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08 Sep 2005 - 23:40 released extended response items on the ISAT.
around the calendar to local businesses. The cost of ad space is based on
the number of square units each ad contains. The company charges $40.00
for Ad Space D. Using this information:
Draw an Ad Space that costs exactly $60 in the gridded space on page 10 of
the answer document.
And here's the illustration:
I like this problem, although wiser heads here at ktm may give me reasons why I shouldn't, in which case I'll revise my opinion. I like it because it's visual & spatial as well as 'numerical' (if that's the right word), and because I've found Christopher to be very challenged by any problem that asks him to combine numerical thinking or problem-solving with spatial 'thinking' or problem solving. And of course I love the Singapore bar models, and this problem reminds me of them. I also like it because it has 2 steps: you have to figure out how much each square costs & then you have to figure out how many squares $60 would buy. I like the open-endedness of this particular problem, too. A child could simply count the number of squares in Ad Space D (40) and then divide 40 dollars by 40 squares to get $1/square. Or he or she could notice that Ad Space D is a standard multiplication array, and multiply 4 by 10 to get 40. I'm sure a lot of kids would start out counting & then notice, mid-stream, that they could have arrived at their answer more efficiently by multiplying instead. Which is good. A little Math Object Lesson buried inside a story problem. I like that! Last but not least, I kind of like the fact that each square turns out to cost exactly one dollar. I don't know why. It reminds me of a genre of problems in Russian Math, in which you go through all kinds of elaborate, painstaking calculations only to end up with an answer of ONE. Or maybe TWO. Or, when things get really fancy, ONE HALF. Interestingly, I'm finding, as I work my way through RUSSIAN MATH, that I'm becoming quite attached to the number one. Every time it crops up as an answer I think: I should have seen that coming. An answer of one always seems like a flag, a sign that there was an easier, more elegant way to do whatever it was I was doing.....but I missed it. Russian Math has all kinds of 'surprise answers,' and I think a surprise answer in the middle of an ISAT could be slightly.....fun? An answer of one is like a little joke.
Susan H says is already happening. We're looking at a massive waste of teachers' and students' time. Last but not least, I'd bet the ranch you learn nothing from the verbal explanation that you didn't already learn by looking at the student's work. Being able to produce a fluent, intelligible verbal explanation of a mathematical solution is almost certainly important for math teachers. It's not important for the rest of us.
but less than 50. Each student will have a partner on the bus. At the
museum, each tour group will have exactly 6 students.
How many students are going to the museum? Show all your work. Explain in words how you got your answer and why
you took the steps you did to solve the problem.
Unless 5th graders in Illinois are doing a lot of prime factor problems, I don't see any reason to include an item like this one on a timed assessment. First of all, no one should have to be doing discovery ON A TEST. And second, this problem has two answers (36 & 42, right?), but the wording implies that it has just one answer, and that one answer is findable. I am DISCOVERING the fact that I don't think red herrings belong in math classes. Certainly not in elementary school math classes. What is the point? You are teaching children to distrust the English language at the precise moment they're learning grammar & composition. An unreliable narrator in a work of fiction can be a terrific device. But an unreliable questioner in an examination is just wrong. I'm against it.
each, and he sold regular pumpkins for $4.00 each. Peter sold 80 pumpkins
and collected $395.00.
How many jumbo pumpkins and regular pumpkins did he sell? Show all your work. Explain in words how you got your answer and why
you took the steps you did to solve the problem.
The problem is fine, assuming these kids have actually been taught some algebra. If they haven't, this is a discovery problem on a timed assessment, and I'm against it. So, assuming they've learned how to set up & solve equations with unknowns, the problem is good. IMO. The demand that the student explain each step in words is not.
