Choosing Carnival Junk Food

It’s standardized test time in Connecticut and my child will be busy participating in this painfully drawn-out process over the next couple of weeks.  While I should clarify that I think testing *can* be useful for diagnostic purposes, I consider the following question from the eighth grade CMT Mathematical Applications section to be yet another example of why I find my state's manner of assessing students quite useless:

Sample Item 8-5 (Numerical):  Buying Tickets

The carnival offers you two different options for buying tickets.

                OPTION A:  $2.00 per person plus $0.75 per ride
                                                                OR
                OPTION B:  $5.00 per person plus $0.25 per ride

If your uncle gave you $10 for the carnival, which option – A or B – would you choose.  Show the mathematics you used to determine your answer.

OPTION CHOSEN: _______

Explanation:

This poor excuse for a word problem is just one example of why we started homeschooling.  While I’m assuming the objective of the question is for the student to show mathematically that option B is the better choice because you can go on 20 rides as opposed to only 10 rides with option A, the question does not indicate that the goal is to go on as many rides as possible.  I can easily imagine any of my children (including the 8-year old) coming up with alternate scenarios that could make either option the better one.  Unfortunately, I can just as easily imagine a scenario where the person responsible for grading 200 tests containing these strange open-ended mathematical responses before the end of their shift would mark their mathematically and numerically accurate answer WRONG.
 

Let's say the student were to choose OPTION A since he really likes to eat junk food at carnivals (just like his favorite uncle who spots him the $10) but hates the rides because they make him dizzy (thereby leaving him with $8 to spend on food instead of $5).  Would that be counted as a correct answer?  I would argue that either option could be the better choice depending on the objective—which was not made clear.  Mathematics is supposed to be clear, precise, and accurate. This question is just silly.

These types of  Everyday Math word problems (I'm being generous here by calling it a word problem) used to make me crazy when my child would come home with them in 4^th grade.  Now here we are in 8^thgrade running in circles all over again.

Meanwhile back in Singapore children are answering this:

Hooke's law for an elastic spring states that the distance a spring stretches is proportional to the force applied. If a force of 150 newtons stretches a certain spring 8 cm, how much will a force of 400 newtons stretch the spring? (New Elementary Math 2 Placement Test)

 *sigh*

words, diagrams, & math

Nick Rowe

how Paul teaches problem solving to middle school kids

I use an organizer that approaches this technique. I think it's adapted from an ELA template called four square. Here's what I have the kids do.

Take a piece of paper and fold into quarters. Open it up and draw a rectangle in the center. Now you've got four quadrants and a rectangle which is not exactly 'four square' but in the interests of marketing I guess four square sounds sexier.

Read the problem twice. Then in the rectangle restate the question in the form 'find blah blah blah in units of xxxxxx'.

Read the problem again and in the upper left quadrant identify and define all the symbols (variables and constants) you'll use in your solution.

Read the problem again and use the lower left quadrant for a diagram/picture.

Read the problem again and use the upper right quadrant to define your strategy. This can be words or preferably a set of equations to solve.

The lower right quadrant is where you show your arithmetic.

Finally, the back of the paper is used to justify your answer.

It works well for entry level problem solving of the kind you would encounter through maybe grade six. If kids master this it should instill some good habits for more complicated things.

foiled again

I was cruising through the word problems at the end of Section 5-8: Problem Solving Using Fractional Equations in Mary Dolciani's Algebra and Trigonometry Structure and Method Book 2, congratulating myself on the fact that not only can I now do what I consider to be reasonably difficult word problems, I enjoy doing them.

Then I hit this one:

19. A commercial jet can fly from San Francisco to Dallas in 3 h. A private jet can make the same trip in 3 1/2 h. If the two planes leave San Francisco at noon, after how many hours is the private jet twice as far from Dallas as the commercial jet? (page 246)

I couldn't do it, and I didn't enjoy not being able to do it.

I'm not even going to look at #20.

Until tomorrow.

SAT/ACT Math and Beyond

Vicky S sent me notice of a workbook Stephen Wilson has posted on his web site: SAT/ACT Math and Beyond: Problems Book by Qishen Huang.

The book is listed here. I've just ordered the solution manual, which Dr. Huang says is highly detailed (460 pages for the manual, 131 for the workbook). That's critical for those of us teaching ourselves, and not easy to find.

Dr. Huang estimates that 20% of Chinese high school graduates can work 90% of these problems, which he says are not as difficult as those on China's SAT equivalent.

