[kenberg]

WellSpyder, on 2015-March-09, 11:14, said:

You might not want to order seven quarters of a pound of fish. But what about sweets? Do you still have shops selling old-fashioned loose sweets by the quarter pound? "I'll have seven quarters of lemon sherberts, please"

Or what about thinking of currency? I might have one dollar and three quarters in my pocket. Or I might have seven quarters. I would expect to be able to buy the same things with them, though...

Yes, I had suggested to Becky that she talk about having seven quarters in change. Still, if she had seven quarter in her purse and someone asked her how much money she had she would not say she had 7/4 of a dollar. I think it is useful though.

What I suspect is going on: She was told as a child that the fraction 7/4 is always to be converted to 1 3/4. She over learned this lesson. 7/4 is marked wrong, 1 3/4 is marked right. 7/4 bad, 1 3/4 good.

It brings to mind a childhood experience. We had a substitute teacher who was suppose to talk to us about rounding off numbers. The rule was that if the number ends in 1,2,3 or 4 you round down, if 5,6,7,8,9 you round up. Ok. But then there was an exercise problem where we were to round 4.49 to the nearest integer. He insisted the answer was 5. First 4.49 gets rounded to 4.5, and then 4.5 gets rounded to 5. Just apply the rule. I insisted that this was nuts. He was bigger than I was.

[barmar]

Using things like quarters and quarts of milk seems like a good way to teach the relationship between mixed numbers and improper fractions. You can go either way:

1. The machine at the laundromat charges $1.75, and only takes quarters. How many quarters do you have to put into it?

2. You put 5 quart bottles of milk into your shopping cart. How much milk do you have?

[kenberg]

Backy and Janet got together for an hour this morning. This was their first face to face on this endeavor. It continues to be instructive and interesting. They talked a bit about improper fractions, and Janet accepted their validity, although Becky was not sure if she really buys ito it or is simply trusting that what Becky says must be right. They reduced some fractions. Janet knew that 4/8 was the same as 1/2 but she did not think of it in general terms. Eventually Becky asked her to reduce 26/39 to lowest terms. Janet was thinking but not moving and so Becky asked what she was thinking. She was thinking of dividing the numerator and denominator by 13 but this seemed wrong. Why? She thought you always had to divide the numerator and denominator by single digit numbers.


My mind often jumps around and I remembered something Becky said about raising kids (we each have kids from a previous marriage): You warn against this and you warn against that but somehow they always find something to do that it never crossed your mind that they would do.

Why wouold someone think you can only divide out by single digit numbers when reducing fractions? I can't answer that with any specifics, but I think we all know that people can get overly submissive to rules, real or imagined. If declarer has AKxxx opposite JT9x, and if lho shows out when he leads the Ace, it really is ok to go to the board and run the Jack, no matter what the rule about nine never says.

Anyway, this is moving along. Janet was tense, and I admire her for taking this on.

Note: Maybe we need a name other than "improper" for fractions such as 7/4. And what is a person supposed to make of irrational numbers. And then their are radicals. The square root of 2 involves a radical and is irrational. Heavens! But then Pi is transcendental, that must be good, and is not at all imaginary or complex.

Added: I was just at a math talk where the speaker quoted Tolstoy, revising Newton:
"Force (the volume of motion) is the product of the mass into the velocity."
See Was and Peace
I grant a little literary license here, the meaning is clear in context.

[hrothgar]

kenberg, on 2015-March-10, 11:00, said:

Note: Maybe we need a name other than "improper" for fractions such as 7/4.

After a bit of thought, I would like to nominate "irresidual" (which I believe I might have just coined)

[Zelandakh]

barmar, on 2015-March-09, 09:50, said:

This is, of course, what makes teaching a class of 30ish students such a challenge -- each student will have different strengths and weaknesses that the teacher has to try to address. With one-on-one tutoring you should be able to zero in on the specific problem areas and come up with a way to clarify them.

Absolutely, and that is one reason I have not become a teacher despite enjoying the one-on-one teaching. But that is also why I am so adamant that the same approach is not right for both. Of course a good teacher will usually be a good tutor too; I am not so convinced the other way around necessarily holds.

