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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
22 comments:
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.
In terms of solving for the sake of solving problems - the famous Russian scholar Lomonosov's saying was in every classroom: math is calistetics for the brain.
Word problems - I make them up by dosens 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 so, 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.
Oh, pardon my typos...
Google Master!
Just the person I'm looking for!
What's the answer to the fourth grade problem - the Magic Kingdom problem??
I think I know, but I need someone else to tell me I'm right.
"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."
Of course I disagree with this. There is no basis for this statement. There are lots of problems that are appropriate for the pre-algebra world, but I see no advantage to delaying the introduction and use of algebra.
"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."
"from first principles"
Ha Ha Ha. And just what are those first principles? The old sink or swim approach to learning. And they thought that traditional math was a filter. What about those kids who don't construct anything? What do they do for them? It doesn't matter because they don't test for "construction".
"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?"
This is fine for pre-algebra, but the question is what do you do when a student can't figure it out? What's the plan? If understanding doesn't magically happen, then what? Assign more problems?
"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."
Not me, but I consider that to be the most important part of pre-algebra word problems - learning to change words into relationships between numbers. It can as simple as looking for key words like "more than", "less than", "of" (for multiply), and "each" (for divide). I guess my point is that you can't just assign problems. There are things you can teach. You can break word problems down into basics. Start with really easy problems in the lower grades. Magically doing it in your head is not good enough. You have to write down the numbers and how they relate. Bar models are great, but the eventual goal is equations, not pictures.
Having a "light bulb" moment of understanding is neither necessary or sufficient, or even possible most of the time. I have had many teachers try to lead us students down the constructivist primrose path. The clues get closer and closer until, in desperation, the teacher tells us the answer.
If students don't construct what you want (even if you could figure this out), then what? Move them to the lower track?
This is fine for pre-algebra, but the question is what do you do when a student can't figure it out? What's the plan? If understanding doesn't magically happen, then what? Assign more problems?
In Russian tradition there is no such thing as "letting a student to figure it out" without first modeling it, learning the simple algorithm, and following the student for a while until these problems become a piece of cake.
All such problems are done first in class - many, many times... In all different variations. Same principle, different story line.
When i started to teach my son addition, we were doing it with objects, then -with imaginary objects, then with just numbers (mental math).
Example (recent): we are driving to Staten Island across the bridge. There are 2 cars in the left line and 3 cars in the right line. How many cars? How many tires altogether?
My son does it in 3 seconds. And makes his own problem for me to solve. "There are 2 cars. In one car, there are 3 people, and in another car, there is 1. How many people are there?"
what do you do when a student can't figure it out? What's the plan?
There are ways of directly teaching how to solve these kinds of word problems. Singapore comes to mind but so also does Fomin's Mathematical Circles and any of the AoS books. And of course, part of the plan is facility with the basics.
When a Russian mathemtician, not math ed but PhD in math from Univ of Moscow with decades experience in teaching kids says "first principles" he probably means something like using the properties of the real numbers as mathematical justification AND the student must write coherent statements that follow one another rather than chicken scratch all over the page.
This is what is so confusing, the mathematicians and the fuzzy math folks often times use the same phrases and terminology but they mean something completely different!
Here is an article by Toom on word problems. He doesn't give specifics but you can see he is not advocating fuzzy math problems:
http://www.de.ufpe.br/~toom/articles/engeduc/MANIPUL.PDF
"In Russian tradition there is no such thing as "letting a student to figure it out" without first modeling it, learning the simple algorithm, and following the student for a while until these problems become a piece of cake.
All such problems are done first in class - many, many times... In all different variations. Same principle, different story line."
I guess this is my point. Trailblazers wants the student to figure everything out themselves from "first principles". So the question is what do they do if the student can't figure it out?
"Generally, the more bare-handed you solve a problem, the better for you."
This is the main point I disagree with and it is what surprised Catherine. The fewer tools you use the better? I don't think so. Math is not some Zen-like process for figuring out problems. The whole point of math is to give you tools so that you don't have to do that. Even with direct, explicit instruction (and lots of tools), there is plenty of room (requirement) for discovery and putting the pieces together.
There are really two issues here. The first is direct instruction versus discovery as the primary approach to teaching math, and the second is what problems should be done in the pre-algebra years and what problems should wait until algebra.
For this second problem, I see too much of the "bare-handed" approach in the pre-algebra years. It wouldn't be too bad if it was coupled with lots of direct instruction, say, using bar models, but with a "first principles" or discovery approach, it's a big waste of time.
Note: One could argue that "bare-handed" doesn't imply no direct instruction. It doesn't necessarily mean discovery. Then the question becomes whether it is better to spend the time using something like bar models (which, actually, would not be "bare-handed") to solve ever more complicated problems, or to start learning algebra.