'Student Friendly' Mathematics Scoring Rubric Assuming I'm reading this correctly (I feel a little distrustful), students must get all computations correct in order to earn the highest possible score of 4. They can earn a score of 3 with minor mistakes in computation, which I feel is fair, though others may disagree. What I reject absolutely is the explanation section:
This is wrong. I don't believe a 4 should depend upon being able to supply an explanation in any case. But here you have a child who can explain why he or she did what she did in a drawing, which is no mean feat (and I'm in a position to know) and even that isn't enough. Pace Anne, you'll notice that it's not OK for a child to explain what he/she has done by offering a mathematical demonstration, as the teachers in Liping Ma's book do. Anne's right about that; it struck me, too. Over and over again, when Liping Ma asks a Chinese teacher why he/she teaches an idea a certain way, the teacher responds by writing out a proof-like mathematical demonstration. That's what makes the book incredibly difficult (and incredibly valuable) to read for most of us; the teachers don't translate math into words, and neither does Ma. For Chinese teachers, math is math.
This drops you to a 3:
A couple of years ago the head of our school board sent out an email explaining the adoption of TRAILBLAZERS that included this line (from memory): In recent years math has become language-based. I think that would come as a surprise to actual mathematicians.
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
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I agree with just about everything you wrote, Catherine (except, I think 48 also works for that field trip problem). I'm a big fan of showing your work, but narrating it seems pointless--and potentially very time consuming on a timed test. -- DanK - 09 Sep 2005
I am pretty sure that cognitive science must have something negative to say about requiring learners to switch back and forth between problem-solving and essay-writing all the time. It must waste a lot of precious energy to be shifting gears like that constantly. Here's an illustrative example that I think will resonate with anyone who has ever programmed computers: I find that if I try to comment as I go, during the process of writing a computer program, then the comments I write turn out afterward to be complete gibberish that even I cannot understand. My belief is that analysis and exposition are two different, and in some sense opposed, intellectual functions. -- CarolynJohnston - 09 Sep 2005
Heavy sigh. Where to begin? The McHarold?'s ad problem is trivial for fifth grade. Can you imagine this problem in fifth grade Singapore math? "It reminds me of a genre of problems in Russian Math, in which you go through all kinds of elaborate, painstaking calculations only to end up with an answer of ONE. Or maybe TWO. Or, when things get really fancy, ONE HALF." I remember those problems in school. You begin to expect nice numbers and double-check your answers when they don't appear. In real life, however, nice numbers go away. Of course, the big question for extended response is whether they are going to take off points if you get the problem correct, but don't explain it well enough - whatever that means. Would this be considered correct? J is the number of jumbo pumpkins sold. R is the number of regular pumpkins sold. 1) J + R = 80 2) 9J + 4R = 395 J = (80 - R) 9(80 - R) + 4R = 395 720 - 9R + 4R = 395 5R = 325 R = 65 J = 15 The math speaks for itself. The next question is whether you can get full credit if your words sound good, but your answer is wrong. Our state's standardized test rubric allows for both of these cases. Finally, this is one of the third grade problems. In a toy store, there are bicycles and tricycles. Bicycles have 2 wheels and tricycles have 3 wheels. There are a total of 19 wheels. How many bicycles and how many tricycles are there in the toy store? ????????????? -- SteveH - 09 Sep 2005
Catherine, I just don't know about that ad space problem. It feels kind of busy to me. It has visual elements and linguistic elements and lots of distractions (like McHarold?'s and the burgers and ad spaces A through D), and you have to jump back and forth a lot, and then there's the injunction to explain yourself on top of it. It feels to me like a challenge problem, not a test problem. -- CarolynJohnston - 09 Sep 2005