And finally, no need to make up a new name for improper fractions - they are also known as top-heavy fractions. Of course, if Janet likes the term "irresidual" then why not? A rose by any other name and all that!

[Elianna]

kenberg, on 2015-March-10, 11:00, said:

Backy and Janet got together for an hour this morning. This was their first face to face on this endeavor. It continues to be instructive and interesting. They talked a bit about improper fractions, and Janet accepted their validity, although Becky was not sure if she really buys ito it or is simply trusting that what Becky says must be right. They reduced some fractions. Janet knew that 4/8 was the same as 1/2 but she did not think of it in general terms. Eventually Becky asked her to reduce 26/39 to lowest terms. Janet was thinking but not moving and so Becky asked what she was thinking. She was thinking of dividing the numerator and denominator by 13 but this seemed wrong. Why? She thought you always had to divide the numerator and denominator by single digit numbers.


My mind often jumps around and I remembered something Becky said about raising kids (we each have kids from a previous marriage): You warn against this and you warn against that but somehow they always find something to do that it never crossed your mind that they would do.

Why wouold someone think you can only divide out by single digit numbers when reducing fractions? I can't answer that with any specifics, but I think we all know that people can get overly submissive to rules, real or imagined. If declarer has AKxxx opposite JT9x, and if lho shows out when he leads the Ace, it really is ok to go to the board and run the Jack, no matter what the rule about nine never says.

Anyway, this is moving along. Janet was tense, and I admire her for taking this on.

Note: Maybe we need a name other than "improper" for fractions such as 7/4. And what is a person supposed to make of irrational numbers. And then their are radicals. The square root of 2 involves a radical and is irrational. Heavens! But then Pi is transcendental, that must be good, and is not at all imaginary or complex.

Added: I was just at a math talk where the speaker quoted Tolstoy, revising Newton:
"Force (the volume of motion) is the product of the mass into the velocity."
See Was and Peace
I grant a little literary license here, the meaning is clear in context.

She sounds like she learned Math, and still thinks of Math, in a very procedural way.

I try to encourage students to NOT memorize procedures, and always go back to first principals to figure out what is going on, until the procedure makes mathematical sense, and they can create the procedure from an understanding of the mathematics involved, not the memorization of that specific problem. It's hard, though, because I don't get them until they're 15 or so, and retraining them when the've experienced success in the past by relying on (temporarily) memorizing.

[kenberg]

Elianna, on 2015-March-10, 20:21, said:

She sounds like she learned Math, and still thinks of Math, in a very procedural way.

I try to encourage students to NOT memorize procedures, and always go back to first principals to figure out what is going on, until the procedure makes mathematical sense, and they can create the procedure from an understanding of the mathematics involved, not the memorization of that specific problem. It's hard, though, because I don't get them until they're 15 or so, and retraining them when they've experienced success in the past by relying on (temporarily) memorizing.

I believe you when you speak of the difficulties. I think it is partly because of early experience, and partly because of essentially different personalities. I think that most people, even if they like the comfort of having strict rules, can with guidance such as you supply learn to appreciate the fundamental logical nature of mathematics. But some fight against it with all of their energy.

[mike777]

Retraining in math at 15 seems not the best use of very scarce resources


If the customer want to pay ./.....ok


If the teacher has unlimited money and time ...ok.


the problem is teachers such as Elianna are thinking of leaving or staying at home.

[Zelandakh]

You are right Mike, it would be far better if the children were taught properly for the long term at an early age. Unfortunately in the real world there are many reasons why a teacher might prefer to get results in the shorter term and that leads to the problem of retraining later on so as to give a student a proper understanding of the subject.

[helene_t]

kenberg, on 2015-March-10, 11:00, said:

She thought you always had to divide the numerator and denominator by single digit numbers.

Maybe this is because some algorithms treat single digit numbers differently. For example when you multiply something with a double digit number, you may multiply with each of the digits seperately and add up the results as in x*47 = (x*4)*10 + x*7.

