# The one that is the smaller is the larger

March 8, 2014 7:55 AM Subscribe

The Teaching of Arithmetic: The Story of an experiment.

*In the fall of 1929 I made up my mind to try the experiment of abandoning all formal instruction in arithmetic below the seventh grade and concentrating on teaching the children to read, to reason, and to recite - my new Three R's. And by reciting I did not mean giving back, verbatim, the words of the teacher or of the textbook. I meant speaking the English language. I picked out five rooms - three third grades, one combining the third and fourth grades, and one fifth grade. I asked the teachers if they would be willing to try the experiment.*Fascinating. I had read some work a while back explaining the problems of founding math on arithmetic, rather than algebraic thinking, and how it made math harder and less enjoyable in the long run for the majority of students, with very little benefit. This reminds me of that research: the emphasis on numbers as concrete entities existing in a vacuum, rather than related to each other and the world they exist within (especially in the third part of the story) is eerily similar. Sadly, I've got three kids in public school here in Washington, and I don't see this changing any time soon. Hell, we recognized gay marriage and legalized weed up here, and both of those were probably a walk in the park compared to the challenge of any fundamental change to the school curriculum.

posted by hank_14 at 8:34 AM on March 8, 2014

posted by hank_14 at 8:34 AM on March 8, 2014

It always kind of bothered me that most of my peers dreaded, in math, the Word Problem, and that it seemed to be because they had so much trouble parsing just what the problem was actually asking, which didn't seem to be measuring what I thought of as "ability to do math".

posted by Sequence at 8:34 AM on March 8, 2014 [3 favorites]

posted by Sequence at 8:34 AM on March 8, 2014 [3 favorites]

The teaching of arithmetic to young children has changed substantially since this was first published. It is also unclear what kinds of followup studies were done to validate the report.

posted by humanfont at 10:03 AM on March 8, 2014

posted by humanfont at 10:03 AM on March 8, 2014

*they had so much trouble parsing just what the problem was actually asking, which didn't seem to be measuring what I thought of as "ability to do math".*

I think this skill is indisputably part of the ability to do math.

posted by escabeche at 10:44 AM on March 8, 2014 [5 favorites]

*It always kind of bothered me that most of my peers dreaded, in math, the Word Problemâ€¦*

I hear this a lot, but it never made sense to me. I always loved word problems; they provide a link albeit sometime pretty tenuous, between mathand the real world. What always filled me with dread were those mimeograph sheets full of what looked like hundreds of arithmetic problems. To my approximately 10 year old self they looked like they would take all night to finish, so I would often say screw these, I'm going outside.

posted by TedW at 10:57 AM on March 8, 2014

The skill of making good estimates is really valuable, in everyday life and in math. An good estimate of 22 * 13 is found with 20*15. Having an estimate not only gives you a check on that calculation, but could let you know that doing that calculation isn't necessary.

Oh, look at that good sale price. How many can I buy? Each costs $7.29. I've got $60. 10 would be about $73, so 9 for about $66. Hmmm, I guess I could get 8. Wait, I forgot the 10% tax. So each will cost about $8. Guess I can get 7.

Someday maybe Siri would get through that. I'd rather not wait.

posted by Twang at 11:28 AM on March 8, 2014 [1 favorite]

Oh, look at that good sale price. How many can I buy? Each costs $7.29. I've got $60. 10 would be about $73, so 9 for about $66. Hmmm, I guess I could get 8. Wait, I forgot the 10% tax. So each will cost about $8. Guess I can get 7.

