed-out work because you aren't allowed to cross out work on the state tests?

If so, is that what cost Christopher 2 points on his answer to number 17 (I'm thinking no. I think he got it wrong.)

And....here's the biggie.

Does the school need to dig out all the state tests Ms. K scored and check to see how many points the kids lost for crossing out incorrect work?

Have I mentioned that each school corrects its own students' tests?

Have I mentioned that the degree of training and attention to inter-rater reliability appears to be practically nil? [UPDATE 12-7-2006: Wrong. Christopher's English teacher says they get quite a bit of training. I doubt it's enough - this would be the state's fault, not the school's - but it's not nil. Of course parents don't know this because we pretty much know nothing of substance that goes on in our schools. We're constantly interviewing each other to try to find out what's coming up next.]

What a mess.

update

I have good news and bad news.

The good news is: Christopher knows how to do combinations.

He learned how to do combinations in class.

Then he remembered how to do combinations on the test, without having studied combinations the weekend before the test.

That's the good news.

The bad news is that apparently I can't understand what the he** he's talking about any more than I can understand what Ms. K is talking about half the time.

I wish I had a digital recording of the whole long series of permutation-combination exchanges Christopher and I have had today.

"Christopher, you got that answer wrong." [re: the answer Ms. K has in fact marked wrong]

"No, I didn't."

"Yes you did."

"No, I didn't, I got it right."

"I called L. She did the problem. You got it wrong. It's a Combination, not a Permutation."

"It's a permutation on the line."

"It's a combination."

"I crossed it out, I got it right."

"It's a combination."

"Christopher! That's what I just said! I said it's a combination. Your answer is a permutation, not a combination."

"It's a combination. I know it's a combination. I got it right."

"Who's on first?"

Now I'm starting to wonder.....maybe the book did cover combinations.

I mean....the whole section on selection without replacement...doesn't that get us into Combination territory?

I have no freaking idea.

I do, however, grasp the events of this our most recent foray into Phase 4 Summative Assessment.

1. Christopher recognized item number 17 as being a Combination problem.

2. Christopher correctly worked the problem, arriving at the correct answer. 220.

3. Christopher then decided that item number 17 was not a Combination, but a Permutation.

4. Christopher crossed out the computations for the Combination.

5. Christopher wrote the Permutation answer in the blank. 1320.

6. Ms. K marked it wrong, correctly citing as her authority in the matter item #6, above.

I'm too old for this.

how CA does it

-- CatherineJohnson - 19 Nov 2006

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Be sure and Ken's D-ed Reckoning post for today titled, "On the responsibilities of school districts." I posted this on another thread, but it seems that it might apply here as well.

-- KarenA - 19 Nov 2006

I'm on my way....

-- CatherineJohnson - 19 Nov 2006

The reading video from readingrockets.com is fantastic.

-- CatherineJohnson - 19 Nov 2006

yeah....

This is incredible:

The problem with the current educational system is that it has no advocacy for the children. In fact, it is a very strong non-advocacy system, which is supported by all major components of the system—the law, colleges of education, local school districts, educational publishers, federal and state grant supports, and teacher unions.
Although it is not possible to detail all the ways in which these various components contribute to the overwhelming incompetence of the system, I'll try to provide a brief summary of the major problems with each component.
The law: Basically, the laws associated with teaching and student performance are two-faced. In one sense, the laws were instituted to protect the students and thereby protect the state's interest in a valuable resource. The other face of the law denies that teachers have any sort of professional skills that are not possessed by the person on the street, asserts that teachers have only "responsibilities," protects schools or teachers from liability, and refuses to recognize rights of students to receive a quality education. Although special education children are modestly protected by laws, the appropriateness of programs is not determined by anything approaching tight standards.
Through laws, states have established a variety of bureaucracies, such as state textbook commissions. These agencies function in a uniformly incompetent manner. Although designed to improve instruction the students receive, the commissions are highly conservative and act as impediments to change.
In summary, there is not help from the law, no hope of malpractice suits (because these suits imply that teachers have professional skills, which the law denies), and no hope of support from state boards of education or state agencies because these agencies are not accountable for achieving their stated mission.
Advocacy for Children Engelmann

-- CatherineJohnson - 19 Nov 2006

THe problem you cited is a "combination" problem. It's 12 "choose" 3, which is 12!/9!3! = 220.

But that's not the real problem. The real problem is knowing the rudiments of probability isn't preparing the student for algebra, even though teachers seem to think it is. Do they cover fractions, decimals, percentages, negative numbers, ratios, percentages and solving simple algebraic equations?

-- BarryGarelick - 19 Nov 2006

Or maybe it's 12.11.10? Choose any of the 12 books, then any of the remaining 11, etc.,

-- VerghisKoshi - 20 Nov 2006

Do they cover fractions, decimals, percentages, negative numbers, ratios, percentages and solving simple algebraic equations?

You'll love this.

The probability chapter is almost at the end of the book.

Chapter 14.

The chapter just before it is on ratio & proportion.

The hardest problems in the Chapter 14 problem sets were probability problems used to practice ratio.