Catherine, A couple of points: on the first problem, I totally agree with Steve H. If you are going to give a problem like this in the 5th grade, then the students should have to deal with fractions. A much better problem would have been to make Ad Space D 30 squares or 50 squares. Why are you giving a word problem in 5th grade that only requires multiplication of whole numbers by 10? The students learned that in the first grade!! This is absolutely what is wrong with our fuzzy math programs. As for extended response, we should be nudging students towards showing their work in the form that they will eventually have to use to show proofs. Mathematics has its own symbols and logic. When you solve a problem, these are all that is necessary. In Liping Ma's book, all of the best teachers in China used proofs to show students why a certain algorithim works. And many of them showed the proof in more than one way. What chance do our children have if we are requiring explanations in written English while the Asian children are being shown rigorous proofs? As for the last problem, again we are only making the students work in whole numbers. Singapore math introduces problems like this in 3B. -- AnneDwyer - 09 Sep 2005
Interestingly, I'm finding, as I work my way through RUSSIAN MATH, that I'm becoming quite attached to the number one. Every time it crops up as an answer I think: I should have seen that coming. An answer of one always seems like a flag, a sign that there was an easier, more elegant way to do whatever it was I was doing.....but I missed it. I just wanted to say that this makes me smile. It takes a real mathematician to be always haunted by the feeling that there was some slicker way to do it. -- CarolynJohnston - 09 Sep 2005
Personally, I'm getting really really sick of all the Big New things. There is never going to be a magic bullet that suddenly causes all the children to learn everything without any effort. The process of trying to find such magic bullets instead of just putting the nose to the grindstone is ruining our schools. -- BernieJohnston - 09 Sep 2005
I remember those problems in school. You begin to expect nice numbers and double-check your answers when they don't appear. In real life, however, nice numbers go away. Russian Math doesn't use problems this way. It will have several problems in a row--these are humongous computations--that all end up being the same simple number (or sometimes not the same simple number). Given the fact that 5th graders don't know fractions, and don't know very much about proportions, I like this problem. It's definitely busy.....and you don't see busy problems like this in Singapore or in Russian Math. -- CatherineJohnson - 09 Sep 2005
What chance do our children have if we are requiring explanations in written English while the Asian children are being shown rigorous proofs? I don't think Singapore Math uses proofs or proof-like demonstrations in 6th grade. I'll check my books again, but I haven't seen it. -- CatherineJohnson - 09 Sep 2005
I just wanted to say that this makes me smile. It takes a real mathematician to be always haunted by the feeling that there was some slicker way to do it. Thank you! Yes, it's true; I've developed that 'mathematician-like' feeling...and it ALWAYS crops up when an answer of 1 appears. -- CatherineJohnson - 09 Sep 2005
Reading through these comments, naturally I'm revising my opinion....I'm thinking each square probably should have cost $1.30 or something.... -- CatherineJohnson - 09 Sep 2005
by the way--I'll get around to blogging this: I finally read Stevenson's & Stigler's book on Japanese math ed! It's incredible. They say nothing about bar models, confirming my suspicion that only Singapore uses them, BUT the Japanese teachers use a physical bar model, made of paper, to do the 'a class has 12 more boys than girls and 36 kids altogether' kind of problem in the 4th grade. Now I'm thinking that Asian countries figured out paper strips used as bar models, and then the Singapore team of mathematicians & math teachers decided to run with them for some reason. -- CatherineJohnson - 09 Sep 2005
I am pretty sure that cognitive science must have something negative to say about requiring learners to switch back and forth between problem-solving and essay-writing all the time. It must waste a lot of precious energy to be shifting gears like that constantly. I'd bet real money on this. People simply don't understand that the brain uses energy exactly like 'the body,' and a child has limited amounts. As a matter of fact, IIRC, the brain consumes FAR more energy than any other aspect of the body.....I'll have to find the SCIENTIFIC AMERICAN article about why we developed our big brains. I THINK the idea was that we were able to develop big brains, which are extremely 'costly' in terms of energy consumed, partly because we learned to walk upright (heck--can't remember how that related) & partly because we learned to EAT ANYTHING WE COULD GET OUR HANDS ON, ESPECIALLY ANYTHING WITH LOTS AND LOTS OF FAT. I love that. -- CatherineJohnson - 09 Sep 2005
