Kitchen.MathProblemHelpLine (r1.1 vs. r1.2)
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 <>  Difference Topic MathProblemHelpLine (r1.2 - 07 Jul 2005 - CarolynJohnston)

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# Entries from MathProblemHelpLine

ItTakesChops 07 Jul 2005 - 22:09 CatherineJohnson

It takes chops to solve this when you're eleven:

Two cars leave simultaneously at 9 a.m. heading toward one another from different cities that are 210 km apart. The average speed of one car is 50 km/h while the other car averages 70 km/h. Come up with an appropriate question and answer it.

This problem appears on page 5, 'Review,' of Mathematics 6: an award-winning textbook from Russia, by Enn Nurk and Aksel Telgmaa.

The 6 in the title stands for 6th grade.

+ + +

update: OK, I solved it.

But I couldn't think of a bar model.

Our Favorite Supplements
RussianMath
RussianMathPart2
RussianMathPart3
WhyILoveCarolyn
Mike McKeown comment

RussianMathPart3 07 Jul 2005 - 21:54 CatherineJohnson

I'm stumped.

The chainring (attached to the pedals) on a one-speed bicycle has 44 teeth while the freewheel (on the back wheel) has 20 teeth. Determine the least number of turns the chainring must make in order for both the chainring and feewheel to return to their original positions. How many turns does the freewheel make during this time?

This is a problem from Mathematics 6: an award winning textbook from Russia by Enn Nurk and Aksel Telgmaa, a 6th grade book.

I've prime factored each number, since that seems to be what's called for.

AnotherWikiPossibility 19 Sep 2005 - 23:07 CatherineJohnson

Another possibility for communal Wiki pages is to do something like the thread for RussianMathPart3: pose a problem or a lesson everyone can comment on.

I'm interested in comments on the fraction lesson J. D . Fisher has posted at Math and Text.

My immediate reaction to J.D.'s post is that it would be terrific for developing teachers' conceptual understanding of mathematics, and possibly for developing teachers' pedagogical content knowledge (pdf file).

But I wouldn't be able to teach it to Christopher, even though he does know that a fraction is (also) a division problem.

(I'll pull my thoughts together on this later--time for a bike ride now.)

I'd love to get other people's reactions.

KitchenTableMathIsAWiki
WikiHowTo

SingaporeWordProblemSampler1 14 Jul 2005 - 03:19 CarolynJohnston

This is a sampler of randomly chosen word problems from the "Primary Mathematics Challenging Word Problems" series. I'm just going to open the books, and write what I see.

For tonight's problem, pick one and give it to your kid, or try one yourself.

For bonus points, spot the problem that was not written by a native English speaker.

Grade 3: String X is 34 cm longer than String Y. String Z is 58 cm longer than string Y. If the total length of Strings X, Y, and Z is 233 cm, find the total length of Strings X and Z. (!!! OK, that was from the 'challenging problems' section of a 'challenging problems' book. Excuses, excuses)

Grade 4: Jane has 70 balloons. 1/10th of them are green, and 3/5 of them are orange. How many more orange balloons than green balloons does she have?

Grade 5: A man bought a dozen sacks of rice at \$18 per sack. Each sack of rice weighed 20 kg. He packed half of the rice into bags of 5 kg and sold them at 6.50 per bag. He sold the rest of the rice at \$1.50 per kg. Find his total profit.

Grade 6: Henry has 3/4 as many paper clips as Joyce. Joyce has 4/5 as many as Claire. If the three girls have 96 paper clips altogether, how many fewer paper clips does Henry have than Claire?

SingaporeWordProblemSampler2 14 Jul 2005 - 03:19 CarolynJohnston

Here's another random sampling of word problems from "Primary Mathematics, Challenging Word Problems".

A KtmGuest (henceforth known as 'Lone Ranger') left the following useful comment on the SummerProgramUpdate thread:

FYI...Singapore Math is organized differently than American elementary math textbooks. The book are arranged in this order 1A, 1B, 2A, 2B...6A,6B. When a student is finished with 6B, that student is ready to begin studying Algebra. Therefore the number on the book does not translate into an American grade level. In fact many people comment that children begin the Singapore program 1 number below their current grade. My child began with level 2B even though she was starting 4th grade.

(thanks, kemosabe).

So who knows what American grades these problems match up to? Just target the problem that suits your kid, and don't worry about whether they're behind what kids are doing in Singapore.

Primary 3: The capacity of a bucket is 9 qt. If 3 qt. 3 c. are added into the bucket, how much more water is needed to make it full?

(I like that last problem because it ties in with this recent post.)

And here is a rather strange one:

Primary 4: 5/9 of a box of chocolates are round, and 2/9 are square. How many more chocolates are round than square? Give your answer as a fraction.

Primary 5: Martin and Gary had 80 stickers altogether. After Martin gave away 35 of his stickers and Gary gave away 1/5 of his stickers, they had the same number of stickers left. How many stickers did Martin have at first?

Primary 6: Linda and Jane set off from City P to City Q at the same time. When Linda reached City Q, Jane was still 140 km away. 2 hours later, Jane also reached City Q. If Cities P and Q were 630 km apart, at what speed was Linda traveling?

SingaporeWordProblemSampler3 14 Jul 2005 - 03:20 CarolynJohnston

Note: solutions to the problems from SingaporeWordProblemSampler2 have been posted here.

So here's a whole new set of problems!

Primary 3: Margo has 3 times as many pears as apples. If she has 84 pears and apples altogether, how many pears does she have?

Primary 4: A cake was cut into 12 equal pieces. Jim ate two pieces and Tom ate four pieces. What fraction of the cake was left?

Primary 5: A bag of potatoes weighs 7/8 kg.. A bag of yams weighs 4/5 as much as the bag of potatoes. Find the total weight of the bag of potatoes and the bag of yams.

Primary 6: Eric has 75% as much money as Joshua. Carl has 60% as much money as Eric and Joshua have together. If Eric has 36 dollars less than Carl, how much money does Joshua have?

I don't know about you all, but I think I perceive these Singapore math problems becoming markedly harder at level 5.

SingaporeWordProblemSampler4 14 Jul 2005 - 03:20 CarolynJohnston

Here are solutions to the problems in SingaporeWordProblemSampler3.

As usual I am going to post a random sample from Primary Mathematics Challenging Word Problems, levels 3 through 6 -- but there are actually extra-challenging word problems in special sections, and today I'll put up some of those.

Primary 3: On Valentine's Day, a teacher gave 37 students 7 candy hearts each. If she had 3 boxes of 100 candy hearts each, how many candy hearts did she have left?

This next one shows how early Singapore Math starts to introduce algebraic word problems.

Primary 4: A farmer had twice as many ducks as chickens. After he had sold 413 ducks and another 19 ducks died, he had half as many ducks as chickens left. How many ducks did he have left?

Primary 5: Laura had 400 stamps. She gave 3/20ths of them to Sam, 5/16s of them to Joe, and 1/15th of the remainder to Jim. How many stamps did she have left?

And now the one we've been dreading.... AAAGH!

I would actually never give this one to a kid who wasn't really fond of puzzles. It's the only way to approach this problem.. fiddle around with it till you get a handle on it.

Primary 6: The ratios of the number of chairs to the number of tables in Halls A and B respectively are 5:2 and 6:1 respectively. The total number of chairs and tables in Hall B is three times that in Hall A. What is the ratio of the number of tables in Hall A to the number of chairs in Hall B?

AnnouncingSolutionsToSingaporeSampler4 17 Jul 2005 - 03:55 CarolynJohnston

This post is just a pointer to the solutions page for the SingaporeWordProblemSampler4, which consisted entirely of problems considered by the Singaporeans to be challenging.

Didn't want to leave anybody hanging if they're still paying attention! But a session of basic algebra with Young Ben has done me in for the evening. I'll be along soon with more Challenging Word Problems from Singapore -- but maybe not quite so challenging as these were.

QuestionAboutReciprocals 31 Jul 2005 - 18:04 CatherineJohnson

My 'benchmark' for the moment when I understand elementary mathematics well enough to move on is:

reciprocals

I find reciprocals utterly mysterious.

They're not quite in the magic category anymore, which is my benchmark for complete and total lack of comprehension. If a maths concept seems like magic, that means I know nothing.

Of late, inside the expanding math section of my brain, reciprocals have put a toe outside the magic category. But not much more than a toe.

### Danica McKellar

Tuesday's SCIENCE TIMES has a profile of Danica McKellar, who played Winnie on Wonder Days. It turns out she's a UCLA math major who published a proof (pdf file) now known as the Chayes-McKellar-Winn theorem.

### car wash problem

McKellar's web site has a mathematics section where she answers reader questions. She's a natural born teacher:

Q: Hi Danica, I heard a question from Mr. Feenie on a "Boy Meets World" episode which he claimed to be unanswerable. After hearing that, I decided to figure it out. If it takes Sam 6 minutes to wash a car by himself, and it takes Brian 8 minutes to wash a car by himself, how long will it take them to wash a car together?

Danica Answers: Hm, unanswerable? That's TV for you. :)

Let's do it: This is a "rates" problem. The key is to think about each of their "car washing rates" and not the "time" it takes them. Alot of people would want to say "it takes them 7 minutes together" but that's obviously not right, after you realize that it must take them LESS time to wash the car together than either one of them would take.

So, what is Sam's rate? How much of a car can he wash in one minute? Well, if he can wash one car in six minutes, then he can wash 1/6 of a car in one minute, right? (think about that until it makes sense, then keep reading). Similarly, Brian can wash 1/8 of a car in one minute. So just add their two rates together to find out how much of a car they can do together, in one minute, as they work side by side on the same car: 1/6 + 1/8 = 7/24 of a car in one minute. That's their combined RATE. (Note: that's a little bit less than 1/3 of a car in one minute). From this point, the way you want to think of it depends on your favorite way of dealing with fractions. You now have their rate. It's 7/24 cars per minute. You can either just take its reciprcal and say: 24/7 minutes for one car, and you're done.

Or, equivalently, you can think of the 7/24 cars/minute RATE as 24 minutes for 7 cars. (think about that until it makes sense, too) So just divide 24 by 7 to find out how many minutes it would take to do just one car. You get around 3.42 minutes for one car, just a little less than 3 and 1/2 minutes. Done! Yes, I think they should work together, it gets done much more quickly that way. :)

By the way, you said when you watched the TV show you decided that YOU would figure it out, right? How did you do?

I love this. McKellar is teaching two things here:

• how to solve a rates problem
• how to read, study, & learn maths (that's metacognition)

First of all, she knows that math novices transfer their normal reading habits to maths books. By normal reading habits I mean that most of us, when we read a book of prose, read straight through at a fast clip, pausing only to underline or make notes in the margin.

You can't read a math book that way; in fact, I've come to feel you can't really read a math book at all. You have to do a math book, or work a math book. McKellar explicitly instructs her reader not to read the solution straight through, but to stop at key points and ponder until he or she gets the point, and is ready to go on to the next point.

She doesn't stop there, either. She also knows the precise spots in her explanation where most novices will need to stop and mull, and she tells them where those spots are.

She's giving novices direct instruction in metacognition.

As to the problem itself, she addresses the most common error novices will make confronting this particular rates problem, which is:

• figure that it takes the 2 boys 14 minutes to wash 2 cars
• so logically it must take them 7 minutes to wash 1 car

Amazing! And all in the space of a few short paragraphs.

I think McKellar's teaching skill here is connected to her acting. There's a large element of 'performance' in teaching, at least in my experience, and to be a good performer you have to know where your audience is, what they want to hear & what they need to hear.

She does.

### back to reciprocals

Here's my reciprocal question.

From this point, the way you want to think of it depends on your favorite way of dealing with fractions. You now have their rate. It's 7/24 cars per minute. You can either just take its reciprcal and say: 24/7 minutes for one car, and you're done.

Or, equivalently, you can think of the 7/24 cars/minute RATE as 24 minutes for 7 cars. (think about that until it makes sense, too) So just divide 24 by 7 to find out how many minutes it would take to do just one car. You get around 3.42 minutes for one car, just a little less than 3 and 1/2 minutes. Done!

I don't understand why you would use the reciprocal to solve this problem.

I understand perfectly well (let's hope) why you would divide 24 by 7.

I didn't even know you could use the reciprocal to find the answer.

### 7 fact families

I haven't had time to sit down and think this through, but I suspect the reciprocal answer to a rates problem is the same concept as the 7 fact family I put together after teaching the Primary Mathematics lesson on ratio & proportion (Primary Mathematics 6A Textbook, p. 21-46):

7 fact families

MathTalkInTheCar 01 Aug 2005 - 16:50 CarolynJohnston

We took the kids to a bar tonight, as it happened. Colin (17) is into playing the bass these days; he has a band that he plays with during the school year. I have a friend at work who is a hot guitar player and who just joined a classic rock band, and he was playing his first gig tonight, and they were letting kids stay through the first set, so we went to see him. It was a long drive for us -- all the way out to Greeley. The place was an authentic roadhouse with motorcycles parked out front, and the food was good -- it was Cajun food, and very authentic given that we were not in Cajun country but in Greeley, Colorado, home of the Feedlot You Can Smell All The Way To Denver.

On the way home, Colin asked us about the difference between the median, the mean, and the mode of a data set, and what each of them is good for. This is, of course, the sort of thing we love to pontificate about. He then told us that he felt he had never really quite gotten the idea of a function, and asked us to explain it.

It's a smart kid who understands what he doesn't understand. Most adults can't do that very well.

Actually, most kids coming into calculus classes are confused by functions. A function is just a black box; you put in an input, and get out an output. What makes it a function is that, when you put in the same inputs, you always get the same outputs. You can't put the same number in the black box and get 2 one time, and 5 the next.

Most texts teach functions using formulas to define the functions; all the functions kids see look like f(x)=3x-5, or g(x)=x/6. But functions don't have to have formulas to go with them; they can defy description by a formula. The only rule is that if you put in the same input multiple times, you get the same output, every time.

The reason kids confuse formulas with functions is that it's hard to define functions that don't use formulas, even though in real life we encounter them all the time. When a function totally defies description with a formula, we often resort to trying to describe it with only a couple of numbers, such as the mean, median, and standard deviation (this is how the whole field of statistics arises).

We played a 'figure-out-the-function' game on the way home from Greeley. Bernie and I would think of a function, and Colin and Ben would give us numbers for inputs, and we would then tell them the output. They'd then try to guess the formula we were using to define the function.

They are both aces at extracting patterns. If anything, Ben would try to generalize from too little data; once he guessed, after one try, that the function was 'add 2'; he'd given me a 2, and I'd come back with 4 (the function I'd thought of was squaring; he got it on the next try). Bernie was giving Colin some functions that are so simple they trip up students with their obviousness, like the function that returns the same number you give it, and the one that returns '3' no matter what you give it. He gave Colin one function that was so bizarre you can't describe it with a pattern.

