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The article here is simply a reproduction of my discussion of guess and check with Mr. Matt Larson, curriculum specialist for math at
Lincoln Public School of Nebraska. This email cites couple of examples to demonstrate my points.
My email to
Matt Larson, curriculum specialist for math
Lincoln Public School, Nebraska
Dear Matt,
Again, just like to point out some of my concerns about the Math education at LPS. And again, I don't have issues with teachers. I simply think this can shed some lights to the curriculum.
The intention of this email is to discuss the Guess and Check strategy taught in the Math.
My opinion is that Guess and Check is over-hyped in the Math curriculum. I understand there are values in Guess and Check, however, the over-hype causes more harm than good. It has shadowed the deterministic value of the math.
To facilitate the discussion, some guess and check problems are used:
Q1. In the zoo there are goats and emus. If there were 10 heads and 36 feet, how many goats are there?
- In solving this question, text book, in general, show this table:
goats emus feet 0 10 20 1 9 22 2 8 24 : : : 8 2 36 - However, most text books failed to point out the importance of the completeness of the guesses - you don't just do a wild guess, you do a systematic guesses to cover all possible guesses. Most text books also fall short on pointing out the pattern shown on the table. And because of this, most text books don't even dare to post a similar question with 1000 of heads.
Q2. Same as #1 but with 1000 head and 3560 feet.
- By constructing the table and observing the pattern:
goats emus feet 0 1000 2000 1 999 2002 2 998 2004 : : : - We see that every time we replace an emu with a goat, the number of feet increased by 2. Since we need 3560 feet, we need replace 780(i.e. (3560-2000)/2) emus with goats. The answer is, therefore, 780 goats and 220 emus. This thinking turns the un-deterministic approach to deterministic approach.
Without observing the importance of the completeness of guesses, students were lead to make wild guesses instead of the systematical thinking, which is the heart and soul of a mathematic training.
Without been introduce to the deterministic approach, students were mis-guided to the in-deterministic view of the universe which is not what a math is all about. Math is deterministic in solving almost all the real life problems.
With these 2 points, I regard the guess and check an abnormal way of solving math problems. Students should be taught to exhaust other means to solve or confine a problem before resort to guess and check.
An example shall shed lights to this point of view:
Q3. There are 12 coins of quarter, dime and nickel. The total value equals $1.75. Find the number of quarter, dime and nickels.
- To solve this question with guess and check we construct a table like:
Quarter Dime Nickel 12 0 0 11 1 0 11 0 1 : : :
- As you can see, this is going to be a long table. However, using deterministic means, we can confine the problem and make it much more manageable:
- By doing 175/25, we can easily confine the number of quarter to within 7 or less. This immediately reduce the number of guesses required by half.
- Using a little bit of algebra, we can easily find all answers in 6 tries:
- 25Q+10D+5N=175
- 5Q+2D+N=35
- 4Q+D+(Q+D+N)=35
- 4Q+D+12=35
- 4Q+D=23
Quarter Dime NIckel 0 23 - Wrong 1 19 - Wrong 2 15 - Wrong 3 11 - Wrong 4 7 1 Correct 5 3 4 Correct
The final conclusion: Math is deterministic( for the most part at least - what good is it if you have to exhaust all guesses to find the answer). Guess and check is not. For the sake of education, students should be warned against guess and check and treated it as the last resort.
Duncan Hsu 2004/09/30