Reading my post entitled "MET Building Futures Workshop" I alluded to a problem we were asked to solve.
"There are 36 students on a bus. There are 8 more boys than girls on the bus. How many boys and girls are there on the bus?"
Here are at least 6 different ways to solve this problem.
PROCEDURE 1 (Keeping the sum constant-36)
( STEP 1) Choose two numbers that add to 36. (e.g. 20 + 16)
(STEP 2) Find the difference between the numbers. (e.g. 20 - 16= 4)
If the difference is less than 8, then add one to the larger number and subtract one from the smaller number. Return to (STEP 2) (e.g. 20+1 = 21 16-1=15 >>> 21-15=6 and so on)
If the difference is greater than 8, then subtract one from the larger number and add one to the smaller number. Return to (STEP 2)
PROCEDURE 2 (Keeping the difference constant-8)
(STEP 1) Choose a number equal to or less than 36. (e.g. 25)
(STEP 2) Add or subtract 8 from this number to obtain your second number. (e.g. 25-8=17)
(STEP 3) If the sum of these two numbers is less than 36, then add one to each number and return to (STEP 3) OR If the sum of these two numbers is greater than 36, then subtract one from each number and return to (STEP 3) (e.g. 25+17=42. 25-1=24. 17-1=16. 16+24=40 and so on)
PROCEDURE 3 (My strategy)
Divide 36 by 2. >>36/2=18. Divide 8 by 2. >> 8/2=4. Add 4 to 18 to find the number of boys. Subtract 4 from 18 to find the number of girls. 18+4=22 boys. 18-4=14 girls. Verify (22+14=36 / 22-14=8)
PROCEDURE 4 (solving algerbraically)
Let X=number of boys.
Let Y=number of girls.
X+Y=36
X-Y=8
Solve for X in X+Y=36. X=36-Y
Substitute X=36-Y in the other equation X-Y=8 >>>
(36-Y)-Y=8
36-2Y=8
-2Y=8-36
-2Y=-28
Y=-28/-2
Y=14
Substitute Y=14 in either equation.>>> X+Y=36 >>> X+14=36 >> X=36-14 >> X=22
PROCEDURE 5 (graphing)
Let X=number of boys
Let Y=number of girls
X+Y=36
X-Y=8
Solve for Y in both equations
X+Y=36 >>>
Y=36-X
X-Y=8 >>>
Y=X-8
Plot Y=36-X and Y=X-8 on a graph. Where the two lines intersect is the solution.
PROCEDURE 6
Add 8 to the number of girls so that the number of girls and the number of boys is now the same.
Adding 8 girls brings the new total to 44 (36+8 added girls)
Divide 44/2=22.
Subtract 8 from 22 to find the number of girls. (22-8=14)
THERE ARE PROBABLY MANY OTHER WAYS TO SOLVE THIS.
If fact, I can think of at least 3 other ways right now.
The point?...
Don't tell your students formulas to solve problems. Let them discover them on their own. And the ones they do not discover, you need not worry because if your classroom is collaborative and cooperative, they will learn these other ways from their peers.
What do they even need teachers for anyways? :)
Strawberry Fields, Central Park, NYC.
Imagine all the people living life in peace...Imagine all the people sharing all the world. You may say that I'm a dreamer but I'm not the only one. I hope someday you will join us and the world will live as one.
6 comments:
hahahhahaa
"What do they even need teachers for anyways? :)"
paul, ur too funny.
That was something that I learned as well, to not give formulas, and let children solve math problems on their own.
I remember learning math with pure formulas. It made my thinking incredibly dull and I was only able to think one way. I don't remmeber teachers asking me "solve this in 2 or more ways". I feel jipped in my previous education, and I know there's so much learning for me to do still, if i want to catch up to the kids. haha
Paul,
thank you for sharing this problem (I'm being honest, not sarcastic at all), I can definitely use this in my practicum. I think that providing problems that generate so many different solution strategies allows children to think outside of the formula box and empowers them to make a decision about what the best strategy is, depending on the situation/problem.
Thanks again!!
Anna
It was pretty neat to see all of the different ways that people would tackle the same problems that she gave. I have mixed feelings...I think you should let them discover the formula, but for the ones that are struggle, I think a little boost in the right direction wouldn't hurt. The cooperative learning environment is very beneficial to the students. I guess they need the teachers more to maintain the classroom now that students are encouraged to socialize in group learning environments. So you're still needed...just not to feed the information.
I don't think the "don't give students the formula" thing only applies to math but to everything involving students. One thing that comes to mind it problem solving in the play ground. It rarely solves the problem by having students "stand in the corner until the end of recess" students need to learning to deal with these issues with their peers. Having teachers step in and tell them what to do next never helps learning, we may need to give little clues to help them to get to where we would like but we should never tell them the specific way....I think I'm rambling...so I'll stop...
Great blog Paul. I totally agree with you in that student exploration is key. We need to teach students the strategies and then let them run with it, allow them the freedom to discover!
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