And...on the subject of workbooks, I've emailed Myrtle, who is using the NEM Workbooks (New Elementary Mathematics Syllabus D 1 and New Elementary Mathematics Syllabus D 2).

Meanwhile, I have done no math at all this summer, because I am busy reading C's massive Summer Assignment list, all 2549 pages of it. As to that, please know that you are in the presence of a woman who has read every last word of Guns, Germs, & Steel. There are few amongst us who can say the same.

Problems with "real world" math problems

I've been complaining for years that all the hands-on, "real world" math activities that dominate Reform programs shortchange kids on the autistic spectrum. They confuse the many who have language delays and gaps in worldly knowledge, and are too full of distractions for those who require streamlined, structured learning environments.

Now a new study by Jennifer A. Kaminski, a research scientist at the Center for Cognitive Science at Ohio State, reported in today's NYTimes, suggests that this kind of hands-on math is bad for everyone.

Quoting the Times: "The problem with the real-world examples, Dr. Kaminski said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems."

As the Times also remarks: "Dr. Kaminski and her colleagues Vladimir M. Sloutsky and Andrew F. Heckler did something relatively rare in education research: they performed a randomized, controlled experiment."

the hard kind of percent problem

The hard kind of percent problem is this one:

John paid $52.50 for a shirt including tax of 5%.
What was the price of the shirt before tax?

I imagine this problem is a cinch (cinch?? sp?) for the gifted, but for the non-gifted, this problem is HARD.

I was thinking this problem is an example of a partitive word problem, but now, re-reading Carolyn's original post on the subject, maybe not.


variations on a theme: 3 kinds of percent problems

As far as I can tell, every percent problem comes in three forms. (PLEASE correct me if I'm wrong.)

price of shirt: $50
tax: 5%
price of shirt including tax: $52.50

1. A shirt sells for $50. Sales tax is 5%. What will the total cost be?
2. A shirt sells for $50. After tax, the shirt sells for $52.50. What is the tax rate?
3. A shirt sells for $52.50 including a tax of 5%. What was the original price of the shirt?

Problem number 1 is easy.

Making the move to problems 2 and 3 is hard for many students.

But 3 is the killer. (I think.)

If you're having to teach or reteach percent to your children, be sure to include lots of number 3s in the mix.

........................

Do math textbooks call numbers 2 and 3 "work backwards problems" problems these days?

They may.

........................

Liping Ma deals with these problems in her chapter on division by fractions.

She describes asking her group of U.S. elementary teachers to create a word problem representing 1 3/4 ÷ 1/2:

Imagine that you are teaching division with fractions. To make this meaningful for kids, something that many teachers try to do is relate mathematics to other things. Sometimes they try to come up with real-world situations or story-problems to show the application of some particular piece of content. What would you say would be a good story or model for 1 3/4 ÷ 1/2?

She goes on to say:

...division by fractions is an advanced topic in arithmetic. Division is the most complicated of the four operations. Fractions are often considered the most complex numbers in elementary school mathematics. Division by fractions, the most complicated operation with the most complex numbers, can be considered as a topic at the summit of arithmetic.

The summit of arithmetic!

I love it!

The U.S. teachers didn't fare well.

Of the 23 U.S. teachers, 212 tried to calculate 1 3/4 ÷ 1/2. Only nine (43%) completed their computations and reached the correct ansewr. For example, Mr. Felix, a beginning teacher, gave this explanation.:

I would convert the 1 3/4 to fourths, which would give me 7/4. Then to divide by 12/2, I would invert 1/2 and multiply. So, I would multiply 7/4 by 2 and I would get 14/4, and then I would divide 14 by 4 to get it back to my mixed number, 3 2/4 or then I would reduce that into 3 1/2.

[snip]

Tr. Bernadette, the experienced teacher who was very articulate about the rationale for subtraction with regrouping, tried a completely incorrect strategy:

I would try to find, oh goodness, the lowest common denominator. I think I would change them both. Lowest common denominator, I think that is what it is called. I do not know how I am going to get the answer. Whoop. Sorry.

Tr. Bernadette sounds a little like me trying to do a simple percent change problem the other night. C. and I looked at the same problem last night and we both said: how did we make this so complicated?

That is the $40,000 dollar question.

The $40,000 dollar answer is: failure to transfer.

Ma goes on to discuss the word problems U.S. teachers came up with to represent 1 3/4 ÷ 1/2, and there we see a clean sweep: only one of the 23 teachers produced a correct model of the problem, and that one model was "pedagogically problematic."