Someday maybe Siri would get through that. I'd rather not wait.

posted by Twang at 11:28 AM on March 8, 2014 [1 favorite]

*Oh, look at that good sale price. How many can I buy? Each costs $7.29.*

See, and in this scenario, I'm the guy who's like "wow, a power of 3, what are the chances, what about a prime power more generally" and then before you know it the cashier is all "Sir? Sir? Did you want a receipt?"

posted by escabeche at 11:51 AM on March 8, 2014 [11 favorites]

I like his 5-question test. Real-life problem solving is not like a linear math drill. It is a complex web of decisions. Children transitioning into adulthood will be well-served by the ability to discard distractions and zero in on the core of a problem. It's a hard lesson to learn while you are already being crushed under the weight of those distractions.

posted by mantecol at 2:36 PM on March 8, 2014

*1. Two boys start out together to race from Manchester to West Concord, a distance of 20 miles. One makes 4 miles an hour and the other 5 miles an hour. How long will it be before both have reached West Concord?*

2. A man can row 4 miles an hour in still water. How long will it take him to row from Hill to Concord [24 miles one way] and back, if the river flows south at the rate of 2 miles an hour?

3. The same man again starts rowing from Hill to Concord in the spring when the water is high and the current is twice as swift as it was before. How long will it now take him to make the round trip?

4. Remus can eat a whole watermelon in 10 minutes. Rastus in 12. I suggest a race between them, giving each half of a melon. How long will it be before the melon is entirely gone?

5. The distance from Boston to Portland by water is 120 miles. Three steamers leave Boston, simultaneously, for Portland. One makes the trip in 10 hours, one in 12, and one in 15. How long will it be before all 3 reach Portland?

It looks easy enough, but I advise you to try it. I will guarantee that high school seniors, preparing for College Entrance Board Examinations in Mathematics, will not average 70 percent. I had some rather ridiculous results. I tried the fourth and fifth examples on a second grade the other day and had an almost perfect score, while a ninth-grade class in arithmetic, which had been taught under the old arithmetical curriculum, made a sorry showing. Out of twenty-nine in the class only six gave me the correct answer to problem five.2. A man can row 4 miles an hour in still water. How long will it take him to row from Hill to Concord [24 miles one way] and back, if the river flows south at the rate of 2 miles an hour?

3. The same man again starts rowing from Hill to Concord in the spring when the water is high and the current is twice as swift as it was before. How long will it now take him to make the round trip?

4. Remus can eat a whole watermelon in 10 minutes. Rastus in 12. I suggest a race between them, giving each half of a melon. How long will it be before the melon is entirely gone?

5. The distance from Boston to Portland by water is 120 miles. Three steamers leave Boston, simultaneously, for Portland. One makes the trip in 10 hours, one in 12, and one in 15. How long will it be before all 3 reach Portland?

It looks easy enough, but I advise you to try it. I will guarantee that high school seniors, preparing for College Entrance Board Examinations in Mathematics, will not average 70 percent. I had some rather ridiculous results. I tried the fourth and fifth examples on a second grade the other day and had an almost perfect score, while a ninth-grade class in arithmetic, which had been taught under the old arithmetical curriculum, made a sorry showing. Out of twenty-nine in the class only six gave me the correct answer to problem five.

posted by mantecol at 2:36 PM on March 8, 2014

I wish he'd given the answers to his five-question test. I think I scored 100%.

posted by Snowflake at 4:58 PM on March 8, 2014 [1 favorite]

posted by Snowflake at 4:58 PM on March 8, 2014 [1 favorite]

This is a pretty radical change, and I can't believe he got away with trying the experiment so casually. On the other hand, as a former physics TA, I think there's something to it. Teach a kid (or even a college student) an algorithm or equation, and they become like the person with a hammer to whom everything looks like a nail. If they stick with the subject they eventually move past that... Or maybe only those who do move past it are able to stick with the subject? In any case, not everyone does.