This could be a good opportunity to discuss the difference between what a number really is (as opposed to the way we happen to write it using the base ten system) and about the distinction between the general applicability of theorems such as

(ax)/(ay) = x/y

as opposed to the way we have implemented calculations in algorithms.

[kenberg]

Probably we all agree on the following, at least in some form. It would be good to have her, and many others, move from "mathematics consists of rules" to "mathematics makes sense".

I mentioned a few posts up about the sub we had in elementary school who rounded off 4.49 to the nearest integer by 4.49->4.5->5, twice applying rules about how to round off one digit at a time. He was absolutely impervious to my contention that 4.49 is closer to 4 then it is to 5. It wasn't that he had some alternative argument. His entire view was that he was following the rule and I should follow the rule.

I was chatting about this with someone yesterday and he swears that he saw the following in a newspaper article about (uh oh) global warming. The author of the article let the reader know that measurements had shown that over a certain period of time the polar ice cap had warmed by 0.1 degree Centigrade. The author was considerate of his American readers, and figured that they would like this in Fahrenheit. So he looked up a formula and explained that this meant an increase of 32+9(0.1)/5 = 32.18 degrees on the Fahrenheit scale. Apparently nothing struck him as unusual about this.

[kenberg]

Anyway, I think that Zel's response to Mike is about right, there are real worls problems. My next door neighbor teaches second grade (that means seven year olds for you non Americans) and I think that the kids are lucky to have her. She is called upon to handle math issues at least in her school and maybe beyond. She is by no means a mathematician or anything like it, but she has good sense and does not dissolve into a pool of water when asked to address an issue.

Most mathematics ( this may not apply with classes for the very best students) in grades 1-6 really is within the reach of anyone. The problems lie in attitude, not intelligence. As kids get into middle school then some items can get thorny and a specialist would be useful. Even in grades 7 and 8 it seems to me that a typical teacher could handle this if s/he put her/his mind to it. But it probably takes some serious time and effort, and there are many things to teach, so a specialist is right.

[barmar]

I see stuff like this all the time on StackOverflow, where much of the content is helping newbie programmers with their problems.

People often don't learn the general principles, but instead they learn by osmosis by copying examples. The result of this is that they sometimes take steps that are totally unnecessary in the particular case they're doing, just because they're often done in examples. For instance, when reading from a database, you usually get multiple sets of results, and you have to perform the same action on each set in a loop (e.g. displaying each set as a row in a table). But sometimes you know you're only getting one result (such as looking up the name and address of the user who is logging in); yet many beginners still use a loop to process this result. They've just learned the idiom "while ($row = fetch()) { do stuff }", and never realized that you can do "$row = fetch()" without looping.

So the woman thinking you can only reduce fractions using single digits may simply have come from the fact that when she learned it, they always used simple examples, and they always had single digits. She made an incorrect generalization from this.

[blackshoe]

barmar, on 2015-March-10, 09:56, said:

Using things like quarters and quarts of milk seems like a good way to teach the relationship between mixed numbers and improper fractions. You can go either way:

1. The machine at the laundromat charges $1.75, and only takes quarters. How many quarters do you have to put into it?

2. You put 5 quart bottles of milk into your shopping cart. How much milk do you have?

1. Too many.
2. Too much.

:-)

[barmar]

blackshoe, on 2015-March-11, 11:36, said:

1. Too many.
2. Too much.

:-)

You got that right. At the beginning of this year they raised the price on my condo's washer and drier from $1 to $1.50. As we used to say every year in collegewhen they announced the new tuition: TDM! (To be fair, it's been over 5 years since it went from 75 cents to $1, and water and heating prices have gone way up.)

Hey, ask the woman if she know what percentage increase that was.

[mike777]

Now percentages are very tricky stuff, now combine them with fractions.

I mean is 2% milk 1% greater than 1% milk? How much greater is 1/2 of 2%


This stuff confuses people when talking about bond math all the time. A coupon or savings account goes from 4% to 3%. Is this a 1% drop? Unemployment goes from 7% to 6% a 1% drop?

[kenberg]

barmar, on 2015-March-11, 11:33, said:

I see stuff like this all the time on StackOverflow, where much of the content is helping newbie programmers with their problems.