I am proud to say that when I read Ma I was immediately able to produce a mathematically correct word problem representing 1 3/4 ÷ 1/2. As I recall my problem went something like this:

Catherine has two dogs, and each dog eats 1/2 can of dog food every morning. If she has 1 3/4 cans of dog food left, how many servings is this?

Six of the U.S. teachers confused dividing by 1/2 with dividing by 2.

The Chinese teachers created two genres of problems:

• quotitive division, which Ma calls the measurement model
• partitive division - finding a number such that 1/2 of it is 1 3/4

Which brings me back to my hard percent problem.

At least, I think it does.


the "measurement" model

The measurement model is more obvious to me. Obvious meaning easy to understand. My own word problem falls in the measurement category: how many 1/2s are in 1 3/4?

How many 1/2s can 1 3/4 be divided into?

How many 1/2-can servings of dog food are there in 1 3/4 cans of dog food?

If I'm measuring the number of 1/2s in 1 3/4, I divide 1 3/4 by 1/2.

1 2/3 ÷ 1/2 = 3 1/2.

I have 3 1/2 servings of dog food.


the "partitive" model

Chinese teachers gave the measurement model short shrift:

Among more than 80 story problems representing the meaning of 1 3/4 ÷ 1/2, 62 stories represented the partitive model of division by fractions--"finding a number such that 1/2 of it is 1 3/4":

Division is the inverse of multiplication. Multiplying by a fraction means that we know a number that represents a whole and want to find a number that represents a certain fraction of that.

After all this time, I still find this passage mystifying.

I was even more mystified when I read the story problem Tr. S came up with:

My story will be: A train goes back and forth between two stations. From Station A to Station B is uphill and from Station B back to Station A is downhill. The train takes 1 3/4 hours going from Station B to Station A. It is only 1/2 time of that from Station A to Station B. How long does the train take going from Station A to Station B?

[isn't this written wrong? shouldn't the numbers be reversed?]

I'm going to reverse them:

My story will be: A train goes back and forth between two stations. From Station A to Station B is uphill and from Station B back to Station A is downhill. The train takes 1 3/4 hours going from Station A to Station B. It is only 1/2 time of that from Station A to Station B. How long does the train take going from Station B to Station A?

update: see below

Here's another:

The mom bought a box of candy. She gave 1/2 of it which weighed 1 3/4 kg to the grandma. How much did the box of the candy originally weight? (Ms. M.)

For a lot of us, the candy problem will be most obviously similar to the "hard" percent problem.

To solve the candy problem:

let x = original weight

1/2x = 1 3/4
1 3/4 ÷ 1/2 = x
3 1/2 = x

The original box weight 3 1/2 pounds.

The hard percent problem has the same form. We know the final price; we know the tax rate. We need to know the price of the shirt sans tax.

let x = price of shirt
1.05x = 52.50
52.50 ÷ 1.05 = x
50 = x

price of shirt = $50

I have to include this for people like me: the reason you take 1.05x is that it is a shortcut.

I spent years of my life taking 5% of $50 to get the sales tax ($2.50), then adding the sales tax amount to the price of the shirt.

Then C's 5th grade teacher explained to me that instead of doing these two steps I could simply multiply the price of the shirt by 1.05 and get the whole thing over with.

The "1" in the 1.05 ensures that the price of the shirt is part of the final value.

That was a revelation.


are partitive problems harder than measurement problems?

I still don't know what kind of language to use in describing these problems (if I'm not going to use the term "partitive," that is, and for the time being I am not).

Can we say that the shirt problem is a part - part - whole problem in which the whole and a part are known and we have to find the other part?

hmmmm.....

Not exactly.

unknown part: price of shirt
unknown part: sales tax in dollars
part: sales tax rate
whole: final price

I have no idea what to call these problems or even how to describe them.

All I know is that they are difficult for kids to learn, and they don't "come naturally" once a student has learned how to find the price of a shirt given the original price and the sales tax rate.

Any thoughts?

....................................

update (from above)

I'm thinking there is a translation problem with the train problem as printed in Ma's book:

My story will be: A train goes back and forth between two stations. From Station A to Station B is uphill and from Station B back to Station A is downhill. The train takes 1 3/4 hours going from Station B to Station A. It is only 1/2 time of that from Station A to Station B. How long does the train take going from Station A to Station B?

As it stands, this problem does not model 1 3/4 ÷ 1/2.

This problem models 1 3/4 x 1/2.

