kitchen table math, the sequel: Linda Seebach on the whale problem

Thursday, October 25, 2007

Linda Seebach on the whale problem

The whale problem again:

A whale swims 40 miles in 1 1/4 hour. How far does he swim in 1 hour?

The math mom I mentioned solved this problem mentally in a couple of seconds, then explained that "1 1/4" is 5 parts.

Linda writes:
What is apparent to someone with good number sense is that 1 and 1/4 is 5/4, that is five of something called "one fourth." There's no work to show, though you could explain that if five somethings equal 50, then one something is 10 and so four somethings are 40.

Since the numbers are easy, you can do it in your head; no need for visual models or bar models or algorithms. But then, I was an algebra person, not a geometry person.

It works just as well when the whale swims 37 miles in 16/13 hours except you probably can't compute (13)(37/16) without writing it down or asking the calculator. Knowing what you need to compute, however, is exactly the same.

I have to say....it's pretty sad that I still can't put it this way (or, rather, don't think to put it this way).

Of course Linda is right; 1 1/4 is 5/4. That's what a person with a good number sense "sees" or, more accurately, grasps.

Remembering back to when I first started reteaching myself K-12 math, seeing that 1 1/4 is 5 one-fourths was an obstacle. I had good fraction knowledge for an American, I think; I could create a correct word problem for 1 3/4 ÷ 1/2, the famous challenge given to 22 math elementary school math teachers in Liping Ma's book. I had no math phobias, SAT math scores were good, took statistics in college.... in short, I am a math literate person, certainly for the U.S., and was before I started studying K-12 math.


An aside: hmmm... It's interesting that I keep using the word "see." I'm going to assume that means I personally do need to "see" this as opposed to "grasp" it.

Interestingly, as I've moved into algebra 2, I don't feel this way about many of the topics I've encountered there (practically all of which are new to me). Will have to give this some thought. At some point, the abstractions of math came to seem natural or "real" to me. At least, I think they did.


Back on topic: In spite of the fact that my own understanding of fractions was perfectly appropriate to my needs, I had trouble with the idea that 1 1/4 is five one-fourths. The "one-fourth" is the unit; the 5 is multiplier and 1/4 is the multiplicand. I remember my neighbor explaining it to me one night in the context of another question. (Wish I could remember it now. I may be able to find it. She was trying to explain something about fractions, I think, by pointing out what Linda has just pointed out --- and I stumbled over the explanation.)

I think bar models are a way of teaching the kind of number sense Linda is describing. Which is why C. is going to carry on doing the bar models in 3rd grade Primary Mathematics.

Sybilla Beckmann's textbook for students in education school teachings bar models, as does Parker and Baldridge's text. (I've found Parker and Baldridge much easier to study and understand than Beckmann's book, but that may not be a fair comparison since I began at the beginning of P&B, and only dipped into Beckmann.)


procedural "versus" conceptual knowledge

Given my experience of (relearning) fractions, I now feel strongly that schools must teach the addition, subtraction, multiplication, and division of fractions to mastery in all kids, bar none.

I think, too, that schools should give kids word problems to solve using all four operations -- and a good number of these word problems should be "real world" word problems, by which I mean the actual real world, as opposed to the constructivist real world, which seems to consist mostly of roller coasters and hay balers (see below).

Non-GATE kids should be given fraction word problems about things like measuring a room for tile or cutting fabric from a bolt to sew a dress (I don't care if the kid actually sews or not. Sewing is something he/she might do one day, especially if he/she has kids and sends them to a public school where the curriculum is project-based.)

This approach, which I think is what I probably had, may not give non math-brain kids a good conceptual understanding of fractions. It didn't give me a good conceptual understanding.

But it did give me all the foundation I needed to use fractions in adult life, and to pick up the study of math later on, when I wanted to.

I now believe that non-GATE kids (not sure about the GATE kids) should be taught bar models, too. No matter what curriculum or approach a teacher/school is using, consistent teaching of bar models should be included.