Ben knew more about functions than I thought, even piping up with "that's the constant function 3" at the appropriate moment. Did they do functions one day for 5 minutes in Everyday Math? Well, he was definitely on the ball that day.

HowToTeachGCFAndLCM 03 Aug 2005 - 16:23 CarolynJohnston

I don't know what the blockage was last week (when I wrote this), but we seem to be cruising again.

It doesn't hurt, of course, that the chapter in Prentice-Hall Mathematics Course 1 that we're doing is something that Ben basically already knows. He did prime factors last year in Everyday Math, using 'factor trees', and learned to stumble through a guess-and-check process for finding the least common multiple of two numbers. So all I have to do is to make sure he's retained the concepts, and to teach him an efficient way to get the answer right every time. Tonight we did efficient prime factorization, and efficient greatest common factors.

In Prentice Hall, as in most math texts, greatest common divisors and least common multiples are taught back-to-back. My recollection of my own school years is that, with the two concepts taught so closely together, and the methods for doing both calculations being so similar, it was easy to get pretty confused about them. That's what I want to avoid. Anyone teaching this topic should (I think) try to clarify the two concepts explicitly.

I think that when you have to teach a tricky topic like this, a really juicy example is typically better than a lot of explanation.

Here is a juicy example. Take 22 and 34. Here are their prime decompositions:

22 = 2 11
34 = 2 17

This is a good example to illustrate the difference between GCF and LCM. There is only one common factor (2), and it's obviously also the greatest one. It's easy to show that any common multiple of both numbers has to have a 2, an 11, and a 17 in it, and the least one is the product of those numbers.

It's a nontrivial example, but it's got no distractions in it. Each number has only two factors, there's no powers of primes in there to trip up a kid, and the GCF is itself a prime. It's easy to generate other examples like this: 15 and 21, for example, is a similar one. You can offer different examples till the kid gets comfortable with the difference.

The next step is to offer an example such as:

30 = 2 3 5
42 = 2 3 7

We've made it a little more complicated, but there are still no powers of primes; all you've done is to add one more prime factor in common, to make the GCF a very simple composite number (6). You would explain to the kid that, this time, there are several common factors; and the GCF is the largest of them.

Here's the approach I taught Ben tonight for finding the GCF. First write each number's prime factorization out, without powers. For example:

24 = 2 2 2 3
20 = 2 2 5

If you find a prime factor in common in both of them (I told him), scratch it out in each expression, and write it somewhere else. So, for this example, Ben would scratch out a 2 in each factorization, and write it elsewhere. Then he would scratch out a second 2 from each factorization, and write 2 next to the first 2.

Then he'd only have 2 3 left from the original factorization of 24, and 5 left from the factorization of 20. There are no factors left in common, so he's done; and he's got the GCF written down (the 2 2s).

I pointed out once again that certainly 4 is a common factor for this example, and there are none bigger.

More tomorrow on how to finish teaching GCFs and LCMs.

HotMath 14 Aug 2005 - 15:09 CatherineJohnson

Thanks to Dan K, I've found a fantastic resource:

[Hotmath provides] explained solutions to the odd-numbered homework problems from most of the popular secondary math textbooks used in California. Thus, teachers can now assign practice problems for homework where teacher-prepared, explained solutions are instantly available, and can mix in even-numbered problems for challenges. Students who do not need to see the worked solutions needn't bother, and students who might abuse the availability of worked solutions will be tested on the even problems.

Here is a sample worked-out problem: algebra problem

And here are the 2 critical paragraphs from the Hotmath 'white paper'. I've begun to come across these studies elsewhere, and I'm inclined to trust these summaries, in part because this discussion jibes with my own experience re-learning maths:

Providing students with worked out examples of math problems has been found to be more effective than simply assigning the same problems for the students to work out on their own. In one experiment (Carroll, 1994), 40 high school students were instructed in how to solve linear equations. In an “acquisition phase” the students were divided into two groups and their instruction differed in the following way: in the “conventional learning” group, students were assigned 44 unsolved problems to work out (in the classroom and at home homework), and in the “worked examples” group students were provided with the same problems, but half of the problems were accompanied by correct solutions. After completion of the assigned problems, both groups were tested on 12 related problems, 10 of which were very similar to the linear equations presented in the acquisition phase, and 2 of which were word problems, used to test whether students could transfer and extend their knowledge to a new context. No worked out examples were available during the test. The test results revealed that students in the “worked examples” group outperformed students in the “conventional learning” group on both types of the test problems. A second experiment, employed a similar methodology but focused on “low achieving” students (students with a history of failure in mathematics, and students identified as learning disabled). Here, the data revealed that students in the “worked examples” group required less acquisition time, needed less direct instruction, made fewer errors, and made fewer types of errors than students in the “conventional learning” group.

Related research (Pass & Van Merrienboer, 1994) sheds light on the cognitive underpinnings of the effects described above. In this study, 60 college-aged students were instructed in geometry concepts. As in the Carroll experiments, students were assigned un-worked problems to solve or worked out examples to review (unlike the Carroll study, the “worked examples” group was assigned no un-worked problems to solve). In this study, the researchers manipulated the nature of the problems presented to the students: within each group, some students received problems which were all similar to each other, while others received a more varied problem set. Furthermore, the researchers measured the “cognitive load” experienced by the students. This research revealed that while students in the worked examples group completed their work more quickly, they perceived the work as less demanding and displayed better transfer performance at test. The effect was most pronounced for the students given highly-variable problems. The researchers suggest that the reduced cognitive load associated with the worked examples enabled students to “take advantage of” the variability in problems by using the available cognitive resources to process the underlying similarity in the problems (i.e., the mathematical concepts being taught), and to integrate the current problem with existing knowledge (Linn, 2000).

The site covers Prentice Hall Pre-Algebra, the book Christopher will be using in the fall, so I'm going to subscribe. Cost is \$49 for 12 months.

I think it's going to be fantastic for Christopher to have an answer source that isn't His Mother.

Especially since it looks like I'm going to have to start some heavy-duty Writing Instruction this year. (That's another story.)

This is going to be an important term for me. It perfectly captures what it is we're trying to do when we push our kids to practice to the point of automaticity.

We're trying to reduce cognitive load.

### update

I've just re-read Dan's original post, and I don't see a reference to hotmath. hmmm. Maybe one of the sites he mentioned pointed me to hotmath. In any case, I'm recommending hotmath, not Dan. (He'll let us know what he thinks, I'm sure.)

StatisticsHelp 19 Aug 2005 - 17:51 CatherineJohnson

I mentioned that Temple and I are writing an op-ed on U.S. high schools.......and I'm stumped by a statistical issue.

What does it mean to say that multivariate analysis shows that a certain factor is highly predictive of a particular outcome, while another factor is less predictive?

What does this form of analysis imply about causality, if anything?

I ask because of an apparently highly influential government report published in 1999: Adelman, C. (1999). Answers in the Tool Box: Academic intensity, attendance patterns, and bachelor’s degree attainment. Washington, D.C.: U.S. Department of Education.

This study finds an extremely high correlation between rigor of high school curriculum and students' eventual completion of a college degree--far higher than the correlations with the factors we're used to hearing about, such as parents' level of educational attainment, socioeconomic status, and race (and substantially higher than high school GPA and SAT scores).

The report itself, which I've barely skimmed, as well as other accounts of it, seem to imply that the relationship is causal. It's not that being smart and motivated in the first place causes a student to take a more rigorous high school curriculum and attend and complete college, but that the more rigorous high school curriculum sets him or her up to succeed in college.

from the American Educator:

Academically well-prepared students are likely to graduate from college regardless of their social background. Unprepared students of all backgrounds are not likely to do so.

The graph below breaks students into quintiles based on their level of academic preparation and their socioeconomic status (SES). As you can see, among the lowest SES students, a bachelor’s degree was earned by 62 percent of those who were well prepared, but only 21 percent of those who were not. Similarly, among the highest SES students, 86 percent of those who were well prepared--but only 13 percent of those who were not--earned a bachelor’s degree.

Percentage of students who graduated from a four-year college by socioeconomic status (SES) and academic preparation.

key words: rigorous high school curriculum predicts graduation from college
how can you tell whether A caused B?
low birth weight paradox
how good are our best students?
statistics and law

StopMakingSense 16 Aug 2005 - 03:31 CatherineJohnson

When I was looking for pictures of teenagers, I found a Teenage Leisure Time project I can't figure out.

Do these charts say what the student says they say?

I can't see it, but maybe when I have a clear head again I will.

For now, my question is: how do these 4 pie charts tell us that people who have Sky TV and people who live in villages watch more TV than people who don't have Sky TV and don't live in villages? (And shouldn't 'sky' be capitalized?)

Plus, I don't see how the pie charts address the assignment:

Pupils were asked to produce a questionnaire to look at student leisure time and come to a conclusion about whether students tend to just watch tv or actually participate in activities.

I'm confused.

This is grim:

The pupil has selected information (data) they need, checked its accuracy and organised it into a suitable form for processing - a database. The student has also consolidated level 4 work as they have understood the need for care in framing questions when collecting, finding and interrogating information. They have interpreted their findings and questioned its plausibility. The use of questionnaires in the outside world was also discussed. The report was also an example of level 4 in that the Communicating Information element involved data from another "variety of sources"

To make further progress the pupil could be given opportunities to develop complex searches and test hypotheses.

Is there any subject-verb agreement here at all?

The Brits have always had unbelievably brilliant writing (and speaking) instruction. I've been trying to find sources on how they do it, and have come up with only constructivist stuff, including something called silent grammar, in which students use Cuisinaire rods to discover grammatical rules.

But where does silence come into this? Gattegno suggested that the teacher, to maximise the learning, only presents the model for language point once and once only. The teacher then uses the tools at his/her disposal to elicit the correct language from the learner. The teacher should be encouraging and open, using body language and facial expressions to show that the language produced is correct or incorrect, as well as correcting the learners using the tools outlined above.

So, silent grammar. Sounds like charades. The teacher waggles his eyebrows, mugs, and flings his arms about, but keeps his jaws clammed shut.

Of course, if British teachers talk like the one I just quoted, it's probably just as well.

### update

Ed says he's always suspected that the students in Britain who learn to write brilliantly are elite students at Oxford & Cambridge.

I have no idea.

FalsePositives 12 Sep 2005 - 03:15 CatherineJohnson

A couple of days ago, Carolyn explained the difference between frequentist statistics and Bayesian. She's a Bayesian, she said.

Well, that explained a lot, because it turns out I'm a Bayesian, too. I just didn't know it. Obviously, that's why Carolyn and I constantly find ourselves traveling the exact same thought path, even though we've never met, and didn't know each other until a year ago.

Of course, a real Bayesian (that would be a Bayesian who knew what she was doing, which would not be me) would probably not conclude that the reason she likes a person well enough to start a vast time-gobbling math-ed web site with her is that you both subscribe to the same school of statistical thought. I'll have to ask Carolyn.

I'm a Bayesian aspirant.

I'm having quite a little midlife run of Self-Discovery here, I must say. First I find out I'm Scots-Irish; next I'm hearing I'm a Bayesian.

I just hope no one's gonna tell me I'm adopted.

### I have a question

My question concerns a passage in a terrific book called What the Numbers Say: A Field Guide to Mastering Our Numerical World by Derrick Niederman & David Boyum. Boyum, it turns out, majored in applied mathematics at Harvard--I didn't know there was such a thing as a major in applied mathematics at Harvard!

Or anywhere else, for that matter.

I wish I'd know that when I was 17.

'Bayes Watch' is Niederman & Boyum's title for this passage:

Years ago a study asked the following question of students and doctors at Harvard Medical School:

If a test to detect a disease whose prevalence is 1/1000 has a false positive rate of 5 percent, what is the chance that a person found to have a positive result actually has the disease, assuming that you know nothing about the person's symptoms or signs?

Ed and I both understand the answer now (neither of us got it right), but we still have a question about the precise calculations. (Don't hit this link unless you want to see the answer.)

### update

I've just checked Niederman & Boyum. They do not specify a zero rate for false negatives. They say nothing about false negatives one way or the other. (Neither does John Kay in false positives, part 2, assuming I'm understanding him correctly).

### Bayes & God

I actually bought this book a couple of years ago, though I haven't read it yet:

I believe it's intended to be a Bayesian proof of the existence of God, although I don't know how the word 'proof' is used either in the book or in the context of Bayesian statistics.

low birth weight paradox (& Monty Hall)
Monty Hall, part 2
Monty Hall, part 3
false positives
false positives, part 2
Doug Sundseth on Monty Hall
John Kay: We are likely to get probability wrong (subscription only)
Monty Hall diagram from Curious Incident
Bayes & the human mind
Bayesian reasoning, intuition, & the cognitive unconscious
most bell curves have thick tails
ECONOMIST explanation Bayesian statistics
Bayesian certainty scale
probability question from Saxon 8/7

Bayesianprobability

FalsePositivesPart2 21 Dec 2005 - 15:31 CatherineJohnson

Another version of the False Positives challenge. This one ran in John Kay's column in the Financial Times yesterday. (Probably only available to subscribers.)

...intuition does not correspond to the mathematics of probability. One person in a 1,000 suffers from a rare disease. A friend has just tested positive for this illness and the test gives a correct diagnosis in 99 per cent of cases. How likely is it that your friend has the disease? Not at all likely. In random groups of 1,000 people an average of 10 would display false positives and only one would be correctly diagnosed with the disease. But most people, including most doctors, think otherwise. “The human mind,” said science writer Stephen Jay Gould, “did not evolve to deal with probabilities.”

Hmmm. Let's see. This problem does give us false negatives, right???

OK, let me think.

[pause]

Good grief. Not only can the human mind not intuit Bayesian probability; apparently the human mind equally cannot produce consistently lucid prose. (Nothing wrong with Mr. Kay's lucidity on a normal day.)

Kay's example, too, appears to assume a false negative rate of 0.

As far as I can tell.

### update

This is funny. I was skimming Amazon reviews of Stephen Jay Gould's Mismeasure of Man, and I found this:

As Oxford academician Richard Dawkins says (see Bryson, "A Short History of Nearly Everything", pp. 330-332) "If only Stephen Gould could think as clearly as he writes!"