I think the problem should read:

My story will be: A train goes back and forth between two stations. From Station A to Station B is uphill and from Station B back to Station A is downhill. The train takes 1 3/4 hours going from Station B to Station A. It is only 1/2 time of [the time it takes to go from] Station A to Station B. How long does the train take going from Station A to Station B?

let x = time it takes to go from Station A to Station B
time it takes to go from station B to Station A: 1 3/4 hours

1/2x = 1 3/4
x = 1 3/4 ÷ 1/2
x = 3 1/2

It takes 3 1/2 hours to go from Station A to Station B

help desk - percent chart

Is there an all-around mode of charting the values in percent problems similar to the charting taught for distance problems?

this one's fun

During a sale, a bookstore sold 1/2 of all its books in stock. On the following day, the bookstore sold 4,000 more books. Now, only 1/10 of the books in stock before the sale are remaining in the store. How many books were in stock before the sale?

source:
SSAT & ISEE 2007 Edition Kaplan
p. 161

C. came pretty close to doing this one on his own.

Of course, close doesn't cut it on a standardized test.

I realize that.

This problem can't be done.

Right?

Not enough information

Or way too much information, as the case may be.

Two trains are loaded with equal amounts of rock salt and ball bearings. Train A leaves Frogboro at 10:00 A.M. carrying 62 passengers. Train B leaves Toadville at 11:30 A.M. carrying 104 passengers. If Train A is raveling at a speed of 5 mph and makes four stops, and Train B is traveling at an average speed of 86 mph and makes three stops, and the trains both arrive at Lizard Hollow at 4:30 P.M., what is the average weight of the passengers on Train B?

source:
Kaplan SSAT & ISEE 2007 edition
p. 155

New Elementary Algebra

I think I'm going to take tonight off and spend my time raiding the Comments for new posts.

from Myrtle:

I've only recently discovered this feature on Google. For example, I've been using this book for supplemental word problems in Algebra:

New Elementary Algebra (1879)

Word problems start on page 139 and go on and on and on.

and:

Fifty Famous Stories Retold for free in Google Books


I've spent my share of time pouring over Google Books these past few weeks. Sometimes they decide I've abused my privileges & they cut me off.


a farmer and his sheep

A farmer has in his pasture 63 head of cattle, consisting of cows, calves and sheep. There are twice as many calves as cows, and twice as many sheep as calves; how many has he of each sort?

Ans. 9 cows; 18 calves; 36 sheep.
New Elementary Algebra
p. 139

mini word problems

Tex asked earlier about the component skills involved in solving word problems, which sent me back to the old site searching for the posts we wrote about mini word problems.

I'm not sure these are directly on-point, but they're useful nonetheless.

I'm still enamored by the concept of "out loud" word problems, which I pretty much forgot about after writing the post saying out-loud problems were a good idea.

March 6, 2006 how do you teach your child word problems?
March 7, 2006 mini word problems
Apri 23, 2007 arithmetic to algebra


component skills

Brainstorm!

We must all re-read exo's posts.

I'm starting a component skill list:

• underline or circle the values you need to "pull out" of the problem (someone else will need to word this properly)
  

• draw a chart
  

more t/k

Ed just got back from Stew Leonard's/Costco, so the next hour of my life is accounted for...

What are the component skills needed for solving word problems?

Is it even useful to ask this question because there can be so many ways to approach the same problem? How would you describe the process of learning how to solve word problems? Actually, I’m sure there are component skills, and I’m just as sure I have no idea how to describe them.

The reason I’m trying to understand this is that I secured an unenthusiastic agreement that our school would attempt to incorporate shorter-term benchmark objectives into my daughter’s annual IEP goals. One of the goals is word problems, and it seems that the benchmarks should be the component skills involved.

The IEP committee chair remarked at the meeting that the teachers are the “experts”, and therefore they should have a good handle on this. On the other hand, maybe they don’t.

A basic idea in learning theory is that complex skills can be analyzed to see what the component parts are, so that these components can be taught individually to develop component skills. In reading, for example, a student learns how to decode words (phonics) and that the page is organized from left to right and top to bottom. As these skills become automatic the teacher and student pay less attention to them and a shift is made to more complex reading tasks. Constructivism, Complete Math and Integrated Content all work against the idea that component skills in math can be identified and taught, and that these form building blocks for subsequent learning.
Mathematically Correct

I’m a little afraid of what they’ll come up with, so I want to be prepared.

One annual goal might be: “When presented with grade-level word problems dealing with a variety of curriculum concepts and skills, the student will read and solve the problems.” What would be the component skills (to be used as benchmark objectives) for this?