This article seems a little meatier than the one posted the other day about learning math concepts through games and toys, in terms of actually describing the content of the proposed curriculum and the test questions by which the results were measured in some detail, and it seems plausible to me. But I would like to see a more formal study. I have seen some of thepapers published by the physics education research community, and I know it is possible to be a lot more quantitative than this in assessing the effectiveness of a curriculum change.

posted by OnceUponATime at 7:13 PM on March 8, 2014

This article seems a little meatier than the one posted the other day about learning math concepts through games and toys, in terms of actually describing the content of the proposed curriculum and the test questions by which the results were measured in some detail, and it seems plausible to me. But I would like to see a more formal study. I have seen some of thepapers published by the physics education research community, and I know it is possible to be a lot more quantitative than this in assessing the effectiveness of a curriculum change.

posted by OnceUponATime at 7:13 PM on March 8, 2014

*Mr. Benezet thereupon puts down 750 for the answer. When he asks how many in the room agree that this is right, practically every hand is raised. By this time the local superintendent was pacing the door at the rear of the room and throwing up his hands in dismay at this showing on the part of his prize pupils. After a time, as Mr. Benezet looks a little puzzled, the children gradually become a little puzzled also. One little girl, Elsie Miller, finally comes to the board, reverses the figures, subtracts, and says the answer is 250 years.*

Mr. B.: All right. If the falls have retreated 2500 feet in 250 years, how many feet a year have the falls moved upstream?

Child: Two feet.

Mr. Benezet registers complete satisfaction and asks how many in the class agree. Practically the whole class put hands up again.

Mr. B.: Well, has anyone a different answer?

Child: Eight feet.

Another child: Twenty feet.

Finally Elsie Miller again gets up, and says the answer is ten feet.

Mr. B.: What? Ten feet? (Registering great surprise)

The class, at this, bursts into a roar of laughter. Elsie Miller sticks to her answer, and is invited by Mr. Benezet to come up and prove it. He says that it seems queer that Elsie is so obstinate when everyone is against her. She finally proves her point, and Mr. Benezet admits to the class that all the rest were wrong.

Mr. B.: All right. If the falls have retreated 2500 feet in 250 years, how many feet a year have the falls moved upstream?

Child: Two feet.

Mr. Benezet registers complete satisfaction and asks how many in the class agree. Practically the whole class put hands up again.

Mr. B.: Well, has anyone a different answer?

Child: Eight feet.

Another child: Twenty feet.

Finally Elsie Miller again gets up, and says the answer is ten feet.

Mr. B.: What? Ten feet? (Registering great surprise)

The class, at this, bursts into a roar of laughter. Elsie Miller sticks to her answer, and is invited by Mr. Benezet to come up and prove it. He says that it seems queer that Elsie is so obstinate when everyone is against her. She finally proves her point, and Mr. Benezet admits to the class that all the rest were wrong.

He better have apologised to little Elsie Miller.

posted by you must supply a verb at 3:46 AM on March 9, 2014 [4 favorites]

*This is like a trick question, right?*

5. The distance from Boston to Portland by water is 120 miles. Three steamers leave Boston, simultaneously, for Portland. One makes the trip in 10 hours, one in 12, and one in 15. How long will it be before all 3 reach Portland?

I mean, is the question just basically asking

"If someone is waiting at the finish line (Portland), how long before they can physically touch all three ships?"Which would be 15 hours, right?

That is: Steamer A arrives after 10 hours, Steamer B arrives 2 hours later, and finally, after 3 more hours (15 hours after it departed) Steamer C finally pulls in--so it takes a total of 15 hours before

**ships are there.**

*all 3*Or am I just reading this question wrong?

posted by blueberry at 2:44 PM on March 9, 2014

You're reading it right. All five questions are like that:

1.

For all of these, the trick is noticing that the only thing that matters is how long it takes the slowest one.

For both of these, the idea is that however much the river slows him down going one direction, it speeds him by the same amount going the other direction, so the rivers' rate of flow is irrelevant.

posted by OnceUponATime at 8:11 AM on March 10, 2014

1.

*... How long will it be before*

4. ... I suggest a race between them, giving

5. ... How long will it be before**both**have reached West Concord?4. ... I suggest a race between them, giving

**each half**of a melon. ...5. ... How long will it be before

**all 3**reach Portland?For all of these, the trick is noticing that the only thing that matters is how long it takes the slowest one.