It's a Core Principle in the Writing Biz (& definitely in the Writing Instruction Biz) that you can't write clearly without thinking clearly. (True in my experience; that's for sure.)

low birth weight paradox (& Monty Hall)
Monty Hall, part 2
Monty Hall, part 3
false positives
false positives, part 2
Doug Sundseth on Monty Hall
John Kay: We are likely to get probability wrong (subscription only)
Monty Hall diagram from Curious Incident
probability question from Saxon 8/7

MontyHallPart2 17 Aug 2005 - 21:49 CatherineJohnson

Here is Kay on the Monty Hall problem:

The Monty Hall problem is named after the host of a 1970s quiz show, Let’s Make a Deal. The successful contestant chooses from three closed boxes. One contains the keys to a car and the other two a picture of a goat. The choice made, Monty opens one of the other doors to reveal – a goat. He taunts the guest to change the decision. Should the guest switch to the other closed box?

When the solution was published in an American magazine, thousands of readers – including professors of statistics – alleged an error. Paul Erdös, the great mathematician, reputedly died still musing on the Monty Hall problem. But the answer is, indeed, yes: you should change.

I'm happy to hear that Paul Erdos stumbled over Monty Hall, seeing as how I still don't understand it.

low birth weight paradox (& Monty Hall)
Monty Hall, part 2
false positives
false positives, part 2
Doug Sundseth on Monty Hall
John Kay: We are likely to get probability wrong (subscription only)
Monty Hall diagram from Curious Incident
probability question from Saxon 8/7

DougSundsethOnMontyHall 18 Aug 2005 - 19:52 CatherineJohnson

Doug Sundseth (welcome, Doug!) posted an explanation of the Monty Hall problem, which I have never been able to understand:

It took me a long time to understand it, too.

The model that finally worked for me was something like this:

You have a 1/3 chance of being right to start with, and a 2/3 chance of being wrong. If you guessed wrong originally, Monte's pick will unambiguously determine the correct choice (he never picks the good door).

There are nine pairs of (your pick):(correct pick), A:A, A:B, A:C, B:A, B:B, B:C, C:A, C:B, and C:C. In three of those, you picked correctly, Monte's information isn't useful, and you shouldn't switch. In the other six, you picked incorrectly and Monte told you which of the other picks was correct; thus you should switch.

If you never switch, you have three chances in nine of being correct. If you always switch, you have six chances in nine of being correct and three chances in nine of switching off the correct choice.

Note that the latter possibility (choosing correctly at random then switching to an incorrect choice) may be more psychologically painful than just guessing wrong and not switching. This may have an undue effect on the choices of contestants.

Doug, thank you!

OK, I've just sat down and quickly thought this through.

[pause]

On my initial reading, I think it makes sense to me. What's particularly useful, for me, is the information that, yes, you could already have chosen the correct door, in which case, if you change your choice, you have moved to the incorrect door.

I think people who haven't studied probability get hung up on the 'what if I'm already right' issue.....and then, when math-savvy people try to explain Monty Hall without addressing, as Doug has, the issue foremost in their minds, the explanation doesn't 'take.'

### metacognition again

I mentioned awhile back that metacognition is a huge issue amongst constructivists, both of the radical & the peer-reviewed , department of psychology cognitive science constructivists.

One of the main reasons for thinking about metacognition as you teach is that students may very well bring quite wrong ideas to the classroom, which they then 'build upon' as they acquire new knowledge. There's a lovely example of this in the National Research Council's book on learning. Many children, when told that the earth is round, picture it as a disk, not a sphere. (more t/k--I need to go take a look at these pages.)

In any case, Doug has addressed an aspect of metacognition that I haven't seen mentioned, which is to tell a student what it is they already know that's right, but incomplete.

I was having the same experience yesterday, puzzling through the 'false positives' problem. The objection both Ed and I were bumping our heads against--if it's 1 in 1000 and 50 in 1000, how can you ever have 1000???--was right; we just weren't seeing what to do about it.

I wonder how often it's the case that an incomplete right answer is the problem, as opposed to a Total Crackpot Misconception that has to be stomped out, obliterated, and disappeared without a trace before a person can learn Thing One about math? (And does this wording give you a feel for the challenge involved in attempting to re-learn elementary math in midlife?)

Or, as Steve H says, A little knowledge goes way too far.

### a new question

This sentence confuses me:

You have a 1/3 chance of being right to start with, and a 2/3 chance of being wrong. If you guessed wrong originally, Monte's pick will unambiguously determine the correct choice (he never picks the good door).

[pause]

hmmm. Interesting. Reading this again, it makes sense.

I'm going to take a paper and pencil break, and see what I come up with working through Doug's explanation myself.

I love it!

### back again

OK, paper and pencil session complete.

I do understand this explanation, with one question: the funky, counterintuitive odds are created by the fact that Monty always opens the wrong door, correct?

That's why you shouldn't go with the 50-50 answer everyone automatically does go with--yes?

Carolyn was telling me the other day that a lot of Bayesian statistical results are counterintuitve (hey! just like the Bayesian proof of the existence of God!).

That's for sure.

### other explanations

Here's a strictly mathematical explanation that will work for some people (and actually works OK for me....although frankly Doug's list helps move me a bit towards 'getting' the Monty Hall problem at a more intuitive level...):

After you pick but before you open any doors, there's a 1/3 chance that you've picked correctly, and a 2/3 chance that you've picked wrong. Assuming that the host can open doors, but can not move prizes, nothing that the host does will change the probabilities described above.

Now the host opens one of the doors, and there's nothing behind it. There's still a 1/3 chance that you've picked correctly, and a 2/3 chance that you've picked wrong. This means that the remaining door has a 2/3 chance of being correct.

This explanation helps me formulate exactly what it is that goes wrong for people: the chance to change your pick seems like a second event, with a second set of probabilities attached.

question: So how often does this happen in life?

How often do we perceive second events where we ought to perceive a continuation of the first?

### update: an intuitive approach to Monty Hall that might work

I'm going to have to live with the Monty Hall problem for awhile....

But here's an interesting approach to rendering the answer intuitively correct:

It was a while ago that I accepted the idea that switching doors was the correct play every time because it improves your chances of winning, but I had trouble convincing my friends that it was the correct answer. However, a friend of mine just came up with this explanation that I think should really make it obvious.

Let's say that you choose your door (out of 3, of course). Then, without showing what's behind any of the doors, Monty says you can stick with your first choice or you can have both of the two other doors. I think most everyone would then take the two doors collectively.

Unfortunately, I don't think this works for me...

### update: Keith Devlin's better version

OK, I think what the person above was trying to say was this:

...one last attempt at an explanation. Back to the three door version now. When Monty has opened one of the three doors and shown you there is no prize behind, and then offers you the opportunity to switch, he is in effect offering you a TWO-FOR-ONE switch. You originally picked door A. He is now saying "Would you like to swap door A for TWO doors, B and C ... Oh, and by the way, before you make this two-for-one swap I'll open one of those two doors for you (one without a prize behind it)."

I agree. Anyone told at the outset that he can pick one door or he can pick two doors would pick the two.

### I give up

from Keith Devlin:

... suppose you are playing a seven door version of the game. You choose three doors. Monty now opens three of the remaining doors to show you that there is no prize behind it. He then says, "Would you like to stick with the three doors you have chosen, or would you prefer to swap them for the one other door I have not opened?" What do you do? Do you stick with your three doors or do you make the 3 for 1 swap he is offering?

OK, I'm switching doors.

But I'm doing so purely on the basis of 4/7 being greater than 3/7. Nothing common sense about it.

Of course, given that my family motto is no common sense-y, it's easy to dump my first pick and jump to Door Number Seven!

low birth weight paradox (& Monty Hall)
Monty Hall, part 2
Monty Hall, part 3
false positives
false positives, part 2
Doug Sundseth on Monty Hall
John Kay: We are likely to get probability wrong (subscription only)
Monty Hall diagram from Curious Incident
probability question from Saxon 8/7

BadMathInEverydayMath 22 Aug 2005 - 02:26 CatherineJohnson

I've just noticed that a ktm guest left a comment on something I'd wondered about myself:

Worse yet, the math is wrong. It's the usual mixup of percents with percentage points.

Look at #5. Food is about 80%, Dandruff about 10%, but 80% is not 60% greater than 10%. 60% of 10% would be 6%. 80% is eight times bigger than 10% or 800%

Here's the original problem:

I'm sure it will come as a shock to no one that I was never taught how to compute a percent increase or decrease; nor was I taught, as far as I can remember, what the question 'How much greater is the percent of men who are willing to alert strangers to smudges on their faces than the percent of women who are willing to do so?' actually means.

As a direct result, I managed to spend my entire adult life utterly confused about the Ultimate Meaning of news stories on Percentage Increase in Federal Spending On Education and the like.

No more! Thanks to Algebra to Go & Russian Math, I now know what both questions mean, and how to answer them, at least in theory. By which I mean that percent increase/decrease and how-many-times-bigger still hold the status of New & Tenuous inside my head.

Yes, I could demonstrate both on a Pop Quiz right this minute.

But I'm not confident I'd be right.

So when I read this question, my first thought was: 10 percent?

Then I thought, Hunh.

I only had to stare at the problem a couple seconds more to arrive at the conclusion that, OK, we're not talking percent increase here.

Which was too bad, because of the two ideas, that's the concept I know best. The idea of how many 'times' bigger (or smaller) one number is than another is something I first learned literally one or two weeks ago. (I know; it's mortifying.)

So 'how many times bigger?' is very new knowledge for me, new enough that I figured the folks at Everyday Math must KNOW.

### I give up, again

I don't know how to figure this. I do think the E-Math folks are asking for a simple subtraction of one percent from the other.

But is that the right way to figure how much larger one number is than the other in this case?

Or would we want to know 'how many times larger' one number is than the other. (I'm thinking I've seen numerous reports and articles in which a simple subtraction was used.....Help!

Well, let's just hope all this confusion will help me understand students' confusion...

### update: Anne Dwyer on the mathematical meaning of words

This is a very interesting problem because it involves interpreting the mathematical meaning of words.

The problems did not ask how many times greater or even percentage increase. The problem asked "how much greater". The author of the problem obviously intended that the student would interpret this as a simple difference problem by subtracting one percentage from another. The author intended that the student would treat each percentage number as a number with the same units that can be subtracted. The real difficulty comes when you write out the answer: 60% greater. The mathematical meaning of the answer is very clear: 60% greater means that you increase something by 60%. This is not what was done in this problem. The answer is incorrect because you cannot use the word "greater" as units to an answer the the problem.

sexism in Everyday Math

EverydayMathThread 19 Aug 2005 - 16:21 CatherineJohnson

Terrific thread about an Everyday Math problem in mistake in Everyday Math?

Welcome Independent George!

MathAndLanguagePart2 22 Aug 2005 - 18:26 CatherineJohnson

Carolyn M is back!

So, while I have her attention, I'm posting my new Russian Math question.

This is from a chapter section called:

### The Quotient of Two Numbers

1. A quotient greater than 1. ... In this case the quotient shows how many times larger the dividend is than the divisor. The number 8 is 1.6 times bigger than the number 5.

2. A quotient equal to 1. ... In this case, the divisor and the dividend are equal numbers.

3. A quotient less than 1. In this case, the quotient shows what part of the divisor the dividend is. And so, the number 6 is 3/4 of the number 8.

This was, believe it or not, the first time I had ever seen a quotient greater than one defined as how many times larger the dividend is than the divisor. Wonderful!

However, I was then surprised to find that a quotient less than one was not defined as how many times smaller the dividend is than the divisor.

Here's my question.

Is it incorrect to say that a quotient less than one tells you how many times smaller the dividend is than the divisor?

Or is it that the authors view the definition they give (what part of the divisor the dividend is) as more important or more fundamental or more profound?

If it is incorrect to say that a quotient less than one tells you how many times smaller the dividend is than the divisor, why is this incorrect?

Thanks!

### Russian Math for everyone

While I'm on the subject, I'll add that the Russian Math book is pure pleasure. It's incredible. I think every 6th grade child in the country should study this book.

And every parent.

### update

When I say this is the first time I've ever read that 'the quotient shows how many times larger the dividend is than the divisor,' I don't mean that I've never heard expressions like '6 is 2 times larger than 3.' I have!

What I mean is that I've never seen a direct, explicit, comparative distinction drawn between 'a quotient larger than 1' and 'a quotient smaller than 1.' I've never seen (I don't think) 'a quotient larger than 1' defined as having a separate & distinct meaning.

This is a standard technique throughout RUSSIAN MATH, which I find incredibly powerful, and which I haven't noticed in U.S. textbooks (or, I think, in the Singapore series).

The RUSSIAN MATH book constantly teaches through difference, which in this case means 'disaggregating' a concept that normally stays aggregated in a U.S. textbook.

In a U.S. textbook--at least in the ones I've used--a quotient is a quotient.

In a 6th grade Russian textbook, all of a sudden a quotient is 3 different things. It's a remarkable book. Incredible.

Another thing: I've never seen a 'quotient larger than one' defined as anything other than 'how many times the divisor goes into the dividend' or 'how many of the divisor are in the dividend.'

Last, but not least, I don't think I ever was made conscious of 'decimal' or 'fractional' phrases like '1.6 times larger.' 'Two times larger,' sure. '1 '1/2 times larger,' yes.

But '1.6 times larger'--never.

This is important, because it looks like we're hardwired to understand 'friendly fractions.' Five year olds know what one-half is.* Five year olds do not know what 6-tenths are, and they certainly do not know what 1 and 6 tenths are!

Morever, our brains do not automatically or easily generalize from 'one half' to '1.6.'

This is another reason to object to the exclusive focus on friendly fractions in constructivist curricula. Friendly fractions are the fractions children know without having to be taught.

*Hunting, Robert P. (1999). Rational-number learning in the early years: what is possible?. In J. V. Copley. (Ed.), "Mathematics in the early years", (pp 80-87). Reston, VA: NCTM.

What Counts: How Every Brain is Hardwired for Math, by Brian Butterworth
The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene
Children's Mathematical Development: Research and Practical Applications by David C. Geary
(fyi: It is possible to buy Geary's book for far less than the \$124 Amazon wants for it, or the \$55 I paid for a used & extensively highlighted copy...)

Carolyn on math and language 7-2-05
Carolyn on math and language again 7-3-05
the language of numbers is not language 7-3-05

FactoringNumbers 09 Sep 2005 - 02:25 CarolynJohnston

A factoid: When you're trying to determine whether a number is composite (i.e., not prime), it's sufficient to check whether all the primes less than the square root of the next greatest square number divide it. If nothing less than the next-larger whole number square root divides the number, then you know the number is a prime.

For example, to check whether 221 is prime, all you have to do is test all the prime numbers less than 15. To check whether 133 is prime, check the prime numbers less than 12.

My question: how do you teach the reasoning behind this rule? I was tackling this tonight with Ben, and I wasn't getting it across very effectively. I tried saying that whenever a number factors into two numbers, one must be less than and one must be greater than the nearest square root. This didn't really click for him.

