Sunday, February 26, 2012

Letter from Huck Finn: Navigating the Connections Between Algebra and Tomato Paste

Out in Left Field proudly presents the third in a series of letters by an aspiring math teacher formerly known as "John Dewey." All personal and place names have been changed to protect privacy.




I student taught for 15 weeks at Aragones Junior High. The school is located in an agricultural region known primarily for its strawberry fields. It is in a residential neighborhood surrounded by hilly farmland. Some of the farms have cows on them, but most grow strawberries. The make-up of the school is about 96% Hispanic students, many of whose parents work in the fields I would be passing on a daily basis. During my time there, the students sometimes appeared to me as adults, sometimes as kids, and other times as people caught in between.

The first weeks of my program I would be observing the three classes I would be teaching; by the fourth week I would begin to teach some classes. On my first day Tina, my supervising teacher, asked me to circulate among the students during the algebra class and answer questions or offer help as needed. The algebra class was a discovery-based class that used the CPM textbook (College Preparatory Math). The problems the students were working on in their groups of four were “guess and check” problems. The book spends more than half of the semester on these types of problems and even provides instruction in how to set up tables of values to maximize the efficiency of an inefficient process. Tina even gave instruction and pointers on how to do this during the class. She’s a good teacher and explains things well, which bolsters a point I’ve made in my previous incarnation as John Dewey: It’s possible to do something horrendous tremendously well.

A moment of full disclosure: I went to school at a time when math was taught in a traditional manner in K-12. Despite my average intelligence and despite claims from various quarters that such method was more destructive than typhoid fever except for very bright people, I managed to learn enough to allow me to major in the subject. The algebra book I used started almost immediately with how to express words as algebraic expressions, and to use that skill to set up and solve equations. The book briefly discussed the “guess and check” technique which at that time was called “trial and error”. It illustrated how a problem could be solved using trial and error, and then how the same problem could be done quickly and efficiently with algebra which then remained the focus of my algebra course.

A boy named Rudy asked me for some help with a problem. Rudy was fairly bright and his group mates seemed to rely on him to get through the problems. The problem was as follows: “In making a batch of soup, the number of cans of tomato paste was five more than twice the number of cans of noodles. A total of 44 cans were emptied into the soup. How many cans of each ingredient did the team use?”

I was mindful of the not-very-optimistic warning I got from the local university which placed me at the school: “You are a visitor/guest in the classroom. If there are any differences of opinion with the teacher, things have to be done as she wants them since it is her classroom.” Thus, I strived to adhere to the guess and check nature of the assignment. I knew from watching a pre-algebra class earlier that students learned how to translate English expressions into algebraic ones. I figured that these students knew how to do that at the very least. But while CPM touts itself as connecting knowledge, connecting to what they learned in pre-algebra was apparently not on the agenda.

“OK,” I said. “If you have twice the number of cans of noodles, how can we write that?”

They stared blankly at me

“How do we write ‘twice’?” I asked. “What number do we use?”

Rudy brightened and said “Two”.

“Yes, two; so if I have four cans of noodles what’s twice that amount?” Rudy thought a bit while the others in the group looked at him.

“Eight.”

“Right! So if I call a can of noodles ‘x’, what’s twice that amount?”

Rudy thought again. “2x?”

Now we were getting somewhere. “OK, so if the number of cans of tomato paste is 5 more than twice the number of cans of noodles, how do we say that?”

Rudy thought for a moment. I expected to hear “2x + 5” but instead, he said: “Paste.”

I asked the others. They all said “Paste.” The others now started to giggle. What I didn’t know was that the students had been setting up guess and check problems with headings like “cans of noodles”, and “tomato paste = twice the number of cans of noodles + 5”. Rudy was stuck on “paste” as an answer and the more he said it, the more I tried to get them to use letters rather than words. The futile conversation with Rudy was making them all laugh. They tried not to, but that only made it worse. They all spoke English well, but for all practical purposes we were speaking different languages. For lack of experience and anything better to say I told them, “If you want to joke around, you’re wasting your time and mine,” and moved to the next group.

Tina told me later that the class wasn't ready yet to translate into expressions using “x”. “You don’t know what ‘guess and check’ is about yet,” she said. The motif of old school teacher meets the modern method then became the basis for her retelling the story to others in the teacher’s lounge. “Guess and check isn’t easy to teach,” she said consolingly. "The book eventually does connect guess and check with equations, and then the light bulbs come on when they put it all together."

I was about to say it might be a lot easier just to teach them algebra. But I thought it probably would be best to keep my mouth shut, so that’s what I did.

11 comments:

Early Knowledge for Kids said...

" I went to school at a time when math was taught in a traditional manner in K-12. Despite my average intelligence and despite claims from various quarters that such method was more destructive than typhoid fever except for very bright people, I managed to learn enough to allow me to major in the subject."

I did well enough at math as well using more traditional methods. These methods definitely did need improvement. But throwing them out altogether and replacing them with new and untried methods has been a complete disaster.

I use Singapore Math to teach my 1st grader. It is close enough to how I was taught but definitely superior in many ways. We need to focus on how to make teaching of subjects better. Eliminating traditional methods altogether and using millions of students as guinea pigs for new teaching ideas is not a better way to do things.

Widow Douglas said...

Bravo!

I actually use word equations for my strugglers and even with my honors students, but VERY briefly and I translate very quickly to variables. I think that some learners need them, particularly those who have trouble with reading comprehension. However, I know that my students are accustomed to translating words into variable representations, so the word equations are mainly to help provide a structure for the benefit of struggling students.

