Question about elementary mathematics and teaching

[Alex Franke]

I started reading the book Knowing and Teaching Elementary Mathematics, which is based on a doctoral thesis. I was a little alarmed by a couple of the statistics and I wonder if any of you all might have some comments on it.

Twenty-three U.S. elementary school teachers were asked to create a "story" to conceptually illustrate the question, "What is 1 3/4 divided by 1/2?" Only one gave a correct story, and the problem was approached in only one way. When it came to calculating the answer on paper, 40% got it right.

To contrast, 90% of Chinese teachers were able to come up with a correct conceptual story, with about 17% providing more than one strategy (e.g. measurement versus product/factors). All of them came up with the corerct answer on paper.

My wife and I were talking about it, and I wonder what the deal is. I suppose that if elementary teachers don't usually teach division by fractions, then they might have assumed the author meant to multiply (which is what nearly all of them did). But, could it be that we're just really missing the boat (assuming this admittedly small sample reflects the population)?

[atgcpaul]

I can get the correct answer but I'm not exactly sure how I would explain it
conceptually. I was taught fractions in 5th grade--25 years ago. I don't
remember how Mr. King taught us how to divide fractions but it worked for
me. Make improper fractions out of any mixed numbers. Flip the denominator
for the numerator on the divisor and multiply the two numbers together.

[Bill in Buena Park]

I was also taught division by fractions in elementary school, and was just this past week trying to explain the concept to my 4th grader. She understood the mechanics of the operation, but she didn't really understand the concept of getting a larger number than the dividend. I reminded her that with division, being multiplication in reverse, you're trying to find out how many of the divisor "goes-inta" the dividend, and the light went on.

[cabinetman]

I was also taught in early math to convert to a fraction. But as a woodworker and cabinetmaker I find the need for a quick on the spot computation without doing all the math with a pad and pencil. In addressing the example in the OP, for a simple fraction, to double the bottom number will divide it in two. For 1¾, I take half of the whole number, and half of the fraction. So, in essence, I'm converting to fractions, but done in my head. I figure the largest common denominator, which would be 8, and half of 1 is 4/8, and half of ¾ is ⅜. Together, I get 7/8.
.

[atgcpaul]

For 1¾, I take half of the whole number, and half of the fraction. So, in essence, I'm converting to fractions, but done in my head. I figure the largest common denominator, which would be 8, and half of 1 is 4/8, and half of ¾ is ⅜. Together, I get 7/8.
.

I think this illustrates what's wrong in America (in, not with). The
interpretation of the question can be misconstrued to say "What is 1 3/4
divided in half?" when the way it is phrased and how I was taught lays it out
just like a formula, "What is 1 3/4 divided by 1/2?" or

? = (1 3/4) / (1/2)

? = 3 1/2

Maybe in Mandarin, when asked the same question, there is less wiggly room
in the interpretation and that's why they had 100% correct. Or in China, they
asked all the math teachers while in America they asked everyone else

Paul

[Alex Franke]

Maybe in Mandarin, when asked the same question, there is less wiggly room in the interpretation and that's why they had 100% correct. Or in China, they asked all the math teachers while in America they asked everyone else

That's a great point -- anyone know Mandarin? I have a friend I can ask at work tomorrow to see if the language itself is part of the problem.

The Americans were elementary school teachers who "were considered 'better than average.' Eleven of them were experienced teachers who were participating in the SummerMath for Teachers Program at Mount Holyoke College. They were considered 'more dedicated and more confident' mathematically. . . .The other 12 were participating in the Graduate Intern Program run jointly by a school district and the University of New Mexico. . . .They were to receive master's degrees at the end of [the] summer"

[Alex Franke]

Flip the denominator for the numerator on the divisor and multiply the two numbers together.

I think the tendency here is to just teach the procedural method, and unfortunately that doesn't provide a very good conceptual understanding.

I reminded her that with division, being multiplication in reverse, you're trying to find out how many of the divisor "goes-inta" the dividend, and the light went on.

What came to my mind for a "story" was "How many half-feet are there in 1 3/4 feet?" and I wonder if I came to this measurement question because of the woodworking...

The Chinese teachers had two other ways to look at it: "If half a length is 1 3/4 feet, how long is the whole?" and "If one side of 1 3/4 square foot rectangle is 1/2 feet, how long is the other side?"

[MilDoc]

Sorry, but there have been a lot of new stories asking teachers basic questions which they could not answer, even in their own fields. IMHO that's why the "best of the best" of our students do so bad when tested against students in other countries.

Jay Leno once asked a history teacher (high school level) what country we fought in the Revolutionary War. Her answer, hesitantly, "The French?"

Leno recently asked several people on the street what the 4th of July was all about. Many could not answer correctly. My wife then went to work the next day and asked the same question. MOST under 35 or 40 couldn't answer. Amazing.

We are in serious trouble here, folks. In this case, the teachers probably couldn't get it right because they DON'T KNOW how to do it.

[Tom Slick]

I asked my wife about it, she's a middle and high school math teacher. Her guess is the mixed numbers tripped them up. She also said making a story often is more difficult than showing the math steps.

