GambillMethodForTeachingAlgebra < Kitchen

18 Oct 2005 - 14:32

the Gambill method for teaching algebra

from:
Principal's Guide to Raising Mathematics Achievement, pp. 81 - 82
by Elaine McEwan

or see:
Gambill method of teaching algebra in a nutshell

....the way in which I want students in my fantasy school to become proficient in algebra: solving a lot of problems. Practice, practice, practice. "Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences. In addition, algebraic skills and concepts are developed and used in a wide variety of problem-solving situations (California State Board of Education, 1999, p. 38).

One of my favorite algebra teachers is Carol Gambill. She taught at the Sewickley Academy in the Pittsburgh area for many years and is now chair of the mathematics department and vice principal of the Littleton, Colorado, Prep Charter School. Carol has developed what her students call the "Gambill Method." I would hire Carol to teach algebra at my fantasy school in a heartbeat if I could. Here is the Gambill Method described in Carol's own words. Compare it to the way your students are learning algebra.

"Most students who enter my eighth-grade Algebra I or Honors Algebra I classes in September each year are ill-prepared to learn algebra because most of them have not fully mastered arithmetic. To make matters worse, I have too few class periods to teach them the entire rigorous course when one adds up the drug education activities, annual class trips, report card day, vacations, snow days, exams, and parent-teacher conferences. These restrictions demand that the students put in extensive quality time outside of class grappling with difficult problems and practicing for accuracy.

"I devised a method that I have used for 15 years with all students, those with disabilities, the average student, and those who are gifted. This method really works. Students make incredible gains during their year with me, because of the system the kids long ago dubbed the Gambill Method. Here's how it works.

"Twenty to thirty problems are assigned for homework every evening, ranging from the easiest to the most difficult of a given section of the text. I always assign the odd problems because their answers are in the back of the book. The answers provide the students with road maps to mastery. If they don't get the correct answer it means they turn back, take a detour, change a flat tire, or find a service station.

"On the day that an assignment is given I do the even problems with students in class using direct instruction. Although I use a traditional algebra textbook (Brown, Dolciani, & Sorgenfrey, 1994), I have developed totally scripted lessons for each algebra unit that require absolute focus and attention, constant oral responses, and intense involvement from every student. Direct instruction assures that all students leave my classroom that day with a thorough understanding and at least partial mastery of the concepts. I tell my students that doing homework does not merely mean writing out the problems, although that most assuredly is a component. I tell them they must master completely every problem of the assignment from the easiest to the most difficult. I would never assign a problem for which I had not given them the answer. The next day, when the students walk in the door I give them their daily quiz over the most difficult four or five problems from that assignment. Their answers and all of their work toward that end must be accurate. Some students work more quickly than others; the first students finished come up and have their papers checked by me, then they become student checkers and grade recorders, and so it progresses, with more and more checkers becoming available as the slower students finish their daily quizzes.

"Within 15 minutes, all students in the class have taken a daily quiz over the previous night's homework assignment. The quizzes have been graded and recorded and are back in the students' hands, thus providing daily immediate feedback to each student on his or her own progression toward mastery, and providing me, of course, with instant knowledge as to whether or not students did their homework.

"I never ask to see the homework of any student unless that student has failed the daily quiz. If I ask to see the homework of a student who has failed, and the student does not have it, he gets an immediate detention for the day. A detention simply means that students must stay after school that day and do under school supervision the assignment that they failed to do on their own. This, in my opinion, is a justifiable, logical consequence.

"Because I am so strict with mastery of homework concepts, I assure the students that I will also do my part to help them be successful. Therefore, I conduct extra help sessions before school and at both lunch periods. It is gratifying to see five to ten eighth-graders gathered around my chalkboard before school, excitedly discussing a difficult algebra problem. The kids love these chalkboard algebra debate sessions. In addition, the students have my telephone number and are invited to call me as a last resort. Please note, however, that I seldom receive more than two calls per school year.) "

"My students win so many academic awards that students from other algebra classes cannot even play in their league. There is much hard work and yet the students love my class, vote it their favorite each year, love math (even those who had despised it up until algebra) and remember their year with me as the one that led them to discover within themselves the power to determine their own destiny in the academic arena. All this is based on a simple system that nurtures and demands daily perfect mastery of each step in the course as it comes along. Other teachers who have adopted the Gambill Method have replicated my results" (Carol Gambill, personal communication, August 13, 1999).