To use an example:
Tim earned $7 for each of the 4 days he raked leaves in his neighbor’s yard. He spent $10 to buy a new rake. How much did he have left? Explain.

In order to solve this problem, you need to know:
-- Reading comprehension – Seems obvious, but how exactly would you assess? By using standard reading comprehension tests?
-- How to apply an appropriate heuristic – ?? With reform math, anything goes. Including guess & check.
-- Which mathematical operation(s) to use – Seems easy to assess.
-- Plug correct numbers into the selected operations - ??
-- Computation, multiplication and addition – Easy to assess

I’m really completely lost in trying to understand this. Wouldn’t a teacher know the skills, the steps? Maybe not, considering the type of teaching commonly in use today.

what is ten percent?

The whole family has gotten into the act on the What is 10% off? issue.

Even Ed is now writing math word problems.

This is serious.

Here are Ed's two from this afternoon:

1.
When you're training for a 10K race, you run a series of 1K laps. Your first lap takes you 4 minutes. Each subsequent lap is 10% slower than the last one.

What is your total time for the 10 laps?

2.
If on average you run 1K in 4 1/2 minutes, how long will it take you to run a 10K race?


I realize these two questions are logically contradictory, but I'm not going to worry about that for now. Ed says sports are a great source of word problems, and he's right.

I decided today to start giving C. the same problem written as a percent & as a fraction.

Then, having fixed on this plan, I decided to throw in a whole-part problem (something he's never done before) to boot:


1.
finding the parts when the whole is given [note: I labeled the problem with these words]

Christopher wants to buy a $50 video game for 20% off.

By what dollar amount is the price reduced? ___________

What will Christopher pay? ___________


Draw and label a bar model of the problem.
Then write the equation and solve it.

2.
finding the parts when the whole is given

Christopher wants to buy a $50 video game for 1/5 off.

By what dollar amount is the price reduced? ___________

What will Christopher pay? ___________


Draw and label a bar model of the problem.
Then write the equation and solve it.

3.
finding the parts when the whole is given

Christopher wants to buy a $50 video game for 1/5 off.

By what dollar amount is the price reduced? ___________

What will Christopher pay? ___________


Draw and label a bar model of the problem.
Then write the equation and solve it.


This is one of those moments where you see exactly how valuable an experienced teacher at the top of his/her game is to kids learning math. Or to kids learning anything.

Because I've worked my way through so much of Saxon, Singapore, & "Russian Math," I have pretty good pedagogical content knowledge. For instance, I now know what "part-whole" versus "whole-part" problems are, a concept I'd never heard of before.

But I still lack "kids-learning-math" knowledge.

I don't have a good sense of the proper use of contrast and comparison in instruction (i.e. having C. do the same problem framed as percent and fraction - good idea or not?); nor do I have a sense of how long it should take for a student C's age to learn these things, which means that when C. doesn't seem to be learning what I'm teaching I can't tell whether he needs more practice or I need to teach differently or both.

I'm making all my mistakes with my own kid.

Still.

By the end of this summer - preferably by the end of tomorrow - he is going to know what 10% off is or I am going to die trying.


hyperspecificity in autism
hyperspecificity in autism and animals
hyperspecificty in the rest of my life
hyperspecificity redux: Robert Slavin on transfer of knowledge

Inflexible Knowledge: The First Step to Expertise
Devlin on Lave
rightwingprof on what college students don't know
percent troubles
what is 10 percent?
birthday and a vacation

a real world problem

C. just came downstairs & told us that the New World Champion of hot dog eating consumed 66 hot dogs in 12 minutes, breaking the previous world record, held by Takeru Kobayashi, of 54 3/4 hot dogs consumed in 12 minutes.

Naturally this inspired a unit multiplier problem AND a bonus percent change problem.

Which is great, because I was planning to teach percent change this summer!

............................

A new world record for eating hot dogs has been set: 66 hot dogs in 12 minutes.
The previous record was 54 ¾ hot dogs consumed in 12 minutes.

Use unit multipliers to determine:

How many hot dogs did the new world champion eat per minute?

How many seconds did the new world champion require to consume one hot dog?

How many hot dogs did the former world champion eat per minute?

How many seconds did the new world champion require to consume one hot dog?

What is the percent increase in number of hot dogs consumed in 12 years, from the former world champion (59 hot dogs in 12 minutes) to the new world champion (66 hot dogs in 12 minutes)?