Cassy T has mentioned that 3rd grade is the big year for Singapore Math; that's the year bar models are introduced. (pls correct me if I'm wrong)

I would ideally have U.S. kids work through either the 3rd grade Challenging Word Problems book, or work all the bar model problems in Primary Mathematics 3.



the hay baler problem

A post from Barry a couple of years back:
Here's a problem that appears in IMP for 9th grade It is known as the "Haybaler Problem"

“You have five bales of hay. For some reason, instead of being weighed individually, they were weighed in all possible combinations of two: bales 1 and 2, bales 1 and 3, bales 1 and 4, bales 1 and 5, bales 2 and 3, bales 2 and 4 and so on. The weights of each of these combinations were written down and arranged in numerical order, without keeping track of which weight matched which pair of bales. The weights in kilograms were 80, 82, 83, 84, 85, 86, 87, 88, 90 and 91. Find out how much each bale weighs. In particular, you should determine if there is more than one possible set of weights, and explain how you know.”

David Klein, a mathematics professor at California State University at Northridge comments on the problem. “The process of solving this problem made me resentful of the stupidity and pointlessness of it. There is nothing ‘real world’ about it. It is completely inappropriate for kids who likely have not been taught how to solve simultaneous linear equations, or exposed at most to two equations in two unknowns. If I had been given such problems at that age, I think that I would have hated math.”

Consistent with much of the philosophy of “real life math”, the goal of the exercise is to explore strategies and to be able to write about it. This is made apparent by the “student guide” that accompanies the problem. It is essentially a scoring sheet, containing categories, with points awarded for each, such as “Restate the problem in your own words” (4 points); describe all the methods you tried before reaching your solution(s) (4 points); describe the process that lead to your solution(s) (4 points); describe all assistance provided and how it helped you (2 points); state the solution (2 points); describe why your solution(s) is correct, include all supporting data (6 points). Out of a total of 50 points, only 2 are given for the solution. In fact more points are given for describing why the solution is correct.

37 comments:

SteveH said...

David Klein, of course, is perfectly correct.

The "Haybaler Problem" is Anti-Math.

Math is a tool that makes solving problems easier. Kids have to solve this problem without that tool.

Catherine Johnson said...

I REALLY dislike the "make math hard" philosophy of math teaching.

Me said...

Catherine, I'm fascinated that you don't see 1 and 1/4 being the same as 5/4.

What I'm wondering is whether there is something about the notation itself that wrongly compartmentalizes this in your brain?

The reason I ask is that I find it hard to believe that if you were following a recipe that called for 1 and 1/4 cups of sugar and the 1 cup measure was unavailable for some reason that you wouldn't automatically realize you could just use the 1/4 cup measure five times without needing a bar model.

Tex said...

Our school just implemented a new k-5 reform math program, and I’m starting to hear complaints from parents and at least one teacher.

Some parents have commented that some of the word problems seem pointless, and they question why so much time is being spent “playing” with problems that can be solved using algebra. Bar models are not being used, at least not routinely. They’re using patterns, guess-and-check and the other typical reform math problem-solving strategies.

SteveH said...

"... 'playing' with problems that can be solved using algebra."

Anti-Math.

Math is all about rules and algorithms that make your life easier. Master math so you don't have to think as hard. Label the variables, define the right number of equations, turn the crank. My brain hardly broke a sweat.

They think that math is some sort of Zen-like skill that has to be developed.

"OK Danny. Be the ball."

(from Caddyshack)

They think that using algorithms is rote or drill-and-kill. They go out of their way to find problems that don't have one answer or that can't be solved by anything other than guess and check. They think they are vindicated.

Unknown said...

I'm glad somebody else mentioned measuring cups. I automatically think of 1 1/4 as 5 quarters, not least because I hate trying to find measuring cups, and I hate using the big glass 2-cup one, too. So I would always measure 1 1/4 cups of flour as 5 quarter cups. (Then when it called for 1 1/3 cups of sugar, I would cause all the pastry chefs to writhe in agony by using 5 and a half quarter cups of sugar, which is not even right, although I'd make it a *small* half quarter cup to help make up for the overage, unless I thought the recipe was likely to be too sweet anyway in which case I'd just use 5. Perhaps cooking is not such a perfect preparation for math.)

Catherine Johnson said...

The reason I ask is that I find it hard to believe that if you were following a recipe that called for 1 and 1/4 cups of sugar and the 1 cup measure was unavailable for some reason that you wouldn't automatically realize you could just use the 1/4 cup measure five times without needing a bar model.

Yes, absolutely, and I have done so many times.

That's a good example of what I'm trying to get at when I say I had "good enough" knowledge of fractions from my own K-12 education.

