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I hope the question of using a curriculum that’s heavy in word problems will get serious attention from researchers.
Singapore Math uses what people call a "problem-solving approach" from the get-go.
I think it's probably more accurate to call it an "applications" approach. Every lesson and every problem set requires students to apply the concept or procedure to something concrete (i.e. "real world").
Saxon is much more abstract. (I’ve wondered whether girls might be happier with Singapore & boys with Saxon, but I have no idea. For what it's worth I remember being struck by Charles Murray's observation that women are underrepresented in disciplines that are highly abstract, i.e. physics, math, music.)
Even when Saxon teaches the classic word problems (mixture problems, coin problems, etc.) he teaches them as representatives of a class. He takes an abstract approach to word problems.
This makes Saxon very different in feel at least from an “everyday math” approach (no caps).
teaching the classic word problems
I’ve had one experience of Saxon’s approach to word problems “working” for me.
I was taught almost none of the classic word problems in my own high school algebra courses. No number problems, no coin problems, no what time do two trains meet problems, no mixture problems, possibly not even any work problems. (And I took two years of algebra. I must have had the original dumbed-down curriculum.)
Learning math now, I've found mixture problems utterly baffling whenever I've encountered them in another book (Saxon first teaches mixture problems in Algebra 2).
Saxon starts with coin problems. You do zillions of coin problems.
I got so I could do coin problems in my sleep....and, somewhere along the line, I realized I could do simple mixture problems by analogy to coin problems.
That was pretty interesting. I remember reading Saxon saying that there's a reason why the classic word problems are classics; my experience of making a leap from coin problems to mixture problems seems like evidence that he's right.
John Saxon on the classic word problems
Books will de-emphasize the teaching of radical expressions, conic sections, paper-and-pencil solutions of trigonometric equations, and the solutions of the old-fashioned fundamental word problems that have been used historically to teach the concepts and skills necessary to solve all problems.
Does anyone know why the classic word problems are classic?
What are the "concepts and skills necessary to solve all problems"?
word problems are easier
I've been told by more than one parent that Trailblazers looks impressive because they see their kids doing word problems, which they always found impossible.
But I wonder whether the reason word problems were so hard for so many had to do with inadequate instruction, premature demands for generalization, etc., not with some intrinsic challenge posed by word problems per se.
In theory, word problems ought to make math easier. The lack of word problems in the Phase 4 class has been a constant issue precisely because the lack of word problems makes the course harder. The kids get very few concrete illustrations of how the math works.
Here's a blogger talking about word problems making math easier:
It sure seems like a lot of folks absolutely hate ‘word problems‘ - you know, math problems that use words in addition to all of those pesky, confusing numbers. I dunno…word problems always seemed to be much easier for me than regular “numbers-only” math…having real-world examples to work with and provide imagery helped me quite a bit.
That's the way Singapore Math uses word problems - to provide real-world examples and imagery that make the concept more manageable.
I think we may all be awed by a curriculum that seems hard given what we had when we were kids.
In reality, the part of Singapore Math that seems "hard" may be the easy part.
3 comments:
I've found mixture problems utterly baffling whenever I encountered them in another book
Me too, I’m working through that Larry the Cable guy algebra text that I blogged about and it just comes down to systems of linear equations. But even cooler than that is that after I struggled to use substitution on a whole slew of mixture and two trains meeting problems the next unit taught matrices and then says something like “go forth and re-solve all the problems that you just did only using this new method.” Sweeeet.
And then I find out that this matrix thing is a worm hole to a previously unknown to me math universe.
What are the "concepts and skills necessary to solve all problems"?
Beats me, did Saxon himself solve all math problems? ALL of them? “All” problems usually means, “All the problems in our curriculum” that we carefully cherry pick from the universe of math problems because they can be solved using the algorithms that you were just taught 3.5 seconds ago.
“You can kill all the animals in the forest with these weapons.” Yeah, because the animals are tethered to trees and drugged into submission, you’ll be sure to kill them without any risk of harm to yourself or failure. Go out in the woods and find a real math problem living in its natural habitat, it’s not so easy to kill.
I love the way Singapore math approaches word problems. I use it for all my students including tutoring college students.
When I teach word problems to my math booster classes, I teach them the Singapore math model of all parts add to a whole. So you draw a rectangle to represent the whole. You draw a rectangle of exactly the same size below the first one but break it into the number of parts listed in the problem. If you are given all the parts in the problem, you add them to get the whole. If you are given the whole, then you will either be given all but one part or you will be given the relationship between the parts.
For example:
Your mom bought 6 apples and 8 pears. How many pieces of fruit did she buy? This gives 2 parts that you add to get the total.
You mom bought 14 apples and pears. She bought 6 apples. How many pears did she buy? This is the same problem, but you are given the total and one part and you have to subtract to get the second part.
Now, in Singapore math, the complexity increases when you are given a total and the relationship between parts.
Your mom buys 14 apples and pears. She buy 2 more apples than pears. How many apples did she buy. You are give a total and the relationship between the two parts. The beauty of the Singapore math system is that the rectangles that you draw, one on top of another, show that you take 2 away from 14 and divide 12 by 2 to get the number of pears and add two to get the number of apples.
'Classic' problems in algebra like the ones you describe (coin, mixture, rate) are done exactly the same way except that you must also practice dimensional analysis. The numbers in the problem are not necessarily directly additive.
For example, a 'classic' rate problem is this: two trains depart from different cities heading towards each other. One is going 55mph and one is going 65mph. If the cities are 330 miles apart, how long will it take them to pass each other?
This is still a word problem with two parts adding to one total. But the total is in miles and the parts are given as a rate. Rates are not additive, so the rates must be converted into miles and the miles from the two different trains have to add to 330. The relationship between the two parts is given by time since when they pass each other, each has been traveling the same amount of time. So 55t+65t=330. Solve for t.
What this shows is that by the time Singapore students get to algebra, they already have a method to solve all the word problem. All they have to do is substitute a letter for one of the boxes.
This thread is old enough that you may never see this comment, but so be it.
On concepts and skills needed to solve word problems:
The skills include closely reading (so some problem solving difficulties go back to language instruction) and parsing a paragraph, so you can figure out what the question is and what unknowns need to be associated with a variable, as well as a few basic algebra methods. The key concept is that relationships given in words can be converted into a compact symbolic form as a way of solving them.
What makes a classic? Who knows, but the classic problems share certain algebraic structures (x+y=A) with other problems (as noted for mixture problems). Also, repetition results in a student knowing the value of a quarter rather automatically, so focus shifts to reading skills and the algebraic structure and important methods such as substitution.
Saxon said "necessary", not "necessary and sufficient". There are some skills that apply to all problems (math, physics, statistics) and those appear in the 'classic' coin problems.
On matrix methods:
Myrtle, I'll try to wrap this supportive comment in snarky irony. That matrix thing is a compact "formalism" that makes self-evident the fact that all linear algebra problems (all problems involving systems of equations) are the same. It comes with some "algorithms" that provide a way (sometimes efficient, sometimes not) to solve them reliably ... and even figure out if they can be solved before trying. You might now guess why a mathematical physicist such as myself thinks the elimination of such things as algorithms from TERC and EM is insane.
Short version: My rule is that all of mathematics consists of turning a new problem into an old one. You turn 3 equations into 2, then 2 into 1, then solve. Matrix methods tell you how to turn N into N-1, and the rest is just labor.
The more I see of Singapore math, the better I understand why my students from Malaysia do so well. It is a lot like US math in a different lifetime, combining the best of old math with some new things.
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