So the class spent a HUGE amount of time going through this problem:

A jar contains red and white marbles. The number of white marbles is 5 more than twice the number of red marbles. If the probability of drawing a red marble at random is 2/7, how many red marbles are in the jar?

This is just unbelievable.

They've never, EVER, been taught how to solve two linear equations; they barely know how to set up a simple equation using just one variable, let alone two.

Plus they've never done word problems prior to this year.

So all of a sudden they're doing a two-variable word problem involving ratio, which they have not been taught since sometime last year when they spiraled through it so fast they've now forgotten it, and this year, with ratio yet t/k.

Ms. K told them she'd put one or two ratio problems on the test (why?), and Christopher said, "That means she'll put 4 or 5," so I kept trying to teach him how to do this problem.....I should have just let it go.

-- CatherineJohnson - 20 Nov 2006

It's 220.

I think the book may teach it in a simple way with Venn diagrams, etc., that I "understood at the time" but have now forgotten....

-- CatherineJohnson - 20 Nov 2006

I think she probably had them do the ratio word problem the "Math Olympiads" way, without two variables but defining one variable in terms of the other from the get-go.

red marbles = x

white marbles = 2x + 5

But I was too addled to teach it that way myself, or to realize that's what she'd done.

-- CatherineJohnson - 20 Nov 2006

Or maybe it's 12.11.10? Choose any of the 12 books, then any of the remaining 11, etc.,

No, the problem asks for how many diferent sets of books can he choose. If you use permutations, you can have A,B,C and A,C,B each count as a set of three, since order matters. Since the sets of three must each be distinct, the number of combinations is called for.

Catherine, what textbook are they using?

-- BarryGarelick - 20 Nov 2006

It's an oldie but semi-goodie (I actually like the writing & explanation, but it's ludicrously "integrated.")

Integrated Mathematics, Course 1 Dressler & Keenan; written in 1966

-- CatherineJohnson - 20 Nov 2006

This is the 3rd edition

-- CatherineJohnson - 20 Nov 2006

Barry

I finally read the article Ken blogged about awhile ago - (I'll find title) -

It cited some interesting research showing that worked examples in algebra were extremely important.

I'm going to have Christopher spend more time "studying" worked examples.

The problem is that I'm not quite sure what "studying" a worked example means....

I feel like he should do something......take notes, maybe?

-- CatherineJohnson - 20 Nov 2006

Worked examples. A worked example constitutes the epitome of strongly-guided instruction while discovering the solution to a problem in an information-rich environment similarly constitutes the epitome of minimally-guided discovery learning.

The worked example effect, which is based on cognitive load theory, occurs when learners required to solve problems perform worse on subsequent test problems than learners who study the equivalent worked examples. Accordingly, the worked example effect, which has been replicated a number of times, provides some of the strongest evidence for the superiority of directly guided instruction over minimal guidance. The fact that the effect relies on controlled experiments adds to its importance. The worked example effect was first demonstrated by Sweller and Cooper (1985) and Cooper and Sweller (1987) who found that algebra students learned more studying algebra worked examples than solving the equivalent problems. Since those early demonstrations of the effect, it has been replicated on numerous occasions using a large variety of learners studying an equally large variety of materials (Carroll, 1994; Miller, Lehman & Koedinger, 1999; Paas, 1992; Paas & van Merrienboer, 1994; Pillay, 1994; Quilici & Mayer, 1996; Trafton & Reiser, 1993). For novices, studying worked examples seems invariably superior to discovering or constructing a solution to a problem.

-- CatherineJohnson - 20 Nov 2006

[1] Kirschner PA, Sweller J, Clark, RE (2006) "Why minimally guided instruction does not work: an analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching", Educational Psychologist 41:2 p75-86. Preprint available online.

-- CatherineJohnson - 20 Nov 2006

Why minimally guided instruction does not work (pdf file)

-- CatherineJohnson - 20 Nov 2006

Here's a blogger who's pro-construvism....it's exactly what Hirsch would predict - he says, "Well, sure, guided instruction is better if the object is to remember stuff."

Of course, if constructivists are willing to concede that direct instruction is better when you're trying to change memory, that would be a help.

-- CatherineJohnson - 20 Nov 2006

moodlers?

-- CatherineJohnson - 20 Nov 2006

The paper has some wonderful stuff on "problem-solving".....(or maybe it's my extension...)

Experts don't really "solve problems."

Or, rather, they solve problems at the edges of their fields.

What makes an expert an expert is that situations that are a problem for us aren't a problem for them.

Constructivism has assumed that all of what experts do is "solve problems."

Apparently there are dozens of medical schools that teach a problem-solving curriculum.

But if you look at what an excellent doctor is doing, he or she is not sitting around looking stumped and trying different alternatives.

-- CatherineJohnson - 20 Nov 2006

Physicians trained via problem-solving ordered lots of unnecessary tests.

swell

No opportunity costs there.

-- CatherineJohnson - 20 Nov 2006

Patients like spending hours of their lives undergoing unnecessary and unpleasant tests.

-- CatherineJohnson - 20 Nov 2006

That's a doozy of a scoring guide. They should just get Scantrons and pay the scoring-guide writers to write multiple-choice questions.