2. How long will it take him to row from Hill to Concord

3. The current is twice as swift as it was before. How long will it now take him to make the

2. How long will it take him to row from Hill to Concord

**and back**, if the river flows south at 2 mph...3. The current is twice as swift as it was before. How long will it now take him to make the

**round trip?**For both of these, the idea is that however much the river slows him down going one direction, it speeds him by the same amount going the other direction, so the rivers' rate of flow is irrelevant.

posted by OnceUponATime at 8:11 AM on March 10, 2014

*Meanwhile, I was distressed at the inability of the average child in our grades to use the English language. If the children had original ideas, they were very helpless about translating them into English which could be understood. I went into a certain eighth-grade room one day and was accompanied by a stenographer who took down, verbatim, the answers given me by the children. I was trying to get the children to tell me, in their own words, that if you have two fractions with the same numerator, the one with the smaller denominator is the larger. I quote typical answers.*

"The smaller number in fractions is always the largest."

"If the numerators are both the same, and the denominators one is smaller than the one, the one that is the smaller is the larger."

"If you had one thing and cut it into pieces the smaller piece will be the bigger. I mean the one you could cut the least pieces in would be the bigger pieces."

"The denominator that is smallest is the largest."

"If both numerators are the same number, the smaller denominator is the largest - the larger - of the two."

"If you have two fractions and one fraction has the smallest number at the bottom. It is cut into pieces and one has the more pieces. If the two fractions are equal, the bottom number was smaller than what the other one in the other fraction. The smallest one has the largest number of pieces - would have the smallest number of pieces, but they would be larger than what the ones that were cut into more pieces."

The average layman will think that this must have been a group of half-wits, but I can assure you that it is typical of the attempts of fourteen-year-old children from any part of the country to put their ideas into English.

"The smaller number in fractions is always the largest."

"If the numerators are both the same, and the denominators one is smaller than the one, the one that is the smaller is the larger."

"If you had one thing and cut it into pieces the smaller piece will be the bigger. I mean the one you could cut the least pieces in would be the bigger pieces."

"The denominator that is smallest is the largest."

"If both numerators are the same number, the smaller denominator is the largest - the larger - of the two."

"If you have two fractions and one fraction has the smallest number at the bottom. It is cut into pieces and one has the more pieces. If the two fractions are equal, the bottom number was smaller than what the other one in the other fraction. The smallest one has the largest number of pieces - would have the smallest number of pieces, but they would be larger than what the ones that were cut into more pieces."

The average layman will think that this must have been a group of half-wits, but I can assure you that it is typical of the attempts of fourteen-year-old children from any part of the country to put their ideas into English.

Oh gods, this. I was a little better-trained in spitting out the right arithmetic vocabulary, but could not seem to convince any teachers, parents, or tutor that learning the names of the components of the problem was not actually explaining to me how to solve it.

posted by desuetude at 10:07 AM on March 10, 2014

I remember an old one from my childhood:

posted by andycyca at 10:08 AM on March 10, 2014 [1 favorite]

Al and Bert are in two cities 200 km apart and start cycling towards each other at 10 km/h. At the same time, a mosquito flies from Al's head towards Bert at a speed of 20 km/h. When it reaches Bert, it goes around towards Al again and goes back and forth until the two friends swat it in a victorious high five when they meet.Assume there's no air friction, the road's density is constant and the cyclists eat their snacks without stopping

What's the total distance traveled by the mosquito?

posted by andycyca at 10:08 AM on March 10, 2014 [1 favorite]

For both of these, the idea is that however much the river slows him down going one direction, it speeds him by the same amount going the other direction, so the rivers' rate of flow is irrelevant.If the 2 mph current doubles (question 3), the guy rowing 4 mph is going to have a bad time.

posted by mpark at 10:26 AM on March 10, 2014 [1 favorite]

Hah! That's what I get for ignoring the numbers and trying to extract general principles! I I am suitably chastened.

posted by OnceUponATime at 10:33 AM on March 10, 2014

posted by OnceUponATime at 10:33 AM on March 10, 2014

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posted by jepler at 8:23 AM on March 8, 2014