One of the reasons I want Carol to teach math in my fantasy school is that she achieves remarkable success with her students. They have repeatedly won a variety of mathematics competition championships at the local, state, and national levels for the past 10 years. They took first place in the Pittsburgh MathCounts Competition for five consecutive years and won a Pennsylvania State MathCounts championship as well. And based on the outstanding achievements of Carol's students on the American Junior High School Math Exam, the NCTM presented her with the Edith Mae Sliffe Award. But awards are not the only or even the most important reason for my wanting Carol to teach in my school. Most important to me is that her students know exactly what is expected of them, and Carol is disciplined and structured enough to be consistent.

daily perfect mastery

checking your work & eating Brussels sprouts

I suddenly realized, reading Carolyn's post, that I never have Christopher check his work (except on tests, which he says he does. I believe him on that, because he's a fairly anxious child, and he wants to get things right on tests).

I do, however, make him sit with me after he's done his work and go over each and every answer to see if he got it right.

He loathes this.

Apparently, even having your parent check your work falls in the Brussels sprouts category. (This approach does have the virtue, however, of not offering him a choice.)

For awhile there, I was letting him go off while I sat there struggling to decipher his handwriting; then we'd have a whole battle of wills when I found wrong answers & told him to come back and re-do them. More, more screaming and shouting!

I finally saw the light on that one.

Now he has to sit with me while I read the correct answers, and he checks his work. If his work is wrong, he re-does it.

This is an excellent reason to buy a Teacher's Edition of the textbook if you don't happen to feel like working all the problems yourself. (Also, having that Teacher's Edition gives me that extra-added legitimacy. Christopher can't really argue with THE TEACHER'S EDITION. What the hey!!!! It's not wrong!!!! It's right!!!!! What the hey!!!!! etc. I have less of this, reading answers out of THE TEACHERS EDITION.)

One thing that's incredibly hard for Christopher, still, is following his own handwriting on the page.

He did well on his first big test because it was mostly multiple choice. (I felt it was challenging multiple choice, so that's fine with me.)

He missed almost every one of the pencil-and-paper problems, because the teacher didn't leave enough space for his big handwriting, and he finally got lost in his own pencil marks. (I went through each one to see how & why he got it wrong.)

It's terrifically difficult for him to find where he was in his own sequence of equations once he's lost his place. I'm certain the working memory demands are just too much. He's still learning this stuff, and to find where he was in his work, he has to remember the original problem, remember the way he solved it, and remember what part of the problem each line of his work was supposed to have solved.

These were long, complicated integer and absolute problems.

Gambill in a nutshell

1.  Teacher Carol Gambill has used her method of teaching
     algebra for 15 years. She has successfully taught
     algebra to students with disabilities, students of
     average ability, and gifted students.

2. Twenty to thirty problems of homework every night.
Problems range from easiest to hardest.

3.  textbook: Brown, Dolciani, & Sorgenfrey, 1994. (Not sure
     which book this is: ALGEBRA Structure and Method Book 1
     or Algebra 1 by Dolciani.)

4. ironclad rule: She assigns only those problems that have the
answers printed in the textbook.

5.  She uses "totally scripted lessons ... for each algebra unit
     that require absolute focus and attention, constant oral
      responses, and intense involvement from every student."

6.  Daily quiz on the 4 or 5 most difficult homework problems.
     First students finished have their quizzes graded & recorded
     by Gambill; these students become graders for
     next wave of students finished, and so on down the line.
     Total class time elapsed: 15 minutes.

7. She looks only at homework only of students who've failed
the quiz.

8.  Students who didn't do the homework have detention that
     day during which they do the homework assignment under
     supervision.

9.  Gambill holds extra help sessions before school & at both
     lunch periods. She gives students her home phone number
     to call as a last resort. It is gratifying to see five to ten
     eighth-graders gathered around my chalkboard before school,
     excitedly discussing a difficult algebra problem. The kids love
     these chalkboard algebra debate sessions.

spaced repetition

from Carol Gambill:

My students win so many academic awards that students from other algebra classes cannot even play in their league. There is much hard work and yet the students love my class, vote it their favorite each year, love math (even those who had despised it up until algebra) and remember their year with me as the one that led them to discover within themselves the power to determine their own destiny in the academic arena.
All this is based on a simple system that nurtures and demands daily perfect mastery of each step in the course as it comes along. Other teachers who have adopted the Gambill Method have replicated my results.