World Hot-Dog Eating Champion Crippled by Jaw Injury
how dogs eat hot dogs (scroll down)

bar model, grade 6 Skill Builders word problem

solution:

The bar model tells you to multiply $8.92 by 2.
Sammy needs $17.84.

large size

source:
Skill Builders Grade 6 word problems
page 34


I may have to force my friend P. to hand over her son for part of the summer. He has a natural interest in math, but is now being potentially moved down to the SPED math track because he can't do word problems.

I'm pretty sure I can fix that.


the Skill Builders series

These little books cost $2.95 apiece.

They can be a bit difficult to track down, so I've got their ISBN numbers here:


Skill Builders World Problems Grade 2
ISBN-10: 1594412790
ISBN-13: 978-1594412790

Skill Builders World Problems Grade 3
ISBN-10: 1932210709
ISBN-13: 978-1932210705

Skill Builders World Problems Grade 4
ISBN-10: 1932210717
ISBN-13: 978-1932210712

Skill Builders World Problems Grade 5
ISBN-10: 1932210776
ISBN-13: 978-1932210774

Skill Builders World Problems Grade 6
ISBN-10: 1932210784
ISBN-13: 978-1932210781


Instructional Fair Workbooks
Skill Builders Word Problems Workbooks (fantastic & cheap - $3.95/book)
Skill Builders Workbooks (all)
mixed review workbooks
Spectrum Math workbooks

Help Desk – Singapore bar modeling

I will be starting to teach my 9-year old daughter how to solve word problems using Singapore bar modeling. I purchased Challenging Word Problems 2 & 3. We have never used Singapore before, only Saxon and Kumon. She has difficulties with reading comprehension and word problems are hard for her.

To prepare myself I’m practicing doing problems in the book. I’m also going to practice with the problems on Thinking Blocks, a wonderful interactive website that enables users to practice with bar diagrams.

I’ve searched for help, and found this scripted prompt that appears useful:
STEP ONE: What do you know already? - Can you draw a bar (or two) to show what you already know?
STEP TWO: What do you need to find out? - How are you going to find that out? - Does the bar(s) you have set up/drawn give you any clues as to what kind of question (operation) this is and what you need to do with the numbers?
STEP THREE: Now DO it and find the answer(s)!

I have virtually no teaching skills, and I’m feeling challenged by the prospect of trying to explain how to solve word problems. Typically I have found myself tongue tied when helping with homework word problems.

Any advice for me? Any resources you would suggest?

Russian teacher in America

In every society certain skills are expected of everybody. For example, if a Brazilian boy tells his mates that he cannot play soccer, he will be considered a nut. Mastery is expected from everyone, so everyone gets it.

Something similar takes place in Russia with respect to word problems. Presence, even abundance of word problems has been normal in Russian school for many decades. The difficulty of problems continually grows from one grade to another, then to olympiads, then to research. Here are a few short statements under which most Russians would subscribe:

• It is good for children to solve multi-step word problems already in elementary school, before starting algebra,.
  

• Children are motivated to solve arithmetical word problems because they are

• generally motivated to overcome difficulties, both physical and intellectual, andare willing to train themselves for that if the tasks are within their possibilitiesand the society approves it.
  

• Solving word problems is a good opportunity for the children to display and train creativity. The fact that the answer is unique and predetermined by thedata by no means contradicts this.
  

• If you can solve a word problem without algebra, it is good for you. Generally, the more bare-handed you solve a problem, the better for you.
  

• It is normal, even necessary that teachers require correct, clear and explicit solutions and answers.
  

• The younger are children, the less they should be allowed to decide what to study.
  

• Solving arithmetical word problems in elementary school should be obligatory for every healthy child.
  

• The government should plan the minimal version of school studies and state theminimal level of difficulty of problems solved in every grade.

• Word problems do not need to be realistic in the literal sense. They are solved for the sake of general intellectual development rather than for a literal application to everyday life.
  

source:
Arithmetical Word Problems in Russia
Andre Toom

This is one of those unpleasant moments when I have to hear a mathematician and math educator say something Math Trailblazers would say.

Of course, when Math Trailblazers talks about solving algebra problems without algebra (or, in the case of Math Trailblazers, adding and subtracting without addition and subtraction algorithms) they're also talking about slowing math learning down to a crawl:

In MATH TRAILBLAZERS, instruction in standard procedures is delayed slightly beyond the traditional time, but problems that would normally be solved by standard procedures are often introduced sooner than is customary. This forces students to use their prior knowledge to devise ways to solve the problems “from first principles,” thus promoting students’ construction of their own understandings.

source:
Arithmetic TIMS Tutor Section 9

update from Google Master:

Oh, but you stopped quoting right at the side-splitting part!