The goal is algebra. The point of algebra is to make solving problems easier. As I have mentioned before, many problems are trivial after you learn algebra. It might be interesting to have students attempt a few of these problems in the pre-algebra years, but I don't see it as an end in itself.
The "first principles" comment came from the Trailblazers quote. They don't know what they are talking about.
" ...the mathematicians and the fuzzy math folks often times use the same phrases and terminology but they mean something completely different!"
They are good at redefining math.
One last comment.
"If you can solve a word problem without algebra, it is good for you."
Perhaps, but I wouldn't build a curriculum around it, and I wouldn't spend much class time trying to achieve it. It is neither necessary or sufficient.
(Aaaaagggggghhhh... Blogger won't let me use the <sub>foo</sub> tag.)
Catherine --
What's the answer to the fourth grade problem - the Magic Kingdom problem??
Problem 3. When Ivan Tsarevich came to the Magic Kingdom, Koschey was as old
as Baba Yaga and Ivan Tsarevich together. How old was Ivan Tsarevich when Koschey
was as old as Baba Yaga was when Ivan Tsarevich came to the Magic Kingdom?
[Geidman.4.1, p.104].
For this one, I cheated and used some algebraic notation.
Let time 1 = "When Ivan Tsarevich came to the Magic Kingdom".
Let the ages of the people at time 1 be K1, B1, and I1.
The first sentence tells us that: K1 = B1 + I1, which we can rephrase to B1 = K1 - I1. [Equation 1]
Parsing the second sentence:
"when Ivan Tsarevich came to the Magic Kingdom" = time 1, so "Baba Yaga was" = B1.
Let "when Koschey was as old as Baba Yaga was" = time 0, and the ages of the people at that time be K0, B0, and I0.
The question is asking, "What was I0 when K0 = B1?"
Recall that the difference between K's age and B's age, and the difference between B's age and I's age, are fixed.
From Equation 1, we know the difference between K's age and B's age; that's I1. So at time 0, we have K0 = B0 + I1. [Equation 2]
Therefore, when we say K0 = B1, we can substitute in K0 = B0 + I1 from Equation 2 to get B0 + I1 = B1, which we can rephrase to B1 - B0 = I1. [Equation 3]
Now, B1 - B0 is simply the amount that B aged between time 0 and time 1, i.e. the number of years between time 0 and time 1.
But since B1 - B0 = I1 (Equation 3), that means that the number of years between time 0 and time 1 is I1.
The amount that I aged between time 0 and time 1 can be expressed as I1 - I0. But we just found the number of years between time 0 and time 1, and it's I1.
This means that I1 - I0 = I1, from which we can see that I0 = 0.
The answer: When Koschey was as old as Baba Yaga was when Ivan Tsarevich came to the Magic Kingdom, Ivan was zero, i.e. just being born!
My first attempt at bar models looked like this:
--- ---------
i + b
-------------
k
and that wasn't helpful to my thinking, so I swapped the order of Ivan's age bar and Baba's age bar, and the bar models look like this:
--------- ---
b + i
-------------
k
and now I can see that when Koschey's age bar is backed off to be as long as Baba's age bar, we have backed off the length of Ivan's age bar to zero.
Slightly off-topic, I know just enough of Russian folklore to have been a bit freaked out by the subtext of the 'Magic Kingdom' problem. There is a particular painting of Koschei that gave me nightmares when I was little (which, of course, only heightened my obsession with the occult and macabre).
"he-he-he...", said Exo. Russian fairy tails when you hear them as an adult are quite scary. (Brothers Grimm (sp?)too) But for kids it's somehow just fine.
You can use any literature characters to make up funny word problems, though)
Exo - I think it's the notion of Baba Yaga bringing infant Ivan Denisovich (age zero) before Koschei that really chills my spine. It's like a ghoulish housewarming party. "Hi, Koschei, welcome to the Magic Kingdom. I love what you've done with the place. Here, I brought you a newborn child!"
A big part of Koschei's appeal to me as a kid was (a) I was discovering something nobody had ever heard of, and (b) since nobody knew these stories, I had free access to the bloody, scary, lurid tales which would have been inevitibly sanitized for my protection had the librarians known about it.
Oh, dang - Ivan Tsarevich, not Denisovich. Ugh.
The remainder of the problems are murkily translated and I am not sure whether they really mean "faster/more than" or "as fast as/as much as". That is, "three times faster than" is not the same as "three times as fast as" but rather is the same as "four times as fast as".
uh-oh -- I didn't get the answer to that question at all.
I think I misread the English - will have to read again.
What a great thread!
What a great thread!
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