It's an interesting question, and I feel I need to "crack it" -- because I'm sure C. is the same way.

For instance, Steve (& you, I think?) says he thinks of a number as the product of its factors.

I don't.

If I think of 42, I think of forty and two, not six and seven.

It's entirely possible that this has to do with my being VERY verbal, as opposed to regular verbal.

When Temple and I wrote our first book we found studies showing that the verbal masks the visual, and this seems to be a general principal of the brain.

You can't be a generalist; you have to be a specialist.

The fact that I've specialized in language and writing (I MUST LEARN GRAMMAR!) may suppress "math" in some way.

Catherine Johnson said...

Some parents have commented that some of the word problems seem pointless, and they question why so much time is being spent “playing” with problems that can be solved using algebra.

My agent is sending all 3 kids to the Bank Street School, which is progressive ed.

She majored in math in college; she knows math.

When her son was around.... age 7?...he came home with algebra problems to do, so his parents started teaching him algebra.

He said, "No! We can't use algebra!"

He new the word algebra, and knew he wasn't supposed to learn any algebra, apparently.

Catherine Johnson said...

I'm going to have to write my nonlinear jump from rural high school to Wellesley post one of these days.

Barry Garelick said...

Do you think the "arrogance" and "jerks" tags should be added to the labels for this one? (I'm thinking of the Hay Baler problem).

Brett Pawlowski said...

Not exactly adding to the conversation here... but I had to laugh when I saw the title. I thought someone was referencing this Onion report: http://www.theonion.com/content/video/in_the_know_are_our_children.

One of the funniest things I've seen in a while - and a dead-on satire of just about every education debate I've ever heard.

Brett Pawlowski said...

Darn - the link didn't paste well. Add our_children to the end of that last link.

Catherine Johnson said...

yes, indeedy, I do

Catherine Johnson said...

that is one of the most insane things I have ever, ever, ever, seen

is this real??

did this really happen???

Catherine Johnson said...

I can't tell if the whole thing was lip-synched or just the end??

Anonymous said...

Could you explain what you mean by "verbal" and "math" and "visual"?

Catherine Johnson said...

well....the research Temple and I read defined "verbal" very simply.

Subjects watched a videotape of a bank robbery.

(I'll probably forget the details a bit.)

Some subjects then wrote accounts of what they'd seen, after which they tried to identify mug shots.

Others did "nothing verbal" (can't remember if they did something else, such as listen to music), then looked at mug shots.

The second group was far better able to identify the burglars.

There was also an effect where they reversed the "masking" effect of verbals....can't remember what they did, though.

Somehow they had the "writing" group just sit around for awhile, not doing any writing or verbalizing, and then had them look at mug shots (I think) --- and iirc they then improved their ability to identify the burglars.

This is called the masking effect.

This seems to be a pretty well settled concept about brain function; one has to specialize.

Being good at one thing will tend to make you worse at the other -- you must choose.

So...when I say "verbal" versus "math," I don't know that that's a proper distinction, but the last quite good study I read of intelligence said that the classic SAT categories -- verbal & math -- really do capture the core division within IQ.

Interestingly, they thought spatial intelligence (rotating figures etc.) might be a third category.

Catherine Johnson said...

I think it's possible that my intense word orientation means that I experience a "number," such as 42, as two words, i.e. forty and two (that's 3 words, I realize, but you know what I mean).

Instructivist said...

This doesn't have to do with whales but is still in the animal realm.

It's a video of an amazing dancing and singing cockatoo:
http://birdloversonly.blogspot.com/2007/09/may-i-have-this-dance.html

Must see!

David said...

Here's my solution to the Hay Baler problem. I think it's a good challenge problem for students who have learned to solve systems of linear equations. The scoring rubric is ridiculous, of course.

Suppose the weights are a < b < c < d < e. The sum of the weights of the ten combinations is equal to four times the sum of the weights of the bales, because each bale belongs to four combinations. So a + b + c + d + e =
(80 + ... + 91) / 4 = 214.

We know that a + b = 80 and d + e = 91, so c = 214 - 80 - 91 = 43. We also know that a + c = 82 and c + e = 90, so a = 39 and e = 47. Finally, b = 41 because a + b = 80 and a = 39; and d = 44 because d + e = 91 and e = 47.

Thus, the weights are 39, 41, 43, 44, and 47 kg.