The following problem is from a recent Russian textbook for the 2-d grade:

Problem 1. Vintik and Shpuntik agreed to go to the fifth car of a train. However, Vintik went to the fifth car from the beginning, but Shpuntik went to the fifth car from the end. How many cars has the train if the two friends got to one and the same car? [Geidman.2.1, p.9].

The following problem is from a well-known russian book for children written by Nosov. Its main character Vitya Maleev did poorly in mathematics in the third grade and promised his teacher to train himself in solving problems from the 3-grade textbook. This is one of them.

Problem 2. A boy and a girl collected 24 nuts. The boy collected twice as many nuts as the girl. How many did each collect?

These are good, simple word problems that need no algebra. They give practice in reading word problems and deciphering what is being asked for, which I believe is what someone (Rory? Steve?) said was step 0 for solving word problems.

The side-splitting part is that I cannot imagine these being given to your average American 2nd or 3rd grader.

and from Exo:

In terms of solving for the sake of solving problems - the famous Russian scholar Lomonosov's saying was in every classroom: math is calisthenics for the brain.

Word problems - I make them up by dozens at a time using fairy tales characters, our family members, friens, everything... The main idea -to identify what is given first, figure out what's asked, and find the solution. A 5-year old can easily do the problems with addition, substruction, easy multiplication, division. My son, after I made him memorize first 3 columns of times table said that the problems became more fun, since he did the easy in his mind.

And multi-step problems... Boy, was I surprised that my 7th graders were hardly able to solve word problems in physics that involved 1 step (substitute the numbers into the formula) and struggled enormously over 2-step problems... When I clearly remember (and my Russian workbook in physics 6-7 proves it) that we were solving problems that required deriving a formula from another, at leat three steps, were not allowed to use calculators, and had to memorize all formulas.

OK.... so now I'm feeling REALLY bad that I never made C. do the Challenging Word Problems books....

Speaking of C., he and his father are downstairs now, working on his paper for social studies.

The kids are all supposed to write a 3-page paper with sources & "create a Civil War artifact" for the Civil War Museum we're all invited to see next Monday or whenever.

(At least, I think we're invited. We were accidentally blind-copied an email, clearly not intended for us, which referred to the teachers having sent out invitations to all parents -- which, judging by my email queue, does not include us.... so ..... maybe we're going to a middle school Civil War Museum and maybe we're not. We shall see.)

In any event, last night Ed was helping C. with his paper and he discovered that C. does not know how to write a paragraph.

More accurately, C. has no idea how to write an informational paragraph summarizing the argument of a piece of historical writing.

He knows how to write a memoir paragraph. I think. Pretty sure.

He does not know how to write a paragraph for a 3-page research paper for social studies.

So.... his assignment is to write a paper, not a paragraph.

Pick a Civil War topic and write a 3-page research paper about it.



update - 6-1-2007

I should add that C's ELA & social studies teachers are very good -- which is leading us to a new perception of what we're struggling with here.

I'm going to save that for a second post.


whole stuff taught wholly
the struggle

Top Ten List

I found a packet in my 5th grader's math binder about a week ago.

Remember, this is math class. 5th Grade Math Class.

Top Ten List of Problem Solving Strategies:
1. Act out or use objects
2. Make a picture or diagram
3. Use or make a table
4. Make an organized list
5. Guess and Check
6. Use or look for a pattern
7. Work backwards
8. Use logical reasoning
9. Make it simpler
10. Brainstorm

Kids use these "strategies" to solve word problems.

I don't even know where to begin with this. This is a Top Ten List? My first thought is, what strategy is left that is so lame brained it couldn't make this list?

Act it out? Presumably, this will help when you have to solve the question on 1/3 of a football team wears glasses, 1/2 of the glasses wearers are blond, if 4 kids are blond glasses wearers, how many kids are on the football team? Or something like that. Get all your blond classmates together and line them up.

No mention of using an algorithm. How about, decide if you are using multiplication/division/addition/or subtraction? That seems like a good starting point.

Brainstorm? guess and check?

This is not from EM. I have no idea where the teacher found this gem.

.......................

Catherine here, parachuting into Lynn's post.