People often don't learn the general principles, but instead they learn by osmosis by copying examples. The result of this is that they sometimes take steps that are totally unnecessary in the particular case they're doing, just because they're often done in examples. For instance, when reading from a database, you usually get multiple sets of results, and you have to perform the same action on each set in a loop (e.g. displaying each set as a row in a table). But sometimes you know you're only getting one result (such as looking up the name and address of the user who is logging in); yet many beginners still use a loop to process this result. They've just learned the idiom "while ($row = fetch()) { do stuff }", and never realized that you can do "$row = fetch()" without looping.

So the woman thinking you can only reduce fractions using single digits may simply have come from the fact that when she learned it, they always used simple examples, and they always had single digits. She made an incorrect generalization from this.

As far as much of html code is concerned, I am one of those who copied and modified. I find it useful to remind myself of this when asking why people just go with what they have seen in math. But html is not exactly fundamental in the way math is, oir at least I don't see it that way. And,as an aside, I recently took a (very) brief course in html just so I could feel like I was more in control.

[mycroft]

At one point I started working through my then partner's Real Analysis book.

This was showing how math works (from the inside).

I am an Engineer - our math was copious, but included many places where the professor said "there's a hole here. You'll never see it so we aren't worrying about it, but..."

I lasted 85 pages before I was happy to go back to the Engineering view of math as "it works, don't worry about it".

This is the step between *my* "math is fundamental" and the real "math is fundamental".

[Zelandakh]

mycroft, on 2015-March-12, 09:25, said:

At one point I started working through my then partner's Real Analysis book.

I lasted 85 pages before I was happy to go back to the Engineering view of math as "it works, don't worry about it".

You did well - some of my fellow maths students didn't last 85 pages of Analysis. Given how most engineers feel about studying maths I am amazed you even tried!

[kenberg]

mycroft, on 2015-March-12, 09:25, said:

At one point I started working through my then partner's Real Analysis book.

This was showing how math works (from the inside).

I am an Engineer - our math was copious, but included many places where the professor said "there's a hole here. You'll never see it so we aren't worrying about it, but..."

I lasted 85 pages before I was happy to go back to the Engineering view of math as "it works, don't worry about it".

This is the step between *my* "math is fundamental" and the real "math is fundamental".

One of my Profs from the 1960s, Larry Markus , was at the board whipping through an argument with various tensors. he remarked "The whole purpose of tensor notation is to allow you to do things far faster than you can understand what it is that you are doing". As with most memorable jokes, there is some truth to it.

My own observations about the utility of mathematics runs like this: "Most mathematicians I have known went into math because they enjoyed it. People pay us because it is useful. It's a good arrangement. People who do not enjoy it will pay us to do something that we do enjoy".

Is a deep understanding of math useful? I guess Norbert Wiener would say yes, Or Alan Turing. Or John von Neumann, or Sergei Brin. Etc. It is my understanding that when von Neumann first began employing operators in hilbert space to study quantum mechanics, a fair portion of the physics community though he was nuts. So you wait a few years for people to catch up.

It's a big wide world with room for different approaches.

Getting back to the more elementary level with fractions, or maybe geometry and algebra, I think that one practical reason for young people to become comfortable with the basic ideas is that it substantially increases their options for making a living. I was at a store in a mall once when I saw the following (numbers made up, but it was along these lines): A shirt is marked at $40, with 30% off, discount taken at the register. A customer asks what the actual price will be. The store clerk has no idea. OK, the customer has enough money to buy a decent shirt, and the store clerk has a job. But maybe he would like to move beyond being a store clerk. I think it would help (yes I know there are other considerations) if he wasn't completely thrown for a loop by the simplest of calculations.

But mostly, for me, I think a person should be able to work out the price with the discount simply because he lives in the world. Iwatched, not for the first time, Born Yesterday with Judy Holliday as Billie. The Supreme Court is mentioned and Billie says "What is it?". You can get through life without knowing, but it seems like it might be intersting to learn a few things here and there. If I have it right, this is Janet's intention. She just got tired of always having to say "well, I don't understand any math stuff".