Gambill method of teaching algebra in a nutshell

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Look here for syntax help.

Interesting Isaac Asimov story.

http://www.themathlab.com/writings/short%20stories/feeling.htm

Jeff Hetzel

-- KtmGuest - 18 Oct 2005

Most students who enter my eighth-grade Algebra I or Honors Algebra I classes in September each year are ill-prepared to learn algebra because most of them have not fully mastered arithmetic.

In my opinion, almost all academic problems can be traced back to day one in kindergarten. Elementray school, the most constructivist laden years, is basically a waste of six years. Once the more challenging material starts rolling in at the middle scool level students have a giant academic hurdle to jump over to perform at grade level instead of taking the path of six small steps they could have taken in elementary school.

Just because the problems start to arise in middle school doesn't mean that middle school is the culprit, at least not entirely.

-- KDeRosa - 18 Oct 2005

The horror of it is that middle schools were created explicitly to tone down academic requirements.

-- CatherineJohnson - 18 Oct 2005

"In my opinion, almost all academic problems can be traced back to day one in kindergarten."

One parent told me years ago that Kindergarten was great, but then it was all downhill after that.

"Just because the problems start to arise in middle school doesn't mean that middle school is the culprit, at least not entirely."

That's why it frustrates me when government tries to fix high schools by restructuring or some such thing. There is usually not one word about going back to the source. It works that same all along the line. I heard a sixth grade teacher once complain about NCLB testing because many students coming into his class didn't know the times table. I would hate to be in his position, but the problem is not testing. He wanted to shoot the messenger.

One of the big reasons Jaime Escalante was successful was that he didn't do it himself. He went back to the lower grades and fixed the pipeline.

-- SteveH - 18 Oct 2005

One parent told me years ago that Kindergarten was great, but then it was all downhill after that.

The reason I don't want to let kindergarten off the hook just yet is because so many kids come into kindergarten ill-prepared for school. These kids need to be identified early and, for them at leas,t need the kindergarten year to bring them as far up to speed as possible. As far as I can tell, most kindergatens treat these kids like all the rest, so when first grade starts they are still unprepared for academics.

The horror of it is that middle schools were created explicitly to tone down academic requirements.

This is why I like the socialism analody to education. being unsuccessful in raising the academic lot of all, they settle on pulling everyone down as far as they can to achieve a rough "equality."

One of the big reasons Jaime Escalante was successful was that he didn't do it himself. He went back to the lower grades and fixed the pipeline.

This is why the movie Stand and Deliver was a bit of a farce. The movie made it look like Escalante had performed a miracle getting these kids up to calculus speed in a single year.

-- KDeRosa - 18 Oct 2005

My son is in kindergarten too. From his experience, I know that it is possible for kids to learn at least single-digit addition and subtraction to mastery by the end of kindergarten. I further suspect that it's possible to gain a solid grounding in multiple-digit addition at the same time.

Kids want to learn things. It gives them power over their environment in a way that nothing else can when you're that age. I don't understand why anyone would want to deny that to children on the grounds that they shouldn't be "pushed" too much.

-- DougSundseth - 18 Oct 2005

One parent told me years ago that Kindergarten was great, but then it was all downhill after that.

LOL

-- CatherineJohnson - 18 Oct 2005

One of the big reasons Jaime Escalante was successful was that he didn't do it himself. He went back to the lower grades and fixed the pipeline.

I didn't know that.

What did he do?

-- CatherineJohnson - 18 Oct 2005

Yeah, you can see that in Carol Gambill's description of her students.

Learning something to mastery is extremely reinforcing.

-- CatherineJohnson - 18 Oct 2005

It's going to be like another serving of brussels sprouts, but at some point all kids doing math have to learn that they need to write down some notation of what variable represents what. I first hit this with my daughter when she had to compute areas of strange shapes. For example, the shape might be a rectangle with a half circle sticking out one end. The problem would give you the dimensions of the rectangular part. You had to recognize that the height of the rectangle was the diameter of the half circle. Then, compute the area of the circle, divide it by two to get the area of the half circle. Add that area to the area of the rectangle. If you don't write something like

Atotal = Asemicircle + Arectangle

and

Acircle = (pi) * 4^2

along the way, you can get very confused in among all your calculations. My daughter is very smart, and so she thinks she can keep it all straight in her head. She hates the idea of writing these little notes with labeled variables. Of course, if she doesn't, though, she leaves out a step like dividing by two to get the area of the semicircle.