I thought that list looked familiar. (hit refresh a couple of times if necessary)

despair

So this morning I finished taking the Singapore Math end-of-5th-grade placement test. (pdf file)

It took me an hour to do the last two problems:

14. A motorist traveled from Town A to Town B. After traveling 1/3 of the distance for the journey at an average speed of 45 km/h, he continued to travel another 480 km to reach Town B. If his average speed for the entire journey was 54 km/h, what was his average speed for the last 2/3 of the distance? (answer: 60 km/h)

15. A car and a truck were traveling to Town Q at constant average speeds. The car overtook the truck when they were 420 km from Town Q. The car arrived at Town Q at 6:30 p.m. while the van was still 120 km away from Town Q. The van arrived at Town Q at 8:30 p.m. What was the average speed of the car? (answer: 84 km/h)

If I hadn't had the answers I would never have gotten the first problem right. I thought I should be able to set it up as a simple averageing problem:

(45 + 2r)/3 = 54

I did manage to work the problem correctly after an endless struggle. But I still don't (really) understand why a simple averaging of rates doesn't work, apart from a vague comprehension that I haven't "accounted for time."

The second one wasn't quite so taxing, but it was depressing nonetheless. I spent quite awhile trying to figure out exactly what I needed to compare to what -- I went down several wrong paths before suddenly realizing that I was missing the figure of 300 km.

Maybe I'll be smarter when I finally get some antibiotics.

I'm not holding my breath. (Coughing, yes. Holding my breath, no.)

If I had the energy I would now be panicking about whether I need to try to put C. back in the Singapore Math books.....actually, I would be alternating between panic and despair. He's only gotten through the first 20 lessons in Saxon Algebra 1/2; teaching a separate coherent curriculum around here is not happening, and is not going to happen.

It takes constant energy and vigilance to get him to do anything other than the work he's assigned at school, and his school assignments aren't teaching him what he needs to know.

AND -- I'm just about done wailing here -- winter illness and 7th grade wiliness have conspired to prevent C. from completing his fancy test prep book. The book has 49 lessons plus 6 "Progress Checks." The state test starts tomorrow and he's completed just 41 lessons and 2 Progress Checks.

He spent a good week or maybe two being too sick to work; then once he was better I was too sick to force him to work.

Now he's off at Stew Leonard's with his dad and I'm going to nap. When he gets back he's doing at least two more lessons; I've already told him he's probably going to have to read the others whether he does any of the problems or not.

I'm just not seeing how we get from here to calculus ---- or even from here to algebra 2.


department of corrections

This is the end of 5th grade test (pdf file).

The test linked to above is the test given at the end of 6A.


update: forty-two & david on distance-rate-time problems


from forty-two:

It helped me with problem 14 to remind myself that you don't need algebra to solve it. When I looked at it strictly arithmetically, it became a lot easier than it was at first look.

My thinking went like this:
Total distance = 480km / (2/3) = 720 km
Total time = 720km * 54km/hr = 13 1/3 hr
time for first 1/3 = (720km/3) * 45km/hr = 5 1/3 hr
time for last 2/3 = 13 1/3 hr - 5 1/3 hr = 8 hr
rate for last 2/3 = 480km / 8hr = 60 km/h

Honestly, I feel dumb, but I'm not sure WHY simple rate averaging doesn't work either. I know it doesn't, because I've beaten my head against enough rate problems to have first-hand experience with the many wrong ways to solve them, but I don't have a concrete reason WHY. I'll have to do some research.

david:

In order to take the average of two speeds, you need equal times, not equal distances.

For example, if I drive at 60 mph for one hour, then I drive at 50 mph for one hour, then my average speed is 55 mph. But if I drive at 60 mph for one mile, then I drive at 50 mph for one mile, then my average speed is less than 55 mph, because I spent more time at the slower speed.

For problem #14, you can use averages, provided that you express the rates as hours per kilometer. The equation is

(1/3)(1/45) + (2/3)(1/x) = 1/54

The idea is to put the unit of distance in the denominator, so we can take averages on the basis of equal distances instead of equal times. (I hope my explanation isn't too unclear.)

I'm going to read these explanations through a few times.

This is one of those cases where doing lots of these problems will inevitably lead to my experiencing these principles as obvious and natural.

procedural fluency preceding conceptual understanding

Happens a lot, I find.

_____________

state test coming right up (2006)
throwing money at the problem
more stuff only teachers can buy
help desk 1
state test coming right up (2007)
help desk 2
my life and welcome to it
inflammatory
canadianteacher.com
progress report
despair
28 out of 30

all the answers are belong to us
email to the math chair
second request
teacher's manual
it would be unusual