I, too, went to school at a time when math was taught in a traditional manner K-12. I don't consider myself of above average intelligence, but like Huck, I managed to get a degree in math without having to retake any courses. My traditional math education served me very well, as it would serve any student, college bound or not.

Anonymous said...

The supervising teacher's not so subtle arrogance that "guess and check" is not easy to teach is so typical of those who now make math harder than it is. This intellectual air makes them look, they think, especially bright. All those who don't "get it" can indeed develop feelings of inadequacy. Hence, a hatred of math develops among new teachers, parents, and finally the children.

These reform teachers are the same ones who condemn traditional math for being unintelligible by girls and minorities (except Asians). According to them, only white males could understand the linear-thinking and analytically-based math procedures.

How can we ever turn this around? Are we to have two distinct Americas--one with those who can think logically because of traditional teaching of specific and efficient steps and one that has to "guess and check" or create from the ground up whenever a new topic is presented to them?

Anonymous said...

I think it's silly to spend so much time on guess and check. I came across a problem last week with one of the students I tutor that we tried to work out, and I failed. At that point, I said that if normal methods fail, if one has a multiple choice problem, that one can try those answers to see if any of them work. At that point, I noticed a crucial piece of information we had not noticed before which made clearly one answer right rather than the other three, and no calculations were necessary. I think the time would be MUCH better spent teaching students how to translate from English into math expressions. I have yet to be in a school, either as a student or teacher, where I've ever seen that done. If story problems are given at all in most schools, it seems they are just thrown at the children in a "sink or swim" fashion.

Lynne Diligent
expattutor.wordpress.com

Anonymous said...

I think it's silly to spend so much time on guess and check. I came across a problem last week with one of the students I tutor that we tried to work out, and I failed. At that point, I said that if normal methods fail, if one has a multiple choice problem, that one can try those answers to see if any of them work. At that point, I noticed a crucial piece of information we had not noticed before which made clearly one answer right rather than the other three, and no calculations were necessary. I think the time would be MUCH better spent teaching students how to translate from English into math expressions. I have yet to be in a school, either as a student or teacher, where I've ever seen that done. If story problems are given at all in most schools, it seems they are just thrown at the children in a "sink or swim" fashion.

Lynne Diligent
expattutor.wordpress.com

bky said...

Guess and check is, in my experience, difficult to teach and to understand, at least to the extent that it is different from flailing around with numbers. There are sophisticated applications of this in numerical algorithms that work by estimating and correcting. You can get a preview of that with a well-known class of problems. Say you want two numbers to have a given sum and difference. One thing that algebra students should be able to do is to set up two equations in two unknowns and solve them. But here you can get some insight into how arithmetic works if you really do guess-and-check in a deliberate way. The non-deliberate way is to come up with guess after guess until you get the right answer.

The deliberate way is to see how far off your first guess is and use that to correct your first guess so that the second guess (or maybe the third or fourth for beginners) is exact.

Example: I want two integers that add up to 100 and whose difference is, say, 8. Since 8 is small compared to 100, start with something we know adds up to 100 and whose difference is small. How about 50 and 50? The difference is 0. Shift one over to get 49 and 51 and the difference goes from 0 to 2. Ah ha -- supposedly. Every time we shift one the difference changes by two, so from the original guess we need to shift by 4 so the difference goes from 0 to 8. Thus 46 and 54.

If the teacher is not teaching kids how to use their guesses to improve their next guess, they are just flailing with numbers.

At any rate, this should be a minor topic, not something that holds kids off from learning real algebra. Translating English into expressions and equations is a good place to start.

LynnG said...

I kept getting hung up on "cans of noodles" -- who puts cans of noodles in their soup? Once I figured out that there were two types of cans, it all made sense.

Katharine Beals said...

Huck Finn responds: "Lynne, I was hung up on what kind of soup would contain tomato paste and noodles."

Anonymous said...

Translating word problems into mathematical language so that Algebra can be applied, instead of guess and check, is a laudable approach.

However, instead of teaching pupils to read, reading instructors often resort to their own particular brand of guess and check, where the pupil is supposed to guess what a passage of text says not by sounding the letters making up the words, but by guessing what is written based on illustrations or other clues. On the whole, a rather sorry and reprehensible abdication of responsibility by teachers to their pupils.

kmtex said...

Unfortunately, the "guess and check" method, along with some of the other "new" ways to have children "flail with numbers" (I really like that phrase!) is being pushed on teachers by curriculum specialists in our school districts, who have been directed by Curriculum and Instructions Superintendants, who in turn are bombarded by state education agencies, who have been pressured by reformers and researchers and federal ageny people.

Believe it or not, as a teacher, I have little to no say in how I teach a concept to my students. When schools spend money on the latest and greatest program, and you aren't following the rest of the herd, then you are in the wrong. Even if you can point to results and say, "Look, this group mastered this concept in half the time and at 96% compared to this other group using this other method," you can be accused of "kill and drill" or rote memorization, and not teaching "conceptually" or with "rigor."

If the fed would get out of my classroom, and corporate figureheads would stop spouting nonsense about things they know little about, and states would devolve responsibility back to local school boards where it belongs, maybe my 6th grade students could write legibly in sentences that actually make sense, could solve math problems and talk about how we could use it to solve an everday life problem, and explain the history of out state or nation.

Instead, I have students whose handwriting I cannot decipher, who cannot solve simple f=ma prblems, and cannot read various scales (graduated cylinders, rulers, thermometers).

Catherine Johnson said...

“Yes, two; so if I have four cans of noodles what’s twice that amount?” Rudy thought a bit while the others in the group looked at him.

“Eight.”

“Right! So if I call a can of noodles ‘x’, what’s twice that amount?”

Rudy thought again. “2x?”


I love that!