Next I tried drawing rectangles to demonstrate the idea. I did 36, and showed the special square rectangle that you can make with that area. I asked him what other rectangles you could draw with that area; we drew 4 by 9, 12 by 3, 18 by 2, and so forth.

I then pointed out that all the rectangles we'd drawn that weren't square had one number less than 6 (the square root) and one greater than 6. I claimed that this is always true. He then said "Oh, I get it!" but I think he's learned that I like it when he says aha, and that it wasn't a genuine aha. It's hard to fake these things.

Anybody have any ideas on how to teach this? I think one problem is that square roots are still a pretty vague concept at this age, and I wonder if this is a trick that 6th grade kids are generally taught. But if they're not taught this trick, then where are they taught to stop checking for divisibility?

MathLessonRepeatingDecimals 22 Sep 2005 - 20:01 CatherineJohnson

My neighbor showed me this yesterday. Naturally no one had ever taught me how to do this, which is par for the course. But she's a statistician & she'd never learned it, either.

I love this. It reminds me of the shenanigans I go through trying to force Microsoft Word to do graphic design.

I've entered this on the Math Lessons page.

other resources

Purple Math

Math Wizz on converting repeating decimal to fraction

update: Saxon meltdown (3-2-06)

Maybe I'm just tired, but I practically had a nervous breakdown tonight trying to convert 0.013333....(repeating decimal) to a fraction.

I just could not get it.

Finally Math Wizz saved me. Of all the websites I looked at, Math Wizz had the simplest, cleanest, & most follow-able explanation.

Math Wizz also has gigantic gifs.

SaxonAlgebraPlacementProblem 11 Oct 2005 - 00:48 CatherineJohnson

I absolutely can't get the answer Saxon gives in the answer key.

Thanks!

keywords: Saxon algebra placement test problem

AsianBrainTeaser 01 Oct 2005 - 17:50 CatherineJohnson

Just got this email from my sister:

FOLLOW THE DIRECTIONS BELOW FIRST
"Everybody has to cross the river." The following rules apply:

• Only 2 persons on the raft at a time.

• The father cannot stay with any of the daughters without their mother's presence.

• The mother cannot stay with any of the sons without their father's presence.

• The thief (striped shirt) cannot stay with any family member if the Policeman is not there.

• Only the Father, the Mother and the Policeman know how to operate the raft.

• To start click on the big blue circle on the right.

• To move the people click on them. To move the raft click on the pole on the opposite side of the river.

everybody has to cross the river

Seeing as how my day is not off to a brilliant start (I completely forgot Jimmy's 'Community Awareness' program, which started at TEN)....I'm not going to add further evidence of stupidness to Ye Old Cognitive Load by attempting this problem. Not today, anyway.

Today, I'm sticking to integers. And harvesting the basil plants to make pesto to freeze for the winter. And maybe, finally, emptying out the overflowing Mail Basket.

And forcing Christopher to do his edhelper.com worksheets. (A great tip from Susan. \$20 a year, and almost certainly worth it--definitely worth it for me, I should say.)

### help wanted

Anyone know of a good source for integer word problems?

RussianMathProblem961 05 Oct 2005 - 17:19 CatherineJohnson

961.

A point with a coordinate of -3 moves along the number line in the following manner: First, it goes 5 units in the positive direction; Then it goes 7 units in the negative direction followed by 10 units in the positive direction and 8 units in the negative direction; Then it goes 3 more units in the negative direction and, lastly, 13 units in the positive direction.

Question: What is the final location of the point on the number line?

source:
Mathematics 6 by Enn R. Nurk and Aksel E. Telgmaa, page 255

RussianMathProblem976 09 Oct 2005 - 22:14 CatherineJohnson

976.
A rectangular park is 400 m longer than it is wide. The ratio of the length of the park to its width is 5:3. How long will it take someone walking at a rate of 2 km/h to go around the park?

### question

How would a child work this problem without using algebra?

Here's the way it's done using a bar model, but I'm not seeing how you would do this without simple arithmetic.

(I hate the way this looks. I need Quark.)

Anyway, using the bar model you see that you have a bunch of equal units, and that two of these units equal 400.

Therefore, each individual unit equals 200.

So the width has to equal 3 x 200, or 600 and the length has to equal 5 x 200 or 10,000.

### ratio problem

Actually, now that I think about it, at this point in RUSSIAN MATH, kids have learned ratios & proportions, so they could just solve it that way.

width = w
length = l = w + 400

5/3 = l/w

5/3 = w + 400/w

What I can't remember is whether the book has taught kids to use two variables...

ExtendedProblem1 09 Oct 2005 - 03:12 CatherineJohnson

Find all the numbers that satisfy all of the following conditions:

1. Positive whole numbers less than 100,
2. Four more than each number is a multiple of 6
3. The sum of the digits of each number is a multiple of 4.

### and what is the best way to do this problem?

We used Doug's number lines (WHICH ARE GOING TO BE GETTING A WORKOUT THIS YEAR, IT'S OBVIOUS). We labeled one number line with multiples of 4, and the other with multiples of 6. We didn't need the multiples of 6, but it made things easier to have all the multiples of 6 sitting there, where we could see them.

PuzzleZoneQuestion 13 Nov 2005 - 14:55 CatherineJohnson

Feb. 9, 2005

A raincoat, hat, and boots were bought for \$280. The raincoat cost \$180 more than the hat, and the hat and the raincoat together cost \$240 more than the boots.

How much did each item of raingear cost?

Hint: You can solve the problem without having to write down any equations.

Answer: If instead of the raincoat, hat, and boots only two pairs of boots were bought, the price would not be \$280, but \$240 less. Thus, the two pairs of boots cost \$280 – \$240 = \$40. Hence, one pair cost \$20.

Now, we find that the raincoat and hat together cost \$280 – \$20 = \$260, the raincoat costing \$180 more than the hat. If instead of the raincoat and hat we could buy two hats, and we would pay not \$260 but \$180 less, that is, \$260 – \$180 = \$80. Hence, one hat costs \$40.

Thus, the prices of the items of raingear were as follows: boots – \$20; hat – \$40; and raincoat – \$220.

I solved this easily using algebra.

But then, when I looked at Puzzle Zone's answer, I didn't have a clue what it meant.

Now that I've spent some time mulling it over, I realize that I still don't completely 'have' the concept of subtraction as comparison (which I was never taught).

Fooling around with bar models helped me figure it out....but even though I can now draw this, I still don't understand—verbally understand—why we are talking about buying two pairs of boots as the key to solving the problem.

At the moment, there's no conceivable way I could explain this in words.

Any suggestions?

### subtraction as comparison

I taught my Singapore Math kids the idea of subtraction as comparison this week, and they were pretty thrilled.

At least, they acted thrilled. They had that nice ah-hah look on their faces.

TwentyMillionths 03 Mar 2006 - 06:02 CatherineJohnson

Place Value Chart
```                0.000000001   one billionth

0.00000001   one hundred-millionth
0.0000001   one ten-millionth
0.000001   one millionth

0.00001   one hundred-thousandth
0.0001   one ten-thousandth
0.001   one thousandth
0.01   one hundredth (1 in the hundredths place)
0.1   one tenth (1 in the tenths place)

1   one (1 in the ones or units place)
10   ten (1 in the tens place)
100   one hundred  (1 in the hundreds place)

1,000   one thousand (1 in the thousands place)
10,000   ten thousand
100,000   one hundred thousand

1,000,000   one million
10,000,000   ten million
100,000,000   one hundred million

1,000,000,000   one billion
10,000,000,000   ten billion
100,000,000,000   one hundred billion

1,000,000,000,000   one trillion...
```

source:
Math Forum

-- CatherineJohnson - 02 Mar 2006

MiniProblems 15 Jul 2006 - 16:33 CatherineJohnson

I've been complaining for months about the lack of word problems in Christopher's math class.

The kids memorize one procedure/rule/formula a day, do a few calculations, and march on. As a direct result, IMO, their knowledge really is rote as opposed to procedural. At least, Christopher's is. And I've had enough math talks with other kids in the class to know some of them are in the same boat.

Today I had a eureka moment reading a Comment left by Kathy Iggy:

The old math books I found (the same ones I used in grade school) have lots of what they call "mini problems" used to illustrate how a recently taught concept would be presented in a word problem. Megan likes these because of their brevity and she doesn't have to struggle with comprehension that much.

For example:

20 yards of ribbon. 1/4 used for dress. How much ribbon used?

That's IT!

mini problems

That's the concept, and the phrase, I've been looking for.

mini problems:word problems :: basic skills:higher order skills .

That's from Ken, and he's exactly right.

[update 4/23/2006: no! he's not right! Actually, he's write about using mini problems to teach word problems; I'm talking about mini problems to teach math - to teach the fundamental concept in a lesson. Awhile back I realized that word problems are the 'real manipulatives.' Now I know what I mean by that.

All concepts should be taught — illustrated — with mini problems. All concepts, every last one.

PRIMARY MATHEMATICS does this; SAXON MATH does it; KUMON does it. I'll post examples.

I've come to feel that the first word problems illustrating a new concept should be so simple children can do them in their heads.

For example, the very first ratio word problem a child does should be something like this:

Christopher bought one pencil for one dollar.
How many pencils can he buy for two dollars?

The question should be written this way, too: on two separate lines, so the child sees instantly that the first sentence is the set-up, and the second sentence is the question. Richard Brown's revision of Mary Dolciani's BASIC ALGEBRA, a book I like very much, does this for many of its word problems. I'll post some of those, too, as I get to it.

mini problems are applications

The problem with word problems is that, in the U.S., they're always hard.

Word problems are so hard people have apparently come to think that if a word problem isn't hard it isn't really a word problem.

I'm wondering if we ought to ditch the phrase 'word problem' (ditto for 'story problem') and adopt the word 'application.'

A better idea: we should think about the point of word problems.

Some word problems are written and assigned to give students practice.

Many word problems are written and assigned to assess whether students have developed flexible knowledge.

I'm talking about a third purpose, which is instruction. I'm talking about word problems designed to teach.

instructional word problems

A word problem is an application. A super-simple, starter word problem explains and demonstrates a mathematical concept by showing students how the concept is applied.

As a matter of fact, an instructional word problem shouldn't even be a 'problem.' It should just be a question, and the answer should be obvious.

A simple, instructional mini-problem should not test the child, should not challenge the child, and certainly should not trick the child.

It should teach.

examples to come

be sure to see Google Master's comment

how do you teach your child word problems?
mini problems (important)
arithmetic to algebra & mini-problems

-- CatherineJohnson - 07 Mar 2006

GregMatteoProblemsThatTeach 11 Mar 2006 - 19:29 CatherineJohnson

For excellent examples of problems that teach, try Math TV at Action Math. The video solutions for some of the fraction and ratio problems even use Singapore methods. Christopher may really enjoy these.

Greg DeMatteo
Norwood Middle School

I'm thrilled to get these — will report as soon as I've watched!

(Sorry. I'm developing a slight THING for animated gifs....)

back again

Action Math is a BLAST!

Here's the first problem:

A new movie theater opened up in town.
The theater contains 40 rows of chairs.
The first row has 10 chairs.
Each additional row has two more chairs than the row before it.
What is the seating capacity of the theater?

This brings up something important.

I think early word problems should be written exactly this way, as a list of sentences, not a paragraph. This helps 'disaggregate' basic working memory and 'environmental dependency' issues from the math problem itself.

The child isn't constantly having to search back and forth in a paragraph to re-isolate the individual sentences.

These videos are incredible!

You MUST go see them.

Wonderful.

I've just gained more conceptual understanding of algebra watching one video than I did in 3 years of high school! (I'm pretty close to serious about that....)

Greg, THANK YOU!

OK, now I'm sick.

If Christopher had had a teacher like the one on this website all year long.....we wouldn't be studying for the state test.

Compare her step-by-step explanation of the solution to the non-explanation of this Extended Response problem Christopher was given early this year:

challenge? or teach?

Virtually all of the problems Christopher has been given this year have been 'Challenge' problems. That's what the Assistant Superintendent in charge of curriculum told me about the Extended Response problems: "These students need to be challenged." (Christopher brought home a whole batch of challenge problems last night; I'll post some of them tomorrow.)

Virtually none of the problems Christopher has been given this year have been Teaching problems.

There is a vast difference between challenging a student and teaching a student.

keyword: actionmath

-- CatherineJohnson - 10 Mar 2006

HowDoYouMakeAModelOfAPlanet 14 Mar 2006 - 07:39 CatherineJohnson

It's 8:30 am Saturday morning, and Christopher just reminded me that he has to 'make a model of the 10th planet' this weekend and bring it in to school on Monday.

He doesn't know how to make a model of the 10th planet, and neither do I. He was thinking of making a clay ball. Good idea! That means a trip to Toys R Us for clay, which will be highly educational.

If any of you knows how to make a model of the 10th planet quickly and efficiently I'm all ears.

The reason we have to do this is that the Irvington Education Foundation funded something called The Jason Project (JASON Fosters the Joy of Teaching!) so now we have to make a model of the 10th planet.

Up until last year I was a dedicated Raiser and Donater of money to the school.

No good deed goes unpunished.

my fevered brain

I was thinking just the other day that my mother used to make papier mache with us all the time, and I've never done it with my own kids.

I'm pretty sure you can model the 10th planet using papier mache.

ktm Brain Trust comes through again!

from Ken:
The easy way out is to go to the art supply store and pick up a styrofoam ball and paint it "tenth planet" color.

[question: has Tenth Planet color been added to Crayola Crayons?]

back to Ken:
The more advanced tenth planet modeler (i.e., crazy) could apply modeling clay or papier mache to the ball to simulate terrain. The modeling clay has the advantage since it doesn't then have to be painted.

and:

or you could always go the more authentic route and use ice

[I'm sitting here reading this stuff out loud to Ed and I am HORSE LAUGHING]

from Susan:
If you have to use stuff around the house, paper mache' would work, with the quick and messy flour, water, salt mixture, and some little ball around the house. It would probably have a nice Flinstone look about it, but it would do.

[question: salt?]

OK, now let's hear from an expert:

Carolyn Morgan, grade 5 teacher:
Styrofoam works. You can suggest terrain by creating little valleys with the side of a knife handle or fingers. One good thing -- it's fast and not at all messy.

while we're on the subject of things I don't know how to do

This reminds me.....I have another question for the ktm Brain Trust.