-- BrendaM - 20 Nov 2006

Worked examples:

Wilfried Schmid, the mathematician from Harvard (and on the NMP) once said that when he learns a new concept he has to try things out, one thinga at a time, following an example, just like kids when they're learning somethng new, until he gets the hang of it.

-- BarryGarelick - 20 Nov 2006

Hi Brenda

And that's just part of it.

The big trauma around here now is the ELA test (haven't got the math scores back).

I see no possible way there could be inter-rater validity on last year's ELA test.

There are zillions of questions requiring written reponses; all teachers score the test (to my knowledge); the training for the tests is nowhere near the level of training given to scorers of the SAT, which itself has a measurement error of 30 points on V & M, 40 points on the written part....

This is a nightmare.

Christopher has had a huge decline in score, and we have no idea - none - what his reading comprehension actually is or how it stacks up compared to other kids his age.

So now I'm in the standardized test-giving business, too.

-- CatherineJohnson - 20 Nov 2006

Barry

That's interesting.

I'm going to have to shift to that.

I've been having him go through problems like this one step at a time, but he doesn't have the steps written out.

I'm obviously including too much "discovery" - i.e. too much "searching" in the process.

(Definitely read the article above when you get a chance - very helpful.)

-- CatherineJohnson - 20 Nov 2006

Plus they've never done word problems prior to this year.

Please tell me this doesn't mean what I think it means. No word problems in grades K-6 ???

-Mark Roulo

-- KtmGuest - 20 Nov 2006

I teach bar models to all my elementary algebra students with two variable algebra problems.

The minute they see the problem 'drawn' as a bar model, they instantly get it. And then they do one example for themselves. And then they can solve the whole set without any problems.

However, this problems requires them to set x/3x+5 equal to 2/7, cross multiply and solve for x. Doesn't it? I got x=10. Did they learn how to set up a proportion with a variable in the numerator and denominator? And did they learn how to solve it?

-- AnneDwyer - 20 Nov 2006

No word problems in grades K-6 ??

Pretty much.

-- CatherineJohnson - 21 Nov 2006

I don't remember K-5 well....I don't think they had a lot of word problems (although they may have - it really escapes me unfortunately).

I could probably count the number of word problems they did all last year on ten fingers.

Half of them were probably on tests, too.

-- CatherineJohnson - 21 Nov 2006

Hi, Anne!

oh boy, yeah!

I love those bar models for simple two-variable problems - thanks for reminding me.

Yes, this problem requires you to do exactly what you've just said - which is NUTS.

No, they didn't learn how to set up a proportion with a variable in the numerator & the denominator; they've never even SEEN such a thing.

This course is insane.

It actually is better since the principal has been actively overseeing it.

But it's still mind-boggling.

As to your last question, I think a bunch of the kids followed her explanation - at least, I wouldn't be surprised.

Christopher was amazingly "comprehending" the several times we went through this.

He couldn't do it himself, and ended up missing the ratio/probability problem on the test.

But he wasn't simply at sea with it the way he was all last year.

-- CatherineJohnson - 21 Nov 2006

It's still insane.

-- CatherineJohnson - 21 Nov 2006

I've been looking again at the question of learning suddenly becoming faster.

I noticed last spring that Christopher suddenly seemed to learn math much faster.

I'm now thinking this speeding-up process happens once you've acquired a "schema" of a field inside your head.

You have a "map of the world" - you have a place to put new things you've just learned.

(I'll get some of this material posted.)

I think Christopher may have a not-half-bad math schema inside his head now.

Remember when we talked about teaching to mastery, and about the fact that the huge gap in learning speed is amongst people learning novel material and a novel field?

The authors of the book were arguing that you could get rid of large "rate of learning" differences once students are no longer learning novel material in a novel field - i.e. once students are learning novel material in a familiar field.

Christopher, I think, is now learning novel material in a familiar field.

And we can see that his disadvantage in the class is much, much smaller this year.

Some of the kids who were way out in front last year are faltering badly (I'm sure they didn't have a year of parental reteaching and/or private teaching last year as Christopher did).

And even the ones who aren't faltering are no longer in a completely different category from Christopher (judging from afar, of course).

At this point, I think I've seen in action the principle that learning novel material in a familiar field really does reduce differences amongst learners very significantly.

-- CatherineJohnson - 21 Nov 2006

No word problems.

Suddenly, we've got ratio word problems on probability.

-- CatherineJohnson - 21 Nov 2006

Here's the concept.

Learning novel material in a novel field is VERY hard, places an immense strain on working memory, and creates large rate-of-learning gaps amongst learners.

Learning novel material in a familiar field narrows those gaps considerably.

-- CatherineJohnson - 21 Nov 2006

I'm sure this is why Wayne Wickelgren tells parents who want to accelerate their children to teach them 1/3 of the course content before they join the course.

That was my plan with Christopher, and then events got in the way.

-- CatherineJohnson - 21 Nov 2006

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Title:	help desk: combinations
TopicType:	WebLog
SubjectArea:	IrvingtonMath
LogDate:	200611191401

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