Anonymous said...

"...the classic SAT categories -- verbal & math -- really do capture the core division within IQ"

Hmm, need to think about this.

In the GRE there are (were?) two sections, verbal and quantitative. IIRC, the verbal had almost nothing to do with writing, but spent a lot of time on patterns and so forth (what I think of as slightly abstract stuff). Never took the SAT, so I don't know if their "verbal" is different.

Most of the people I know in the hard sciences did very well in both, but they aren't talkative/writerive types at all.

Very odd.

Anonymous said...

Interesting stuff. I commented below (in the GATE kids/mental bar models post) that I had immediately recognized 1 1/4 as 5/4, but in this post, as you were discussing how you see 42 as 40 and 2, it occurred to me that I automatically do that breakup when I multiply.

Ex: What is 42 x 6? Well, I don't have my times tables memorized up to 42, but my brain automatically divides the problem, and to solve it, I immediately see 40x6 = 240, and 2x6 = 12, so 42x6 = 252.

Is this weird?

Instructivist said...

"...I immediately see 40x6 = 240, and 2x6 = 12, so 42x6 = 252.

Is this weird?"

Not weird at all. It's making use of expanded notation. Expanded notation is practiced in school.

Anonymous said...

my cooking expertise consists
mainly of remembering where
i left the can-opener.
and rephrasing "one and one-fourth"
as "five fourths" comes so readily
to me that i tend to forget
ever having learned it;
rational number arithmetic
is my mother language, for heck sake.

but i know this:
if you tell 'em
"a dollar and a quarter",
pretty much any student
will see right away
that it has the same value
as "five quarters".
once you mention money,
they just suddenly get interested
and turn their thinking apparatus
back on (they turned it off
because you were talking about math).

it's maddening.

Catherine Johnson said...

I remember GRE verbal as being the same as SAT verbal, but it's been a long time.

I took a sample SAT test last year; the verbal section is terrific. It's mostly reading; now they have a grammar section, too.

I'll try to find that study.

I don't think it's all that meaningful that high-scorers tend to score high across the board....but someone who knows statistics will have to explain why!

I think the study I read saying that verbal/math are divisions of "g" (which is general intelligence) is "factored" --- damn. Don't remember the term for it.

"Factor analysis"??

Why don't I go find the reference.

Catherine Johnson said...

Here it is -- and I'd forgotten that this article says the fluid versus crystallized distinction doesn't work. (This is Bouchard; he's someone I would listen to...)

Wendy JohnsonT, Thomas J. Bouchard Jr., The structure of human intelligence: It is verbal, perceptual,
and image rotation (VPR), not fluid and crystallized INTELLIGENCE 17 February 2005

Abstract
In a heterogeneous sample of 436 adult individuals who completed 42 mental ability tests, we evaluated the
relative statistical performance of three major psychometric models of human intelligence—the Cattell–Horn fluidcrystallized
model, Vernon’s verbal–perceptual model, and Carroll’s three-strata model. The verbal–perceptual
model fit significantly better than the other two. We improved it by adding memory and higher-order image
rotation factors. The results provide evidence for a four-stratum model with a g factor and three third-stratum
factors. The model is consistent with the idea of coordination of function across brain regions and with the known
importance of brain laterality in intellectual performance. We argue that this model is theoretically superior to the
fluid-crystallized model and highlight the importance of image rotation in human intellectual function.

Catherine Johnson said...

They're calling the distinction verbal-perceptual, but I remember that as having translated to verbal-math (SAT distinction).

I'll check and see if I'm misremembering.

Catherine Johnson said...

aack!

I've got it wrong!

good lord

at least I think I do....sigh

Here's a section talking about Vernon's model, which was supported by this study:

He stressed
the importance of general intelligence in contributing to all mental abilities, but observed that, once a
general intelligence factor is extracted from any collection of ability tests, the correlations among the
residuals fall into two main groups. He labeled one of these v:ed to refer to verbal and educational
abilities, and the other k:m to refer to spatial, practical, and mechanical abilities. The v:ed group, he
noted, generally consists of verbal fluency and divergent thinking, as well as verbal scholastic
knowledge and numerical abilities. The k:m group generally consists of perceptual speed, and
psychomotor and physical abilities such as proprioception in addition to spatial and mechanical
abilities. These broad groupings of residuals have practical importance and empirical support:
working with large military samples throughout the 1950s, Humphreys (1962) found that v:ed and
k:m added significantly to the prediction of work performance beyond g, but more narrow
dimensions did not.

