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[H]ere are some strategies that might actually be useful:
1) Identify what sort of answer the problem needs. (If the question is of the form "What percentage of the students are blond", an answer of "7 students" cannot be correct.)
2) Identify the units in which a correct answer will be expressed.
3) Write down the equations that you know from the stated problem.
4) Compare what you now have with other problems you have done and work the problem.
5) Make sure the form of the answer you got matches what you identified about the form of a correct answer in steps 1 and 2.
6) Think about whether the answer makes sense. (If you've calculated that the sales tax on a $10,000 car is $6.5 million, you might have made a mistake somewhere.)
7) Recheck your answer by working the problem backwards.
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8 comments:
These are the sorts of conceptual difficulties my students have the biggest problems with. I couldn't tell you how many times, when working with proportions, that I would ask "If you can buy 10 bananas for $2, does it make sense that 1 costs $5?" only to see a sea of blank faces, or people actually responding "yeah!"
They knew they had to divide, but their lack of confidence with fractions led them to divide the wrong way. I'll definitely keep this list in mind
Hi Obi-wandreas!
That's interesting.
I think it may be possible to use unit multipliers to get around the "Am I supposed to divide or multiply or what?" issue... I've tried it with C. a bit, and it has seemed successful. (Don't bet money on this!)
I did find, definitely, that saying to C. & his friend M., when they were in 6th grade, "If you can buy 1 pencils for 50 cents, how much would 2 pencils cost?" instantly made sense to them. This was in conversation, walking down the street.
This is starting from the other side... but it seems logical to me that you could then "flip" and ask them whether 3 pencils for five bucks would make sense.
Doug,
Your second point is a gem of a starting point. Thanks!
I like to break problem solving into two groups: pre-algebra and algebra. Many pre-algebra problems would be trivial once the student learns some algebra. Without algebra, however, they just have to "figure it out". I have very mixed feelings about this.
When I was in school, I recall thinking that my ability to solve one problem wasn't useful for solving another. Bar models and drawing pictures are useful, but only for some problems. You can work really hard at this, but the time might be best used to move right along to algebra. I'm a big fan of starting algebra early with very simple equations. Don't wait until the problems are difficult.
In my son's fifth grade EM workbook, they like to use things like triangles and squares in place of letter variables. They are making this more confusing than it has to be.
I see too many complicated pre-algebra problems. My son's school gave them the following problem.
You have some black and white marbles. The total number of marbles is 860, and there are 230 more black marbles than white marbles. How many marbles of each color do you have?
This is a trivial problem when you use algebra. It's not clear what advantange there is in having pre-algebra students spend a whole class in groups trying to guess and check their way through it. There may be some benefit for a few, but it's not clear that it is time well spent.
I really like Doug's list, but number 3 assumes that they are able to do algebra. If so, then I would expand on it. I would call them governing equations.
I would make a list of all sorts of governing equations for the students. There is d=rt, there are mixture and work rate per person equations, there are percentage equations (markup, discount). With each equation, I would define the units of each variable and give some examples of how the units have to match up throughout the equation. For example, in the d=rt equation, if rate = miles per hour, then the distance has to be in miles and the time has to be in hours. I would print and hand out this list.
Students can then practice looking at problems and picking out which governing equation they can use. Then they have to find the numbers that go with each variable and check for consistency of units. The goal is to get one equation in one unknown, or two equations in two unknowns, etc. For each type of governing equation, you have to start with easy problems and work towards more complicated variations. You don't have to "figure out" a problem before you do it. You just do legal math things and let the math figure it out for you. Let the equations give you the understanding.
If you can't find a governing equation for a problem, then you have to make up your own equations by reading the problem. You start by defining unknowns. You don't have to think a lot. It doesn't matter if you have 50 variables defined. Just start giving unknown numbers a name. For the marble problem, 'B' is the number of black marbles and 'W' is the number of white marbles.
The next job is to come up with equations that link the variables together. You don't have to think a lot. You don't have to figure out the problem in your head. You don't even have to know whether the equations are useful. All you have to do is to is to write legal equations.
In the marble problem, it's easy to see that W + B = 860. You can "see" where you are going in this problem, but it isn't a requirement (possibility) in complicated problems. All you need to know is that the equation is corect. Another equation would be B = W + 230. You don't have to think much. Let the math solve the problem. You have two equations and two unknowns. The problem is done. Solving just requires turning the crank. OK, students have to learn how to solve systems of equations, but this is just a skill; this is not the difficult part of solving word problems.
Problem solving is very teachable. It's not some sort of Zen-like skill. It can be broken down into very specific steps that don't require any sort of magical understanding. But it does require practice. There are many problem variations that can be quite confusing, so you have to build up to them.
My motto: "Let the math do the work."
Steveh
I am trying to figure out the best words to convey the message to you that your explanations and comments are just amazing. My initial thought after reading your above comment was that you should start a blog or write a math textbook of some kind.
I particularly like your comment about the overly-complicated pre-algebra problems. This just hits the nail on the head. It makes math seem overly complex and thereby frustrating, which leads, I think, to a lot of kids just turning off out of frustration.
I have reached more non-math parents using Steve's words than I anything I've ever tried.
They also think they're my words which makes me look really smart. Little do they know....
Well, I don't feel so smart sometimes, like after talking math with the curriculum advisor at my son's (ex) private school. I keep wanting to find just the right argument or angle.
I talked about how they have to use the same algebra textbook in 8th grade as the private high schools use. She said that they do and that the kids do well in high school. I couldn't believe that it's the same textbook. They use a watered-down Prentice-Hall algebra text. I didn't have the details so that shut me up.
I talked about how much better Singapore Math is and how I compared it side-by-side with Everyday Math. She "likes" Singapore Math, but she's checked out EM at the What Works Clearinghouse and it seems to be more adaptable to the various levels and learning styles of the kids. There was an underlying message that curricula like Saxon and Singapore math are too rigorous. You can't argue generalities and the parents aren't allowed in the meetings where they get down to the details.
The kids do well when they get to high school. That about covers it.
A parent friend of mine did find a couple of angles. She is much more aggressive than I am. In a big meeting, she asked them why it was that all of the people who were against EM were scientists and engineers, and why all of the people who were for EM were educators. The second (which will only work for a private school), is why should parents send their kids to the private school and get the same math curriculum as the public school. Elitism is a good angle. I posed the question a little differently. I told them that parents look to private schools for better academics. If they would adopt Saxon or Singapore Math, that would be a clear signal to many parents.
The end reault will probably be that they will keep EM and try to find some way to give the better students more in the later grades. The fundamental fallacy is that they don't think average kids can handle something like Saxon or Singapore. They don't think these curricula are worse that the reform math curricula. They think that they are too much and will hurt many kids. With reform math, less becomes more.
" In a big meeting, she asked them why it was that all of the people who were against EM were scientists and engineers, and why all of the people who were for EM were educators."
Because scientists and engineers have to do something useful with mathematics. An engineer or scientist cannot sit around trying to reinvent the wheel when it comes to mathematics. They must know the math cold or at least be able to look up the correct algorithm to use for a particular problem. A scientist or engineer must use their time on the science or the engineering not the math.
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