How come the odds of rolling snake eyes are 1/36 but the odds of rolling a 3 are 1/18?

simple papier mache recipe
papier mache instructions ("papier mache is the ultimate recyclers dream")
papier mache at Enchanted Learning
papier mache by Rozani

-- CatherineJohnson - 11 Mar 2006

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# Entries from MathProblemHelpLine

ItTakesChops 07 Jul 2005 - 22:09 CatherineJohnson

It takes chops to solve this when you're eleven:

Two cars leave simultaneously at 9 a.m. heading toward one another from different cities that are 210 km apart. The average speed of one car is 50 km/h while the other car averages 70 km/h. Come up with an appropriate question and answer it.

This problem appears on page 5, 'Review,' of Mathematics 6: an award-winning textbook from Russia, by Enn Nurk and Aksel Telgmaa.

The 6 in the title stands for 6th grade.

+ + +

update: OK, I solved it.

But I couldn't think of a bar model.

Our Favorite Supplements
RussianMath
RussianMathPart2
RussianMathPart3
WhyILoveCarolyn
Mike McKeown comment

RussianMathPart3 07 Jul 2005 - 21:54 CatherineJohnson

I'm stumped.

The chainring (attached to the pedals) on a one-speed bicycle has 44 teeth while the freewheel (on the back wheel) has 20 teeth. Determine the least number of turns the chainring must make in order for both the chainring and feewheel to return to their original positions. How many turns does the freewheel make during this time?

This is a problem from Mathematics 6: an award winning textbook from Russia by Enn Nurk and Aksel Telgmaa, a 6th grade book.

I've prime factored each number, since that seems to be what's called for.

AnotherWikiPossibility 19 Sep 2005 - 23:07 CatherineJohnson

Another possibility for communal Wiki pages is to do something like the thread for RussianMathPart3: pose a problem or a lesson everyone can comment on.

I'm interested in comments on the fraction lesson J. D . Fisher has posted at Math and Text.

My immediate reaction to J.D.'s post is that it would be terrific for developing teachers' conceptual understanding of mathematics, and possibly for developing teachers' pedagogical content knowledge (pdf file).

But I wouldn't be able to teach it to Christopher, even though he does know that a fraction is (also) a division problem.

(I'll pull my thoughts together on this later--time for a bike ride now.)

I'd love to get other people's reactions.

KitchenTableMathIsAWiki
WikiHowTo

SingaporeWordProblemSampler1 14 Jul 2005 - 03:19 CarolynJohnston

This is a sampler of randomly chosen word problems from the "Primary Mathematics Challenging Word Problems" series. I'm just going to open the books, and write what I see.

For tonight's problem, pick one and give it to your kid, or try one yourself.

For bonus points, spot the problem that was not written by a native English speaker.

Grade 3: String X is 34 cm longer than String Y. String Z is 58 cm longer than string Y. If the total length of Strings X, Y, and Z is 233 cm, find the total length of Strings X and Z. (!!! OK, that was from the 'challenging problems' section of a 'challenging problems' book. Excuses, excuses)

Grade 4: Jane has 70 balloons. 1/10th of them are green, and 3/5 of them are orange. How many more orange balloons than green balloons does she have?

Grade 5: A man bought a dozen sacks of rice at \$18 per sack. Each sack of rice weighed 20 kg. He packed half of the rice into bags of 5 kg and sold them at 6.50 per bag. He sold the rest of the rice at \$1.50 per kg. Find his total profit.

Grade 6: Henry has 3/4 as many paper clips as Joyce. Joyce has 4/5 as many as Claire. If the three girls have 96 paper clips altogether, how many fewer paper clips does Henry have than Claire?

SingaporeWordProblemSampler2 14 Jul 2005 - 03:19 CarolynJohnston

Here's another random sampling of word problems from "Primary Mathematics, Challenging Word Problems".

A KtmGuest (henceforth known as 'Lone Ranger') left the following useful comment on the SummerProgramUpdate thread:

FYI...Singapore Math is organized differently than American elementary math textbooks. The book are arranged in this order 1A, 1B, 2A, 2B...6A,6B. When a student is finished with 6B, that student is ready to begin studying Algebra. Therefore the number on the book does not translate into an American grade level. In fact many people comment that children begin the Singapore program 1 number below their current grade. My child began with level 2B even though she was starting 4th grade.

(thanks, kemosabe).

So who knows what American grades these problems match up to? Just target the problem that suits your kid, and don't worry about whether they're behind what kids are doing in Singapore.

Primary 3: The capacity of a bucket is 9 qt. If 3 qt. 3 c. are added into the bucket, how much more water is needed to make it full?

(I like that last problem because it ties in with this recent post.)

And here is a rather strange one:

Primary 4: 5/9 of a box of chocolates are round, and 2/9 are square. How many more chocolates are round than square? Give your answer as a fraction.

Primary 5: Martin and Gary had 80 stickers altogether. After Martin gave away 35 of his stickers and Gary gave away 1/5 of his stickers, they had the same number of stickers left. How many stickers did Martin have at first?

Primary 6: Linda and Jane set off from City P to City Q at the same time. When Linda reached City Q, Jane was still 140 km away. 2 hours later, Jane also reached City Q. If Cities P and Q were 630 km apart, at what speed was Linda traveling?

SingaporeWordProblemSampler3 14 Jul 2005 - 03:20 CarolynJohnston

Note: solutions to the problems from SingaporeWordProblemSampler2 have been posted here.

So here's a whole new set of problems!

Primary 3: Margo has 3 times as many pears as apples. If she has 84 pears and apples altogether, how many pears does she have?

Primary 4: A cake was cut into 12 equal pieces. Jim ate two pieces and Tom ate four pieces. What fraction of the cake was left?

Primary 5: A bag of potatoes weighs 7/8 kg.. A bag of yams weighs 4/5 as much as the bag of potatoes. Find the total weight of the bag of potatoes and the bag of yams.

Primary 6: Eric has 75% as much money as Joshua. Carl has 60% as much money as Eric and Joshua have together. If Eric has 36 dollars less than Carl, how much money does Joshua have?

I don't know about you all, but I think I perceive these Singapore math problems becoming markedly harder at level 5.

SingaporeWordProblemSampler4 14 Jul 2005 - 03:20 CarolynJohnston

Here are solutions to the problems in SingaporeWordProblemSampler3.

As usual I am going to post a random sample from Primary Mathematics Challenging Word Problems, levels 3 through 6 -- but there are actually extra-challenging word problems in special sections, and today I'll put up some of those.

Primary 3: On Valentine's Day, a teacher gave 37 students 7 candy hearts each. If she had 3 boxes of 100 candy hearts each, how many candy hearts did she have left?

This next one shows how early Singapore Math starts to introduce algebraic word problems.

Primary 4: A farmer had twice as many ducks as chickens. After he had sold 413 ducks and another 19 ducks died, he had half as many ducks as chickens left. How many ducks did he have left?

Primary 5: Laura had 400 stamps. She gave 3/20ths of them to Sam, 5/16s of them to Joe, and 1/15th of the remainder to Jim. How many stamps did she have left?

And now the one we've been dreading.... AAAGH!

I would actually never give this one to a kid who wasn't really fond of puzzles. It's the only way to approach this problem.. fiddle around with it till you get a handle on it.

Primary 6: The ratios of the number of chairs to the number of tables in Halls A and B respectively are 5:2 and 6:1 respectively. The total number of chairs and tables in Hall B is three times that in Hall A. What is the ratio of the number of tables in Hall A to the number of chairs in Hall B?

AnnouncingSolutionsToSingaporeSampler4 17 Jul 2005 - 03:55 CarolynJohnston

This post is just a pointer to the solutions page for the SingaporeWordProblemSampler4, which consisted entirely of problems considered by the Singaporeans to be challenging.

Didn't want to leave anybody hanging if they're still paying attention! But a session of basic algebra with Young Ben has done me in for the evening. I'll be along soon with more Challenging Word Problems from Singapore -- but maybe not quite so challenging as these were.

QuestionAboutReciprocals 31 Jul 2005 - 18:04 CatherineJohnson

My 'benchmark' for the moment when I understand elementary mathematics well enough to move on is:

reciprocals

I find reciprocals utterly mysterious.

They're not quite in the magic category anymore, which is my benchmark for complete and total lack of comprehension. If a maths concept seems like magic, that means I know nothing.

Of late, inside the expanding math section of my brain, reciprocals have put a toe outside the magic category. But not much more than a toe.

### Danica McKellar

Tuesday's SCIENCE TIMES has a profile of Danica McKellar, who played Winnie on Wonder Days. It turns out she's a UCLA math major who published a proof (pdf file) now known as the Chayes-McKellar-Winn theorem.

### car wash problem

McKellar's web site has a mathematics section where she answers reader questions. She's a natural born teacher:

Q: Hi Danica, I heard a question from Mr. Feenie on a "Boy Meets World" episode which he claimed to be unanswerable. After hearing that, I decided to figure it out. If it takes Sam 6 minutes to wash a car by himself, and it takes Brian 8 minutes to wash a car by himself, how long will it take them to wash a car together?

Danica Answers: Hm, unanswerable? That's TV for you. :)

Let's do it: This is a "rates" problem. The key is to think about each of their "car washing rates" and not the "time" it takes them. Alot of people would want to say "it takes them 7 minutes together" but that's obviously not right, after you realize that it must take them LESS time to wash the car together than either one of them would take.

So, what is Sam's rate? How much of a car can he wash in one minute? Well, if he can wash one car in six minutes, then he can wash 1/6 of a car in one minute, right? (think about that until it makes sense, then keep reading). Similarly, Brian can wash 1/8 of a car in one minute. So just add their two rates together to find out how much of a car they can do together, in one minute, as they work side by side on the same car: 1/6 + 1/8 = 7/24 of a car in one minute. That's their combined RATE. (Note: that's a little bit less than 1/3 of a car in one minute). From this point, the way you want to think of it depends on your favorite way of dealing with fractions. You now have their rate. It's 7/24 cars per minute. You can either just take its reciprcal and say: 24/7 minutes for one car, and you're done.

Or, equivalently, you can think of the 7/24 cars/minute RATE as 24 minutes for 7 cars. (think about that until it makes sense, too) So just divide 24 by 7 to find out how many minutes it would take to do just one car. You get around 3.42 minutes for one car, just a little less than 3 and 1/2 minutes. Done! Yes, I think they should work together, it gets done much more quickly that way. :)

By the way, you said when you watched the TV show you decided that YOU would figure it out, right? How did you do?

I love this. McKellar is teaching two things here:

• how to solve a rates problem
• how to read, study, & learn maths (that's metacognition)

First of all, she knows that math novices transfer their normal reading habits to maths books. By normal reading habits I mean that most of us, when we read a book of prose, read straight through at a fast clip, pausing only to underline or make notes in the margin.

You can't read a math book that way; in fact, I've come to feel you can't really read a math book at all. You have to do a math book, or work a math book. McKellar explicitly instructs her reader not to read the solution straight through, but to stop at key points and ponder until he or she gets the point, and is ready to go on to the next point.

She doesn't stop there, either. She also knows the precise spots in her explanation where most novices will need to stop and mull, and she tells them where those spots are.

She's giving novices direct instruction in metacognition.

As to the problem itself, she addresses the most common error novices will make confronting this particular rates problem, which is:

• figure that it takes the 2 boys 14 minutes to wash 2 cars
• so logically it must take them 7 minutes to wash 1 car

Amazing! And all in the space of a few short paragraphs.

I think McKellar's teaching skill here is connected to her acting. There's a large element of 'performance' in teaching, at least in my experience, and to be a good performer you have to know where your audience is, what they want to hear & what they need to hear.

She does.

### back to reciprocals

Here's my reciprocal question.

From this point, the way you want to think of it depends on your favorite way of dealing with fractions. You now have their rate. It's 7/24 cars per minute. You can either just take its reciprcal and say: 24/7 minutes for one car, and you're done.

Or, equivalently, you can think of the 7/24 cars/minute RATE as 24 minutes for 7 cars. (think about that until it makes sense, too) So just divide 24 by 7 to find out how many minutes it would take to do just one car. You get around 3.42 minutes for one car, just a little less than 3 and 1/2 minutes. Done!

I don't understand why you would use the reciprocal to solve this problem.

I understand perfectly well (let's hope) why you would divide 24 by 7.

I didn't even know you could use the reciprocal to find the answer.

### 7 fact families

I haven't had time to sit down and think this through, but I suspect the reciprocal answer to a rates problem is the same concept as the 7 fact family I put together after teaching the Primary Mathematics lesson on ratio & proportion (Primary Mathematics 6A Textbook, p. 21-46):

7 fact families

MathTalkInTheCar 01 Aug 2005 - 16:50 CarolynJohnston

We took the kids to a bar tonight, as it happened. Colin (17) is into playing the bass these days; he has a band that he plays with during the school year. I have a friend at work who is a hot guitar player and who just joined a classic rock band, and he was playing his first gig tonight, and they were letting kids stay through the first set, so we went to see him. It was a long drive for us -- all the way out to Greeley. The place was an authentic roadhouse with motorcycles parked out front, and the food was good -- it was Cajun food, and very authentic given that we were not in Cajun country but in Greeley, Colorado, home of the Feedlot You Can Smell All The Way To Denver.

On the way home, Colin asked us about the difference between the median, the mean, and the mode of a data set, and what each of them is good for. This is, of course, the sort of thing we love to pontificate about. He then told us that he felt he had never really quite gotten the idea of a function, and asked us to explain it.

It's a smart kid who understands what he doesn't understand. Most adults can't do that very well.

Actually, most kids coming into calculus classes are confused by functions. A function is just a black box; you put in an input, and get out an output. What makes it a function is that, when you put in the same inputs, you always get the same outputs. You can't put the same number in the black box and get 2 one time, and 5 the next.

Most texts teach functions using formulas to define the functions; all the functions kids see look like f(x)=3x-5, or g(x)=x/6. But functions don't have to have formulas to go with them; they can defy description by a formula. The only rule is that if you put in the same input multiple times, you get the same output, every time.

The reason kids confuse formulas with functions is that it's hard to define functions that don't use formulas, even though in real life we encounter them all the time. When a function totally defies description with a formula, we often resort to trying to describe it with only a couple of numbers, such as the mean, median, and standard deviation (this is how the whole field of statistics arises).

We played a 'figure-out-the-function' game on the way home from Greeley. Bernie and I would think of a function, and Colin and Ben would give us numbers for inputs, and we would then tell them the output. They'd then try to guess the formula we were using to define the function.

They are both aces at extracting patterns. If anything, Ben would try to generalize from too little data; once he guessed, after one try, that the function was 'add 2'; he'd given me a 2, and I'd come back with 4 (the function I'd thought of was squaring; he got it on the next try). Bernie was giving Colin some functions that are so simple they trip up students with their obviousness, like the function that returns the same number you give it, and the one that returns '3' no matter what you give it. He gave Colin one function that was so bizarre you can't describe it with a pattern.