So, the bitter vegetables to swallow are:

Label your variables and the intermediate values that you are computing, and

Use lots and lots of paper to show your work. Paper is cheap, comes from a renewable source, and is recyclable.

-- DanK - 18 Oct 2005

Kids want to learn things.

We often think that, in addition to learning, kids should also have fun. What this fails to understand is that, before they become cynical, LEARNING IS FUN for kids. They love to learn stuff, and they love to show off what they know.

-- DanK - 18 Oct 2005

It's going to be like another serving of brussels sprouts, but at some point all kids doing math have to learn that they need to write down some notation of what variable represents what.

It's not that.

This is an 'occupational therapy' issue; his letters are literally too big for the space she's given him to use.

I think I can show it now that I've got the html.

-- CatherineJohnson - 18 Oct 2005

25 - 16 ÷ 4 · 2 + 6

-- CatherineJohnson - 18 Oct 2005

There are no units to label. (Actually, he got that problem right.)

The teacher gave him a technique, that's very good, where you draw a karat below the two numbers you're calculating, and write the answer to that calculation directly below its point.

That way you can tell which numbers you did (though I'm wondering whether he ought to just cross them out as he goes).

Actually, the karat (carat?) allows you to check your work better.

But the carats/karats add several more lines to the calculation.

-- CatherineJohnson - 18 Oct 2005

Here's the extra credit problem:

84 ÷ (5² + 4 - 15) + 6 · 2 + 3 · 2 - 48 ÷ (11 + 15 - 23) - 72 ÷ (2² + 2³)

He had enough time to do this problem, and he probably could have done the problem.

But without quadrille paper, it was impossible.

-- CatherineJohnson - 18 Oct 2005

Wit and wisdom!

Although, and I need to write about this soon, some learning is more fun than most.

The RUSSIAN MATH book is incredibly fun--radically more fun than SAXON, though SAXON has its moments.

I taught my first SINGAPORE MATH class yesterday, and I brought in some of the problems from the "Brain Maths" books (also from Singapore) and the kids loved them!

I was shocked.

I'd rounded them up just to have a 'Do Now' sheet, and the kids didn't want to stop doing them.

They're hard, too.

-- CatherineJohnson - 18 Oct 2005

Of course, they're probably starved for right answers.

-- CatherineJohnson - 18 Oct 2005

@ Dan: "...but at some point all kids doing math have to learn that they need to write down some notation of what variable represents what."

7 ± 2

Every variable definition you're keeping in short-term memory takes up a slot that could be used for something else.

-- DougSundseth - 18 Oct 2005

Every variable definition you're keeping in short-term memory takes up a slot that could be used for something else.

WIT AND WISDOM!

This is definitely true.

I ran into it doing the RUSSIAN MATH problems. The book has many massively long, complicated computations (like the extra-credit problem above) and I found it very difficult to retrace my steps.

I was trying to write neatly, and was using quadrille paper, and I still found my digits slanting off to the right, getting out of alignment, etc.

And when I was dealing with lots of minus signs & absolute value signs & fractions and decimals (RUSSIAN MATH throws all this stuff into the same computations) the Working Memory load gets seriously out of hand.

-- CatherineJohnson - 18 Oct 2005

I used to tell people that my son is a sponge for knowledge, but the school is feeding him with a teaspoon. When he was in Kindergarten, I used to leave out math worksheets on the table and he would walk by, sit down, and do them. He was thrilled. Adds and subtracts, less than, greater than, etc. I mentioned this to his Kindergarten teacher and she didn't look pleased.

He loves geography, but his first grade teacher said that "Yes, he has a lot of superficial knowledge."

Some parents would comment to him that perhaps he wasn't looking forward to school at the end of the summer. He didn't know what they were talking about. He loved school.

-- SteveH - 18 Oct 2005

"...his first grade teacher said that 'Yes, he has a lot of superficial knowledge.'"

I'd have been speechless. Then, after suppressing (I hope) my immediate inclination to make a scatological comment, I'd have started looking for a new teacher or a new school.

That is among the most offensive things I've ever heard attributed to a teacher. It shows that not only does the teacher not value education, but that he has a deep contempt for children.

I sincerely hope that I'm never in the position that you were then.

