Barry Garelick, who wrote various letters under the name Huck Finn, published here, is at work writing what will become "Conversations on the Rifle Range". This will be a documentation of his experiences teaching math as a longterm substitute. OILF proudly presents episode number 15:
I had come to the point in the chapter on systems of linear equations in my algebra 1 class where the book presented mixture, rate and current, and number problems. To prep them for the onslaught, I included a word problem into one of the warmup problems I had them do as I checked in their homework.
The problem was: “The length of a rectangle is 3 units more than the width. The perimeter is 58 units. Find the length and width."
Students asked me “How do you do this problem?” as I came around to check their homework. I offered one hint: “You can solve it using the substitution method”.
"What does this problem have to do with the substitution method?" a boy named Lonnie asked.
I answered his question when I went over the warmup questions. “If you solve the problem to find length and width you will have two equations in two unknowns which you can solve by substitution."
Many students shouted at once.
“I can’t answer you when you all shout at once. Raise your hands.”
I called on Lonnie. "How do you get the two equations?" he asked.
I drew a rectangle on the board. “How do I express length (L) in terms of width (W) if length is 3 more than width?”
One person ventured an answer, a very smart and quiet girl named Anna. “L = 3 + W” she said.
“Now give me an equation for the perimeter.” After some struggling to remember the formula for it, they came up with 2L + 2W = 38. From there it was obvious how substitution played a role. With this framework now in place, I segued to rate and current problems (which I discussed in Chapter 7), some of which leant themselves to solving by substitution, and others solving by elimination.
After we worked through a few problems I said: “These problems seem hard now, but the problems in the book are really all the same. If you learn how to solve one, you’ve solved them all.”
Which is one of the many complaints that math reformers have against traditional math, though I hasten to say here that such an approach is traditional math done poorly. And Holt Algebra 1 is guilty of providing problems that are exactly alike in structure and vary only in the given values in the problem. I would also like to see problems that up the ante in difficulty, so that subsequent problems force students to extend their thinking from the worked examples.
“I expect that you will find these very easy after a while,” I continued. “So easy, you’ll find them boring. And when that happens, I’ll give you some that are a bit more interesting.”
They didn’t like this idea, and Naomi, a seventh grader, asked "When will we stop doing word problems?"
"We won't,” I said.
“So the answer is ‘never’?”
“Correct,” I said.
And with that I launched into the other types of word problems. Over the next two days we covered the various word problems. For mixture problems, they learned to put the information into a particular format, whether working with nickels and dimes summing to a certain amount, coffee beans selling for various prices per pound, or mixtures of acids of varying strengths. For example:
“A grocery store sells a mixture of peanuts and raisins for $1.75 per pound. Peanuts cost $1.25 per pound and raisins cost $2.75 per pound. Find the amount of peanuts and raisins that go into a 6 pound mixture.”
The information is placed into a table and information arranged in such a way that the associated equations become obvious.
$1.25/lb

$2.75/lb

$1.75/lb


Weight

x

Y

6

Cost

1.25x

2.75y

1.75*6= 10.50

What I wanted to do was to introduce harder problems of my own eventually, once they caught on and found them easy. I wanted to progress to variations such as: “A grocer blends teas worth $0.66 and $0.48 a pound. If he interchanges the amounts, he saves $6 in a blend of 100 pounds. Find the ratio of the weight of the two teas in the original blend.” This is varied enough that they cannot simply plug the values in to a handy table.
But I never got to do this. Looking back, this would have involved giving them a bunch of word problems interspersed between some of the lessons. As it was, the class had started a little behind where other classes were, so I was playing catchup. Not to mention writing lesson plans only about a week ahead of where I was. And as it was, not as many students achieved the facility with these problems that I wanted.
And then there was the eye opening revelation that came when doing number problems. These are the type such as: “The sum of the digits of a twodigit number is 3. When reversed, it forms a number that is 9 more than the original number. Find the number.” When going through these in class the day before the quiz, I asked a girl who got the answer correct (the original number is 12) how she did it.
“I guessed,” she said.
“What do you mean you guessed?”
“I just can’t do it the way you showed us. But I can get the answer really quick if I try certain numbers out till I get the right answer.”
An argumentative girl named Sandra who did very little work and had a knack for hijacking conversations (e.g., “What’s your favorite color, Mr. G?”) came to the other girl’s defense. Her teacher last year (Mrs. Perren, the math department chair) embraced some of the math reform philosophy and Sandra apparently absorbed what her teacher told her. "Guess and check is a perfectly legitimate mathematical procedure that requires deductive reasoning to narrow down the choices," she told me.
My response: "You need to show the equations for the problems on tomorrow's quiz or it will be marked wrong. Guess and check will not be allowed."
“Even if we have the right answer?” she asked.
“Correct,” I said.
This experience would serve me well, I thought. If I ever got to interview for a teaching job and I was asked to describe how I would work within the Common Core standards, I could say “Getting the right answer isn’t enough; students have to show their reasoning” or some such language.
In the meantime, with my position firmly stated, I went over once more how to solve the number problems using algebra.
6 comments:
"This experience would serve me well, I thought. If I ever got to interview for a teaching job and I was asked to describe how I would work within the Common Core standards, I could say “Getting the right answer isn’t enough; students have to show their reasoning” or some such language."
Nice! :)
But why to insist on equations? Aren't pictures, "visual fractions," and "area models" good enough? We need no stinkin' equations in Common Core! Another :)
Thanks for the smiley, Ze'ev. And not to torpedo my own work, but it occurs to me that the interviewer could say "And what do you think of 'guess and check' as a reasoning strategy?" Oops. Guess I better seek employment in a car wash!
Answer: I believe it is important for students to work realworld problems, and the real world doesn't come with an answer key to check against. An engineer often has no way to check their calculations. If an aircraft engineer is wrong, the plane crashes. People in the real world who use math have to have a reliable method for attaining the answer.
They also will be fired if they spend all day guessing and checking instead of reaching a quick and reliable answer.
Guessandcheck is simply not a realworld method.
Guess and check is only an acceptable way to reach an answer if the possible options are VERY limited. Adults use it all the time (in our heads) when there are only 2 or 3 possible answers. So you can introduce it to children on that basis, using a few examples that show that this is an OK method ONLY if it gets you to the answer very quickly.
@Auntie Ann
When I used to teach word problems in Beginning Algebra I would tell students, "You may be able to solve these word problems without equations, but I'm not so much interested in the answers as I am in the writing of the equations. If I were only interested in answers the questions would be A LOT harder. And if you're interested in just answers, go take an engineering class because they're mostly interested in answers  but your answers better be right!"
And the equations are a great way to convince someone else that your answers are right!
You might provide students with an example that's prepared in advance, for addressing this question, with solution (5+5/7, 22/7). One in which guessing the solution is highly unlikely.
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