Ben knew more about functions than I thought, even piping up with "that's the constant function 3" at the appropriate moment. Did they do functions one day for 5 minutes in Everyday Math? Well, he was definitely on the ball that day.

HowToTeachGCFAndLCM 03 Aug 2005 - 16:23 CarolynJohnston

I don't know what the blockage was last week (when I wrote this), but we seem to be cruising again.

It doesn't hurt, of course, that the chapter in Prentice-Hall Mathematics Course 1 that we're doing is something that Ben basically already knows. He did prime factors last year in Everyday Math, using 'factor trees', and learned to stumble through a guess-and-check process for finding the least common multiple of two numbers. So all I have to do is to make sure he's retained the concepts, and to teach him an efficient way to get the answer right every time. Tonight we did efficient prime factorization, and efficient greatest common factors.

In Prentice Hall, as in most math texts, greatest common divisors and least common multiples are taught back-to-back. My recollection of my own school years is that, with the two concepts taught so closely together, and the methods for doing both calculations being so similar, it was easy to get pretty confused about them. That's what I want to avoid. Anyone teaching this topic should (I think) try to clarify the two concepts explicitly.

I think that when you have to teach a tricky topic like this, a really juicy example is typically better than a lot of explanation.

Here is a juicy example. Take 22 and 34. Here are their prime decompositions:

22 = 2 11
34 = 2 17

This is a good example to illustrate the difference between GCF and LCM. There is only one common factor (2), and it's obviously also the greatest one. It's easy to show that any common multiple of both numbers has to have a 2, an 11, and a 17 in it, and the least one is the product of those numbers.

It's a nontrivial example, but it's got no distractions in it. Each number has only two factors, there's no powers of primes in there to trip up a kid, and the GCF is itself a prime. It's easy to generate other examples like this: 15 and 21, for example, is a similar one. You can offer different examples till the kid gets comfortable with the difference.

The next step is to offer an example such as:

30 = 2 3 5
42 = 2 3 7

We've made it a little more complicated, but there are still no powers of primes; all you've done is to add one more prime factor in common, to make the GCF a very simple composite number (6). You would explain to the kid that, this time, there are several common factors; and the GCF is the largest of them.

Here's the approach I taught Ben tonight for finding the GCF. First write each number's prime factorization out, without powers. For example:

24 = 2 2 2 3
20 = 2 2 5

If you find a prime factor in common in both of them (I told him), scratch it out in each expression, and write it somewhere else. So, for this example, Ben would scratch out a 2 in each factorization, and write it elsewhere. Then he would scratch out a second 2 from each factorization, and write 2 next to the first 2.

Then he'd only have 2 3 left from the original factorization of 24, and 5 left from the factorization of 20. There are no factors left in common, so he's done; and he's got the GCF written down (the 2 2s).

I pointed out once again that certainly 4 is a common factor for this example, and there are none bigger.

More tomorrow on how to finish teaching GCFs and LCMs.

HotMath 14 Aug 2005 - 15:09 CatherineJohnson

Thanks to Dan K, I've found a fantastic resource:

[Hotmath provides] explained solutions to the odd-numbered homework problems from most of the popular secondary math textbooks used in California. Thus, teachers can now assign practice problems for homework where teacher-prepared, explained solutions are instantly available, and can mix in even-numbered problems for challenges. Students who do not need to see the worked solutions needn't bother, and students who might abuse the availability of worked solutions will be tested on the even problems.

Here is a sample worked-out problem: algebra problem

And here are the 2 critical paragraphs from the Hotmath 'white paper'. I've begun to come across these studies elsewhere, and I'm inclined to trust these summaries, in part because this discussion jibes with my own experience re-learning maths:

Providing students with worked out examples of math problems has been found to be more effective than simply assigning the same problems for the students to work out on their own. In one experiment (Carroll, 1994), 40 high school students were instructed in how to solve linear equations. In an “acquisition phase” the students were divided into two groups and their instruction differed in the following way: in the “conventional learning” group, students were assigned 44 unsolved problems to work out (in the classroom and at home homework), and in the “worked examples” group students were provided with the same problems, but half of the problems were accompanied by correct solutions. After completion of the assigned problems, both groups were tested on 12 related problems, 10 of which were very similar to the linear equations presented in the acquisition phase, and 2 of which were word problems, used to test whether students could transfer and extend their knowledge to a new context. No worked out examples were available during the test. The test results revealed that students in the “worked examples” group outperformed students in the “conventional learning” group on both types of the test problems. A second experiment, employed a similar methodology but focused on “low achieving” students (students with a history of failure in mathematics, and students identified as learning disabled). Here, the data revealed that students in the “worked examples” group required less acquisition time, needed less direct instruction, made fewer errors, and made fewer types of errors than students in the “conventional learning” group.

Related research (Pass & Van Merrienboer, 1994) sheds light on the cognitive underpinnings of the effects described above. In this study, 60 college-aged students were instructed in geometry concepts. As in the Carroll experiments, students were assigned un-worked problems to solve or worked out examples to review (unlike the Carroll study, the “worked examples” group was assigned no un-worked problems to solve). In this study, the researchers manipulated the nature of the problems presented to the students: within each group, some students received problems which were all similar to each other, while others received a more varied problem set. Furthermore, the researchers measured the “cognitive load” experienced by the students. This research revealed that while students in the worked examples group completed their work more quickly, they perceived the work as less demanding and displayed better transfer performance at test. The effect was most pronounced for the students given highly-variable problems. The researchers suggest that the reduced cognitive load associated with the worked examples enabled students to “take advantage of” the variability in problems by using the available cognitive resources to process the underlying similarity in the problems (i.e., the mathematical concepts being taught), and to integrate the current problem with existing knowledge (Linn, 2000).

The site covers Prentice Hall Pre-Algebra, the book Christopher will be using in the fall, so I'm going to subscribe. Cost is \$49 for 12 months.

I think it's going to be fantastic for Christopher to have an answer source that isn't His Mother.

Especially since it looks like I'm going to have to start some heavy-duty Writing Instruction this year. (That's another story.)

This is going to be an important term for me. It perfectly captures what it is we're trying to do when we push our kids to practice to the point of automaticity.

We're trying to reduce cognitive load.

### update

I've just re-read Dan's original post, and I don't see a reference to hotmath. hmmm. Maybe one of the sites he mentioned pointed me to hotmath. In any case, I'm recommending hotmath, not Dan. (He'll let us know what he thinks, I'm sure.)

StatisticsHelp 19 Aug 2005 - 17:51 CatherineJohnson

I mentioned that Temple and I are writing an op-ed on U.S. high schools.......and I'm stumped by a statistical issue.

What does it mean to say that multivariate analysis shows that a certain factor is highly predictive of a particular outcome, while another factor is less predictive?

What does this form of analysis imply about causality, if anything?

I ask because of an apparently highly influential government report published in 1999: Adelman, C. (1999). Answers in the Tool Box: Academic intensity, attendance patterns, and bachelor’s degree attainment. Washington, D.C.: U.S. Department of Education.

This study finds an extremely high correlation between rigor of high school curriculum and students' eventual completion of a college degree--far higher than the correlations with the factors we're used to hearing about, such as parents' level of educational attainment, socioeconomic status, and race (and substantially higher than high school GPA and SAT scores).

The report itself, which I've barely skimmed, as well as other accounts of it, seem to imply that the relationship is causal. It's not that being smart and motivated in the first place causes a student to take a more rigorous high school curriculum and attend and complete college, but that the more rigorous high school curriculum sets him or her up to succeed in college.

from the American Educator:

Academically well-prepared students are likely to graduate from college regardless of their social background. Unprepared students of all backgrounds are not likely to do so.

The graph below breaks students into quintiles based on their level of academic preparation and their socioeconomic status (SES). As you can see, among the lowest SES students, a bachelor’s degree was earned by 62 percent of those who were well prepared, but only 21 percent of those who were not. Similarly, among the highest SES students, 86 percent of those who were well prepared--but only 13 percent of those who were not--earned a bachelor’s degree.

Percentage of students who graduated from a four-year college by socioeconomic status (SES) and academic preparation.

key words: rigorous high school curriculum predicts graduation from college
how can you tell whether A caused B?
low birth weight paradox
how good are our best students?
statistics and law

StopMakingSense 16 Aug 2005 - 03:31 CatherineJohnson

When I was looking for pictures of teenagers, I found a Teenage Leisure Time project I can't figure out.

Do these charts say what the student says they say?

I can't see it, but maybe when I have a clear head again I will.

For now, my question is: how do these 4 pie charts tell us that people who have Sky TV and people who live in villages watch more TV than people who don't have Sky TV and don't live in villages? (And shouldn't 'sky' be capitalized?)

Plus, I don't see how the pie charts address the assignment:

Pupils were asked to produce a questionnaire to look at student leisure time and come to a conclusion about whether students tend to just watch tv or actually participate in activities.

I'm confused.

This is grim:

The pupil has selected information (data) they need, checked its accuracy and organised it into a suitable form for processing - a database. The student has also consolidated level 4 work as they have understood the need for care in framing questions when collecting, finding and interrogating information. They have interpreted their findings and questioned its plausibility. The use of questionnaires in the outside world was also discussed. The report was also an example of level 4 in that the Communicating Information element involved data from another "variety of sources"

To make further progress the pupil could be given opportunities to develop complex searches and test hypotheses.

Is there any subject-verb agreement here at all?

The Brits have always had unbelievably brilliant writing (and speaking) instruction. I've been trying to find sources on how they do it, and have come up with only constructivist stuff, including something called silent grammar, in which students use Cuisinaire rods to discover grammatical rules.

But where does silence come into this? Gattegno suggested that the teacher, to maximise the learning, only presents the model for language point once and once only. The teacher then uses the tools at his/her disposal to elicit the correct language from the learner. The teacher should be encouraging and open, using body language and facial expressions to show that the language produced is correct or incorrect, as well as correcting the learners using the tools outlined above.

So, silent grammar. Sounds like charades. The teacher waggles his eyebrows, mugs, and flings his arms about, but keeps his jaws clammed shut.

Of course, if British teachers talk like the one I just quoted, it's probably just as well.

### update

Ed says he's always suspected that the students in Britain who learn to write brilliantly are elite students at Oxford & Cambridge.

I have no idea.

FalsePositives 12 Sep 2005 - 03:15 CatherineJohnson

A couple of days ago, Carolyn explained the difference between frequentist statistics and Bayesian. She's a Bayesian, she said.

Well, that explained a lot, because it turns out I'm a Bayesian, too. I just didn't know it. Obviously, that's why Carolyn and I constantly find ourselves traveling the exact same thought path, even though we've never met, and didn't know each other until a year ago.

Of course, a real Bayesian (that would be a Bayesian who knew what she was doing, which would not be me) would probably not conclude that the reason she likes a person well enough to start a vast time-gobbling math-ed web site with her is that you both subscribe to the same school of statistical thought. I'll have to ask Carolyn.

I'm a Bayesian aspirant.

I'm having quite a little midlife run of Self-Discovery here, I must say. First I find out I'm Scots-Irish; next I'm hearing I'm a Bayesian.

I just hope no one's gonna tell me I'm adopted.

### I have a question

My question concerns a passage in a terrific book called What the Numbers Say: A Field Guide to Mastering Our Numerical World by Derrick Niederman & David Boyum. Boyum, it turns out, majored in applied mathematics at Harvard--I didn't know there was such a thing as a major in applied mathematics at Harvard!

Or anywhere else, for that matter.

I wish I'd know that when I was 17.

'Bayes Watch' is Niederman & Boyum's title for this passage:

Years ago a study asked the following question of students and doctors at Harvard Medical School:

If a test to detect a disease whose prevalence is 1/1000 has a false positive rate of 5 percent, what is the chance that a person found to have a positive result actually has the disease, assuming that you know nothing about the person's symptoms or signs?

Ed and I both understand the answer now (neither of us got it right), but we still have a question about the precise calculations. (Don't hit this link unless you want to see the answer.)

### update

I've just checked Niederman & Boyum. They do not specify a zero rate for false negatives. They say nothing about false negatives one way or the other. (Neither does John Kay in false positives, part 2, assuming I'm understanding him correctly).

### Bayes & God

I actually bought this book a couple of years ago, though I haven't read it yet:

I believe it's intended to be a Bayesian proof of the existence of God, although I don't know how the word 'proof' is used either in the book or in the context of Bayesian statistics.

low birth weight paradox (& Monty Hall)
Monty Hall, part 2
Monty Hall, part 3
false positives
false positives, part 2
Doug Sundseth on Monty Hall
John Kay: We are likely to get probability wrong (subscription only)
Monty Hall diagram from Curious Incident
Bayes & the human mind
Bayesian reasoning, intuition, & the cognitive unconscious
most bell curves have thick tails
ECONOMIST explanation Bayesian statistics
Bayesian certainty scale
probability question from Saxon 8/7

Bayesianprobability

FalsePositivesPart2 21 Dec 2005 - 15:31 CatherineJohnson

Another version of the False Positives challenge. This one ran in John Kay's column in the Financial Times yesterday. (Probably only available to subscribers.)

...intuition does not correspond to the mathematics of probability. One person in a 1,000 suffers from a rare disease. A friend has just tested positive for this illness and the test gives a correct diagnosis in 99 per cent of cases. How likely is it that your friend has the disease? Not at all likely. In random groups of 1,000 people an average of 10 would display false positives and only one would be correctly diagnosed with the disease. But most people, including most doctors, think otherwise. “The human mind,” said science writer Stephen Jay Gould, “did not evolve to deal with probabilities.”

Hmmm. Let's see. This problem does give us false negatives, right???

OK, let me think.

[pause]

Good grief. Not only can the human mind not intuit Bayesian probability; apparently the human mind equally cannot produce consistently lucid prose. (Nothing wrong with Mr. Kay's lucidity on a normal day.)

Kay's example, too, appears to assume a false negative rate of 0.

As far as I can tell.

### update

This is funny. I was skimming Amazon reviews of Stephen Jay Gould's Mismeasure of Man, and I found this:

As Oxford academician Richard Dawkins says (see Bryson, "A Short History of Nearly Everything", pp. 330-332) "If only Stephen Gould could think as clearly as he writes!"