-- DougSundseth - 18 Oct 2005

"...his first grade teacher said that 'Yes, he has a lot of superficial knowledge.'"

ditto that

It's shocking.

I heard the same thing, only a more extended version, at a school meeting one day (about someone else's child).

-- CatherineJohnson - 18 Oct 2005

test

-- CatherineJohnson - 18 Oct 2005

caret = ^. 1 karat = 1 carat = 1/24 pure (gold). can i have a carrot?

-- VlorbikDotCom - 19 Oct 2005

"...his first grade teacher said that 'Yes, he has a lot of superficial knowledge.'"

"I'd have been speechless. Then, after suppressing (I hope) my immediate inclination to make a scatological comment, I'd have started looking for a new teacher or a new school."

This was three years ago (public school) and it caught me off-guard. By the way, the teacher must have been in her 60's) My comment was just to relate to her that my son and I love to look at and study maps. It was a naive comment on my part. I didn't say anything to her comment. (I wish I could do that over now.) Ironically, later in the year (during a thematic unit on "Sands from around the world"), my son had to show the student teacher where Kuwait was on the world map. It was also funny that for a play the class did on Germany (once again thematic) that year, my son was the one who got the part that told the audience all of the countries that surround Germany.

He goes to a private school now. Expectations are higher, but they use Everyday Math. Choices are pretty limited around here.

-- SteveH - 19 Oct 2005

The real meat in her method is here:

"Although I use a traditional algebra textbook (Brown, Dolciani, & Sorgenfrey, 1994), I have developed totally scripted lessons for each algebra unit that require absolute focus and attention, constant oral responses, and intense involvement from every student. Direct instruction assures that all students leave my classroom that day with a thorough understanding and at least partial mastery of the concepts. "

How is she keeping their attention? How is she ensuring that every student stays intensely involved with the lesson, and leaves with partial mastery of the subject? That's what I'd like to know.

-- CarolynJohnston - 19 Oct 2005

I'm sure it would have caught me off guard too. My reponse here was, I'm sure, much more glib than my response would have been when talking to the teacher. Who expects that sort of attitude from someone who presumably chose to spend a lifetime educating children? I have little doubt that your reaction was better than mine would have been. (Especially three years ago; my eyes are a bit further open now than then.)

Now I would be in the principal's office within the day (though I still don't know what I could usefully say to the teacher in that situation). I've come to believe (with thanks to KTM, Joanne Jacobs, Kimberly Swigert, Moebius Stripper, et al.) that such attitudes are toxic to children given over to the care of teachers.

-- DougSundseth - 19 Oct 2005

"Now I would be in the principal's office within the day ..."

Of course, now I would simply tell the teacher that my wife and I consider it to be fundamental knowledge - not that it would do much good.

In our case, talking to the principal would not have helped much. Our state adopted "Standards-Based Education 7 or 8 years ago. (It took me quite a while to realize that Standards-Based Education really meant low and fuzzy standards.) Our town hired a new superintendent that cleaned house. You were either on the bus or off the bus. Life became very difficult for traditionally-oriented teachers, many of whom decided that they would rather be off the bus into retirement. Now they use MathLand. Twenty-five percent of the kids in our town now go to other schools. And, our public schools are ranked as "High Performing" according to our state's standardized tests. After all of these years I still don't fully understand what is going on.

-- SteveH - 19 Oct 2005

Our town hired a new superintendent that cleaned house. You were either on the bus or off the bus. Life became very difficult for traditionally-oriented teachers, many of whom decided that they would rather be off the bus into retirement. Now they use MathLand. Twenty-five percent of the kids in our town now go to other schools. And, our public schools are ranked as "High Performing" according to our state's standardized tests. After all of these years I still don't fully understand what is going on.

This is one of the worst stories I've heard so far (worst direct story; I don't know the in's and out's of many places...)

-- CatherineJohnson - 19 Oct 2005

How is she keeping their attention? How is she ensuring that every student stays intensely involved with the lesson, and leaves with partial mastery of the subject? That's what I'd like to know.

Check this article out: Student-Program Alignment and Teaching to Mastery

Zig has the answers to all your questions and a few more you probably haven't even thought of yet.

-- KDeRosa - 19 Oct 2005

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Title:	the Gambill method for teaching algebra
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the Gambill method for teaching algebra

daily perfect mastery

checking your work & eating Brussels sprouts

Gambill in a nutshell

spaced repetition

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