Catherine Johnson said...

ok, I give up

I have no idea what "perceptual" intelligence is, beyond the idea that it is somehow "mechanical."

Don't know whether this invalidates the notion that I'm more "verbal" than "mathematical."

I may just be more stimulus bound (and I DO know what that means, at least in theory!)

Stimulus bound is a bad thing...

It means that you have trouble abstracting from the surface or the environment.

That could be an explanation of my being "captured" by the words "forty two."

The study Temple and I looked at found verbal masking of visual -- and I'm assuming "visual" must be part of "perceptual," as this study defines it, but at this point I'm lost.

Catherine Johnson said...

What is 42 x 6? Well, I don't have my times tables memorized up to 42, but my brain automatically divides the problem, and to solve it, I immediately see 40x6 = 240, and 2x6 = 12, so 42x6 = 252.

Yes, that I do.

I'm pretty down with the distributive property!

Catherine Johnson said...

I'm thinking about the quarters example....

It's definitely much more natural and obvious for me to think of $1.25 as 5 quarters than it is for me to think of 1 1/4 as 5 1/4s.

Interesting.

I was a check out girl every summer for 3 years back before there were calculators in cash registers.

Me said...

It's definitely much more natural and obvious for me to think of $1.25 as 5 quarters than it is for me to think of 1 1/4 as 5 1/4s.

If you can figure out why, I think you'll have made an important breakthrough.

I'm beginning to suspect there's something about the way we teach fractions that causes problems; not the concept of fractions per se.

Catherine Johnson said...

I'm beginning to suspect there's something about the way we teach fractions that causes problems; not the concept of fractions per se.

That was my thought....because why should it be simple for me to think of $1.25 as 5 quarters?

(I don't think that's simple for C., btw - must check.)

This does seem to get back to number lines, rulers, etc....

maybe

I'm thinking kids should spend a lot of time doing what Wu says, which is to locate fractions on number lines - and to see that 2 "1/8"s on a number line are the same as 1 "1/4" etc.

Anonymous said...

The whole question about what is the structure of intelligence and what are its components is fascinating.

How about people who score at the top in verbal, mathematical and/or visual-spatial ability? By way of example one friend of mine scored 800 on both sections of the GRE. She seemed equally gifted in both.

The Johnson O'Connor Research Foundation (http://www.jocrf.org) does some interesting aptitude testing that provides for building profiles of strengths appropriate for different occupations etc. A family member doing research in this area recruited some of us to take the battery.

One boy (my nephew) scored extremely high in both verbal and visual-spatial/mathematical skills, as well as skills related to three-dimensional modeling and design. His profile was such that they could suggest few "appropriate" occupations for him (they said that if a person is extremely gifted in some area, but their work does not provide for it, they are unlikely to be satisfied with their work).

I scored well in verbal, musical and mathematical ability, pretty high in mechanical and spatial reasoning but at the bottom of the heap -- 2nd percentile or something -- in manual dexterity and ability to use tools. Gave me a good excuse to take mending to a tailor. They suggested I should be an airline pilot, a research scientist, or a writer, and that I absolutely did NOT have the right aptitude set for a teacher;-( (too late now).

I'm not convinced that being outstanding in one area necessarily excludes the possibility of being outstanding in the other, however. Maybe it's just unusual for the two to go together?

For those who don't know of it, a fascinating blog on some of these topics is http://intelligencetesting.blogspot.com and a companion one, http://ticktockbraintalk.blogspot.com, both manned by Kevin McGrew ( a proponent of the Cattell-Horn-Carroll school of thought), who is witty, erudite and has wide-ranging interests. He features a write-up on math learning in a recent entry on the first one, dealing with the work of David Geary. A companion Yahoo group he runs is here: http://groups.yahoo.com/group/IAPCHC
Another one with lots of interesting information, from other sources is this : http://tech.groups.yahoo.com/group/cognitiveneuroscienceforum

Anonymous said...

To David, you are stupid. We found the solution to the problem. 3 Eleventh graders from Lynn, Alabama.you will never know the answer....

Anonymous said...

....HaHaHa....

Anonymous said...

And HaHa...check your addition next time,go back to kindergarten....