It's a Core Principle in the Writing Biz (& definitely in the Writing Instruction Biz) that you can't write clearly without thinking clearly. (True in my experience; that's for sure.)

low birth weight paradox (& Monty Hall)
Monty Hall, part 2
Monty Hall, part 3
false positives
false positives, part 2
Doug Sundseth on Monty Hall
John Kay: We are likely to get probability wrong (subscription only)
Monty Hall diagram from Curious Incident
probability question from Saxon 8/7

MontyHallPart2 17 Aug 2005 - 21:49 CatherineJohnson

Here is Kay on the Monty Hall problem:

The Monty Hall problem is named after the host of a 1970s quiz show, Let’s Make a Deal. The successful contestant chooses from three closed boxes. One contains the keys to a car and the other two a picture of a goat. The choice made, Monty opens one of the other doors to reveal – a goat. He taunts the guest to change the decision. Should the guest switch to the other closed box?

When the solution was published in an American magazine, thousands of readers – including professors of statistics – alleged an error. Paul Erdös, the great mathematician, reputedly died still musing on the Monty Hall problem. But the answer is, indeed, yes: you should change.

I'm happy to hear that Paul Erdos stumbled over Monty Hall, seeing as how I still don't understand it.

low birth weight paradox (& Monty Hall)
Monty Hall, part 2
false positives
false positives, part 2
Doug Sundseth on Monty Hall
John Kay: We are likely to get probability wrong (subscription only)
Monty Hall diagram from Curious Incident
probability question from Saxon 8/7

DougSundsethOnMontyHall 18 Aug 2005 - 19:52 CatherineJohnson

Doug Sundseth (welcome, Doug!) posted an explanation of the Monty Hall problem, which I have never been able to understand:

It took me a long time to understand it, too.

The model that finally worked for me was something like this:

You have a 1/3 chance of being right to start with, and a 2/3 chance of being wrong. If you guessed wrong originally, Monte's pick will unambiguously determine the correct choice (he never picks the good door).

There are nine pairs of (your pick):(correct pick), A:A, A:B, A:C, B:A, B:B, B:C, C:A, C:B, and C:C. In three of those, you picked correctly, Monte's information isn't useful, and you shouldn't switch. In the other six, you picked incorrectly and Monte told you which of the other picks was correct; thus you should switch.

If you never switch, you have three chances in nine of being correct. If you always switch, you have six chances in nine of being correct and three chances in nine of switching off the correct choice.

Note that the latter possibility (choosing correctly at random then switching to an incorrect choice) may be more psychologically painful than just guessing wrong and not switching. This may have an undue effect on the choices of contestants.

Doug, thank you!

OK, I've just sat down and quickly thought this through.

[pause]

On my initial reading, I think it makes sense to me. What's particularly useful, for me, is the information that, yes, you could already have chosen the correct door, in which case, if you change your choice, you have moved to the incorrect door.

I think people who haven't studied probability get hung up on the 'what if I'm already right' issue.....and then, when math-savvy people try to explain Monty Hall without addressing, as Doug has, the issue foremost in their minds, the explanation doesn't 'take.'

### metacognition again

I mentioned awhile back that metacognition is a huge issue amongst constructivists, both of the radical & the peer-reviewed , department of psychology cognitive science constructivists.

One of the main reasons for thinking about metacognition as you teach is that students may very well bring quite wrong ideas to the classroom, which they then 'build upon' as they acquire new knowledge. There's a lovely example of this in the National Research Council's book on learning. Many children, when told that the earth is round, picture it as a disk, not a sphere. (more t/k--I need to go take a look at these pages.)

In any case, Doug has addressed an aspect of metacognition that I haven't seen mentioned, which is to tell a student what it is they already know that's right, but incomplete.

I was having the same experience yesterday, puzzling through the 'false positives' problem. The objection both Ed and I were bumping our heads against--if it's 1 in 1000 and 50 in 1000, how can you ever have 1000???--was right; we just weren't seeing what to do about it.

I wonder how often it's the case that an incomplete right answer is the problem, as opposed to a Total Crackpot Misconception that has to be stomped out, obliterated, and disappeared without a trace before a person can learn Thing One about math? (And does this wording give you a feel for the challenge involved in attempting to re-learn elementary math in midlife?)

Or, as Steve H says, A little knowledge goes way too far.

### a new question

This sentence confuses me:

You have a 1/3 chance of being right to start with, and a 2/3 chance of being wrong. If you guessed wrong originally, Monte's pick will unambiguously determine the correct choice (he never picks the good door).

[pause]

hmmm. Interesting. Reading this again, it makes sense.

I'm going to take a paper and pencil break, and see what I come up with working through Doug's explanation myself.

I love it!

### back again

OK, paper and pencil session complete.

I do understand this explanation, with one question: the funky, counterintuitive odds are created by the fact that Monty always opens the wrong door, correct?

That's why you shouldn't go with the 50-50 answer everyone automatically does go with--yes?

Carolyn was telling me the other day that a lot of Bayesian statistical results are counterintuitve (hey! just like the Bayesian proof of the existence of God!).

That's for sure.

### other explanations

Here's a strictly mathematical explanation that will work for some people (and actually works OK for me....although frankly Doug's list helps move me a bit towards 'getting' the Monty Hall problem at a more intuitive level...):

After you pick but before you open any doors, there's a 1/3 chance that you've picked correctly, and a 2/3 chance that you've picked wrong. Assuming that the host can open doors, but can not move prizes, nothing that the host does will change the probabilities described above.

Now the host opens one of the doors, and there's nothing behind it. There's still a 1/3 chance that you've picked correctly, and a 2/3 chance that you've picked wrong. This means that the remaining door has a 2/3 chance of being correct.

This explanation helps me formulate exactly what it is that goes wrong for people: the chance to change your pick seems like a second event, with a second set of probabilities attached.

question: So how often does this happen in life?

How often do we perceive second events where we ought to perceive a continuation of the first?

### update: an intuitive approach to Monty Hall that might work

I'm going to have to live with the Monty Hall problem for awhile....

But here's an interesting approach to rendering the answer intuitively correct:

It was a while ago that I accepted the idea that switching doors was the correct play every time because it improves your chances of winning, but I had trouble convincing my friends that it was the correct answer. However, a friend of mine just came up with this explanation that I think should really make it obvious.

Let's say that you choose your door (out of 3, of course). Then, without showing what's behind any of the doors, Monty says you can stick with your first choice or you can have both of the two other doors. I think most everyone would then take the two doors collectively.

Unfortunately, I don't think this works for me...

### update: Keith Devlin's better version

OK, I think what the person above was trying to say was this:

...one last attempt at an explanation. Back to the three door version now. When Monty has opened one of the three doors and shown you there is no prize behind, and then offers you the opportunity to switch, he is in effect offering you a TWO-FOR-ONE switch. You originally picked door A. He is now saying "Would you like to swap door A for TWO doors, B and C ... Oh, and by the way, before you make this two-for-one swap I'll open one of those two doors for you (one without a prize behind it)."

I agree. Anyone told at the outset that he can pick one door or he can pick two doors would pick the two.

### I give up

from Keith Devlin:

... suppose you are playing a seven door version of the game. You choose three doors. Monty now opens three of the remaining doors to show you that there is no prize behind it. He then says, "Would you like to stick with the three doors you have chosen, or would you prefer to swap them for the one other door I have not opened?" What do you do? Do you stick with your three doors or do you make the 3 for 1 swap he is offering?

OK, I'm switching doors.

But I'm doing so purely on the basis of 4/7 being greater than 3/7. Nothing common sense about it.

Of course, given that my family motto is no common sense-y, it's easy to dump my first pick and jump to Door Number Seven!

low birth weight paradox (& Monty Hall)
Monty Hall, part 2
Monty Hall, part 3
false positives
false positives, part 2
Doug Sundseth on Monty Hall
John Kay: We are likely to get probability wrong (subscription only)
Monty Hall diagram from Curious Incident
probability question from Saxon 8/7

BadMathInEverydayMath 22 Aug 2005 - 02:26 CatherineJohnson

I've just noticed that a ktm guest left a comment on something I'd wondered about myself:

Worse yet, the math is wrong. It's the usual mixup of percents with percentage points.

Look at #5. Food is about 80%, Dandruff about 10%, but 80% is not 60% greater than 10%. 60% of 10% would be 6%. 80% is eight times bigger than 10% or 800%

Here's the original problem:

I'm sure it will come as a shock to no one that I was never taught how to compute a percent increase or decrease; nor was I taught, as far as I can remember, what the question 'How much greater is the percent of men who are willing to alert strangers to smudges on their faces than the percent of women who are willing to do so?' actually means.

As a direct result, I managed to spend my entire adult life utterly confused about the Ultimate Meaning of news stories on Percentage Increase in Federal Spending On Education and the like.

No more! Thanks to Algebra to Go & Russian Math, I now know what both questions mean, and how to answer them, at least in theory. By which I mean that percent increase/decrease and how-many-times-bigger still hold the status of New & Tenuous inside my head.

Yes, I could demonstrate both on a Pop Quiz right this minute.

But I'm not confident I'd be right.

So when I read this question, my first thought was: 10 percent?

Then I thought, Hunh.

I only had to stare at the problem a couple seconds more to arrive at the conclusion that, OK, we're not talking percent increase here.

Which was too bad, because of the two ideas, that's the concept I know best. The idea of how many 'times' bigger (or smaller) one number is than another is something I first learned literally one or two weeks ago. (I know; it's mortifying.)

So 'how many times bigger?' is very new knowledge for me, new enough that I figured the folks at Everyday Math must KNOW.

### I give up, again

I don't know how to figure this. I do think the E-Math folks are asking for a simple subtraction of one percent from the other.

But is that the right way to figure how much larger one number is than the other in this case?

Or would we want to know 'how many times larger' one number is than the other. (I'm thinking I've seen numerous reports and articles in which a simple subtraction was used.....Help!

Well, let's just hope all this confusion will help me understand students' confusion...

### update: Anne Dwyer on the mathematical meaning of words

This is a very interesting problem because it involves interpreting the mathematical meaning of words.

The problems did not ask how many times greater or even percentage increase. The problem asked "how much greater". The author of the problem obviously intended that the student would interpret this as a simple difference problem by subtracting one percentage from another. The author intended that the student would treat each percentage number as a number with the same units that can be subtracted. The real difficulty comes when you write out the answer: 60% greater. The mathematical meaning of the answer is very clear: 60% greater means that you increase something by 60%. This is not what was done in this problem. The answer is incorrect because you cannot use the word "greater" as units to an answer the the problem.

sexism in Everyday Math

EverydayMathThread 19 Aug 2005 - 16:21 CatherineJohnson

Terrific thread about an Everyday Math problem in mistake in Everyday Math?

Welcome Independent George!

MathAndLanguagePart2 22 Aug 2005 - 18:26 CatherineJohnson

Carolyn M is back!

So, while I have her attention, I'm posting my new Russian Math question.

This is from a chapter section called:

### The Quotient of Two Numbers

1. A quotient greater than 1. ... In this case the quotient shows how many times larger the dividend is than the divisor. The number 8 is 1.6 times bigger than the number 5.

2. A quotient equal to 1. ... In this case, the divisor and the dividend are equal numbers.

3. A quotient less than 1. In this case, the quotient shows what part of the divisor the dividend is. And so, the number 6 is 3/4 of the number 8.

This was, believe it or not, the first time I had ever seen a quotient greater than one defined as how many times larger the dividend is than the divisor. Wonderful!

However, I was then surprised to find that a quotient less than one was not defined as how many times smaller the dividend is than the divisor.

Here's my question.

Is it incorrect to say that a quotient less than one tells you how many times smaller the dividend is than the divisor?

Or is it that the authors view the definition they give (what part of the divisor the dividend is) as more important or more fundamental or more profound?

If it is incorrect to say that a quotient less than one tells you how many times smaller the dividend is than the divisor, why is this incorrect?

Thanks!

### Russian Math for everyone

While I'm on the subject, I'll add that the Russian Math book is pure pleasure. It's incredible. I think every 6th grade child in the country should study this book.

And every parent.

### update

When I say this is the first time I've ever read that 'the quotient shows how many times larger the dividend is than the divisor,' I don't mean that I've never heard expressions like '6 is 2 times larger than 3.' I have!

What I mean is that I've never seen a direct, explicit, comparative distinction drawn between 'a quotient larger than 1' and 'a quotient smaller than 1.' I've never seen (I don't think) 'a quotient larger than 1' defined as having a separate & distinct meaning.

This is a standard technique throughout RUSSIAN MATH, which I find incredibly powerful, and which I haven't noticed in U.S. textbooks (or, I think, in the Singapore series).

The RUSSIAN MATH book constantly teaches through difference, which in this case means 'disaggregating' a concept that normally stays aggregated in a U.S. textbook.

In a U.S. textbook--at least in the ones I've used--a quotient is a quotient.

In a 6th grade Russian textbook, all of a sudden a quotient is 3 different things. It's a remarkable book. Incredible.

Another thing: I've never seen a 'quotient larger than one' defined as anything other than 'how many times the divisor goes into the dividend' or 'how many of the divisor are in the dividend.'

Last, but not least, I don't think I ever was made conscious of 'decimal' or 'fractional' phrases like '1.6 times larger.' 'Two times larger,' sure. '1 '1/2 times larger,' yes.

But '1.6 times larger'--never.

This is important, because it looks like we're hardwired to understand 'friendly fractions.' Five year olds know what one-half is.* Five year olds do not know what 6-tenths are, and they certainly do not know what 1 and 6 tenths are!

Morever, our brains do not automatically or easily generalize from 'one half' to '1.6.'

This is another reason to object to the exclusive focus on friendly fractions in constructivist curricula. Friendly fractions are the fractions children know without having to be taught.

*Hunting, Robert P. (1999). Rational-number learning in the early years: what is possible?. In J. V. Copley. (Ed.), "Mathematics in the early years", (pp 80-87). Reston, VA: NCTM.

What Counts: How Every Brain is Hardwired for Math, by Brian Butterworth
The Number Sense: How the Mind Creates Mathematics by Stanislas Dehaene
Children's Mathematical Development: Research and Practical Applications by David C. Geary
(fyi: It is possible to buy Geary's book for far less than the \$124 Amazon wants for it, or the \$55 I paid for a used & extensively highlighted copy...)

Carolyn on math and language 7-2-05
Carolyn on math and language again 7-3-05
the language of numbers is not language 7-3-05

FactoringNumbers 09 Sep 2005 - 02:25 CarolynJohnston

A factoid: When you're trying to determine whether a number is composite (i.e., not prime), it's sufficient to check whether all the primes less than the square root of the next greatest square number divide it. If nothing less than the next-larger whole number square root divides the number, then you know the number is a prime.

For example, to check whether 221 is prime, all you have to do is test all the prime numbers less than 15. To check whether 133 is prime, check the prime numbers less than 12.

My question: how do you teach the reasoning behind this rule? I was tackling this tonight with Ben, and I wasn't getting it across very effectively. I tried saying that whenever a number factors into two numbers, one must be less than and one must be greater than the nearest square root. This didn't really click for him.