Here it is! Food for Thought: Dietary change was a driving force in human evolution -- CatherineJohnson - 09 Sep 2005
I added the rubric-- -- CatherineJohnson - 09 Sep 2005
Also, I'm thinking that the 'unreliable narrator' in the factoring problem is another cases of the English-language-artsing of mathematics. I can't think of a good example (if I see one in Russian Math I'll post it), but a real 'trick question' is a problem where the mathematics can carry you off in a right direction or a wrong direction. In problems like the factoring one above, there is no mathematical reason to look for one answer. Solely a verbal reason. -- CatherineJohnson - 09 Sep 2005
These aren't trick questions; they're simply dishonest. -- CatherineJohnson - 09 Sep 2005
& partly because we learned to EAT ANYTHING WE COULD GET OUR HANDS ON, ESPECIALLY ANYTHING WITH LOTS AND LOTS OF FAT.-- catherine Yes, we learned to eat cheesecake. Cheesecake is brain food. -- CarolynJohnston - 09 Sep 2005
IT IS!!!!!! -- CatherineJohnson - 09 Sep 2005
I'VE gotta post some of this article (EXCEPT I HAVE TO GET MY SINGAPORE MATH DESCRIPTION WRITTEN!) It's the SCI AMER explanation of fast food; it really is. (The reason for the high-fat, high-calorie food is that it's so energy dense you don't have to spend much energy to get it!) -- CatherineJohnson - 09 Sep 2005
When I first got out of college and had about $90 in my checking account, I actually put together an Excel spreadsheet to sort the calorie/dollar ratios of various fast-food items. Hey, I was an econ major. I'm all about optimizing marginal utility. -- IndependentGeorge - 09 Sep 2005
What'd you find out?? -- CatherineJohnson - 10 Sep 2005
And how big is your brain? -- CatherineJohnson - 10 Sep 2005
about the size of both your fists pushed together.
twice the size of your heart.
who moved my cheesecake? -- VlorbikDotCom - 10 Sep 2005
When I first got out of college and had about $90 in my checking account, I actually put together an Excel spreadsheet to sort the calorie/dollar ratios of various fast-food items I'm pretty sure that Kraft Macaroni and Cheese wins that award hands down, although I guess it's not technically fast food. -- CarolynJohnston - 10 Sep 2005
ramen noodles, baby! -- 25\cents a pack
and cooking is optional! -- VlorbikDotCom - 10 Sep 2005
-- CarolynJohnston - 10 Sep 2005
Well, I live in Chicago, so the leader was the 9-piece chicken nuggets meal from Harold's Chicken Shack (which provided somewhere in the neighborhood of 1200 kCal for $2.50). I think the mainstream fast food leader was the $3.25 Whopper Extra Value Meal (the Big Mac value meal in my neighborhood cost $3.95; I'm not sure why Burger King was so cheap, but I wasn't complaining). As for ramen - $0.25 per pack? Do I look like a Rockefeller to you? Maybe your fancy-schmancy, brand-name Ramen that came with the dehydrated vegetables cost $0.25 per pack, but I went with the generic, for $0.10 per pack! I could write an entire book on how to eat cheaply. I actually learned how to prepare a decent meal from ramen noodles. Basically, you add some frozen carrots/peas (8 oz bag for $1.09), about 1 tablespoon of soy sauce (if you buy it, i believe it comes out to around $0.10 per tbsp; I used the little packets that come with Chinese takeout), and a teaspoon of oil. I actually got fancy and used the rather expensive sesame oil, but it was worth it. The funny thing is, money's no longer so tight, but I still eat variations of the same theme. Replace the ramen with penne pasta, sesame oil with olive oil (surprisingly calorie dense - I think it's somewhere around 200 kCal per tablespoon), mince a clove of fresh garlic (maybe $0.05 per clove - at most), and add a touch of shredded parmesan, and you've got a great meal for about $0.75 for a generous 600 kCal serving. -- IndependentGeorge - 10 Sep 2005
Also, generic PBJ on Wonderbread. From memory, I believe it was $0.15 per 300 kCal serving. I think I need to create a new spreadsheet. -- IndependentGeorge - 10 Sep 2005
Ok, off-topic philosophical question: tenderloins are generally the best cuts of meat because they are worked the least, and therefore have less connective tissues amongst the muscle fibers. So when zombies go on brain-eating rampages, do they go after the people who think the least? Yet another reason to do well in school: in the event of radiological disaster, zombies will eat your brains last. -- IndependentGeorge - 10 Sep 2005