I looked at the problem exactly the same way Cabinetman and calc'd 7/8 but was incorrect was Paul pointed out. As Paul alludes to, it may be a recognition/assumption issue with many of the teachers reading it as "1 3/4 divided IN HALF". We tend to assume when we see 1/2 that the problem is looking for half of something.
In real life why would you divide by half? In fact for the rest of our out-of-school lives we will only deal with fractions in coarse measurement, everything else will be in decimal.
The metric system, which China uses, doesn't use fractions so that's one less factor to confuse in the original question.

FYI: in California mixed number multiplication/division is introduced in about grade 6.

[Alex Franke]

I agree that it's a lot harder to come up with a story, but I wonder if that's just because we're taught procedural math (to get the result, you take steps X, Y, Z, etc.) instead of a more conceptual math. I remember that I had a difficult time in some of my math classes because I couldn't always remember the steps or formulas -- instead I would often re-derive them because it was easier for me to understand how to solve a problem I knew how the math used to solve the problem worked in the first place.

In real life why would you divide by half? In fact for the rest of our out-of-school lives we will only deal with fractions in coarse measurement, everything else will be in decimal.

How many half-foot pen blanks can I get out of this 1.75-foot length of zebrawood?

The metric system, which China uses, doesn't use fractions so that's one less factor to confuse in the original question.

I can't imagine they wouldn't use fractions, even with the metric system. How would the Chinese teachers have known how to solve it?

FYI: in California mixed number multiplication/division is introduced in about grade 6.

This is exactly what I was pondering last night... if elementary school math teachers never even teach division by fractions, and they were asked a question concerning division by fractions, then I think it would be natural for them to assume the researcher meant something that they do teach -- like cutting something in half. If it's not part of the elementary school curriculum, then why would they even need to know it?

But this of course would beg the question, "Why is it not in the curriculum?"

[Alex Franke]

How many half-foot pen blanks can I get out of this 1.75-foot length of zebrawood?

(commenting on my own post here...) I realize that I should have made the point that this is exactly what I'm talking about with respect to conceptual versus procedural math. "Divide by 1/2" to a lot of people seems like a pretty obscure thing, but the conceptual "story" examples seem a lot more concrete. . . .We can pretty naturally and quickly figure out how many half-foot pen blanks we can get out of a 1 3/4' length of wood -- without even thinking about it. But without the connection between the conceptual understanding and the procedural method, "what is 1 3/4 divided by 1/2" seems a lot more obscure and difficult... Even more obscure and disconnected from reality is "convert 1 3/4 into a fraction, take the reciprocal or 1/2, multiply the numerators and denominators, then simplify."

Are kids (and their teachers) missing this connection in elementary school?

[Tom Slick]

...
How many half-foot pen blanks can I get out of this 1.75-foot length of zebrawood?

Been using your grade rod for measuring pen blanks? Who uses fractions of a foot other than surveyors? In practicality the questions would be "how many six inch pen blanks can I get from twenty one inches of zebrawood?" Which would be an easier 21/6 = 3 and a half.

I can't imagine they wouldn't use fractions, even with the metric system. How would the Chinese teachers have known how to solve it?

The "foreigners" I know will say "point five liter" or "five hundred milliliter, "one point seven five meters" or "one hundred seventy five centimeters" although they will also say "half an apple". Our system uses 2/3 cup, 1/2 inch, 3/4 mile, etc. My point being we automatically associate fractions to measurements and 1/2 usually meaning half of something or dividing by 2. In Chinese, 1/2 may not automatically associate to half of a measurement, it may just be a fraction.

But this of course would beg the question, "Why is it not in the curriculum?"

Basic fraction concepts, such as 1/4, 1/2, 3/4 are introduced as early as grade 2 but mixed numbers and their arithmetic come in later.

Another thing to look at is errors in the study. The sample size was tiny, in fact statistically insignificant. The teachers tested may not teach these concepts and with math if you don't use it you lose it, especially with a elementary school teacher which is the "jack of all trades" when it comes to teaching.

[Tom Slick]

As a follow-up, I wonder if the question was "what is 1 3/4 divided by 5/8" if as many would have answered it incorrectly.

[charliex]

To put the example in question format would be redundant. "What is 1 3/4 divided by 1/2?" is already a word problem. What is 1 3/4 of an apple pie divided by 1/2? does not make a lot of sense. That also leads to the 7/8 answer. My understanding of the question that is asked would get the answer of 3 1/2.

I've been reviewing my math skills for about a week now and I think that I have forgotten more than I ever knew about the subject. The statement made earlier about "use it or lose it" is all too true. I was an ET and radio station engineer for a number of years (25+ years back) and when I looked at my old formula book I hardly recall any of it. I'm struggling with Algebra 1 for Dummies and a new calculator right now. I need a better understanding of Trig. so I don't find working with angles so confusing. Compound blade and miter angles drive me nuts sometime.

[Mr__Bill]

Mrs. Franke has baked two pies. She gives a quarter of a pie to Mr. Franke and has one and three quarters of a pie left. She will give a half a pie to each boy. How many half pie pieces does she have for the boys. Express your answer as whole numbers and fractions there of.

Pies cost a half dollar each. Alex has one and three quarters dollars. How much pie can Alex get. Express your answer as whole numbers and fractions there of.


Bill
blackberry pie