Next I tried drawing rectangles to demonstrate the idea. I did 36, and showed the special square rectangle that you can make with that area. I asked him what other rectangles you could draw with that area; we drew 4 by 9, 12 by 3, 18 by 2, and so forth.

I then pointed out that all the rectangles we'd drawn that weren't square had one number less than 6 (the square root) and one greater than 6. I claimed that this is always true. He then said "Oh, I get it!" but I think he's learned that I like it when he says aha, and that it wasn't a genuine aha. It's hard to fake these things.

Anybody have any ideas on how to teach this? I think one problem is that square roots are still a pretty vague concept at this age, and I wonder if this is a trick that 6th grade kids are generally taught. But if they're not taught this trick, then where are they taught to stop checking for divisibility?

MathLessonRepeatingDecimals 22 Sep 2005 - 20:01 CatherineJohnson

My neighbor showed me this yesterday. Naturally no one had ever taught me how to do this, which is par for the course. But she's a statistician & she'd never learned it, either.

I love this. It reminds me of the shenanigans I go through trying to force Microsoft Word to do graphic design.

I've entered this on the Math Lessons page.

other resources

Purple Math

Math Wizz on converting repeating decimal to fraction

update: Saxon meltdown (3-2-06)

Maybe I'm just tired, but I practically had a nervous breakdown tonight trying to convert 0.013333....(repeating decimal) to a fraction.

I just could not get it.

Finally Math Wizz saved me. Of all the websites I looked at, Math Wizz had the simplest, cleanest, & most follow-able explanation.

Math Wizz also has gigantic gifs.

SaxonAlgebraPlacementProblem 11 Oct 2005 - 00:48 CatherineJohnson

I absolutely can't get the answer Saxon gives in the answer key.

Thanks!

keywords: Saxon algebra placement test problem

AsianBrainTeaser 01 Oct 2005 - 17:50 CatherineJohnson

Just got this email from my sister:

FOLLOW THE DIRECTIONS BELOW FIRST
"Everybody has to cross the river." The following rules apply:

• Only 2 persons on the raft at a time.

• The father cannot stay with any of the daughters without their mother's presence.

• The mother cannot stay with any of the sons without their father's presence.

• The thief (striped shirt) cannot stay with any family member if the Policeman is not there.

• Only the Father, the Mother and the Policeman know how to operate the raft.

• To start click on the big blue circle on the right.

• To move the people click on them. To move the raft click on the pole on the opposite side of the river.

everybody has to cross the river

Seeing as how my day is not off to a brilliant start (I completely forgot Jimmy's 'Community Awareness' program, which started at TEN)....I'm not going to add further evidence of stupidness to Ye Old Cognitive Load by attempting this problem. Not today, anyway.

Today, I'm sticking to integers. And harvesting the basil plants to make pesto to freeze for the winter. And maybe, finally, emptying out the overflowing Mail Basket.

And forcing Christopher to do his edhelper.com worksheets. (A great tip from Susan. \$20 a year, and almost certainly worth it--definitely worth it for me, I should say.)

### help wanted

Anyone know of a good source for integer word problems?

RussianMathProblem961 05 Oct 2005 - 17:19 CatherineJohnson

961.

A point with a coordinate of -3 moves along the number line in the following manner: First, it goes 5 units in the positive direction; Then it goes 7 units in the negative direction followed by 10 units in the positive direction and 8 units in the negative direction; Then it goes 3 more units in the negative direction and, lastly, 13 units in the positive direction.

Question: What is the final location of the point on the number line?

source:
Mathematics 6 by Enn R. Nurk and Aksel E. Telgmaa, page 255

RussianMathProblem976 09 Oct 2005 - 22:14 CatherineJohnson

976.
A rectangular park is 400 m longer than it is wide. The ratio of the length of the park to its width is 5:3. How long will it take someone walking at a rate of 2 km/h to go around the park?

### question

How would a child work this problem without using algebra?

Here's the way it's done using a bar model, but I'm not seeing how you would do this without simple arithmetic.

(I hate the way this looks. I need Quark.)

Anyway, using the bar model you see that you have a bunch of equal units, and that two of these units equal 400.

Therefore, each individual unit equals 200.

So the width has to equal 3 x 200, or 600 and the length has to equal 5 x 200 or 10,000.

### ratio problem

Actually, now that I think about it, at this point in RUSSIAN MATH, kids have learned ratios & proportions, so they could just solve it that way.

width = w
length = l = w + 400

5/3 = l/w

5/3 = w + 400/w

What I can't remember is whether the book has taught kids to use two variables...

ExtendedProblem1 09 Oct 2005 - 03:12 CatherineJohnson

Find all the numbers that satisfy all of the following conditions:

1. Positive whole numbers less than 100,
2. Four more than each number is a multiple of 6
3. The sum of the digits of each number is a multiple of 4.

### and what is the best way to do this problem?

We used Doug's number lines (WHICH ARE GOING TO BE GETTING A WORKOUT THIS YEAR, IT'S OBVIOUS). We labeled one number line with multiples of 4, and the other with multiples of 6. We didn't need the multiples of 6, but it made things easier to have all the multiples of 6 sitting there, where we could see them.

PuzzleZoneQuestion 13 Nov 2005 - 14:55 CatherineJohnson

Feb. 9, 2005

A raincoat, hat, and boots were bought for \$280. The raincoat cost \$180 more than the hat, and the hat and the raincoat together cost \$240 more than the boots.

How much did each item of raingear cost?

Hint: You can solve the problem without having to write down any equations.

Answer: If instead of the raincoat, hat, and boots only two pairs of boots were bought, the price would not be \$280, but \$240 less. Thus, the two pairs of boots cost \$280 – \$240 = \$40. Hence, one pair cost \$20.

Now, we find that the raincoat and hat together cost \$280 – \$20 = \$260, the raincoat costing \$180 more than the hat. If instead of the raincoat and hat we could buy two hats, and we would pay not \$260 but \$180 less, that is, \$260 – \$180 = \$80. Hence, one hat costs \$40.

Thus, the prices of the items of raingear were as follows: boots – \$20; hat – \$40; and raincoat – \$220.

I solved this easily using algebra.

But then, when I looked at Puzzle Zone's answer, I didn't have a clue what it meant.

Now that I've spent some time mulling it over, I realize that I still don't completely 'have' the concept of subtraction as comparison (which I was never taught).

Fooling around with bar models helped me figure it out....but even though I can now draw this, I still don't understand—verbally understand—why we are talking about buying two pairs of boots as the key to solving the problem.

At the moment, there's no conceivable way I could explain this in words.

Any suggestions?

### subtraction as comparison

I taught my Singapore Math kids the idea of subtraction as comparison this week, and they were pretty thrilled.

At least, they acted thrilled. They had that nice ah-hah look on their faces.

TwentyMillionths 03 Mar 2006 - 06:02 CatherineJohnson

Place Value Chart
```                0.000000001   one billionth

0.00000001   one hundred-millionth
0.0000001   one ten-millionth
0.000001   one millionth

0.00001   one hundred-thousandth
0.0001   one ten-thousandth
0.001   one thousandth
0.01   one hundredth (1 in the hundredths place)
0.1   one tenth (1 in the tenths place)

1   one (1 in the ones or units place)
10   ten (1 in the tens place)
100   one hundred  (1 in the hundreds place)

1,000   one thousand (1 in the thousands place)
10,000   ten thousand
100,000   one hundred thousand

1,000,000   one million
10,000,000   ten million
100,000,000   one hundred million

1,000,000,000   one billion
10,000,000,000   ten billion
100,000,000,000   one hundred billion

1,000,000,000,000   one trillion...
```

source:
Math Forum

-- CatherineJohnson - 02 Mar 2006

MiniProblems 15 Jul 2006 - 16:33 CatherineJohnson

I've been complaining for months about the lack of word problems in Christopher's math class.

The kids memorize one procedure/rule/formula a day, do a few calculations, and march on. As a direct result, IMO, their knowledge really is rote as opposed to procedural. At least, Christopher's is. And I've had enough math talks with other kids in the class to know some of them are in the same boat.

Today I had a eureka moment reading a Comment left by Kathy Iggy:

The old math books I found (the same ones I used in grade school) have lots of what they call "mini problems" used to illustrate how a recently taught concept would be presented in a word problem. Megan likes these because of their brevity and she doesn't have to struggle with comprehension that much.

For example:

20 yards of ribbon. 1/4 used for dress. How much ribbon used?

That's IT!

mini problems

That's the concept, and the phrase, I've been looking for.

mini problems:word problems :: basic skills:higher order skills .

That's from Ken, and he's exactly right.

[update 4/23/2006: no! he's not right! Actually, he's write about using mini problems to teach word problems; I'm talking about mini problems to teach math - to teach the fundamental concept in a lesson. Awhile back I realized that word problems are the 'real manipulatives.' Now I know what I mean by that.

All concepts should be taught — illustrated — with mini problems. All concepts, every last one.

PRIMARY MATHEMATICS does this; SAXON MATH does it; KUMON does it. I'll post examples.

I've come to feel that the first word problems illustrating a new concept should be so simple children can do them in their heads.

For example, the very first ratio word problem a child does should be something like this:

Christopher bought one pencil for one dollar.
How many pencils can he buy for two dollars?

The question should be written this way, too: on two separate lines, so the child sees instantly that the first sentence is the set-up, and the second sentence is the question. Richard Brown's revision of Mary Dolciani's BASIC ALGEBRA, a book I like very much, does this for many of its word problems. I'll post some of those, too, as I get to it.

mini problems are applications

The problem with word problems is that, in the U.S., they're always hard.

Word problems are so hard people have apparently come to think that if a word problem isn't hard it isn't really a word problem.

I'm wondering if we ought to ditch the phrase 'word problem' (ditto for 'story problem') and adopt the word 'application.'

A better idea: we should think about the point of word problems.

Some word problems are written and assigned to give students practice.

Many word problems are written and assigned to assess whether students have developed flexible knowledge.

I'm talking about a third purpose, which is instruction. I'm talking about word problems designed to teach.

instructional word problems

A word problem is an application. A super-simple, starter word problem explains and demonstrates a mathematical concept by showing students how the concept is applied.

As a matter of fact, an instructional word problem shouldn't even be a 'problem.' It should just be a question, and the answer should be obvious.

A simple, instructional mini-problem should not test the child, should not challenge the child, and certainly should not trick the child.

It should teach.

examples to come

be sure to see Google Master's comment

how do you teach your child word problems?
mini problems (important)
arithmetic to algebra & mini-problems

-- CatherineJohnson - 07 Mar 2006

GregMatteoProblemsThatTeach 11 Mar 2006 - 19:29 CatherineJohnson

For excellent examples of problems that teach, try Math TV at Action Math. The video solutions for some of the fraction and ratio problems even use Singapore methods. Christopher may really enjoy these.

Greg DeMatteo
Norwood Middle School

I'm thrilled to get these — will report as soon as I've watched!

(Sorry. I'm developing a slight THING for animated gifs....)

back again

Action Math is a BLAST!

Here's the first problem:

A new movie theater opened up in town.
The theater contains 40 rows of chairs.
The first row has 10 chairs.
Each additional row has two more chairs than the row before it.
What is the seating capacity of the theater?

This brings up something important.

I think early word problems should be written exactly this way, as a list of sentences, not a paragraph. This helps 'disaggregate' basic working memory and 'environmental dependency' issues from the math problem itself.

The child isn't constantly having to search back and forth in a paragraph to re-isolate the individual sentences.

These videos are incredible!

You MUST go see them.

Wonderful.

I've just gained more conceptual understanding of algebra watching one video than I did in 3 years of high school! (I'm pretty close to serious about that....)

Greg, THANK YOU!

OK, now I'm sick.

If Christopher had had a teacher like the one on this website all year long.....we wouldn't be studying for the state test.

Compare her step-by-step explanation of the solution to the non-explanation of this Extended Response problem Christopher was given early this year:

challenge? or teach?

Virtually all of the problems Christopher has been given this year have been 'Challenge' problems. That's what the Assistant Superintendent in charge of curriculum told me about the Extended Response problems: "These students need to be challenged." (Christopher brought home a whole batch of challenge problems last night; I'll post some of them tomorrow.)

Virtually none of the problems Christopher has been given this year have been Teaching problems.

There is a vast difference between challenging a student and teaching a student.

keyword: actionmath

-- CatherineJohnson - 10 Mar 2006

HowDoYouMakeAModelOfAPlanet 14 Mar 2006 - 07:39 CatherineJohnson

It's 8:30 am Saturday morning, and Christopher just reminded me that he has to 'make a model of the 10th planet' this weekend and bring it in to school on Monday.

He doesn't know how to make a model of the 10th planet, and neither do I. He was thinking of making a clay ball. Good idea! That means a trip to Toys R Us for clay, which will be highly educational.

If any of you knows how to make a model of the 10th planet quickly and efficiently I'm all ears.

The reason we have to do this is that the Irvington Education Foundation funded something called The Jason Project (JASON Fosters the Joy of Teaching!) so now we have to make a model of the 10th planet.

Up until last year I was a dedicated Raiser and Donater of money to the school.

No good deed goes unpunished.

my fevered brain

I was thinking just the other day that my mother used to make papier mache with us all the time, and I've never done it with my own kids.

I'm pretty sure you can model the 10th planet using papier mache.

ktm Brain Trust comes through again!

from Ken:
The easy way out is to go to the art supply store and pick up a styrofoam ball and paint it "tenth planet" color.

[question: has Tenth Planet color been added to Crayola Crayons?]

back to Ken:
The more advanced tenth planet modeler (i.e., crazy) could apply modeling clay or papier mache to the ball to simulate terrain. The modeling clay has the advantage since it doesn't then have to be painted.

and:

or you could always go the more authentic route and use ice

[I'm sitting here reading this stuff out loud to Ed and I am HORSE LAUGHING]

from Susan:
If you have to use stuff around the house, paper mache' would work, with the quick and messy flour, water, salt mixture, and some little ball around the house. It would probably have a nice Flinstone look about it, but it would do.

[question: salt?]

OK, now let's hear from an expert:

Carolyn Morgan, grade 5 teacher:
Styrofoam works. You can suggest terrain by creating little valleys with the side of a knife handle or fingers. One good thing -- it's fast and not at all messy.

while we're on the subject of things I don't know how to do

This reminds me.....I have another question for the ktm Brain Trust.

How come the odds of rolling snake eyes are 1/36 but the odds of rolling a 3 are 1/18?

simple papier mache recipe
papier mache instructions ("papier mache is the ultimate recyclers dream")
papier mache at Enchanted Learning
papier mache by Rozani

-- CatherineJohnson - 11 Mar 2006

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